Stochastic dynamics of reacting biomolecules

Wst(q,tp)

= 271"-1 ( / 2 - 7 r r i ) ~

(3-5)

R is the direct product of the area of the interior of the square (minus the

114

Motion of test particles in a 2-d potential

Fig. 3.8

landscape

Billiard with a small opening.

area of the interior of the circle) and the interval [0;27r). In other words, this is a rectangular parallelepiped (l,l,2n) with a cylindrical (radius r\ and length 2ir) hollow in it. This equation follows from the microcanonical distribution wst (q,p) = C05 ( -p2 - -pi j = Cop^S (p - p0), which indicates that the distribution in R should be uniform. When the opening FG is sufficiently small, the lifetime of the cluster is large and specified as

•K

where v = (0,1). Therefore, when (3.5) is taken into account, we have ir (I2 - irrl) hvo

(3.6)

It is seen that rav —> oo if l0 —>• 0, i.e. gets closer to the case of the stable state. The dissociation process in this model was calculated numerically for the values n/l = 0.2; l0/l = 0.034. We assume a uniform initial distribution in the area 0.45 < qi/l < 0.6, 0.73 < q^/l < 0.76, 0 <

Stratonovich

115

method

i.o

0,8

0,2

0,0 i

0

•

'

•

SO

•

100

i

i

190

i

i

i

200

i

230

i

i—

300

y

Fig. 3.9

Cluster decay.

The empirical distribution function FN {y) = P (T/rav > y) for the lifetime was obtained for N trajectories (Fig.3.9). It was approximated by the exponential function. The empirical mean lifetime r'av, which was obtained by direct averaging of the lifetimes, was VQT^/1 = 84, while (3.6) gives the value V0T^V/1 — 80.8. It can be assumed that the relative disparity between the above values gets smaller with decreasing length of the opening. The decay curve is nearly exponential because the system quickly forgets the initial conditions due to discontinuity of transformation of angles at the moment of impact. 3.3.3

The calculation namic theory

of the rate of cluster

dissociation

using

dy-

The dynamic method proposed earlier for investigating the escape from a potential well is elaborated for calculating the spontaneous decay constant of a cluster or a metastable molecule. The purely dynamic method uses the fact that the dynamic process is a fluctuation consequence of self-stochastization, i.e., dynamic chaos. If the escape (decay) mean time is much longer than the characteristic time in which a stationary or quasistationary probability distribution is established in the phase space, then it can be assumed that almost all the time the system exhibits a distribution close to an equilibrium distribution in a stable system obtained by slightly deforming the initial distribution. Here we assume that the equilibrium distribution in the deformed system represents a microcanonical distribution: w(q,v) = CS{T(v) +

U(q)-E),

where T (v) = T\ (p(v)) - is the kinetic energy expressed in terms of the velocity

116

Motion of test particles in a 2-d potential

landscape

v = q. It is equivalent to the distribution wx (q,p) — C\8 (H (q,p) — E), since the Jacobian of the transformation p = p (v) is constant and can be included into the normalization constant. A stationary distribution in the deformed system can be used to find the decay constant a of the initial system, if this constant is small. The constant a is the escape probability per unit time given by:

w2 (x) fa (x) dTa

o=

(3.7)

r

Here x = (q,p) or x = (q,v), fa (x) — x (x), T is a closed hypersurface in the a

phase space, T_ is the part of the surface that corresponds to the condition of escape from a region lying inside T: fadTa > 0. Consider dissociation of a cluster consisting of atoms with their own coordinates and velocities. Then (3.7) can be rewritten as:

a = / w (q,v)r](vjnj)vjnjdSodv,

(3-8)

So

where So is a closed hypersurface in the coordinate space, n^-is the unit vector of the outer normal to So, VjTij is the scalar product of multidimensional vectors, and 77 (y) = (1 + signy) /2. Introduction of the factor r\ (VJUJ) selecting the space points where £) • VjTij > 0, is analogous to selecting the exit part T_ of the surface T in (3.7), and VjJijdSodv corresponds to fadTa. Thus we define a hypersurface So- If one of the atoms goes beyond this surface, the cluster is assumed to be dissociated. Then the probability of such an escape per unit time is given by the product of the velocity vector and the outer normal to this surface integrated with the probability distribution over all the velocities and S0. It is natural to place the hypersurface So, the crossing of which by an image point symbolizes the cluster dissociation, on top of the potential barrier. Computer experiment [Stratonovich, Chichigina, 1996; Chichigina, 1997] used a cluster consisting of three equal atoms interacting by Lennard-Jones potentials. The calculated cluster mean lifetime agrees well with the theoretical value calculated using (3.8). It means that the closed system consisting of only three particles can be described by thermodynamic methods. Such a possibility is related to the dynamic instability of the motion.

Stratonovich

3.3.4

117

method

Mean time of escape from a potential of noise. Metastable approximation

well under the

action

Formula (3.8) can be used not only in the absence of the external noise but for any metastable state inside the potential well U(x) to which the probability distribution w (x, v) can be related. This method is much simpler than the solution of FockerPlanck equation. For example, Gibbs distribution can be used as w (x, v) in the case of the external noise characterised by the temperature T. Assume that at the edge of the potential well U = oo everywhere apart from the area in the coordinate space that corresponds to the exit where Uout — 0. Large escape time Tav, i.e. metastability of the state can be achieved due to smallness of the opening (exit). The smallness of the opening allows one to neglect its geometry which substantially simplifies the problem. Consider 2D motion in the potential well U (xi,x2) in the area Cl. The escape takes place if the particle crosses a small fragment of the length 2A that is perpendicular to Oxi axis. As we consider the first escape from the area f2, v\ > 0 and we can restrict ourselves to integration over only positive values of v\ and omit 77-function in (3.8). In this case the final expression for a is: OO

OO

A

2

/

1

a=

[ A [A f A I dx2Viexp C / / / I -00

0

- A

2 \

Uout+^t±^t\ Ty

^

( ^

'

where Uout = 0, and the normalizing constant for Gibbs distribution is determined by the relationship: OO

C=

OO

f dv2 f dVl f dx1dx2exp -00

(

U

^

x

* ) + ^

+

-2X j

(310)

0

Integrations for a and C can be divided into two parts: the first related tp the coordinate the second related to the velocities: a = a'a", where A

"hi

dx2 =

-

-A -A

C = / dx1dx2expl

*'

-n and also 2kBT mix The final expression for the decay constant is:

2

j ,

118

Motion of test particles in a 2-d potential

landscape

2A J2kBT where C" can be sometimes calculated analytically. For example, C = kBTn/n the symmetrical parabolic well U = K (X\ + x\) and

3.4

for

Test particle motion in a three-minima potential landscape

We studied the transition of a particle from one potential well into another under the influence of random forces. The particle is initially located in the origin (XQ = 2/0 — 0) corresponding to the minimum of the barrier. Assume that at certain moments (from 20 to 100 times during the period of free oscillations determined by the frequency spectrum - see Section 3.2) the X and Y components of the velocity change by the values vx£, vy£. Here £ is a random quantity uniformly distributed in the interval [—0.5,0.5], vx, vy are additional velocity amplitudes (it was assumed that vx = vy = 1, which is insufficient for overcoming the potential barrier if the particle starts from the potential minimum). Due to noise action (such a noise can be considered as "white") the system is nonconservative, and we have to introduce friction. Coefficient h in (3.1) was determined by a numerical experiment based on conservation of the mean total energy of the system during the time of experiment. In calculations h = 0.05. The Lennard-Jones potential (3.3) was superimposed on the potential field (3.2). The dependence of the mean time (averaged over 25 realizations) of the first ten transitions from one well to another (tlO) on the depth and characteristic radius of the potential ULJ was determined. We considered five variants:

N 1 2 3 4 5

d 0 100 50 100 50

ro 0 10 10 13 13

Figure 3.10 shows phase trajectories of a particle and topograms of the potential landscape for the cases 1-3. The transition from one well into another can be either direct (via the potential barrier between the wells (3.2)) or indirect (via the bypass of the potential (3.3)).

Test particle motion in a three-minima

potential

119

landscape

y(t)

m.n -

x(t)

• IS

-10

-5

0

5

10

IS

b)

f% • IS

Fig. 3.10 vx = l;vy

-10

-5

0

5

/

0

15

m

Phase trajectories of a T P influenced by noise and friction, a) h\ = 0.1; /12 = 0.1; = l\d = 0; r0 = 0; b) hi = 0.1; h2 = 0.1; vx = 1; vy = 1; d = 100; r 0 = 10;

In the second case the particle can stay at the bypass for a long time or come back to the well from which it started. It was demonstrated that the TP lifetimes in each well are nearly equal. The distortion of the potential (3.2) by the potential (3.3) is the stronger the larger is the characteristic radius and the relative depth at the fixed position of the center of the latter. This influences, in turn, the time often transitions and can both increase and decrease this time in comparison with the absence of the potential (3.3). Note that the depth of the potential is the decisive factor. Specifically, a relatively deep potential (3.3) can diminish substantially the potential barrier separating it from the wells (3.2) and, thus, facilitate the transition from one well into another via the bypass. However, if the bypass is rather far, the particle can stay there for a long time accumulating the energy for escape. Eventually the time of ten transitions increases substantially. On the other hand, shallow potential can not influence the potential barriers and the straight way becomes more preferable. Therefore, deep and wide (large value of ro) Lennard-Jones potential decreases (variant 4) and narrow one increases (variant 2) the time of the first ten transitions

120

Motion of test particles in a 2-d potential

landscape

in comparison with the case of the absence of the potential (3.3) (variant 1). Shallow wide and narrow potentials (cases 3 and 4) do not change the time substantially. In our opinion the problem of three minima is directly related to the proton transfer in the system of hydrogen bonds disturbed by substrate binding (see Chapter 7). In Chapter 7 we consider the motion of TPs of different shape in the potential relieves with several minima for a specific case of the ACE force field.

3.5

The problem of a test particle transition in the potential field with periodically changing parameters

Below we consider two cases of the periodic changes of the parameters of three-well and two-well potential relieves. a) Assume that the center of the Lennard- Jones potential oscillates harmonically along the Y axis so that yc = 20 + 3 sin(fct) and the particle is initially placed in the center of one of the two wells (4.2). We determined the dependence of the time when the particle leaves this well for the first time (moves from the centre of the well by the distance equal to half of the distance between well) on the oscillation frequency k. Figure 3.11 shows the results of calculations: the calculated time is plotted versus the parameter 2nk. In practical calculations we considered the times that were less than 1000 (calculations were terminated if by that time the particle was still in the same well). The curves exhibit two pronounced minima. (This result is predicted by the theoretical analysis of the resonance vibrational frequencies under the conditions of linearization.) If the frequency of the potential center oscillations is close to the frequencies of the free oscillations (see problem 2), the escape time is the shortest. The second minimum (longer escape times) corresponds to double free oscillation frequency (Fig. 3.11). Two intervals of resonance frequencies correspond to two mechanisms of excitation of TP vibrations: additive (with the frequency close to the natural one) and parametric (with the frequency close to double natural frequency). If we introduce noise the result does not change dramatically. We can find the same ranges of frequencies where the escape times are minimal. If we introduce only friction it appears that a) there is a threshold value of the friction coefficient under which the particle does not leave the well at any frequency from the considered range during the observation time; b) if the friction coefficient does not reach the threshold value the ranges of resonance frequencies become more narrow (the range of double resonance frequencies may disappear) and the corresponding times increase proportionally to the increase of the friction coefficient. It is the noise but not the variation of Lennard-Jones potential that influences the time of the first escape. (In the case of random noise the escape time is the mean time determined

Periodically changing

parameters

121

in

IT ;i

2nk

* 13

b

Fig. 3.11 Plots of the T P escape times versus the frequency of oscillations of Lennard-Jones potential at different noise intensities (vx,vy) and friction coefficients (hi,hi): a) vx = 0; vy = 0; hi = h2 = 0; b) vx = 0.1; vy = 0.1; hi = h2 = 0.01; c) vx = 0.2; vy = 0.2; hi = h2 = 0.015; d) vx = 0.5; vy = 0.5; hi = h2 = 0.025;

by means of 25 realizations of a random process. A scatter of values of these times was less then 10% for the frequencies close to resonance ones and increased far from them). b) The problem similar to the previous one was solved for the case of the oscillating potential field (3.2) in the absence of the potential (3.3). The initial position of the particle in the proximity of one of the minima was XQ — 6 + £, yo = £, (where £ is a random quantity uniformly distributed at the interval [—1,1]); the initial velocity was equal to zero. It was assumed that the centers of the potential wells oscillate in counterphase (one of the constants in (3.2) was oscillating as g — 7,5±sin (kt)). Thus, the value and direction of the force acting up on the particle varied in time and were determined by the current position of the particle. The time of the first escape (crossing Y axis) depending on the oscillation frequency was determined. The calculations at each value of the potential oscillation frequency used 100 random initial positions of the particle and the results were averaged. In this case we can select several ranges (not only two as in the previous case) of the resonance frequencies being aliquot to the natural one (see Fig. 3.12a). The escape

122

Motion of test particles in a 2-d potential

landscape

-—t—'

r TfT•

i. -1

i

\ r\ I L tf i .

as

l.s

is

3.5

ft5

/.5

2.5

3.5

b

Fig. 3.12 Plots of the escape times versus the frequency of oscillations of the centers of the potential wells: a) xo = —5; j/o = 0; vx = 0; vy = 0; hi = h2 = 0; b) 10 = — 5; yo = 0; vx = 0; Vy - 0; h\ = h2 = 0.01

time quadratically increases with increasing frequency. If friction and (or) noise are introduced the results are similar to those obtained in Section 3 (see Fig. 3.12b). Note that the escape times larger than 500 were not determined because of the limited computational time. Thus, the horizontal fragments of the curves in Fig. 3.12 do not correspond to the real escape times (there were no transitions during the computation time at the corresponding frequencies). Thus, the harmonic variation of the parameters of the potential landscape on both the frequency close to one of the T P natural frequencies and the frequencies close to the main parametric resonance result in abrupt (by two, two and a half orders of magnitude) decrease of the particle lifetime in the area of one of the minimums. The relative variation of the parameter or its amplitude (relative change of the distance between the minimums) is only 0.1 How can we relate these effects to the processes taking place in the active site of an enzyme? The vibrational frequency of the proton in H-bond between donor A and acceptor B is 7 • 10 13 — 1014Hz (Sokolov, 1981) and depends on the distance between A and B atoms. The vibrational frequencies of large clusters that form the potential landscape in A-H-B system are much lower (less than 10 12 Hz). Thus, one can hardly expect any resonances. On the other hand, time T can get shorter due to slow decrease in the distance between the minima and the corresponding lowering of the potential barrier in the potential landscape. The problem of resonance with quantum-mechanical transitions is discussed in Chapter 8. In any case harmonic or quasi-harmonic parametric processes influence substantially the spectrum of proton oscillations (broadening of the spectrum). As the proton moves in 2D or 3D space, the spectrum of oscillations can be influenced also by the low-frequency modulations due to Fermi resonance. This possibility is mentioned by Sokolov (1981) and the corresponding mathematical models are presented in (Netrebko et al., 1994, 1996;

Periodically changing

parameters

123

Chikishev et al., 1996). On the other hand, the resonance phenomena can be expected for the vibrations of ligands bound by weak (e.g., hydrogen) bonds in the pocket of the active site. In particular, the natural frequencies of tryptophan bound by several H-bonds to the atoms of CT active site belong to the range 2 — 4 • 1012Hz. Therefore, the oscillations of groups of atoms and clusters that influence the AS potential field can be the sources of coloured additive or parametric noise. We mention that a similar situation holds for the calculation of conformational jumps of the angles $ and $ in helical peptide structures (see e.g. Fig. 6.2).

References O.A. Chichigina(1997): "Computer simulation of processes of spontaneous dissociation of 3-particle clusters", Vestnik Moskowsk. Univ., Ser. Fiz. 3. Fizika, Astronomiya 5, 6-9. A.Yu. Chikishev, W. Ebeling, A.V. Netrebko, N.V. Netrebko, Yu.M. Romanovsky, L. Schimansky-Geier (1996): "Stochastic cluster dynamics of macromolecules", Nonlinear Dynamics and Structures in Biology and Medicine: Optical and Laser Technologies, V. V. Tuchin, Editor, Proc. SPIE 3053, 54-70. N.V. Netrebko, Yu.M. Romanovsky, E.G. Shidlovskaya, V.M. Tereshko(1990): "Damping in the models for molecular dynamics", Proc.SPIE 1403, 512- 514. A. Netrebko, N. Netrebko, Yu. Romanovsky, Yu. Khurgin, E. Shidlovskaya (1994): "Complex modulation regimes and vibration stochastization in cluster dynamics models of macromolecules", Izv. Vuzov: Prikladnaya Nelineinaya Dinamika, 2, 2643. (In Russian) A. Netrebko, N. Netrebko, Yu. Romanovsky, Yu. Khurgin, W. Ebeling (1996): "Stochastic cluster dynamics of enzyme-substrate complex" (in Russian), Izv. Vuzov: Prikladnaya Nelineinaya Dinamika. 3, 53-64. Y. Sinai (1963): Dokl. Akad. Nauk SSSR 153, 1261-1264. N.D. Sokolov (ed.) (1981): "Hydrogen bond" (in Russian), Nauka, Moscow R.L. Stratonovich (1963): "Topics in the theory of random noise", Vol. I,II, Gordon & Breach, New York/London R.L. Stratonovich (1995): " On dynamical theory of spontaneous decay of complex molecules", JETP 8 1 , 729-735.

124

Motion of test particles in a 2-d potential

landscape

R.L. Stratonovich, O.A. Chichigina (1996): " Dynamical calculation of the spontaneous decay constant of a cluster of identical atoms". Soviet Phys JETP 83, 708-715. M. Toda (1974): "Instability of trajectories of the lattice with cubic nonlinearity". Phys. Lett. A 48, 335-345. S. Weiner, P. Kollman, D. Case, U. Singh, C. Chio, G. Alagona, S. Profeta, Jr., P. Weiner (1984): "A new force field for molecular mechanical simulation of nucleic acids and proteins", J. Am. Chem. Soc., 106, 765-784.

Chapter 4

Microscopic simulations of activation and dissociation W. Ebeling, V. Yu. Podlipchuk, M.G. Sapeshinsky, and A.A. Valuev 4.1

Discussion of the Heat Bath Model

The stochastic reaction theory as represented in brief in Chapters 1 — 2 concentrates mainly on the explanation of the high-energy events leading to overcoming the potential barriers between wells. Eyring, Kramers et al. developed several reaction theories interpreting thermal activation processes. For recent reviews of this problem see e.g. (Hanggi, Talkner, and Borkovec, 1991; Agudov and Malakhov, 1993; Talkner & Hanggi, 1995). The model developed by Kramers in 1940 represents the simplest model of the statistical reaction theory. As discussed already in Chapters 1 — 2 it is based on special solutions of the Fokker-Planck equation for the reacting molecules or on the corresponding Langevin equation with a white noise source. The present chapter aims to investigate reacting sites on a more microscopic basis. We will study reacting molecules which are imbedded into an atomistic fluid heat bath. We start with the hypothesis that the effects of hard collisions which act on the site may have - under certain conditions - a drastic effect on transition rates. In this way we want to make another step from a phenomenological theory of reactions at active site of complex molecules in solutions to a microscopic theory. In this way we want to contribute to the development of an appropriate statistical theory of reactive events at complex molecules. In some sense the microscopic approach which models the surrounding of an active center by an ensemble of discrete molecules instead by a heat bath is equivalent to introducing a coloured noise source into the Langevin equation instead of a white noise source. This corresponds to the fact that the spectrum of the realistic molecular forces acting on the reacting site is not white but has a much more complicated spectrum (Jenssen and Ebeling, 2000). Thus, before coming to the microscopic study we formulate here again our problem in the context of a coloured noise heat bath. According to Kramers phenomenological approach, which was explained in some detail in Chapters 1 — 2, the dynamics of the reacting molecule is modelled by a Langevin equation for the active site

125

126

Microscopic simulations

d2q

dq

of activation and

dU (q)

dissociation

/—- , ,

,

x

where m is the mass of the active particle (in the following m — 1), 7 is the effective friction constant (having the dimension of a frequency), D the diffusion constant. In Kramers original approach £(£) corresponds to (^-correlated noise (white noise). The potential U(q) is assumed to have two minima, therefore the system is bistable. The corresponding Fokker-Planck equation was solved by Kramers with a special boundary condition modelling the transition over a potential barrier. As we have shown in Chapters 1 — 2, the result for the log of the transition rate is (assuming moderate or large values of the friction)

ln* = l n , - ^ ,

(4.2)

where following Kramers (for moderate and for large 7) the frequency v is given by -Y 2

+

\

V 2

T -0


"i

ss-

(")

Here u>o is the angular frequency inside the minimum, where the transition starts and Wfc is the angular frequency at the potential maximum (the transition state). Equations (4.2-4.3) are in agreement with the Arrhenius law. In the limit of very weak friction (7 < < wo) the theory is to be generalized including energy diffusion (Straub & Berne, 1986; Hanggi, Talkner, & Borkovec, 1991). In the generalized Kramers theory the rate is given by a different expression (see chapter 1) since at small 7 the transition process is controlled by energy diffusion (Straub & Berne, 1986). As a consequence log A; shows in addititon to the linear Arrhenius term also a weak logarithmic dependence on AU /kg'T. According to chapter 1 we find in some approximation (additivity of the characteristic times) which includes energy diffusion as well as as quantum effects ,

/2TT7

2kBT

h \

AU

/A A.

