Download Article A Model of Substitution Trajectories in

A Model of Substitution Trajectories in Sequence Space and
Long-Term Protein Evolution
Dinara R. Usmanova,1,2,3 Luca Ferretti,4,5 Inna S. Povolotskaya,2,3 Peter K. Vlasov,2,3 and
Fyodor A. Kondrashov*,2,3,6
1
Moscow Institute of Physics and Technology, Institutskiy Pereulok 9, g.Dolgoprudny, Russia
Bioinformatics and Genomics Programme, Centre for Genomic Regulation (CRG), Barcelona, Spain
3
Universitat Pompeu Fabra (UPF), Barcelona, Spain
4
Systematique, Adaptation et Evolution (UMR 7138), UPMC University Paris 06, CNRS, MNHN, IRD, Paris, France
5
CIRB, Collège de France, Paris, France
6
Institucio Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain
*Corresponding author: E-mail: [email protected].
Associate editor: Hideki Innan
2
Abstract
The nature of factors governing the tempo and mode of protein evolution is a fundamental issue in evolutionary biology.
Specifically, whether or not interactions between different sites, or epistasis, are important in directing the course of
evolution became one of the central questions. Several recent reports have scrutinized patterns of long-term protein
evolution claiming them to be compatible only with an epistatic fitness landscape. However, these claims have not yet
been substantiated with a formal model of protein evolution. Here, we formulate a simple covarion-like model of protein
evolution focusing on the rate at which the fitness impact of amino acids at a site changes with time. We then apply the
model to the data on convergent and divergent protein evolution to test whether or not the incorporation of epistatic
interactions is necessary to explain the data. We find that convergent evolution cannot be explained without the
incorporation of epistasis and the rate at which an amino acid state switches from being acceptable at a site to being
deleterious is faster than the rate of amino acid substitution. Specifically, for proteins that have persisted in modern
prokaryotic organisms since the last universal common ancestor for one amino acid substitution approximately ten
amino acid states switch from being accessible to being deleterious, or vice versa. Thus, molecular evolution can only be
perceived in the context of rapid turnover of which amino acids are available for evolution.
Key words: molecular evolution, fitness landscape, epistasis.
Introduction
Article
Whether or not epistasis, a situation when fitness is dependent on the interaction of alleles, plays a major role in molecular evolution is the subject of scrutiny and debate (de
Visser et al. 2011; Lehner 2011; Breen et al. 2012; Hansen
2013; McCandlish et al. 2013; de Visser and Krug 2014).
Two types of approaches are being used to reveal the type
and amount of epistasis in protein evolution: First, studies
aiming to reconstruct recent evolutionary trajectories revealing potential epistatic interactions among substitutions that
occurred recently (Weinreich et al. 2006; Bridgham et al. 2009;
Lozovsky et al. 2009; Romero and Arnold 2009; Lunzer et al.
2010; Khan et al. 2011; Zhang et al. 2012; Covert et al. 2013);
second, studies quantifying the degree of epistasis among
substitutions that occurred across long evolutionary periods
(Miyamoto and Fitch 1995; Huelsenbeck 2002; Kondrashov
et al. 2002; Choi et al. 2005; Bazykin et al. 2007; Wang et al.
2007; Rogozin et al. 2008; Rokas and Carroll 2008; Bollback
and Huelsenbeck 2009; Povolotskaya and Kondrashov 2010;
Gloor et al. 2010; Soylemez and Kondrashov 2012; Naumenko
et al. 2012; de Juan et al. 2013; Wellner et al. 2013).
Such studies are often statistical in nature and usually
cannot identify specific interactions, yet they provide a
broader outlook on the nature of the fitness landscape
across different areas of the sequence space. Many of these
studies claimed that different aspects of molecular evolution
are not compatible with evolutionary models devoid of epistatic interactions. However, the dynamics of protein sequence divergence has not been subject to modeling with
explicit parameters of epistasis.
Modeling protein (or DNA) sequence evolution is often
concerned with estimating the rate at which two sequences
diverge typically describing sequence divergence as a Markov
chain process. Initially, such models considered the neutral
divergence of DNA sequences. The most general neutral, siteindependent, general time-reversible (GTR) model (Tavare
1986), which was created following more restricted models
(Jukes and Cantor 1969; Kimura 1980; Felsenstein 1981),
allows different substitution rates for each nucleotide pair
(see O’Meara 2012 for review). Within the existing Markov
chain models, the probabilities of each site being occupied by
each of the four nucleotides are estimated across time using a
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Mol. Biol. Evol. 32(2):542–554 doi:10.1093/molbev/msu318 Advance Access publication November 17, 2014
Model of Substitution Trajectories . doi:10.1093/molbev/msu318
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FIG. 1. Three categories of Markov chain models of protein evolution. The general time reversal models estimate the probability that a site is occupied
by a specific nucleotide, Z. The probability of finding specific nucleotides at a site changes with time and the rate of change is described by a 4 4
matrix, R, because each of the four nucleotide can change into the other three nucleotides with a certain rate ri!j. The ri!j rates typically reflect the rate
of mutation and, therefore, Z½t þ ¼ Z½t eR models the neutral rate of change of nucleotides across sites. As selection influences the rate of
substitution in sites it is introduced as a parameter !, with Z½t þ ¼ Z½t e!R models. In that case ! 4 1 reflects the action of positive selection
and accelerates the rate of evolution and ! < 1 reflects negative selection slowing down the rate of change of Z. As the action of selection may be
different in different sites, some models attempt to capture the resulting rate variation across sites by assigning a different ! to different sites. The
covarion models reflect the possibility that the rate of evolution of a site is itself subject to change with time. They introduce extra parameters allowing
for sites to switch among the different ! categories.
