1. Introduction

Biological systems consist of structural hierarchies, each level of which undergoes evolution due to a multiplicity of factors (Sarkar Reference Sarkar1998; Okasha Reference Okasha2006). Each of these factors, for instance, the generation of variation (through point mutation, gene duplication, other types of mobile DNA proliferation, etc.) or the operation of natural selection, is subject to theoretical analysis. Because natural selection has long been regarded as a principal factor of evolution (Wright Reference Wright1921; Fisher Reference Fisher1922, Reference Fisher1930; Haldane Reference Haldane1924, Reference Haldane1932), an elaborate theory of natural selection has been constructed over the years. As Frank (Reference Frank2012) and many others (e.g., Sarkar Reference Stephens, Illari, Russo and Williamson2011) have repeatedly emphasized, this theory must be clearly distinguished from the theory of evolution because it (intentionally) ignores the other factors relevant to evolution.

Two approaches can be distinguished about what a theory of natural selection should accomplish. The first treats this theory as consisting of the construction of a constellation of predictive models that attempt to illuminate ongoing evolution and evolutionary history. This approach goes back to Haldane’s (e.g., Reference Haldane1924) work in the 1920s that may be regarded as providing the foundation for a theory of natural selection in the context of understanding evolution (Sarkar Reference Sarkar2004, Reference Sarkar, Matthen and Stephens2007). The second approach goes back principally to Fisher’s seminal Genetical Theory of Natural Selection. It explores natural selection in full generality independent of its role in evolution, thereby treating it, in Fisher’s (Reference Fisher1930, vii) words, “as an independent principle worthy of scientific study.” The latter approach is the one relevant to the analysis that follows.

From the perspective of the second approach, a central result—certainly, historically the most interesting one—is one that Fisher (Reference Fisher1930, vii) called the “fundamental theorem of Natural Selection” (FTNS). Roughly, the claim is that the rate of change of mean fitness of a population at any time is equal to its genic (i.e., additive genetic) variance at that time. The controversies over the scope, significance, history, and interpretations of the FTNS are beyond the scope of this article.Footnote ¹ Strictly, the FTNS is only applicable to models with constant fitnesses. It is thus not relevant, for example, to ubiquitous frequency-dependent selection (Sarkar Reference Illari, Russo and Williamson2014). Nevertheless, the dynamics of systems under natural selection can be modeled generally in this “Fisherian” framework in which the changes in phenotypic distributions within populations are predicted from fitnesses and related parameters, with standard statistical measures, especially the mean and the variance, used to track temporal changes in the relevant distributions. This perspective will be called the standard dynamical (SD) interpretation of natural selection.Footnote ²

Two points about the SD interpretation of evolution will be explicitly relevant to the analysis below:

1. • The SD interpretation is supposed to characterize the dynamics of all systems under natural selection, what has sometimes been called “Darwinian dynamics” (Michod Reference Michod2000). In particular, it does not restrict natural selection to act at only one level of the biological hierarchy.

2. • In general, under the SD interpretation, one aspect of the FTNS will remain relevant to all models: the rate of change of the mean fitness of a population will depend critically on the available variance in fitness. This dependence has a natural interpretation: the power of natural selection to effect change depends on the amount of variability available to it (as, appropriately, measured by the variance).

These points are relevant because the SD interpretation of natural selection has recently come under challenge from Frank (Reference Frank2012).

Instead of the SD interpretation, Frank argues for an interpretation of natural selection in which the dynamics of information (as quantified by a suitable measure) provides a more appropriate framework for understanding natural selection. As Frank (Reference Frank2012, 2377) puts it: “information is a primary quantity with intuitive meaning in the study of selection, whereas the genetic variance just happens to be an algebraic equivalence for the measure of information. The history of evolutionary theory has it backwards, using statistical expressions of variances and covariances in place of the equivalent and more meaningful expressions of information. To read the fundamental equations of evolutionary change, one must learn to interpret the standard expressions of variances and covariances as expressions of information.” This assessment is based on an apparently elegant formal result that Frank (Reference Frank2012) has recently obtained that is strikingly analogous to the FTNS. This result states that the rate of change of the mean of the logarithm of fitness (i.e., of the Malthusian parameter) is equal to a measure that has an information-theoretic (IT) interpretation, that is, the Jeffreys divergence, . From Frank’s perspective, this result is apparently more fundamental than the FTNS and provides the basis for preferring the IT over the SD interpretation of natural selection.

The purpose of this analysis is to bring this interpretation into question. Section 2 will set the stage for a comparison of the SD and IT interpretations by introducing a simple model of a system dynamically changing through natural selection. This is a haploid model bereft of all the complications of inheritance that typically make biological evolution difficult to analyze—and interesting. Section 3 will analyze this model from the SD perspective and prove and interpret a simple instance of the FTNS. Section 4 will reanalyze this model from Frank’s (Reference Frank2012) IT perspective. The core of this article is section 5 in which these two analyses are critically compared. Some final remarks, which put this analysis in a broader context of the role of information theory in the sciences, are relegated to section 6. None of the formal results reported here are at variance with the results used by Frank. The difference is one of interpretation.

2. A Haploid Model

At stake is a haploid model with an infinite population, discrete generations, and selection acting at a single locus with n alleles. Let be the frequency of the ith allele in the present (i.e., 0th) generation. Then . Let be the fitness of the ith allele, and let fitnesses remain constant over generations. Let be the frequency of the ith allele in the next (i.e., first) generation. Then

or

where

is the mean fitness in the present generation. Then the mean fitness in the next generation, , is given by

3. Standard Dynamics

We begin by computing the change in mean fitness between the 0th and first generation:

Now, the variance in the fitnesses, , is given by

From equations (3) and (4), we get

Thus, the rate of change of fitness due to natural selection is proportional to the variance in fitness in any generation. Equation (5) is perhaps the simplest version of the FTNS.

