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The application of statistical physics to evolutionary biology

Edited by Simon A. Levin, Princeton University, Princeton, NJ (received for review March 7, 2005)
This article has a correction. Please see:
Abstract
A number of fundamental mathematical models of the evolutionary process exhibit dynamics that can be difficult to understand analytically. Here we show that a precise mathematical analogy can be drawn between certain evolutionary and thermodynamic systems, allowing application of the powerful machinery of statistical physics to analysis of a family of evolutionary models. Analytical results that follow directly from this approach include the steadystate distribution of fixed genotypes and the load in finite populations. The analogy with statistical physics also reveals that, contrary to a basic tenet of the nearly neutral theory of molecular evolution, the frequencies of adaptive and deleterious substitutions at steady state are equal. Finally, just as the free energy function quantitatively characterizes the balance between energy and entropy, a free fitness function provides an analytical expression for the balance between natural selection and stochastic drift.
Even very simple mathematical models of the evolutionary process can be surprisingly difficult to understand analytically. For example, the Wright–Fisher process with viability selection, a relatively basic set of rules modeling stochastic reproduction and selection, gives rise to a probability distribution of allele frequencies that was revealed only through Kimura's application of diffusion theory and his solution of the resulting set of differential equations (1, 2). Similarly, Fisher's well known geometric model of evolution (3), in which adaptive change is represented by stepwise movement of a point toward the center of a hypersphere, has been studied by moment approximation and simulation (4–6), but exact analytical expressions for many quantities of interest remain inaccessible. In view of the analytic difficulties presented by our most basic models of evolution, new approaches that render large families of models more accessible could prove important both in furthering our understanding of the evolutionary process and in producing basic theoretical results useful in population genetic analysis of sequence data. Here we show that statistical physics furnishes one such approach.
Historical efforts to apply the methods of physics to the problems of biology can be divided into two rather different pursuits. In one, organisms, populations, or ecosystems are viewed as systems that, despite their emergent complexity, are subject to physical laws operating at lower levels of organization (7, 8). Attempts are then made to move from a lower level of organization, at which a physical system is analyzed, to higher levels of organization, at which biological systems are observed. This is not the approach we adopt here. In a very different pursuit, a parallel is drawn between a well understood physical system and a reduced or abstracted biological system; if it is sufficiently complete, the parallel allows the application of tools developed in the physical sciences to the analysis of biological systems. Successful examples of this approach include Kimura's application of diffusion theory (2) and Hopfield's analogy between neural networks and spin glasses (9, 10). Here we show that a very precise mathematical analogy can be developed between certain evolutionary and thermodynamic systems. This is a useful finding, because it allows us to apply the powerful tools of statistical physics to the analysis of simple evolutionary models, yielding several results.
In the present work, we concentrate on the family of models that depict the evolutionary process as a succession of mutant fixations, each of which occurs on the genetic background of the population's previous common ancestor. These models neglect linked polymorphism and the possibility of temporally overlapping fixations. Such effects are treated in other families of population genetic models (e.g., refs. 11–13), but we reserve for future work the extension of the methods developed here to those important problems. The successive fixation models examined here provide a decent approximation to the realistic population dynamic in systems in which the fixation probability of a mutation is not affected by other segregating alleles, with the obvious exception of the allele from which the new mutant was derived. (This condition holds when the product of the population size and the mutation rate is small, i.e., Nμ << 1.) Perhaps more importantly, as we will consider in the Discussion, the models examined here provide natural null models for nearly neutral evolution, with which alternative models involving more complex processes can be compared.
A general result that derives from the application of statistical physical methods to simple evolutionary models contradicts a basic tenet of the nearly neutral theory of molecular evolution. We therefore briefly review here the history and significance of that basic assumption. If the majority of evolutionary substitutions are truly neutral, the molecular divergence between two species is expected to be proportional to the number of generations that have elapsed since their separation (14). This prediction is contradicted by the observation that the rate of evolution appears to be roughly constant across organisms with dramatically different generation times (ref. 15, p. 38). To explain this relatively constant rate of evolution (among several other observations), Ohta (16) suggested that “... the majority of the amino acid substitutions in evolution, although subject to random genetic drift, are not completely neutral but rather very slightly selected against.” If organisms with shorter generation times also have larger populations, Ohta reasoned, the reduced probability of fixation of slightly deleterious mutations in larger populations could offset the larger number of generations per year, resulting in a rate of evolution that does not depend on generation time.
