[Paleopsych] PLoS Computational Biology: Evolution of Genetic Potential

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Evolution of Genetic Potential
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Volume 1 | Issue 3 | AUGUST 2005

Research Article

Evolution of Genetic Potential

    Lauren Ancel Meyers^1,^2^*, Fredric D. Ancel^3, Michael Lachmann^4

    1 Section of Integrative Biology, Institute for Cellular and Molecular
    Biology, University of Texas, Austin, Texas, United States of America,
    2 Santa Fe Institute, Santa Fe, New Mexico, United States of America,
    3 Department of Mathematical Sciences, University of Wisconsin,
    Milwaukee, Wisconsin, United States of America, 4 Max Planck Institute
    for Evolutionary Anthropology, Leipzig, Germany

    Organisms employ a multitude of strategies to cope with the dynamical
    environments in which they live. Homeostasis and physiological
    plasticity buffer changes within the lifetime of an organism, while
    stochastic developmental programs and hypermutability track changes on
    longer timescales. An alternative long-term mechanism is "genetic
    potential"--a heightened sensitivity to the effects of mutation that
    facilitates rapid evolution to novel states. Using a transparent
    mathematical model, we illustrate the concept of genetic potential and
    show that as environmental variability decreases, the evolving
    population reaches three distinct steady state conditions: (1)
    organismal flexibility, (2) genetic potential, and (3) genetic
    robustness. As a specific example of this concept we examine
    fluctuating selection for hydrophobicity in a single amino acid. We
    see the same three stages, suggesting that environmental fluctuations
    can produce allele distributions that are distinct not only from those
    found under constant conditions, but also from the transient allele
    distributions that arise under isolated selective sweeps.

    Editor: Eddie Holmes, Pennsylvania State University, United States of

    Received: April 15, 2005; Accepted: July 22, 2005; Published: August
    26, 2005

    DOI: 10.1371/journal.pcbi.0010032

    Copyright: © 2005 Meyers et al. This is an open-access article
    distributed under the terms of the Creative Commons Attribution
    License, which permits unrestricted use, distribution, and
    reproduction in any medium, provided the original author and source
    are credited.

    Abbreviation: MHC, major histocompatibility

    *To whom correspondence should be addressed. E-mail:
    laurenmeyers at mail.utexas.edu

    Citation: Meyers LA, Ancel FD, Lachmann M (2005) Evolution of Genetic
    Potential. PLoS Comput Biol 1(3): e32


    Variation is the fuel of natural selection. Understanding the
    mutational processes that underlie evolution has long been a central
    objective of population genetics. Today, amidst a computational
    revolution in biology, such understanding is pivotal to progress in
    many biological disciplines. For example, neutral mutations make the
    molecular clock tick, and this clock is fundamental to reconstructing
    phylogenies, measuring recombination rates, and detecting genetic
    functionality. In this manuscript, the researchers provide an original
    perspective on a long-standing question in evolutionary biology: to
    what extent do mutation rates evolve? They argue that to cope with
    environmental fluctuation, populations can evolve their phenotypic
    mutation rate without changing their genetic mutation rate. That is,
    populations can evolve "genetic potential"--a heightened sensitivity
    to the effects of mutation. The researchers use a simple mathematical
    model of amino acid evolution to illustrate the evolution of genetic
    potential, and show that as environmental variability decreases,
    evolving populations reach three distinct states. In a rapidly
    fluctuating environment, organisms evolve the flexibility to cope with
    variation within an individual lifetime; in moderately variable
    environments, populations evolve the ability to evolve rapidly; and in
    fairly constant environments, populations evolve robustness against
    the adverse effects of mutation.


    Recent work in evolutionary biology has highlighted the degeneracy of
    the relationship between genes and traits [1]. For any particular
    trait value, there will exist a large set of genotypes that give rise
    to that value. A mutation from one such genotype to another will be
    neutral, having no noticeable impact on the physiology, behavior, or
    fitness of organisms. Metaphorically, one can imagine a population
    moving via mutation through a region of genotype space that maps to a
    neutral plateau in phenotype space. Near the periphery, mutations are
    likely to produce different (usually worse and occasionally better)
    phenotypes, whereas near the center of the neutral plateau, mutations
    have little impact on the phenotype. Evolutionary theory suggests that
    populations can harness this variation to achieve phenotypic stability
    under steady conditions through either mutational insensitivity [2,3]
    or mutational hypersensitivity [4], or to facilitate phenotypic
    exploration during adaptation [5,6].

    A separate body of evolutionary theory addresses adaptation under
    fluctuating conditions [7,8]. The rate of the fluctuations will
    influence the resulting response. If the environment changes rapidly
    relative to the average generation time, populations may evolve
    mechanisms such as physiological plasticity and learning by which
    individual organisms can respond to their conditions [9,10]. As
    environmental change slows down, viable strategies include stochastic
    or directed heterogeneity in developmental pathways that give rise to
    phenotypic variation on the order of once per generation [11]. For
    even slower rates of change, mutations may produce novel phenotypes at
    a sufficiently high rate. Hypermutable lineages can produce novelty
    every few generations, as has been observed in viruses and mutator
    strains of bacteria [12,13]. When environmental fluctuations are rare,
    populations may experience extended epochs of directional selection
    and thus have sufficient time to achieve genetic robustness for any
    given state. Immediately following an environmental shift, however,
    such populations may pass through transitional periods of
    within-individual or between-generation plasticity before completely
    losing the previously favored phenotype in favor of a currently
    favored phenotype. This evolutionary transformation--from a trait that
    is acquired through phenotypic plasticity to a genetically determined
    version of the same trait--is known as the Baldwin Effect [9,14].

