Mathematical models of chemical reactions: theory and applications of deterministic and stochastic models

=

da(t)fdt = db(t)jdt = d
VPFK V;n -

VPK VPK

(7.12)

ro•.

The enzyme rate of VPFK and VPK (PK stands for pyruvate kinase) are complicated functions of a, b and concentrations of other components that are not involved explicitly in the model (ATP, fructose-6-phosphate) V;n consists of the mean input flux superimposed by the input flux oscillation. While other models of forced glycolysis (e.g. Richter (1984 based on the Selkov model)) demonstrates the occurrence of entrainment (the frequency locking of an autonomous oscillator by externally imposed frequencies) Hess and Markus emphasised the failure to see entrainment leading to quasiperiodic or even chaotic dynamics, and the coexistence of complicated periodic attractors. Currently the biological function of complex temporal patterns are not well-understood. Some neurobiological examples will illustrate this problem.

7. 3 Fluctuation and oscillation phenomena in neurochemistry 0. The neurochemistry of synaptic transmission has been thoroughly investigated over the decades. Though the approach and techniques of neurochemistry is static rather than dynamic, kinetic models may prove useful for explaining both the chemical details and the regulatory aspects of synaptic transmission. First, a short review of synaptic transmission is given. Second, postsynaptic membrane noise as a source of information will be mentioned; then oscillatory and other temporal patterns connected to integrated synaptic activity will be analysed. 1. Information is transferred from one neuron to another at specialised points of contact, at synapses. Synapses are most often made between one axon of cell and the dendrite of another (but there are other kinds of synaptic junction too). On the arrival of an impulse at the nerve terminal, a chemical transmitter substance is released from the presynaptic cell, crosses the synaptic cleft separating the two cells, and interacts with specific receptor sites on the postsynaptic membrane. The receptors are large protein molecules embedded in the semifluid matrix of the cell membrane. The binding of the transmitter to the receptors implies the opening and closing of 'membrane channels' controlling the flow of specific ions through the membrane. Most of what is known about the mechanism of synaptic transmission

186

Mathematical models of chemical reactions

Jo
Synaptic vesicle

Vesicular· grid

Postsynaptic density Synaptopore

Fig. 7.5 Schematic illustration of the chemical synapse.

comes from experiments on a particular synapse: the neuromuscular junction of the frog, where acetylcholine (ACh) serves as the transmitter. The schematic illustration of a synapse is given in Fig. 7.5. The metabolism of the ACh can be considered as a recycling process constituted by the following 'elementary' steps: (I) ACh synthesis. The precursors of the ACh are the choline (Ch) and the

acetyl coenzyme A (Ac-CoA). The reversible formation of ACh is catalysed by the choline acetyl transferase (ChAT; EC 2.3.1.6). Since the enzyme has both cytoplasmic and membrane-bound forms, consequently the synthesis can take place by two different mechanisms. The synthesis can be influenced by the formation reactions of the precursors and by their transport to the place of the synthesis. Both the sodium-dependent high affinity choline uptake (Barker & Mittag, 1975) and the transport of the mitochondrial Ac-CoA to the cytoplasm (Jope, 1979) may be the rate-limiting step of the synthesis. (2) ACh release. According to the classical, vesicular hypothesis (Del Castillo & Katz, 1954) packets of transmitter molecules ('quantums') contained in synaptic vesicles are released by exocytosis. The vesicular hypothesis was in accordance with morphological studies demonstrating the existence of synaptic vesicles (De Roberties & Bennett, 1954) and explained the discrete nature of the postsynaptic response, namely the occurrence of miniature-endplate potentials (Fatt & Katz; 1952). Alternative hypotheses (e.g. Israel & Dunant, 1979; Tauc, 1979) are based on experimental observations of the release of cytoplasmic ACh. (3) Cleft processes. ACh, when released, diffuses the synaptic cleft separating

Applications

187

the presynaptic and postsynaptic membranes. However, the synaptic cleft cannot be considered as an 'empty tube', since it contains different types of chemical molecules. Chemical reactions taking place in the cleft can strongly modify the mechanism of the information transfer. (4) Transmitter-receptor interaction. While one class of channels in neural membranes responds to voltage differences across the cell membrane (i.e. they are voltage-gated), a second type of channel is chemically gated. Chemically gated channels are activated by the transmitter-receptor interaction taking place at the postsynaptic membrane surface. Conformational changes in membrane proteins form the basis for gating since they serve to open and close the channel by slight movements of critically placed portions of the molecule that unblock and block (at least partially) the pore. When channels are open, the resulting electric current can be measured. The opening and closing of chanels have a random character, therefore fluctuation in the electric current measured reflects certain biochemical details of the transmitter-receptor interaction. (5) Hydrolysis of the postsynaptically bound ACh. The most rapid step of the ACh metabolism is the enzymatic decomposition of ACh. Acetyl choline esterase (EC 3.1.1. 7) is a very active enzyme. Rapid hydrolysis is the precondition of rapid recovery of the resting state after transmitting an impulse. (6) Diffusion and (re)uptake of choline. Choline diffuses from the postynaptic cell towards the presynaptic cell. Choline resulting from different sources can be taken up by different mechanisms, but thin is beyond our present scope. 2. The possibility of the measurement of postsynaptic membrane noise (Katz & Miledi, 1972; De Felice, 1981) gives a deeper insight into the details of the chemical mechanism of transmitter-receptor interaction. According to a simple assumption the transmitter-receptor complex has one 'closed' and one 'open' form. This assumption can be questioned and an algorithm can be given to determine the number of 'non-completely closed' conformations of the complex, and the kinetic rate constants of the conformation changes. The measurable fluctuations are in the electrical potential or current of postsynaptic cells, and are induced by fluctuations of quantities of chemical components, particularly transmitter, receptor and intermediate molecules. Consequently, it is possible to give a relation between the quantities describing the electrical fluctuations in a postsynaptic cell and the stoichiometric and kinetic parameters of transmitter-receptor interaction. Assumptions of the model: (I) Transmitter-receptor interaction can be described by the following

simple model scheme:

188

Mathematical models of chemical reactions (7.13)

where x is the number of transmitters binding per one receptor molecule, T is the transmitter, R is the receptor, T.R is the conformation of the complex having zero ion-conductance, T.R; are conformations to be associated with non-zero ion-conductance, T.R 1 is the conformation with maximal conductance. (2) The quantity of transmitter is maintained at a constant level. (3) The conductance is a linear function of the T.R; conformation quantity vector (denoted by N): g=LN,

(7.14)

where g is the h-dimensional conductance vector (h is the number of ionic species considered), the dimension of the vector N is I + 2, the dimension of the matrix L is h x (I + 2). (4) The conductances of the individual ions are summarized by g = grl. Here g is the total conductance, 1 is an h-dimensional vector with all coordinates equal to unity. (5) A particular form of the function g(N) can be specified assuming one of the channel-models (Chen, 1976): I

g(N)

=

L g,exp[- (I -

j)E)kTJN;,

(7.15)

j=O

where g 1 is the conductance vector of one channel being in the completely open state, Ni is the quantity of the complex T.Ri, T is the temperature and Ei is the activation energy of the conformation change T.Ri-+ T.Ri+ 1• This channel model is strongly motivated by the transition state theory of reaction kinetics. This model for describing transmitter-receptor interaction' can be identified with the stochastic model of closed compartment systems. Adopting the fluctuation-dissipation theorem the conductance spectrum of the postsynaptic cell is determined by three qualitatively different factors: (I) the length of the compartment system (implicitly); (2) the precise form of the function connecting the conductance and the quantities of the different conformations (e.g. eqn. 7.15). (3) the rate constants. The analysis of this model in Erdi & Ropolyi (1979) is an illustration of how noise analysis may be used as a tool for obtaining kinetic information. 3. At least three different neurochemical and neurophysiological oscillatory phenomena appear at different hierarchical levels of the cholinergic synaptic

Applications

189

transm1ss1on system. (Erdi & T6th, 1981; Erdi, 1983). 'Rapid' and 'slow' oscillation of free, presumably cytoplasmic, acetylcholine (ACh) have been described (Dunant et a/, 1977; Israel et a/., 1977). Another oscillatory phenomenon is the series of miniature-endplate potentials (Fatt & Katz, 1952) that correspond to the postsynaptic effects of pockets of transmitter, at least in the spirit of the classical vesicular hypothesis. A complete theory of the hierarchical mechanism of dynamic synaptic activity would require the study of co-operation and competition among the three oscillators. Here only one model, established for explaining the 'slow oscillation' will be presented. The 'isolated' model shows periodic behaviour. The skeleton model of the 'slow' oscillation in free acetylcholine can be formulated in terms of mass-action kinetics (Csaszar et a/., 1983). In principle, instead of setting up a lumped, skeleton model, a more complex model could be defined to take into account the details of subprocesses (synthesis, storage and release of ACh, cleft processes, transmitter-receptor interaction, diffusion, re-uptake). Experimental information, however, certainly would not be sufficient to parametrise such kinds of models. According to the assumptions of the four-component model the state of the system is characterised by: -

cytoplasmic ACh concentration: X, ACh concentration at the postsynaptic membrane surface, choline (Ch) concentration near postsynaptic cell, Ch concentration near presynaptic cell. The subprocesses of the model:

(I) transmitter release, cleft processes, transmitter-receptor interaction;

(2) ACh hydrolysis (the most rapid subprocess); (3) metabolic products (mostly Ch) diffuse to the vicinity of the presynaptic cell; (4) (re)uptake of Ch; (5) autocatalytic synthesis of ACh. A four-component formal chemical model k1

X-+ Z,

k1

Z-+ W,

kJ

W-+ Y,

k4

Y-+ X,

2X

ks

+ Y-+ 3X

can be associated with the synaptic activity containing the five subprocesses. This mass-conserving model has a single nonlinearity and can exhibit sustained oscillations in numerical computations. Fixing the 'total mass', but changing the initial concentration the limit cycle behaviour can be visualised (Fig. 7.6(a)). Fixing the rate constants also, but changing only the 'total mass', a family of limit cycles (limit shells) can be derived (T6th, 1985) (Fig. 7.6(b)).