It is important to note that Ohta's (16) suggestion that most substitutions are slightly deleterious does not lead necessarily to the (rather absurd) notion that all organisms are experiencing an ineluctable decline from an original state of perfect adaptation. The alternative to such steady decay is simply that each adaptive fixation compensates for many slightly deleterious fixations, resulting in a longterm steady state in which most substitutions are slightly deleterious (ref. 6; ref. 15, p. 38; and ref. 16). In effect, many baby steps downhill in the fitness landscape are offset by a larger leap uphill. This scenario is compatible with Ohta's (16) demonstration that in a reasonably large population, fixation of slightly deleterious mutations is possible, but fixation of more significantly deleterious mutations is not.
Models and Analysis
When a mutant appears in a population, its lineage faces two possible fates. The lineage may grow to take over the population, such that the mutant becomes the population's most recent common ancestor, or, as happens more frequently, the lineage may go extinct. When a single mutant with genotype j appears in a population with a wildtype genome i, the probability that it fixes depends on its fitness f_{j} , the fitness of the wildtype f_{i} , and the effective population size N. The probability of fixation also depends on the sampling process that describes the replacement of parents by offspring. For the Moran birth–death process, the exact probability of fixation is (see ref. 17, equation 3.50). In Supporting Text, which is published as supporting information on the PNAS web site, we show that for the Wright–Fisher process, the equation where a = 2 in a haploid population, and a = 1 for a diploid population with multiplicative fitness within loci, provides a closer approximation to the probability of fixation than the canonical formula (18). This expression also has properties that will prove convenient in analyses presented below.
Given the probability of fixation, the succession of mutant fixations can be depicted as a Markov process. In the models we study, the state of the population is described by the fixed genotype or, equivalently, by the genotype of the most recent common ancestor. The state of the Markov system at time t is described by a probability vector = (P _{1}(t),..., P_{M} (t)), where P_{i} (t) denotes the probability that the most recent common ancestor of the population at time t had genotype i. The transition matrix W, describing the rate at which genotype j replaces genotype i as the most recent common ancestor of the population, is given by where C_{i,j} is 1 if genotype j is one mutation away from genotype i and is otherwise 0; μ _{i,j} is the rate of mutation from genotype i to genotype j; and a, defined above, depends only on the population's ploidity. Thus, the Markovian evolutionary dynamic in discrete time with discrete states takes the form . Although here we use the Markov process that is discrete in both states and time, in Supporting Text, we show that our derivations apply also to the corresponding Markov systems that are continuous in states or time.
As a first step in developing the analogy between evolutionary and thermodynamic systems, we examine their equilibria. The sites in a gene that can be in one of four states (A, G, C, or T) are analogous to the degrees of freedom in statistical physics, such as the positions and momenta of particles in a gas (Table 1, State variable). The Markov system we have described is irreducible, i.e., there is a finite path of nonvanishing probability between any two states, or genotypes, in the system. This implies [by the Perron–Frobenius theorem (19)] that whatever the initial state, the distribution of probability that the population's most recent common ancestor is of any given genotype approaches a unique steady state. This stationary distribution is defined by the requirement that , or equivalently, that the probability flow in and out of each state is balanced, i.e., for any genotype i.