    In this paper we show that genetic degeneracy may give rise to an
    alternative outcome under fluctuating conditions: the evolution of
    genotypes with heightened sensitivity to mutation. We introduce the
    term "genetic potential" to describe this state. Metaphorically,
    populations with genetic potential lie near the edge of neutral
    plateaus. Although the rate of mutation is unchanged, the likelihood
    that mutations produce beneficial variation increases. Heightened
    sensitivity to mutations has been recognized as a critical and
    transient phase of adaptive evolution [5,15,16]. Here we argue that
    genetic potential can be a stable condition for a population evolving
    under changing selection pressures. Using a simple mathematical model,
    we show that as environmental variability increases, natural selection
    at first moves populations between genetically robust states, then
    increasingly favors genetic potential, and ultimately produces
    mechanisms for environmental robustness within individual organisms.

    We then present a more biological example of this phenomenon using a
    model of amino acid evolution. There is evidence that, within viral
    pathogens, the physiochemical properties of amino acids found within
    epitopes--regions of proteins that directly interact with the host
    immune system--can rapidly evolve [17,18]. Likewise, highly evolvable
    codons have been identified in bacteriophage experiencing shifting
    hosts [19] and in enzymes experiencing shifting substrates [20].
    Motivated by these observations, we model codon evolution at a single
    amino acid site under fluctuating selection for hydrophobicity. As in
    the first model, natural selection produces three distinct outcomes
    with increasing environmental variability. Each outcome corresponds to
    distinct expectations about the distribution of amino acids and their
    codons at selected sites.

    Under infrequent environmental change, populations evolve from one
    mutationally robust phenotype to another, briefly passing through
    genotypes that can easily mutate to either state. One might therefore
    be tempted to equate genetic potential with confinement to the
    intermediate steps on a path from robustness for one phenotype to
    robustness for another (Figure 1). While this is true in our simple
    model, the codon model illustrates that fluctuating environments may
    drive populations towards significantly greater genetic potential than
    found during these transient stages of isolated selective sweeps.

Figure 1. Evolution of Genetic Potential

    The gray regions represent neutral networks--sets of genotypes that
    give rise to each phenotype. The degree of shading indicates the
    likelihood that mutations will impact phenotype, where darker regions
    are robust to mutations. Under constant conditions, populations evolve
    toward the most robust regions of neutral networks. Under variable
    conditions, populations may evolve toward genotypes that easily mutate
    from one phenotype to the other. These regions of genetic potential do
    not always lie on the evolutionary path between the equilibrium states
    for constant environments (arrow).


Description of Models

    The simple model. We consider the evolution of a trait in an
    environment that alternates between two states (E[A] and E[B]),
    spending exactly l generations per state between shifts. The simple
    model includes three phenotypes--one optimal phenotype for each of the
    two environments (A and B) and a third that has intermediate quality
    in both environments (V)--and a minimal amount of degeneracy in the
    relationship between the genotype and the phenotype. In particular,
    there is a single genetic locus, and five allelic possibilities at
    that locus (Figure 2A). Three of the alleles, g[0], g[1], and g[2],
    give rise to phenotype A, the fourth, g[3], gives rise to phenotype V,
    and the fifth, g[4], gives rise to phenotype B. The mutational
    structure is a pentagon in which g[i] can mutate to g[(i - 1) mod 5]
    or g[(i + 1) mod 5] for i [isin.gif] {0,1,2,3,4}.

Figure 2. Mutational Networks

    (A) Five alleles lie on a mutational pentagon with genetic degeneracy
    for the A phenotype. Colors indicate phenotypes with blue for A,
    yellow for B, and gray for V. Edges indicate that an allele on one
    side can mutate to the allele on the other side. Arrows illustrate the
    dynamics in equation 2.

    (B) Each vertex represents an amino acid. The size of the vertex
    indicates the number of codons coding for the amino acid. Edges
    indicate point mutations between hydrophobicity classes. Mutations
    that preserve hydrophobicity class, including those that preserve the
    amino acid, are included in the model but not depicted here. The color
    of the vertex corresponds to the hydrophobicity class: blue indicates
    hydrophobic, yellow indicates hydrophilic, green indicates
    intermediate, and red indicates stop codons [21]. This network was
    drawn with PAJEK [50].

    The fitness function changes with the environment such that

    where w[A] and w[B] are the fitnesses in environments E[A] and E[B],
    respectively, s > 0 is the fitness advantage for the specialized
    phenotype (A or B) in its preferred environment, and 0 =< k =< 1
    determines the intermediacy of the V phenotype.

    We can write the full model as a set of difference equations

    for i [isin.gif] {0,1,2,3,4}, where µ is the mutation rate and w[t]
    denotes the fitness in the current environment (Figure 2A). The number
    of individuals with genotype g[i] at time t is denoted by g[i,t]. The
    changing environment is governed by the following rule:

    To simplify the analysis, this model tracks changes in the absolute
    population sizes of the various genotypes rather than their relative
    frequencies. Since the dynamics scale linearly with the total
    population size, one can achieve the same population dynamics by
    replacing the absolute sizes with relative frequencies and normalizing

    Variations on the simple model. There are exactly 14 unique mutational
    networks consisting of five alleles on a pentagon, with at least one
    encoding A and at least one encoding B (see Materials and Methods).
    These include, for example, the pentagon with four consecutive alleles
    coding for A and one for B and the pentagon with alleles alternating
    in phenotype-A-B-A-V-B-. We are presenting analysis of the -A-A-A-V-B-
    model because it gives rise to some of the most interesting and
    generic dynamics found among these 14 models.

