Thermodynamics of evolution

Ilya Prigogine is professor of physical chemistry and theoretical physics at the University Libre de Bruxelles and is director of the Center for Statistical Mechanics and Thermodynamics at the University of Texas, Austin. Gregoire Nicolis and Agnes Babloyantz are both at the University Libre de Bruxelles: Nicolis is charge de cours and Babloyantz is chef de travaux.

The functional order maintained within living systems seems to defy the Second Law; nonequilibrium thermodynamics describes how such systems come to terms with entropy.

The physicochemical basis of biological order is a puzzling problem that has occupied whole generations of biologists and physicists and has given rise, in the past. to passionate discussions. Bio logical systems are highly complex and ordered objects. It is generally accepted that the present order reflects structures acquired during a long evolution. More over, the maintenance of order in actual living systems requires a great number of metabolic and synthetic reactions as well as the existence of complex mechanisms controlling the rate and the timing of the various processes. All these features bring the scientist a wealth of new problems. In the first place one has systems that have evolved spontaneously to extremely organized and complex forms. On the other hand metabolism, synthesis and regulation imply a highly heterogeneous distribution of matter inside the cell through chemical reactions and active transport. Coherent behavior is really the characteristic feature of biological systems (see the box on page 24).

In contrast to this is the familiar idea that the evolution of a physicochemical system leads to an equilibrium state of maximum disorder. In an isolated sys tem, which cannot exchange energy and matter with the surroundings, this tendency is expressed in terms of a function of the macroscopic state of the system: the entropy. It amounts to saying that entropy S increases monotonically until it becomes a maximum. This celebrated second law of thermodynamics implies that in an isolated system the formation of ordered structures is ruled out.

Consider now a system in contact with an energy reservoir at temperature T. Its state is now described by a new function, the tree energy Fdefined by

Unfortunately this principle cannot explain the formation of biological structures. The probability that at ordinary temperatures a macroscopic number of molecules is assembled to give rise to the highly ordered structures and to the coordinated functions characterizing living organisms is vanishingly small. The idea of spontaneous genesis of life in its present form is therefore highly improbable, even on the scale of the billions of years during which prebiotic evolution occurred.

The conclusion to be drawn from this analysis is that the apparent contradiction between biological order and the laws of physics--in particular the second law of thermodynamics--cannot be resolved as long as we try to under. stand living systems by the methods of the familiar equilibrium statistical mechanics and equally familar thermodynamics.

Coherent behavior, dissipation and life

Nonequllibrium open systems

From the simplest bacterial cell to man, living organisms are maintained and reproduced thanks to a continuous exchange of energy and matter with the surroundings. (Viruses present an exception to this rule. On the other hand it is well known that they cannot multi ply unless they infect a cell; that is. until they become open systems.) This trivial statement has several less obvious and even far-reaching consequences. The thermodynamic theory of open systems, systems exchanging both energy and matter with the environment, has long been developed by Théople DeDonder and the Brussels school (for a historical account, see reference 1). Ludwig von Bertalanffy (1940)² and Erwin Schrödinger³ have insisted on the importance of this feature for biological systems. One of us has formulated¹ an extended version of the second law that applies both to isolated and to open systems. The main point is that the Clausius-Carnot inequality governing the variation of entropy during a time interval dt takes the form

Thus open systems differ from isolated systems in the presence of flow terms in the entropy variation, terms related to exchanges of energy and of matter with the outside world. Although d_iS is never negative, the flux term deS has no definite sign. As a result, during evolution a system may reach a state where entropy is smaller than at the start (see the box on page 25). Moreover, this state, which from the standpoint of equation 2 would be highly improbable, can be maintained indefinitely provided the system can reach a steady state such that dS= 0 or

Thus, in principle at least, if we supply a system with a sufficient amount of negative entropy flow, we can maintain the system in an ordered state. Equation 5 also implies that this supply must occur under nonequilibrium conditions, otherwise d_iS(and, by equation 5, also d_eS) would vanish identically. A simple illustration¹ of this nonequilibrium order is seen in a thermal diffusion experiment in a system subject to a temperature gradient. Assume that initially the system has uniform composition. The separation of matter that is achieved then corresponds to a decrease of entropy from the initial value.