Because a number of distinct notions of the steadystate distribution have been used in the population genetic literature (1, 4, 20), we pause to clarify both the meaning of the steady state defined by Eq. 4 and its relation to the classical notion of mutationselectiondrift balance. A well known example of mutationselectiondrift balance is Kimura's Ushaped distribution, which describes the frequency distribution of an allele in a model with two neutral alleles that mutate into each other (1). When we wish to generalize the notion of Kimura's steady state to a system with many different genotypes at various mutational distances from one another, it is useful to view the Ushaped distribution as a superposition of two processes. The first process is that of fixation, which in the long run switches between the two alleles. Although the population may never be homogeneous at a locus, fixation is mathematically and conceptually well defined in terms of the state of the population's most recent common ancestor. The second process describes the frequency of the mutant allele conditional on the wild type being fixed. When both processes are combined, we obtain the Ushaped distribution. However, when we consider a system with many genotypes at various mutational distances from one another and exhibiting various selection coefficients, calculating the generalization of the Ushaped distribution becomes very difficult. Moreover, such a generalization may not be very interesting, simply because very few of the alleles that appear in such a distribution appear together in the population at the same time (in general, only alleles that are one, or very few, mutations from each other appear in the population simultaneously). A natural alternative to this exceedingly complex steadystate distribution is the distribution of time spent with each allele fixed or, equivalently, the probability distribution of finding the population with a given allele fixed. In principle, any population genetic measure of interest, such as the average fitness (see below), average heterozygosity, or the effective number of alleles, can be calculated conditional on this steadystate distribution. For example, in the limit where no more than two alleles exist simultaneously in the population (Nμ << 1), the sojourn time formalism (17) can be used to calculate heterozygosity conditional on any given fixed genotype, and the steadystate distribution can be used to calculate the average heterozygosity across the fixed genotypes. This form of steady state has recently been used in the literature (4, 6), and it is precisely the steady state we calculate here. In the case of Kimura's system with two neutral alleles, this steady state is simply (1/2, 1/2). Naturally, the distribution becomes substantially more complicated when one considers multiple genotypes at various mutational distances from one another and exhibiting various selection coefficients.
Quite surprisingly, the steadystate distribution of fixed genotypes can be found for any given fitness scheme. Here we assume that mutation is symmetric, i.e., that μ _{i,j} = μ _{j,i} ;in Supporting Text, we generalize our results to cover cases of asymmetric mutation. Consider a single edge in the fitness landscape, specifically, the edge connecting genotypes i and j, which are separated by a single mutation. From Eqs. 1 , 2 , 3 , we see that for this edge, where v = N  1 for the Moran process, v = 2(N  1) for the haploid Wright–Fisher process, and v = 2N  1 for the diploid Wright–Fisher process with multiplicative fitness within a locus. When the rates of transition between any two states satisfy such a relation, in which their ratio can be described as a ratio of a function F evaluated at the two states, statistical physics tells us that the steadystate distribution takes a simple form (19). Because Eq. 5 shows that it follows that the unique solution to Eq. 4 is given by where x_{I} = ln(f_{i} ). We introduce the variable x, which we refer to as the additive fitness, for two reasons. First, the additive fitness x, unlike f, exhibits a desirable property when we focus on the relationships among multiple alleles. As a simple example, consider three alleles that have fitness values f _{1}, f _{2}, and f _{3}. The usual population genetic selection coefficient of allele 2 relative to allele 1 would be s _{1,2} = f _{2}/f _{1}  1. Similarly, the selection coefficient of allele 3 relative to allele 2 would be s _{2,3} = f _{3}/f _{2}  1. Consideration of allele 3 relative to allele 1 shows that the selection coefficient, s, does not behave additively. That is, s _{1,3} ≠ s _{1,2} + s _{2,3}. However, if we use x instead of f, the relationships are simplified. Instead of the usual population genetic selection coefficient s_{i,j} = f_{j} /f_{i}  1, we now have Δx_{i,j} = x_{j}  x_{i} and, conveniently, Δx _{1,3} = Δx _{1,2} + Δx _{2,3}. Moreover, this additivity extends to an arbitrary number of alleles.
The second reason for introducing the variable x is that the substitution places the steadystate distribution (Eq. 7 ) in a form familiar from statistical physics, emphasizing that this steady state is precisely analogous to the description of a physical system at thermal equilibrium. The analogy is summarized in Table 1. When a physical system is at equilibrium with a thermal bath at temperature T, the probability that the system would be in a microscopic state i is given by the Boltzmann factor, where β = 1/k _{B} T, k _{B} is the Boltzmann coefficient, and E_{i} is the energy of state i (21). Similarly, we find that the probability of the population being fixed with genotype i at steady state is given by where v is analogous to β, and the additive fitness, x_{i} , is analogous to the energy E_{i} . Energy is an additive quantity in physical systems, so it is not surprising that its counterpart in the evolutionary process should be the additive fitness, x. (Because the sign of energy is defined such that the dynamics tend to reduce it, whereas evolutionary systems tend to increase fitness, a minus sign must be introduced in translation.) The analogy between v and β indicates that the population size affects the evolutionary system as the inverse of the temperature affects a physical system. When the temperature is zero, a physical system at equilibrium is always at the lowest energy state. Analogously, when the population is infinite, the evolutionary system at steady state is always at the genotype with the highest fitness. When the temperature is not zero, the probability of finding the physical system at any microscopic state depends only on a state's energy. Analogously, in an evolutionary system with a finite population, the probability of finding the population with a given fixed genotype depends only on the fitness associated with that genotype.
