Robustness and epistasis in mutation-selection models.

Abstract

We investigate the fitness advantage associated with the robustness of a phenotype against deleterious mutations using deterministic mutation-selection models of a quasispecies type equipped with a mesa-shaped fitness landscape. We obtain analytic results for the robustness effect which become exact in the limit of infinite sequence length. Thereby, we are able to clarify a seeming contradiction between recent rigorous work and an earlier heuristic treatment based on mapping to a Schrödinger equation. We exploit the quantum mechanical analogy to calculate a correction term for finite sequence lengths and verify our analytic results by numerical studies. In addition, we investigate the occurrence of an error threshold for a general class of epistatic landscapes and show that diminishing epistasis is a necessary but not sufficient condition for error threshold behaviour.

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arXiv:0901.0663v1 [q-bio.PE] 6 Jan 2009
Robustness and epistasis in mutation-selection
models
Andrea Wolff and Joachim Krug
Institut f¨ ur Theoretische Physik, Universit¨ at zu K¨ oln, Z¨ ulpicher Str. 77, 50937 K¨ oln,
Germany
E-mail: awolff@thp.uni-koeln.de and krug@thp.uni-koeln.de
Abstract.
phenotype against deleterious mutations using deterministic mutation-selection models
of quasispecies type equipped with a mesa shaped fitness landscape. We obtain analytic
results for the robustness effect which become exact in the limit of infinite sequence
length. Thereby, we are able to clarify a seeming contradiction between recent rigorous
work and an earlier heuristic treatment based on a mapping to a Schr¨ odinger equation.
We exploit the quantum mechanical analogy to calculate a correction term for finite
sequence lengths and verify our analytic results by numerical studies. In addition, we
investigate the occurrence of an error threshold for a general class of epistatic landscape
and show that diminishing epistasis is a necessary but not sufficient condition for error
threshold behavior.
We investigate the fitness advantage associated with the robustness of a
1. Introduction
In current evolutionary theory, the concept of robustness, referring to the invariance
of the phenotype under pertubations, is of central importance [1, 2]. Here we address
specifically mutational robustness, which we take to imply the stability of some biological
function with respect to mutations away from the optimal genotype. To be precise,
suppose the genotype is encoded by a sequence of length L, and the number of
mismatches with respect to the optimal genotype is denoted by k. Robustness is then
quantified by the maximum number of mismatches k0, that can be tolerated before the
fitness of the individual falls significantly below that of the optimal genotype at k = 0.
This situation arises e.g. in the evolution of regulatory motifs, where the fitness is a
function of the binding affinity to the regulatory protein [3, 4]. Assuming that the fitness
is independent of k both for k ≤ k0and for k > k0, the fitness landscape has the shape
of a mesa parametrized by its width k0and height w0[5].
We consider deterministic mutation-selection models of quasispecies type, which
describe the dynamics of large (effectively infinite) populations [6]. We analyse the
stationary state of mutation-selection balance, focusing on the dependence of the
population fitness on the parameters k0and w0. This allows us to identify the conditions
under which a broad fitness peak of relatively low selective advantage outcompetes a