4 4 matrix of nucleotide substitution rates (fig. 1). Amino
acid substitution models are analogous in a sense that a
20 20 matrix of amino acid substitution rates can be used
to estimate the probabilities of a site being occupied by each
of the 20 amino acids.
The first level of complication of these models arises from
the impact of selection on sequence divergence. When the
matrix of substitution rates reflects only the rate of mutation
the Markov chain models reflect neutral sequence divergence.
The typical way to introduce selection to these models is to
assume a multiplier of the mutation rate, !, which models
selection by slowing down the rate of substitutions at sites
under selection (fig. 1). For DNA sequence models the rate of
evolution can be estimated separately for synonymous and
nonsynonymous sites, with the matrix of nonsynonymous
substitution rates multiplied by !, which is the single parameter determining the strength of selection. Amino acid
sequence divergence models often use a precomputed
matrix of amino acid substitution rates, such as BLOSUM
or PAM (Dayhoff et al. 1978; Henikoff S and Henikoff JG
1992; Whelan and Goldman 2001).
The second level of complication of these models comes
from the realization that the strength of selection may be
different at different sites. A class of Markov chain models
with variable rate of evolution across sites has been created
(Nei and Gojobori 1986; Yang 1994) in which ! follows a
specific distribution, typically a Gamma distribution.
Practically, a discrete Gamma distribution is used and sites
are, therefore, classified into a number of categories with a
different rate multiplier !i. Within this approach some sites
may be completely invariable, in that case for those sites
!i = 0.
The final level of complication considers the possibility
that the strength of selection at a site changes with time.
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Usmanova et al. . doi:10.1093/molbev/msu318
Such covarion models (Fitch and Markowitz 1970) introduce
parameters, which describe the rate of change between
different substitution rate multipliers. The first such models
allowed a site to switch between invariable and variable
states (Tuffley and Steel 1998; Huelsenbeck 2002) the
second ones to switch between several !i categories
(Galtier 2001). Wang et al. (2007) combined both approaches
into a more general covarion model.
The models that incorporate selection do so by varying the
rate of evolution across different sites without varying the
probability of different substitution across sites. Biologically
speaking we know that the fitness impact of a L!C
substitution in one site may be radically different from that
of the L!C substitution at a different site. However,
assuming a different substitution matrix for different sites is
impractical as it overparametrizes the model. Therefore, for
the time being the assumption that sites differ from each
other only in their rate of evolution, which in a covarion
model is time-dependent, remains widely accepted. This
assumption may not interfere with the models ability to
reliably estimate the overall rate of evolution. However,
such models may not be appropriate when differences in
the rate of the same type of substitutions across different
sites contribute substantially to the sequence divergence
process.
A widely utilized verbal model of molecular evolution has
been formulated by Maynard Smith (1970), which compares
protein evolution to a word game, whereby two words (proteins) with meaning (function) are connected by a series of
one letter (amino acid) changes (substitutions) such that a
continuous pathway between these two words is created. The
example used by Maynard Smith is that of the word “WORD”
evolving into the word “GENE” through meaningful intermediates comprising a trajectory of substitutions in sequence
space: WORD$WORE$GORE$GONE$GENE. This trajectory reveals an important property of time-dependence, or
epistasis, of substitutions in different sites. For example, the
substitution of R!N at the third site is meaningful only after
D!E and W!G substitutions occurred in the fourth
and first sites, respectively. If R!N at the third site was
to be the first substitution in WORD, it would lead to the
sequence of WOND, which does not have meaning in the
English language. The existing Markov chain based models
may accurately estimate the rate of evolution of sequences
where such time-dependence is common. However, such
models cannot be used to study the time-dependence
itself. Here, we develop a mathematical model of long-term
protein evolution focusing on the evolutionary dynamics
of evolving sequences through sequence space in a similar
way to that described by Maynard Smith (1970). Instead
of focusing on the rate of amino acid substitution as the
estimated parameter of the model we focus on the
prevalence of epistatic interactions between amino acid
states as the parameter of interest. We then attempt to fit
our model to several recent observations of long-term
evolution. We do not investigate issues related to phylogeny
inference.
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Results
The Concept of the Model
Verbal Model
We take Maynard Smith’s (1970) analogy as the basis for our
model of sequence evolution, which investigates the impact
of interactions of sites across the protein sequence. An exhaustive description of all possible interactions is impossible
for a sequence even of moderate length (L) as it requires to
consider all 20L sequences in sequence space. This vast
number can be reduced by focusing on evolutionary trajectories within the sequence space. For the trajectory considered by Maynard Smith (1970) WORD!WORE!GORE!
GONE!GENE the entire sequence space, in English, is
264 = 456,976. However, to understand the local restrictions
imposed by the interactions of letters in different places
across the word it is sufficient to describe the fitness landscape one substitution away from each of the five words in
the trajectory (fig. 2). The current sequence as well as the
potential fitness impact of all single letter substitutions can be
shown in a matrix, which we call the fitness matrix. In the case
of a four letter word in English, a cell of the fitness matrix may
have three different states, A, B, or C. First, C, or the “current”
letter represents the current state of the letter in a specific
position. For example, the second letter of WORD is O and,
therefore, the (2,O) cell of the matrix has the state C. Second,
A stands for a letter that is currently “available” for evolution.
In the second letter of the word WORD, the (2,A) cell has the
state A because O!A substitution in the second letter creates the word WARD, which has meaning in English. Finally, a
substitution may be “blocked” (B), whereby if the substitution
were to occur it would not create a meaningful word, but this
letter at the same site may be present in another word. All
substitutions in WORD in the second site other than O!A
are of such type. For example, O!J substitution creates a
meaningless WJRD; however, the letter J at the second site can
be part of an actual word, AJAR and, therefore, the (2,J) cell is
occupied by state B (see table 1 for a list of all abbreviations).
