Evolution with recombination as Gibbs sampling.

This work presents a population genetic model of evolution, which includes haploid selection, mutation, recombination, and drift. The mutation-selection equilibrium can be expressed exactly in closed form for arbitrary fitness functions without resorting to diffusion approximations. Tractability is achieved by generating new offspring using n-parent rather than 2-parent recombination. While this enforces linkage equilibrium among offspring, it allows analysis of the whole population under linkage disequilibrium. We derive a general and exact relationship between fitness fluctuations and response to selection. Our assumptions allow analytical calculation of the stationary distribution of the model for a variety of non-trivial fitness functions. These results allow us to speak to genetic architecture, i.e., what stationary distributions result from different fitness functions. This paper presents methods for exactly deriving stationary states for finite and infinite populations. This method can be applied to many fitness functions, and we give exact calculations for four of these. These results allow us to investigate metastability, tradeoffs between fitness functions, and even consider error-correcting codes.

Developmental structuring of phenotypic variation: A case study with a cellular automata model of ontogeny.

Developmental mechanisms not only produce an organismal phenotype, but they also structure the way genetic variation maps to phenotypic variation. Here, we revisit a computational model for the evolution of ontogeny based on cellular automata, in which evolution regularly discovered two alternative mechanisms for achieving a selected phenotype, one showing high modularity, the other showing morphological integration. We measure a primary variational property of the systems, their distribution of fitness effects of mutation. We find that the modular ontogeny shows the evolution of mutational robustness and ontogenic simplification, while the integrated ontogeny does not. We discuss the wider use of this methodology on other computational models of development as well as real organisms.

The evolution of frequency-dependent cultural transmission.

In a model of vertical and oblique cultural transmission of a dichotomous trait, the rates of transmission of each form of the trait are functions of the trait frequency in the population. Sufficient conditions on these functions are derived for a stable trait polymorphism to exist. If the vertical transmission rates are monotone decreasing functions of the trait frequency, a complete global stability analysis is presented. It is also shown that a unique protected polymorphism can be globally stable even though the sufficient conditions are not met. The evolution of frequency-dependent transmission is modeled using modifier theory, and exact conditions are derived for a transmission modifier to invade a population at a stable polymorphism. Finally, the interaction between frequency-dependent selection and frequency-dependent transmission is explored.

The Surprising Creativity of Digital Evolution: A Collection of Anecdotes from the Evolutionary Computation and Artificial Life Research Communities.

Evolution provides a creative fount of complex and subtle adaptations that often surprise the scientists who discover them. However, the creativity of evolution is not limited to the natural world: Artificial organisms evolving in computational environments have also elicited surprise and wonder from the researchers studying them. The process of evolution is an algorithmic process that transcends the substrate in which it occurs. Indeed, many researchers in the field of digital evolution can provide examples of how their evolving algorithms and organisms have creatively subverted their expectations or intentions, exposed unrecognized bugs in their code, produced unexpectedly adaptations, or engaged in behaviors and outcomes, uncannily convergent with ones found in nature. Such stories routinely reveal surprise and creativity by evolution in these digital worlds, but they rarely fit into the standard scientific narrative. Instead they are often treated as mere obstacles to be overcome, rather than results that warrant study in their own right. Bugs are fixed, experiments are refocused, and one-off surprises are collapsed into a single data point. The stories themselves are traded among researchers through oral tradition, but that mode of information transmission is inefficient and prone to error and outright loss. Moreover, the fact that these stories tend to be shared only among practitioners means that many natural scientists do not realize how interesting and lifelike digital organisms are and how natural their evolution can be. To our knowledge, no collection of such anecdotes has been published before. This article is the crowd-sourced product of researchers in the fields of artificial life and evolutionary computation who have provided first-hand accounts of such cases. It thus serves as a written, fact-checked collection of scientifically important and even entertaining stories. In doing so we also present here substantial evidence that the existence and importance of evolutionary surprises extends beyond the natural world, and may indeed be a universal property of all complex evolving systems.