The existence of nonequilibrium constraints is far from exceptional in biological systems. The biosphere as a whole is subject to a solar energy gradient. At the cellular level, products of metabolism are either rejected to the environment or diffuse within the cell to fulfill other functions. As a rule, then, a cell or a biochemical reaction chain is subject to the gradients in chemical potentials of the various reacting constituents.

The arguments we have advanced here cannot, of course, suffice for solving the problem of biological order. One would like not only to show that the second law is satisfied as a whole (d_iS >= 0) but also to indicate how the state of low entropy and high coherence is maintained. The remaining part of our discussion will be devoted to this question, and we shall discuss in some detail the approach we have developed during the last few years.

The evolutionary feedback

It is important to insist on the complementarity between this increasing dissipation behavior, which we believe is essential for prebiological evolution, and the commonly observed tendency of physicochemical systems near equilibrium toward a state of minimum dissipation or entropy production, which we shall discuss later. Existing thermal data appear to indicate that this second type of behavior holds for living systems after some initial period, which in higher organisms extends only over a fraction of the embryonic life. One is tempted to argue that only after synthesis of the key substances necessary for its survival (which implies an increase in dissipation) does an organism tend to adjust its entropy production to a low value compatible with the external constraints. At this point, contact can probably be made with Darwin's idea of the "survival of the fittest," because a low rate of entropy production is likely to give to an organism a selective advantage. 

Dissipative structures

The best known examples of this duality in behavior are instabilities in fluid dynamics, such as the onset of thermal convection in a fluid layer heated from below. For a critical value of the external constraint (temperature gradient), that is, beyond a critical distance from equilibrium, an instability arises that causes the spontaneous emergence of convection patterns. Be low the instability threshold, the energy of the system is distributed in the random thermal motion of the molecules. But beyond instability it appears partly as the energy of macroscopic convection patterns.

In a quite different domain, a spectacular example of emergence of order far from equilibrium has been worked out recently by Hermann Haken.⁵ He shows that the generation of coherent light by a laser may be interpreted as a nonequilibrium phase transition. Be low instability is the incoherent regime; beyond the transition threshold, corresponding to a critical value of the radiation field, the system switches spontaneously to the coherent state.

We have investigated systematically the behavior of nonlinear chemical net works of biological interest.^{4, 6}  At first one would expect that those systems that, contrary to the two previous examples, are purely dissipative would always show a tendency to a disordered regime. The surprising result was that in fact they share most of the properties of hydrodynamic instabilities with the additional important feature that the variety of the regimes beyond instability is much greater in chemical kinetics.

This is not really surprising when we realize that in chemical kinetics non linearity may arise in a practically un limited number of ways through auto catalysis, cross-catalysis, activation inhibition, and so on. In contrast. the Stokes-Navier equations of fluid dynamics assume a universal form. Beyond instability, which again arises when a critical distance from equilibrium is reached, the reaction systems may become spontaneously inhomogeneous and present an ordered distribution of the chemical constituents in space. Under different conditions the concentrations of the chemicals may show sustained oscillations. Finally, other systems may exhibit a multiplicity of steady states combined with hysteresis. (We will discuss examples later.)

In all these phenomena. a new ordering mechanism, not reducible to the equilibrium principle (equation 2), appears. For reasons to be explained later, we shall refer to this principle as order through fluctuations. The structures are created by the continuous flow of energy and matter from the outside world; their maintenance requires a critical distance from equilibrium, that is, a minimum level of dissipation. For all these reasons we have called them dissipative structures.⁴

Stability properties of the aulocatalytic system described on page 25 (system 7). The non- equilibrium steady-state solutions are shown as a function of the amount of substance B present and of the diffusion constant for substance Y. In domain I the steady-state is stable with respect to fluctuations in mixture composition. Fluctuations increase monotonically in domain II (see figure 2) and, in domain III, they are present as amplified oscillations. The steady-state, then, is unstable in both these regions. Figure 1

Model dissipative systems

Here the source terms V_idescribe the effect of chemical reactions. We assume that the rates of the various chemical transformations are given by the law of mass action, so that V_i({x_j}) will be a nonlinear (usually algebraic) function of the x_j's.