Mathematical models of chemical reactions

190 y

40 M

38

=

300

36 34

32

30 28

26 24 22

20 18 161 14 12

10 6 4 2

o~.,~.~.,~ ••~,.-.~,.--.~..-.~••-~.~.-••~.~.~••~.~••-••~.~••-.~~w-

270

280

290

300

(a)

0

40 (b)

Fig. 7.6(a) The Y versus W selection curve seems to tend from outside and from inside to a stable limit cycle. (b) The relatively stable limit cycles are changed as a function of the total mass of the system; the system has limit cycles within a certain (ange of M, the shape and location of these closed curves vary as the constraint is varied.

Adopting the neurochemical assumption according to which the choline uptake (not exclusively, however) controls the process, an independent choline oscillation is assumed. Therefore the kinetic equation is:

Applications dx(t)fdt = -k 1 x(t) + k 4 y(t) + k 5 x 2 (t)y(t) dz(t)/dt = k 1 x(t)- k 2 z(t) dw(t)fdt = k 2 z(t) - k 3 w(t) dy(t)fdt = k 3 w(t)- k 4 y(t)- k 5 x 2 (t)y(t) + acos(wt

191

(7.16)

+
Numerical studies (Erdi & Barna, 1986a, b) gave the evidence of occurrence of 'abnormal' dynamic behaviour. Phase space plots and stroboscopic plots are obtained. The former are identified with two-dimensional projection of trajectories, the latter can be constructed by points taken at regular intervals of the period (figs 7.7 and 7.8).

Fig. 7.7 Phase plane diagram in different time regimes. (a) Approaching to the 'phantom attractor'. (b), (c) Intermediate regime, (d) Near the true attractor.

In general terms, many neural disorders are characterised by changing the 'normal' temporal patterns to 'abnormal' behaviour. This abnormal dynamics might be interpreted as a 'dynamical disease' occurring in a physiological control system operating within a range of control parameters (Guevara et a/., 1983).

192

Mathematical models of chemical reactions

"/~::....

/

,_

I

l

'•·

1.~.

i.

(b)

.. .

.

'::'

.

..

1.". · , .

'"-":.<,.~

·/···. .. ······-·

..

.. ·········· ...

I

(c)

,•... · ~·

,_ ..

(d)

··.:·.

. :.i)

/

\.·····

Fig. 7.8 Stroboscopic representation of perturbed variable. (a) Quasistable 'butterfly' shape. (b) Decay of the quasistable shape and start to tend to a new shape. (c) Tending to a new attractor. (d) Arriving at a fixed point of the Poincare plot.

7.4 Population genetics I. Population genetics describes interactions among alleles. The system (in the simplest case it is a one gene locus) is composed of alleles A 1 , ••• , A •. The state of the system is characterised by the frequencies x 1 , .•• , x. of the alleles. The frequency of the genotype A;Aj is 2x;x;; (i=Fj). Adopting the traditional assumptions the survival (or fitness) of A;Aj is W;; (w;; ~ 0; w;; = wj;). A scalar quantity h

w*(t)

=L

w,.x,(t)x.(t)

r.j~

(7.17)

I

can be assigned to the whole population (mean fitness), and the normalisation LX; = I is valid. Selection equations in a deterministic framework can be· given by the differential equation: h

dx;(t)fdt

= X;(t)( L w;;xj(/) - w*(t)). j~l

(7.18)

Applications

193

The reconciliation of the Darwinian theory of natural selection and of Mendelian population genetics has been based on this model (Fisher, 1930; Wright, 1931). According to Fisher's theorem, the mean fitness w* increases steadily along the trajectory of (7 .18). Technically it means that w* is Lyapunov function. Kimura (1958) stated that the trajectories of (7.8) point in the direction of maximal increase of w*. Technically it would mean that (7 .18) is a gradient system. However, the system is not a gradient system, at least not in the traditional sense. Maximal increase implies that the direction of the trajectories is orthogonal to the contour lines of w*, and this is not the case in general. The 'spirit' of Kimura's maximum principle have been saved by a redefinition of orthogonality by Shahshahani ( 1979) who introduced a new Riemannian metric at every point of the state-space. Denoting the state space by s.={x=(xl> ... ,x.);

•

LX;= l,x) ~0},

(7.19)

i= I

the Shahshahani metric is given at pEs. by (x, y)p =

L (x;y;)/P;

(7.20)

i= I

(x andy are elements of the tangent space). Using this metric or (equiva-

lently) making an appropriate transformation of co-ordinates, Akin (1979) presented a new geometrical framework of population genetics. One important prize from the work is the validity of Kimura's principle. It can be shown more generally (Sigmund, 1984) that a differential equation dx;(t)/dt = x;(/;(x(t)) - LX;(t)/;)

(7.21)

;

is a Shahshahani gradient system, if and only if dx(t)/dt = f(x(t)) is a usual gradient system. The basic selection model has been generalised in many different ways (Akin, 1979; but see Hofbauer, 1985). Mutations from Ai to A; with probability bii have been taken into account leading to the model: (7.22) For the case of equal mutation rate (i.e if bii is independent ofJ) the system is Shahshahani gradient system. For any other bii wii can be found such that the Shahshahani gradient property fails and periodic behaviour occurs via Hopf bifurcation. In terms of genetics this means that the frequency distribution of the alleles does not converge to a stable equilibrium point, but may exhibit oscillations. Even the occurrence of chaos seems probable in slightly

194

Mathematical models of chemical reactions

modified versions of the model. While the existence of oscillatory and even chaotic behaviour has become more or less acceptable in the chemical and biomathematical communities, it is a serious challenge for population geneticists, since it contradicts their 'adaptive landscape' concept. However, if chemical kinetics could adopt the geometric approach of population dynamics, the interplay between the two disciplines could be productive. 2. Many stochastic models are used in population genetics to describe the effects such as random drift selective force, and mutation pressure (e.g. Karlin & McGregor, 1964; Crow & Kimura, 1970; Maruyama, 1977). Diffusion models are mostly used, and some of them can be interpreted in terms of the CCS model of chemical reactions. The models are in terms of the probability distribution of gene frequencies. The tractable models are more or less restricted to the one locus situation. Let x, be the frequency of A genes at the time t. Pure random drift is superimposed by the reversible mutation u

A ¢a

The infinitesimal generator

A is:

- _ x(l - x) d 2 A = 4N dx2

+ {v -

(u

d

+ v)x} dx'

i.e. it is the infinitesimal generator of a simple, often-investigated diffusion of a single particle. Many other models have been defined, and the problems of obtaining solutions are similar to those in the case of chemical reactions.

7.5 Ecodynamics 7.5.1 The theory of interacting populations 7.5.l.l Boulding ecodynamics Observing the theory of dynamics of populations from the point of view of chemical kinetics, it is well-known that the celebrated Lotka-Volterra model has both chemical and ecological interpretations. It is quite reasonable that the formal theories of chemical kinetics and of 'mathematical' ecology are highly overlapping. Ecodynamics (Boulding, 1978, 1983) motivated by the general system theory tries to become a general theory of ecosystems. The ecosystem is the basic concept of Ecodynamics. It can be defined as a set of interacting populations of different species, the population of a species being defined as a set of objects, or elements of some kind, which are similar enough to be

Applications

195

interesting as an aggregate. This lands us immediately in the problem of taxonomy, which is the big skeleton in the closet science. If we get our taxonomies wrong, as the alchemists did-earth, air, fire and water are far too heterogeneous to be elementswe will never get very far. Physics and chemistry succeeded because they somehow got the right, or at least very useful, taxonomies, though even chemical elements exploded somewhat into isotopes and the Newtonian taxonomy of matter and energy got into trouble with Einstein. (Boulding, 1983)

Having determined the state variables by adopting an appropriate taxonomy, dynamic laws have to be defined: To define an ecosystem, then, we have to take a bold and dangerous step and just assume that we can define significant species. We have the fact that the total number or quantity of a species, whether of hydrogen atoms, water molecules, horses, automobiles, or persons who believe in the second law of thermodynamics, follows the principle of what I have called the 'bathtub theorem'. It states that the total quantity or population of anything increases by the number of additions less the number of subtractions. Thus, the number of water molecules in a given area increases when hydrogen burns in oxygen (water molecules are 'born') and diminishes when the water molecule is dissociated (as, for instance, by an electric current). The same principle goes for biological organisms of any kind, for human artefacts, and even for human ideas. Additions consist of births and in-migrants; subtractions of deaths and out-migrants. The rate of additions and subtractions to a population, however, is going to depend upon its own size and the size of the other populations of the ecosystem. (Boulding, 1983)

The 'ideology' of ecodynamics have partially been induced from the experiences of many ecological model experiments (and of course it was deduced from the principles of the general system theory). Some illustrations will be given. 7.5.1.2 Compartmental ecokinetics Compartmental analysis is a well-known technique of ecokinetics (an example: Blau eta/., 1975). Let the ecosystem be a fish in an aquarium. The uptake and clearance of compounds by fish can be modelled in the simplest way by a twocompartment model treating the fish as a single homogeneous compartment. Assuming that: (I) Uptake of chemical (through gills, skin etc.) can be taken into considera-

tion by a single rate constant k 12 • (2) The clearance of the chemical by different mechanisms can be characterised by a single rate constant k 21 • (3) Constant rate of chemical addition and removal into and from the water compartment are involved. (4) Kinetics is zero-order or first order.

196

Mathematical models of chemical reactions Water compartment

Fish compartment

Constant rate of chemical addition (a)

Constant rate chemical removal (r)

Fig. 7.9 Two-compartment model of bioconcentration test ecosystem.

The two-compartment model is shown in Fig. 7.9. The kinetic differential equation system with the initial conditions is: dc 1 {t)/dt =a- k 12 c 1 (t) + k 21 c2 (t)- r dcz(t)Jdt = ktzCt(t)- kztCz(t) c 1 (0) = Cw and c2 (0) = 0.

