Monotonicity of Fitness Landscapes and Mutation Rate Control^{1}^{1}1This work was supported by EPSRC grant EP/H031936/1
Abstract
The typical view in evolutionary biology is that mutation rates are minimised. Contrary to that view, studies in combinatorial optimisation and search have shown a clear advantage of using variable mutation rates as a control parameter to optimise the performance of evolutionary algorithms. Ronald Fisher’s work is the basis of much biological theory in this area. He used Euclidean geometry of continuous, infinite phenotypic spaces to study the relation between mutation size and expected fitness of the offspring. Here we develop a general theory of optimal mutation rate control that is based on the alternative geometry of discrete and finite spaces of DNA sequences. We define the monotonic properties of fitness landscapes, which allows us to relate fitness to the topology of genotypes and mutation size. First, we consider the case of a perfectly monotonic fitness landscape, in which the optimal mutation rate control functions can be derived exactly or approximately depending on additional constraints of the problem. Then we consider the general case of nonmonotonic landscapes. We use the ideas of local and weak monotonicity to show that optimal mutation rate control functions exist in any such landscape and that they resemble control functions in a monotonic landscape at least in some neighbourhood of a fitness maximum. Generally, optimal mutation rates increase when fitness decreases, and the increase of mutation rate is more rapid in landscapes that are less monotonic (more rugged). We demonstrate these relationships by obtaining and analysing approximately optimal mutation rate control functions in 115 complete landscapes of binding scores between DNA sequences and transcription factors. We discuss the relevance of these findings to living organisms, including the phenomenon of stressinduced mutagenesis.
keywords:
Adaptation, Fitness landscape, Mutation rate, Population Genetics, Phenotypic PlasticityContents
1 Introduction
Mutation is one of the most important biological processes that influence evolutionary dynamics. During replication mutation leads to a loss of information between the offspring and its parent, but it also allows the offspring to acquire new features. These features are likely to be deleterious, but have the potential to be beneficial for adaptation. Thus mutation can be seen as a process of innovation, which is particularly important as the number of all living organisms is tiny relative to the number of all possible organisms. A question that naturally arises with regards to mutation is whether there is an optimal balance between the amount of information lost and potential fitness gained.
The seminal mathematical work to investigate biological mutation is by Ronald Fisher Fisher30, who considered mutation as a random motion in Euclidean space, the points of which are vectors representing collections of phenotypic traits of organisms. Using the geometry of Euclidean space, Fisher showed that probability of adaptation decreases exponentially as a function of mutation size (defined using the ratio of mutation radius and distance to the optimum), and concluded therefore that adaptation is more likely to occur by small mutations. Several studies, however, suggested that large mutations can be quite frequent in nature, thereby prompting reexamination of the theory Orr05. Thus, Kimura Kimura80 extended the theory to take into account differences in probabilities of fixation for mutations of small and large size. Subsequently Orr Orr98 considered the effect of mutation across several replications. Interestingly, while he had a critical role in developing mathematical theory around discrete alleles, Fisher in his geometric model uses Euclidean space, which is uncountably infinite and unbounded. That this is an important issue became apparent only after the realisation that biological evolution occurs in a countable or even finite space of discrete molecular sequences Smith70. However, subsequent geometric models based on Fisher’s, while they have explicitly modelled discrete mutational steps (e.g. Orr02), continue to assume that they occur within the same infinite Euclidean space. This issue may contribute to the fact that the predictions of such models have at best only been partially verified in actual biological systems McDonald11; Bataillon11; Kassen06; Rokyta08. One of the contributions of the current work is that we consider mutation using the geometry of other spaces, and in particular the geometry of a Hamming space, which is finite and leads to a radically different view about the role of large mutations.
Mutation size as considered by Fisher is closely related to mutation frequency measured in biology in terms of the number of mutations per replication per DNA base. Mutation rates in biology vary over several orders of magnitude Drake98. Nonetheless, mutation rate for any particular species is typically believed to be minimised, within bounds set by physiology Drake91, or more likely population genetics Lynch10. Despite this, mutation rates are known to vary within and among populations of a single species Bjedov03 and recently, populationgenetic models have been developed proposing that variable mutation rates may be in fact adaptive in biology Ram12.
Independent of such biological concerns, researchers in evolutionary computation and operations research have a longer history of considering variable mutation rates in genetic algorithms (GAs) (e.g. see Eiben_etal99; Ochoa02; Falco_etal02; CervantesStephens06; Vafaee_etal10 for reviews). In particular, Ackley suggested in Ackley87 that mutation probability is analogous to temperature in simulated annealing, which decreases with time through optimisation. A gradual reduction of mutation rate was also proposed by Fogarty Fogarty89. In a pioneering work, Yanagiya Yanagiya93 used Markov chain analysis of GAs to show that a sequence of optimal mutation rates maximising the probability of obtaining global solution exists in any problem. A significant contribution to the field was made by Thomas Bäck Back93, who studied the probability of adaptation in the space of binary sequences and suggested that mutation rate should depend on fitness values rather than time. More recently, numerical methods have been used to optimise a mutation operator Vafaee_etal10 that was based on the Markov chain model of GA by Nix and Vose NixVose92. The complexity of this model, however, restricted the application of this method to small spaces and populations. It is these insights regarding mutation rate variation from evolutionary computation and operations research which we develop here towards the particular issues presented by biological systems.
We develop theory in the following directions:

Generalise Fisher’s geometric model of adaptation for metric spaces, and in particular for discrete spaces of sequences, such as the Hamming spaces with arbitrary alphabets.