Equations 6 form a nonlinear system of partial differential equations. No general theory is available for these systems, so we shall illustrate the points we wish to make with particular models.

The development of irreversible thermodynamics of open systems by the Brussels school had, by the 1950's, led to the. investigation of nonlinear processes (see reference 4 for a survey of the subject). Thus, we had already come to recognize the important role of autocatalytic processes in the understanding of biological order.^4,7It was only then that we noticed a remarkable paper by A. M. Turing (1952)⁸ who had actually constructed a chemical model showing instabilities. His work had previously escaped our attention because it dealt with the more specific subject of formation of morphogenetic patterns. The work we have undertaken since then has demonstrated the relation of this type of behavior to thermodynamics as well as its wide applicability to biology.

We shall briefly summarize here the results obtained from the model chemical reaction chain (see references 4 and 6 for details)

  Note the autocatalytic third reaction; this reaction step introduces the non linearity that will be largely responsible for the unusual behavior of the system:

1. At or near thermodynamic equilibrium, the system of equation 7 has a unique steady-state solution that is al ways stable with respect to arbitrary perturbations.
2. Suppose that the system is driven far from equilibrium. In the extreme case that the concentrations of D and E are maintained vanishingly small in the system, the distance from equilibrium becomes very large. Still, for time-independent boundary conditions, we can define a steady state corresponding to the continuation of the equilibrium-like behavior. The branch of states defined in this fashion will be referred to as the "thermodynamic branch." This time however, the study of the conservation equations (equation 6) reveals that, owing to the nonlinearity and to the nonequilibrium constraints, states on this branch are not necessarily stable. Depending on the type of perturbations to which the system is subject and on the values of the parameters--such as A (concentration of A) and B, the diffusion coefficients DA, DB, DX and DY -- different situations will be reallized. This is illustrated by the non-equilibrium phase diagram of figure 1. For simplicity, the parameters A, DA, Dx are held constant and DB is taken so large that the distribution of B remains uniform.
3. In domain II of the diagram, the thermodynamic branch becomes unstable with respect to perturbations in the chemical composition. Beyond the transition threshold initial deviations from the steady state are amplified and finally drive the system to a new steady state corresponding to a regular spatial distribution of X and Y, This state is represented in figure 2.


The result is a low-entropy spatial dissipative structure arising beyond a nonequilibrium transition that spontaneously breaks the initial symmetry of the system. Its behavior is quite different from that of systems on the thermodynamic branch. In addition to the coherent character exhibited by such states, one can show that the final con figuration depends to some extent on the type of initial perturbation. This primitive memory effect makes these structures capable of storing information accumulated in a remote past.

4. For DB » DA » DY ~= Dx, the thermodynamic branch may again become unstable. But this time the system is driven beyond instability to a new state that not only depends on space but also is periodic in time. In the course of one period, wavefronts of composition appear and propagate in the reaction volume, first outwards and then, after reflecting on the boundaries, to the interior. Moreover, at each point in space the system performs quasidiscontinuous concentration oscillations in time. This space--time dissipative structure there fore provides a primitive mechanism for propagating information over macroscopic distance, in the form of chemical signals.
5. For DA finite, both structures discussed in c and d are localized inside the reaction volume, Their boundaries, as well as the periods and other characteristics, are determined from the values of the parameters A, B, DX and so on, independently of the initial conditions. For DA, DB --> ¥ the structures tend to occupy the whole reaction volume. Finally, when DX, DY are very large (with respect to the chemical rates) the space dependencies disappear in case d, and the result is a system that, beyond instability, oscillates with the same phase everywhere. The amplitude and period of the oscillations are determined by the system itself. More over, the periodic motion is stable in the sense that all perturbations introducing an initial deviation from this state are damped. This type of solution is well known in mathematics and in analytical mechanics as a "limit cycle." Note that the existence of localized states and of wavelike solutions shown in c and d raises a number of fascinating mathematical problems related to the existence and stability of periodic solutions for nonlinear parabolic partial differential systems.