Given a trajectory of substitutions in sequence space, it is
possible to track not only the substitutions that have occurred but also the associated changes in the fitness impact
of other substitutions. The first substitution in the given trajectory is the D!E substitution in the fourth letter. This is
reflected in the A!C switch of the cell (4,E) and the reciprocal C!A switch of the (4,D) cell. Thus, a C$A switch
represents one substitution. Furthermore, each substitution
may cause a number of A!B and B!A switches. In this case,
the D!E substitution at site 4 causes two B!A switches in
cells (2,E) and (2,I). This occurs because these substitutions,
O!E and O!I, now lead to meaningful words (WERE and
WIRE, respectively), whereas the same substitutions prior to
the D!E substitution did not lead to a meaningful combination of letters (WERD and WIRD). If we consider the second
substitution in the sequence W!G in the first letter (leading
to WORE!GORE), another B!A switch occurs in the cell
(2,Y), which leads to the word GYRE, and three A!B
switches at the same site in the cells (2,A), (2,E), and (2,I),
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FIG. 2. The fitness matrices of words encountered in the trajectory of substitutions WORD!GENE described by Maynard Smith (1970). The fitness
matrix of a specific sequence reflects both the current (C) sequence, with the state C in the corresponding cell of the matrix, as well as the fitness impact
of all possible single letter substitutions. For example, in the first word in the trajectory, “WORD” there are 16 available (A) substitutions, out of 100 total
possible ones, that would lead to another word in English (having high fitness). All other 84 states are blocked (B), meaning that if such a substitution
were to occur would not lead to a meaningful sequence of letters. A substitution that actually occurred in the trajectory is reflected by a bidirectional
C$A switch in two cells of the matrix. With every substitution the potential impact of other substitution also changes (changes between the current
and the previous fitness matrix are shown in orange).
which lead to the meaningless combinations of GARE, GERE,
and GIRE, respectively.
The fitness matrix approach to investigating the trajectory
of substitutions in the WORD$GENE analogy can be further
dissected to reveal various parallels with protein evolution.
Some letters (sites) tend to have more substitutions available
at any given moment (the first letter compared with the
second in the example), some substitutions can transition
rapidly between allowed and blocked, whereas others
switch between these states at a much slower rate, etc.
However, investigating many or long trajectories of substitutions using the full fitness matrix may still be troublesome,
although this has been done experimentally for a small
number of proteins with very short trajectories (McLaughlin
et al. 2012; Gong et al. 2013; Roscoe et al. 2013; Thyagarajan
and Bloom 2014). Therefore, for ease of mathematical treatment we follow site-independent models and assume that all
sites across the sequence have the same properties. Our
model aims to study the probability that a cell in the fitness
matrix is one of these three states (C, A, or B) and the rate of
A$B switches with every sequence substitution (A$C
switch). The GTR models investigate the rate of nucleotide
substitutions on average across the entire sequence without
tracking individual sites yet they can reconstruct the expected
sequence divergence between the evolving and the ancestral
sequence. Similarly, our model investigates the rate of
switches between different states of the fitness matrix cells
across the entire matrix without tracking individual sites and
retains the ability to determine the expected divergences between the evolving and the ancestral sequence.
Fitness Matrix
The protein fitness matrix has L columns and 20 rows, where
L is the length of the protein and 20 is the number of all
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Table 1. Abbreviations.
Possible states of cells in the fitness matrix
C
Current
An
Available and neighboring
Available and far
Af
Bn
Blocked and neighboring
Bf
Blocked and far
F
Forbidden
Constants
L
Length of protein sequence
m
Number of non-F amino acids per site
a
Fraction of A cells among A and B cells
c
Number of switches A$B occurring
with one amino acid substitution
d
Fraction of n cells among n and f cells
’
Number of n$f switches occurring
with one amino acid substitution
Rates of switches
sC!An
sAn!C
sAn !Bn ¼ sAf !Bf ¼ sA!B
sBn !An ¼ sBf !Af ¼ sB!A
sAn !Af ¼ sBn !Bf ¼ sn!f
sAf !An ¼ sBf !Bn ¼ sf !n
Markov process variables
Z = (ZC, ZAn ZAf, ZBn, ZBf )
Vector of probabilities that a cell is in
a particular state
R
Rate matrix of switches
Empirical observations (Povolotskaya and Kondrashov 2010)
D
Protein distance
U
Amino acid usage
Nt
Number of substitutions toward the
reference sequence
Number of substitutions away from the
Na
reference sequence
Rate of convergent evolution
Kc
Kd
Rate of divergent evolution
K4
Rate of synonymous evolution in
4-fold sites
amino acids. In addition to the three aforementioned states A,
B, and C we introduce another state F, or a “forbidden” amino
acid, which confers low fitness in all possible genetic backgrounds. In the WORD$GENE analogy, forbidden states do
not appear because for each letter of the alphabet at each of
the four sites there exists at least one word in English in which
that letter is used at the site in question. The fitness matrix as
we define it assumes a binary distribution of fitnesses, such
that a genotype can have a high, 1, or low, 0, fitness without
intermediate values (C and A correspond to high fitness
whereas B and F to low fitness).
Because we consider trajectories of amino acid substitutions we must also take into account properties of the genetic
table. Some amino acid substitutions may be available from a
fitness perspective; however, they cannot occur at the present
moment because more than one nucleotide substitution is
required on the DNA level. We, therefore, segregate the available (A) and blocked (B) cells into mutationally “neighboring”
546
FIG. 3. Switches between five states in the fitness matrix. The current
amino acid state can switch into an available amino acid that is one
nucleotide substitution away (C$An), which reflects one amino acid
substitution. With every C$An switch amino acid states that were
previously available to evolution become blocked (An/f!Bn/f ) and vice
versa, other amino acid states that were blocked become available
(Bn/f!An/f ). Furthermore, with every C$An switch ’ amino acid states
that were previously in the mutational neighborhood become unreachable with one nucleotide mutation (An!Af or Bn!Bf switches) and
vice versa (Af!An or Bf!Bn) switches. F never changes because it
reflects those amino acid states that can never be found in a protein
sequence.