Define problems of optimal mutation rate control within such spaces, and study how different problem formulations (e.g. time horizon, objective function) affect the solutions.

Extend the theory to more biologically realistic (i.e. rugged) fitness landscapes.
Some relevant results have already been reported. For example, results for general Hamming spaces were first reported in Belavkin_etal11:_ecal11; Belavkin11:_itw11. We develop these results towards biology in Section 2. Various optimisation problems were considered in Belavkin11:_dyninf; Belavkin11:_qbic11, deriving theoretical optimal mutation rate control functions. We address how such control functions may also be obtained numerically in Section LABEL:sec:metaga. In Section LABEL:sec:monotonic, we develop theory to consider a fitness landscape as a memoryless communication channel between fitness values and distance from an optimal sequence. We introduce the ideas of local and weak monotonicity of a landscape. This allows us to formulate hypotheses about monotonicity and mutation rate control in biological fitness landscapes. We test these hypotheses by numerically obtaining optimal mutation rate control functions for 115 published complete landscapes of transcription factor binding Badis09. Our results presented in Section LABEL:sec:TFlandscapes show that all the optimal mutation rate control functions in these biological landscapes do indeed converge to nontrivial forms consistent with the theory developed here. We also observe differences among optimal mutation rate control functions, variation that relates to variation in the landscapes’ monotonic properties. We conclude in Section LABEL:sec:discussion by discussing how mutation rate control as considered here may be manifested in living organisms.
2 A Generalisation of Fisher’s Geometric Model of Adaptation
In this section, we consider an abstract problem, in which organisms are viewed as points in some metric space and adaptation as a motion in this space towards some target point (an optimal organism). In such formulation, maximisation of biological fitness corresponds to a minimisation of distance to the target, and geometry of the metric space allows us to solve the optimisation problem precisely. These abstract results will be used in the following sections to develop the theory further bringing it closer to biology.
2.1 Representation and assumptions
Let be a set of all possible organisms. Environment defines a preference relation on (a total preorder), so that means is better adapted to or has a higher replication rate in a particular environment than . Throughout this paper we shall consider only the case of countable or even finite , although the theory can be easily extended with certain care to the uncountable case. It is wellknown from game theory (e.g. NeumannMorgenstern) that in the countable case the preference relation always has a utility representation: there exists a real function such that if and only if . In the biological context, the utility function is called fitness, and it is usually defined to have nonnegative values (i.e. if is the replication rate of ). Having positive fitness values is not essential, because the preference relation does not change under a strictly increasing transformation of , such as adding a constant to or multiplying it by a positive number (i.e. representation is equivalent to for any and ). Thus, our interpretation of fitness simply as a numerical representation of a preference relation on organisms is distinct from population genetic definitions of fitness (e.g. see Orr09). We shall assume also that there exists a top (optimal) element such that , which is the most adapted and quickly replicating specie in the current environment. Note that a finite set always contains at least one top (optimal) element as well as at least one bottom element .
Generally, one can consider also the set of all environments (including other organisms), because different environments impose different preference relations on , which have to be represented by different fitness functions . In this paper, however, we shall assume that a particular environment has been fixed, and therefore consider only one preference relation and one fitness function.
During the replication, organism can mutate into with probability , and the products define the selectionmutation matrix — the infinitesimal generator of the replicatormutator dynamics (generally nonlinear Markov evolution). Mutation can have different effects on fitness of the offspring. Mutation can be deleterious, if , neutral, if , or beneficial, if . We shall analyse how the probability of beneficial mutation can be related to the ‘geometry’ of mutation.
Fitness is defined by the interaction of an organism with its environment, and therefore it is a property of a phenotype. Thus the set , which is the domain of the fitness function, can be thought of as not just the set of all organisms, but the set of all possible phenotypes. Reproduction of organisms, however, involves passing of information about the phenotypes in the form of codes, which can be elements of some other set. Consider a representation of phenotypes by points of a topological vector space (e.g. a space of traits, a space of DNA sequences and so on). In information theory, a mapping is called a code, and we shall assume here that it is uniquely decodable: implies . That is, is an injection of into a possibly larger space . In biological terms, each genotype has either one or no phenotype, and each phenotype has precisely one genotype. In addition, we shall assume that the image of is closed under the operation of addition in , which implies that for all , , there exists such that . Thus, mutation in can be represented in by addition of codes and , as shown on the following diagram: