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The dependence of the mutator probability q on the genome length L.: The the single peak fitness model (smooth line) with J = 1.05, μ1 = 1, μ2 = 10, a = α1 = 0.001. For the L = 5000 the single peak model’s numerical result coincides with the analytical result for L = ∞ with the relative accuracy about 0.1%.
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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-sel...
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... L = 5000, the accuracy of our analytical result is approximately 0.1%, as seen in Fig. 4. Figure 4 and Eq. (14) show that the fraction of the wild allele does not approach to 0 even for a large ...
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Citations
... Our research aims to investigate the impact of random transitions between two general fitness landscapes on the evolution of populations. This problem resembles 57003-p1 somehow the mutator phenomenon, where the mutation in a specific gene in the genome results in a drastic change of the mutation rates of other genes or their fitness landscape [32], so we again have an evolution with two fitness landscapes. In the case of the mutator, there is a wellformulated mathematical model with a system of 2(L + 1) equations (where L is the genome length) [32]. ...
... This problem resembles 57003-p1 somehow the mutator phenomenon, where the mutation in a specific gene in the genome results in a drastic change of the mutation rates of other genes or their fitness landscape [32], so we again have an evolution with two fitness landscapes. In the case of the mutator, there is a wellformulated mathematical model with a system of 2(L + 1) equations (where L is the genome length) [32]. ...
... However, the accurate consideration of evolution on fluctuating fitness landscapes requires a complicated system of functional equations, it is a much harder mathematical tool than the one used in the mutator model [32]. The significant difference between the two phenomena lies in the fact that we have macroscopic transition rates for the mutator model, while our model of evolution on fluctuating fitness landscapes assumes stochastic transitions. ...
We consider the Crow-Kimura model in case of random transitions between different fitness landscapes. The epochs (system has constant in time fitness landscape) length is given by an exponential distribution. To solve the model exactly we need a large system of functional equations. We solve the model approximately at the limits of slow or fast transitions, calculating the first-order corrections via the transition rate or its inverse. We consider the case of slow transitions and find that the mean fitness equals the average fitness for the evolution on the static fitness landscapes, minus some quantity, which we call a load. While calculating the load, we speculate about the analogy with information thermodynamics. We also look at the model with few genes and identify exact transition points to the transient phase.
... From our perspective, the investigation of a new (mutator) phase 19) was especially important. In our paper, 20) one can find a more detailed discussion of the different approaches to mathematical modelling of mutator phenomena. In Ref. 20, we solved the mutator model in the bulk approximation calculating the mean fitness, attending mainly the unidirectional transitions from normal allele to mutator allele. ...
... From our 18 perspective, the investigation of a new (mutator) phase [28] was particularly important. In our previous paper [30], one 19 can find a more detailed discussion of the different approaches to mathematical modelling of mutator phenomena. 20 Let us focus on the quasispecies models by Crow and Kimura and by Eigen, which were successfully applied to 21 many problems in microbiological evolution [31]. ...
... 1-3). In [30], we solved the mutator model in the bulk approximation calculating the mean fitness, attending mainly 28 the uni-directional transitions from normal allele to mutator allele. The dynamics of the model has been obtained in [32], 29 looking at the symmetric transition rates between the normal allele to mutator. ...
... The dynamics of the model has been obtained in [32], 29 looking at the symmetric transition rates between the normal allele to mutator. 30 In the current article, we examine the role of epistasis for recombination. While we look at the recombination ...
We investigate the evolutionary model with recombination and random switches in the fitness function due to change in a special gene. The dynamical behaviour of the fitness landscape induced by the specific mutations is closely related to the mutator phenomenon, which, together with recombination, plays an important role in modern evolutionary studies. It is of great interest to develop classical quasispecies models towards better compliance with the observation. However, these properties significantly increase the complexity of the mathematical models. In this paper, we consider symmetric fitness landscapes for several different environments, using the Hamilton–Jacobi equation (HJE) method to solve the system of equations at a large genome length limit. The mean fitness and surplus are calculated explicitly for the steady-state, and the relevance of the analytical results is supported by numerical simulation. We consider the most general case of two landscapes with any values of mutation and recombination rates (three independent parameters). The exact solution of evolutionary dynamics is done via a solution of a fourth-order algebraic equation. For the more straightforward case with two independent parameters, we derive the solution using a quadratic algebraic equation. For the simplest case, when there are two landscapes with the same mutation and recombination rates, we derive some effective fitness landscape, mapping the model with recombination to the Crow–Kimura model.
... In the situation, when the mutation rule is defined by several Hamming classes, we have a similar mathematical problem as in the case of Parrondo games with the parameter . One can also consider to introduce the conditional mutation rate in the mutator model for cancer [46]. It is interesting to find the dynamics of our model, while it is a much harder problem than the solution of the dynamics of the Crow-Kimura model [25]. ...
Cancer is related to clonal evolution with a strongly nonlinear, collective behavior. Here we investigate a slightly advanced version of the popular Crow–Kimura evolution model, suggested recently, by simply assuming a conditional mutation rate. We investigated the steady-state solution and found a highly intriguing plateau in the distribution. There are selective and nonselective phases, with a rather narrow plateau in the distribution at the peak in the first phase, and a wide plateau for many Hamming classes (a collection of genomes with the same number of mutations from the reference genome) in the second phase. We analytically solved the steady state distribution in the selective and nonselective phases, calculating the widths of the plateaus. Numerically, we also found an intermediate phase with several plateaus in the steady-state distribution, related to large finite-genome-length corrections. We assume that the newly observed phenomena should exist in other versions of evolution dynamics when the parameters of the model are conditioned to the population distribution.