(n) states, An and Bn, or nonneighboring states labeled with
the f subscript (for “far”), Af and Bf. Thus, we define six distinct
states: C, An, Af, Bn, Bf, F with An and Af forming set A whereas
Bn and Bf forming set B. Conversely, An and Bn form set n, all
amino acid states that can be reached by a single substitution
and Af and Bf form set f, those amino acid states that cannot.
Each cell of the fitness matrix has one of these six states and
the model considers the rate of switches among them (fig. 3).
Evolution and Switches between States in the Fitness Matrix
Three types of switches of the cell state are possible in the
fitness matrix. First, an amino acid substitution causes one cell
to switch from state C to state An and another cell to have the
opposite switch—from state An to state C. Second, in the
same site the sets of n-labeled and f-labeled cells also
change, with some amino acids that were previously more
than one nucleotide substitution away become neighboring
amino acids and vice versa. Third, with every substitution
some previously blocked amino acids became available and
some available amino acids became blocked. Thus, the dynamics of accumulating substitutions in a sequence can be
described with the change of the state of cells of the fitness
matrix with a total of ten different switches: C!An, An! C,
An!Af, Af!An, Bn!Bf, Bf!Bn, An!Bn, Bn!An, Af!Bf,
and Bf!Af (fig. 3).
Markov Process for States of a Single Cell in the Fitness
Matrix
We introduce an approach to analyze the state of a single cell
of the fitness matrix. We describe a non-F cell with a vector Z
that contains probabilities that the cell is in one of the five
possible states, Z = (ZC, ZAn, ZAf, ZBn, ZBf ). The probability a
state F of a cell is not included because non-F cells never
become F and vice versa. Vector Z changes with time because
states switch between each other. We measure the rate of the
switches si!j as the expected number of switches occurring
with every substitution in a site. Therefore, sC!An ¼ 1.
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Moreover, we consider the fitness impact of an amino acid
state to be an independent parameter from its mutational
neighborhood, that is, whether a site is in state A or B is
independent of whether or not it is n or f. Therefore,
sAn !Bn ¼ sAf !Bf ¼ sA!B , sBn !An ¼ sBf !Af ¼ sB!A , sAn !Af
¼ sBn !Bf ¼ sn!f , sAf !An ¼ sBf !Bn ¼ sf !n . The switches
form a Markov process with the rate matrix, R:
0
1
½Bf ½C
½An ½Af ½Bn B
C
B ½C
C
1
0
0
0
B
C
B
C
B ½An sAn !C
C
s
s
0
n!f
A!B
B
C
R¼B
C;
B ½Af 0
sf !n
0
sA!B C
B
C
B
C
B ½Bn 0
sB!A
0
sn!f C
@
A
½Bf 0
0
sB!A
sf !n
where diagonal entries are determined by the constraint that
rows sum to 0.
The transition probabilities between different states can
then be obtained by taking the exponent of the product of
the rate matrix and time measured as the number of substitutions per site, or eRt . The distribution of probabilities to
obtain a cell of a specific particular state at time t equals:
Z½t ¼ Z½0 eRt :
ð1Þ
Figure 4 shows an example of evaluation Z[t] for
Z[0] = (1,0,0,0,0) and for switching rates in the same value
range as that obtained for real proteins families (see next
section).
Constants
To estimate si!j rates in the rate matrix, several constants are
necessary. First, we introduce m, the average number of non-F
cells per column. Biologically m is the number of amino acids
per site that can confer nonzero fitness in at least one genetic
background. Second, we introduce , the fraction of cells with
state A among all cells with states A and B. Third, we introduce , the number of switches A$B that happen with one
amino acid substitution. The and constants have an affinity in that describes the relative amount of A and B states
and describes the rate of change between them. Two more
constants are estimated from the genetic code. The fraction
of cells with state n among all cells with state n and f, = 0.4
calculated considering two amino acids as neighbors if any of
their codons are separated by one nucleotide substitution.
The number of switches n$f, ’0 = 3.3, also calculated as an
average across all codons for all 20 amino acids. The and ’
constants are related in that stands for the number
of neighboring amino acids whereas ’ stands for how
rapidly this set of neighboring amino acids changes.
However, we are interested only in switches between non-F
states. Thus, we define the number of possible n$f switches
per site as
’ ¼ ’0
m1
:
20 1
FIG. 4. Numerical evaluation of components of Z[t]. The initial condition is Z[0] = (1,0,0,0,0) with constants = 0.06, = 5, m = 7.3.
ð2Þ
The constant 1 appears in the numerator and denominator because at each site one amino acid is C, which is neither n nor f.
Using these constants we calculate the number of cells
with different states in one column of the fitness matrix
and the stationary probabilities of non-F states to which components of vector Z converge independently of the initial
state (table 2). We then derive rates of switches as the
number of switches divided by the number of cells from
which a switch could have occurred:
sAn !C ¼
sA!B ¼
sB!A ¼
;
¼
NAn þ NAf ðm 1Þ
;
¼
NBn þ NBf ð1 Þðm 1Þ
sn!f ¼
sf !n ¼
1
1
;
¼
NAn ðm 1Þ
’
’
;
¼
NAn þ NBn ðm 1Þ
’
’
:
¼
NAf þ NBf ð1 Þðm 1Þ
Thus, our model has five constants, with two of them, and ’, characterizing the genetic code whereas the other
three, m, and , characterize the fitness landscape. To estimate realistic ranges of m, , and , we use empirical data on
evolution of protein sequences.