We see that energy diffusion which comes into play at small values of the friction and small values of the Arrhenius term AU/ksT leads to deviations of the Arrhenius plot from a straight line what is demonstrated in the following figure. We see in Fig. 4.1 that the generalized Kramers theory (including energy diffusion) shows at small 7 an Arrhenius plot deviating from a straight line. At

7/wo = 0.1 we find a strongly expressed minimum of the transition time, at

Discussion

of the Heat Bath Model

4.0 6.0 (Delta U / k T)

Fig. 4.1 Arrhenius plot for the transition time log(r) = — log(fe) according to the theoretical expression eq.(4.4) including stochastic transitions and energy diffusion for 3 different values of the friction (7 = 0.1; 1.0; 10;w 0 = ub = l;h = 0).

7/wo = 1-0 this minimum is still weakly expressed and only at large friction 7/wo = 10 we find a straight line. Summarizing so far these results we may state that energy diffusion effects, observed at small friction in the generalized Kramers theory, already lead to deviations from the Arrhenius behaviour. Now we have to look for other effects. The crucial point in the Kramers theory of the transition rates, and its generalizations, is the assumption about a white noise character of the forces acting on the active site. In real molecular systems, and in particular under the influence of nonlinear collision effects, the 'real noise' may have a very complicated spectrum. In the next section we will show in detail by MD-simulations that the spectrum of the forces acting on a molecule in a bath has a maximum at finite frequencies and shows some similarity to coloured noise (harmonic noise). Here we will discuss only the question what are the consequences of introducing coloured noise sources into the phenomenological theory. Simple models of systems with harmonic noise were studied by several workers [Straub k Berne, 1986; Ebeling k Schimansky-Geier, 1989]. Straub and Berne studied the following generalization of (4.1) [Straub k Berne, 1986]:

m

cPq dt2

dU (q) dq

mey (t).

(4.5)

128

Microscopic simulations

of activation

and

dissociation

Here it is assumed that the bath acts directly only on the nonreactive auxiliary variable y. Schimansky-Geier, Ziilicke and Hesse investigated a similar model neglecting the term of order m/M [Schimansky-Geier & Ziilicke, 1990; Hesse & SchimanskyGeier, 1991]. It is easy to show, that the spectrum of the y-coordinate is harmonic [Schimansky-Geier & Ziilicke, 1990]

s

m M =

2Dft 2 „2 , . , , 2 , w r + (w

^ 22) -n

(4-7)

The peak of the spectral intensity is located at

ujmax = V« 2 - r 2 /2,

(4.8)

2DQ.A Syy {Umax) = p2(Q2 _ ( l / 4 ) p 2 ) '

^4'9^

and its height is

We consider now the oscillations of q around one minimum of the potential U(q), where it can be approximated by a harmonic force

U(q) = \u% (q ~ qof •

(4.10)

As shown by Hesse and Schimansky, the energy distribution for the oscillations around qo is approximately given by [Hesse & Schimansky-Geier, 1991]

P (E) - Z-^exp ( - ^ f - )

(4- 11 )

with the effective temperature

Teff ~ Syy (wo) •

(4.12)

We see that in this approach the effective temperature is no more a constant as for the white noise case but through WQ and Syy a function of the potential parameters itself. For the effective temperature and for the mean value of the energy, a resonance at uJma.x— wo- is found. The resonance condition is

Discussion

of the Heat Bath Model

n2-T2/2

= ujl.

129

(4.13)

In the vicinity of resonance large amplitudes and therefore possibly also enhanced transitions are observed. In this way it was shown by Schimansky et al. (1990, 1991), that coloured noise may drastically change the transition rates. Strictly speaking the Schimansky-Geier model does not apply to a molecule imbedded into a thermal bath in equilibrium, since the model does not obey detailed balance. Anyhow we may expect that this easily tractable model will reflect the qualitative properties of the stochastic forces in more realistic systems. We mention that the transition rates in real molecular systems are also influenced by the changes of the average forces, which are due to the equilibrium correlations. As well known from the modern theory of liquids (in particular we have in mind the BBGKYtheory), the interaction potentials are in dense thermal systems to be replaced by potentials of average forces, which are defined by the equilibrium distribution functions. Since the potential landscapes are changed by this equilibrium effect, this is another reason to expect that for real molecular systems the Arrhenius factor AU/kgT is to be replaced by an effective Arrhenius factor (AU/kBT)eff. Evidently the physical assumptions leading to a Fokker-Planck equation are rather severe, since in real systems the forces acting on a molecule do not correspond to the Langevin model and further we know from Gibbs statistics that the potential energy of a molecule must not obey an individual Boltzmann distribution. Having in mind that in spite of the great success of the classical reaction theory, several phenomena remain unexplained [Popielawski, 1989; Frauenfelder & Wolynes], a more careful study of the microscopic forces acting at special molecules seems to be necessary. The standard form of the equipartition theorem of statistical mechanics is saying that any quadratic contribution AH — aq2 to the Hamiltonian leads in thermal equilibrium to a contribution ksT/2 to the internal energy. This follows immediately from the Boltzmann distribution

P (q) = const exp ( —j^-f ) •

( 4 - 14 )

In other words, there is no way to accumulate in average more energy than ksT/2 on a linear degree of freedom. However this is not true for nonlinear excitations. Therefore we have to look for nonlinear effects capable to accumulate energy at certain sites. In the fifth chapter we will study in detail the nonlinear excitations and the statistical thermodynamics of nonlinear 1-d systems with Toda interactions. We will analyze in the following analytically and numerically the basic nonlinear dynamical effects, and in particular the soliton fusion, which is one of the effects supporting energy localization at definite sites.

130

Microscopic simulations

of activation

and

dissociation

In order to study the effects of nonlinear excitations, correlations and "real noise" in microscopic models, we simulated microscopic collisions on an active site by methods of molecular dynamics. The system we are going to study is a Hamiltonian system; the active site is coupled to a thermal bath consisting of molecules in equilibrium. For a special case, if the bath consists of an infinite set of harmonic oscillators, the method of Zwanzig allows the derivation of a generalized Langevin equation [Hanggi, Talkner & Borkovec, 1991]. This new Langevin equation obeys a fluctuation-dissipation theorem. The random force is stationary and Gaussian, it contains harmonic contributions as well as the memory kernel of the friction term [Hanggi, Talkner & Borkovec, 1991]. For systems with a bath consisting of hard molecules the situation is much more complicated and a complete theoretical treatment is still out of range. In our empirical MD-study we will concentrate on a special physical effect which might be specific for hard interactions. As pointed out already, we have observed in earlier work rather long tails in the energy distribution due to hard excitations and a harmonic structure of the force-spectrum [Ebeling & Podlipchuk, 1996]. These findings suggested, that under special conditions a considerable reaction rate enhancement could be possible. For simplicity we will restrict ourselves to the investigation of classical 2-d models of molecular systems. The rather simple model which we aim to study here is a system consisting of one "soft" molecule simulating the reactive site. This reactive molecule consists either of one soft atom, possessing two stable states in an external field, or of two soft atoms bound together by strong chemical bonds. The reactive molecule is imbedded into a thermal bath consisting of hard molecules forming a solvent. Note that we pay special attention to the areas of energy localization and highenergy "tails" of the distributions. It is assumed that the complicated reactions, such as DNA denaturation [Dauxois et al., 1993] and the enzymatic activity [Ebeling et al., 1989; Romanovsky et al., 1994], are related to the nonlinear excitations leading to energy localization in the special reaction centers. We studied the problem of elementary excitations in biomolecules and their possible role in the activation processes based on the simple models [Ebeling et al., 1989; Romanovsky et al., 1994].

4.2

Molecular dynamics of transitions between potential wells

In order to model a monomolecular reaction of Kramers-type we assume a soft 2 — d molecule (the active site) imbedded into a molecular bath of rather hard 2 — d molecules. We further assume, that the active molecule is moving in an external bistable 2 — d potential. Our main task is to study the stochastic transitions of the active molecule between the two wells in the 2 — d potential landscape (see Fig. 1.1 in Chapter 1). The external potential acting on the active molecule is denned as a 2 — d generalization of the Kramers potential

Molecular dynamics

of transitions

V(x,y,z)

between potential

= ex(x4-2x2)

131

wells

+ eyy2.

(4.15)

We see that ex corresponds to the height of the barrier between the wells and ey to the width of the cleft. In our model the active molecule which is wandering between the two wells is imbedded into a 2 — d molecular system consisting of N relatively hard compressible disks. In other words we have a kind of 2 — d solution of an active molecule in a solvent. The active molecule is modelled as a soft disk. The disks interact with each other by collisions due to thermal motions. The interaction energies are described in an appropriate way by finite-range LennardJones potentials [Hafskjold & Ikehoshi, 1995; Ebeling et al., 1995]. These potentials are obtained by a modification of the standard Lennard-Jones potentials: y L (r) = 4 e ( ( ^ ) 1 2 - ( ^ ) 6 ) .

(4.16)

Here e is the debth of the attraction well, this quantity will be used in the following as the energy unit. Further erj, will be used as the length unit. The Lennard-Jones potential is infinite-range. For the purpose of molecular dynamics, it is quite useful to work with potentials of finite range [Ebeling et al., 1995]. We have used as in our previous work for the hard interaction between solvent molecules the potential:

Vhh(r)

=

Vhh(r)

= 0ifr>1.5ahh.

\

Ahhe((^r-l)exP((^-h-A,

r

/

\ Ohh

*

'

(4.17)

Further we assumed that the interactions of the active molecule with the surrounding bath molecules are rather weak. This weak interaction is described by a potential with a weak r~2 singularity: Vwh(r) = Vwh{r)

\

= Oifr>

Awhe((^f-l)exp((—-h-i), r / V crwh 2, / 1.5awh.

(4.18)

The shape of the potential (4.18) is shown in Fig. 4.2 in comparison with a standard Lennard-Jones potential. In the following we shall call these special potentials FLJ-potentials ("finite range Lennard-Jones-potentials"). The parameters were chosen in such a way that the minimum is at exactly the same place

V(r) = - e

at r = 2*aL,

(4.19)

as for the Lennard-Jones case. This condition corresponds to the choice A^ = 28.05, <Jhh — 1.028
132 2 Morse potential 1.5

Lennard-Jones potential

FRJ, n=2

1

FRJ, n=8

u °5 o -0.5 -1 0.9

1

1.1

1.2

1.3

1.4

1.5

Fig. 4.2 Shape of the Morse potential, finite-range potentials for solvent-solvent interactions (n=8), for solut-solvent interactions (n=2) and for the corresponding Lennard-Jones interaction.

The potential parameters for the interaction of soft with hard molecules are chosen as, Awh — 69.75, crwh — 1.0024o,i . This choice guarantees that the minimum is at the same place as the minimum of the potential given by eq. (4.19) (see Fig. 4.2). The potential (4.17-4.18 ) has a finite range with a cutoff at 3/2, at this distance the potential itself and its derivative disappear. The finite range is of some advantage in MD-calculations for a smaller number of molecules in a cell. In most of our simulations we considered N = 100 i.e. 99 hard and 1 soft molecules. The density p was chosen always in such a way that in the static equilibrium configuration the distances correspond to the minimum of the potential. In the further calculations we used the zero point of the Lennard-Jones potential (4.19) as length unit and the potential depth e of an argon molecule as the energy unit. The corresponding temperature unit is T = 119-ftT. In these units we studied 2 — d molecular systems with the density p = 0.7. The dimensionless temperatures T were varied in the range 5 < T < 375. For this case we carried out molecular dynamical simulations for 100 particles in a box with periodic boundary conditions. A typical trajectory of the active molecule is shown in Fig. 4.3. We observe irregular transitions between the two wells. After a sufficiently long run the average time between transitions was obtained. The result is shown in Fig. 4.4. We represented the log of the average time between transitions from one of the Kramers wells to the other one, as a function of the Arrhenius exponent (AU/T). The temperature was in all runs fixed at the value T = 5. The heigth of the the potential well was changed between AU = 10 and AU — 50. Since in logarithmic scale the average time in the Arrhenius and also in the first (large friction) Kramers approximation should be a straight line, we have shown that there are significant deviations from the linear behaviour. In some sense the shape of the curves reminds the behaviour of the generalized Kramers theory represented in Fig. 4.1. This would mean that the transition processes are not

Dynamics

of Recombination

Reactions

133

Fig. 4.3 Transitions between the 2 wells of the bistable Kramers potential caused by molecular collisions.

Fig. 4.4 The log of the average time (j/-axis) log(r) = — log(fc) between transitions from one of the Kramers wells to the other one, as a function of the Arrhenius exponent AU/ksT. Line 1 - MD simulation; line 2 - theoretical expression eqs.(4.2-4.3); 7 Ri uo ~ ^b] line 3 - theoretical expression eq.(4.4); 7 3> <*>o; the value for 7 = 12.3 was taken from the MD simulation.

overdampded as often assumed but operate in a moderate or small friction regime. Further we see from the results of the simulations that the effective temperature corresponding to the local slope depends on the potential itself. This is what we observed already for a harmonic noise. In other words, the realistic microscopic noise is similar to a coloured noise.

134

Microscopic simulations

of activation

and

dissociation

100-

50-

o-50-

-100-100

-50

0

50

IOC

Fig. 4.5 Typical trajectories in the space of the relative distances x\ — X2 and j/i — j/2 between two Morse molecules. The arrows show events which lead to a dissociation of the molecule.

4.3

Dissociation of Morse Molecules

Our second model is a 2 — d diatomic soft Morse-molecule imbedded into a 2 — d solution consisting of rather hard atoms. Similar problems for low pressure systems were studied by Borkovec and Berne (see Hanggi, Borkovec & Talknerl). We study here the case of systems with liquid densities and assume that the two atoms forming the molecule are bound together by the Morse potential VM(r) = e((exp(-6(r - aL • 2*)) - l ) 2 - l ) .

(4.20)

The interactions between the atoms forming the molecule and the solvent molecule are described in the same way as above by the FLJ-potential (4.18). The molecular forces between the hard solvent molecules were approximated again by the FLJ-potential (4.17). The shape of the potentials is shown in Fig. 4.2 (Morse potential = dotted line, the finite-range potential for hard repulsion = full line, for the weak repulsion = dash-dotted lines). For comparision also the corresponding Lennard-Jones interaction is represented (by another dashed line). For this model we carried out again molecular dynamical simulations for N = 100. particles in a box with periodic boundary conditions. In Fig. 4.5 several typical trajectories in the space of the relative distances Xi — x-i and j/i — j/2 between two Morse molecules are shown. The arrows correspond to events which lead to a dissociation of the molecule. By averaging over many dissociation events we obtained the mean dissociation times, which are represented in Fig. 4.6. Here the logarithm of the dissociation time is represented as a function of the Arrhenius exponent. The full line corresponds to a fixed value of the potential barrier AU — 50 and the dashed line to fixed temperature T = 7.5 and variable potential barriers AU.

Dynamics

of Recombination

135

Reactions

.3 I

1

1

1

1

1

1

1

0

1

2

3

4

5

6

7

Fig. 4.6 The log of the dissociation time for a Morse molecule as a function of the Arrhenius exponent. The full line corrsponds to a fixed value of the potential barrier AU = 50 and the dashed line to fixed temperature T = 7.5 and variable potential barriers AU. All energies and temperatures are given in the Argon-units used in this paper (i.e. T = 1 corresponds to 119 K).

Again we do not find an appropriate description by the Arrhenius law which would require that both curves are identical and have the slope one. Again we could interprete this by coloured noise effects. The microscopic picture we have here is more rich and allows a more detailled study. The price to pay is however, that microscopic simulations require a lot of computer time, so that the coloured noise picture seems to be a reasonable compromise.

4.4

Dynamics of Recombination Reactions

In order to model the fusion reaction of two molecules we defined a potential with a well inside a high wall VF(r)

=

^ e x p ( - ^ ) ( ( ^ - l ) e x VF{r) = 0

if

r > 1.5

aF.

P

( ( ^ - ^ ) , (4.21)

The parameters aoF,crF and AF are choosen in such a way that the minimum of the potential (4.21) has the same depth and coordinate as for Lennard-Jones potential and the hight of the potential is equal to a given value (see below). We notice some similarity with the potentials used by Henderson et al. (1995) for the description of association phenomena. These authors modelled association by an attracting Gaussian potential inside a Lennard-Jones core. Our model corresponds to an association where the atoms merge completely, this is what we call 'fusion'. The shape of our potential is shown in Fig. 4.7 for different values of the reaction

136

Microscopic simulations

of activation

and

dissociation

60 50 40 30 U

20 10 0 -10 0.2

0.4

0.6

r

0.8

1

1.2

Fig. 4.7 Potential model for a fusion reaction with different 10,20,30,40,50, again measured in the Argon units).

1.4

reaction barriers (AC/

=

Fig. 4.8 The log of the average fusion time (time for crossing the barrier). The lower full line shows the time for reaching the maximum. The upper full line shows the time up to the end of one oscillation inside the well. The dashed line shows the time up to the moment where 100 time-steps are performed. Shot-dashed line shows the average life-time of the soft molecule inside the potential well.

barriers (AU — 10,20,30,40,50). All simulations were started in the position where the two atoms are initially at a distance corresponding to the outer well. The temperature was fixed at the value T = 5, the heigth of the barrier was changed (AU = 10,20,30,40). The average time was recorded for the events:

Spectrum of atomistic

collisional

forces

137

(i) reaching the top of the barrier, (ii) reaching the inner well and finishing at least one oscillation inside the well, (iii) after 100 time-steps inside the inner well. We see that the criteria (ii) and (iii) are more or less equivalent and may be used for defining the characteristic reaction time. The log of the average fusion time (with respect to the criteria given above) was represented in Fig. 4.8. The lower full line corresponds to the condition (i) i.e. it shows the time for reaching the maximum. The upper full line shows the time corresponding to the condition (ii), i.e. it represents the time up to the end of one oscillation inside the well. The dashed line shows the time up tho the moment where 100 time-steps are performed corresponding to the condition (iii). Already a first inspection shows, that the curves show a larger deviation from the Arrhenius law which increases with increasing height of the potential barrier.

4.5

Spectrum of atomistic collisional forces

We have shown in the previous sections that the molecular collisions acting on a reacting molecule generate a kind of coloured noise. In order to make this picture more precise we will study in this section the time correlations by means of molecular dynamics simulations. In particular we will calculate within our models the correlation functions as coordinate-coordinate, velocity-velocity and force-force for soft molecules solved in a solvent consisting of hard molecules. We simulate the thermal equilibrium of one soft molecule with r~2 - repulsion imbedded into a bath of molecules with r~ 8 - repulsion. Here the soft molecule should model an activation center in a solution formed of hard molecules. Following an earlier work [Ebeling & Podlipchuk] we will show that in thermal equilibrium the spectrum is similar to that of a coloured noise. Further we will show that a region of densities and temperatures exists, where the correlation functions of the solute molecules are quite different from those of the hard molecules of the solvent. Finally we will show that the diffusion coefficient of the soft molecules is much higher than that of the solvent molecules. In the previous sections we developed rather simple models of reactions in solutions which is rather well suited for MD-investigations. Here we aim to work out the model of coloured noise. Further we want to contribute to the study of the kinetic properties starting with the investigation of the time-correlation functions. Our present simulation refers to 3 — d molecules interacting via the finite-size Lennard-Jones model introduce above. For this case we carried out molecular dynamical simulations for 32-108 particles in a box with periodic boundary conditions. The molecular forces between hard molecules were approximated by the finite-range Lennard-Jones potentials introduced in the previous sections. We remember that the shape of this potential was shown in Fig. 4.2 for the exponent n = 8 in com-

138

Microscopic simulations

of activation and

dissociation

parison with a standard Lennard-Jones potential. As above we we shall call these special potentials finite - range Lennard-Jones-potentials (FLJ-potentials). The parameters were chosen in such a way that the minimum is at exactly the same place

Vmin = - c

at

r = 21/6L

(4.22)

as for the Lennard-Jones case. This condition corresponds to the choice n = 8, A = 28.05e, and a = 1.028L for the interaction of hard molecules with hard molecules. The potential parameters for the interaction of soft with hard molecules are chosen as n = 2, A = 69.75e, a = 1.0024L. This choice guarantees that the minimum is always at the same place. Our potentials have a finite range with a cutoff at 3L/2, at this distance the potential itself and its derivative disappear. The finite range is of some advantage in MD-calculations for a smaller number of molecules in a cell. In most of our simulations we considered N = 32 i.e. 31 hard and 1 soft molecules, a few calculations were made for N = 108. The density was chosen always in such a way that in the static equilibrium configuration the distances correspond to the minimum of the potential. This might correspond to a liquid-like phase. In the present calculations we used the zero point of the Lennard-Jones potential as length unit and the potential depth e as the energy unit. In these units we studied 3-dimensional molecular systems with the dimensionless densities p = n/L3 = 1, which correspond to the liquid state. The dimensionless temperatures T were in the simulations fixed at T = 10 which is not far above the characteristic temperature kT = 10/D (here D = 1,2,3 according to the dimension of the coordinate space) where the maximum of the energy localization effect is observed [Ebeling et al., 1995]. In the mentioned work we discussed the main physical excitations of the system which are expected to occur, as phonon- and soliton-like excitations. In Chapter 5 we will consider these effects again for Toda chains. In the following we will show that in the transition region also pecularities in the time correlation functions occur. For the simulations we have used an energyconserving algorithm [Norman et al., 1993; Ebeling et al., 1995, 1997]. In our earlier work [Ebeling et al., 1993] the distribution functions of the energy were calculated and studied especially in the region of the high-energy tails. We found that in thermal equilibrium there exists a characteristic region of temperatures and densities where energy is mainly concentrated on soft sites. Here we concentrate on the temperature-density region where long tails in the energy distribution were observed. For physical reasons one expects that high energy events could lead to special noise effects and kinetic properties. The result of our MD-simulations is demonstrated in Figs. 4.9-4.11. Figure 4.9 shows the time-dependence of the force-force correlation functions. What we observe is a typical well at about t — 0.5—1.5 in dimensionless followed by a slow sinusoidal damping. The spectral density (the Fourier transform) of the force-force correlations (Fig. 4.10) shows a peak at a frequency of about 25 (in

Spectrum of atomistic

collisional forces

139

Fig. 4.9 Autocorrelation functions force-force < f(t),f(t + At) > . 1-solute (potential-eq.(4.18)); 2-solvent (potential-eq.(4.17)). Two 2-dimensional Morse molecule. AU = 1; T = 1.8; p = 0.8.

dimensionless units). The behaviour of solute and solvent molecules is distinctly different. In Fig. 4.11 the time dependence of the mean square deviations ((Ar) 2 ) is represented as a function of time. According to Einstein we expect

<(r(t) - r(0)) 2 ) = 2dDt,

(4.23)

where D is the coefficient of diffusion of the solute molecules (or self-diffusion of the solvent molecules). We see that the diffusion coefficient of the solute molecules is by more than a factor 1.3 higher than that of the solvent molecules. Let us summarize: By MD-simulations for solutions of molecules with finitesize Lennard-Jones potentials we have considered a single soft molecule embedded into a surrounding, otherwise uniform hard system. It turns out that the soft molecule is able to trap and superpose narrow pulse-like excitations impinging from all directions within a characteristic excitation time. This energy superposition may lead not only to considerable concentrations of potential energy at the soft molecule as shown elsewhere (Ebeling et al, 1995) but also to larger diffusion coefficients and to a rather narrow-peaked power density of the force correlations.

140

Microscopic simulations

of activation and

dissociation

l«W

Fig. 4.10 Spectrum of force-force < / / > w time correlations. 1-solute (potential-eq.(4.18)); 2solvent (potential-eq.(4.17)). Two 2-dimensional Morse molecule .AC/ = 1; T = 1.8; p = 0.8.

4.6

Discussion of activation processes in an atomistic heat bath

We have shown in this chapter by molecular dynamical simulations most for 2—d and also for a few 3 — d systems, that larger deviations from a white noise spectrum and from an Arrhenius law occur. At first we observe some curvature of the Arrhenius slopes in particular at small values of the threshold similar as shown in Fig. 4.1 for the generalized Kramers theory. We may conclude from this that the transition processes in liquids are not overdamped processes. Further we observe that the slope of the Arrhenius curve is in general smaller than one (see Figs. 4.4, 4.6, 4.8) and shows in all regions some curvature. In other words, the effective Arrhenius factor (At7/fcfiT) e // if different from the original one and is not a constant but a potential-dependent function. This might be due to several reasons. At first we mention the effects of energy diffusion typical for small friction as demonstrated in Fig. 4.1. Another reason are deviations of the potential of average force from the vacuum potential or the deviations of the effective temperature from the real temperature in the solution. The latter is eventually caused by the fact, that the noise generated by molecular collisions is not of white noise type. I has been shown by Schimansky-Geier et al. (1990, 1991) that harmonic noise leads to an effective temperature, which is determined by the spectrum of the noise. In the previous section based on earlier work [Ebeling &Podlipchuk, 1996] we have investigated

Discussion

of activation processes in an atomistic heat bath

141

Fig. 4.11 The mean square deviation as a function of time. Line 1 - solvent (potential-eq.(4.17)); line 2 - solute (potential-eq.(4.21), AU = 30); line 3 - solute (Kramers potential, AU = 30); line 4 - solute (Morse potential, AU = 30); line 5 - solute (Morse potential, AU = 1); T = 1.8; p = 0.8. The mean slope of the curves corresponds to the effective diffusion constants.

the spectrum of the force-force correlations; we obtained a distribution which is similar to a harmonic noise distribution. This is a point which needs a more careful investigation including the study of the average forces, but in any case one reason for the the change of the slope could be the deviation from a white noise spectrum. Another observation which seems to be relevant is, that the log of the transition time is not a straight line and depends in a different way on AU and on T (see Figs. 4.4, 4.6 and 4.8). A conclusive interpretation of this quite complicated behaviour of the transition times is not possible so far. However our previous results on the energy distribution lead us to the hypothesis, that the reasons for the deviations from the Arrhenius-type behaviour are in part also due to the existence of hard (soliton-like) excitations in the system. In order to clarify this point, let us discuss in brief the main physical excitations of the molecules which are expected to occur in a dense system: In a thermal regime we have reasons to expect a whole spectrum of excitations, as phonon- and soliton-like excitations. In the case of a purely linear coupling we should find sinusoidal oscillations and waves, acoustical and optical phonons etc.. In nonlinear systems we will find also local excitations with local peaks of the energy density. From the theory of infinite chains of molecules with special interactions (as e.g. the Toda interactions, see for details Chapter 5), we know that there exist soliton-solutions. Solitonic excitations are absolutely stable local

142

Microscopic simulations

of activation and

dissociation

excitations. In realistic systems of molecules such excitations cannot persist already for purely mathematical reasons. However there might be hard localized excitations which remind a soliton behaviour. We consider such excitations as candidates for local energy spots and eventually as to be responsible for an enhancement of the rate of chemical reactions. At the present moment this interpretation is just a plausible hypothesis. However we have every reason to expect that further theoretical, numerical and experimental work will contribute to a solution of the problems, which were raised here. In Chapter 5 we will come back to this problem investigating in detail the properties of chains (rings) of masses with exponential repulsion (Toda chains).

References N.V. Agudov, A.N. Malakhov (1993): "Nonstationary diffusion through a piecewise continuous profile. Exact solutions and time characteristics", Izv. Vuzov Radiophysics 36, 148-165. T. Dauxois, M. Peyrard, A. Bishop (1993): "Dynamics and thermodynamics of a nonlinear model for DNA denaturation", Phys. Rev 47, 648-695. W. Ebeling, & L. Schimansky-Geier (1998): in "Noise in Nonlinear Dynamical Systems", (F. Moss, P.E.V. McClintock, eds.), Cambridge, Cambridge University Press. W. Ebeling, M. Jenssen & Yu. M. Romanovskii (1989): "100 years Arrhenius law and recent developments in reaction theory" , in: Irreversible Processes and Selforganization, eds. W. Ebeling and H. Ulbricht, Teubner, Leipzig, p. 7-24. W. Ebeling & V.Yu. Podlipchuk (1996): "Molecular Dynamics of Time - Correlations in Solutions", Z.physik.Chem. 193, 207-212. W. Ebeling, Yu. Romanovsky, Yu. Khurgin, A. Netrebko, N. Netrebko, E. Shidlovskaya (1994): "Complex regimes in the simple models of molecular dynamics of enzymes", Proc. SPIE 2370, 434-447. W. Ebeling, V. Podlipchuk & A.A. Valuev (1995): "Molecular Dynamics Simulation of the Activation of Soft Molecules Solved in Condensed Media", Physica A 217, 22-37. W. Ebeling, A. A. Valuev, V.Yu. Podlipchuk (1997): "Microscopic models and simulations of local activation processes", J. Molec. Liquids 73,74, 445-455. W. Ebeling, V.Yu. Podlipchuk, M.G. Sapeshinsky (1998): "Microscopic models

Discussion

of activation processes in an atomistic heat bath

143

and simulations of local activation processes", Int. J. Bifurc. & Chaos, 8, 755-760. W. Ebeling, M.G. Sapeshinsky (2001): "Microscopic models and simulations of local activation processes", Prog. XXV ICPIG, 255-256. W. Ebeling, M.G. Sapeshinsky (2001): "Microscopic models and simulations of local activation processes", Prog. IUVSTA IVC-15, AVS-48, ICSS-11, 160. B. Hafskjold, T. Ikehoshi (1995): "Partial specific quantities computed by nonequilibrium molecular dynamics", Fluid Phase Equilibria 104, 173-184. P. Hanggi, P. Talkner & M. Borkovec (1991): "Reaction-rate theory: fifty years after Kramers", Rev. Mod. Phys. 62, 251-341. J. Hesse, L. Schimansky-Geier (1991): "Inversion in harmonic noise driven bistable oscillators", Z. Phys. B 84, 467-470. D. Henderson, S. Sokolowski, A. Trokhimchuk (1995): "Association in a LennardJones fluid from a second-order Percus-Yevick equation", Phys. Rev. E 52, 32603262. N.D. Sokolov (ed.) (1981): "Hydrogen bond" (in Russian), Nauka, Moscow. M. Jenssen, W. Ebeling (2000): "Distribution functions and excitation spectra of Toda systems at intermediate temperatures", Physica D 141, 117-132. G.E. Norman, V.Yu. Podlipchuk, A.A. Valuev (1992): "Molecular Dynamics Method: Theory and Applications", J. Moscow Phys. Soc. 2, 7-17. G.E. Norman, V.Yu. Podlipchuk, A.A. Valuev (1993): "Theory of molecular dynamics method", Molecular simulation 9, 417-427. L. Schimansky-Geier & Ch. Ziilicke (1990): "Effect of Harmonic Noise on Bistable Systems", Z.Phys. B 79, 451-458. J.E. Straub, B.J. Berne (1986): "Energy diffusion in many-dimensional Markovian systems", J. Chem. Phys. 85, 2999-3006. P. Talkner, P. Hanggi (1995): "New trends in Kramer reaction theory", Kluwer Academic Publ., Dordrecht.

This page is intentionally left blank

Chapter 5

Excitations on rings of molecules A. Chetverikov, W. Ebeling, M. Jenssen, and Yu. Romanovsky

5.1

Solitary excitations in Toda systems

In the present chapter we investigate the effect of solitary excitations on reaction rates. Solitary waves as excitations of nonlinear chains of molecules found a remarkable interest in the last twenty years (Toda, 1983; Toda & Saitoh, 1983; Bolterauer & Opper, 1985; Trullinger et al., 1986; Mertens & Biittner, 1979, 1986; Theodorakopoulos, 1984; Schneider & Stoll, 1986; Ebeling & Jenssen, 1988, 1991; Jenssen, 1991; Jenssen & Ebeling, 2000). Of special interest for the study of Toda systems is the existence of exact solutions for the dynamics and the statistical thermodynamics. On this basis it was shown in the papers (Bolterauer & Opper, 1982; Schneider & Stoll, 1986; Theodorakopoulos & Bacalis, 1992; Jenssen & Ebeling, 2000) that phonon excitations determine the spectrum at low temperatures and strongly localized soliton excitations are the most relevant at high temperatures. In this chapter we will study the dynamics and activation processes on rings of masses connected by Toda or Morse interactions. Such configurations may be considered as special models of macromolecules, which allow a full theoretical treatment. We postulate a dissipative dynamics which includes white noise and friction. The noise strength is chosen such that it corresponds to an intermediate temperature region where both phonon and soliton excitations are present. In the Toda model such excitations are adequately described in terms of mutually interacting cnoidal waves. We will show here, that there exists a temperature region around a transition temperature TtT between the phonon and the ideal soliton regime, where the interaction of nonlinear excitations has a remarkable influence on several physical phenomena, a first one being energy localization at special sites, and another one the excitation of a broadband coloured noise spectrum with an 1//— region at low frequencies. This way we aim to give a contribution towards a new theory of energy activation processes as, e.g., chemical reactions in biomolecules. Here we will concentrate first on the investigation of classical Toda models of molecular systems. Toda forces (exponential repulsion) were introduced in section 1.3. The extension to more realistic forces (e.g. Morse models) follows in the last paragraph of the present chapter. 145

146

Excitations

on rings of molecules

1.0

0.9 0.8 cv/kB 0.7

0.6 t_T/c

0.5

10'10

10'5

'"I

105

1010

1015

Fig. 5.1 Specific heat per molecule in a Toda chain as a function of the temperature. At low temperatures the phonons dominate and at high temperatures the solitonic excitations. At the transition temperature with cy ~ 3fes/4 the most interesting properties are observed.

In our earlier work we have shown that soft Toda springs are able to trap and superpose pulse-like excitations (solitons) impinging from the hard host lattice (Ebeling & Jenssen, 1988; Ebeling et al., 1989; Jenssen, 1991; Ebeling & Jenssen, 1991, 1992; Jenssen & Ebeling, 1996, 2000). In thermal equilibrium there exists a finite temperature T(oc where the fusion of solitons leads to a maximal localization of potential energy at soft springs (Ebeling & Jenssen, 1991, 1992). We will show in this chapter that a system consisting of few soft springs which are imbedded into a bath of hard springs might be considered as a very rough model of an active reaction site imbedded into a bath of relatively passive atoms. The physical reason is that based on nonlinear excitations energy localization is possible at the soft sites (springs or molecules). In this and in the next paragraphs we concentrate on the study of systems with Toda interactions in thermal equilibrium including noise and friction effects following in large the work (Jenssen & Ebeling, 2000). At first we introduce the model and discuss its general properties. Then the properties of the equilibrium distribution function of the stochastic forces are investigated in particular regarding the transition region. Energy localization at soft sites will be shown to be a special consequence of the properties of the host system in the transition region. By numerical integration of Langevin equations we will study the time correlations of the equilibrium fluctuations of the stochastic forces. The influence of system size and varying degree of thermal coupling will be discussed. Our 1 — d model of a macromolecule consists of N point masses m; connected to the next neighbours at both sides by Toda springs. The actual distance between the mass i and the mass i + 1 is i?j, the equilibrium distance is assumed to be c;, therefore the spring elongation reads r, = Ri - (j{. The spring energy is described

Solitary excitations

in Toda systems

147

by Toda potentials

V{ri;ui,bi)

muj,

= —^-(exp ( - 6 ^ ) - 1 + 6;r;)

(5.1)

Here 6j is the stiffness of the spring i and w, is the linear oscillation frequency around the equilibrium position. We assume then the following equations of motion

Pi =

mi-iUi-! bt-i

c_bi_,ri_r

_

Pi+i

Pi

mi+i

rrii

rmujtc_biri h

1/2

+ 11

(5.2)

{2kBTmi)1l\i{t)-1\'2Pi

(5.3) The second term on the r.h.s. describes noise and damping generated by a surrounding heat bath with



(5.4)

The validity of an Einstein relation between noise strength and damping strength is assumed which guarantees the existence of a thermal equilibrium, independent on the friction parameter 7; > 0. We note that 7; = 0 corresponds to the conservative case. Toda springs are exponentially hard with respect to compression and rather weak with respect to expansion. Because molecules have similar properties we can consider the mass m, together with the spring r* fixed to the right side as a crude 1 — d model of a single molecule. For simplicity let us assume in the following that all the masses mi — m, all friction constants 7i = 7 and all equilibrium distances a; = a are equal. Further we assume that in the rest state at T — 0 the average distance of the particles is Ri = a, i.e. r* = 0. The linear term in (1) describes the dilatation interaction and implies the artificial assumption of an infinite range potential. However, interaction is confined to the next neighbours and consequently mu)}cr £i =

(5.5)

can be considered as the equivalent for the depth of the potential well in the Toda system. This quantity will be used as energy unit in the following. We will differ between uniform Toda chains where all the spring parameters bi = bo and un — UJQ are equal, and nonuniform chains where different spring parameters will occur. In the following paragraph we restrict our study to uniform systems and omit for simplicity all the indices.

148

Excitations

on rings of molecules

5.2

Statistical and stochastic theory of Toda rings

Let us begin with the discussion of the statistical properties of a canonical ensemble of "Toda rings" with pressure P and temperature T. With the abbreviations X

abkBT

Y =

P bkBT

the distribution function of a single molecule reads

f(p,r) = Zl exp(^ z-i._. 1

—

j

bx*+y y/2irnikBT exp (X) T(X + Y)

l

'

;

with the effective potential Veff(r)

= V(r) + Pr

Here kB denotes the Boltzmann constant. V(r) is the Toda potential as defined above. All thermodynamical functions follow by derivation of the partition function with respect to the corresponding parameters. Due to the Einstein relation the equilibrium distribution function does not depend on the friction strength 7. The mean specific volume is given by

v = j(lnX-y{X

+ Y)) + a

(5.7)

0

From this the pressure P is determined by: P = 6feBT*-1(lnX-6(v-(T))-e/a

(5.8)

where the digamma-function * is defined as usual by the logarithmic derivative of the T-function. The specific internal energy of the system is given by

e = E/N = ^kBT

+ (u)

(5.9)

The mean potential energy reads

(u) = ^ + ±(\nX-^(X 0 oa

+ Y))

(5.10)

Statistical

and stochastic theory of Toda rings

149

where the pressure P is given above. At low temperatures the equipartition theorem holds and we find

<«> = \kBT

(5.11)

In the high-temperature limit the "Toda rings" behave like rings of hard spheres and the potential energy disappears. The specific heat per molecule at constant volume reads

cv = kB(\+X

+

Y--^Y))

(5.12)

where the trigamma-function & is defined as usual by the second logarithmic derivative of the T-function.This function tends to ks in the low-temperature regime due to the thermal energy ksT of each phonon. On the other hand, cy shows the properties of a 1-d hard-sphere gas at high temperatures, i.e. cy tends to l/2fcs. For constant total length all the thermodynamic functions can be obtained from the phenomenology of an ideal soliton gas in the high-temperature limit, each soliton possessing a mean thermal energy of fcsT/2. However, all attempts to reconstruct the thermal properties of Toda systems from a phonon/soliton phenomenology failed in the transition region between the phonon regime and the regime of non-interacting solitons. Here we introduce a transition temperature Ttr corresponding to cy = 3A;B/4 (Fig.5.1). In the region around Ttr thermal excitations should be described more generally as cnoidal waves which contain both phonons and solitons as special cases in the low- and high- amplitude limit, respectively. Loosely speaking, cnoidal or solitary waves can be considered as deformed phonons with more or less steep compression humps and shallow dilatation valleys between them. In the transition region the interaction between solitary-wave excitations may lead to interesting physical effects as will be demonstrated in the following sections. In the remainder of this section we will discuss the time-independent properties of the stochastic forces. According to the definition of the Toda potential the force F acting on a molecule is given by

F=-(exp(-6r)-l) a The equilibrium distribution follows as tl„.

a

Xx+Y

/

aFY^-1

(5.13)

( aXF\

,C1^

150

Excitations

f(F)*E/o

on rings of molecules

10

F*a/e

Fig. 5.2 Distribution of the force acting on one molecule in a Toda chain for 3 different temperatures corresponding to the phonon regime (dotted line), the transition regime (solid line) and the solitonic regime (dashed line).

In the low-temperature limit we find the well-known symmetrical shape of the distribution function around the mean value F = 0 which is obtained for harmonic lattices (Fig. 5.2). With increasing temperature the distribution becomes biased and the maximum is shifted towards the left. As can be seen from the distribution, for X + Y < 1 the probability density diverges at F = — e/a whereas the integral over the F axis yields 1. The value F = —e/a corresponds to an infinite dilatation r. This behaviour reflects the tendency of the molecules of the chain to drift apart in the result of collisions, i.e., due to the presence of thermal solitons. As can be seen from our expression for cy, this behaviour is observed for cy < 0.89/CB, i.e., both in the transition region and in the region of noninteracting solitons. In the latter case, if T tends to infinity, the distribution function tends to get a rectangular shape consisting of a narrow peak at F = —e/a and a shallow but long-range tail for large F values which is typical for hard collisions (Fig. 5.2). In contrast, at the transition temperature we observe a relatively broad distribution with an exponential decay for large F. This behaviour reflects the dominance of "soft" collisions with relatively long interaction range. By integration we find for the mean value of the force F

(5.15)

(F) = P and the relative mean square deviation reads

52 =

(Ff

p2

+?I*BT

(5.16)

Energy accumulation

at nonuniformities

151

10 5

10 4

10 3

5

2

10 2

10

kBT/e

1 10- 10

T

10"5

1

10 5

1

10 10

1

1015

Fig. 5.3 Dependence of the mean square fluctuation of the molecular forces on the temperature; the minimum corresponds to the transition temperature.