    The codon model. The previous model offers a transparent illustration
    of evolutionary dynamics under different rates of environmental
    change. Although somewhat simplistic, we believe that the qualitative
    predictions of the model will hold for a wide range of more plausible
    genotype-phenotype maps. To demonstrate this, we consider the
    evolution of a single amino acid site under fluctuating conditions. In
    this model, the genotypes are the 64 codons in the standard genetic
    code and the phenotypes are hydrophobicities of the corresponding
    amino acids [21]. The environment alternately favors hydrophobic and
    hydrophilic amino acids. There are three classes of amino
    acids--hydrophobic, intermediate, and hydrophilic--and all amino acids
    in a class share the same fitness. The fitnesses are determined as in
    equation 1, with the fitnesses of all three stop codons equal to zero.

    Each codon is mutationally connected to the nine others to which it
    can mutate via point mutation. This gives rise to the genetic network
    depicted in Figure 2B and the dynamics given by

    for 1 =< i =< 64, where µ is the overall mutation rate, b is the
    transition/transversion ratio (2b is the transition/transversion rate
    ratio), F[i] is the set of three transition point mutations of codon
    i, and G[i] is the set of six transversion point mutations of codon i.

Analysis of the Simple Model

    We provide an intuitive perspective on evolution in fluctuating
    environments using the simple model and then demonstrate the
    generality of the results in the codon model. The first results assume
    a mutation rate µ = 0.01, and fitnesses 1, 1.5, and 2 for the
    unfavored, intermediate, and favored phenotypes, respectively. In a
    constant environment, a population will equilibrate on genotypes that
    encode the optimal phenotype. In environment E[A], the equilibrium
    relative frequencies of g[0], g[1], g[2], g[3], and g[4] are 0.291,
    0.412, 0.292, 0.003, and 0.002, respectively, and in environment E[B],
    they are 0.005, 0.000, 0.000, 0.010, and 0.985, respectively. When
    there is degeneracy, as there is for phenotype A, the populations
    evolve genetic robustness, that is, more mutationally protected
    genotypes appear in higher frequency. In particular, g[1], which lies
    in the center of the three genotypes that code for A, appears in
    higher frequency than either genotype on the edge of the neutral
    network for A (g[0] and g[2]) at equilibrium in E[A]. In the absence
    of degeneracy (phenotype B), we observe a mutation-selection balance
    around the single optimal genotype. These findings are consistent with
    and provide a transparent example of the extensive theory on
    mutation-selection balance, quasi-species, and the evolution of
    genetic robustness in neutral networks [2,22-24].

    Under rapid environmental fluctuations, populations do not have time
    to reach a stable allele distribution. As the environment becomes more
    variable, the distributions of alleles go through three distinct
    phases. Figure 3 shows the frequency of every allele averaged over
    each environmental condition after the population has reached steady
    oscillations. For relatively stable environments, the populations
    swing back and forth between near equilibrium conditions for E[A] and
    E[B], thereby alternating between genetic robustness for A and a
    mutation-selection balance around the single allele for B. At
    intermediate rates of fluctuation, populations hover near g[4] and
    g[0], where the genotypes for A abut the genotype for B. Thus,
    mutation between the two phenotypes occurs frequently. We call this
    outcome genetic potential because of the enhanced potential for
    mutations to give rise to novel (beneficial) phenotypes. Finally, for
    highly variable environments, the populations converge on the
    phenotype V, which has unchanging, intermediate fitness in both
    environments. Phenotype V corresponds to organismal
    flexibility--individual organisms tolerate both conditions, but
    neither one exceptionally well. There are a variety of mechanisms that
    can give rise to an intermediate phenotype including homeostasis,
    somatic evolution, physiological plasticity, and behavioral plasticity
    [7,8]. As originally predicted by Dempster [25], the ascent of V under
    rapid fluctuations only occurs if the fitness of V is greater than the
    geometric mean fitness over time for either A or B.

Figure 3. Allele Distributions under Environmental Fluctuations

    The graphs show the stationary allele distributions averaged over an
    E[A] epoch (top) and an E[B] epoch (bottom) as a function of the
    variability of the environment. As environmental variability
    decreases, the population moves from the intermediate phenotype to the
    genetic boundary between the A and B phenotypes, and eventually to an
    oscillation between the center of the network for A and the gene for
    B. Diagrams above the graphs illustrate the frequency distributions in
    each of the three phases. Vertex areas are proportional to the average
    frequencies for each allele. (For the data depicted in this figure, s
    = 1, k = 0.5, and µ = 0.01.)