Before developing the analogy further, we pause to consider its use in addressing a few basic questions about the nature of the steadystate distribution of fixed genotypes. At this steady state, the evolutionary system is constantly changing, but without adaptation (6, 22). As discussed in the Introduction, a basic tenet of nearly neutral theory is that many small deleterious substitutions are compensated by fewer more considerable adaptations. This turns out to be incorrect. An important property that derives from the steadystate solution is that along any edge in the landscape This property, which plays a central role in statistical physics, is called detailed balance (DB) (19). For any system at steady state, the flow of probability in and out of each state is balanced. For a system that satisfies DB, this balance holds between any two states. Thus, at steady state, the rate at which one genotype fixes and replaces the other as wild type is precisely equal to the rate at which the opposite fixation occurs; this implies that the numbers of adaptive and deleterious substitutions in the evolutionary system are equal. The implications of this equality for studies of nearly neutral evolution will be considered in the Discussion.
Because the steadystate distribution of fixed genotypes includes types that are not maximally fit, the population suffers a cost. This cost is referred to as the fixeddrift load (4), and it is generally calculated as the proportional reduction of the population's average fitness due to stochastic fixation of suboptimal alleles. Calculation of load bears on important population genetic problems, such as the evolution of sex and the extinction risk of small populations (4, 23, 24). The analogy developed here reduces the calculation of fixeddrift load to several straightforward steps, into which appropriate expressions may be substituted according to the particular population genetic model or fitness landscape under consideration. For brevity, we illustrate these steps using a simple model, corresponding to Fisher's geometric model in one dimension; application of the same approach to complex models, such as Fisher's multidimensional model, as well as a detailed comparison of the analytic results with simulations, will be demonstrated elsewhere. Consider the simple landscape in which fitness ranges continuously between 0 and 1, and the density of genotypes that correspond to any fitness is uniform, i.e., ρ _{g} (f) = 1 for f ∈ (0,1). Assume further that mutation between genotypes is symmetric. According to Eq. 7 , at steady state, the probability density of having fitness f is Therefore, the average fitness is and the fixeddrift load is That the fixeddrift load is approximately 1/v becomes intuitive below, when we consider how the efficacy of selection at steady state depends on the effective population size. Eq. 13 shows that the load is independent of the properties of mutation other than symmetry; this follows from the more general result that the steadystate distribution of fixed genotypes (Eq. 7 ) is itself independent of the properties of mutation. This distribution (and, consequently, quantities that can be expressed as functions of this distribution alone) depend only on the fitness function and the population size. In Supporting Text, we show that these results still hold when we incorporate the well established asymmetries in mutation, such as those responsible for GC or AT bias (25). The independence of steady state from the details of mutation considerably simplifies the analysis of evolutionary behavior at equilibrium.
What can we say about the evolutionary process before steady state is attained? In physics, it is useful to find an energy function (or a Lyapunov function in mathematics), i.e., a function of the system's state that changes monotonically as the dynamics progress in time. In Supporting Text, we use detailed balance to show that where S = 〈ln(P)〉, is an energy, or Lyapunov, function of the evolutionary dynamic. [The proof that G monotonically increases under the evolutionary dynamics is precisely analogous to the proof of Boltzmann's H theorem (21).] Because of its close parallel with free energy, we refer to Eq. 14 as the free fitness function (Table 1, Free fitness, free energy). The monotone increase of free fitness and its eventual maximization at steady state is analogous to the maximization of free energy at thermal equilibrium, which is a manifestation of the second law of thermodynamics. At thermal equilibrium, the minimization of free energy balances between a physical system's tendencies to lower energy and increase entropy (S). At the evolutionary steady state, the maximization of free fitness balances between the evolutionary tendencies in finite populations to increase both fitness and entropy. Although other Lypunov functions can be found, this function is uniquely defined by being extensive (26, 27), i.e., if we consider a system of n genetic sites that contribute to fitness independently, the free fitness of this system equals the sum of free fitness values across sites.