Representing this quantity as a function of temperature we see, that the relative mean square deviation has a minimum near to the transition temperature Ttr (Fig. 5.3). Finally we notice that in the case of fixed total length the distribution functions given above are exactly valid only in the thermodynamic limit, i.e., for an infinite number of molecules. This is due to the fact, that we fixed the volume only in the thermal average by eq.(6). Hence the nature of fluctuations around the mean values may be modified for decreasing number of molecules in the chain. In particular, in a finite Toda chain the probability density of the force F cannot diverge at F — —e/a. With the constraint of periodic boundary conditions the expansion tendency of a finite Toda ring around Ttr may lead to a strong coherence of the molecular motion affecting the time correlations of the fluctuations, as will be shown in section 5.4.

5.3

Energy accumulation at nonuniformities

Now we will extend our model to the case of nonuniform rings which are considered as dilute solutions of n chemical sorts each represented by Ns soft "Toda atoms" with stiffness ba(s = l,...,n) in a bath of iVo hard "Toda atoms" (bs < bo) at constant volume. In the limit 60 —> 00 the bath becomes a l - d hard-core gas. In the following we assume Ns << No- Then the pressure is determined by the bath of hard molecules and remains the same as before (eq. (5.7-5.8)). The internal energy of the system is given by

E=^kBTlNo + J2Ns)+N(u°)+J2Ns(u°)

( 5 - 17 )

152

Excitations

on rings of molecules

5

4

3

/k B T 2

1

kBT/E 0 10"5

1

10

5

10

10

10 1 5

Fig. 5.4 At the transition temperature the mean potential energy of weak Toda springs imbedded into a bath of hard springs shows a pronounced maximum.

The mean potential energies of hard and soft molecules, respectively, read (Jenssen, 1991)


(us) = ?- + ^(\nXs-y(Xs

(5.18)

+ Ys))

(5.19)

where the pressure P is given by eq. (5.8). At low temperatures the equipartition theorem holds and we find

(u0) = (us) =

l

-kBT

(5.20)

In the high-temperature limit the "Toda molecules" behave like hard spheres and the potential energies disappear. Especially we find for the case v = a

{us) = ^ K )

(5-21)

Us

In previous work this exact result was reconstructed from a gas of noninteracting solitons which dominate the dynamics at high temperatures (Jenssen, 1991). There was also shown that at intermediate temperature the mean potential energy of soft molecules reaches a maximum in units of thermal energy fcgT (Fig.5.4). This effect was explained by superposition of thermal excitations at the soft molecules corresponding to multiple elastic collisions. Now we will study this maximum in more detail. From the localization condition

Energy accumulation

»

at

153

nonuniformities

=

.

(5-22)

we find using eqs. (5.10, 5.21)

lnXs - * [X. + Y.) - bh°Z{*Sty\

+, ' ,

osW'(A0 + YQ)

(5-23)

Y)

eaW(Xo + Y0)

+ <E±—El (l-X,*'{Xa+

¥.)) = 0

Equation (5.23) can be solved numerically yielding the temperature T\oc defining the maximum localization point. According to eq.(5.5) es can be interpreted as the depth of the potential well or binding energy of the soft molecules. For a given stiffness bs it defines the linear oscillation frequency of the soft molecules. With decreasing es the localization point T\oc is shifted to lower temperatures but (us(Tioc)) /kBTi0C increases slightly (cf. curves in Fig. 5.4). This means that mainly low-amplitude or phonon-like excitations will be localized at the soft sites with decreasing linear oscillation frequency. However, the time scale for high-amplitude or soliton-like excitations is given by the stiffness parameter. Hence the amount of collimated energy (us(Tioc)) /ksTioc increases significantly with decreasing bs (cf. upper curve in Fig. 5.4) due to the enhanced superposition of thermal solitons (Ebeling and Jenssen, 1988, Jenssen, 1991, Jenssen and Ebeling, 2000). In order to study this effect in more detail we will set es = e in the following. For sufficiently low stiffness b3 of the diluted molecules the localization temperature Tioc depends on the properties of the surrounding hard Toda chain only (Fig. 5.5). This behaviour was already observed in molecular-dynamics simulations of more realistic 3 — d models of dense fluids (see Chapter 4). In particular, for a vanishing ratio of stiffness parameters bs/bo the localization condition (5.22) can be simplified yielding

I n X . - ^ . + y

^

j

^

l

^

^

-

^

.

)

^

(5.24)

Solving this equation together with eq.(5.7) and inserting the solution in eq.(5.12) we obtain for the specific heat at the localization point

cv{Tloc)

~ 0.73

(5.25)

yielding a good agreement with the definition of the transition temperature Ttr given above. This result confirms the statement that energy localization at soft

154

Excitations

on rings of molecules

sites is a special consequence of solitary-wave interaction in the transition region. Finally we obtain from eqs.(5.17, 5.18, 5.23) for the mean potential energy of soft molecules at localization temperature a linear relation to the ratio of stiffness parameters

(us(Tioc)) = const.-±kBTloc

(5.26)

Energy localization at soft sites in thermal equilibrium is due to the pressure P produced by the bath of hard molecules in the transition region and can be attributed to the superposition of solitary waves corresponding to multiple elastic collisions. According to eqn.(5.15) the pressure P acts as a random force with a non-zero mean on the soft molecules. Hence the distribution functions (5.6) of the soft molecules are raised up to higher energies. If we interpret the soft molecule as a reactive site with a certain activation barrier, this force P may lead to a considerable enhancement of the transition rate in the region of Tj oc as was shown within the frame of elementary transition-state theory (Ebeling & Jenssen, 1991).

5.4

Fluctuations in Toda rings and time correlations

Beside the static localization effect discussed in the last section we expect novel effects concerning the nature of fluctuations around the equilibrium functions due to the interaction of nonlinear excitations in the transition region. So we will now proceed to the investigation of time correlation functions (ACF) and the derived spectra. In this section we restrict ourselves to the investigation of uniform Toda systems. Since the analytical theory of the ACF is restricted to the harmonic case, we rely on computer experiments in the following. Correlation functions from molecular-dynamic simulations of Toda systems were calculated, e.g., by Schneider (1986), mainly in order to identify thermally activated solitons in the spectra. Here we ask the question if Toda systems may transform the uncorrelated fluctuations of the surrounding heat bath, which are modelled by the Gaussian white noise (,(t) in eq.(5.4), into some kind of coloured noise that may favour low-frequency activation processes. This question is of central importance especially for the understanding of transition processes in complex biomolecules (Welch et al., 1982; Havsteen, 1989; Ebeling et al., 1989; Chikishev et al., 1998; Regeida et al., 1999). In contrast to the equilibrium properties following from the distribution function (5.6), the time correlation functions depend on the friction parameter 7, because its inverse determines the characteristic relaxation time of the system (5.2-5.3). In the conservative case 7 = 0 the system follows the Hamiltonian dynamics. On the other hand, for large 7 the Hamiltonian dynamics is modified and the "Toda atoms"

Fluctuations

in Toda rings and time

155

correlations

0.2

kBTlx/e 0.1

/ /

1

b/b s .

i

10

,

i

i

100

i

i

.I

1000

Fig. 5.5 Temperature of maximal energy localization at weak Toda sites imbedded into a bath of hard Toda springs. The temperature Tioc ~ 0.26e corresponds to the transition, since cv(Tioc) ~ 0.73fcB.

reproduce the white noise of the surrounding heat bath. So we performed our simulations of the Langevin eqs. (5.2,5.3) for the case of weak thermal coupling with 7 = 10_3wo- For integration we used a Runge-Kutta algorithm of 4th order using a constant stepsize of 10_3o;o including white noise sources according to eq.(5.4). First the system was heated up to a temperature around Ttr. After attaining the thermal equilibrium we calculated the one-particle ACF of the forces (5.13) from the trajectory

ACF(t) = (6F(t0 + t)5F(to))

(5.27)

From this we calculated the spectrum (FF) defined as the Fourier transform ofeq.(5.27). For low temperatures the Toda system behaves like a harmonic lattice and the dynamics is determined by the phonons. Fig. 5.6 shows the force spectrum calculated from a long-term simulation of a molecular ring consisting of N = 10 identical "harmonic molecules" subject to periodic boundary conditions. As expected, the excitation spectrum shows 5 distinct peaks at the well-known normal-mode frequen-. cies. Naturally, the white noise of the surrounding heat bath is reproduced at low frequencies. For N —> oo the thermal energy will be distributed equally among an infinite number of phonons, the distinct peaks disappear and we observe a whitenoise spectrum over the entire frequency range. On the other hand, for high temperatures the Toda spectrum behaves like an ideal hard-core gas characterized by the simple white noise again. Let us study now the characteristic features of nonlinear excitations. Fig. 5.7 shows typical realizations of the force acting on a Toda site in dependence on time and

156

Excitations

on rings of molecules

(FFVmax((FFX„) 1-3

0.001-

(l)/COo

0.00010.01

"1 10

Fig. 5.6 The spectrum of the force-force time-correlations for a uniform ring of 10 harmonic oscillators. We observe 5 phonon peaks at u = 2u>osin(iir/10) for i = 1,2, ...5 and a white noise tail at small frequencies.

Fig. 5.8 shows the corresponding excitation spectrum of the one-particle ACF. The curves were obtained from a simulation of a uniform Toda ring made up of TV = 10 "Toda atoms" in the transition-temperature region. The trajectory of the molecular force F shows excitations on many time scales. The high-energy events corresponding to high-compression peaks or solitary waves occur most likely in clumps as it is typical for beating phenomena (Fig. 5.7). The excitations on many time scales correspond to a very specific spectrum (Fig. 5.8). Most strikingly, in the double-logarithmic presentation we observe a straight line with a slope near to — 1 clearly indicating a broadband coloured noise of 1 / / type at low frequencies. In particular, the 1 / / spectrum implies a hierarchy of beatings where periods with more energetic compression pulses are more probable to appear at longer time intervals. The hierarchical order of the fluctuations is due to the Hamiltonian dynamics of the finite-size Toda ring. Finally let us try to get a hint on this underlying dynamics by comparison of the spectra of the harmonic and the Toda ring in more detail (Fig. 5.9). We notice the complete failure of the first phonon peak in the Toda spectrum. The time-dependence of the force-force ACF is shown in Fig. 5.10. In the harmonic ring the (linear) excitations of the first phonon correspond to a standing wave with two knots which can be decomposed into 2 waves of the same frequency but running in opposite directions. Indeed, on the Toda ring we observe during the simulations mostly 2 distinct soliton-like wave peaks running in opposite directions and merging into mainly 1 wave peak 2 times each turn (Fig. 5.10). The frequency of these solitary waves corresponds to the broad peak in Figs. 5.7 and 5.9 which has its maximum accurately at the second-phonon frequency. Furthermore, the speed of the nonlinear waves is varied with the fluctuations of their

Fluctuations

in Toda rings and time

157

correlations

200

F*o7e

t/10000*coo

200

F*o7e 100

t/10000*coo 1,165

Fig. 5.7 Timedependence of the force acting on one site in Toda chain with N = 10 for 2 time scales. We observe a hierarchy of fluctuations on different time scales and beating-like phenomena.

amplitudes and hence we do not have a fixed phase relation leading to standingwave phenomena. Instead, we observe a beating-like modulation of the amplitude of compression peaks due to the strong interaction of mainly 2 solitary or cnoidal waves each of wavelength N and a frequency fluctuating around that of the 2nd harmonic normal mode. The internal dynamics of the Toda ring and the hierarchy of nonlinear excitations is accompanied by long-term correlated random rotations of the whole ring (Fig. 5.11). These coherent fluctuations of the ring correspond to a diffusion regime. With respect to the spectrum of fluctuations there is some analogy to the traffic-jam models which are classical examples of 1 / / noise (Helbing, 1997; Helbing et al., 2000).

158

Excitations

on rings of molecules

(FFX/aV 0,01 —

TTT]

0.01

0.1

oVcoo

100

Fig. 5.8 Spectrum of the force-force correlation function for Toda rings with N = 10 and N = 20. The temperature is in the transition regime, the friction is rather small 7 = O.OOlwo- We observe a 1 / / tail at small frequencies.

(FF)„,/max((FF)„) 1

(0/(Oo

0,1

10

Fig. 5.9 Comparison of the force-force spectra of an N— particle Toda ring with the corresponding ring of harmonic oscillators. We observe the complete failure of the first-phonon peak in the Toda system and instead a soliton-determined peak at the frequency of the second (harmonic) phonon peak.

In this analogy the "Toda atoms" correspond to "cars" moving around a circular highway and the compression peaks make up the jams of different size. However, our "Toda atoms" do not possess a "fuel tank" and are driven only by the Brownian motion of the surrounding heat bath. Consequently, the long-term average of the velocities tends to zero. The case of driven Toda atoms which may use internal energy sources was treated by Erdmann et al. (2000). We stress that the passive Toda rings treated in the present work are in thermal equilibrium and the 1 / / tail of the spectrum reflects the character of equilibrium fluctuations due to nonlinear excitations.

Fluctuations

in Toda rings and time

correlations

159

O

<

Fig. 5.10 The autocorrelation functions of the force acting on a Toda site in rings of 10 particles (full line) and 20 particles (dotted line). We observe a distinct soliton peak corresponding to two solitary waves running in opposite directions around the ring. The time at which the maximum appears corresponds to the time needed for one full rotation, which nearly linearly increases with the size of the ring N.

Prom the above considerations it follows not only a dependence of the effect from the temperature and the degree of thermal coupling, but we presume also a great importance of the boundary conditions and hence a dependence on the size of the Toda ring. We already speculated about coherent molecular motion as a result of the repulsion tendency of the molecules within a finite Toda ring when discussing the distribution function of molecular forces. Clearly, such a coherence gets lost with increasing number of molecules. It follows from Fig. 5.10 that the distinct solitarywave peak corresponding to the second-phonon frequency will not only shift to the right on the time axis but also gets less pronounced with increasing N. Hence the thermal energy gets more equally distributed over a broad spectrum of cnoidal waves with varying wavelength and frequencies instead of being contained in mainly 2 strongly interacting pulses. Therefore in the infinite Toda ring we expect a white noise at low frequencies again, the noise effects we discuss here are typical finite-size effects. The most important observation is a kind of l//-noise. The two main types of l//-type noise systems discussed so far in the literature are: (i) flicker-noise observed in many simple physical systems (Klimontovich, 1995), (ii) long-correlated noise observed in many complex systems in nature and society (Bak, 1996). As shown by Klimontovich, flicker-noise may be interpreted as a diffusion regime

160

Excitations

on rings of molecules

Total moment * a(0(/£ 40

t/10000*0)o

Fig. 5.11 The total momentum of the Toda ring corresponding to the simulations shown in the previous figures fluctuates slowly what corresponds to slow stochastic rotations of the ring.

of the dynamics of finite systems (Klimontovich, 1995). Typical properties of the flicker-noise of Klimontovich-type are: (i) This type of flicker noise appears at frequencies uifi bounded from above by the diffusion time L2/D and from below by the observation time t0bs

(1/ioto) < wfi <

(D/L2)

(5.28)

where D is the diffusion constant (corresponding here to the rotational stochastic motion shown in Fig. 5.11) and L is the length of the system (corresponding here to 10a). (ii) the amplitude of the flicker noise is proportional to l/(Nw), where N is the particle number. A quite different type of l//-noise has been observed in large and complex manyparticle systems far away from thermal equilibrium. This type of l//-noise is of central importance in the theory of self-organized criticality (Bak, 1996). In this section we presented a relatively simple dynamical system fluctuating in thermal equilibrium that transforms the uncorrelated, white noise of the surroundings into noise of l//-type. In many respect the type of noise we have observed corresponds to a flicker-noise of the type investigated by Klimontovich since it is

Spatio-temporal

excitations

on rings

161

clearly connected with a diffusion-type of dynamics as shown in Fig. 5.7 and since it seems to decrease with a dependence 1/N. A closer inspection of this point however requires more extensive simulations; therefore we have to leave this question to further investigations. Even at the present stage of the investigations we may draw some general conclusions: The l//-noise observed in Toda systems is not connected to a fine-tuning of temperature, structural parameters, particle number, or thermal coupling, but occurs in a wide range of these quantities with varying intensity. Furthermore, it can be expected to persist for a wide class of more realistic molecular potentials in the transition region. However, it was not observed in previous investigations for Lennard-Jones molecules at higher dimensions so far (see Chapter 4). This point needs further clarification. From the investigations carried out so far for our models we hypothetically derive the following preconditions for fluctuating equilibrium systems to transform white noise into l//-type noise: (i) Nonlinear, asymmetrical interactions between molecular units with a steep repulsive and a flat attractive branch. (ii) Quasi-Id configurations of a finite many-body system with periodic boundary conditions (ring). (iii) Weak thermal coupling to a surrounding heat bath in the transition-temperature region. We hope that the further detailed investigation of the effects demonstrated in this section paper will support the understanding of complex molecular motions and energy activation processes. A first application in this direction is presented below.

5.5

Spatio-temporal excitations on rings

The volume/density fluctuations discussed in the previous section are closely related to the structure factor which is a central quantity for the description of excitations (Schneider & Stoll, 1986; Mertens & Biittner, 1986; Ortner, Schautz & Ebeling, 1997; Ebeling & Ortner, 1998; Ebeling, Chetverikov and Jenssen, 2000). We will use here the definition (Ebeling and Ortner, 1998): 1 r°° •wt S(w, k) = ^ J e (p(k, t)p(-k, 0)> dt,

(5.29)

P(*'') = ^ E e x p H M * ) )

(5-30)

where

162

Excitations

on rings of molecules

is the Fourier component of the density of masses in the chain. The static structure factor is defined as the integral over u> which is denned by

s

w = jj E <exp Wri - r*))>

(5-31)

This quantity may be expressed by the partition function. Several analytical and numerical approaches to estimate the structure factor in the volume edited by Trullinger, Zakharov & Pokrovsky (1986). The dynamical structure factor allows judging time behaviour of spatial (collective) structures of specific scales, generated on a ring. Setting quantity k, defining a frequency composition S(w, k) and estimating breadth of spectrum peaks, it is possible to judge a stability of structures (nonlinear waves) and velocity of its motion on a ring. We study here rings of N masses with Toda interactions. Most of our simulations were carried out for rings with N — 10. This has the advantage that the number of excitations is still rather small and so we can observe their shifts in some detail. Further we are interested in finite size effects, which play an important role in the soliton-determined region. Let us first discuss the linear case of only N phonons i = —N/2 + 1-7- N/2 with wave numbers and frequencies ki0 = (2Tri/N(T),

wi0 = 2w0|sin(fcio(7/2)|

(5.32)

For simplicity it is supposed that N = 2M is an even number. The positive and negative signs denote waves traveling either to the right or to the left, the excitation with k = 0 corresponds to very slow rotation of the ring as a whole, and k — n denotes the non-traveling wave. The motion of any mass is strictly sinusoidal. The speed of the longest traveling wave of i — 1 is near to the long wave limit vs — UWQ which is the sound velocity. The group velocity of the phonon excitations is always smaller than the sound velocity

vig = vs cos ——

(5.33)

Including nonlinear interactions, in the simplest case only quadratic terms, the phenomena due to nonlinear multiwave interactions in a medium with dispersion are very specific. Suppose that N/2 waves are propagating left/right on the ring in accordance with the dispersion relation D(u>i,ki), including two components of the standing wave with a wave number ki — ir. Then due to the nonlinearity combination waves will arise with frequencies and wave numbers

kc = ^2niki,

wc = ^ n i W i

(5.34)

Spatio-temporal

excitations

on rings

163

Wave numbers kc coincide with wave numbers ki always because of equally spaced components of a wave numbers spectrum on a ring. But combination frequencies OJC do not coincide with Wi in a general case due to dispersion and may be distinguished in the frequency spectrum. If nonlinear interactions cause the waves to have forms very different from sinusoidal, the quasiharmonic approximation will not work. In the sufficiently (but not very strongly) nonlinear case e.g. for the Toda ring the situation is the following. In the unlimited chain we could observe N cnoidal waves which might be described by elliptic functions. In some sense the cnoidal waves are nonlinear modifications of the phonons, they are still periodic functions but are not sinusoidal. A wave length of cnoidal waves grows when their energy increases and nonlinearity develops, and then they convert into N proper solitons. The velocity of the solitons depends on their energy but it is always larger than the sound velocity. However, the wave length of nonlinear waves on a ring is limited by the ring length, so they are not cnoidal waves, strongly speaking, when the energy becomes sufficiently high. Thus, each phonon converts to the nonlinear ("like cnoidal") wave with only one hump on the ring finally. But in relatively long rings they may be considered as solitons if their widths are much less in comparison with the ring length. Specific role is played by the first phonon with the longest wavelength. It has the space structure topologically like the one hump nonlinear wave from the very beginning. Therefore it is transformed to the soliton at lower temperature than the other "many hump" phonons. We now consider these linear and nonlinear excitations in thermal systems: In accordance with section 3 and refs. [4-7], the excitations at low temperatures are determined by the phonons and the excitations at high temperatures are soliton-like. Between these limits there is a temperature, where the specific heat turns from ks to ks/2. Around the transition (or localization) temperature we observed in earlier work very specific physical properties and therefore we will pay here special attention to it. We calculate the dynamic structure factor (DSF), since this quantity contains a rich information about spatial and temporal excitations as well. We simulated a dynamics by a Langevin equation including a white noise at given temperature, then we switched off the noise and the friction (7 = 0) and started the calculation of the dynamic structure factor under Newtonian dynamics. This procedure ensures that the dynamical effects studied refer to a Hamiltonian system in thermal equilibrium at temperature T. In our numerical calculations we use the same dimensionless variables as in previous sections. In particular, we suppose m = 1 and use dimensionless time T = t(cj0/2n) and coordinates r' = R/a0. Our choice for w0 is the linear oscillation frequency around the minimum and a0 is the equilibrium distance. The dynamic structure factor of a ring of N identical linear oscillators may be estimated analytically. Taking into account that the time of integration Tint of the integrals defined above are limited in numerical modeling we get for this case the analytic expression