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Robustness and epistasis in mutation-selection models
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higher but narrower peak [7], a phenomenon that has been referred to as the survival
of the flattest [8, 9, 10, 11, 12]. Our central aim is to obtain analytic results for the
robustness effect that become exact in the limit of long sequences. In particular, we
want to clarify whether the selective advantage is a function primarily of the relative
number of tolerable mismatches x0= k0/L, or of the total number of mismatches k0.
Robustness in the sense described above is a special case of epistasis, which refers
more generally to any nonlinear relationship between the number of mutations away
from the optimal genotype and the corresponding fitness effect [13]. A simple way to
parametrize epistasis is to let the loss of fitness vary with the number of mismatches
as kα, such that the non-epistatic case α = 1 separates regimes of synergistic (α > 1)
and diminishing (α < 1) epistasis [14, 15]. An important problem in previous work
on mutation-selection models has been to identify the conditions under which epistatic
fitness landscapes display an error threshold, a term that refers to the discontinuous
delocalization of the population from the vicinity of the fitness peak as the mutation
rate is increased beyond a critical value [6, 16]. Improving on earlier work that found
that only landscapes with diminishing epistasis (α < 1) have an error threshold, we
derive here the more stringent condition α ≤ 1/2 on the epistasis exponent.
1.1. Organization of the article
We base our work on two complementary analytic approaches. First, recent progress
in the theory of mutation-selection models [5, 16, 17, 18, 19, 20, 21, 22] provides
an expression for the population fitness in terms of a maximum principle (MP) that
becomes exact when the limit L → ∞ is performed keeping the ratio x0= k0/L fixed.
Second, Gerland and Hwa (GH) [3] have used a drift-diffusion approximation to map
the mutation-selection problem onto a one-dimensional Schr¨ odinger equation which is
then analyzed with standard techniques.
Our work was initially motivated by the observation of a discrepancy between the
two approaches: Whereas the MP predicts that the selective advantage of a broad mesa
should vanish when the limit L → ∞ is taken at fixed k0, in the GH approach a finite
selective advantage is retained in this limit, which depends on the absolute value of k0
rather than on x0. After introducing the model and briefly reviewing the results of the
MP approach in section 2, we therefore provide a detailed discussion of the drift-diffusion
approximation used by GH in section 3. We emphasize that it amounts to a harmonic
approximation, and show how it can be improved in such a way that the results based
on the MP are recovered.
The mapping to one-dimensional quantum mechanics is nevertheless useful, as it
allows us to derive the leading finite size correction to the population fitness. As a
consequence we find excellent agreement between the analytic predictions and numerical
solutions of the discrete mutation-selection equations. In section 4 we consider the
selection transition in a two-peak landscape first studied by Schuster and Swetina
[7, 23], in which the population shifts from a high, narrow fitness maximum to a lower

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Robustness and epistasis in mutation-selection models
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but broader peak with increasing mutation rate.
is an indicator for the superiority of robustness over fitness in certain parameter
regimes.In section 5 we use the MP approach to derive the critical value of the
epistasis exponent α and verify our prediction by numerical calculations.
some conclusions are presented in section 6. Details of the derivation of the improved
continuum approximation and the generalization to arbitrary alphabet size can be found
in two appendices.
The occurrence of this transition
Finally,
2. Mutation-selection models and the maximum principle
We consider the simplest case of binary sequences and adopt continuous time dynamics
of the Crow-Kimura type, in which the mutation and selection terms act in parallel
[6]. Point mutations occur at rate µ, and the (Malthusian) fitness is assumed from the
outset to be a function wkonly of the Hamming distance k to the optimal sequence at
k = 0. The population structure is described by the fraction Pk(t) of individuals with
k mismatches, which satisfies the evolution equation
dPk
dt
with 1 ≤ k ≤ L − 1 and obvious modifications for k = 0 and k = L. The nonlinearity
introduced by the mean fitness ¯ w(t) =
unnormalized population variables [6, 24]. At long times the population distribution
therefore approaches the principal eigenvector P∗
solution of the eigenvalue problem
= (wk− ¯ w)Pk+ µ(k + 1)Pk+1+ µ(L − k + 1)Pk−1− µLPk.(1)
?
kwkPk can be eliminated by passing to
kof the linear dynamics, which is the
ΛP∗
k= (wk− µL)P∗
k+ µ(k + 1)P∗
k+1+ µ(L − k + 1)P∗
k−1
(2)
with the maximal eigenvalue Λ.
the mean population fitness ¯ w, and it is the main quantity of interest in this paper.
Depending on the context we will refer to Λ as the mean population fitness, the
population growth rate, the principal eigenvalue of the mutation-selection matrix defined
by (2) or the ground state energy of the corresponding quantum mechanical problem,
to be defined in subsection 3.2.
A considerable body of work has been devoted to the solution of (2) for large L. In
order to obtain nontrivial behavior in the limit L → ∞, it is necessary to either scale
the mutation rate ∼ 1/L or the fitness ∼ L. We adopt here the first choice and take
L → ∞, µ → 0 with γ = µL fixed. If, in addition, the fitness landscape wkis assumed
to depend only on the relative number of mismatches, such that
This eigenvalue is equal to the long-time limit of
wk= f(x),x = k/L(3)
the principal eigenvalue in (2) is given, for L → ∞, by the solution of a one-dimensional
variational problem as [16, 17, 18, 19, 20, 21]
?
Λ = max
x∈[0,1]{f(x) − γ[1 − 2
x(1 − x)]};(4)