Fitting Observations to Model
Our model is aimed at testing recent statements regarding
the epistatic nature of long-term protein evolution.
Specifically, the model is used to fit previously obtained
data on protein sequence divergence from Povolotskaya
and Kondrashov (2010) and Breen et al. (2012). We thus
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Table 2. Number of Cells with Various States in the Fitness Matrix.
Nj, number of cells with state j in one column of
fitness matrix
Z1
j , equilibrium frequency of state j for non-F cell
C
1
An
adðm 1Þ
Af
að1 dÞðm 1Þ
Bn
ð1 aÞdðm 1Þ
Bf
ð1 aÞð1 dÞðm 1Þ
1
m
ad m1
m
að1 dÞ m1
m
ð1 aÞd m1
m
ð1 aÞð1 dÞ m1
m
provide a brief description of the previously published data
that we investigate here with the present model.
Polarization of amino acid substitutions with one or more
outgroup sequences reveals the directionality of evolution.
For example, if one sequence has an Alanine (A) and another
sequence in the orthologous site contains a Threonine (T)
and closely related outgroup sequences also contain a
Threonine then an T!A substitution is inferred. Such polarized substitutions can then be related to a fourth, reference,
sequence when the reference sequence is an outgroup to all
three sequences involved in the polarization. Following our
example, if the orthologous site in the reference sequence is
occupied by a T then the T!A substitution can be inferred
to be a substitution away from the reference sequence.
Conversely, if the orthologous site in the reference sequence
is occupied by an A then the T!A substitution can be inferred to be a substitution toward the reference sequence.
The ratio of the sum of all toward substitutions (Nt) and the
sum of all away substitutions (Na), Nt/Na, can then be taken as
a measure of the relative rate of divergence of the sister sequences and the reference sequence. The observation that
sister and reference sequences that have already diverged
considerably continue to do so, that is, Nt/Na < 1 for large
values of sequence divergence (protein distance, D), has been
claimed to be a consequence of epistatic interactions between sites in the evolving sequences (Povolotskaya and
Kondrashov 2010).
The rate of divergence of sequences from each other can
be estimated in an analogous manner to the Nt/Na measure
by deconstructing Nt/Na into two independent rates of evolution—the rate of divergent evolution (Kd) and the rate of
convergent evolution (Kc). The rate of divergent evolution is
estimated as Na divided by the number of sites in which an
away substitution could have occurred. The estimate of the
number of such divergent sites is equal to the number of sites
in which the ancestral state of the two sister sequences
matches the reference sequence. Similarly, Kc is Nt divided
by the number of convergent sites, in which a toward substitution could have occurred. The number of sites that are
occupied by a different amino acid in the ancestor of the
sister sequences and the reference sequence and are separated with only a single nucleotide substitution is the target
number of convergent sites. Kd was estimated to be substantially slower than the rate of synonymous divergence in 4-fold
sites (K4), independently of D (Povolotskaya and Kondrashov
2010). Alternatively, Kc is comparable to K4 when D is near 0
and rapidly declines as D increases reaching a plateau slightly
above Kd (fig. 5). The dependence of Kc/K4 but not Kd/K4 on
D, and similar observations (Kondrashov et al. 2002; Choi et al.
2005; Rogozin et al. 2008; Rokas and Carroll 2008; Bollback
548
F
ð20 mÞ
—
FIG. 5. Relative rate of protein evolution. Kc/K4 is shown by and Kd/K4
by w (from Povolotskaya and Kondrashov 2010). We fit the observed
Kc/K4 to that calculated by the solution of equation (4) varying as a
parameter. The optimal fit was found for ~ 5 (thick solid line). Two
near fits for = 4 and = 6 are depicted with thin solid lines. Thick
dashed line shows Kc/K4 for significantly higher and thick dotted line
for significantly lower values of .
and Huelsenbeck 2009; Gloor et al. 2010; Naumenko et al.
2012; Soylemez and Kondrashov 2012), has been claimed to
show support for epistasis in protein evolution but has not
been modeled.
The final observation that at present requires more formal
modeling deals with amino acid usage (U), the number of
different amino acids in an orthologous site. In a large multiple sequence alignment, U ~ 9 (Breen et al. 2012), or approximately half of all possible amino acids. However, the same
proteins exhibit a short-term rate of nonsynonymous evolution (Kn) up to an order of magnitude lower than Kn ~ 0.5
that is expected if an amino acid state has the same effect in
all species. The reason why amino acids are accepted in the
long-term yet rejected in the short term may be due to epistatic interactions being common (Fitch and Markowitz
1970; Maynard Smith 1970; Povolotskaya and Kondrashov
2010; Breen et al. 2012); however, this parameter has also
not been put into a formal context of a macroevolutionary
model.
We obtained quadruplet alignments from Povolotskaya
and Kondrashov (2010) where each quadruplet alignment
consisted of two sister sequences, one outgroup sequence
and one reference sequence. Such quadruplet alignments
were available for 572 clusters of orthologous groups
(COGs), functional gene families that were predicted to
have been present in the last universal common ancestor,
LUCA (Mirkin et al. 2003). Reference sequence from the alignments corresponds to the sequence at t = 0 in our model.
Model of Substitution Trajectories . doi:10.1093/molbev/msu318
MBE
Distance
As a sequence diverges from the sequence it was at t = 0, we
can estimate the protein distance between them as a function
of time. The protein distance (D) between two sequences is
defined as 1 minus the sequence identity. The sequence identity between the sequence at time t and at t = 0 equals the
probability that a cell that was in state C at t = 0 is at state C at
time t, that is, ZC[t] when Z[0] = (1,0,0,0,0). This gives:
agreement with Kc/K4 from Povolotskaya and Kondrashov
(2010). Second, we explore Kc/K4 [t] when t!1. From
table 2:
D½t ¼ 1 ZC ½t j Z½0¼ð1;0;0;0;0Þ :
ð3Þ
We then consider the value of D at equilibrium, at t = 1 as
In
Povolotskaya
and
D1 ¼ 1 ZC 1 ¼ 1 1=m.