(7.23)

In practical cases dc 1{t)/dt = 0 used to be assumed. The solution of the model can be easily given, but it is not very important for us at this moment. The fish itself can be represented by two compartments where one of these compartments corresponds to a particular tissue in which the chemical compound may be preferentially concentrated. Two possible threecompartment models are shown in Fig. 7.10. The two models differ only in that the second model a rate constant for clearance from the third compartment to the water compartment also was involved. Model discrimination by solving the inverse problem is a difficult task. 'Note that this discrimination takes place only among the set of postulated models. That is, the model selected may be the best of the originally postulated models but totally inadequate for characterizing the system' (Blau et a/., 1975). (Model discrimination and structural identifiability of compartment systems have been mentioned in Subsection 4.8.4). 7.5.1.3 Generalised Lotka-Volterra models The examination of competitive interactions among different species has been one of the main topics of mathematical biology. The most often used mathematical model is still a generalisation of the Lotka- Volterra model; systems of polynomial ordinary differential equations expressible in terms of formal chemical reactions have also been investigated. The main problem is to find criteria for the coexistence of species. All species in the communities

Applications

197 Fish compartment

Metabolite on particular tissue

Fish compartments

Metabolite

Fig. 7.10 Three-compartment model of bioconcentration test ecosystem.

can coexist in three distinct ways; species may coexist at equilibrium, in persistent oscillation or in chaotic motion (see, for example, May & Leonard, 1975; Takeuchi & Adachi, 1983a, b; Freedman & Waltman, 1985). The generalised Lotka-Volterra system is: dx;(t)fdt = X;(t)[b;- Laiix;(t)], i = I, 2, .. . n,

(7.24)

j

and some of the most often used models are the: Two-species-competing system: dx 1 (t)/dt = x 1 (t)[l- x 1 (t)- ax 2 (t)] dx 2 (t)fdt = x 2 (t)[l- bx 1 (t)- x 2 (t)];

(7.25)

Two-prey, one-predator system: dx 1 (t)/dt = x 1 (t)[l- x 1 (t)- ax 2 (t)- ey(t)] dx 2 (t)fdt = x 2 (t)[l- bx 1 (t)- x 2 (t)- uy(t)] dy(t)/dt = y(t)[ -1 + dex 1 (t) + dux 2 (t)]; Two-prey, two-predators system:

(7.26)

198

Mathematical models of chemical reactions

dx 1 (t)fdt = x 1 (t)[I - x 1 (t)- ax 2(t) - e 1y 1 (t) - e 2y 2(t)] dx 2(t)Jdt = x 2(t)[I- bx 1 (t)- x 2(t)- u 1YJ(t)- u 2J2(t)] dYI (t)Jdt = y 1 (t)[ -I + d 1 e 1 x 1 (t) + d 1 u 1 x 2 (t)) dy2 (t)/dt = Y2(t)[- I + d2e 2x 1 (t) + d 2u 2x 2(t)].

(7.27)

Here X;(t), y;(t) are the population sizes for prey and predators, respectively, a> 0 and b > 0 represent competitive effects between two prey, e; > 0 and u; > 0 are coefficients of decrease of prey due to predation, and the d; are the transformation rates of i'h predator. The possible equilibria for (7.26), for example, are denoted by (E+ + +), (E+ +0), (E+o+), (Eo++), (E+ 00), (E0 +0 ), Here (E+o+) denotes the equilibrium in which prey x 1 and predatory remain positive, and prey x 2 is extinct. The two-species-competing system (7.25) cannot show oscillatory behaviour (in consequence of the Hanusse-Tyson-Light-P6ta theorem: it is different from the Lotka-Volterra model). The conditions of stability of equilibria were studied by Takeuchi & Adachi (l983b). Accordingly, the (E++) is globally stable if and only if a< I, b I, b > I (E+o) and (E00 ) are locally stable. Equation (7.26) is derived from (7.25) including predator. How does this inclusion affect the possibility and mechanism of coexistence? (Takeuchi & Adachi I983a, b) proved that the addition of a predator to a system with a stable equilibrium may give a bifurcation to a stable limit cycle at increasing rates of predation. They showed a + b < 2 implies (under certain technical conditions) the global stability of nonnegative (E + + +>· Supposing ab ~ I, in certain parameter regions Hopf bifurcation might occur. The assumption leading to a limit cycle can be further classified. According to numerical studies in the case of a > I, b > I the limit cycle can collapse into one of two single-prey equilibria when there is a further increase in the control parameter. In the case when a~ I, b ~ I (or a~ I, b ~ I) there is the possibility of turning into (spiral) chaos (Gilpin, I979). The combined analytical and numerical analysis of the two-prey, twopredators, system (Takeuchi & Adachi, I983a, b) showed similar the following qualitative results: a stable equilibrium also can bifurcate to a stable limit cycle and the limit cycle turns into a nonperiodic oscillation of bounded amplitude with increasing cycle time. Remarks

'Predator-mediated coexistence' of competing species has been found both in spatially homogeneous (see Takeuchi, I985) and in inhomogeneous systems (Hutson & Vickers, I983; Hutson & Moron, I985). The term means that the introduction of a predator may allow two prey to coexist which would not otherwise do so.

Applications

199

7.5.1.4 The advantages of stochastic models: illustrations

Differences between deterministic and stochastic ecological models have been emphasised many times, explicity or implicitly. The main qualitative deviation in the behaviour of ecodynamic models can be interpreted in terms of survival and extinction.

u

a>1

b>1

Fig. 7.11 In the regions (A), (8), and (C) E+ ++exists. In (A) (or (B)) there exists at least one Hopf bifurcation parameter e* (or u*) for any straight line u = constant (or e = constant) if dfde(A 0 A 1

-

ADI,~.•

¥- 0; or dfdu(A 0 A 1

-

ADiu~u•

¥-0

where A 0 = xf + x~. A, =(I - ab)xfx~ + du 2 x~y* = d(e 2 + u 2 - (a+ b)eu)xfx~y*

+ de*y*xf,

A2 is satisfied.

Macroscopic ecological level equations have emerged from microscopic genetic equations, and there is a feedback from the macroscopic structure to modify the microscopic dynamics. The appearance and eventual stabilisation of a new mutant might have a crucial role in the evolutionary process. Allen & Ebeling (1983) have argued for the necessity of a stochastic description of simple ecosystems connected to evolution. They considered interactions among a 'prey' species, X;, and a 'predator' species, Yk, where i and k denote phenotypes. 'Elementary' steps at the microscopic level modifying the number of interacting species: (a) normal replication of species X;: X;--+ X;+

I; with probability a;x;(l - x)N),

200

Mathematical models of chemical reactions

(b) capture and destruction of prey X;~ X;-

X;

by predator Yk:

I; with probability (s;k/V)X;Jk,

(c) normal reproduction of predator Yk: Yk

~ Yk

+ 1; with probability (S;k/ V)x;yk,

(d) death of a predator Yk: Yk

~ Yk

- I; with probability DkYk,

(e) 'diffusion' of a prey or predator from the system: X;~ X;+

I, Yk

~ Yk +

I; with probability D;x;0 and Dky2,

(f) 'diffusion' of a prey or predator of the system: X;~ X;-

I, Yk - Yk

~

1; with probabilities D;x; and Dkyk,

(g) 'abnormal' reproduction of X;: X;Xj ~ X;Xj

+ I with probability Miix;(I - xjJN)

(h) 'abnormal' reproduction of the predator Yk: YkYI ~ YkY1 + 1; with probability Mk;X;YJ! V.

To described the temporal evolution of the probability distribution of a new mutant population starting from a pre-existing situation a linearised birth and death equation can be derived: dP(x 2, t)Jdt = A(x?, yy .. .)((x 2 - I)P(x 2 - x 2P(x2, t)) + D(x?, y? ... ) x ((x 2 + I)P(x 2 + I, t)- x 2P(x 2, t)),

-

I, t)

where x 2 denotes the numbers of the mutants, the function A contains all the processes tending to increase x 2 , linearised in x 2 a, and the function D contains all the terms tending to decrease x 2 • Let us denote with A' and D' the value of these functions, in general. If A' < D' then the probability of extinction is I, while for A'> D', this probability·is D'JA'. The increase or decrease in efficiency of the mutant type is d A' ID' - I. The probability of survival P,.,(t) is

=

d

P,.,(t)

=

I + d- exp(- (A' - D')t) .

Evaluating this expression for a time t = n/A', where n is the number of reproduced generations during t: P,.,(n)

=I+

d d- exp- (ndj(l +d))·

The general form of the function P,.,(n) is given by Fig. 7.12. Is shows that,

201

Applications 1-------------------------L

Step function resulting from deterministic analysis

f

Probability of survival

10%

20% 30% 40%

SO%

60%

70%

ao•to

90% 100% 110•t.

d

Fig. 7.12 Probability of survival of a mutant of fitness d, for a number of generations, n.

while deterministic analysis results in very sharp distinction between favourable and unfavourable mutations, the stochastic analysis leads to a much 'smoother' picture. Models of population growth in random environment have been constructed by inserting 'white noise' fluctuations in the deterministic growth equations (May, 1972; Capocielli & Ricciardi, 1974; Ricciardi, 1977; Nobile & Ricciardi, 1984a, b). The structure of these models is similar to those described in Section 5.8 ('external fluctuations') and we shall not discuss them here. As was mentioned in Subsection 5.6.2, stochastic versions of the Lotka-Volterra model lead to qualitatively different results from the deterministic model. The occurrence of a similar type of results is not too surprising. A simple model for random predator-prey interactions in a varying environment has been studied, staring from generalised LotkaVolterra equations (De, 1984). The transition probability of 'extinction' is to be determined. The standard procedure is to convert the problem to a Fokker-Pianck equation (adopting continuous approximation) and to find an approximation procedure for evaluating the transition probabilities of extinction and of survival. Stochastic models are appropriate when we are interested in the behaviour of mutant populations present in very small quantities only. In more general terms, the role of 'genetic drift' due to randomness during evolution has been emphasised (Crow & Kimura 1970) in the framework of 'neutral theory of evolution'. The relative importance of natural selection and genetic drift, and more generally of the deterministi!= and stochastic factors in evolution. remains the most important unsolved problem in our understanding of the mechanisms that bring about biological evolution. (Dobzhansky eta/., 1977, p. 164)

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Mathematical models of chemical reactions

7.5.2 An ecological case study A relatively detailed example will be given to illustrate the methods offormal reaction kinetics in ecological modelling. (It is considered to be a case study, and the choice of this model is quite arbitrary.) The model is of the eutrophication process taking place in the largest lake in Central Europe, Lake Balaton, Hungary. There is a wide recreational area all around the lake, which is very popular with tourists from abroad and from within the country as well. The phosphorus input to the lake has greatly increased in recent years, and as a consequence of this the growth of the phytoplankton has also increased. In order to understand is phenomenon, and to help to reverse the process of eutrophication, a modeling study was started in the late nineteenseventies. In what follows we briefly describe the main characteristics of a deterministic and a stochastic model of the eutrophication process in Lake Balaton, using the concepts of formal reaction kinetics. The phenomenon to be described here does not fit precisely into the framework of mass-action type reaction kinetics. However, the differences are of the same nature that are met within the area of chemical applications too. First, one has time-dependent input (here: forcing) functions. Second, the rate of some of the elenentary reactions are not of the mass-action type. (Sometimes Michaells-Menten-type rate functions have been chosen, sometimes even more complicated ones consistent with present knowledge in ecology. After a general introductory discussion of the stochastic model the two models will both be defined. Relevant, and ecologically meaningful, differences between the two models will be discussed at the end.