... Let us briefly overview the main directions of research in this field. In our recent paper, 5) we have discussed the scope of mathematical approaches to this problem in further detail. There have been phenomenological population genetics studies 6,7) and several works which considered the infinite population of binary sequences with a finite genome length. ...
... There have been phenomenological population genetics studies 6,7) and several works which considered the infinite population of binary sequences with a finite genome length. 8,9) Our previous papers 5,10) are devoted to the analytical investigation of the infinite population model at a large genome limit. In Ref. 5 we calculate the mean fitness for the general fitness landscape. ...
... 8,9) Our previous papers 5,10) are devoted to the analytical investigation of the infinite population model at a large genome limit. In Ref. 5 we calculate the mean fitness for the general fitness landscape. In Ref. 10 we calculated how the mean number of mutations in the total population (both wild-types and mutator-types) changes with the time in the large genome length limit when there are back transitions in the mutator genes. ...
We considered the infinite population version of the mutator phenomenon in evolutionary dynamics, looking at the uni-directional mutations in the mutator-specific genes and linear selection. We solved exactly the model for the finite genome length case, looking at the quasispecies version of the phenomenon. We calculated the mutator probability both in the statics and dynamics. The exact solution is important for us because the mutator probability depends on the genome length in a highly non-trivial way.
... To describe evolution of biological organisms, one may use either discrete time models (e.g., the discrete time Wright-Fisher (WF) model [1][2][3], the Moran model [4], or the Eigen model [5,6]) or continuous time models (e.g., the continuous time Eigen model [6][7][8], the Crow-Kimura model [9][10][11][12], and the Moran model [4]). The former is proper for the evolution with nonoverlapping generations, and the latter is proper for the evolution with overlapping generations. ...
... We have a transposed matrix compared with Eq. (5) [4]. Equation (12) gives the steady state distribution of the WF model, whereas the meanings of π i and p i are different in these two cases. ...
The discrete time mathematical models of evolution (the discrete time Eigen model, the Moran model, and the Wright-Fisher model) have many applications in complex biological systems. The discrete time Eigen model rather realistically describes the serial passage experiments in biology. Nevertheless, the dynamics of the discrete time Eigen model is solved in this paper. The 90% of results in population genetics are connected with the diffusion approximation of the Wright-Fisher and Moran models. We considered the discrete time Eigen model of asexual virus evolution and the Wright-Fisher model from population genetics. We look at the logarithm of probabilities and apply the Hamilton-Jacobi equation for the models. We define exact dynamics for the population distribution for the discrete time Eigen model. For the Wright-Fisher model, we express the exact steady state solution and fixation probability via the solution of some nonlocal equation then give the series expansion of the solution via degrees of selection and mutation rates. The diffusion theories result in the zeroth order approximation in our approach. The numeric confirms that our method works in the case of strong selection, whereas the diffusion method fails there. Although the diffusion method is exact for the mean first arrival time, it provides incorrect approximation for the dynamics of the tail of distribution.
We consider the mutator model with unidirected transitions from the wild type to the mutator type, with different fitness functions for the wild types and mutator types. We calculate both the fraction of mutator types in the population and the surpluses, i.e., the mean number of mutations in the regular part of genomes for the wild type and mutator type, which have never been derived exactly. We identify the phase structure. Beside the mixed (ordinary evolution phase with finite fraction of wild types at large genome length) and the mutator phase (the absolute majority is mutators), we find another new phase as well—it has the mean fitness of the mixed phase but an exponentially small (in genome length) fraction of wild types. We identify the phase transition point and discuss its implications.
We investigate the mutator model for the asymmetric transition rates between the wild-type and mutator type. When the mutator gene changes its type, both mutation rate of genome and fitness landscape are changed. We look at smooth symmetric fitness landscapes both for the wild type (normal allele of special gene) and mutator type (mutator allele). In some cases involving small degree of transition rates asymmetry (or not too large genome length) where we have smooth large genome length limit, the large system of ODE can be replaced by Hamilton–Jacobi equation. We derive here the analytical results for the mean fitness and population distribution in steady state including the finite size corrections. We have observed that some interesting oscillations arise involving longer genomes, and we cannot map the large system of ODE to the single partial differential equation HJE using a simple ansatz. We assume that the found counter-intuitive phenomenon should exist for evolution on fluctuating landscapes also, for asymmetric transition rates. Actually, the asymmetry of transition (mutation) rates is an important characteristics of evolutionary dynamics.
Evolution on changing fitness landscapes (seascapes) is an important problem in evolutionary biology. We consider the Moran model of finite population evolution with selection in a randomly changing, dynamic environment. In the model, each individual has one of the two alleles, wild type or mutant. We calculate the fixation probability by making a proper ansatz for the logarithm of fixation probabilities. This method has been used previously to solve the analogous problem for the Wright-Fisher model. The fixation probability is related to the solution of a third-order algebraic equation (for the logarithm of fixation probability). We consider the strong interference of landscape fluctuations, sampling, and selection when the fixation process cannot be described by the mean fitness. Such an effect appears if the mutant allele has a higher fitness in one landscape and a lower fitness in another, compared with the wild type, and the product of effective population size and fitness is large. We provide a generalization of the Kimura formula for the fixation probability that applies to these cases. When the mutant allele has a fitness (dis-)advantage in both landscapes, the fixation probability is described by the mean fitness.