Unified reduction principle for the evolution of mutation, migration, and recombination.

Modifier-gene models for the evolution of genetic information transmission between generations of organisms exhibit the reduction principle: Selection favors reduction in the rate of variation production in populations near equilibrium under a balance of constant viability selection and variation production. Whereas this outcome has been proven for a variety of genetic models, it has not been proven in general for multiallelic genetic models of mutation, migration, and recombination modification with arbitrary linkage between the modifier and major genes under viability selection. We show that the reduction principle holds for all of these cases by developing a unifying mathematical framework that characterizes all of these evolutionary models.

Some topics in theoretical population genetics: Editorial commentaries on a selection of Marc Feldman's TPB papers.

This article consists of commentaries on a selected group of papers of Marc Feldman published in Theoretical Population Biology from 1970 to the present. The papers describe a diverse set of population-genetic models, covering topics such as cultural evolution, social evolution, and the evolution of recombination. The commentaries highlight Marc Feldman's role in providing mathematically rigorous formulations to explore qualitative hypotheses, in many cases generating surprising conclusions.

Generation of variation and a modified mean fitness principle: Necessity is the mother of genetic invention.

Generation of variation may be detrimental in well-adapted populations evolving under constant selection. In a constant environment, genetic modifiers that reduce the rate at which variation is generated by processes such as mutation and migration, succeed. However, departures from this reduction principle have been demonstrated. Here we analyze a general model of evolution under constant selection where the rate at which variation is generated depends on the individual. We find that if a modifier allele increases the rate at which individuals of below-average fitness generate variation, then it will increase in frequency and increase the population mean fitness. This principle applies to phenomena such as stress-induced mutagenesis and condition-dependent dispersal, and exemplifies "Necessity is the mother of genetic invention."

Resolvent positive linear operators exhibit the reduction phenomenon.

The spectral bound, s(αA + ßV), of a combination of a resolvent positive linear operator A and an operator of multiplication V, was shown by Kato to be convex in ß ∈ R. Kato's result is shown here to imply, through an elementary "dual convexity" lemma, that s(αA + ßV) is also convex in α > 0, and notably, ∂s(αA + ßV)/∂α ≤ s(A). Diffusions typically have s(A) ≤ 0, so that for diffusions with spatially heterogeneous growth or decay rates, greater mixing reduces growth. Models of the evolution of dispersal in particular have found this result when A is a Laplacian or second-order elliptic operator, or a nonlocal diffusion operator, implying selection for reduced dispersal. These cases are shown here to be part of a single, broadly general, "reduction" phenomenon.

An evolutionary reduction principle for mutation rates at multiple Loci.

A model of mutation rate evolution for multiple loci under arbitrary selection is analyzed. Results are obtained using techniques from Karlin (Evolutionary Biology, vol. 14, pp. 61-204, 1982) that overcome the weak selection constraints needed for tractability in prior studies of multilocus event models.A multivariate form of the reduction principle is found: reduction results at individual loci combine topologically to produce a surface of mutation rate alterations that are neutral for a new modifier allele. New mutation rates survive if and only if they fall below this surface-a generalization of the hyperplane found by Zhivotovsky et al. (Proc. Natl. Acad. Sci. USA 91, 1079-1083, 1994) for a multilocus recombination modifier. Increases in mutation rates at some loci may evolve if compensated for by decreases at other loci. The strength of selection on the modifier scales in proportion to the number of germline cell divisions, and increases with the number of loci affected. Loci that do not make a difference to marginal fitnesses at equilibrium are not subject to the reduction principle, and under fine tuning of mutation rates would be expected to have higher mutation rates than loci in mutation-selection balance.Other results include the nonexistence of 'viability analogous, Hardy-Weinberg' modifier polymorphisms under multiplicative mutation, and the sufficiency of average transmission rates to encapsulate the effect of modifier polymorphisms on the transmission of loci under selection. A conjecture is offered regarding situations, like recombination in the presence of mutation, that exhibit departures from the reduction principle. Constraints for tractability are: tight linkage of all loci, initial fixation at the modifier locus, and mutation distributions comprising transition probabilities of reversible Markov chains.