Anaylsis of the Codon Model

    The codon model gives rise to similar oscillations (Figure 4). Here we
    have assumed a transition/transversion ratio b = 2, mutation rate µ =
    10^ -5, and fitnesses 1, 1.5, and 2 for the unfavored, intermediate,
    and favored phenotypes, respectively. (We address the impact of
    mutation rate in the Discussion.) Whereas in the simple model only one
    of the three phenotypes had multiple genotypes, in this model all
    three phenotypic classes have genetic degeneracy, and thus can evolve
    genetic robustness (Figure 4A). For highly variable environments,
    codons for amino acids with intermediate hydrophobicity dominate, and
    in particular, those that are least likely to mutate to one of the
    other two classes (Figure 4B). In a moderately variable environment,
    the populations exhibit genetic potential, hovering near the edges of
    the neutral networks for the two extreme classes, thereby enabling
    rapid evolution upon environmental transitions (Figure 4C). In
    relatively constant environments, we find alternating genetic
    robustness for the two extreme classes (Figure 4D).

Figure 4. Codon Distributions under Environmental Fluctuations

    (A) gives the robustness for each codon, that is, the fraction of all
    possible point mutations that leave the hydrophobicity class
    unchanged. The codons have been ordered to reflect roughly the
    mutational adjacency of the hydrophobicity classes.

    (B-D) show the average codon frequency distribution for each epoch
    type after the population has reached stationary oscillation. These
    show frequencies for environmental epochs of exactly l generations
    (thick lines) and epochs of random duration--Poisson distributed with
    mean l (thin lines). Black corresponds to epochs favoring
    hydrophobicity and gray corresponds to epochs favoring hydrophilicity.
    The rate of environmental fluctuations is decreasing from (B) to (D)
    (l = 10, 10^2, and 10^6, respectively).

    The genetic potential of a population can be estimated by the
    probability that a currently favored codon in the population will
    mutate to a currently unfavored or intermediate codon. This indicates
    the capacity to bounce back (via mutation and selection) if and when
    the environment reverts. For populations that have equilibrated in a
    constant environment and have recently experienced an environmental
    shift, genetic potential will decrease as the population becomes
    increasingly robust to the effects of mutation (Figure 5). For
    populations that have evolved under moderately fluctuating conditions,
    genetic potential remains noticeably higher. This suggests that the
    regular oscillations of such populations involve distributions of
    codons that are quite different (more mutable) than those found during
    the early stages of adaptation in an isolated selective sweep.

Figure 5. Faster Environmental Fluctuations Yield Greater Genetic Potential

    Genetic potential is the likelihood that a mutation to a gene coding
    for the currently favored phenotype will produce the intermediate or
    unfavored phenotype. Thick lines correspond to populations that have
    reached stable oscillations when l = 100, and thin lines correspond to
    populations that experience a single environmental shift after having
    equilibrated in a constant environment. The maximum genetic potential
    after a single shift is significantly less than the minimum under
    persistent fluctuations.

    This difference also appears in the distributions of amino acids. We
    calculated the genetic potential in each generation of a population
    experiencing fluctuations every l = 10^2 generations. Figure 6 (left)
    depicts the amino acid distributions for the generations that have the
    highest genetic potential in E[A] and E[B]. We then compared these two
    distributions to the evolving amino acid distribution in a population
    that equilibrates in one of the two environments and then faces an
    environmental shift. Figure 6 (right) shows the steady state
    distributions for this population and the transitional distributions
    that are most similar (i.e., smallest average squared difference in
    relative frequencies) to those depicted in Figure 6 (left). The
    distributions of amino acids in regions of genetic potential are
    strikingly different than those realized in populations evolving after
    an isolated environmental shift.

Figure 6. Amino Acid Distributions Reflect Genetic Potential

    The left figure illustrates amino acid distribution in the generations
    with greatest genetic potential during each of the two epochs for l =
    100. Vertex area is proportional to the relative frequency of an amino
    acid. The right figure gives the amino acid distributions at
    equilibrium in the two environments (far left and right networks), and
    the transitional amino acid distributions that are most similar to
    those depicted for l = 100 (left). Similarity is measured as mean
    squared difference in frequencies across all amino acids. The amino
    acid networks were drawn with PAJEK [50].


    We have provided an intuitive framework for studying the evolutionary
    implications of heterogeneous environments. Although much is known
    independently about the evolution of genetic robustness [3] and
    organismal flexibility [7,8], this model demonstrates that the extent
    of environmental variability may determine which of these two states
    evolves, and suggests the possibility of an intermediate state of
    heightened mutability. The transition points among the three states
    will be functions of both the environment and the mutation rate. In
    particular, increasing (decreasing) the mutation rate (within a
    moderate range) has the same qualitative effect as increasing
    (decreasing) the duration of an environmental epoch. As the mutation
    rate decreases, populations take longer to achieve genetic robustness,
    and therefore evolve genetic potential (rather than robustness) over
    large ranges of environmental variability. For example, at a mutation
    rate of µ = 10^ -5 in the codon model, populations evolve genetic
    potential when environment varies at rates of 10^1 < l < 10^6
    generations, approximately (Figure 4). If the mutation rate increases
    to µ = 10^ -2, the qualitative results are similar, with populations
    evolving genetic potential when the environmental variability is in
    the more limited range of 10^0 < l < 10^3 generations, approximately.
    If, instead, the mutation rate decreases to µ = 10^ -9, then
    adaptation to genetic robustness proceeds at an exceedingly slow pace,
    yielding genetic potential throughout the extended range of 10^2 < l <
    10^10 generations, approximately. To understand the comparable roles
    of mutation and environmental variability, note that the model
    includes three time-dependent processes--mutation, environmental
    change, and population growth. If one of these rates is changed, the
    other two can be modified to achieve identical system behavior on a
    shifted time scale. Since the dynamics only weakly depend on the force
    of selection, we can change the mutation rate and then scale the rate
    of environmental change to produce the original qualitative results.
    The connection between environmental variability and mutation has been
    noted before, with theory predicting that the optimal mutation rate
    under fluctuating environmental conditions is µ = 1/l [26,27].