The monotone increase of free fitness bears an interesting relationship to Fisher's fundamental theorem of natural selection (3), which states that a population's average fitness increases at a rate proportional to the additive genetic variance in the population. [To be precise, Fisher's theorem has been interpreted (28–30) as stating that in an infinite population, the fitness component associated with a constant environment, both ecologically and genetically, increases at a rate proportional to the additive genetic variance.] Unlike the models considered here, Fisher's theorem applies under conditions of a changing environment and includes an explicit representation of a population's standing genetic variance. Despite its extraordinary generality, however, Fisher's theorem does make the important simplifying assumption that the population size is infinite; it omits the effects of drift in finite populations. The free fitness function, although it lacks the generality of Fisher's theorem on other counts, does account for the evolutionary effects of finite population size. When the population size is infinite, the second term in the free fitness function vanishes, and the function's monotone increase becomes an increase in average fitness, in accordance with Fisher's theorem. But when the population is finite, the second term in the free fitness function is nonvanishing. Indeed, this term, which is simply the entropy of the probability distribution of genotypes, becomes proportionally more important as the population size becomes smaller. Whereas increases in the first term of the free fitness function are associated with adaptation, increases in the second term indicate augmented importance of stochastic, or nonadaptive, change.
Fisher (ref. 3, p. 39), as well as others after him (e.g., ref. 6), has wondered about the relative efficacy of the forces “destroying adaptation” and that “building it up.” To put the tradeoff between fitness and entropy in quantitative terms, we consider how population size affects selective discrimination between genotypes at steady state. The scale of selective discrimination, i.e., the difference in fitness that causes a unit of change in the logarithm of the probability of a genotype at steady state, derives from the relation For a biologically realistic population size (N >> 1), Eq. 15 implies that where s_{j,i} is the selection coefficient. Therefore, the scale of selective discrimination satisfies the relation vs ≈ 1. That is, free fitness determines the tradeoff between fitness and entropy such that a selection coefficient of 1/v causes a unit change in the steadystate probability of a genotype. This relation, which is a generalization of classical results from neutral theory (14, 31), distills how population size determines the efficacy of selection. This is analogous to a thermodynamic system, in which the temperature determines how differences in the energy associated with different states affect their probabilities at thermal equilibrium (Table 1, Equilibrium scale).
Discussion
The suggestion that most amino acid substitutions are slightly deleterious was originally offered to resolve contradictions between the predictions of neutral theory and empirical observations (15). In exploring how this novel premise might realign predictions and data, a number of theoretical studies simulated molecular evolution under predefined distributions of mutant selection coefficients (see, for example, ref. 32). The results presented here suggest that a priori assumption of a particular distribution of mutant selection coefficients is inappropriate for steadystate models of nearly neutral evolution. The distribution of mutant selection coefficients is determined by the operation of the evolutionary dynamic on the landscape of all possible genotypes and therefore cannot be assumed a priori. In fact, the distribution determined by the evolutionary dynamic will differ in important ways from distributions assumed a priori. For instance, we found that the steadystate distribution of mutant selection coefficients will take a form that causes the number of fixations involving an additive fitness change Δx to equal the number of fixations involving an additive fitness change Δx. This property will not, in general, be exhibited by distributions of mutant selection coefficients assumed a priori.
Although we find that the premise that most substitutions are slightly deleterious may be incorrect, many of the deductions that rely on this premise remain in tact. For example, steadystate nearly neutral evolution may still provide an explanation for the constancy of the rate of molecular evolution across organisms with very different generation times. According to detailed balance, on average, for one adaptive substitution with a given selection coefficient to occur, a corresponding deleterious substitution must also take place. Because deleterious mutations are much less likely to fix than adaptive ones, fixation of deleterious mutations remains the ratelimiting step in molecular evolution, and Ohta's (16) rationale for the molecular clock may still hold. Other classical arguments, such as Ohta's explanation for the surprisingly weak relationship between polymorphism and population size, can be rephrased along similar lines.