7.5.2.1 Arguments for a stochastic model The exact definition of the model will be done very easily using the metalanguage of formal reaction kinetics. The stochastic model of the reaction to be defined below will be the stochastic model of the phenomena of eutrophication. Therefore the only questions worth treating here are: (I) What are the special ecological reasons that make it plausible to use a stochastic model? (2) Why are the arguments for the usual continuous time discrete state stochastic model so good for reaction kinetics here? (3) What are the consequences of the stochastic model defined here, both from the scientific and from the management point of view?

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Deterministic models (most commonly systems of nonlinear differential equations) have been used to model the dynamics of lake ecosystems (Park eta/., 1974; Chen & Orlob, 1975; Joergensen, 1976; Di Toro eta/., 1979). These models have been satisfactory to describe the main trends of ecological phenomena; however, deterministic models are not capable of reflecting the inherently random behaviour of ecosystems. This source of uncertainty is especially important in large multiparameter models. The cases when there are qualitative differences between a deterministic model and the corresponding stochastic one have been enumerated previously. A single viewpoint should be emphasised here. Models in general are aimed at describing the investigated phenomena, and making predictions of the behaviour of the system under different conditions. In terms of Schofield & Krutchkoff's (1974, p. I) statement: 'it is possible for the instantaneous pollution concentrations to be at ecologically dangerous levels, while the means or deterministic concentrations remain at acceptable levels.' We are interested in how far the evolution of a variable is reproducible under essentially the same conditions. In general the method of confidence intervals (Zacks, 1971, Ch. I 0) may be suggested for approaching a problem of this kind. As it follows from the nature of the stochastic model, an increase in input load of phosphorus does not necessarily lead to an acceleration of the eutrophication process. The same input load, however, may lead to a major increase in algae biomass (as was probably the case in 1982). It should be mentioned here that spatial inhomogeneities and fluctuations have been disregarded in the present model, although it is easy to incorporate some of them in the following-usual-way: let us divide the lake into four different basins and let us suppose that they are homogeneous. If linear diffusion is supposed to take place between them that can be modelled by first-order elementary reactions then we have a homogeneous kinetic model of a phenomenon taking place in an inhomogeneous medium. Here only the one-basin model will be treated. Finally, let us mention an argument for the stochastic model coming from the endeavour to revitalise the science of ecological modelling. Large multiparameter systems aimed at a detailed description of lake eutrophication seem to have become less popular in recent years. This may either be the consequence of either the mathematical and computational problems involved or be due to the lack of measurements, or both. It is hoped that stochastic models constitute a possibility in this regard. Very few stochastic lake models seem to exist in the literature. Ecological\ problems related to those investigated here are described stochastically by ! the following. tools: time series, random walk, diffusion processes, differential/ equations with random parameters (taken in the wide sense), and Markovia¢ pure jump processes. As for the different types of stochastic models: time series, random walk

204

Mathematical models of chemical reactions

and diffusion models seem to be too simple to cope with the ecological details of a process. It is a question of judgement whether the fluctuation of the state variables are considered to be caused by the parameters (including initial conditions, external forcing functions and parameters in the narrow sense) or whether assumptions are made on the fluctuations of the state variables themselves. Here the second alternative has been chosen because of the importance of inner fluctuations. So one had to choose between stochastic differential equations (apparently never used in ecological modelling) and Markovian jump processes. The latter type was chosen because (I) a certain amount of experience has accumulated in connection with these models in physics, in some areas of biology, and-last but not least-in reaction kinetics; (2) randomness enters the picture without introducing further ecological parameters that require estimation (see, however, page 000); (3) a model of this type corresponding to a deterministic model of the usual type (to a system of-non-mass-action-kinetic differential equations); can quickly be constucted; (4) these models are easy to simulate (see Subsection 5.3.8 and Jernigan & Tsokos, 1980); (5) estimation of the parameters from the measurements will be done more easily if such a model is adequate (cf. T6th & Erdi, 1977b; Erdi & Ropolyi, 1979). 7.5.2.2 A common description of the deterministic and stochastic models

Formal components and formal elementary reactions The formal chemical components of the model are as shown in Table 7.1. Table 7.1

Formal components of the lake eutrophication model.

Ecological meaning Autumn-spring phytoplankton Summer phytoplankton Autumn phytoplankton Blue-green algae Bacterioplankton Dead organic matter Dissolved inorganic phosphorus Dissolved inorganic nitrogen Organic matter in the sediment Exchangeable phosphorus in the sediment

Notation A( I)

A(2) A(3) A sa B

OM p N

s PS

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Fig. 7.13 Vol'pert graph of the eutrophication process.

The structure of the model is usually visualised using the Vol'pert graph of the reaction shown in Fig. 7.13. The formal elementary reactions considered could have been given as a huge set of 40 elementary reactions-mainly with kinetics of the mass-action type. Instead of this formal and awkward definition only some of the formal elementary reactions will be discussed in detail here. As to the others the reader is referred to Herodek et a/. ( 1982) where all the necessary biological background is available, including the reasons for the selection of the forms of the rates of elementary reactions, the external forcing functions and the parameter values. We hope that the biological background there and the background in formal reaction kinetics accumulated from the present book is sufficient to exactly define the model. First, a short description of the deterministic model is given.

The deterministic model

The deterministic model is the (nonautonomous, nonpolynomial) induced kinetic differential equation of the reactions in Fig. 7.13. This model was described in detail by Herodek et a/. (1982). Now we give a formal description of a small part of the model. As an example let us consider the time evolution of summer phytoplankton. Our assumption is that it takes part in the elementary reactions No. 2, 6, 13, 26, 34, therefore the equation for a2 is:

Here

Mathematical models o.fchemical reactions

206

R 2 (a 2 (t), p(t), n(t), t) = PMAX2 •TEMP 2 (9(t))•U 2 (p(t), n(t), /0 (t))•a 2 (t),

corresponding to the elementary reaction A(2) + N + P-+ 2A(2) +(I - nr)•N +(I - pr)•P. PM AX corresponds to the reaction rate constant, called maximal production rate in the present context, TEMP expresses a kind of dependence on time

through temperature (a phenomenon common in nonisothermal reactions), it is called here temperature limitation, /0 is an external forcing function: it is the global radiation on the water surface, U 2 is a joint limitation factor describing light- and nutrient-dependence of primary production, or to use the language of reaction kinetics: it expresses the deviation from mass-action type kinetics. All the remaining rates can be (and have been) defined in the same way; see Herodek et a/. (1982). The right-hand side of the induced kinetic differential equation consists of analogous terms. More precisely, each subprocess contributes, as usual, additively to the corresponding derivatives. Let us emphasise again that 9 and /0 are not state variables, they are forcing functions known from measurements. Results obtained from the numerical solution of the differential equations are evaluated by Herodek et a/. (1982). It may be worth emphasising that recent results by Kutas & Herodek ( 1987) show the usefulness of the model in aiding decision. Comparisons between the two models will be given below.

The stochastic model

To define the stochastic counterpart of a deterministic model of a reaction, the elementary reactions and the reaction rates of the elementary reactions have to be known. As this has been given above, the stochastic model may be considered to have been defined. Several remarks still may be appropriate. Defining the state-space as usual is quite realistic for the needs of reaction kinetics, because a change of state has a physical meaning: an elementary reaction has taken place and the change itself means changing by a single molecule. In the ecological context, how.ever, the unity of change has no natural biological meaning, it is an artificial unit to measure the changes. (This is the case with the deterministic model as well!) The meaning of this measure is arbitrary, and may be varied from one simulation to another. This is an extra parameter (or ten extra parameters if different components are measured in different units). The model obtained is complicated enough not to be amenable to analytical investigations; therefore it had to simulated by the standard procedures described in Section 5.3. As the process is inhomogeneous in time

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207

and measured forcing functions are to be taken into consideration, and some of the rates are not of the mass-action type, a separate simulation program has been written for this special problem in the language SIMULA '67 which is especially suitable for simulation purposes. Qualitatively different consequences

Similarities and differences between the stochastic and deterministic models are as usual (approximate consistency in mean, decreasing relative variance with the increase of the number of the introduced 'functional particles', etc.). The most astonishing and ecologically relevant result found was that there is a marked difference between the stochastic mean and the deterministic result, and there is an even greater difference between realisations and the deterministic value of the biomass of blue-green algae. At the locations of maxima the maximum curve is ten times higher than the deterministic one. This reveals a very high instability of the system, not only in the model, but also in reality, as in two consecutive years with similar meteorological conditions and in a slightly different input situation the summer biomasses showed very high differences. This cannot easily be described by any deterministic model; however, if the measurements could be simulated with the deterministic model the same could be done with the stochastic one too. It may be useful to know that simulation of a one-year-long period of this really complex phenomenon takes 2-3 minutes on an IBM 3031 computer, only three times as much as to simulate the deterministic model. 7.5.2.3 Exercise I. Verify that the deterministic model by Herodek et a/. (1982) fulfils the

conditions by Vol'pert (1972) which assure nonnegativity of the solutions starting from nonnegative initial values. 7.6 Aggregation, polymerisation, cluster formation 1. Many studies in different disciplines deal with kinetic phenomena where 'elementary units' (e.g. monomers) of a population interact to form greater structures. Cell aggregation, polymerisation showing gelation and an abstract growth model formulated in terms of chemical kinetics, will be presented here as characteristic examples. 2. The phenomenon of sorting out of embryonic cells has been modelled (Rogers, 1977) by adopting the celebrated Smoluchowski model. The original model described the coagulation of colloidal particles based on the principles of Brownian motion. Three main assumptions are postulated: the cells are discrete, randomly mobile and differentially cohesive and adhesive. Starting from a population containing two cell types, say a and b, assuming

208

Mathematical models of chemical reactions

that forces of adhesion of bb are essentially larger than for the other two, a model can be given for the temporal evolution of the concentration of different b-cell aggregates. Let c 1 (t), c2 (t), ... denote the concentrations of single b-cells, double b-cell aggregates and so on. Assuming mass-action type interactions among the cells the system of differential equations can be derived: dck(t)/dt

= 1/2

L

'X

L akjcj(t).

auc;(t)cit) - ck(t)

i+j~k

j~

(7.28)

I

It means that an increase in ck is due to the formation of groups of k b-cells by merging a group of i b-cells with a group of j b-cells and a decrease is given due to the formation of aggregates of size k + j (where j = 1, 2, ... ). Equation (7.28) has no known closed form solution. In the particular case of aij =A =constant, introducing the transformed time t At/2, and writing nk(t) instead of ck(t) we get

=

dnk(t)/dt

=

L

00

L nj(t).

n;(t)nj(t)- 2nk(t)

i+j~k

j~

I

The series of solutions of these equations is given by nk(t)

= n0 [(n 0 t/- 1/(1 + n 0 t)k+ 1].