164

S{u,k) _

Excitations

1

JZ

on rings of molecules

{sm(±(k-km0)N)\

/|sin(i(W-Wroo)Tint)|

\±(k-km0)N\ J \

||(w-« m o)r int | J

(5.35) Here Smax is the maximum of S(uj,k) in the range of (u>,k) considered, am — const/w„0. The last follows from the fact that the dynamical structure factor is determinated by averaged displacements of particles < rf > and each phonon gives contribution to < rf > proportional to l / w ^ 0 because of all phonons have the same potential energy < um > = | w ^ 0 < r*hm >= \kBT. Here < r*hm > denotes the averaged displacement in the m—th phonon. The phonon with k = LJ = 0 ("zero" phonon) corresponding to rotation of the ring as a whole should be discarded in our consideration because of am=o —> oo. As it is supposed that N = 2M = 10 is an even number, then M linear phonons traveling to the right and M linear phonons traveling to the left with frequencies ujio may be excited in the chain taking into account that the standing wave with k = IT may be represented as a superposition of two opposite traveling waves. As it follows from the above equation, the dynamical structure factor of a linear oscillators ring as a function of a frequency looks as a single spectral line of the frequency w^o for the case k = fcjo- But for k ^ k^ we can see 5 peaks corresponding to the 5 different frequencies w;o- It follows that the initiation of excitations with wave numbers near to ki may be observed if A; = ki. Further we may observe a shift of wave numbers ki which could be caused e.g. by nonlinear effects. At low temperatures when nonlinear effects are weak, the interaction of the weakly deformed phonons may be described by the quadratic terms in the forces (the cubic term in the potential). This approximation leads to a potential of Fermi-Pasta-Ulam type (Ebeling, Chetverikov and Jenssen, 2000). The Fig. 5.12 shows the result of simulations for this approximation. Due to the quadratic nonlinearity adjacent phonons having frequencies uiio and Wi_|_i,o give rise to combination components, i.e. excitations with frequencies Au>i — u)i+ito — w;o and intensities S(Auii) ~ ajOi+i. Because of the dispersion relation of the ring u — oj(k) the lowest component of them is A W M - I , and the highest one is Awi which is near to woi. The wave numbers of the excitations A/c, are approximately equal to k\ and they are very much pronounced in the spectrum S(w, k = kit0), especially if N is not big. In this case there are only ./V = 2M traveling phonons with M frequencies on the ring and we are able to study their shifts in detail. As phonons convert to cnoidal waves, it is reasonable to suppose that under increasing of temperatures first two soliton-like structures with only one maximum, traveling in opposite directions, will be excited on the ring appearing from two lowest traveling long wave phonons. They are equivalent and we will consider for definition the one with a positive wave number. For simplicity we call this onehump cnoidal wave the "soliton". Its energy is small and its velocity is near to 1 in

Spatio-temporal

excitations

0.01

0.1

on rings

165

o.i

S E u,

a to a; 3

0.01

0.001

0.0001

le-05 1

10

w

Fig. 5.12 Dynamic structure factor under weak nonlinearity for the first phonon wave number k — k\fl at T = 0.5 * 1 0 - 3 e . We observe a spectral peak of the first phonon and four combination components.

accordance with the dispersion relation but always a bit larger than 1. Consequently, the recurrence frequency of the soliton on the ring is near to the basic frequency tOio but grows when energy is increased. At transient temperature the frequency corresponding to the soliton wso; is already twice the initial value uji0 (Fig.5.13) and continues to grow further as well the velocity of the soliton. Simultaneously a shift of the wave numbers and an increase of the frequencies of higher cnoidal waves in the range W20 -=- w50 — 2 take place. Near Ttr all of them including the soliton" interact intensively giving rise to combination components with frequencies lower than the frequency of the soliton wso; = W\ due to dispersion effects (Fig. 5.13). Also the harmonics 2u>soi, Zu>soi and others become important. However, higher waves are less stable already than the soliton as they begin to change their space structures to convert into nonlinear waves with the only hump at high temperatures. It is proved by results of the study of a ring chain with relatively strong connection to surrounding (7 = 10~3w0, Fig.5.14). The noise spectrum corresponding to coloured noise is observed at T « Ttr with pronounced frequency peak of the soliton, a decaying tail corresponding to higher waves and harmonics and the only distinguishable combination. So structures corresponding to higher waves are easily destroyed under influence of noise. This is always observed by analyzing the dynamical structure factor under increasing coupling of the chain to the surrounding. With increasing AT the process of forming of the high energetic soliton as result of synchronization and degradation of some low frequency phonons becomes not so distinct because of density increasing of the spectrum of phonons. Here however we did not study in details rings with big JV. When temperature increases in the region T > Ttr, energy, velocity and fre-

166

Excitations

on rings of molecules

Fig. 5.13 Dynamic structure factor for the wave number k = /s^o near at T = 0.13e near the transition temperature. We observe spectral peaks of the soliton (w s o ;), its harmonics and combinations with w < u/soi.

Fig. 5.14 Dynamic structure factor for the wave number k = kio at T = 0.13e of the ring imbedded into a heat bath with 7 = 10 -3 u>o.

quency UJSOI of the soliton grow too. Furthermore high cnoidal waves convert into soliton like structures also. The soliton becomes more narrow, hence its harmonics in the spectrum S(w) grow as well and combination components decrease owing to moderation of the waves interaction (Fig.5.15). The results described refer to the behaviour of ring excitations on a space scale corresponding to the main resonance scale of the ring defined by its length. We

Spatio-temporal

excitations

on rings

167

Fig. 5.15 Dynamic structure factor for the wave number k = fcio at high temperature T = 24.5e. Spectral peaks of both the soliton and its harmonics are observed pronounced. Combinations do not disappear still.

highlight them due to the exclusive role of the soliton as an asymptotic structure of all nonlinear excitations on the ring at high temperatures. However the range of small frequencies and wave numbers is a field of specific interest because of the phenomena of 1 / / noise and flicker noise found in earlier work (Klimontovich, 1995, Jenssen and Ebeling, 2000). In Fig.5.16 we represent the dynamical structure factor for the case of small wave number k = 0.1 for the Hamiltonian system (7 = 0) at the transient temperature. We observe both a spectrum with 1 / / tail at low frequencies and the typical high frequency spectrum of nonlinear waves and combinations described above. But the like spectrum in the small frequencies region is found at small temperatures as well, in spite of the fact that the average velocity of particles is set to zero at the beginning of the equilibrium stage and that the average shift of all particles excludes a rotation of the ring as a whole. It means that we can not exclude even the possibility that the weak numerical noise in calculations of the dynamical structure factor plays a role in virtue of < r% > ~ 1/w2 —>• 00 for excitations with w « k « 0 even if they have very small energy. Moreover it might be that chaos phenomena in this Hamiltonian system gives a contribution to a low frequency part of the DSF. But if a ring contacting with a surrounding is considered (7 ^ 0), the most essential contribution to the 1 / / part of S(LJ) is provided by a very slow rotation of the ring as whole (Fig.5.17.). It should be associated definitely with excitations with a frequency and wave number near to zero that usually are excluded in thermodynamical considerations. Besides the frequency dependence l / w a , a > 1 is observed at the low frequency edge of this part of the spectrum. It may be supposed that it is due to a diffusion

168

Excitations

on rings of molecules

Fig. 5.16 Dynamic structure factor for the small wave number k = 0.1 at T = 0.13e. We observe both peaks in the region of traveling nonlinear waves and 1 / / spectrum at small frequencies

Fig. 5.17 Dynamic structure factor for the small wave number k — 0.1 at T = 0.13e of the ring imbedded into a heat bath with 7 = 10~3W0'

type flicker noise predicted by Klimontovich (1995) as mentioned above. It takes

place in the frequency range defined by (5.28). In accordance with the work of Klimontovich the dynamic structure factor as a function of a frequency and wave number is expressed at small ui and k by Dk2

S{u,k) = C 1 w + (Dk2)'

<

D k

\

(5.36)

Spatio-temporal

S

excitations

on rings

169

o.oi -

Fig. 5.18 Dynamic structure factor for the small wave number k = 0.1 at X « 24.5e of the ring imbedded into a heat bath with 7 = 10—3CJQ-

Fig. 5.19 Dynamic structure factor for the small wave number k = 0.2 at T R S 24.5e of the ring imbedded into a heat bath with 7 10 Ju>o- (Narrow dips in the range considered are not of physical nature)

where C is a constant. The maximum value of dynamic structure factor is realized at w = ujextr ~ 0.725Dk2. As it follows from an estimation, cjextr is small enough at T w Ttr. Therefore first the phenomenon has been studied at high temperature T = 24.5 « 190T tr . Indeed we find nonmonotonic function S(u) with the maximum shifted as k2 (Figs. 15.18 and 15.19). Returning to the system under the transient temperature, we can see a slope

170

Excitations

on rings of molecules

corresponding to nicker noise which is shifted as k2 indeed. But much further studies are required to confirm the relation S(CJ, k) w Du>2/k2 at small Dk2 and its connection with the flicker-noise of Klimontovich-type. 5.6

A ring model of enzymes

As we have underlined already several times in the previous Chapters, the detailed investigation of the dynamics of molecular processes is of central importance to a better understanding of the activation processes in enzymes. In fact many activation phenomena in complex molecular systems as, e.g., enzymatic reactions or protein-folding processes remained unexplained so far. We summarized also several approaches to extend the original Kramers theory to more complicated reactions and to treat the coupling of non-reactive molecular systems to the reactive site explicitely. As a particular simple 1-d models of excitations in molecular systems we investigated in the previous section Toda systems. We have shown that it is the special form of the Toda interactions which admits exact solutions for the dynamics and statistical thermodynamics. In respect to activation problems the most important result is, that in the transition-temperature region the strong interaction between solitary waves may lead to an enhancement of transition rates. We have studied several mechanisms in Chapter 3 and 4. The most effective one is the energy localization at "soft" reaction sites embedded in a hard Toda chain (investigated the present Chapter) or in a two-dimensional bath of hard molecules (Chapter 4). A complete theory of activation processes assisted by nonlinear excitations is not yet available. In some earlier work (Ebeling, Sapeshinsky and Valuev, 1998) as well as here in the forth chapter we estimated reaction rates by means of molecular dynamics simulations. Further we gave analytical estimates according to transition-state theory assuming equilibrium distributions of coupled reaction systems (Ebeling & Jenssen, 1991). Several estimates resulted in considerable rate enhancements at intermediate temperatures. One of the factors leading to enhancement follows from the non-zero average force (pressure) which acts on a soft reactive molecule and is of static character. This force affects the exponential factor of the rate constant significantly by lowering the activation barrier. Another dynamic effect of enhancement can be interpreted by a superposition of solitary waves which becomes possible at "soft" molecules at intermediate temperatures. It is not restricted to 1-d Toda lattices but persists also in more realistic 2-d and 3-d models of dense fluids consisting of solvent and solute molecules with Morse- or Lennard-Jones interactions. Superposition of solitons corresponds to multiple collisions in these systems. In higher dimensions a weak localization of potential energy was observed also at the bindings of the bath molecules and was connected to a transition between different lattice configurations(Ebeling et al., 1995).

A ring model of enzymes

171

Beside this static effect a considerable rate enhancement can be expected from a transformation of the uncorrected noise of a surrounding heat bath into a broadband coloured noise with a long tail at low frequencies. This behavior was proven above for a finite-size Toda ring with weak thermal coupling in the transitiontemperature range. Such a system seems to be an ideal host for the excitation of special active sites possessing resonance frequencies inside this low frequency band. The finite-size Toda ring with moderate coupling to a surrounding heat bath in the transition-temperature region is perhaps the simplest classical model of a ring-shaped biomolecule in solution. In the following investigation based on an earlier work (Ebeling, Jenssen and Romanovsky, 1989) the real molecular forces in proteins and DNA will be fitted to Morse potentials. We will show that the transition-temperature region corresponds to the range of physiological temperatures (Muto et al., 1989). Indeed, statistical analysis of experimentally observed single-molecule trajectories of the enzyme cholesterol oxidase revealed significant and slow fluctuations in the reaction rate (Lu et al., 1998). This effect was described as a molecular memory phenomenon, in which an enzymatic turnover is not independent of its previous turnovers because of slow fluctuations of protein conformation. Such long-term correlated conformational changes which are relevant in context with protein-folding processes and enzyme reactions might be explained by dynamical mechanisms similar to those observed in our simple Toda model. In the models discussed before we found coherent molecular motions, in particular mainly 2 soliton-like waves of similar amplitude and frequency running in opposite directions. The fluctuations of their amplitudes and frequencies lead to some kind of nonlinear beating phenomenon that is connected to a region of the spectrum that is similar to l//-noise. In particular we consider a molecular ring in the diffusion regime as an ideal host for the excitation of special active sites possessing resonance frequencies inside this low frequency band. A possible field of applications is the dynamics of biomolecules with encymatic activity (Welch et al., 1982; Havsteen, 1991; Chikichev et al., 1998; Lu et al., 1998). Consider a particular enzyme molecule. Cluster models of enzymes sometimes give examples of the chains of masses interconnected by H-bonds. One of this models was discussed in Chapter 1 (see Fig. 1.2). Recall that the successful operation of the molecular scissors (CT) depends on the proton transfer in the hydrogen bond from oxygen (of Serl95) to nitrogen (of His57). Note that these amino acids are located on different subglobules. It is seen from Fig. 1.2 (see Chapter 1) that each subglobule consists of 6 rods interacting with each other by H-bonds. Therefore, the vibrations of the rods and the subglobules may change the parameters of the selected H-bond ( N...H-0) in the enzyme active site and substantially influence the probability of the proton transfer. This problem is discussed in details in Chapter 8. Here we make an attempt of estimating the statistical regularities of the cluster (rod) vibrations using a simple model. We consider a chain of masses forming a

Excitations

on rings of molecules

V(r0<0), kJ/mole

a)

30

* .

20 10

••

r.

.1

0 10

20

t, ps O.l-i

00.1-

r0, nm

: x*.: :-S ^ /A M r*. r**x^*\f~y/~fi^ v v './

..

•;.•

:j

v'

v

•;

V

v

1/

20

10 t, ps

b)

V(r 0 <0), kJ/mole 30 20 H 10

V .A ft

>*-* 10

S\

-A20

t, ps

r0, nm O.H

:ywyvvwvvv^ A

'•

T

10

t, ps

20

Fig. 5.20 Simulations for the Morse potentials (a) and for quadratic potentials (b). Represented is the potential energy V and the elongation ro of the active spring after an initial excitation of the weak spring.

ring with H-bonds described by the Morse potentials. Based on the ideas and results discussed above, we propose the following ring model for an enzyme: T h e dynamic systems consists of n interacting spring interaction by forces with the potential V(rn, where rn denotes the deviation from the equilibrium position. We will use as model cases the Toda potential

A ring model of enzymes

V(r„) = {an/bn) [exp(-6„r n ) - 1 + brn]

173

(5.37)

and in most cases the more realistic Morse potential V{rn) = Dn [(exp(-A„r„) - l ) 2 - l]

(5.38)

Both potentials reduced after expansion up to quadratic terms into harmonic ones. For the Toda case the frequency is given by

ul = anbn

(5.39)

Ji = 2DnA2n

(5.40)

and in the Morse case we have

The equations of motion are defined by

—j- =vn-

u„-i

(5.41)

nil

^ = - 7 n « n + [V'(rn+1 - V'(rn)} + Fn(t) at n = 0,l,...N-l

(5.42)

(5.43)

Here Fn(t) is an external force acting on the mass n which will be used to model external impacts. For the simulations which we will discuss now we assumed N = 12 masses connected by Morse springs, where n = 0 corresponded to a weak spring and n = 1 — 11 to hard springs. In order to be as realistic as possible we assumed Dn = D — 17kJ/mole,An = 0.5nm _ 1 for n = 1 — 11. These parameters should be a rough approximation for the situation of hydrogen bonds. For the weak spring which should model the active site we assumed AQ — A/50 = O.Olnm -1 and WQ = w 2 /10. The masses were adapted to the case of a-chymotrypsin corresponding to M„ = 2083a.u. + / - 10%. The motion of the active spring is exposed to the solvent modelled by a friction 70 = 2 . 8 6 1 0 n s - 1 . The internal motions of the springs n = 1 — 11 are assumed to be undamped 7„ = 0. An initial impact modeling an adsorption event is different from zero only for a small moment

Fu = -F0{t) = C, ifT <= 2TT/UJ0

(5.44)

Excitations

174

on rings of molecules

V(r 0 <0), kJ/mole

a)

30 20 10

i / V / V / w A . . ^ /\

J^

-.

20 t, ps 0.1-

r0, nm

.^yV^Vw^vv^^ i

10

20 t, ps

V(r 0 <0), kJ/mole

b)

30 20 10

-~T—•

r-

20

10 t, ps r0, nm O.l-i

0-

VV>Wy

0.1-

-r 10 t, ps

20

Fig. 5.21 Simulations for the Morse potentials (a) and for quadratic potentials (b). Represented is the potential energy V and the elongation ro of the active spring after an initial excitation of the hard spring.

The height of the pulse is adjusted in such a way to provide a maximum total energy of 65kJ/mole. Fig. 5.20 shows the result of the simulations for a ring of Morse springs in comparison with the case of linear springs. We see that in the case of Morse springs the initial compression energy is rather weakly damped and survives at least 10 ps. Opposite to this the pulse decays in a linear chain very quickly. In Fig. 5.21 the same experiment is demonstrated for the case that the initial compression refers to the hard spring which connects the

A polymer reaction model including entropy

effects

175

masses 3 and 4. This situation could possibly model an excitation by the absorption of an energyrich molecule (ATP). In the linear case the initial energy never reaches the active site. In the nonlinear Morse model the energy is transferred always to the soft site. In this way we have demonstrated that the nonlinearities of interactions in a macromolecule may provide a localization and repeated recurrence of compressional energy at a functionally relevant part. In a recent paper a systematic investigation of the coherent motions and the cluster formation in Morse rings including the action of negative friction has been undertaken (Dunkel et al., 2001,2002). Note that all the effects discussed above are clearly seen at low decay decrements. In Chapter 8 we will estimate the Q-factors and damping coefficients for the oscillations of the elements or clusters at the amplitudes of tenths of angstroms.

5.7

A polymer reaction model including entropy effects

The Kramers theory describes reactions as stochastic transitions over a barrier of potential energy AU (activation processes). In reality the reaction processes occur in a heat bath of given temperature. Then as we have explained already in Chapter 1, the role of the energy barrier is taken over by the free energy barrier AF = AU — TAS. In this way the entropic changes along the raction path are included. Since macromolecular reactions are mostly connected with conformational changes this point may play a very important role. In order to elucidate the possible role of entropy changes let us consider in the following a toy model for the microscopic mechanism of reactive transitions in a polymer which are coupled to entropic changes. The physical effects which we would like to study is the coupling between a simple Kramers transition over a barrier with entropic transitions inside the reacting polymer. The model is closely related to recent studies of the activation of polymers (Sung and Park, 1998; Park and Sung, 1968) and also to the model explained in section 4.1. At first we develop a special microscopic model for the coupling of a Kramers-type bistable reaction site to a polymer conformation. We repeat here again that our special interest is devoted to the microscopic dynamics of the transitions in enzymes and in particular in the exceptionally high transition rates. Let us introduce now the model: We consider a reacting molecule with the coordinates XQ and the momenta po with the internal reaction coordinate q following an overdamped dynamics. The reacting corresponds to the breaking of a bond which is modelled as a motion in the cubic potential

U(q) = \aq2 - \aq*

(5.45)

176

Excitations

on rings of molecules

The bond is breaking, if q reaches the value of the barrier qbar = a/b. The center of mass of our molecule is moving freely in a liquid. Further we assume, that a linear polymer is swimming in the liquid with a Gaussian distribution of the end-to-end-distance 2/3

3L 2 \

exp(-^j

(5-46)

We assume that the free energy of the polymer is a known function of the end to end distance F(T, L) and the value corresponding to the thermodyamic equilibrium is the average over L.