Thus, as is well known from work on the FTNS, the power of natural selection is limited by the variability available in the population. The relevant variation is the additive variance in fitness: the analysis above did not have to incorporate additivity explicitly because the model is a haploid (there is thus no potential for a dominance effect, i.e., of an interaction between alleles at a locus) and for a single locus (there is thus no potential for epistasis or interaction between loci). Moreover, the genotype-phenotype map was assumed to be trivial with fitnesses directly assigned to the genotype. It is straightforward to extend the analysis to show that the variance in fitness decreases with selection each generation and that it becomes equal to 0 at a state of equilibrium at which the mean population fitness is at its maximum.

4. The Information-Theoretic Alternative

To obtain the IT interpretation, we begin from equation (2). First, define the Malthusian parameter, . Then

Note that, even though the are constant, the are not constant from generation to generation because changes. Now, from equation (1), we get

where . Next, define

Then, using equation (6)

Thus,

Now, the crucial IT measure for this analysis is the Kullback-Leibler divergence or relative entropy, . Given two discrete probability distributions with distribution functions q and ,

It can be roughly interpreted to measure the divergence or “relative entropy” of the -distribution from the q-distribution. Now, we can define the symmetric Jeffrey’s divergence, , by

Note that, because, , we have . Although was originally introduced as a suitable prior for Bayesian inference (Jeffreys Reference Jeffreys1946), it has since become a standard information-theoretic measure.

Returning to our haploid model, let the -distribution be that of the alleles in the first generation and the q-distribution be that of the alleles in the 0th generation. Then, from equations (9) and (10), using equation (12), we get

This result is due to Frank (Reference Frank2012); its elegance is obvious. But so are the difficulties.

5. Analysis and Interpretation

The critical problem is the definition of as given by equation (8).

1. 1. Note, first, what is not (contrary to what Frank [Reference Frank2012, 2381] claims): it is not the change in mean log fitness (Malthusian parameter) that would be given by

   
   In equation (8), is multiplied by , the values in the 0th generation rather than the first generation.

2. 2. This problem can be avoided if it is assumed that the are constant. However, this would contradict equation (6). Moreover, the only way to ensure constancy of the is to require constancy of , that is, of the mean fitness. But any such suggestion would be conceptually incoherent in the context of a theory of natural selection. We would be requiring natural selection not to increase mean fitness (in any generation).

3. 3. Now, Frank (Reference Frank2012) does not present equation (8) as a definition; rather, according to him, it is a derivation from prior results. These results are first established for any quantitative character , and then is treated as a special case. However, these prior derivations explicitly require that the are constant (2389). When this restriction is dropped, we are not left with anything that approximates equation (13) (2389–90). What is ironic is that Frank continually refers to as the change due to selection, but, as we just saw, it has to presume that there is no change in mean fitness that is better interpreted as an absence of selection.

I suggest that , as defined by equation (8), does not have any natural biological interpretation.

Even if equation (8) can be rehabilitated, a final interpretive worry will persist. Because refers to both the present (0th) and the next generations, equation (8) violates a minimal condition of dynamical sufficiency (sensu Lewontin Reference Lewontin1974): it does not allow the prediction of the future based entirely on parameter estimates available at the present generation. I am not suggesting that the model being analyzed is itself dynamically insufficient: it obviously is not, as its analysis using standard dynamics (sec. 3) showed. Rather, the problem is with the IT interpretation, and this feature of should be taken as providing an additional reason to prefer the SD interpretation over the IT interpretation of the model.

A final benefit of the SD interpretation, although perhaps obvious, should also be explicitly noted. The parameters used are standard statistical ones—means, variances, and so on; their interpretation is straightforward in a sense in which informational parameters are not.

6. Final Remarks

Suppose that the problems addressed in section 5 can be successfully addressed. There still remains the question whether the information-theoretic reformulation of the results of section 3 is any more than rewriting familiar results in new notation. Frank’s (Reference Frank2012) enthusiasm for the IT interpretation was already noted in section 1. He went on to argue that “a general theory of information … provide[s] the foundation of the deepest concepts in communication, physics and statistics” (2378). If, indeed, “information” plays such a role, there is ample reason to presume that the IT interpretation is more than merely an exercise in notational dexterity.

But there are many compelling reasons for caution, if not downright skepticism, on this point. There is no question that information theory has been valuable in the study of communication and related fields—that is what the theory was designed to do when it was developed in the 1940s and 1950s. It has some pedigree in statistics (e.g., Kullback Reference Kullback1968), but what is perhaps the most influential use of “information” in statistics today, that is, in the context of the Akaike Information Criterion (AIC; Akaike Reference Akaike, Petrov and Csaki1973), is at best marginally related to information theory. By the time we get to physics, we are largely in the realm of speculation. Within biology the most widespread use of “information” is in molecular genetics—but, even here, there are skeptics (Sarkar Reference Sarkar and Sarkar1996a, Reference Sarkar1996b).

Until there are more plausible grounds to induce confidence in an information-theoretic unification of the sciences, it seems to me that it is best to continue to embrace the standard dynamical interpretation of natural selection, recognize the complexity of so-called Darwinian dynamics when selection operates simultaneously at multiple levels of organization, and glean insights from results such as the fundamental theorem of natural selection with the standard statistical interpretation of its parameters. This may be a very conservative vision of evolutionary theory, but conservatism in intellectual matters is often warranted.