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Evolutionary model with recombination and randomly changing fitness landscape
Journal Pre-proof Evolutionary model with recombination and randomly changing fitness landscape David B. Saakian, Edgar Vardanyan, Tatiana Yakushkina
PII: DOI: Reference:
S0378-4371(19)31744-3 https://doi.org/10.1016/j.physa.2019.123091 PHYSA 123091
To appear in:
Physica A
Received date : 13 June 2019 Revised date : 9 September 2019 Please cite this article as: D.B. Saakian, E. Vardanyan and T. Yakushkina, Evolutionary model with recombination and randomly changing fitness landscape, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123091. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Journal Pre-proof *Highlights (for review)
. We solved the mutator model with recombination. . For the same muitation rates the model is mapped to the Crow-Kimura (CK) model.
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. We defined the effective fitness.
Journal Pre-proof *Manuscript Click here to view linked References
Evolutionary model with recombination and randomly changing fitness landscape David B. Saakian1,2 ,∗ Edgar Vardanyan3,4, and Tatiana Yakushkina5 1
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Laboratory of Applied Physics, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam 2 Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia 4 Yerevan State University, Yerevan, Armenia and 5 National Research University Higher School of Economics, Moscow, Russia (Dated: September 9, 2019)
PACS numbers: 87.23.Kg
I.
INTRODUCTION
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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.
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The role of changeable fitness landscapes in evolutionary processes of various nature has been extensively studied for more than 50 years, and a growing body of literature addresses different aspects of this problem. Fitness variations are often induced by mutator genes, which responsible for mechanisms that maintain genomic stability. Alterations in such genes result in a type with increased spontaneous mutation rate called a “mutator” phenotype. Over decades, experimental research on bacteria, viruses, and simple model organisms extends our understanding of mutator phenomenon. The early studies describe particular mutator genes and their evolutionary parameters, that provided evidence for mutator effect: E. coli [1–3], bacteriophage [4], and even for prokaryotes as drosophila [5, 6] or yeast [7, 8]. However, in many cases, the genes that cause the mutator effect and the following evolutionary path are still to be revealed. One of the areas of application for the mutator dynamics is cancer evolution, which significantly advanced with the studies on clonal evolutionary dynamics [9–11] and genome instability [12]. As most of the cancers are characterized by multiple mutations, the changes in mutator genes have been considered [13, 14] as one of the causes of malignant type development.
∗ Electronic
address: [email protected]
Recombination is one of the key factors of evolution and generally assumed as one of the origins of biological life complexity. Its investigation [25–28, 30–32] is certainly one of main directions of evolution research. It is especially important to clarify the advantages of evolutionary dynamics due to recombination. Since [28], and it is widely accepted [29] that recombination leads to an increase in the mean fitness of the population in the steady-state for negative epistasis, see reviews [30],[31] . The current article considers the synergy of recombination and mutator phenomena. There have been many attempts to mathematically formalize and explain evolutionary dynamics for systems with mutator genes. One of the first approaches to the mutator phenomenon was the phenomenological one [15– 18]. From this perspective, the probability of back mutations is generally assumed to be insignificantly low. This class of models is beneficial for investigation of system dynamics in general fitness case; however, it is not accurate for describing the steady-state solutions. Later, the quasispecies models have been constructed and examined to provide an adequate description of the steady-state of the evolving population. This research direction arose from [19], which introduced a simple Crow-Kimura model with linear fitness function. For this case, the authors provided some analytical estimates as well as numerical analysis. In [20], the solution and the phase structure for the Eigen model with mutator gene and the multiplicative Wrightian fitness function have been obtained. As it has been first found by M. Eigen [21], there are selective
Journal Pre-proof 2 Evolutionary dynamics of the system is shaped by several processes acting together: • mutation with the rate per genome equal to 1 for wild-type and µ for mutator type,
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• recombination with the rate c for wild-type sequences and µC for mutator sequences: at any position in the genome main part, an allele can be replaced by 1 with the probability ¯lτ /L or by −1 with the probability 1 − ¯lτ /L, τ = 1, 2,
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• replication with the fitness f1 (xl ) for normal sequences and f2 (xl ) for the mutator sequences, where xl = 1 − 2l/L stands for the average gene state corresponding to the number of accumulated mutations l, • transitions from the normal state to the mutator state with the rate α1 and back with the rate α2 ,
• limited resource supply: the population size is fixed.
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and non-selective phases in the model, similar to ferromagnetic and paramagnetic phases in the Ising model. From our perspective, the investigation of a new (mutator) phase [20] was particularly important. In our previous paper [22], one can find a more detailed discussion of the different approaches to mathematical modelling of mutator phenomena. Let us focus on the quasispecies models by Crow and Kimura and by Eigen, which were successfully applied to many problems in microbiological evolution [23]. From the classical viewpoint, this method considers the evolution with constant mutation rates on the static fitness landscape. Most of the studies use symmetric fitness landscapes, where fitness is a function of the total number of accumulated mutations compare to the reference genetic state. However, the version of the model with the multiple reference sequences is also solvable. Both CrowKimura and Eigen models can be solved in the dynamics with O(1/L) accuracy (L is the genome length) [23]. In the statics, the finite genome size corrections have derived as well. Here, we demonstrate a smooth large L limit ( which is discussed below together with the illustration on Figs. 1-3). In [22], we solved the mutator model in the bulk approximation calculating the mean fitness, attending mainly the uni-directional transitions from normal allele to mutator allele. The dynamics of the model has been obtained in [24], looking at the symmetric transition rates between the normal allele to mutator. In the current article, we examine the role of epistasis for recombination. While we look at the recombination phenomenon in the mutator-like model (the mutator allele brings to the change of fitness landscape), our results are very relevant for the understanding of the recombination phenomenon in standard evolution models. First, we obtain an effective fitness landscape for our mutator model, then look at the standard evolutionary model with such effective fitness landscape. From this approach, we derive an unusual and counter-intuitive result.