Obviously, the occurrence of instabilities far from equilibrium is not a universal phenomenon in chemical kinetics. Coherent behavior requires some very particular conditions on the reaction mechanism, whereas the equilibrium order principle is always valid (for short- range forces). But this variety of behavior in chemistry is welcome, if we want to account for the variety of situations observed in systems driven far from equilibrium.

We have insisted on the diversity of situations that may arise in nonlinear systems far from equilibrium. Now we shall see that despite this diversity, there is a general thermodynamic theory underlying the mechanism by which a system is driven to the new regime beyond the instability of the thermodynamic branch,

Within the framework of our macroscopic description, we would like to relate instabilities to the thermodynamic properties of the systems, such as entropy, entropy production, and so on. Consider again the decomposition of dS given by equation 3. Explicit calculation of d_iS/dt is possible for systems that can be described macroscopically in terms of a limited number of local variables. (From the microscopic point of view, this condition implies that, locally, the momentum distribution function of the system should not deviate appreciably from the Maxwellian form.) Note that this restriction is compatible with large deviations from chemical equilibrium. The result is

where the J_p are the rates of irreversible processes (chemical reaction velocities. diffusion, heat flow, and so on). X_p are the corresponding forces (such as differences in chemical potentials and temperature gradients). Remarkably, in this bilinear form the entropy production is expressed entirely in terms of macroscopic quantities of direct physical interest such as the flows and the forces.

Near equilibrium, equation 8 becomes quadratic in the X_p's. One of us^1,4has shown that in this limit, and provided the system is subject to time-independent boundary conditions,

This implies a minimum entropy production at the steady state and the stability of this state; the proof is based on a classical analytic result known as Lyapounov's theorem, which demonstrates stability once one can construct a definite function (the Lyapounov function) whose time variation is also definite with opposite sign. This result is in agreement with the analysis of model systems discussed in the previous section,

Inequality 9 breaks down for states far from thermodynamic equilibrium. but we can still obtain a general inequality in this domain if we decompose dP into two terms, according to equation 8:

We have introduced explicitly the variation in P due to a variation of the flows and a variation of the forces, and can now show that in the whole domain where equation 8 is valid, and provided the system is subject to time-independent boundary conditions,

This in the extension of the minimum entropy-production property to the non linear domain of irreversible processes However, in contrast to inequality 9, inequality 11 does not imply the stability of the steady state, primarily because d_XPis not the differential of a state function in the general case. Instead, we use inequality 11 to derive a stability condition for such states, and we find that stability will be ensured whenever

Here dJp and dXp are the excess flows forces due to the deviation of the

Relation 12 provides a universal thermodynamic stability criterion for nonequlibirum states. One can show that the inequality is always satisfied in the neighborhood of equilibrium as well as in the absence of a feedback process of the autocatalytic type. An alternative interpretation of the stability criterion in terms of a Lyapounov function is also possible.⁴ Consider the entropy S as a function of the nonequilibrium state, with S₀ its value in the reference state whose stability is studied, We expand Saround S₀:

Now in the domain where relation 8 is valid, one shows⁴ that d²Swhich is a quadratic form, obeys the inequality

On the other hand, in mechanical equilibrium, we show that

Thus, according to Lyapounov's theorem, the reference state will be stable provided inequality 12 is satisfied; that is, provided the excess entropy (d²S)/2 increases in time.