    Our results suggest an alternative perspective on the evolution of
    mutation rates. Theory suggests that the optimal mutation rate should
    correspond to the rate of environmental change [26,28], yet the extent
    to which mutation rate can evolve is unclear [12,13,29]. Here we
    suggest that the genotypic mutation rate need not evolve as long as
    the phenotypic or effective mutation rate evolves. By evolving toward
    genotypes with higher genetic potential, populations increase the rate
    of phenotypically consequential mutations without modifications to the
    underlying genetic mutational processes.

    We would like to emphasize that our second model is intended as one
    possible example of fluctuating selection among many thought to exist
    in nature. Whether or not one has much confidence in the particular
    evolutionary scenario, the qualitatively similar outcomes for the
    simple and complex models presented here suggest that the results may
    hold for a large class of systems in which there is redundancy in the
    relationship between genotype and phenotype. Hydrophobicity is just
    one of several physicochemical properties thought to play a role in
    the shifting functional demands on amino acids [17-20]. Another
    example is phase-shifting bacteria that have mutational mechanisms,
    for example, inversions in promoter regions [30] and slip-stranded
    mispairing within microsatellites [12], that lead to variation in
    functionally important phenotypes. The remarkable suitability of the
    phase-shifting variants to the diverse conditions experienced by the
    bacteria suggests that phase shifting may have evolved as a mechanism
    for genetic potential. We hypothesize that the major
    histocompatibility complex (MHC), which is the component of the immune
    system responsible for recognizing and binding foreign particles, may
    also have evolved genetic potential as a by-product of the flucuations
    arising out of coevolution with pathogens [31]. Studies suggest that
    several components of the immune system exhibit high overall rates of
    genetic change. In particular, there are specific amino acid sites
    within the MHC complex that seem to have experienced rapid
    evolutionary change [32]. One possible explanation is that each MHC
    gene as a whole, and these sites in particular, have a history of
    rapid adaptation to changing distributions of potential antigens. We
    therefore predict that such sites may have evolved genetic potential.

    Evolvability has been defined as a population's ability to respond to
    selection [6,33]. Although the term has only recently taken root,
    ideas concerning the evolution of evolvability itself date back to the
    Fisher-Wright debate over the evolution of dominance [34,35] and
    include the large body of theory on the evolution of mutation rates
    and recombination [36,37]. Developmental biologists have begun to
    identify genetic architectures that promote diversification [38] and
    buffering mechanisms, such as heat shock proteins, that allow the
    accumulation of cryptic variation [39]. Although one can think of
    genetic potential as an abstraction of all mechanisms that increase
    the likelihood that a mutation will have a phenotypic effect, the
    genetic potential that evolves in our models is a very simple form of
    evolvability that exploits redundancy in the map from genotype to

    Genetic potential evolves in our models because prior and future
    environments are identical. If, instead, the environment continually
    shifts to completely novel states, the evolutionary history of a
    population may not prepare it for future adaptation. We speculate that
    some degree of genetic potential may still evolve if there exist
    genotypes on the periphery of neutral networks with broad phenotypic

    Biologists often refer to phenotypic plasticity, learning, and other
    forms of organismal flexibility as "adaptations" for coping with
    environmental heterogeneity [7,8]. Should genetic potential be seen as
    an alternative "solution," or should it be viewed as simply a product
    of fluctuating selection? Although we remain agnostic, we note that
    this question might be asked of all forms of adaptive variation.
    Whether or not genetic potential should be viewed as an evolved
    strategy, we emphasize that it is not simply the truncation of the
    adaptive path a population follows from the equilibrium state in one
    constant environment to the equilibrium state in the other. In the
    codon model, intermediate rates of environmental fluctuations push the
    population into regions of the codon network where genetic potential
    is consistently higher than the regions of network through which a
    population crosses after an isolated environmental shift (Figures 1,
    5, and 6).

    A long-standing technique for identifying selected genes is to compare
    the frequencies of nonsynonymous and synonymous substitutions
    (K[a]/K[s]) [40]. Genes experiencing frequent selective sweeps should
    have relatively large amounts of variation in sites that modify amino
    acids. Such genes might be in the process of evolving a new function
    or, more likely, involved in an evolutionary arms race, for example,
    epitopes in human pathogens [31,41] or genes involved in sperm
    competition [42]. In the latter case, our model suggests that, in
    addition to an elevated K[a]/K[s], such genes should employ a distinct
    set of codons with high genetic potential. Note that this type of
    genetic potential is not equivalent to codon bias, but rather implies
    changes in the actual distribution of amino acids.