An important simplifying assumption in our analysis was that mutations are sufficiently rare that the fixation probability of a mutation is not affected by other segregating alleles. Although future work may determine whether the methods presented here can be generalized to incorporate interactions among segregating sites, we must presently address the applicability of theory that omits such important processes as hitchhiking (15, 33) and background selection (12). Most straightforwardly, the results presented here can be viewed as an approximation that becomes acceptably accurate under certain population parameters. For example, in small populations, mutations are rare, and therefore our simplified evolutionary dynamics and the results we have derived from them become decent approximations of reality (4). More generally, however, the theory considered here may allow us to more adequately understand the behavior of models that frequently serve either as null models or as theoretically tractable heuristics in studies of more complicated molecular evolutionary dynamics. For instance, revising the premise that slightly deleterious substitutions predominate may affect how the null hypothesis of steadystate nearly neutral evolution is simulated in studies measuring rates and effects of adaptation in the genome. Or, to offer another example, closedform expressions for the effect of population size on key population genetic statistics may facilitate evaluation of the suggestion that the effects of hitchhiking or background selection can be approximated theoretically by a rescaling of population size in neutral theory (12, 34).
Besides adopting important simplifying assumptions, the theory treated here reduces the complexity of dynamics by treating the averages of large numbers of degrees of freedom. Although such averages are clearly meaningful in the classical systems of statistical physics, in which the number of degrees of freedom is generally on the order of Avogadro's number, one may reasonably ask whether averages are actually useful in genetic systems, in which the number of degrees of freedom is so much smaller. Although population genetics initially concerned itself with the dynamics at one or a few sites of interest, the recent flood of genomic data has allowed measurement of averages over many sites. For instance, it is now common to compare average evolutionary rates or average levels of codon bias across many genes in the genome (35, 36). Such studies encounter tremendous noise, and predictions are far from precise (especially by the standards of statistical physics), but averaging across sites within genes and comparing large numbers of genes often allow detection of important trends.
The applicability of the theory treated here should also be discussed in light of Eq. 9 , which gives the probability that any given genotype is fixed in the population. The time scales of evolution do not allow exhaustive exploration of large and rugged fitness landscapes; instead, populations are often confined to exploration of a local fitness peak. Such metastable states do not reflect an equilibrium in the strict sense because, given sufficient time, the system would eventually leave each local peak. Nonetheless, under the approximation that the local landscape is bounded by valleys that cannot be traversed, Eq. 9 and the results that follow from it would apply.
A number of authors (3, 37–39) have noted parallels between statistical physics and evolution. In 1930, R. A. Fisher wrote that “... the fundamental theorem [of natural selection] bears some remarkable resemblance to the second law of thermodynamics. It is possible that both may ultimately be absorbed by some more general principle” (3). In accordance with this suggestion, we have shown that the mathematical description of evolution of a finite population in a constant environment is analogous to that of a thermodynamic system. Our treatment does not address how evolutionary systems, which are themselves physical systems, are subject to the laws of statistical physics [as, for example, Schrödinger suggests (40)]. The analogy we have developed does, however, show that the methods used to describe systems in statistical physics can be applied to evolutionary systems in a useful way. This analogy leads to a general analytical form for the steadystate distribution of fixed genotypes, thus reducing the solution of a large family of evolutionary models, including Fisher's geometric model, to several straightforward substitutions. The close parallels between statistical physics and evolutionary dynamics also prove useful in elucidating basic evolutionary relationships, such as that between genetic load and effective population size, and in revealing new generalizations about dynamic behavior at steady state, such as the equality of the number of adaptive and deleterious substitutions. Finally, the analogy permits derivation of an energy function for evolutionary dynamics of a finite population. The form of this energy function is precisely that of free energy, and the maximization of free fitness is precisely analogous to the second law of thermodynamics.
Acknowledgments
We thank Ilan Eshel for numerous helpful discussions. We also thank Tsvi Tlusty, Nati Linial, Erel Levin, Itamar Pitowsky, Sergiu Hart, Yossi Rinott, Dmitri Petrov, and two anonymous reviewers for valuable comments at various stages of this work. This work was supported by the Center for Complexity Science of the Yeshaya Horowitz Association (G.S.). A.E.H. was supported by National Institutes of Health Grant 28428 (to Marcus W. Feldman).
Footnotes

↵ ‡ To whom correspondence should be addressed. Email: gsella{at}math.huji.ac.il.

Author contributions: G.S. and A.E.H. performed research.

This paper was submitted directly (Track II) to the PNAS office.
 Copyright © 2005, The National Academy of Sciences
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