F{T) = f dLF(T, L)W(L) * G(L)

(5.47)

where G(L) is a geometric factor. Let us assume now that the reacting molecule can be bound to the two ends of the polymer. We may imagine that the molecule is hold between the ends of the polymer like a tool in a lathe; the binding energy is Eb- In difference to the ring model explained in the previous chapter, the ring has now a free form, this causes strong entropy effects. In the free state where the reacting particle is far from the polymer, the potential barrier is rather high, so that transitions are practically excluded. In the bound state the reacting molecule is coupled to the polymer and its barrier depends on the end-to-end distance L of the polymer. We may assume that the constants a, b are dependent on the endto-end distance: a(L),b((L) with the condition a(L) —» a,b(L) —> b. We call our model in the following a Kramers polymer. In the bound configuration of polymer and reacting molecule the latter is hold between the ends of the polymer like a tool in a lathe. Due to the parameter dependence on L the potential is getting L—dependent

C/(g;L) = i 6 ( L ) g 4 - i a ( L ) g 2

(5.48)

This may lead to a lowering of the potential barrier

«-W

(M,)

Correspondingly we may find an enhancement of the rates. The main effect however is of entropic nature. In the state with the end-to-end distance L the entropy of the polymer is much smaller than in the thermal equilibrium and its free energy is higher F(T,L) > F(T). In this way we obtain the free energy barrier

A polymer reaction model including entropy

effects

SF(T, L) = [F(L) - F{T, L) + Eb] + AU(L)

177

(5.50)

In the case that the inequality [F(L) — F(T, L) + Eb] < 0 is fulfilled, this corresponds to an effective lowering of the barrier by entropic effects. Cutting the bond gives freedom to the polymer which will assume in a short time the equlibrium distribution of L and the equilibrium value of the free energy. The bound polymer has so to say an reservoir of free energy which can be used for overcoming the barrier. As a first estimate we get for the rates in the strongly damped case

If the polymer is sufficiently large the dependence on the end-to-end distance may be rather strong and the reservoir of free energy may be high. It was the aim of this section to study entropic effects on a simple artificial model. We have seen that under certain circumstances we may expect strong effects of the polymer confirmation changes on rates of bond breaking in an associated reacting molecule.

References P. Bak (1996): "How Nature Works", Springer, New York. J. Bernasconi, T. Schneider, Eds. (1981): "Physics in One Dimension", SpringerVerlag, Berlin-Heidelberg-New York. H. Bolterauer, M. Opper (1981): "Solitons in the Statistical Mechanics of the Toda Lattice", Z. Phys. B 42, 155-165. H. Buttner, F. Mertens (1979): Solid State Comm. 29, 663-673. A.Yu. Chikishev, W. Ebeling, A.V. Netrebko, N.V. Netrebko, Yu.M. Romanovsky, L. Schimansky-Geier (1996): "Stochastic cluster dynamics of macromolecules", in: Nonlinear Dynamics and Structures in Biology and Medicine: Optical and Laser Technologies, V.V. Tuchin, editor, Proc. SPIE 3053, 54-70. A.Yu. Chikishev, W. Ebeling, A.V. Netrebko, N.V. Netrebko, Yu.M. Romanovsky, L. Schimansky-Geier (1998): "Stochastic cluster dynamics of macromolecules". Int. Journal of Bifurcation & Chaos 8, 921-926.

178

Excitations

on rings of molecules

T. Dauxois, M. Peyrard & A.R. Bishop (1993): "Dynamics and Thermodynamics of a Nonlinear Model for DNA Denaturation", Phys. Rev E 47, 648-695. J. Dunkel, W. Ebeling, U. Erdmann (2001): "Thermodynamics and transport in an active Morse ring chain", Eur. Phys. J. 24, 511-524. J. Dunkel, W. Ebeling, U. Erdmann, V.A. Makarov (2002): "Coherent motions and clusters in a dissipative Morse ring", Int. J. Bifurcations & Chaos, in press (2002). W. Ebeling, Yu.M. Romanovsky (1985): "Energy Transfer and Chaotic Oscillations in Enzyme Catalysis". Z. Phys., Chem. Leipzig 266, 836-843. W. Ebeling, M. Jenssen (1988): "Soliton dynamics and energy trapping in enzyme catalysis", Z. Phys. Chem. 1, 269-279; Physica D 32, 183-193. W. Ebeling, M. Jenssen (1988): "Trapping and fusion of solitons in a nonuniform Toda lattice", Physica D 32, 183-193; Physica A 188, 350-355. W. Ebeling,W., M. Jenssen & Yu. M. Romanovskh (1989): "100 years Arrhenius law and recent developments in reaction theory" , in: In: Irreversible Processes and Selforganization, eds. W. Ebeling and H. Ulbricht, Teubner, Leipzig, p. 7-24. Ebeling,W. & Schimansky-Geier,L. (1989): In "Noise in Nonlinear Dynamical Systems", F. Moss, P.E.V. McClintock, eds., Cambridge, Cambridge University Press. W. Ebeling,W. & M. Jenssen (1991): "Soliton-Assisted Activation Processes" Ber. Bunsenges. Phys. Chem. 95, 356-362. W. Ebeling, J. Ortner (1998): "Quasiclassical theory and simulations of strongly coupled plasmas", Physica Scripta T75, 93-98. W. Ebeling, M.Jenssen (1999): "Brownian particles with Toda interactions - a model of nonlinear molecular excitations", in: Proc. Int. Conf. on Nonlinear Optics Saratov 1998, SPIE 3726, 112-123. W. Ebeling, A. Chetverikov & M. Jenssen (2000): "Statistical thermodynamics and nonlinear excitations of Toda systems" Ukrainian. J. Phys. 45, 479-487 W. Ebeling, U. Erdmann, J. Dunkel, and M. Jenssen (2000): "Nonlinear dynamics and fluctuations of dissipative Toda chains", J. Stat. Phys. A 101, 443-457. U. Erdmann, W. Ebeling, L. Schimansky-Geier, F. Schweitzer (2000): "Brownian

A polymer reaction model including entropy effects

179

particles far from equilibrium", Eur. Phys. J. 15, 105-113. H. Frauenfelder & P.G. Wolynes (1985): "Rate theories and the puzzles of protein kinetics", Science 229, 337-345. H. Frauenfelder, N.A. Alberding, A.Ansary et al., J.Phys. Chem. 94, 1024 (1990) P. Gruner-Bauer, F.G. Mertens (1988): Z. Phys. B 70, 435-445. B. Havsteen (1989): "A new principle of enzyme catalysis: coupled vibrations facilitate conformational changes". J. Theor. Biol. 140, 101-109. B. Havsteen (1991): "A stochastic attractor participates in chymotrypsin catalysis. A new facet of enzyme catalysis". J. Theor. Biol. 151, 557-571. D. Helbing (1997): "Verkehrsdynamik", Springer, Berlin. D. Helbing (2001): "Traffic and related self-driven many-particle systems", Rev. Mod. Phys. 73, 1067-1141. M. Jenssen (1991): Phys. Lett. A 159, 6-16. M. Jenssen, W. Ebeling (1991): In: "Far-from-Equilibrium Dynamics of Chemical Systems", eds. J. Popielawski and J. Gorecki, World Scientific, Singapore. M. Jenssen, W. Ebeling (2000): "Distribution functions and excitation spectra of Toda systems at intermediate temperatures", Physica D 141, 117-132. M. Karplus (1982): "Dynamics of proteins". Ber. Bunsenges. 386-240.

Phys. Chem. 86,

Yu. L. Klimontovich (1995): 'Statistical theory of open systems" Kluwer Academic Publ., Dordrecht. J.A. Krumhansl, J.R. Schrieffer (1975): Phys. Rev B 11, 3535-3545. H.P.Lu, L.Xun, X.S.Xie (1998): Science 282, 1877-1887. J.A. McCammon, S.C. Harvey (1987): "Dynamics of Proteins and protein acids", Cambridge University Press. F.G. Mertens, H. Biittner (1986): "Solitons on the Toda lattice", In: Trullinger et

180

Excitations

on rings of molecules

al., eds., I.e. V. Muto, A.C. Scott, P.L. Christiansen (1989): Phys. Lett. A 136, 33-43. J. Ortner, F. Schautz, W. Ebeling (1997): "Quasiclassical molecular dynamics simulations of the electron gas: dynamic properties", Phys. Rev. E 56, 4665-4670. P.J. Park, W. Sung (1998): "Polymer release out of a spherical vesicle through a pore", Phys. Rev. £ 5 7 , 730-734. R. Regeida, A. Romero, A. Sarmiento, K. Lindenberg (1999): J. Chem. Phys. I l l , 1373-1383. Yu.M. Romanovsky (1997): "Some problems of cluster dynamics of biological macromolecules". In: Stochastic Dynamics, L. Schimansky-Geier, T. Poeschel, Eds. Ser. Lecture Notes on Physics, Springer Verlag, Berlin, p. 140-152. T. Schneider (1986): "Classical statistical mechanics of lattice dynamic model systems", In: Trullinger et al., eds., I.e. W. Sung, P.J. Park (1996): "Polymer translocation through a pore in a membrane", Phys. Rev. Lett. 77, 783-786. N. Theodorakopoulos (1984): Phys. Rev. Lett. 53, 871-881. M. Toda (1983): "Nonlinear Waves and Solitons" Kluwer, Dordrecht. M. Toda, N. Saitoh (1983): J. Phys. Soc. Japan 52, 3703-3713. S.E. Trullinger, V.E. Zakharov, V.L. Pokrovsky, Eds (1986): "Solitons", North Holland, Amsterdam. G.R. Welch, B. Somogyi, S. Damjanovich (1982): "The role of protein fluctuation in enzyme action". Prog. Biophys. Molec. Biol. 39, 109-146.

Chapter 6

Fermi resonance a n d K r a m e r s p r o b l e m in 2-d force field S.V. Kroo, A.V. Netrebko, Yu.M. Romanovsky, L. Schimansky-Geier 6.1

2-d potential landscape and Fermi resonance

In the previous Chapters we presented several problems related to the Kramers problem or specification of Arrhenius law. Among them we considered the problem of a "test particle" transfer from one potential well into another. In particular we studied in Chapter 6 a potential landscape with the property that the returning force in x— direction Fx = — ^ depends only on x, and the corresponding force Fy = — ^f- depends only on y. However, the configuration of the real potential landscapes may be quite different. In Chapters 1 and 4 we studied already in brief the effects of a nonlinear coupling between x and y. In particular we pointed out that Fermi resonance may lead to an enhancement of transition rates. Here we continue to study this problem in more detail. In a recent paper (Shidlovskaya, Schimansky-Geier & Romanovsky, 2000) the problem of the substrate oscillations in the active site of an enzyme (Figs. 1.3, 1.10, 1.11 and 6.1) was considered. Examples of proton oscillations in two-well and threewell potential relieves can be found in (McDonald, Thorson & Choi, 1993; Netrebko et al., 1994). In particular, the distortion of the symmetry of the two-well potential in the system of H-bonds in the active site of serine hydrolases (chymotrypsin and acetylcholinesterase) results from substrate binding. The allowance for the rotation of the cluster planes relative to each other leads to the relationship between the returning forces. In this case the potential landscapes can be asymmetric (Fig.6.2) (Rubin, 1987). Note the recent studies of the 2-d systems consisting of the chains of long molecules subjected to the impacts of the water molecules. The static characteristics of such systems were calculated in (Shaitan, Pustoshilov, 1999; Shaitan, Ermolaeva, Saraikin, 1999) (see also Chapter 10). The physical pendulum is a classical example of a mechanical system in which each component of the returning force depends on both coordinates due to nonlinear coupling of both degrees of freedom (Fig.6.3). Let us assume that there are limitations on the angle, then the potential land181

182

Fermi resonance and Kramers

problem

4) proton transfer 2.P5 X

His57-,^^N'&

•

r N2^

-H-0-Serl95 2.0 k.

_

.2.8 A

-C— N ~ d

S;o:

\ /

substrate \ /

\2.8 A\

\ N - /

V

Aspl02 \ 2) H-bonJ break \

0

\C-c-c>" I I I Ser214

5) peptide bond break

') "^placement

Fig. 6.1 Catalytic triad or "charge relay system" in serine proteases. Substrate is fixed in the active site by a system of H-bonds allowing oscillations of small amplitude.

4>

I

-60

0

VN

60

120

V

180

'•P(N-C=t), rpaa b) Fig. 6.2 (a) Plane conformation of methylamide-N-acetyl-L-alanine molecule; (b) Diagram of the potential surface of methylamide-N-acetyl-L-alanine molecule with conformation energy (in kJ/mole) level lines.

scape U(
183

Basic 2-d cluster model

////

\

\

/

\ 1 1

Fig. 6.3

y

y

X

Pendulum with two vibrational degrees of freedom.

analyzed the Raman spectra of gaseous carbon dioxide by means of perturbation method. In this molecule the fundamental frequency of one of the modes (1330 c m - 1 ) is close to the doubled frequency of another fundamental mode (667.5 c m - 1 ) which results in appearance of the additional bands in Raman spectra. The classical analogy of this effect (redistribution of energy in classical pendulum shown in Fig.6.3) was demonstrated by Volkenstein (1947). Several authors (e.g., (Volkenstein, 1975) showed that Fermi resonance is possible also in protein structures because the frequency of N-H stretching vibration coincides with the doubled frequency of amide II vibration. At present Fermi resonance is demonstrated for different systems with self-modulation (see, for example, (Pippard, 1985, 1989). Continuing the principal study of Fermi resonances in section 1.4. we consider now in detail an example of the a simple mechanical system (Fig.6.4) in which Fermi resonance takes place (Netrebko et al., 1994; Shidlovskaya, Schimansky-Geier & Romanovsky, 2000). The discussion follows the schema: 1. Fermi resonance in a conservative system. 2. Induced oscillations and Fermi resonance in a system with losses and external harmonic force. 3. System with losses under the action of noise. 4. Test particle (TP) escape from a 2-d potential well in the case of resonance redistribution of energy between the degrees of freedom.

6.2

Basic 2-d cluster model

Consider dynamics of a classical particle bound by four linear springs to four immobile walls (Fig.6.4) The dynamics of the system in y\, ?/2 coordinates is given by

m

tli.

-

dt

2

_

d U

(Vl'Vl)

dyi

_

d

Udist (2/1 , V2 )

dyi

•_ i o

(a i N

184

Fermi resonance and Kramers

problem

t*

Fig. 6.4 Schematics of the basic 2-d cluster model; m is the cluster mass; ki (i = 1...4) are the rigidities of the springs simulating the H-bonds under linear approximation.

where the second terms in the right-hand sides are determined by the local distortions of the potential landscape in the presence of rigid sphere. In the case of linear forces (springs) and the absence of distortions and external action the potential function of the system is given by:

U(yi,y2) = fy(y/vl + (l-V2)2-i)

uV / +k{\Jyl

+ fy (Jvl

V + {i + yi?-i)

+ {I ~ Vif - l) +

u\ I +!%{\/y22 + (i +

V

(6 2)

-

yi)2-i)

here 21 is the distance between the walls. Note that in spite of the linear forces the dynamics of the system is nonlinear. We use this simplest model for the study of the dynamical aspects of behaviour of the unchanged part of the substrate bound inside the pocket of the active site. Immobile walls in Fig.6.4 simulate the nearest massive immobile clusters of the active site. The force coefficients are determined by the number and the directions of i7-bonds. The distortions of the potential landscape in the area of small oscillations of the substrate result from subglobular or cluster motility. This simplest model can be used for the description of some dynamical features typical of more complicated cluster models. It can be considered as a part of two or several crossing cluster chains in models of /3-strands and /^-sheets (Fig. 1.2). The model can be used for the study of low-frequency processes in systems of more heavy elements of the secondary structure (12 /3-strands connected with each other form two /3-sheets in CT molecule) (Netrebko et al., 1996, Romanovsky, 1997). As a matter of fact we should consider not 2-d but 3-d case, in particular, possible rotating motions of aromatic rings of the substrate inside the CT active site (Shidlovskaya, Schimansky-Geier & Romanovsky, 1999). We assume that the behaviour of the system in 3-d (nonfractal) potential landscape is similar to that in 2-d landscape but the behaviour in 1-d system is essentially different.

185

Basic 2-d cluster model

The system of equations can be linearized in the case of small oscillations 3/i) 2/2 < / (e < 0.1, where e = ao/Z, a 0 is the maximum vibrational amplitude and I is the length of H-bond). Two uncoupled modes with the natural frequencies w\ = \/(k2 + k4)/m , w2 — -\/{ki + k3)/m are observed. Nonlinear features of the model and coupling between two degrees of freedom can be considered analytically by solving the system (6.1) using the averaging method in the case of small vibrations e < 0.1. Under this approximation the resonance relation between two fundamental frequencies can be used. Even in this simplest case the motions can be quite complicated and redistribution of energy between the vibrational modes can be observed like in the classical model of the Fermi resonance in the unharmonic potential landscape. Let us substitute (6.2) in (6.1) and introduce the new variables: Vi V2

£2/2-

The expansion of the equations in the Taylor series in the point of equilibrium yields the expressions as follows (hereafter we omit asterisks in y\ and y^)•

dt2 +e2

+ co2yx

=

e (fci - k3) yiy2 + 2 (k2 - h)y2

- \ (fci + fc3) y\ + (fci +k2 + k3 + k4) 2/12/1

+

+ ... , (6.3)

^ 1 , 2 +62

=

e (k2 - k4) 2/12/2 + 5 (fci - k3) y\

- \ {k2 + kA) 2/| -f (fci + k2 + k3 + k4) y2y\

+

+ ... ,

where fej, i = 1...4 - are the force coefficients of springs, w\ = \Jk2 + k4 and w2 = y/ki + k3 are the natural frequencies. Equations (6.3) use the dimensionless units: I = 1 corresponds to the length of H-bond and m — 1 is the cluster mass. Below we consider the case k\ ^ £3 or k2 ^ k4 and the most interesting case of the internal (autoparametric) resonance when the value of one natural frequency coincides with the doubled value of another one: a>2 = 2wi or u\ = 2u)2. Note that under approximation of small vibrational amplitudes analogous equations can be written for nonlinear springs (for example, when the H-bonds are simulated by Morse or Lennard-Jones potentials). For example, if each spring in Fig.6.4 corresponds to one or several H-bonds described by Morse potential

Ui (r) = Doi ( l

-di(r-iy

Doi,

then the following system can be obtained instead of (6.3):

(6.4)

186

Fermi resonance and Kramers

Vi + (k2 + k4) yi=e +£Z

(fci - k3) yxy2 +

ki 2

*y - (a2 - aA) yj

2

+

2

M ^ i + it A yxy\ + (t <*t) ylV 2 - (ft +ft) + yf

j/2 + (Ai + k3) yi=e

+e<

hi2 -

problem

_«2.

Uk2 - k4) yxy2 +

fcl

k 2

3y2 - (a x - a 3 ) y | +

^ % 2 3 + ( E * i ) J/iIfc + (E<*) y\y2 - (ft + ft) y\

+ •

The terms with the factors a* and ft are missing new in the system (6.3). In this case 3 7 ki = 2dD0i, an - -kid, Pi — 7,kiQ, C = dl. The typical values for an H-bond are (see, for example, (Dashevskii, 1987)) I = 0.17 -r 0.18 nm, d = 30 -f 5 nm" 1 . That is why C ~ 10. As e = 0.1, only the terms of the order of e and e2 must be taken into account in (6.3) (the terms of higher orders must be taken into account in the case of large shifts). 6.3

Analytical study

We use approximate asymptotic methods (Bogolyubov, Mitropolskii, 1974) to demonstrate a periodic intermode transfer of the vibrational energy in the conservative system (6.3) at the frequency ft. Following the method of slowly varying amplitudes and phases we present yi (i=l, 2) in a way as follows: yt = Ai(t) cos ipi, Ipi = Wit + tfi (t) . Here Aj(t) and
((aAiA2 cos ipi cos ip2 + M 2 c o s 2 ^2) sini/ ) i) T ,

A'2 —

((cAiA2Cosij)icosil)2 + dA\cos2ipi)

sini/j2)T,

LO2


V>2 =

-

AiU)!