    A similar argument also underlies the recent use of codon
    distributions for detecting genetic loci under directional selection
    [43]. Codon volatility--the probability that a codon will mutate to a
    different amino acid class, relative to that probability for all
    codons in the same amino acid class--is a measure of genetic
    potential. Genes with significantly heightened volatility will be more
    sensitive to mutation. Our model suggests a different explanation for
    codon volatility than that presented in [43]: volatility may indicate
    a history of fluctuating selection rather than an isolated
    evolutionary event. If true, then we would not expect the codon
    distribution to reflect a transient out-of-equilibrium distribution as
    the population is moving from one constant environment to another
    [16]. Instead, we expect the distribution to reflect the stationary
    level of genetic potential that corresponds to variability in the
    selective environment for that gene. On a practical level, therefore,
    the isolated selective sweep model assumed in [43] may misestimate the
    expected volatility at such sites. Codon volatility, however, can
    arise as a by-product of processes other than positive (or
    fluctuating) selection. It has been noted that codon volatility may
    instead reflect selection for translation efficiency, relaxed negative
    selection, strong frequency-dependent selection, an abundance of
    repetitive DNA, or simple amino acid biases [44-48]. Therefore, the
    presence of codon volatility by itself may not be a reliable indicator
    of either recent directional selection or fluctuating selection.

    We would like to emphasize that the goal of this study was not to
    develop a new method for detecting positive (or fluctuating)
    selection, but rather to develop a theoretical framework for
    considering the multiple outcomes of evolution under fluctuating
    conditions. We conclude by suggesting an empirical method to identify
    loci that have evolved genetic potential under such conditions as
    distinct from those that have experienced a recent selective sweep.
    Suppose that a gene experiences fluctuations at a characteristic rate
    across many species. Furthermore, suppose that multiple sites within
    the gene are influenced by such fluctuations. For example, there may
    be fluctuating selection for molecular hydropathy, charge, size, or
    polarity, and several sites within the gene may contribute to these
    properties. Such sites should evolve in tandem and equilibrate on
    similar levels of genetic potential, and thus exhibit similar codon
    (and amino acid) distributions across species. In contrast, if a gene
    experiences isolated selective sweeps, then the variation at all sites
    should correspond to both the history of selective events and the
    species phylogeny, and the amino acid distributions at sites should
    correlate only when sites functionally mirror each other. Thus, one
    can seek evidence for the evolution of genetic potential as follows.
    First, identify genes that are rapidly evolving, perhaps by
    calculating K[a]/K[s] ratios. Such sites have been identified, for
    example, in human class I MHC genes, the HIV envelop gene, and a gene
    from a human T cell lymphotropic virus (HTLV-1) [31,32]. Within these
    genes, search for sites for which there is minimal correlation between
    the species tree and the amino acid distribution. Our model predicts
    that some of these sites should share similar distributions of amino
    acids across species.

Materials and Methods

Mathematical analysis of models.

    For the two models, we calculate the deterministic, infinite
    population allele frequency distributions in constant and fluctuating
    environments. Let M[A] and M[B] be the normalized transition matrices
    that govern changes in the allele frequencies in E[A] and E[B] epochs,
    respectively. The entries in these matrices are defined by equations 2
    and 4. The left leading eigenvectors for M[A] and M[B] give the
    equilibrium frequency distributions of alleles in each of the two
    constant environments, respectively. Under fluctuating conditions with
    epoch duration of l generations, we iteratively apply the matrices,
    and then compute the left leading eigenvector of . This vector, which
    we call v[B], gives the allele frequency distribution at the end of an
    E[A] epoch followed by an E[B] epoch.

    We are interested not only in the final allele distributions, but also
    in the dynamics throughout each epoch. Thus, we calculate the average
    frequency of each allele across a single E[A] epoch by

    where G is the total number of alleles in the model (G = 5 for the
    simple model and G = 64 for the codon model) and the subscript k
    indicates the kth entry in the vector. Similarly, the average
    distribution across an E[B] epoch is given by

    where v[A] is the allele frequency distribution at the end of an E[B]
    epoch followed by an E[A] epoch and is equal to the left leading
    eigenvalue of

    For the codon model, we compare these calculations that assume a
    regularly fluctuating environment to numerical simulations that assume
    a Poisson distribution of epoch lengths. In each generation of the
    simulations, the environmental state switches with probability 1/l and
    the codon frequencies are then multiplied by the appropriate
    transition matrix.

Proof of 14 unique pentagonal networks.

    We use an elementary group theoretic result known as Burnside's Lemma
    [49] to prove that there are 14 distinct mutational networks
    consisting of five alleles on a pentagon that map to the set of
    phenotypes {A, B, V} and contain at least one of each specialist
    phenotype (A and B) (Figure 7). We assume that all rotations and
    reflections of a network are equivalent to the original network, and
    that A and B are interchangeable. For example, the six networks with
    phenotypes -A-A-A-B-B-, -B-A-A-A-B-, -B-B-A-A-A-, -B-B-B-A-A-,
    -A-B-B-B-A-, and -A-A-B-B-B- are equivalent.

Figure 7. Pentagonal Mutational Networks

    These are the 14 possible pentagonal mutational networks consisting of
    five alleles producing phenotypes A, B, or V, with at least one
    encoding A and one encoding B.

    Let X be the set of all pentagons with vertices labeled {A, B, V}
    having at least one A vertex and at least one B vertex. The size of X
    is the number of all pentagons with labels {A, B, V} minus the number
    of pentagons with labels {A, V} or {B, V}, that is, |X| = 3^5 - (2 ·
    2^5 - 1) = 180.