Order through fluctuations

So far stability has been related to deviations (arising, for example, from external perturbations) from the reference state. A much deeper insight is provided by the result of recent investigations⁹that in the whole domain of validity of relation 8, the excess entropy d²S I also determines the probability of a small fluctuation around the reference state:

This expression provides a generalization to nonequilibrium situations of the celebrated Einstein formula describing the distribution of small fluctuations around equilibrium. Now fluctuations -- the spontaneous deviations from some average regime--are a universal phenomenon of molecular origin and are always present in a system with many degrees of freedom. Thus, a system in an average state close to but below the transition threshold will always have a nonvanishing probability of reaching the unstable region through fluctuations, When this happens, certain types of fluctuations will be amplified and they will subsequently drive the sys tem to the new regime. In the thermal instability problem mentioned previously, these fluctuations would generate small convection currents that would be damped below the transition point but give rise to a macroscopic cur rent beyond the instability.

In all these situations a new order principle appears that corresponds essentially to an amplification of fluctuations and to their ultimate stabilization by the flow of matter and energy from the surroundings. We may call this principle "order through fluctuations."

An additional important element should be pointed out: The formation of a fluctuation of a given type is fundamentally a stochastic process. The response of the system to this fluctuation is a deterministic process obeying the macroscopic laws as long as the system can damp the fluctuation. Now in the domain of formation of a new structure, fluctuations are amplified and drive the average values to the new regime. Thus, in this region the macroscopic description in terms of averages breaks down and the evolution acquires an essentially statistical character.

We have then a picture of a system evolving through instabilities: In the neighborhood of a stable regime, evolution is essentially deterministic in the sense that the small fluctuations arising continuously are damped, But near the transition threshold the evolution be comes a stochastic process in the sense that the final state will depend on the probability of creating a fluctuation of a given type. Of course, once this probability is appreciable, the system will eventually reach a unique (apart from small fluctuations) stable state, once the boundary conditions are specified.

This state will then be the starting point for further evolution.

Localized steady-state dissipative structure emerges in domain II of the stability diagram.  This new steady state (see figure 1) is found numerically and cannot be obtained in a continuous extrapolation of the equilibrium behavior. In domain III concentration waves may propagate. Numerical values of the parameters here and in figure 1 are D_X = 1.05 x 10^-3, D_Y = 5.25 x 10^-3, D_A= 197 x 10^-3, D_B -> infinity, and all forward kinetic constants are equal to unity. In this domain B  = 26.0, <A>= <X> = 14.0, <Y> = 18.6. Figure 2

Some biological examples

Dissipative structures have been produced in the laboratory in an organic oxidation reaction.¹⁰ More recently, D. Thomas¹¹ demonstrated that an inhomogeneous pH distribution may arise spontaneously inside an artificial membrane wherein two different types of enzymes have been reticulated in a spatially homogeneous fashion. How ever, the problem that concerns us here is primarily the usefulness of the theory outlined in the previous sections in the understanding of biological phenomena. It will be convenient to discuss separately two types of problems:

Is it possible to understand the functional order observed in actual living systems? This question refers to the physicochemical basis of maintenance of life.

How did the structures observed in living beings (nucleic acids, proteins, cells as a whole) arise from an inert pre biotic mixture of simple molecules? This is the problem of prebiotic evolution or of the origin of life.

We shall discuss the first point briefly, postponing for a while the problem of evolution. A number of typical phenomena can be analyzed in terms of the theory outlined in the previous section: regulatory processes, excitable systems and cell aggregation.

The existence of elaborate control mechanisms to ensure that the various chemical reactions in living cells happen at the proper rate and at the right time is well known. The first type of control mechanism ensures that there is no excessive synthesis, or lack of small metabolites, for instance, of energy-rich molecules such as ATP (adenosine triphosphate), The usual way this mechanism operates is to affect the rate at which a particular protein (enzyme) catalyzing one reaction step acts, One of the best studied biochemical chains from this point of view is glycolysis, a process of great importance for the energetics of living cells. Experiments show that the concentrations of the chemicals participating in the reaction present undamped oscillations in time, with perfectly reproducible periods and amplitudes. On the other hand, starting from known data on the elementary reaction steps, one can construct mathematical models for glycolysis.^12,13 A detailed study of the rate equations shows that the experimental results may be interpreted quantitatively as oscillations of the limit-cycle type arising beyond the instability of a time-independent solution that belongs to the thermodynamic branch. In other words, glycolysis is a temporal dissipative structure. This result is expected to extend to a whole series of regulatory processes at the metabolic level.