From genotypes to organisms: State-of-the-art and perspectives of a cornerstone in evolutionary dynamics.

Understanding how genotypes map onto phenotypes, fitness, and eventually organisms is arguably the next major missing piece in a fully predictive theory of evolution. We refer to this generally as the problem of the genotype-phenotype map. Though we are still far from achieving a complete picture of these relationships, our current understanding of simpler questions, such as the structure induced in the space of genotypes by sequences mapped to molecular structures, has revealed important facts that deeply affect the dynamical description of evolutionary processes. Empirical evidence supporting the fundamental relevance of features such as phenotypic bias is mounting as well, while the synthesis of conceptual and experimental progress leads to questioning current assumptions on the nature of evolutionary dynamics-cancer progression models or synthetic biology approaches being notable examples. This work delves with a critical and constructive attitude into our current knowledge of how genotypes map onto molecular phenotypes and organismal functions, and discusses theoretical and empirical avenues to broaden and improve this comprehension. As a final goal, this community should aim at deriving an updated picture of evolutionary processes soundly relying on the structural properties of genotype spaces, as revealed by modern techniques of molecular and functional analysis.

Proof of the Feldman-Karlin conjecture on the maximum number of equilibria in an evolutionary system.

Feldman and Karlin conjectured that the number of isolated fixed points for deterministic models of viability selection and recombination among n possible haplotypes has an upper bound of 2(n)-1. Here a proof is provided. The upper bound of 3(n-1) obtained by Lyubich et al. (2001) using Bézout's Theorem (1779) is reduced here to 2(n) through a change of representation that reduces the third-order polynomials to second order. A further reduction to 2(n)-1 is obtained using the homogeneous representation of the system, which yields always one solution 'at infinity'. While the original conjecture was made for systems of selection and recombination, the results here generalize to viability selection with any arbitrary system of bi-parental transmission, which includes recombination and mutation as special cases. An example is constructed of a mutation-selection system that has 2(n)-1 fixed points given any n, which shows that 2(n)-1 is the sharpest possible upper bound that can be found for the general space of selection and transmission coefficients.

The evolutionary reduction principle for linear variation in genetic transmission.

The evolution of genetic systems has been analyzed through the use of modifier gene models, in which a neutral gene is posited to control the transmission of other genes under selection. Analysis of modifier gene models has found the manifestations of an "evolutionary reduction principle": in a population near equilibrium, a new modifier allele that scales equally all transition probabilities between different genotypes under selection can invade if and only if it reduces the transition probabilities. Analytical results on the reduction principle have always required some set of constraints for tractability: limitations to one or two selected loci, two alleles per locus, specific selection regimes or weak selection, specific genetic processes being modified, extreme or infinitesimal effects of the modifier allele, or tight linkage between modifier and selected loci. Here, I prove the reduction principle in the absence of any of these constraints, confirming a twenty-year-old conjecture. The proof is obtained by a wider application of Karlin's Theorem 5.2 (Karlin in Evolutionary biology, vol. 14, pp. 61-204, Plenum, New York, 1982) and its extension to ML-matrices, substochastic matrices, and reducible matrices.

Physical limitations of work extraction from temporal correlations.

Recently proposed information-exploiting systems extract work from a single heat bath by using temporal correlations on an input tape. We study how enforcing time-continuous dynamics, which is necessary to ensure that the device is physically realizable, constrains possible designs and drastically diminishes efficiency. We show that these problems can be circumvented by means of applying an external, time-varying protocol, which turns the device from a "passive," free-running machine into an "actively" driven one.

Norm statement considered harmful: comment on 'evolution of unconditional dispersal in periodic environments'.