(7.29)

The successive maxima of nk occur at n 0 = (k- 1)/2. (Fig. 7.14) 2. Reversible polymerisation is the combined process of formation and breakage of chemical bonds in a system of reaction polymers (see, for example van Dongen & Ernst (1984)).

314

N No

1/2

1/4

0 0

2

3

4

Fig. 7.14 The variation with time, N 0 , = CN0 t/2, of the number of k-aggregates of ceiJs of the internalising type, Nk. The successive maxima of Nk occur at No,= (k- 1)/2.

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Reversible reactions where an arbitrary ak denotes a cluster containing k monomeric units of a;. Two sets of rate constants are assigned, Kij describes the bimolecular coagulation process, while Fij is assigned to the unimolecular fragmentation process. The temporal evolution of the concentration ck of k-mers is given by the infinite set of differential equations:

L

dck(t)fdt = 1/2

(Kijc;(t)cj(t)- Fijci+j(t))

i+ j~k X

- L (Kijck(t)cj(t) -

Fkjck + j(t))

(7.30)

j~l

with the initial distribution ck(O)

=

M()kl

=

()kl·

(Here M is the initial concentration of monomers; for the sake of convenience M = I.) Particular theories of polymerisation make different structural assumptions that result in different Kij and Fij. According to the detailed balance condition Fijc; + j( oo)

= kijcij( oo )cu( oo ).

(7 .31)

The normalisation condition requires that the total fragmentation rate of a k-mer be proportional to the number of bonds: 1/2

L

Fij

= g(k-

1),

i+ j~k

here g is the proportional constant. Since Fij and kij are interconnected, the definition of Kij implies the form of Fij as well as the stationary size distribution. The latter is expressed in terms of c 1 ( oo) as: (7.32)

where Nk is a combinatorial factor. The kind of mechanisms that lead to gelation characterised by infinite clusters are not clear. The infinite cluster contains of course a finite fraction G(t) of the total mass (M(t) + G(t) = 1). Pre-gel and post-gel states separated by a gelation transition can be analysed in terms of a kinetic equation. Sol-gel transitions are similar to phase transition phenomena. It is not surprising that scale in variance principles elaborated in the theory of phase transition can be adopted for polymer systems. Modern percolation theory (see, for example Stauffer (1979)) offer a conceptual framework to treat cluster formation. 3. Certain growth models can also be formulated in terms of formal reactions. Let us consider a process in which an introduced seed grows on a lattice by creating particles at nearby sites, which then in turn grow.

Mathematical models of chemical reactions

210

Furthermore, multiply occupied sites are 'highly unfavourable'. The model scheme contains these two steps

e

=

2X;-+ X;.

Denoting with P({x;(t)}, t) P(x 1 , x 2 , particles at time t the CDS model is

••

x., t) the probability of finding {x;}

N

ofotP({x;(t)}, t) = k

L xiP(x]' ... ' X; -

I' ... XN, t)

i.j= I

N

- P({x;}, t)) + e L

X;(X;

+ I)P(x 1 ,

... ,

x., t)

i= I

I)P({x;}, t). (7.33) The superiority of the stochastic over the deterministic approach was emphasised by Elderfield (1985), who calculated the parameters of fluctuation. In the model of diffusion limited aggregation, growth occurs by capture of a diffusing particle. Two sets of components, X; (as previously) and diffusers (D;) are postulated. The reaction scheme is -

X;(X;-

D;+ ~-+X;+~ 2X;-+X; D;-+Di D;+X;-+X;.

The stochastic version of the model was studied by Elderfield (1985), who exactly calculated the parameters of the fluctuation. Growth phenomena (e.g. cluster formation) are adequately described by cellular automata (e.g. Wolfram, 1986). Cellular automata are examples of systems constituting many identical components. While the components are simple, their collection is capable of complex dynamic behaviour. Cellular automata provide an alternative approach to spatia-temporal structure formation, as opposed to models based on differential equations. A one-dimensional cellular automation consists of a line of sites, with each site carrying a value (0 and I in the simplest case). The value of each position is updated in discrete time steps by an identical deterministic rule depending on the neighbourhood of sites around it. Reaction-diffusion systems, growth phenomena with inhibition effects and a wide variety of biological systems are well modelled by cellular automation models.

7. 7 Chemical circuits I. General remarks. The theory of dynamic systems has three large fields of

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applications: mechanics, chemical kinetics (and analogously population dynamics) and electronics. As we mentioned earlier (Chapter 2), chemical kinetics can not be 'mapped' into mechanics, though some analogous phenomena can be found. Electronics applies 'linear' and 'nonlinear' elements for finding networks capable of showing complex behaviour. The question what can electronics offer to chemical kinetics (from the theoretical point of view) has two sides: ( 1) How can chemical phenomena be modelled by using electric and electronic devices? (2) How can phenomena well-known in circuit theory be modelled in terms of chemical kinetics? The first question was answered by 'Network Thermodynamics' (e.g. Oster et a/., 1973;. Schnakenberg, 1977; Peusner, 1985) adopting the 'bond graph' technique. This approach benefits from the formal correspondence between certain interpretations of nonequilibrium thermodynamics and of electrical network theory. Adopting the notions of 'chemical impedance', 'chemical capacity' and 'chemical inductance', chemical reactions as well transport processes can be represented by networks obeying Kirchhoff's current and voltage laws. The second problem seems to be more interesting. Though ionic reactions and transport processes in solutions are much slower than electron transport in solids, the investigation of the realisability of'electric circuits' by chemical tools might be important from the point of view of theoretical biology. Furthermore, the possibility of constructing molecular computing devices can increase our interest in a theory of 'dynamic chemotronics'. A few concept introduced more or less in connection with circuits will be studied in terms of chemistry. 2. The term chemical amplification can be interpreted in at least two different senses. First, a trace concentration can be amplified by chemical methods to yield a higher order of magnitude of product concentrations. In contrast to the traditional analytical procedure, chemical amplification may be performed before (and not after) the measurement step. Second, the concept of amplification can be introduced as an abstract (and perhaps a bit artificial) analogy, where the x(t) concentrations are considered as the input signals, and f(x(t)) dx(t)/dt are the output signals. The relationship of;/ox; > 0, expressing the occurrence of an autocatalytic reaction can be identified with 'positive feedback' phenomena. Some methods cf chemical amplification can be used in analysis (Blaedel & Boguslaski, 1978). Both inorganic and enzyme catalysis can be applied for carrying out sensitive and selective analysis by amplification. Another possibility is the use of 'cyclic reactions':

=

212

Mathematical models of chemical reactions

An interesting approach for achieving amplification is through a reaction mechanism that casts the analyte itself into a cycling role, so that a stoichiometric amount of product is produced and accumulated each time that the analyte is cycled or turned over. If the cycling is efficient without loss of the analyte to side reactions, and if the measured product is stable enough to permit accumulation, the amplification achieved by this approach may be large. To date, mainly cofactors for enzymecatalysed reactions have been amplified in this way, but in principle cycling is also generally applicable to other substances. (Blaedel & Boguslaski, 1978)

Searching for analogies between elements of electronic circuits and those of chemical reactions, Rossler ( 1974b) associated chemical concentrations to voltages across capacitors. Based on this approach he presented a (Michaelis-Menten-like) analogy of non-inverting as well inverting amplifiers. 3. A formal chemical analogy of linear RC-oscillators leading to harmonic oscillation was introduced (Seelig, 1971; Seelig & Gobber, 1971 ). A threecomponent system dx(t)/dt

= Kx(t) + j;

x(t), j e IR 3

was introduced. As it is well-known, linear oscillators are structurally unstable and cannot be realised by physical devices. 4. The most famous electric oscillator, namely the van der Pol-Lienard oscillator given by i(t) = u[y- x 3 /3- x)] y(t) = - (1/u)x,

is not a mass-action type kinetic equation. The Brusselator model was qualified as a simple 'functional analogue' of the van der Pol equation, since it can also bifurcate to limit cycle oscillation. Furthermore, the periodically perturbed version of both oscillators might exhibit bifurcation from limit cycle to quasiperiodicity and even chaos (Cartwright & Littlewood, 1945; Tomita & Kai, 1979). 5. Multivibrators have three different classes. A bistable multivibrator has two stable states switching by perturbation of an external signal. A monostable multivibrator has one stable state; it is attained by every perturbation. An astable multivibrator has two locally stable states. They occur periodically without getting an external signal. Rossler (1972; but see 1974a, 1974b) suggested an astable multivibrator in terms of MichaelisMenten-like kinetics. Applying the 'inverse' of the usual approximative techniques of enzyme kinetics it can be decoded as (Erdi & T6th, 1976) 81-+ AI

2AI ~AI + 82 A 3 -+A 1.

AI-+ A2

A2-+ 83

A 1 -:-A 2 ~A 3

A realisation of its CDS model obtained by simulation experiments can be seen (Fig. 7.15).