^2^2

((aAiA2

cos^i cos^2 + bA\ cos2 ^2) cosV , i) T ,

((c.Ai.^cosV'i cosV>2 + cL4 2 cos 2 ^i) cos^2) T ,

Analytical

study

187

where a = ki - k2, b = (k2 - k4)/2, c = k2 - k4, d = (ki - k3)/2. Brackets stand for time averaging. Averaging over the period T\ = 2TT/WI (in the case when UJ2 = 2wi) yields: ' dA1/dt = - C A i ^ 2 s i n $ , dA2/dt = | C A 2 s i n $ ,

(6.5)

d$/dt = 2C (-A2 + J ^ i J cos$, where e

C

k\ — 4

(6.6)

(6.7)

$ = 2(/>i - 02The following integral of the system (6.5) can be easily obtained:

Al + 4Al = Al

(6.8)

Multiply the last equation in (6.5) by cos $ in order to obtain the relationship between the phase $ and amplitudes A\ and A2:

dZ/dt

=2C-

\-z

8

4 l . + ^i 8A1

(6.9)

Z = sin $ As dA2 = 1 CA\Z, dt ~ 4 then ZdZ 1 - Z2

-8Al + A\ dA2. ^2A2

Using (6.8) to exclude A\ we obtain:

dQln(l-Z 2 ))

<M2

4dAl

A2 + AI-4AV

- - In (1 - sin 2 $ ) + InD = In A2 + In (A20 - 4A\) where D is the integration constant. Finally we have:

188

Fermi resonance and Kramers

D — A2A\

COS

problem

$.

(6.10)

Quadratures for A\(t), A2(t), and $(£) can be obtained but the corresponding expressions are quite cumbersome. That is why consider first the stability of the stationary solution of the system (6.5) in order to reveal the possible periodic changes of A±, A2, and $. Stationary values of <&,Ai,A2 are given by algebraic equalities: sin $ = 0

(cos $ = 1) ,

8Al = A\.

Using the integrals of the system (6.8) and (6.10), we obtain: M =^
,A2

$ = 0.

2V3'

Thus, the stationary solutions depend on AQ and, hence, on the initial conditions. Linearization of the system (6.5) near the values Ai, A2, $, yields the following system for small deviations ai, a2, $:

dai/dt — da2/dt =

-CAiA2$, \CA\&,

(6.11)

d$/dt = 2 C { | ^ 0 ! - 2a2\ . Characteristic equation for the complex frequency p follows from (6.11):

so that p —

P

0

0 -Cs/2

v AC

JCAQ.

CAl-± °3V2 -CAl\ p

= 0,

or p J + CMjS = 0

Substituting C from (6.6) we obtain:

fe

n^A.T1*1 4 sjk2 + ki

(6.12)

Resonance energy exchange is characterized by modulation frequency fii that is proportional to the difference between the force coefficients of the "high-frequency" mode \ki — k^\ and inverse proportional to the root of their sum v f c i + ^ 3 (i-e- ^ l is inverse proportional to w\ and u>2). This conclusion is valid at various initial conditions, in any degree of freedom, and at any values of the parameter k, corresponding to the resonance condition except for k\ = k%. In the latter case we have to take into account additional terms in the expansion in the powers of e.

Analytical

189

study

Let us calculate the modulation frequency fl in (6.5) without employing the linearization procedure. Using the system integral D we can write:

.

yJ{A\A2f-D* A\A2

Substituting this expression into the second equation of the system (6.5) with the allowance of (6.8) we obtain:

dA -^2

a

dt

= = _c_

4A2

r^77->. T^J ^AUAl-4Alf-D*.

The modulation period is given by the expression:

^

^AUAl-AAlf-D*

where A2min and A2max are the minimum and maximum values of the envelope A2. These quantities can be determined as follows. Since |cos$| < 1 and D — A2A\cos§, \D\ A*{Al-lAl)

< 1.

Therefore, _ A0 2min — ~^

(-K a\ COS I — + — I ,

A Ao 0

fir

a\

where cos a = 3\/^\D\/AQ. Going back to determining the modulation period and introducing the variable rj = A\ — 4A\ we obtain:

T •

2 C

\2 /"* 2 m i n J J-v3

I

dr) + AW

- 4L>2

2max

The denominator of the integrand equals zero at: „ 2 2 a0 Al B 0 = - - j 4 g c o s - ^ + -£, 2

A2

/7T

a0\

B2 = - ^ c o s ( - -

Al

y

)

+

^ ,

190

Fermi resonance and Kramers

problem

where

As a 0 = 2a, B1 T =

=

54£>2 cosa 0 = —75 1, B0 < B± < B2. A o A 2 - 4A22max a n d B2 = A 2 4A22min,

dy

2 f CJ

=

2_

2

C

^B2 - B0

V(£o - V) (Bi - V) {B2 - r?)

K [mh

where K(m) is the elliptic integral of the first type:

J y/(l-x2)(l-mx2) B2-Bi B2 - B0

2^oV 2 VZctgtf

m ;

+1

Finally, we have: T

=

2TT

^/(2n-l)!! = > A>C^cos(f-f)^oV 2«n! 7

ao = arccos

VVZctg{a0/Z) + 17 '

(T-0

When linearizing the system we assume that cos$ = 1, hence, D — A\A2 -^9= and a 0 = 0. In this case T = A and fi = .4oC = e A 0 i J * L ^ M , 3V3 ^oO 4 ^ 2 + /.4 which coincides with (6.12). 6.4

Numerical study

Below we present the results of numerical solution of the equations of the base system. There are two resonance regimes of effective energy exchange between two vibrational modes at 2 : 1(0.5 : 1) ratio of the natural frequencies in the given 2-d conservative system. Figures 6.5 and 6.6 show the time-dependencies of the coordinate yi and the envelopes of the coordinates y\ and y2 of the imaging point. It was assumed that the cluster mass is m — 100 a.u., k\ = k, k^ = 2k, k2 = 0, 5A;, ki = 0,25A;, where k = blN/m is the force coefficient of a single H-bond. Note that the cluster mass is close to that of the substrate fragment bound to the groups of the CT AS pocket. The values of fc; were chosen to meet the conditions of the intermode energy transfer. In fact, the real values of ki depend on

Numerical

study

191

yi 0,1-

—^fll m»-

i

-0,1-

100

200

Fig. 6.5 Vibrations in y\ axis at the carrier frequency ui\. Initial conditions: yi(0) = 0 and o J/2(0) = 0.1. Time scale: 1 = 0.1 ps; amplitude scale: 1 = 2 A-

0.12-1

0.08

0,04

4000

8000

Fig. 6.6 Positive envelopes of the amplitudes y\ and 2/2 (in dimensionless units). Initial conditions: 2/i(0) = 0 . 1 and 2/2(0) = 0. Carrier frequencies are not shown. Scales and parameters are the same as in Fig.6.5.

the plane in which the oscillations take place and the specific conformation of the enzyme-substrate complex. The latter can slowly (relative to the ligand oscillation period) vary in time. As the formula (6.12) is obtained using linearization, it remains valid only at small deviations from the stationary state. To illustrate this fact, let us find numerical solution of the system (6.5) setting the initial conditions in a way as follows. Let the initial phase $ be always equal to zero and the initial amplitudes change in such a way that the energy of the system (integral AQ) always remains constant. In this case the stationary values of A\, A2 , and $ do not depend on the initial conditions: Ax = A0

$ = 0. 2^3'

The deviation of the system from the stationary state can be characterized by the parameter d that is equal to the ratio of the initial and stationary value of one

192

Fermi resonance and Kramers

problem

l.l - i

l.o -

_>

0.9 -

./

0.8 -

l-^•

/

0.7 -

\

1

/

•

1

0.6-

°-H—'—i—' 0.0

0.2

i—'—i—'—i—'—i—'—i— 0.4

0.6 d

0.8

1.0

1.2

Fig. 6.7 A plot of the ratio fi/fi; versus the parameter d(fi) is the frequency determined by the numerical solution of the equations (6.5), H; is determined by the formula (6.12)). AQ = 0,5; £=0,01; fci = 1; fc2 = 0,5; k3 = 2; fc4 = 0,25.

of the amplitudes, e.g. Ai (d = 1 corresponds to the stationary state): d = Ax/A^l. Fig.6.7 shows the dependence of the ratio fl/fij on the parameter d(fl) is the frequency determined by the numerical solution of the equation 6.5); fi; is given by the formula (6.12). Thus, the frequency determined by the numerical solution of equations (6.5) under small deviations from the stationary value coincides with the frequency given by the formula (6.12). We compared the results of numerical solution of equations (6.1) and (6.5). A good coincidence of the redistribution frequencies is possibly limited by the accuracy of the numerical scheme. For the case of the Morse potentials we also observed the regimes of selfmodulated oscillations representing a particular case of the Fermi resonance (although at different frequencies) (Netrebko et al., 1994, Ebeling et al., 1994). It is demonstrated that there are three resonances instead of two corresponding to the case of linear springs. Their positions are substantially different from the positions of two peaks in the two-resonance case. Even if a
Stochastization

of the

193

vibrations

yi

yi

yi

a)

b)

c)

Fig. 6.8 Various trajectories of the imaging point and the boundaries of the areas of possible stochastization.

6.5

Stochastization of the vibrations

Note that the described strongly modulated regimes are observed in the case of very small vibrational amplitudes (in molecular dynamics the amplitudes of such processes do not exceed 10% of the H-bond length which corresponds to 0.1^-0.3/9^4). If the amplitudes are larger than 0.51 (or e > 0.5) the stochastic behaviour can be observed. The possible stochastization areas can be calculated for different potential relieves using the Toda method (see chapter 3). Figure 6.8 shows the areas of possible stochastization and the trajectories of the imaging point for the case of the Fermi resonance in the system (6.1) with linear bonds at large amplitudes (0.5/ — 1.0Z). The motion is regular if the imaging point does not cross the areas of possible stochastization (Fig.6.8a). Stochastization is possible if the imaging point passes through Toda area (Fig.6.8c). At the same time the latter condition is not sufficient for chaos formation (Fig.6.8b). Note that Toda areas for the system with Morse potentials (6.4) differ from those for the system with the corresponding quadratic potentials (Fig.6.9) (Nunez-Yepez et al., 1990). Chaotic behavior can be observed also in the case of the small amplitudes e < 0.1, if we introduce distortions into the potential landscape or introduce a solid cluster into the oscillation area (Netrebko et al., 1994).

6.6

Basic model including damping and external harmonic action

Up till now we studied a conservative system. However, the real oscillations of clusters decay in time and they are subjected to the action of external (in the general case, random) forces. Let us consider the system (6.3) in the case of external regular excitation and damping in only one degree of freedom y\. Leaving only the terms linear in e and assuming that m = l w e obtain:

194

Fermi resonance and Kramers

problem

0.5-

yi °°-D

0.0-

•0.5-

-1.0-

I CDC

) VJ

> )DD r yi c)

Fig. 6.9 Stochastization areas for a system with (a) quadratic and (b) Morse potentials. Exponential instability is possible for the shaded areas, (c) Superimposed boundaries of the stochastization areas for the same potentials.

d2 2/1 dt2

,

2

( « i - f c a ) 2/12/2 +

d2V2 ^

, 2 + ^ 2 / 2 =

£

(k2

—»—2/2

hi)

eF0cos(wt + <$>)-e5^,

2/12/2+2

( fc i

(6.13)

~ k3) 2/i

Here Fo and u are the amplitude and frequency of the external periodic force and 5 is the damping decrement. We take into consideration only the terms of the order of e. Thus, the following relationships for 8 and FQ are obtained:

|&i-fc3|4i42~.Fo~<MiWi,

(6.14)

where A\ and A\ are the maximum amplitudes of the envelopes y\ (t) and 2/2(0For ui = uii we search for a solution of (6.13) in the form: yt = Ai cos (cjit + ifi). In the resonance case u\ = w = W2/2 one can average the equations for slowly varying envelopes (6.13) and obtain an expression for the modulation frequency: \/3 \h-k3\F0 fii = e ' 4v^2 k2 + k4 S '

(6.15)

The maximum amplitudes of the envelopes are given by:

A?1 = ^ 5w

and

Al2 = ^ A \ . 2v^

(6.16)

Computer simulation

of the nonautonomous

system with

damping

195

Fig. 6.10 Positive envelopes of the amplitude y\ and 3/2 for the case FQ = 0.05, a = 0.5; initial conditions: yi(0) = 0 and 2/2(0) = 0. The scales and parameters are the same as in Fig.6.5. An intermode energy exchange at the frequency Cliis observed.

If fci —¥ &3 we have to take into consideration the second order approximation. On the other hand, if the external force is large, the relationship (6.14) is not valid because the nonlinear term becomes larger than the linear one: 1*1 -kz\A\ |*i-fc3|

»<5w, 1

F0

» 6w.

That is why one can expect complication of the modulation regime in the case of large FQ. 6.7

Computer simulation of the nonautonomous system with damping

Figures 6.10 and 6.11 show the solutions of the system (6.13) at small amplitudes of the external force under various initial conditions. It follows from these solutions that the induced oscillations with the envelope modulated at the frequency Oi are established in the system after a certain transient process. The modulation regime becomes more complicated if the value Fo increase 10 times: "double modulation" appears at the frequencies fix ~ e and O2 ~ e 2 (Fig.6.12). This result can be obtained by averaging with the allowance for the terms of the order of e2. The same result can be envisaged for the system in which the Morse potential is used instead the quadratic one. To specify the behavior of the system under large perturbations we solved the initial nonlinear equations (6.1) with the allowance of the periodic force and damping: d2y1 _ dt2

dU(yuy2) dyi

eF0 cos (ut + $ ) - eS

dyi dt '

196

Fermi resonance and Kramers

10.000

20,000

problem

30,000

40.000

50,000

Fig. 6.11 Positive envelopes of the amplitude y\ and 1/2 for the case Fo = 0.05, a = 0.5; initial conditions: 2/1 (0) = 0.3 and 2/2(0) = 0. In spite of the higher initial energy the type and parameters of oscillations are the same.

.

10-000

"b e a t i n g i " at Q -t

20.000

30.000

4 0 . 0 0 0

50.000

b) Fig. 6.12 Positive envelopes of the amplitude y\ and 2/2 for the case Fo = 0.5, a = 0.5; initial conditions: 2/1 (0) = 0 and 2/2(0) = 0. The scales and parameters are the same as in Fig.6.5. An intermode energy exchange at the frequency fii and modulation at the frequency 02 ~ e 2 are observed.

d2y2 dt2 where U{y\,y2)

=

dU(yi,y2) 0y2 '

is determined by the relationship (6.2).

Computer simulation

of the nonautonomous

system with

damping

197

y2ioo

T

T

-1

0.25

w/w,

0.00

w/w.

4

0 (C)

Fig. 6.13 Positive envelopes of the amplitudes j/i and j/2 (a), trajectory of the imaging point (b) and the normalized spectral power density of the time-dependence of coordinates (c). I — 100, 8 = 1, 3/i(0) = y 2 (0) = 0, F o = 3 0 .

If w = wi there are two regimes determined by the values of Fo, 6, and I. The first regime exhibits a regular energy exchange between the modes and is analogous to that considered above. Figures 6.13 and 6.14 show time-dependencies of the positive envelopes y\ and 2/2 at two values of Fo. Note that the maximal amplitude of oscillations along 7/2 axis can be larger than that for oscillations along y\ axis (Fig.6.14). There is no energy exchange between the modes under stationary conditions for the second regime (Fig.6.14). Further studies are needed to determine the criterion by which one can differ between the first and the second regimes. Note that there is no stochastization of motion in this system even for the deviations from the equilibrium that are larger than the length of the spring. Figure 6.16 shows the trajectories of the stationary motion of the imaging point for two values of Fo. If the system (6.13) uses noise instead of the harmonic force, one can also expect the regime of energy redistribution between the modes "j/i" and "2/2"- For the high quality system (5 ) we must use < F (wi) > ^

I/^FK)

2S

= \/5'F(WI)AW,

instead of F0 for estimating the statistical parameters A\, A\, and fii. Here SF (WI)

198

Fermi resonance and Kramers

problem

y 2 100-

S» 0.50

Fig. 6.14 Positive envelopes of the amplitudes y\ and J/2 (a), trajectory of the imaging point (b) and normalized spectral power density of the time-dependence of coordinates (c). I = 100, S = 1, yi(0) = y 2 (0) = 0, F 0 = 20.

is the spectral density of F(t) at the frequency uii and Awi — uii/Q = 28 is the spectral bandwidth of our vibrational system. Using (6.15) and (6.16) we obtain:

/fl~I,/2*H, u>

,A

°' ~ ^ v f • 2UJ

v^j,2 "

l|fci-fc 3 |./35F(a>i)

n

11

V o

£

4 A;2 + A;4

The increase of T/SF (U)/S leads to the proportional increase in RMS values of ^4°! -4°' a n d ^ i - The corresponding distributions must be estimated by Rayleigh formula. Figures 6.17 and 6.18 demonstrate the action of the white noise F(t) in only one direction (yi and y2, respectively). It is seen that a transient process is followed by the regime of stochastic modulations. The envelopes of the amplitudes y\ and 2/2

Kramers problem for 2-d potential

199

landscape

y,ioo 4U -

-

^ y

20-

fh

r i

i

0

i

I

'

I

200 0

1000

'

3000

-100

-50

50

0

100

(b)

(a) 1.00 -

1.00 —i

0.75 -

0.75 -

Syi0.50-

S y 0.500.25 -

0.25 -

w/w

'

1

1

1

'

w/w 1

1

1

1

1

1

I

(c)

Fig. 6.15 Positive envelopes of the amplitudes y\ and 3/2 (a), trajectory of the stationary motion of imaging point (b) and normalized spectral power density of the time-dependence of coordinates (c). I = 100, <5 = 1, j/i (0) = 2/2 (0) = 0, F 0 = 10.

change in opposite phase like in the case of the nonstochastic regimes considered above. It is likely that the real systems exhibit such types of motion. Note that the colour noise action seems to be more probable. To our opinion, the Fermi resonance in the systems with damping and external force action is important for molecular dynamics of macromolecules.

6.8

Kramers problem for 2-d potential landscape

We meet the problem of the particle escape from the minimum of 2-d potential landscape each time when it is necessary to estimate the time of transition of a ligand from one minimum to another or the time of the escape of the reaction products from the AS pocket. In 1-d case such estimates for noninteracting particles can be done using Kramers formula. This problem was discussed in details in Chapter 1. Substantial differences between 2-d and 3-d cases may result from the possible energy exchange between the vibrations in different directions. In the case of the 2d potential landscape the consideration is reduced to solving the problem of escape into 4-d space (see, for example, (Tikhonov, Mironov, 1977)). This problem was considered in Chapter 3 under the assumption that the Fermi resonance conditions

200

Fermi resonance and Kramers y,100

problem

y,100

-100

50

100

1.00

Sy 0.50 -

Sy 0.50

0.25

w/w,

0.00

w/w.

x_ui_

S„ 0.50

S» 0.50

'2

0.25

0.25 -

w/w,

0.00

w/w,

.I, M, 2

(a)

(b)

Fig. 6.16 Trajectories of the stationary motion and the normalized spectral power density of the time-dependence of coordinates at / = 100, S = 1, FQ = 80 (a) and FQ = 85 (b).

are not met. However, the equations for the escape probability and the mean escape time are rather complicated and can be solved only numerically. We estimated above the value of the mean amplitudes Ai and A^ and the beat frequency (see formulas (6.15) and (6.16)) for the case of the white noise in one ofthe coordinates (yi). This estimates show that Fermi resonance leads to oscillations in 2/2 even if £2 = 0. Thus, the particle can escape in the direction j / 2 - Rough estimate of the barrier U* = U(yl) (Fig.6.19) crossing in the direction 2/2 can be done in a way as follows. TP reaches the threshold value y\ in the case of the maximum and minimum values ofthe envelopes A2(t) and Ai(t), respectively, because Ai{t) and A%{t) change in the opposite phase. The number of escapes per unit time for the function

Kramers problem for 2-d potential

0

T ' 1 ' 1 ' 1 2000 3000 4000 5000

1000

201

landscape

1 ' 1 ' 1 ' 1 2000 3000 4000 5000

1000

Fig. 6.17 T P trajectory at the white noise in the y\ coordinate under the condition of the Fermi resonance (wi = 2^2).

0.4 •

0.2 •

0.0 .no-

-0.2-

-0.4 •

I 1000

'

I 2000

'

I 3000

'

I 4000

'

I 5000

T^

I 1000

'

I 2000

'

I 3000

'

I 4000

'

I 5000

Fig. 6.18 T P trajectory at the white noise in the j/2 coordinate under the condition of the Fermi resonance (a>i = 2u)2).

y2{t) = A2{t) cos u2{t) can be estimated by the formulas (1.14) and (1.15): w

2

(

yf

(6.17)

If the vibrational amplitude in the direction of the random force is (Ai) ~ 0,3pA, the escape is possible only in the perpendicular direction, and the boundary value is y2 ~ 0, 5pA, then according to (6.16): <^> = ^ ui2

n2 = — exp 2n

yf 2 (AD

W2

= ^=0,025, /

2TT 6 X P I

0,25

2-0,025

= - ^ e ~ 5 =0,001w 2 2TT

If the noise of the same intensity acts only along the y2 axis, we have:

202

Fermi resonance and Kramers

problem

n2 = 0,012w2. Thus, the escape provided by the energy redistribution is an order of magnitude less probable than that realized under the direct noise action. If noncorrelated noises act up in both directions, there is no correlation in the energy redistribution processes, A\{i) and A2{i) do not change in the opposite phase, and one can not use (6.17). In this case it is necessary to solve the problem in 2-d space. Note once more that the above estimates are valid only if the quality factors of the system are very high for both coordinates. The coincidence of the characteristic frequencies of the external action and the resonance frequencies is hardly possible but one should keep in mind that the force constants ki may change as a result of, e.g., slow modulation of the length and orientation of the il-bonds by which the ligands are bound in the enzyme AS. The estimates show that the resonance frequencies of the substrate bound in CT AS range from 1012 to 1013Hz and vary slowly with the motion of the clusters constituting CT subglobules. To specify Kramers conclusion on the dependence of the escape time r on h/u we carried out a computer experiment. The system of equations for the motion of a particle (m = 1) under the action of the delta-correlated noises £i, £2 in the potential field U(x, y) is:

We studied the dependence of the time that is necessary for a particle to reach a boundary G or to acquire a certain potential energy U* on the friction coefficient h that is unambiguously related to the noise amplitude. We considered five different variants of the shape of the potential profile and configurations of the boundary G (topograms of the potential surfaces are presented in Fig.6.19(a-d)): 1) 2) 3) 4)

U U U U

= = = =

x2 x2 x2 x2

+ y2, (G): U* = 25; i.e. x2 + y2 < 25. + y2 , (G): U* = 16; i.e. x2 + y2 < 16. + y2 , (G): U* = 25; or x < 4. + 2/2/4, (G): U* = 25; or x < 4.