The mathematical symbol for the norm, which is heavily overloaded with multiple definitions that have both universal and specific properties, lends itself to confusion. This is manifest in the proof of an important theorem for population dynamics by Schreiber and Li on how dispersal increases population growth in a periodic environment. Here the theorem is placed in context, the proof is clarified, and the confusing but inconsequential errors corrected.

Simon-Ando decomposability and fitness landscapes.

In this paper, we investigate fitness landscapes (under point mutation and recombination) from the standpoint of whether the induced evolutionary dynamics have a "fast-slow" time scale associated with the differences in relaxation time between local quasi-equilibria and the global equilibrium. This dynamical hevavior has been formally described in the econometrics literature in terms of the spectral properties of the appropriate operator matrices by Simon and Ando (Econometrica 29 (1961) 111), and we use the relations they derive to ask which fitness functions and mutation/recombination operators satisfy these properties. It turns out that quite a wide range of landscapes satisfy the condition (at least trivially) under point mutation given a sufficiently low mutation rate, while the property appears to be difficult to satisfy under genetic recombination. In spite of the fact that Simon-Ando decomposability can be realized over fairly wide range of parameters, it imposes a number of restriction on which landscape partitionings are possible. For these reasons, the Simon-Ando formalism does not appear to be applicable to other forms of decomposition and aggregation of variables that are important in evolutionary systems.

Evolvability suppression to stabilize far-sighted adaptations.

The opportunistic character of adaptation through natural selection can lead to evolutionary pathologies--situations in which traits evolve that promote the extinction of the population. Such pathologies include imprudent predation and other forms of habitat overexploitation, or the tragedy of the commons, adaptation to temporally unreliable resources, cheating and other antisocial behavior, infectious pathogen carrier states, parthenogenesis, and cancer, an intraorganismal evolutionary pathology. It is known that hierarchical population dynamics can protect a population from invasion by pathological genes. Can it also alter the genotype so as to prevent the generation of such genes in the first place, that is, suppress the evolvability of evolutionary pathologies? A model is constructed in which one locus controls the expression of the pathological trait, and a series of modifier loci exist that can prevent the expression of this trait. It is found that multiple evolvability checkpoint genes can evolve to prevent the generation of variants that cause evolutionary pathologies. The consequences of this finding are discussed.

PERSPECTIVE: COMPLEX ADAPTATIONS AND THE EVOLUTION OF EVOLVABILITY.

The problem of complex adaptations is studied in two largely disconnected research traditions: evolutionary biology and evolutionary computer science. This paper summarizes the results from both areas and compares their implications. In evolutionary computer science it was found that the Darwinian process of mutation, recombination and selection is not universally effective in improving complex systems like computer programs or chip designs. For adaptation to occur, these systems must possess "evolvability," i.e., the ability of random variations to sometimes produce improvement. It was found that evolvability critically depends on the way genetic variation maps onto phenotypic variation, an issue known as the representation problem. The genotype-phenotype map determines the variability of characters, which is the propensity to vary. Variability needs to be distinguished from variations, which are the actually realized differences between individuals. The genotype-phenotype map is the common theme underlying such varied biological phenomena as genetic canalization, developmental constraints, biological versatility, developmental dissociability, and morphological integration. For evolutionary biology the representation problem has important implications: how is it that extant species acquired a genotype-phenotype map which allows improvement by mutation and selection? Is the genotype-phenotype map able to change in evolution? What are the selective forces, if any, that shape the genotype-phenotype map? We propose that the genotype-phenotype map can evolve by two main routes: epistatic mutations, or the creation of new genes. A common result for organismic design is modularity. By modularity we mean a genotype-phenotype map in which there are few pleiotropic effects among characters serving different functions, with pleiotropic effects falling mainly among characters that are part of a single functional complex. Such a design is expected to improve evolvability by limiting the interference between the adaptation of different functions. Several population genetic models are reviewed that are intended to explain the evolutionary origin of a modular design. While our current knowledge is insufficient to assess the plausibility of these models, they form the beginning of a framework for understanding the evolution of the genotype-phenotype map.