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a,(t)

Fig. 7.15 Chemical multivibrator.

6. Logical circuits are one of the most important groups of digital devices. The fundamental logical elements are the AND, OR and NOT circuits. Combination of these elements can lead to complex logical networks. The realisability of logical functions by chemical bistable systems was mentioned by Rossler (l974b): The functions AND and OR are trivially realized by algebraic addition, in the presence of a threshold. Now any flip-flop is a threshold device. Hence any pair of 'convergent' reactions (leading to the same product) realizes these functions in a straightforward manner. If both influxes are required for the setting (or resetting, respectively) of the flip-flop, AND is realized, and if either is sufficient, OR.

7. The possible fabrication of a molecular electronic device (MED) resulted in serious interest in elaborating the principles of molecular computation (Conrad, 1985). Switching is possible even at molecular level by means of conformational changes. Three promising mechanisms of switching, namely electron tunnelling in short periodic arrays, soliton switching and soliton valving was reviewed (Carter, 1984). 7.8 Kinetic theories of selection 7.8.1 Prebiological evolution 7.8.1.1 Introductory remarks Several, not completely disjoint phases of biological evolution can be distinguished: (I) during chemical evolution the building blocks of biopolymers are formed spontaneously from small molecules; (2) the building blocks just mentioned were self-organised into structures having the ability to exhibit self-replication;

214

Mathematical models of chemical reactions

(3) the evolution of species can be identified with the conventional subject of evolutionary theories. The second phase, namely the self-organising mechanism of the molecular self-replication has been investigated by Eigen (1971) and his coworkers (Eigen & Schuster, 1979) unifying the detailed results of biochemical experiments with mathematical models of chemical kinetics. Theoretical studies have initially been motivated by the 'test-tube experiments' on RNA evolution (for an early review see Spiegelman (1971)). The molecular mechanism of RNA replication is still always being studied (Biebricher eta/., 1983). Eigen's approach is based on the concepts of 'quasispecies' and 'hypercycles'. Quasispecies is considered as a set of molecules acting as a group. Not single molecules, but quasispecies are the target of evolution. Hypercycles are mentioned in the next Subsection. 7.8.1.2 The hypercycle: the basic model According to the founders of the concept (Eigen & Schuster, 1979, page v): A. Hypercycles are a principle of natural self-organization allowing an integration and coherent evolution of a set of functionally coupled selfreplicative entitites. B. Hypercyles are a novel class of nonlinear reaction networks with unique properties, amenable to a unified mathematical treatment. C. Hypercycles are able to originate in the mutant distribution of a single Darwinian quasi-species through stabilization of its diverging mutant genes. Once nucleated hypercycles evolve to a higher complexity by a process analogous to gene duplication and specialization. The original mathematical model is based on three assumed properties: 'metabolism', 'self-reproduction' and 'mutability'. The simplest deterministic equation derived is: X;= (A;Q;- D;)

L W;kXk + F;.

+

(7.34)

k~i

By X; the quantity of the population variable is described. Metabolism, according to its present interpretation spontaneous formation and decomposition, is expressed by the terms (A;Q;x;) and (D;x;) respectively. Selfreproduction is reflected by the fact that the rate of formation depends on x. Mutability is introduced by the quality factor Q;. It expresses the measure of the exactness of copying. The term L W;kxk describes the contribution to the i~k

increase of X; by mutations. A conservation relation IA;(1 - Q;)x; = ;

L L w;kxk ; k '#:;

(7.35)

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holds for the 'error copies' obtained by mutation. F; describes the changes of the individuals by non-chemical interactions. Specifically, F;

=F,(x;/ L xk)

(7.36)

k

expresses that each species conributes to the total flow F, in proportion to its presence. Using the assumption of 'constant overall organisation'

L xk = constant = en

(7.37)

k

and consequently (7.38) where E; =A;- D; is the 'excess productivity'. Introducing the notation W;;

=A;Q; -

D;

(7.39)

for the (intrinsic) selective value, and E*(t)

='[_Ekxk/'[_xk k

k

for the average excess productivity the differential equation (7.34) can be transformed for the constraint of 'constant overall organisation' as follows: X;= (W;;- E*(t))x;(t)

+

L W;kxk(t).

(7.40)

k.;. i

Remarks I. Generalisations of the model are possible within the framework of the hypercycle concept:

-

kinetics of the 'metabolism' might have more complex nonlinearity; self-reproduction can be realised via cyclic catalytic process; more detailed description of the mutation can be given; different kinds of constraint can be given instead of the 'constant overall organisation': discrete state-space stochastic models can be defined to describe phenomena, e.g. in finite populations.

Stochastic effects due both to finite population size and to other sources have been reviewed recently (Schuster (1985); for the details see McCaskill (1984) and Sigmund (1984). 'Random drift' or 'random selection' and 'random replication' are the most important stochastic phenomena which play a role in the evolution of polynucleotides. The former is important in small populations and in particular in the limit of kinetic degeneracy. The

216

Mathematical models of chemical reactions

term degeneracy means that there is no difference in the rate constants of the same type of reactions of the competing 'subunits'. Replication was considered as a multitype branching process, which is a particular class of stochastic process. This approach makes it possible to analyse the mutationevoked error propagation. (For further details see the reference given above.) 2. From the methodological point of view the hypercycle model is the most impressive example of the possibility of unifying detailed biochemical examinations and mathematical models of chemical reactions. Both the prebiotic relevance of the model and its mathematical structure were studied, sometimes critically (e.g. King, 1981; Miiller-Herold, 1983; Szathmiuy, 1984). It was suggested that the hypercyclic evolution is not as effective as conventional autocatalysis, and the model should be defined by multiple time singular perturbation theory. Rossler (1981, l983b) presented the outline of the possibility of a 'deductive prebiology' based on the particular properties of chemical dynamic systems. As he emphasises, the creation of a chemical system with huge (say 10 10 000 ) state variables is very easy, in contrast to electronic systems. Just purchase some ten substances at a pharmacist's: a liter of distilled water, a gram of ammonium chloride, and some grams of sodium carbonate plus some other salts. This mixture, when shaken continually at 40"C, is bound to implement an equilibrium between some I 0 10 000 substances. With infinite observation time granted, the expectation value of all these state variables become non-zero. Mathematically this whole machine exists the very moment the ten substances have been poured together. (Rossler, 1983b)

Rossler argued that the very long-term qualitative dynamic behaviour of an evolving chemical system cannot be analysed. The attractor can be different from the present known attractors: dynamic systems describing evolution internally select a sequence of subsystems in which each generates a viable successor even if the environment changes during the process. 7.8.2 The origin of asymmetry of biomolecules

I. Substances that can exist in pairs of optical antipodes are found in nature almost exclusively only in one form: amino acids in proteins belong to the Ltype, most of the sugars are o-type. The question of the formation of this asymmetry can be treated by two approaches: (i) biological asymmetry is the clear implication of certain well-defined physical asymmetries; (ii) asymmetry spontaneously emerges by some self-organising mechanism. 2. There are some physical factors which can be considered as possible candidates to cause biological asymmetry. The most often mentioned physical phenomenon is the weak parity violating interaction. It seems to be very appealing to explain biological asymmetry by the the general asymmetry of the physical world.

Applications

217

In a series of papers Kondepundi & Nelson (1983, 1984, 1985) have investigated the consequences of the energy difference A£ between Land D enantiomers causing differences also in the rate constants expressed by the factor g AE/k 8 T = (kLfk 0 ) - 1. (The subscripts L and D of the rate constant k refer to the enantiomers; k 8 is the Boltzmann constant.) The crucial point of this approach is the relative effect of the factor g and the fluctuations. The order of magnitude of g is about 10- 17 • It is an open question whether small or perhaps very large volumes are necessary. Very different estimations are known for the time being, but g seems to be too small to imply biological asymmetry. 3. Theories suggesting the spontaneous formation of biological asymmetry are based on two-component chemical systems, which are symmetric for the two components (mirror-image reaction systems). Fluctuations in the initial conditions are amplified by the kinetic mechanism leading to macroscopic asymmetry. (For a review of critical evaluation of models for the amplification of chirality see Fajszi & Czege (1981).) The measure of the asymmetry can be expressed by

=

11

=(L -

D)/(L

+ D).

One of the simplest models capable of chemical amplification is dL(t)fdt

=

kLL(t)- r(L(t)- D(t)) + r(L(t) - D(t)).

dD(t)/dt = k 0 D(t)

(7.41)

In this model there is no interaction among the enantiomers, r is the velocity of racemisation. In case of kL = k 0 = k the 11 versus g function is

11= (rfk)

+

g j(rfk) 2

+ g2

(7.42) •

The simplest model taking into consideration heterochiral interactions was given by Franck ( 1953): dL(t)/dt dD(t)Jdt

= (k = (k -

k 2 D(t))L(t) k 2 L(t))D(t).

(7.43)

The symmetric stationary state L* = D* = kjk 2 is unstable; small differences in the initial conditions determine which component survives and which dies out. (For a similar but more complex reaction scheme see Seelig (1971).) Since biological asymmetry due to physical asymmetry has not been verified, more realistic selection kinetic mechanisms are searched for to get a more detailed picture of the origin of optical chirality. It may be useful to make comparisons (see Table 7.2) between the models of this chapter in order to see what we have done and obtained when applying the mathematical models of chemical reactions in some areas of science.

Table 7.2

Comparison of models.