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Robustness and epistasis in mutation-selection models
4
see subsection 3.5 and Appendix A for a heuristic derivation, and Appendix B for the
generalization to arbitrary alphabet size. Moreover, if f(x) is differentiable the leading
order correction to (4) takes the form [19, 20]
γ
2L?xc− x2
where xcis the value at which the maximum in (4) is attained.
In the first part of this paper we focus on mesa landscapes of the form
?
0:k > k0,
∆Λ =
c
[1 −
?
1 − 2f′′(x∗)(xc− x2
c)3/2/γ], (5)
wk=
w0> 0:0 ≤ k ≤ k0
(6)
where w0 is the selective advantage of the functional phenotype and k0 denotes the
number of tolerable mismatches. Within the class of scaling landscapes (3), this is
realized by setting
f(x) = w0θ(x − x0),(7)
where θ is the Heaviside step function and x0= k0/L. Provided x0< 1/2, application
of the maximum principle (4) yields
?
0
The value wc
the population delocalizes from the fitness peak and the location xcof the maximum in
(4) jumps from xc= x0to xc= 1/2.
The expression (5) is clearly not applicable to the discontinuous mesa landscape
(7). In fact we will show below that the leading order correction ∆Λ is of order L−2/3
or L−1/2rather than L−1in this case.
Λ =
w0− γ(1 − 2?x0(1 − x0))
0of the selective advantage marks the location of the error threshold at which
:
:
w0> wc
w0< wc
0= γ(1 − 2?x0(1 − x0))
0.
(8)
3. Continuum limit and the drift-diffusion equation
3.1. Derivation and status
A natural approach to analyzing (1) and (2) for large L is to perform a continuum limit
in the index k. To this end we introduce ǫ = 1/L as small parameter and replace the
population variable Pkby a function
φ(x) = lim
L→∞PxL.(9)
The fitness is taken to be of the general form (3). Expanding the finite differences in
(2) to second order in ǫ then yields the stationary drift-diffusion equation
fφ − ǫγd
dx(1 − 2x)φ +ǫ2γ
2
d2
dx2φ = Λφ.(10)
This is identical to the equation obtained by GH [3], who however write it in terms of
the unscaled variable k = Lx.

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Robustness and epistasis in mutation-selection models
5
Before proceeding with the analysis of (10), some remarks concerning the accuracy
of the second order expansion are appropriate. In the absence of selection (f = 0) the
principal eigenvalue in (10) is readily seen to be Λ = 0, and the corresponding (right)
eigenfunction is a Gaussian centered at x = 1/2,
φ0(x) ∼ exp[−(1 − 2x)2/2ǫ].(11)
This is just the central limit approximation to the binomial distribution
?L
which solves (2) for wk = 0 and Λ = 0.
approximation of (12) is accurate in a region of size
imprecise for deviations of order L. An improved approximation is provided by the
theory of large deviations [25], in which the ansatz
P0
k= 2−L
k
?
(12)
It is well known that the central limit
√L around k = L/2, but becomes
Pk∼ exp[−Lu(x)](13)
is made to obtain an expression for the large deviation function u(x). In the context of
mutation-selection models, this approach has recently been introduced by Saakian [20],
who showed that it allows to derive the exact relation (4) in a relatively straightforward
manner (see Appendix A). Equivalent results can be obtained by continuing the
expansion in (10) to all orders in ǫ and treating the resulting equation in a WKB-type
approximation, which essentially corresponds to the ansatz (13), see [22].
We conclude that the drift-diffusion approximation (10) can be expected to be
quantitatively accurate only near the center x = 1/2 of the sequence space. We will
nevertheless adhere to this approximation in following three subsections, because it
allows us to make contact with the work of GH and to formulate the eigenvalue problem
(2) in the familiar language of one-dimensional quantum mechanics. In subsection 3.5
we then show how to go beyond the second order approximation.
3.2. Mapping to a one-dimensional quantum problem
The key step in reducing (10) to standard form is to symmetrize the linear operator on
the left hand side, thus eliminating the first-order drift term. This can be achieved by
the transformation
?
with φ0(x) from (11), which leads to the stationary Schr¨ odinger equation
−ǫ2γ
2
with the effective potential
V (x) =γ
2(1 − 2x)2− f(x).
The latter consists of the superposition of a harmonic oscillator centered around x = 1/2
with the (negative) fitness landscape. As pointed out in [5], the inverse sequence length
φ(x) =φ0(x)ψ(x),(14)
d2
dx2ψ + V (x)ψ = −(Λ − ǫγ)ψ(15)
(16)