Biological evolution in a multidimensional fitness landscape

Solution of the Crow-Kimura model with a periodically changing (two-season) fitness function

Since the origin of life, both evolutionary dynamics and rhythms have played a key role in the functioning of living systems. The Crow-Kimura model of periodically changing fitness function has been solved exactly, using integral equation with time-ordered exponent. We also found a simple approximate solution for the two-season case. The evolutionary dynamics accompanied by the rhythms provide important insights into the properties of certain biological systems and processes.

Network Analysis of Protein Adaptation: Modeling the Functional Impact of Multiple Mutations

The evolution of new biochemical activities frequently involves complex dependencies between mutations and rapid evolutionary radiation. Mutation co-occurrence and covariation have previously been used to identify compensating mutations that are the result of physical contacts and preserve protein function and fold. Here, we model pairwise functional dependencies and higher order interactions that enable evolution of new protein functions. We use a network model to find complex dependencies between mutations resulting from evolutionary trade-offs and pleiotropic effects. We present a method to construct these networks and to identify functionally interacting mutations in both extant and reconstructed ancestral sequences (Network Analysis of Protein Adaptation). The time ordering of mutations can be incorporated into the networks through phylogenetic reconstruction. We apply NAPA to three distantly homologous β-lactamase protein clusters (TEM, CTX-M-3, and OXA-51), each of which has experienced recent evolutionary radiation under substantially different selective pressures. By analyzing the network properties of each protein cluster, we identify key adaptive mutations, positive pairwise interactions, different adaptive solutions to the same selective pressure, and complex evolutionary trajectories likely to increase protein fitness. We also present evidence that incorporating information from phylogenetic reconstruction and ancestral sequence inference can reduce the number of spurious links in the network, whereas preserving overall network community structure. The analysis does not require structural or biochemical data. In contrast to function-preserving mutation dependencies, which are frequently from structural contacts, gain-of-function mutation dependencies are most commonly between residues distal in protein structure.

The rich phase structure of a mutator model

We propose a modification of the Crow-Kimura and Eigen models of biological molecular evolution to include a mutator gene that causes both an increase in the mutation rate and a change in the fitness landscape. This mutator effect relates to a wide range of biomedical problems. There are three possible phases: mutator phase, mixed phase and non-selective phase. We calculate the phase structure, the mean fitness and the fraction of the mutator allele in the population, which can be applied to describe cancer development and RNA viruses. We find that depending on the genome length, either the normal or the mutator allele dominates in the mixed phase. We analytically solve the model for a general fitness function. We conclude that the random fitness landscape is an appropriate choice for describing the observed mutator phenomenon in the case of a small fraction of mutators. It is shown that the increase in the mutation rates in the regular and the mutator parts of the genome should be set independently; only some combinations of these increases can push the complex biomedical system to the non-selective phase, potentially related to the eradication of tumors.

Mathematical Models of Quasi-Species Theory and Exact Results for the Dynamics

We formulate the Crow-Kimura, discrete-time Eigen model, and continuous-time Eigen model. These models are interrelated and we established an exact mapping between them. We consider the evolutionary dynamics for the single-peak fitness and symmetric smooth fitness. We applied the quantum mechanical methods to find the exact dynamics of the evolution model with a single-peak fitness. For the smooth symmetric fitness landscape, we map exactly the evolution equations into Hamilton-Jacobi equation (HJE). We apply the method to the Crow-Kimura (parallel) and Eigen models. We get simple formulas to calculate the dynamics of the maximum of distribution and the variance. We review the existing mathematical tools of quasi-species theory.

Modularity enhances the rate of evolution in a rugged fitness landscape

Biological systems are modular, and this modularity affects the evolution of biological systems over time and in different environments. We here develop a theory for the dynamics of evolution in a rugged, modular fitness landscape. We show analytically how horizontal gene transfer couples to the modularity in the system and leads to more rapid rates of evolution at short times. The model, in general, analytically demonstrates a selective pressure for the prevalence of modularity in biology. We use this model to show how the evolution of the influenza virus is affected by the modularity of the proteins that are recognized by the human immune system. Approximately 25% of the observed rate of fitness increase of the virus could be ascribed to a modular viral landscape.

Evolution dynamics of a model for gene duplication under adaptive conflict

We present and solve the dynamics of a model for gene duplication showing escape from adaptive conflict. We use a Crow-Kimura quasispecies model of evolution where the fitness landscape is a function of Hamming distances from two reference sequences, which are assumed to optimize two different gene functions, to describe the dynamics of a mixed population of individuals with single and double copies of a pleiotropic gene. The evolution equations are solved through a spin coherent state path integral, and we find two phases: one is an escape from an adaptive conflict phase, where each copy of a duplicated gene evolves toward subfunctionalization, and the other is a duplication loss of function phase, where one copy maintains its pleiotropic form and the other copy undergoes neutral mutation. The phase is determined by a competition between the fitness benefits of subfunctionalization and the greater mutational load associated with maintaining two gene copies. In the escape phase, we find a dynamics of an initial population of single gene sequences only which escape adaptive conflict through gene duplication and find that there are two time regimes: until a time t single gene sequences dominate, and after t double gene sequences outgrow single gene sequences. The time t is identified as the time necessary for subfunctionalization to evolve and spread throughout the double gene sequences, and we show that there is an optimum mutation rate which minimizes this time scale.

Evolutionary advantage via common action of recombination and neutrality

We investigate evolution models with recombination and neutrality. We consider the Crow-Kimura (parallel) mutation-selection model with the neutral fitness landscape, in which there is a central peak with high fitness A, and some of 1-point mutants have the same high fitness A, while the fitness of other sequences is 0. We find that the effect of recombination and neutrality depends on the concrete version of both neutrality and recombination. We consider three versions of neutrality: (a) all the nearest neighbor sequences of the peak sequence have the same high fitness A; (b) all the l-point mutations in a piece of genome of length l≥1 are neutral; (c) the neutral sequences are randomly distributed among the nearest neighbors of the peak sequences. We also consider three versions of recombination: (I) the simple horizontal gene transfer (HGT) of one nucleotide; (II) the exchange of a piece of genome of length l, HGT-l; (III) two-point crossover recombination (2CR). For the case of (a), the 2CR gives a rather strong contribution to the mean fitness, much stronger than that of HGT for a large genome length L. For the random distribution of neutral sequences there is a critical degree of neutrality ν(c), and for μ<μ(c) and (μ(c)-μ) is not large, the 2CR suppresses the mean fitness while HGT increases it; for ν much larger than ν(c), the 2CR and HGT-l increase the mean fitness larger than that of the HGT. We also consider the recombination in the case of smooth fitness landscapes. The recombination gives some advantage in the evolutionary dynamics, where recombination distinguishes clearly the mean-field-like evolutionary factors from the fluctuation-like ones. By contrast, mutations affect the mean-field-like and fluctuation-like factors similarly. Consequently, recombination can accelerate the non-mean-field (fluctuation) type dynamics without considerably affecting the mean-field-like factors.