Field of application

Model type

Methods

Results

Biochemical control theory

MichaelisMenten-like equations

Qualitative, numerical

Conditions for oscillation, periodic perturbation of oscillations may lead to chaos

Neurochemical fluctuations

CDS

Periodicity and aperiodicity in neurochemical systems

ceo

Numerical integration, phase space analysis

'Abnormal' temporal patterns occur due to the impairment of the control system

Population genetics

Deterministic selection equations

Stroboscopic method

Examination of 'generalised' gradient systems

Ecokinetics

ceo

Compartmental analysis

Model discrimination

Generalised Lotka-Volterra

ceo

Bifurcation analysis

Equilibrium oscillation

Algorithm for calculating the transport coefficients

--+

limit cycle

--+

nonperiodic

Stochastic ecology

CDS, stochastic differential equation

Eutrophication in Lake Balaton

CDS

Simulation

Explanation for the irregular behaviour of blue algae

Aggregation of cells

Infinite dimensional deterministic

Analytic

Determination of duration necessary to reach the maximal degree of aggregation

Harmonic oscillation

ceo

Analytic

Linear oscillators are structurally unstable

Multi vibrators

CDS

Simulation

Prebiological evolution

Deterministic, stochastic

Qualitative

The origin of asymmetry of biomolecules

Combination of deterministic and stochastic models

Shift in the extinction-survival property due to presence of small mutant populations

Selection of specific biomolecules

Kinetic mechanisms might lead to selection due to small fluctuations

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Index

'abnormal' dynamic behaviour 191 absorption of drugs 12 acetylcholine 186, 187, 188, 189 activator xxii, 175-6 activity 4, 8, 9 activity coefficient 4 acyclicity 31, 45, 46, 67, 78 adenosine 5'-diphosphate xv, 31, 185 adenosine 5'-triphosphate xv, 31, 185 aggregation 71, 207, 218-19 algae 204 algebraic description of a reaction 29 algebraic structure 21 algebraic structure of reaction networks I amplification of fluctuations 129 analogue computer 15 anomalous fluctuation 138 asymmetry of biomolecules 216-17 atom xv, 4, 22, 24, 29, 30 atom-free stoichiometry 26, 28 atomic balance 22, 24, 26 atomic component 24 atomic matrix xvi, 22, 24, 29 atomic reaction 24, 29 atomic stoichiometry 24 atomic structure 21, 22, 24, 26 autonomous ordinary differential equation 18, 19 autonomous polynomial differential equation 35 Balaton 202 Belousov-Zhabotinskii reaction 39, 58, 59, 60, 61, 81, 163, 173 bifurcate II bifurcation parameter xxii, 56, 123 bifurcation phenomenon 13, 166, 178 bifurcation point xxii, 54

Biltzmann equation 3, 5 biochemical control theory 177, 218-19 biochemistry 5 biological memory 8-9 biomathematical model 12 biomathematics 107, 196 biomodality 138 biomolecular reaction 2, 3, 55, 107 biomolecule 121-3, 216 birth and death process 112, 140, 141, 142 bistable reaction 135 blowing up 159 Bodenstein principle 88 Boulding ecodynamics 194 boundedness 32 Brownian motion xx, 98, 100, 101, 116, 130, 207 Brusselator 166 canonic reaction 65, 67, 74 ceo model 19, 159 CDS model 19, 159 cell 5, 21, 174 cellular automata 210 cellular model 169 central limit theorem 171 centre-manifold approach 154, 155 change of scale 87 chaos II, 19, 20, 38, 49, 59, 60, 61 chaotic behaviour 12, 60, 183 chaotic chemical reaction 62 Chapman-Enskog approximation 3 Chapman-Kolmegorov equation 71, 96,99 charge 27 charge transfer 12 chemical amplification 211 chemical circuit 210-13

Index chemical thermodynamics 18 chemistry 3, 7, 93 chronon 15 closed compartmental system 30 closed generalized compartmental system 30 cluster formation 6, 207 cluster growth 6 coexistence 44, 196-8 cold flame 54 cold plasma 5 collision 2, 3, 4, 5, 23 coloured noise 151, 152 combinatorial model I 04 compartmental analysis 12 compartmental ecokinetics 195 compartmental system 12, 30, 69, 73, 107 competition 12, 177 competitive phenomenon 13, 196 complete 29 completeness 28 complex xvii, 12, 30 complex vector xvii, 23 component xx, I, 22, 24, 26, 27, 29, 30, 34, 163 composition I, 7 concentration xiv, xxiv, 5, 7, 8-9, 10, 26, 33 concentration vector xiv, xvii, 10 condensed matter 6 condensed phase reaction 4 conformation 5, 79 conformation change 8-9, 127, 188 comformational change 5 connectedness of the state-space 29 conservative 27, 28, 29, 30, 32 conservative reaction 30, 71 conservativity 27, 28, 30, 32, 67, 78 consistency in the thermodynamic limit 160 constituent 33 constitutive functional xv, 17, 18, constitutive quantity xv, 7, 16, 17, 18 construction of material 28 continuous component xvii, 17, 78 continuous flow stirred tank reactor 50 continuous mechanics 16 continuous semigroup xx continuous state model 7, 33 continuous state-space 16, 97, 163 continuous time 10, 15,.33, 91, 163

253 cooperation 12, 177 cooperative phenomenon 13 core 73 correlation function xviii, 116 cosmology 16 covalent bond xiv, 5 creation of mass 27 critical point 6, 95, 135, 172 critical species 50 cross-catalytic mechanism 81 cross-effect 34 curve 19 cyclic phenomenon 16 DCD model 19 decomposition (of density functions and discrete distributing) 140 deficiency xvii, 29, 43, 78 deformation tensor 8-9 deoxyribonucleic acid xviii, 127, 128 dependence 33, 35 dependent 23, 24 designing oscillatory reactions 56 destruction 27, 28 detailed balance 3, 45, 46 determination 14 deterministic 8-9, 18, 19, 21 deterministic dynamic system 48 deterministic model 6, 7, II, 33, 34, 46, 149, 162, 163, 206 deterministic process 18 dichotomous noise xix, !52 difference equation 19 differential-difference equation 10, 103 differential equation 7, 12, 19, 35, 54, 55 differential system 19 diffusion 4, 6, 8-9, 14, 16, 97, 162-75 diffusion matrix xxii, 97 diffusion process 97, 100, Ill digital computer 15 dilute gas 4, 8-9 direct problem 38, 63 directed edge 30 discrete 15, 16, 19, 21 discrete state I 0, 91 discrete state-space 7, 10, 16, 18, 99 discrete time 15, 18 discrete-time model 15 dissipative process 7, 116 dissipative structure 13 drift 201

254 drift vector 97 dynamic system xv, 14, 18, 85 ecodynamics 194-6 ecokinetics 218-19, 195, 197 ecological case study 202 ecological model 12 ecological system 12 econometrics 77 ecosystem 195-7, 199-201 electric field 5 electrolytes 4, 8-9 electron 24, 27 elementary reaction xiii, 2, 22, 23, 24, 25, 26, 27, 29, 30, 33, 34, 35 elementary reaction vector xvi, xvii, 22, 23,24 elliptic differential operator xxiii, 164 empty reaction 65 endpoint xvii, 172 energetic interaction 16 energy level 12 entropy, as 'chemical component' 126 entry point xvii, 72 enzyme xv, xxiii, 4, 5, 26, 27, 182, 187 enzyme kinetics 5, 119, 147, 177-9 enzyme reaction 4, 182 enzyme-substrate interaction xv, xxiv, 26, 179 enzyme system 8-9 epidemiology 13 equilibrium 7, 8-9, II, 16 equilibrium distribution 4, 118 equilibrium fluctuation 93, 117, 126 equilibrium measurement 10 equilibrium point xvi, 137 ergodic properties 12 estimation of reaction rate constant 71, 157 eutrophication 218, 219 eutrophication process 202 evolution 199, 201, 213 evolution law 19 exotic behaviour II exotic chemical system II exotic kinetics II exotic reaction 49 experimental observation 25 explodator 56, 58 external fluctuation 95, 146, 156, 201 external noise 10, 96, 147, 153 fading memory 6

Index fast reactions 125-8 feature sensitivity analysis 84 Feinberg-Horn-Jackson graph 70 Field-Kt!ros-Noyes-model xvi, 39 field theory 16 first integral xvi, 86 first-order endpoint xvii, 72 first-order Markov process 18 first-order phase transition 134 first-order reaction 23, 30 fluctuation 6, 7, 8-9, II, 91, 185 fluctuation around the equilibrium point 7, 84 fluctuation-dissipation theorem 7, 92, 94, 95, 115, 116, 125, 148, 157 fluctuation phenomenon 10, 93, 128 fluctuation theory II Fokker-Pianck equation 98, 110, Ill, 136, 149, 150, 154, 155 formal 14, 30 formal linear combination 24, 30 formal reaction kinetics 202 forward rate 3 Fourier transform xxi, 117 'free-energy-like' function Ill functional 17 functional differential equation 8-9 gain-loss equation II gas-phase reaction 3, 6 Gaussian process 139 general solution 25, 26 generalised compartmental system 30, 69-70, 72 generalised Lotka-Volterra model xvii, 196, 198, 218-19, 201 generation 15 genetic model 12 Gierer-Meinhardt model xxii, 175 glycolitic oscillation 182 glyoxylate cycle 31 gradient system xvii, 80, 82, 193 group of general linear transformations xvi, 87 group of transformations xxii, xxiii, 165 growth phenomena 209-10 harmonic oscillation 218-19 heterogeneous reaction 4 hierarchial level 12 historyxv, 16, 17,18,21 homogeneity 14

Index homogeneous reaction kinetics 49 homogeneous spatial distribution 7 hot plasma 5 hyperchaos II hypercycle 214, 215 identifiability 62 incidence relation 31 independence 22, 24 independence of components 28 independence of elementary reactions 28 independent component 22, 25 independent elementary reactions 22, 24, 28 induced kinetic differential equation xvi, 35, 39, 55, 86, 88 industrial chemical system II infinite system 7 infinitesimal transition probability xx, II, 99, 103 infinitesimal transition probability matrix xiii, 102 inhibitor xxiii, 175-6 inner fluctuation 204 instability 13, 91, 129 instability point 7, 8-9, 128 interacting populations 194 internal component 29 internal fluctuation 10, 95, 156 internal state 5 internal structure 3, 6 inverse problem 38, 63, 72 invertibility, 18 ion 23, 24 ion reaction 4 irreversible 23 irreversible phenomenon 16 irreversible process 16 isola 53 isomerisation 79 isotope exchange 12 iterated map 19