A second type of control mechanism in living cells affects the rate of synthesis of the various protein molecules that exist in a cell. Usually this mechanism works on a group of more than one enzyme molecule, François Jacob and Jacques Monod have proposed several ingenious models: Either the products of the metabolic action of the enzymes act on the genetic material to inhibit the synthesis, or the initial metabolites added to the medium have the effect of switching on the action of a part of the genetic material. Again, one can construct mathematical models for this process.^14,15The study of rate equations reveals that the activated and in activated regimes belong to two different branches of solutions which, under certain conditions, are separated by an instability.

A number of vital biological processes, in particular the functioning of the nervous system, rest on the ability of certain cell membranes to switch abruptly from a rest state of low ionic permeability to an excited state of high permeability. The former is a polarized state arising from the maintenance of different ionic-charge densities on the two sides. In the excited state the ionic-charge- density difference tends to diminish in an almost discontinuous fashion (all-or-none transition). This depolarization can be interpreted quantitatively¹⁶ as a transition arising beyond the instability of the polarized state and belonging to the "nonthermodynamic" branch. Here the constraint driving the system far from equilibrium is the difference in charge density on the two sides.

Certain unicellular organisms develop a kind of organization composed of individual cells aggregated in colonies; a primitive form of differentiation between cells is also observed in these colonies. Among the best studied families showing this behavior are the slim molds. Their aggregation is mediate by a cyclic AMP (adenosine monophosphate) that can be secreted by the cells. The initiation of this aggregation can be interpreted¹⁷as an instability of the uniform distribution (corresponding to the absence of aggregation) of the individual cells, which again belongs to the thermodynamic branch. One is tempted hope that these aggregation phenomena will provide valuable indications of how higher organisms develop. In this case the interpretation in terms of dissipative structures would provide a much needed unifying principle for all these extremely diverse and complex processes.

This is the first part of a two-part article. In the second part, to appear next month, we contrast prebiological with biological evolution, as described in terms of the "survival of the fittest" principle. Of special interest to us will be instabilities that may lead to increased entropy production. We shall describe Eigen's recent theory of competition among biopolymers and, with simple mathematical models, shall illustrate the relation between stability and evolution.

References

1. I. Prigogine. Etude thermodynamique des Phenomenes Irreversibles, Desoer. Liege (1947).
2. L. von Bertalannfy, General System Theorv. Braziller, New York (1968).
3. E. Schrödinger, What Is Life?. Cambridge University Press. London (1945).
4. P. Glansdorff, I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations, Wiley, New York (1971).
5. R. Graham, H. Haken, Z. Phys. 237, 31 (1970).
6. I. Prigogine, C. Nicolis, Quart. Rev. Biophys. 4, 107 (1971).
7. I. Prigogine, Physica 31, 719 (1965).
8. A. M. Turing, Phil. Trans. Roy. Soc. (London) B237, 37 (1952).
9. G. C. Nicolis, I. Prigogine, Proc Nat. Acad Sci. (US) 68, 2102 (1971).
10. A. Zhabotinski, Biphisics9, 306 (1969); M. Herschkowitz-Kaufman, Comptes Rendus 270C, 1049 (1970). A. T. Winfree. Science 175, 634 (1972).
11. D. Thomas, A. Goidheter. private communication (to be published).
12. E. E. Sel'kov, Eur. J. Biochem. 4, 79 (1968).
13. A. Goldbeter, R. Lefever, Biophys. J. October 1972.
14. D. Cherniavskii L. Grigorov M Polvakova in Oscillatory Processes in Biological and Chemical Systems, Nauka. Moscow (1967).
15. A. Babloyantz, C. Nicolis, J. Theor. Biol 34, 185 (1972).
16. R. Blumenthal, J. P. Changeux, R. Lefever, J. Membrane Biol. 2, 351 (1970).
17. E. F. Keller, L. Segel, J. Theor. Biol 26, 399 (1970).