[Paleopsych] PLoS Computational Biology: Evolution of Genetic Potential
Premise Checker
checker at panix.com
Thu Sep 1 00:27:18 UTC 2005
Evolution of Genetic Potential
http://compbiol.plosjournals.org/perlserv/?request=get-document&doi=10.1371/journal.pcbi.0010032
[Links omitted.]
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
America
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
Synopsis
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.
Introduction
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.
thumbnail
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).
Results
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}.
thumbnail
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
appropriately.
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).
thumbnail
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.
thumbnail
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.
thumbnail
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].
Discussion
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
phenotype.
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
lability.
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.
thumbnail
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) =  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,
Acknowledgments
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.
References
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
online
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.
18-29.
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
online
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
online
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
online
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
online
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
More information about the paleopsych
mailing list