We study the adaptation dynamics of an initially maladapted population evolving via the elementary processes of mutation and selection. The evolution occurs on rugged fitness landscapes which are defined on the multi-dimensional genotypic space and have many local peaks separated by low fitness valleys. We mainly focus on the Eigen's model that describes the deterministic dynamics of an infinite number of self-replicating molecules. In the stationary state, for small mutation rates such a population forms a {\it quasispecies} which consists of the fittest genotype and its closely related mutants. The quasispecies dynamics on rugged fitness landscape follow a punctuated (or step-like) pattern in which a population jumps from a low fitness peak to a higher one, stays there for a considerable time before shifting the peak again and eventually reaches the global maximum of the fitness landscape. We calculate exactly several properties of this dynamical process within a simplified version of the quasispecies model.
Deep Dive into Adaptation dynamics of the quasispecies model.
We study the adaptation dynamics of an initially maladapted population evolving via the elementary processes of mutation and selection. The evolution occurs on rugged fitness landscapes which are defined on the multi-dimensional genotypic space and have many local peaks separated by low fitness valleys. We mainly focus on the Eigen’s model that describes the deterministic dynamics of an infinite number of self-replicating molecules. In the stationary state, for small mutation rates such a population forms a {\it quasispecies} which consists of the fittest genotype and its closely related mutants. The quasispecies dynamics on rugged fitness landscape follow a punctuated (or step-like) pattern in which a population jumps from a low fitness peak to a higher one, stays there for a considerable time before shifting the peak again and eventually reaches the global maximum of the fitness landscape. We calculate exactly several properties of this dynamical process within a simplified version o
Consider a maladapted population such as a bacterial colony in a glucose-limited environment, or a viral population in a vaccinated animal cell. In such harsh environments, the less fit members of the population are likely to perish and only the highly fit ones can survive to the next generation. In this manner, the fitness of the population increases with time and the initially maladapted population evolves to a well-adapted state. In the last century, there has been a concerted effort to put this verbal theory of Darwin [1] on a solid quantitative footing by performing long-term experiments on microbial populations and studying theoretical models of biological evolution.
One of the questions in evolutionary biology concerns the mode of evolution. In the experiments on microbes, it is found that the fitness of the maladapted population can increase with time in either a smooth continuous manner [2] or sudden jumps [3]. The latter mode is consistent with evolution on a fitness landscape defined on genotypic space with many local peaks separated by fitness valleys. On such a rugged fitness landscape, a low fitness population initially climbs a fitness peak until it encounters a local peak where it gets trapped since a better peak lies some mutational distance away. In a population of realistic size, it takes a finite time for an adaptive mutation to arise and the fitness stays constant during this time (stasis). Once some beneficial mutants become available, the fitness increases quickly as the population moves to a higher peak where it can again get stuck. Such dynamics alternating between stasis and rapid changes in fitness go on until the population reaches the global maximum.
This punctuated behavior of fitness is also seen in deterministic models that assume infinite population size. An example of such a step-like pattern for average fitness is shown in Fig. 1. A neat and unambiguous way of defining a step is by considering the fitness of the most populated genotype also shown in Fig. 1. Since large but finite populations evolve deterministically at short times [4], it is worthwhile to study the punctuated evolution in models with infinite number of individuals. In this article, we will briefly describe some exact results concerning the dynamics of an infinitely large population on rugged fitness landscapes [5,6]. We will find that the mechanism producing the step-like behavior is not due to “valley crossing” as in finite populations but when a fitter population “overtakes” the less fit one as described in the subsequent sections.
We consider an infinitely large population reproducing asexually via the elementary processes of selection and mutation. Each individual in the population carries a binary string σ = {σ 1 , …, σ L } of length L where σ i = 0 or 1. The 2 L sequences are arranged on the multi-dimensional Hamming space. The information about the environment is encoded in fitness landscape defined as a map from the sequence space into the real numbers and is generated by associating a non-negative real number W (σ) to each sequence σ. Fitness landscapes can be simple possessing some symmetry properties such as permutation invariance, or complex devoid of any such symmetries [7,8]. Fitness functions with single peak are an example of simple fitness landscapes while rugged landscapes with many hills and valleys belong to the latter class.
The average population fraction X (σ, t) with sequence σ at time t follows mutation-selection dynamics described by the following discrete time equation [8,9]
The last two factors in the numerator of the above equation give the population fraction when a sequence σ ′ copies itself with replication probability W (σ ′ ) since fitness is defined as the average number of offspring produced per generation.
After the reproduction process, point mutations are introduced independently at each locus of the sequence σ ′ with probability µ per generation. Thus, a sequence σ is obtained via mutations in σ ′ with probability
where the Hamming distance d(σ, σ ′ ) is the number of point mutations in which the sequences σ and σ ′ differ. The denominator of (1) is the average fitness of the population at time t which ensures that the density X (σ, t) is conserved. The stationary state of the quasispecies equation ( 1) has been studied extensively in the last two decades for various fitness landscapes. These numerical and analytical studies have shown that for most landscapes, there exists a critical mutation rate µ c below which the population forms a quasispecies consisting of fittest genotype and its closely related mutants while above it, the population delocalises over the whole sequence space. This error threshold phenomenon can be easily demonstrated for a single peak fitness landscape defined as
where σ 0 is the fittest sequence. In the limit µ → 0, L → ∞ keeping U = µL fixed, the frequency of the fittest sequence in the steady state of (1) is given by
which is an
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