Kondrashov (2010), the distance limit has been estimated
as the time when Nt/Na = 1. In principle, we can use D1 to
estimate one of the constants, the number of non-F amino
acids m, as
m¼
1
:
1 D1
ð4Þ
From Povolotskaya and Kondrashov (2010) the divergence
equilibrium (Nt/Na = 1) was estimated as D1 &0:90 0:95.
However, from equation (4) it is clear that when D1 is large a
difference of just 5% leads to a large scatter of m with
m = 10 20. Thus, although it appears that m is likely to be
high, consistent with a large U observed by Breen et al. (2012),
this approach is not suitable for an accurate estimation of m.
Sequence Divergence and Convergence
The ratio Kd/K4 relates the rate of divergence of nonsynonymous substitutions to the rate of 4-fold synonymous
evolution. In terms of our model, a nucleotide substitution
leading to amino acid with state A fixes with the same probability as a synonymous substitution. Thus, Kd/K4 equals to
the proportion of substitutions that lead to an amino acid
with state A, in other words it is the ratio of the number of
cells with state A and the number of cells with all non-C
states. Therefore,
Kd
NA
m1
:
¼
¼ 19
K4 20 NC
ð5Þ
The ratio Kc/K4 relates the rate of convergent amino acid
substitutions to the 4-fold synonymous rate of evolution. Kc
measures the rate of substitution toward the reference
sequence, or toward cells in the fitness matrix which were
C at t = 0. Amino acids with state A fix with the same probability as synonymous substitutions. But when calculating Kc
only amino acids in the mutational neighborhood are taken
into account. Thus, Kc/K4 equals the probability that a cell
that was in state C at time t = 0 is in state An at time t divided
by the probability that a cell is in state n:
Kc
ZAn ½t
½t ¼
:
ZAn ½t þ ZBn ½t j Z½0¼ð1;0;0;0;0Þ
K4
ð6Þ
We then explore equation (6) as t!0. At t = 0 ZAn = 0 and
ZBn = 0 whereas Kc/K4 is not defined, because there are no
convergent sites. Applying L’H^opital’s rule, we find
dZAn ½0 dZAn ½0
dZBn ½0
Kc
K4 ½t ! 0 ¼ dt =ð dt þ dt Þ ¼ 1. That is in a good
Kc 1
Z1
½D ¼ 1 An 1 ¼ :
K4
ZAn þ ZBn
ð7Þ
From data we estimate Kc/K4!0.06, which implies
= 0.06. Kd/K4 is approximately constant and equals 0.02,
substituting it and into equation (5) we get m = 7.3.
From equation (2), we calculate ’ = 1.1.
Estimating the Degree of Epistasis in Protein Evolution
In the previous sections, we estimated values for all constants
used in the model except for , which reflects the amount of
epistasis in evolution. To obtain it, we fit the empirically obtained Kc/K4 from Povolotskaya and Kondrashov (2010) to
the estimated Kc/K4[D] that we obtain from our model. We
evaluate equation (1) numerically varying as a parameter.
For a given , we use the functions ZC(t), ZAn(t), ZAf(t), ZBn(t),
and ZBf(t) obtained from equation (1) to calculate Kc/K4[t]
in equation (7) and D[t] in equation (3). We repeat this
process varying the parameter until we obtain a good
fit between the observed and predicted Kc/K4 across D
(fig. 5). Mathematically speaking we minimize relative errors
of fit:
2
ND 1 X
Kc =K4 emp ðDi Þ Kc =K4 theor ðDi Þ
min
; ð8Þ
ND
Kc =K4 theor ðDi Þ
i¼1
where ND is the number of data points for Kc/K4[D]. ¼ 5
1 provides the best fit (fig. 5).
Evolution Relative to a Reference Sequence, Nt/Na
As we have estimated all parameters of the model, we can
now estimate Nt/Na:
Nt Nsites
Nsubst
¼ tsites tsubst ;
Na Na
Na
ð9Þ
where Nsites
tðaÞ is amount of convergent (divergent) sites in protein, and Nsubst
tðaÞ defines how many substitutions from one
convergent (divergent) site toward (away) the reference are
possible. Convergent sites are those from which a single nucleotide substitution can lead to reference amino acids that is
currently available. Thus, Nsites
t ~ZAn . For every convergent site,
only one substitution can lead toward the reference amino
¼ 1. Sites with the amino acid which matches
acid: Nsubst
t
subst
reference are divergent, so Nsites
equals the average
a ~ZC . Na
number of available and neighboring amino acids per column
¼ NAn ¼ ðm 1Þ .
in the fitness matrix: Nsubst
a
Therefore, we get:
Nt
ZAn ½t
½t ¼
ZC ½t ðm 1Þ Na
¼
Kc
ZAn ½t þ ZBn ½t
½t :
19 ZC ½t j Z½0¼ð1;0;0;0;0Þ
Kd
ð10Þ
Equation (10) allows us to convert Kc =Kd into Nt =Na and
vice versa. We calculate Nt =Na using rates of divergent and
549
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Usmanova et al. . doi:10.1093/molbev/msu318
kind, S(N,n). There are m!/(m n)! ways to associate these
subsets with n different colors:
P½n ¼
SðN; nÞ
m!
; usage ¼ EðnÞ
mN ðm nÞ!
ð11Þ
We estimate m = 9 (see next section) as the mean value of
m distribution among different COGs (fig. 7) and using equation (11) calculate the dependence of usage on N (fig. 8). Due
to the high value, the amino acid usage is considerable even
for N ~ 10, number of substitutions per site, with usage almost
approaching its maximal value with 30 substitutions per site.
FIG. 6. Observed and predicted relative rates of sequence divergence.
The observed values of Nt/Na shown by and the predicted fit with our
model using optimal parameters shown with w.
convergent evolution from Povolotskaya and Kondrashov
(2010) and estimate Z from equation (1) when Z[0] =
(1,0,0,0,0) and with the five estimated constants. Figure 6
shows the comparison of data (fig. 3 from Povolotskaya
and Kondrashov 2010) and the estimate obtained with our
model.
Usage
NA = (m 1) = 0.06(7.3 1) = 0.38 indicates how many
amino acid substitutions are available at a given moment.
Thus, on average less than 1 amino acid substitution per
site is acceptable at a time. However, when longer time periods are taken into account, up to eight different amino acids
may be found at the same site across different species (Breen
et al. 2013). These two observations can be reconciled if , the
number of A$B switches associated with one C$An switch,
is relatively high. For the data from Povolotskaya and
Kondrashov (2010), our model predicts &5. Such a high implies that per site there are five times more switches between the available and blocked states of amino acids than
there are actual amino acid substitutions at the same site. It
follows that the distribution of available amino acids changes
substantially even for a very small number of substitutions in
that site. Therefore, we can model all non-F amino acids (m)
as having an equal probability to occur at a site after a substitution and the usage expected to be observed after a certain amount of sequence divergence is a simple combinatorial
problem.
Let N be the number of substitutions observed per site
with every substitution being a random choice from m.
Reformulating the problem, we have a pool of objects of m
different colors, we then take N random objects from the pool
and calculate n, the number of different colors out of N objects taken from the pool. The value of n varies across trials
but we consider amino acid usage to be equal expectation of
n distribution. The probability to get n different colors in set
of N objects is given by the number of possible sets with n
different colors divided by the total number of possible sets,
which is mN. The number of ways to partition N objects into n
nonempty subsets equals the Stirling number of the second
550
Considering the Distribution of Parameters across Gene
Families
In the previous sections, we used data which were obtained
by taking average values across all 572 COGs (Povolotskaya
and Kondrashov 2010). Here, we use data on the same variables that were obtained for each COG separately. For some
protein families, there are not enough data to calculate Kc/K4
and Kd/K4 with an acceptable degree of precision. Therefore,
we selected 119 COGs for which Kd/K4 and Kc/K4 were defined across distances 0.1–0.8 without strong outliers. For
each COG, we obtained Kd/K4 as an average Kd/K4 across
different D between the trios and the reference sequence in
bins of 0.1 of D. We presume that is a limit of Kc/K4 when
D!1. Then, we calculate m using equation (5) and ’ using
equation (2). Finally, we fit Kc/K4 with numerical evaluation of
equation (1) and find . Distributions of these parameters for
the 119 COGs are shown on figure 7.
Discussion
The model presented here makes three assumptions. The first
assumption is that fitness is either 0 or 1. Binary fitness landscapes, while unrealistic in certain situations, are nevertheless
useful when recapitulating the multidimensionality of the
sequence space (Gavrilets 1997; Gavrilets and Gravner 1997;
van Nimwegen et al. 1999; Aita et al. 2003; Gravner et al.
2007). When fitness values are either 1 or 0 mutations are
either neutral or lethal and all permitted substitutions have
an equal probability of occurrence. Thus, the impact of
slightly deleterious and beneficial substitutions on molecular
evolution cannot be taken into account, even though their
contribution may be considerable (Ohta 1998; Andolfatto
2005; Popadin et al. 2007; McCandlish et al. 2013). The
reason for excluding the effects of such substitutions is 2fold. First, the practical considerations of modeling a complex
and multidimensional fitness landscape in which substitutions are not equal in effect. The nonepistatic impact of
slightly deleterious and beneficial substitutions on evolution
in the absence of epistasis is well characterized (Crow and
Kimura 1970), whereas the effects of epistasis on long-term
molecular evolution have not been subject to the same level
of scrutiny. Our work develops a null model that may be
expanded to incorporate these effects. Second, it appears
certain that neither the fixation of slightly deleterious or beneficial alleles can serve as the basis for explaining the measurements considered here, at least without the impact of
epistasis. The fixation of slightly deleterious alleles is
Model of Substitution Trajectories . doi:10.1093/molbev/msu318
MBE
FIG. 7. Distribution of estimated parameters for 119 COGs. The distribution of the number of nonforbidden amino acids per site (m), proportion of
available amino acids over all available and blocked states (), and the rate of A$B switches () are shown.
FIG. 8. Estimating amino acid usage. Usage calculated as the most probable number of amino acids to be observed at a site as a function of the
number of accumulated substitutions per site. The solid line represents
the number of nonforbidden amino acids at a site (m).
compatible with a large U under some circumstances
(McCandlish et al. 2013). However, slightly deleterious alleles
alone cannot lead to large sequence divergence in fast-evolving proteins and are incompatible with reaching maximum
sequence divergence distances substantially slower than the
rate of divergence of neutral sequences (Kondrashov et al.
2010). There is a tradeoff between the contribution of slightly
deleterious alleles toward amino acid usage and sequence
divergence; it is not possible to have a high U and a high D
at the same time solely due to the contribution of fixation of
deleterious alleles (Breen et al. 2013). Furthermore, periodic
fixation of nonepistatic slightly deleterious alleles cannot lead
to a decline in the rate of convergent evolution, in contrast to
the observed Kc/K4 relationship (Povolotskaya and
Kondrashov 2010). Therefore, slightly deleterious alleles may
contribute to molecular evolution but a model based on the
accumulation of slightly deleterious alleles alone cannot explain all of the available data (also see Kondrashov et al. 2010).
A similar argument is applicable to the impact of beneficial
alleles on observations of molecular evolution considered
here. Large sequence divergence observed between sequences
is compatible with most differences between sequences
having been fixed by positive selection. However, the high
values of Kc/K4, especially for small D, are incompatible with
sequence divergence being largely driven by the accumulation
of beneficial alleles.
The second assumption is that all sites in the protein are
expected to have the same properties (described by , , ’, m,
and ). This includes genetic properties, such as the number
of mutational neighbors at each site, or the properties of the
protein, such as the number of available, blocked, and forbidden amino acids at a site. Indeed, it may be possible to alleviate this assumption by introducing a variation of our model
in a manner similar to that has been done for models of the
rate of protein evolution (Nei and Gojobori 1986; Yang 1994).
We hope that the present work will be a stepping stone in this
direction. The third assumption is that all of these five parameters are independent of each other. This may not be
entirely the case, for example amino acids in the mutational
neighborhood of each other are more likely to be available
(Freeland et al. 2000).
Our model is not unique in modeling interactions between
different alleles to describe the fitness landscape. However,
the model presented here differs substantially from models
developed previously. Previous efforts were principally concerned with the issue of the shape of the fitness landscape
551
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Usmanova et al. . doi:10.1093/molbev/msu318
(Kauffman and Levin 1987; Macken and Perelson 1989;
Gavrilets 1997; Gavrilets and Gravner 1997; Ohta 1998;
Kondrashov FA and Kondrashov AS 2001; Gravner et al.
2007; Ferrada and Wagner 2010; Crona et al. 2013;
Lobkovsky et al. 2013). Our model, on the other hand,
considers patterns of long-term divergent and convergent
molecular evolution as a factor of the epistatic interactions
arising on a multidimensional fitness landscape. Our model
shares the same conceptual basis as covarion models
(Fitch and Markowitz 1970; Tuffley and Steel 1998; Galtier
2001; Huelsenbeck 2002), which allow the rate of evolution
at a site to change over time. Our model approaches the
same issue but from a perspective that, in our opinion,
more accurately reflects the biological basis for interaction
between sites.
Our analysis has revealed several important aspects of protein evolution. The crucial aspect of our model is that we can
find a limited parameter space that is consistent with all of
the available observations of protein evolution. The three
estimated parameters that reveal the nature of the fitness
landscape are the number of amino acids that in principle
can confer nonzero fitness (m), the fraction of them which
allowed for substitution in one moment (), and the number
of switches between amino acids in available and blocked
states per a single amino acid substitution ().
We estimate an m between 5 and 15 for most protein
families (fig. 7), which indicates that few states are forbidden,
that is, correspond to low fitness regardless of the amino acid
composition of other sites. Here, a modest number of universally forbidden amino acids implies that most amino acid sites
can accept most of the possible amino acids given the right
combination of states at other sites (Fitch and Markowitz
1970; Maynard Smith 1970; Povolotskaya and Kondrashov
2010). Conversely, this implies that a single amino acid site
can accept many different amino acids, consistent with the
previous observations of a high amino acid usage (Breen et al.
2012).
We estimate = 0.06, the fraction of available amino acids
at a given time as just a small fraction of all amino acid states.
Therefore, at any given time many sites cannot accept any
substitutions. This estimate is consistent with the observation
of slow rate of sequence divergence of sequences in our data
set, with Kd/K4 ~ 0.02 and with mutational studies (Guo et al.
2004). Evidently, for faster evolving protein families is likely
to be higher, however, as we considered old gene families,
which tend to be conservative, our estimate is unlikely to be
representative of the entire diversity of protein rates of evolution found across all taxa.
Within the confines of our model, it is not possible to
maintain = 0 while maintaining a fit to all of the observed
parameters of protein evolution. Specifically, appears to be
the key parameter that allows the model to fit the observation of the decline in the rate of convergent evolution, Kc/K4
(fig. 5). Because the model is scaled to the number of amino
acid substitutions that occur in a protein the = 5 estimate
cannot be taken as a direct indication that epistatic interactions are intragenic. Individual proteins do not evolve in isolation and cases of intergenic epistatic interaction have been
552
documented (Lehner 2011). Indeed, our model does not
imply a causative interaction between amino acid substitutions and switches between available and blocked states of
amino acids in the fitness matrix. Indeed, a general interpretation of = 5 is that on average five switches between
available and blocked states occur on the same timeframe
as a single amino acid substitution. These switches may
either be a consequence of changes in the same protein or
in the rest of the genome. If most interactions are intragenic
then the best fit of = 5 implies that the fitness matrix
of a protein is changing faster than its sequence
implying that the inherently epistatic nature of the fitness
landscape is an inseparable and defining factor of molecular
evolution.
Our model can be used to analyze the patterns of molecular evolution of specific gene families, as was done here
(fig. 7). We observed that all parameters, including , vary
substantially across different gene families, implying that the
nature of the fitness landscape is a feature that may differ
across protein families. The application of this model to a
broader set of proteins than considered here can lead the
way toward a more general characterization of fitness landscapes and evolutionary trajectories in nature.
Acknowledgments
The work was supported by grants from Agence Nationale de
la Recherche (ANR-12-JSV7-0007), HHMI International Early
Career Scientist Program (55007424), the EMBO Young
Investigator Programme, MINECO (BFU2012-31329 and
Sev-2012-0208),
and
an
ERC
Starting
Grant
(335980_EinME). All authors participated in the design of
the model. D.R.U. and L.F. performed the mathematical analysis. D.R.U and I.S.P. obtained and analyzed data on protein
evolution. D.R.U. and F.A.K. wrote the draft.
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