    We define the group G of all actions on X that produce equivalent
    pentagons (as specified above). G is made up of (1) the identity, (2)
    the four rotations and five reflections of the pentagon, (3)
    interchanging all As and Bs, and (4) all the combinations of the above
    actions. Thus G is equal to the 20-member group {i, r, r^2, r^3, r^4,
    s[0], s[1], s[2], s[3], s[4], a, ar, ar^2, ar^3, ar^4, as[0], as[1],
    as[2], as[3], as[4]} where i is the identity, r is a single (72°)
    rotation, s[i] is a reflection through vertex i, and a is replacement
    of all As with Bs and all Bs with As. (Note that the reflections are
    rotations of each other, for example, r^2s[0] = s[1].)

    The number of distinct mutational networks is equal to the number of
    orbits of G on X. Burnside's Lemma tells us that this number is

    where F(g) = {x [isin.gif] X | gx = x} is the set of fixed points of
    g. For each of the twenty elements of G, we exhaustively count F(g).

    The identity fixes all elements of X, that is, F(i) = X. Each of the
    various rotations of a pentagon (through 72°, 144°, 216°, and 288°)
    has the property that its iterations move a given vertex to every
    other vertex of the pentagon without changing the letter assigned to
    that vertex. The same is true of the square of the product of any
    rotation and an A-B flip. Hence, any fixed point of one of these
    elements of the group G would necessarily have the same label at each
    vertex of the pentagon. Since every labeled pentagon in X has at least
    one A label and at least one B label, then no element of X has the
    same label at each vertex. Thus, the fixed point set of every rotation
    and of every product of a rotation and an A-B flip must be empty, that
    is, F(r^n) = F(ar^n) = &#58067; for all n. By a similar argument, the
    simple A-B flip also has no fixed points. Every reflection fixes 12
    elements of X, for example,

    and every product of a reflection and an A-B flip fixes eight elements
    of X, for example,

    In sum, all eight group elements that involve rotations fix no
    elements of X, all five reflections fix 12 elements of X, and all five
    combinations of a reflection and an A-B exchange fix eight elements of
    X. Thus,


    We thank Carl Bergstrom and Jim Bull for their valuable insights and
    comments on the manuscript.

    Competing interests. The authors have declared that no competing
    interests exist.

    Author contributions. LAM and ML conceived and designed the
    experiments. LAM performed the experiments. LAM, FDA, and ML analyzed
    the data and contributed reagents/materials/analysis tools. LAM and ML
    wrote the paper.