5) U = (x2 + y2) * ( f | - ^

+ ^

+

9c

° 1 S f Q ) ) , 0 < a <2TT, (G):U* = 25;

or x < 2\/2. Figure 6.19e shows the plots of the mean time (averaging over 1000 realizations) versus the friction coefficient for the above variants.

Kramers problem for 2-d potential landscape

0.01

0.10

1.00

10.00

203

100.00

e)

Fig. 6.19 (a)-(d) Topograms of the potential relieves for the cases 1, 3-5: the escape is possible at (a) ( / > [ / * = 25; (b), (c) U > U* = 25 or x > 4; and (d) U > U* = 25 or x > 2%/2. (e) plots of the mean escape time versus the ratio h/uj (u; = 1).

The character of the escape time dependence on the friction coefficient agrees well with the Kramers' results for the 1-d case (see paragraph 1.1): minimum is achieved at the friction coefficient equal to the natural frequency of free oscillations, detuning from this value leads to exponentially growing escape time. Potential surfaces 3 and 4 represent elliptical paraboloids cut by the plane x = 4; curve in the cross section is parabola; the value of the potential energy in the minimum is 16. For these cases the two-dimensionality of the potential is critical because the point mass can reach the boundary at lower energy (in comparison with the case 1 - uncut potential). In the case 5 the shape of the potential surface is chosen in a special way and corresponds to the systems in which the conditions of the Fermi resonance are met.

204

Fermi resonance and Kramers

problem

However, the potential surface obtained as a result of summation of the potentials U = ki ((x — Xi) + (y — y{) J appears to be "inconvenient" for numerical calculations. Firstly, its type substantially depends on the length of the springs. It is natural that oscillations are quasi-harmonic only in the vicinity of the equilibrium and they are not such if the amplitude is larger than 0.1/j. Hence, one can hardly speak about energy redistribution. Secondly, the potential landscape is still symmetrical relative to X and Y axes but it looses symmetry in other directions. Thirdly, the cross section of this surface by a plane can give more than one minimum or the minimum can be far from X or Y axes. All these facts forced us to consider potential 5 that is obtained by means of approximating the sum potential landscape of four springs with the coefficients hi = 0,25, fc2 = 1, A3 = 0,5, and ki = 2, under the condition that they are either extended or compressed. The conditions of the energy redistribution (multiplicity of the natural frequencies in the directions X and Y) are maintained in this potential but it is free of the mentioned shortcomings. Figures 6.20a and 6.20b show the relieves for the potentials U5 and U2. Vibrational process and, consequently, energy redistribution resulting in fast approaching the boundary are observed in this case at small friction coefficient. Vibrational motion collapses at large friction coefficients and the time of approaching the boundary is comparable with that for the cases 3 and 4. Additional studies of the case 5 were performed: 5-1) 5-2) 5-3) 5-4)

U = (x2 + y2) * f(a), (G): x < 2yf{2). U = (x2 + y2) * / ( a ) , (G): x > - 4 . U=(x2+y2)*f (a), (G):y<8. U = (x2 + y2) * f(a), (G): y > - 4 ^ 2 ) .

Here there is an exit in only one direction. Figure 6.20c shows the results of the calculations. Here the curves with triangles correspond to the cases 5-1 and 5-3, the curves with circles correspond to the cases 5-2 and 5-4. It is seen that the escape time depends on the two-dimensionality of the potential landscape in the case of the vibrational process (the curves seize to be symmetrical relative to 1 - the frequency of natural oscillations). Moreover, we considered the case (curve without markers) of the boundary 5-1 and the external force acting up only in the Y direction. Under such conditions the particle never crosses the boundary provided that there is no energy transfer from one mode into another. In the considered case the vibrational process allows energy redistribution and at the small friction coefficients the escape time was nearly the same as in the cases 5-1 - 5-4. However, if the friction coefficient is rather high (the system is overdamped) there is no vibrational motion in the system and the escape time rapidly increases. Note that we studied qualitative effects, so in the last cases the amplitude of the noise action was increased four times and the number of realizations was decreased down to 100. Such changes lead to breaking curves and decreasing escape time.

Kramers problem for 2-d potential

Y

i

Y,

a)

b)

205

landscape

0.01

0.10

i.oo

10.00

100.00

c)

Fig. 6.20 (a) and (b) Topograms of the potential relieves for the case 5: the escape is possible only at (a) y\ > 2\/2 or j/i < - 4 and (b) y2 > 8 or 2/2 < —4y/2. (c) Plots of the mean escape time versus h/ui (w = 1) for the potential relieves (triangles) 5a and (circles) 5b; the curve without markers corresponds to the case when the noise acts up in Y2 direction and the escape is possible in Y\ direction.

However, the qualitative character of the results remains unchanged (it was proved by additional computations at certain values of h in which the number of realizations was increased to 1000 and the noise amplitude was decreased). In the computer experiments the inevitable quantization error leads to limiting time of the T P motion preceding its escape from the potential well. If the lifetime of the given state is rather large, one must use the theoretical description (see paragraph 3.3) valid at the low escape probability. In the notation of the paragraph 3.3 the dimensionless time is represented as:

^_KZI2R [k^T\ (Umin\ * - 2A \lUmin[eXP\kBT)

,' \ •

This result does not depend on the friction coefficient h. However, the particle moves slowly at large h which means the violation of the condition of smallness of the correlation time in comparison with the escape time. On the other hand, the noise action becomes negligible at small h and one can hardly apply the statistic approach instead of the dynamic one. Table shows escape times at several depths of the potential well.

Umin/ksT T

5 118

6 294

7 740

8 1183

Note that the results of the above calculations (see Fig.6.19e) agree well with the case when f/ m f n /fcsT = 5.

206

Fermi resonance and Kramers

problem

How can we use (at least qualitatively) these results for solving the problem of the products escape from the active site? It is likely that they can be used for evaluation of the situation in the CT active site (Shidlovskaya, Schimansky-Geier & Romanovsky, 2000) where only one ligand resides after peptide bond breaking. The situation is much more complicated in the case of ACE.

References N.N. Bogolyubov, Yu.A. Mitropolskii (1974): "Asymptotic methods in nonlinear vibration theory" (in Russian), Nauka, Moscow. V.G. Dashevskii (1987): "Conformational analysis of macromolecules" (in Russian), Nauka, Moscow. W. Ebeling, Yu. Romanovsky, Yu. Khurgin, A. Netrebko, N. Netrebko, E. Shidlovskaya (1994): "Complex regimes in the simple models of molecular dynamics of enzymes", Proc. SPIE 2370, 434-447. E. Fermi (1931): "Uber den Ramaneffekt des Kohlendioxids", Zeitschrift fur Physik, 250-259. K.M. McDonald, W.R. Thorson, J.H. Choi (1993): "Classical and quantum proton vibration in a nonharmonic strongly coupled system", J. Chem. Phys. 99, 46114621. A. Netrebko, N. Netrebko, Yu. Romanovsky, Yu. Khurgin, E. Shidlovskaya (1994): "Complex modulation regimes and vibration stochastization in cluster dynamics models of macromolecules" (in Russian), Izv. Vuzov: Prikladnaya Nelineinaya Dinamika 2, 26-43. A. Netrebko, N. Netrebko, Yu. Romanovsky, Yu. Khurgin, W. Ebeling (1996): "Stochastic cluster dynamics of enzyme-substrate complex" (in Russian), Izv. Vuzov: Prikladnaya Nelineynaya Dinamika 3, 53-64. H. N. Nunez-Yepez, A.L. Salas-Brito, C.A. Vargas, L. Vincente (1990): "Onset of chaos in an extensible pendulum", Phys. Lett. A145, 101. A.B. Pippard (1983): "The Physics of Vibration", Cambridge University Press, Cambridge, London, New York, Melbourne, Sidney Yu.M. Romanovsky (1997): "Some problems of cluster dynamics of biological macromolecules", In: Stochastic Dynamics, L. Schimansky-Geier, T. Poeschel, Eds. Ser.

Kramers problem for 2-d potential

landscape

207

Lecture Notes on Physics, Springer Verlag. Berlin, p. 140-152. A.B. Rubin (1987): "Biophysics" (in Russian), Vyshaya Shcola, Moscow. K.V. Shaitan , M.D. Ermolaeva, S.S. Saraikin (1999): "Nonlinear dynamics of the molecular systems and the correlations of internal motions in the oligopeptides", Ferroelectrics 220, 205-220. K.V. Shaitan , P.P. Pustoshilov (1999): "Molecular Dynamics of a Steric Acid Monolayer", Biophysics 44, 429-434. E. Shidlovskaya , L. Schimansky-Geier , Yu.M. Romanovsky (2000): "Nonlinear vibrations in 2-dimensional protein cluster model with linear bonds", Z. Phys. Chem. 214, No 1, 65-82. V.I. Tikhonov, M.A. Mironov (1977): "Markov's processes" (in Russian), Sovietskoe Radio, Moscow. M.V. Volkenstein (1974): "Molecular biophysics" (in Russian), Nauka, Moscow.

This page is intentionally left blank

Chapter 7

Molecular scissors. Cluster model of acetylcholinesterase A.Yu.Chikishev, S.V.Kroo, A.V.Netrebko, N. V.Netrebko, Yu.Romanovsky 7.1

The role of acetylcholinesterase in the synaptic transfer

To illustrate the possible applications of the problems discussed in the previous Chapters we consider the interaction of a hydrolytic enzyme acetylcholinesterase (ACE) with its substrates. ACE plays an important role in the synaptic transfer of the nerve pulses. In addition, ACE offers a good example of how diversified can be the problems of the Brownian motion as applied to functioning of the molecular machine. ACE, as well as CT, is a serine hydrolase. It catalyses ether bond breaking in the neuromediator molecule (AC). In Chapter 1 we discussed a possibility of the spontaneous bond breaking in the aqueous environment. Note that the synaptic (chemical) transfer of the electric excitation from one neuron to another or from the nerve terminus to the target cells that control the work of the muscle fibers is impossible without ACE. Consider in brief the process of the synaptic transfer (see, for example [Alberts et al., 1994]). A synapse represents an intermembrane contact of two excitable cells. According to the mechanism of the pulse transfer from neuron to neuron the synapses are classified as chemical, electric, and mixed. The chemical synapses are the dominating ones in the synaptic apparatus of the central nervous system of animals and human beings. Nerve-muscle transfer uses only chemical synapses. The synapses consist of the three main elements: presynaptic membrane, postsynaptic membrane, and the synaptic cleft. (Fig.7.1). Presynaptic membrane covers the nerve terminus that can be considered as a neurosecretory apparatus. Here the mediator is stored and released. The mediator excites or inhibits a target cell. Acetylcholine, adrenaline, noradrenaline, dofamine, and some other substances can be mediators in the interneuron synapses. Acetylcholine plays the role of the mediator in the skeletal muscles of all vertebrates and human beings. A presynaptic terminus accommodates 50-nm "bubbles" containing acetylcholine. The neuromediator is released from the bubbles under the action of the propagating potential and goes into the synaptic cleft. Each nerve pulse releases a certain 209

210

Molecular scissors.

Cluster model

1 7 8 2

Fig. 7.1 The scheme of the nerve pulse transfer in the chemical synapse. 1 presynaptic neuron terminus; 2 postsynaptic neurone; 3 presynaptic membrane; 4 postsynaptic membrane; 5 synaptic cleft; 6 bubbles with neuromediator; 7 released neuromediator; 8 acetylcholine receptors; 9 enzyme acetylcholinesterase.

amount of AC molecules. The repetition rate of the pulses is about 1 kHz. The width of the cleft is about 50 nm. Mediator diffuses quickly through the cleft and acts up on the membrane of the target cell. The part of the cell membrane that is most close to the nerve terminus is called postsynaptic. Acetylcholine receptors are located at the postsynaptic membrane. In response to the action of acetylcholine the receptors change the penetrability with respect to Na+ and K+ ions which results in generation of the action potential in the postsynaptic muscle fiber or in the body of another neuron. It is known that postsynaptic membranes contain large amounts of acetylcholinesterase. This enzyme represents a kind of molecular scissors "cutting" acetylcholine. Recall that under natural conditions the nerve pulses arrive at the postsynaptic cells at a rather high rate so that the postsynaptic membrane depolarized by the preceding portion of acetylcholine becomes insensitive to the action of the next portion. Normal exciting action of the subsequent pulses is possible only if the previous portion of the mediator is removed or deactivated. There are several pharmacological agents capable of inhibiting acetylcholinesterase activity. These inhibitors are used to avoid muscle relaxation under narcosis and in the case of such diseases as myastenia. On the other hand, it is known that people can be poisoned by insecticides based on these inhibitors. The spasms caused by these intoxications result from the prolonged activation of the acetylcholinergetic synapses especially in the vegetative nerve system. The major part of toxins acts on acetylcholinesterase. Thus the study of the mechanism of this molecular machine is an important

ACE computer model based on X-ray data

211

problem for medicine and for interpreting the principles of the enzymatic catalysis.

7.2

ACE computer model based on X-ray data

Consider the geometry of ACE. The coordinates of the heavy atoms 0 , C, N, and S were determined by X-ray analysis [Sussman et al., 1991] with the accuracy of 2.8pA. The results of the analysis can be found in the Protein Data Bank. File format is described in [Bernstein et al., 1977]. In this work we used the data file with the atomic coordinates of ACE from Torpedo californica. ACE represent an ellipsoid with the dimensions 45 x 60 x Q5pA. This molecule consists of 537 amino acids. The AS pocket represents a deep and narrow cleft that goes inside the protein globule. AS catalytic triade consists of the amino acid residues Ser 2 0 0 , Glu327, and His440 and is located at the bottom of the cleft. 14 aromatic residues form the walls of the cleft and prevent the positively charged substrate (AC) from the interaction with the negatively charged acidic residues. The calculations of the electrostatic field of the enzyme based on the X-ray data showed that an extremely large dipole moment of ACE is directed along the axis of the cleft. (There are contradictory data regarding the value of ACE dipole moment: 505 D [Ripoll et a l , 1993], 1500 D [Antosiewich, Gilson k McCammon, 1994], 1 D = 3.33564 • 10~ 30 C-m.) The positively charged substrate can be pulled into the cleft due to electrostatic interaction. In addition, the total charge of the enzyme is negative: it is about — 12e without allowance for the ionic surrounding that results from the interaction of the protein with the solvent. The problem of the release of the reaction products from the active site is mentioned in [Ripoll et al., 1993]. Indeed, one of the reaction products, choline, is positively charged and its diffusion out of the cleft is hindered by the electrostatic attraction to the enzyme. This difficulty gave birth to the hypothesis of the "back door" allowing the escape of the reaction products from the active site. As the indole ring of Trp84 is situated inside the cleft and the residue as a whole is located at the outer surface of the protein, the authors proposed a possibility of a conformational change opening an additional exit from the cleft of the active site. Gilson et al. [Gilson et al., 1994] studied the possibility of the "back door". The conformations of the protein molecule under which a water molecule could leave the active site were studied. Simulation of ACE dynamics in water showed that an exit really appears for a short time. The channel is formed in a thin wall of the active site in the vicinity of TrpM. Its entrance is near Gly441, Tyr442, and TrpM. The channel goes around Trp84 and appears at the surface near Glu445. Displacement of Trp84, Val129, and Gly441 residues at the mean deviation of their atoms of about 1.3pA opens the channel. No other channels were found in spite of the fact that all the conformations were analyzed. Fig.7.2 shows the entrance to the cleft, the catalytic triad, the bottom of the cleft, and the "back door". The calculation of the electrostatic potential around the enzyme showed that

212

Molecular scissors.

I- gorge entry

active site

I III IN'

Cluster model

|., :

| - gorge bottom

^ S S ~ 'back door'

Fig. 7.2 ACE molecule: (a) the axis of the cleft is perpendicular to the figure plane; (b) the axis of the cleft belongs to the figure plane.

the potential minimum is located near the bottom of the active site in the vicinity of He444. Thus, the electric field does not allow the positively charged choline to leave the cleft through the "back door". The problem of the escape of the reaction products remains unsolved. Let us consider in details the geometry of the enzyme and substrate molecules. X-ray data regarding both AC and ACE are available. Fig.7.3 demonstrates the layers of 5-angstrom thickness cut by the planes shown in Fig.7.2. Fig.7.4 shows two positions of the substrate molecule: near the cleft entrance and in the vicinity of the catalytic triad. It follows from Fig.7.2-7.4 that the cleft of the active site can accommodate only one AC molecule. Assume that the "back door" does not exist then it is evident that the substrate molecule can approach the catalytic triad only if the previous substrate molecule or its fragments have left the AS cleft. Note that diffusion of the positively charged molecule from the cleft is hindered by the electrostatic attraction to ACE molecule which can limit the rate of the enzymatic reaction.

7.3

Electrostatic field of ACE molecule

Consider the motion of the substrate and the reaction products in the vicinity of the enzyme. One has to take into account their interaction with solvent molecules, van-der-Waals interaction with each other and with the enzyme molecule, and the electrostatic interaction. In order to compare the influence of these three factors, let us calculate the electrostatic potential of the enzyme in the solvent.

Electrostatic field of ACE

Layer H

Layer G

Layer A Fig. 7.3

7.3.1

213

molecule

Layer F

Layer B

The layers of ACE molecule confined by the planes shown in Fig.7.2.

Charge distribution

inside the

molecule

Amino acids are classified as acidic (can have negative charge), alkali (can have positive charge), and neutral [Volkenstein, 1988]. There are three acidic (Asp, Glu, Tyr) and three alkali (His, Lys, Arg) amino acid residues. The charge of the amino acid residue depends on pH of the solvent. At pH 7.0 that corresponds to natural environment of the protein [Pasynsky, 1963] only four residues are ionized: Asp, Glu, Ly, and Arg. All the further calculations are performed for pH 7.0. Thus, we know the charges and coordinates of all the ionized atoms of the protein and can use them for calculating electrostatic parameters of the enzyme. Before going to the results let us discuss the choice of the coordinates. In this work the origin coincides with the atom CD1 of the residue He444 that is located at the bottom of the cleft. Z axis goes through the center of the entrance to the active site and, thus, represents the axis of the cleft. The center of the entrance to the active site is defined as a mean value of the coordinates of Glu73.CA, Asn280.CB, Asp285.CG, and Leu333.0. X and Y are oriented in such a way that the atom Glu73.CA belongs to the plane YZ. ^-coordinate of the center of entrance and

214

Molecular scissors.

Cluster model

a) Fig. 7.4

b)

(a) Substrate enters the cleft; (b) substrate is in the vicinity of the catalytic triad.

z, A

20-

o-

-20-

-16

-12

-8

-4

0

4

8

12

16

Integral charge density

Fig. 7.5

Distribution of the enzyme integral charge density along Z axis.

y-coordinate of Glu73.CA are positive. Fig.7.5 shows the integral charge density distribution along Z axis, i.e. along the axis of the cleft. The curve shows the sum charge between the planes that are perpendicular to the axis of the cleft; one of the planes is fixed and crosses Z at z = 0. The charge of the "upper" part of the enzyme is negative and equals -14e, the charge of the "lower" part is +2e. Thus, the total charge of ACE is negative and equals -12e. The number of the ionized atoms is 110. Asymmetry of distribution of the charge shows that the molecule must have a dipole moment. Below we determine the dipole moment for different ionic strengths of the solvent for comparison with the experimental data. The agreement of the results can prove indirectly the correctness of the calculated potential.

Electrostatic field of A CE molecule

7.3.2

Calculation

of the

215

potential

Different ions surround ACE molecule in the interstitial liquid. Thus, it is necessary to take into account the influence of the ionic atmosphere on each charge. Assume that the solvent contains different ions. Let their valences be Zi and the concentrations Co,. If a system of charges is introduced in bulk of the solvent the concentrations of ions Cj become spatially inhomogeneous. The concentrations are given by the Boltzmann distribution Ci \f) = c0i exp '

kBT

where