255 kinetic initial value problem 64 kinetic logic 47, 67 kinetic model 14, 19, 50, 61 kinetic theories of selection 213 kinetic theory of phase transition 6 kinetics 35 Kolmogorov equation 5, 97, 98, 99 Krameri-Moyal-Stratonovich equation 98, Ill, 115 Kurtz-type infinitesimal transition probabilities xx

jump process 10, 110, 112

Lake Balaton 202, 218-19 lake models 203 Iangevin equation 148 Laplace transform xx, 105, 108 Laplacian operator xxiii law of atomic balance 22, 24, 26, 29, 30 law of definite proportions 28 law of large numbers 171 law of mass balance 27 law of multiple proportions 28 length xv, 23, 30 ligand migration 121, 122 limit cycle 12, 131, 154, 189, 198, 212 Lindeberg condition 97 linear 33 linear combination 23, 24, 25, 26, 29, 30 linear nonequilibrium thermodynamics 116, 127 linearly dependent 23 linearly independent 24, 25 linkage class xvii, 43 local sensitivity coefficient 83 logical circuit 213 Lohmann mechanism 31 Lohmann reaction 30 long-term behaviour 12 Lorenz equation xvi, xvii, xviii, 59, 62, . 64-5 Lotka-Volterra mechanism 30 Lotka-Volterra model 27, 54, 55, 129, 194 Lotka-Volterra reaction 30, 44, 62, 113 lumping xiv, 6, 34, 78-9

kinetic condition 12 kinetic differential equation xiv, xvi, 7, 19, 35, 64, 65, 67 kinetic equation 7, 8-9, 33 kinetic experiment 50, 54, 60 kinetic gradient system 80

macrophysics 16 macroscopic 3, 5, 137, 194 macroscopic approach I macroscopic average value 7 macroscopic continuum 8-9 macroscopic effect 7

256 macroscopic structure 13, 95 macroscopic system 6, 10, 92 macroscopic theory 16 macroscopic variable II Markov chain 18 Markov process xx, 10, 18, 19, 96 Markovian jump process 10 Markovian property 18 mass 27 mass action I, 2, 3, 177, 178, 179 mass action kinetic II, 32, 34, 35 mass balance 27 mass-conserving 27 master equation 10, II, 104, 106, 107, 108, Ill, 120, 130, 136-7, 168 mathematical analysis 33 mathematical model 14 mathematical system theory 16 maximum of the stationary distribution xxii, xxiii, 143 mechanics 16, 84-6 mechanism 32, 41 membrane channel 123, 124, 185 membrane noise 95, 123, 124, 125, 187 memory 6, 8-9, 17 memory-free 18 mesoscopic 8-9, 94, 163 metalanguage 12, 177, 202 Michaelis-Menten reaction xv, 26, 39, 119, 178, 180 microbiology 13 microphysics 16 microreversibility 3 microscopic I, 3, 10, 167 mobility 4, 8-9 molecular computation 213 molecular weight distribution 17 molecularity xiii, 23 monomolecular reaction 2 Monte Carlo methods 112 morphogenesis 174, 176 motion xv, 14, 116 multimodality II, 143, 144 multistability 49, 53 multistage reactor 12 multistationarity II, 49, 50, 133, 136, 143, 144 multivibrator 212,218-19 mushroom 53 mutarotation of simple sugars 2 natural selection 20 I nature of determination 14, 18, 19

Index nature of time 16 negative cross-effect 7, 64, 66 network I, 12 network thermodynamics 211 neurochemical fluctuation 218-19 neurochemical oscillation 188-90 neurochemistry 185 noise-induced transition 96, 147, 149, 151 nonequilibrium statistical mechanics 3, 5, 8-9, 93 nonequilibrium thermodynamics 16, 115, 116, 167 nonlinear 2, 57 nonlinear equation II nonlinear phenomenon II nonlinear science I nonlinear system 13, 147, 148 nucleation process 169 occupation number 6 oil 16 order of an elementary reaction 23 order of a reaction (with respect to a component) 2 ordinary differential equation 2, 10, 18 Oregonator model 26, 27 origin of asymmetry of biomolecules 218-19 Ornstein-Uhlenbeck process 152, 157 oscillation 185, 193 oscillatory reaction II, 38, 54, 56 oscillatory solution 165 overall reaction 22 overshoot-undershoot 57 oxyacid 6 parameter estimation 74 parameter sensitivity 83 particle xix, 16, 24 particle number xxiii, 7 pattern formation 163, 174 Pauli equation 5 periodic perturbation 183 periodic solution II, 19, 38, 165 periodicity 54, 56 phase transition 6, 8-9, 13, 131, 209 phenomenological description 6 phenomenological kinetics 5 phenomenological theory 3 phosphorus 204 physical chemistry 22, 34 physicochemical background 2

Index physics 93 phytoplankton 204 plasma 4, 5, 8-9, 147 plasmachemistry 5 Poincare map, plot 19, 20, 192 Poisson distribution 138, 139, 142 Poisson process xxi, I 00, I 0 I polymer 4, 6, 8-9, 17 polymerisation 207, 208 polynomial differential equation 7, 64 population dynamics 16, 134 population genetics 192, 194, 218-19 population growth 20 I positive equilibrium point xvi, 54 potential xvii, xx, 80 prebiological evolution 213, 218-19 prebiotic chemical model 12 predator 197, 200 prediction 203 pressure xx, 2, 21 prey 197, 200 primary structure 4, 5 principle of quasistationarity 88 probability xx, 5, II, 18 process 16, 18 process-time 14 process without after-effect 10, 96 product xvi, xix, xxiv, 26 product complex vector xvi, xvii, 23 protein 4, 5 pseudo-wave 173 qualitative analysis 12 qualitative property 7, 12, 19, 164 qualitative theory 12, 37, 57 quantitative sociology 13 quantum mechanics 3, 15 quantum statistical mechanics 3 quantum theory 15 quasicomponent 6, 76 quasithermodynamic behaviour 41, 45 radical 4, 23, 27 random environment 201 random variable 14 rate 35 rate constant xvii, xix, xxi, 2, 7, 108, 117, 119, 128, 141 rate equation 2 rate law 3 rate of reaction 2 reactant complex vector xv, xvi, xvii, 23

257 reaction 21, 22, 26, 27, 31, 33 reaction-diffusion system 163, 166 reaction kinetics I, 7, 16, 21 reaction rate 2, 4, 7, 8-9, 16, 17, 157 reactor 12, 21 recurrence 12 reference category 5 regular behaviour II relaxation time xxii, 134, 136 release 186 reservoir 26 reverse rate 3 reversibility 3 reversible 23 Riemannian manifold xvi, 85 Rossler equation xvi, 63 salt 6 scaling theory 137 secondary structure 5 second-order endpoint xvii, 72 second order mechanism 23 second-order phase transition 133 sediment 204 selection 12, 213 self-replication 213 semigroup operator 99 sensitivity 83-4 separation 35 short complex 23, 32 simulation method 112-15, 168 site xv, 16, 17, 18 size 7, 8-9 small fluctuation 7 small system 91, 95, 110 sociobiological model 12 socioeconomic situation 13 solid 4, 6 solid phase 6 solid phase reaction xiii, 6 solid state 8-9 solitary wave 166 solution 35 source of information 7 spatial course 16 spatial homogeneity 14 spatial structure 163, 172-4 spatial transport 12 spatio-temporal phenomenon 16, 101, 163 species xiii, I, 5, 21 spectral density function xx, 117 spectrum xx, 117, 119, 124

Index

258 spectrum line xix, 126 stability 96 stable stationary point II, 12 stablesteady state xxii, 47 state xix, xx, xxi, 6, 14, 16, 17, 18,77 state-space xv, 14, 16, 19, 29, 96-7, 159 state variable 6, 16, 17, 18 stationarity 12 stationary distribution xxii, xxiii, II, 109, 133, 142, 169 statistical mechanical treatment 5 statistical mechanics 6 steady state approximation 178 stochastic description I 0 stochastic differential equation 147, 157 stochastic dynamic system xxii, 147, 148 stochastic ecology 218-19 stochastic model xiii, 6, 7, 10, 19, 29, 91, 99, 105, 162, 167, 202, 206 stochastic process xx, xxi, xxii, 5, 6, 18, 91, 96 stochastic reaction kinetics, logical status 91 stochastic thermodynamics 93 stochastic thermostatics 93 stoichiometric coefficient xiii, xvi, xvii, xxi, 23, 33, 34 stoichiometric matrix xvi, xvii, 22, 23, 29 stoichiometric space xvi, xvii, 23, 32 stoichiometry I, II, 21, 26, 28, 29, 35 stress tensor 8-9 structural characterisation 62 structural identifiability 82 structurally stable 7 structure 14, 26, 32 structure of the state space 14, 16 subconservative 28, 29 subconservativity 28, 31, 32, 67 substrate xvi, xx, xxiv, 4, 27 superconservative 28, 29 superconservativity 32 supply 26 surface reaction 4 sustained oscillation II symmetry 6, 84-8, 164 synapse 185 synergetics 13, 95 system 17, 18 system theory 16, 73 temporal change 2

temporal evolution 5, 10 temporal process 14, 16 ternary interaction 12 test-tube 21 theoretical reaction kinetics 12 theory of dissipative structures 13 theory of irreversible processes 16 thermodynamic 21 thermodynamic flow 16 thermodynamic fluctuation theory II, 139 thermodynamics II, 15, 16, 91, 93, 95 thermomechanics 6, 8-9, 17 time xiv, 15, 19 time delay 8-9 time evolution xiv, xv, 21, 26, 33 time homogeneous Markov process 19 topochemistry 8-9 traditional belief 40 trajectory 19, 40, 55 transformation group xvi, xxii, xxiii, 84, 85 transient bimodality 144, 145 transition probability xix, 19 transition state theory 4, 5 transmitter 125, 186, 188 transmitter-receptor interaction 187 transport process 4, 8-9, 14, 16, 21 transport theory 3 travelling wave solution 165 trigger wave 173 trimolecular reaction 2 Turing model 175 two-cell model xxii, xxiii, 168 unimodal distribution II unimodality 138, 140 unique stationary point II uniqueness 67 unstable stationary point II, 12, 167 vector of concentrations 34 velocity I, 2, 7 vesicular hypothesis 186 Vol'pert graph 30, 31, 205 Vol'pert's theorem 45 weak reversibility 29, 43, 44 well-mixed 21 white noise xxi, 148, 150 Wiener process xx, 148, 157 Wiener-Khintchine relation 116, 117 without after-effect I 0

Index XYZ models 19 zero complex xvi, xviii, xxi, 30 zero deficiency 69

259 zero deficiency theorem 42, 62 zero divergence 86 zeroth approximation 7, 8-9