     1. Huynen MA, Stadler PF, Fontana W (1996) Smoothness within
        ruggedness: The role of neutrality in adaptation. Proc Natl Acad
        Sci U S A 93: 397-401. Find this article online
     2. van Nimwegen E, Crutchfield JP, Huynen MA (1999) Neutral evolution
        of mutational robustness. Proc Natl Acad Sci U S A 96: 9716-9720.
        Find this article online
     3. De Visser JAGM, Hermisson J, Wagner GP, Meyers LA, et al. (2003)
        Perspective: Evolution and detection of genetic robustness.
        Evolution 57: 1959-1972. Find this article online
     4. Krakauer DC, Plotkin JB (2002) Redundancy, antiredundancy, and the
        robustness of genomes. Proc Natl Acad Sci U S A 99: 1405-1409.
        Find this article online
     5. Ancel LW, Fontana W (2000) Plasticity, evolvability, and
        modularity in RNA. J Exp Zool 288: 242-283. Find this article
     6. Schlichting C, Murren C (2004) Evolvability and the raw materials
        for adaptation. In: Taylor I, editor. Plant adaptation: Molecular
        biology and ecology. Vancouver: NRC Canada Research Press. pp.
     7. Meyers LA, Bull JJ (2002) Fighting change with change: Adaptive
        variation in an uncertain world. Trends Ecol Evol 17: 551-557.
        Find this article online
     8. Schlichting CD, Pigliucci M (1998) Phenotypic evolution--A
        reaction norm perspective. Sunderland (Massachusetts): Sinauer
        Associates. 387 p.
     9. Ancel LW (1999) A quantitative model of the Simpson-Baldwin
        effect. J Theor Biol 196: 197-209. Find this article online
    10. Kawecki TJ (2000) The evolution of genetic canalization under
        fluctuating selection. Evolution 54: 1-12. Find this article
    11. Bull JJ (1987) Evolution of phenotypic variance. Evolution 41:
        303-315. Find this article online
    12. Moxon ER, Rainey PB, Nowak MA, Lenski RE (1994) Adaptive evolution
        of highly mutable loci in pathogenic bacteria. Curr Biol 4: 24-33.
        Find this article online
    13. Miller JH (1998) Mutators in Escherichia coli. Mutat Res 409:
        99-106. Find this article online
    14. Baldwin JM (1896) A new factor in evolution. Am Nat 30: 441-451.
        Find this article online
    15. Fontana W, Schuster P (1998) Continuity in evolution: On the
        nature of transitions. Science 280: 1451-1455. Find this article
    16. Plotkin J, Dushoff J, Deasai M, Fraser H (2004) Synonymous codon
        usage and selection on proteins. Arxiv.org E-Print Archives
        Available: http://arxiv.org/PS_cache/q-bio/pdf/0410/0410013.pdf.
        Accessed 3 August 2005.
    17. Yang W, Bielawski JP, Yang Z (2003) Widespread adaptive evolution
        in the human immunodeficiency virus type 1 genome. J Mol Evol 57:
        212-221. Find this article online
    18. Bush R, Bender C, Subbarao K, Cox N, Fitch W (1999) Predicting the
        evolution of human influenza A. Science 286: 1921-1925. Find this
        article online
    19. Crill WD, Wichman HA, Bull JJ (2000) Evolutionary reversals during
        viral adaptation to alternating hosts. Genetics 154: 27-37. Find
        this article online
    20. Matsumura I, Ellington AD (2001) In vitro evolution of
        beta-glucuronidase into a beta-galactosidase proceeds through
        non-specific intermediates. J Mol Biol 305: 331-339. Find this
        article online
    21. Kyte J, Doolittle RF (1982) A simple method for displaying the
        hydropathic character of a protein. J Mol Biol 157: 105-132. Find
        this article online
    22. Eigen M, McCaskill JS, Schuster P (1989) The molecular
        quasispecies. Adv Chem Phys 75: 149-263. Find this article online
    23. Wagner GP, Booth G, Bagheri-Chaichian H (1997) A population
        genetic theory of canalization. Evolution 51: 329-347. Find this
        article online
    24. Wagner A, Stadler PF (1999) Viral RNA and evolved mutational
        robustness. J Exp Zool 285: 119-127. Find this article online
    25. Dempster E (1955) Maintenance of genetic heterogeneity. Cold
        Spring Harb Symp Quant Biol 20: 25-32. Find this article online
    26. Lachmann M, Jablonka E (1996) The inheritance of phenotypes: An
        adaptation to fluctuating environments. J Theor Biol 181: 1-9.
        Find this article online
    27. Leigh EG (1973) The evolution of mutation rates. Genetics 73:
        1-18. Find this article online
    28. Meyers LA, Levin BR, Richardson AR, Stojiljkovic I (2003)
        Epidemiology, hypermutation, within-host evolution, and the
        virulence of Neisseria meningitidis. Proc R Soc Lond B Biol Sci
        270: 1667-1677. Find this article online
    29. Drake JW, Charlesworth B, Charlesworth D, Crow JF (1998) Rates of
        spontaneous mutation. Genetics 148: 1667-1686. Find this article
    30. Lederberg J, Iino T (1956) Phase variation in salmonella. Genetics
        41: 743-757. Find this article online
    31. Nielsen R, Yang Z (1998) Likelihood models for detecting
        positively selected amino acid sites and applications to the HIV-1
        envelope gene. Genetics 148: 929-936. Find this article online
    32. Yang Z, Wong WSW, Nielsen R (2005) Bayes empirical Bayes inference
        of amino acid sites under positive selection. Mol Biol Evol 22:
        1107-1118. Find this article online
    33. Wagner GP, Altenberg L (1996) Perspective: Complex adaptations and
        the evolution of evolvability. Evolution 50: 967-976. Find this
        article online
    34. Fisher RA (1922) On the dominance ratio. Proc R Soc Edinb 42:
        321-341. Find this article online
    35. Wright S (1934) Physiological and evolutionary theories of
        dominance. Am Nat 68: 24-53. Find this article online
    36. Sniegowski PD, Gerrish PJ, Johnson T, Shaver A (2000) The
        evolution of mutation rates: Separating causes from consequences.
        Bioessays 22: 1057-1066. Find this article online
    37. Feldman MW, Otto SP, Christiansen FB (1997) Population genetic
        perspectives on the evolution of recombinations. Annu Rev Genet
        30: 261-295. Find this article online
    38. Schlosser G, Wagner GP, editors. (2004) Modularity in development
        and evolution. Chicago: University of Chicago Press. 600 p.
    39. Rutherford SL, Lindquist S (1998) Hsp90 as a capacitor for
        morphological evolution. Nature 396: 336-342. Find this article
    40. Yang Z, Bielawski J (2000) Statistical methods for detecting
        molecular adaptation. Trends Ecol Evol 15: 496-503. Find this
        article online
    41. Endo T, Ikeo K, Gojobori T (1996) Large-scale search for genes on
        which positive selection may operate. Mol Biol Evol 13: 685-690.
        Find this article online
    42. Torgerson DG, Kulathinal RJ, Singh RS (2002) Mammalian sperm
        proteins are rapidly evolving: Evidence of positive selection in
        functionally diverse genes. Mol Biol Evol 19: 1973-1980. Find this
        article online
    43. Plotkin JB, Dushoff J, Fraser HB (2004) Detecting selection using
        a single genome sequence of M. tuberculosis and P. falciparum.
        Nature 428: 942-945. Find this article online
    44. Dagan T, Graur D (2005) The comparative method rules! Codon
        volatility cannot detect positive Darwinian selection using a
        single genome sequence. Mol Biol Evol 22: 496-500. Find this
        article online
    45. Hahn MW, Mezey JG, Begun DJ, Gillespie JH, Kern AD, et al. (2005)
        Evolutionary genomics: Codon bias and selection on single genomes.
        Nature 433: E5-E6. Find this article online
    46. Nielsen R, Hubisz MJ (2005) Evolutionary genomics: Detecting
        selection needs comparative data. Nature 433: E6. Find this
        article online
    47. Sharp PM (2005) Gene "volatility" is most unlikely to reveal
        adaptation. Mol Biol Evol 22: 807-809. Find this article online
    48. Zhang J (2005) On the evolution of codon volatility. Genetics 169:
        495-501. Find this article online
    49. Martin G (2001) Counting: The art of enumerative combinatorics.
        New York: Springer-Verlag.
    50. Batagelj V, Mrvar A (1998) PAJEK--Program for large network
        analysis. Connections 21: 47-57. Find this article online

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