The system dynamics is described by master equations for the probabilities Pl and Ql : the total frequency of all wild-type sequences with l mutations (normal mutation rate) and all mutator type sequences with l mutations (increased mutation rate µ) correspondingly. In these two cases, we have different in general recombination rates c and µC.
II.
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dPl (t) = Ldt
SOLUTION OF MUTATOR GENE MODEL WITH RECOMBINATION A.
dQl (t) = Ldt
The mathematical model
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We consider a simplified evolution model with two types of biallelic genes and symmetric fitness landscape. The genome includes a mutator gene, having a normal and mutator allele, and the main part with L genes. Each gene in a sequence is encoded by ±1: +1 stands for the wild-type allele for the gene, −1 — for the mutant allele. Thus, the genetic space consists of 2L+1 different sequences, which can be divided into Hamming classes according to the number of accumulated mutations in the main part. Depending on the mutator gene, the genetic sequence can be in one of two states: the wild-type state with the mean number of mutations ¯l1 ; and the mutator state with the mean number of mutations ¯l2 .
R(t) =
α2 Ql + Pl (f1 (xl ) − (1 + α1 )) + ¯l1 L−l+1 1+c + Pl−1 L L L − ¯l1 l+1 1+c − Pl+1 L L ¯ ¯l1 l1 l l 1− cPl 1− + − Pl R(t), L L LL α1 Pl + Ql (f2 (xl ) − (µ + α2 )) + ¯ L−l+1 l2 µ Ql−1 1+C + L L L − l¯2 l+1 1+C − Ql+1 L L ¯ l¯2 l2 l l 1− CQl 1− + − Ql R(t), L L LL X (Pl (t)f1 (xl ) + Ql (t)f2 (xl )) , l
¯l1 =
X l
Pl l, ¯l2 =
X
Ql l,
l
xl = 1 − 2l/L, 0 ≤ l ≤ L. We can think of this model as a combination of the basic mutator model [22] and standard recombination model
(1)
Journal Pre-proof 3
b. Different mutation rates, but same recombination rates µ 6= 1, c = C. c. Same mutation and recombinations rates µ = 1, c = C. d. A generalization of Eq. (1) for several fitness landscapes with µ = 1, c = C. To solve Eq. (1), we use the following ansatz [22]: Pl = P (x, t) = v1 (x)eLu(x,t) , (2)
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where x is obtained from xl = 1 − 2l/L in a large genome limit. Applying the expressions (2) to the system (1), we derive: 1 − xs v2 α2 + v1 −u′t + f1 (x) − α1 − c + 2 1 + x 2u′x 1 − x −2u′x c1 e + c2 e −1 = 0, 2 2 1 − xs v1 α1 + v2 −u′t + f2 (x) − α2 − Cµ + 2 ′ ′ 1−x 1+x C1 e2ux + C2 e−2ux − 1 = 0. (3) µ 2 2 , c2 = 1 + c 1+s , Here, we denoted c1 = 1 + c 1−s 2 2 1+s C1 = 1 + C 1−s , C = 1 + C . While deriving 2 2 2 Eq. (3), we dropped the terms of the order O(1/L). We can consider Eq. (3) as a system of linear equations with respect to the variables v1 , v2 . The consistency of a system of linear equations is guaranteed by a zero determinant det A = 0 condition, where we denoted the matrix on the right-hand side of Eq. (3) as A. By solving detA(q) = 0 for q ≡ ut , we obtain a Hamilton–Jacobi equation (HJE). In Appendix A, we derive the HJE for the general case. Here, we provide the solution for the case C = c: u′t
Having the HJE derived, we can calculate the mean fitness using the minimax principle [23]. To apply it, we take the minimum of the Hamiltonian with respect to p = u′x and then the maximum with respect to x. The latter can be derided following the idea of the study [33] that the fitness equals to the value of the fitness function at the surplus point. Thus, we get the expressions for the mean fitness value in the steady-state: f1 (x) + f2 (x) α1 + α2 1+µ R = max − +A 2 2 2 q 1 2 ± (f1 (x) − f2 (x) − (α1 − α2 ) − (1 − µ)A) + 4α1 α2 , 2 x r p 2 2 2 c s 1 − xs c A = 1 − x2 − −c . (5) 1+ 2 4 2 Then, taking into account p = 0 at the point x = s, we obtain from Eq. (4) an additional equation:
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Ql = Q(x, t) = v2 (x)eLu(x,t) ,
Analytical results for the two-landscape case with different mutation rates
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a. Different mutation and recombination rates µ 6= 1, C 6= c (see Appendix A).
B.
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[32], where the set of terms in the equations (1) is examined. The parameters of this model (1) define the balance between replication, mutation, and recombination, and, thus, the evolutionary behaviour of the system. We will consider the following scenarios for the model:
f1 (s) + f2 (s) α1 + α2 R(s) = − (6) 2 2 q 1 2 (f1 (s) − f2 (s) − (α1 − α2 )) + 4α1 α2 , + 2
where s = 1 − h 2l L i is the surplus of the population, l is the number of mutations. Thus, we get a system of two equations for the variables x and s. However, for the considered case, we can not introduce the effective fitness. C. Numerical results for the two-landscape case with different mutation and recombination rates and different fitnesses
Figures 1–3 describe the case referred as (a) in our list, where µ 6= 1, C 6= c, and the fitness functions for mutator and normal types are different. The numerics confirms the validity of our results for mean fitness and surplus. R 0.5 0.4 0.3
+ H± (x, p) = 0, (4) 0.2 f1 (x) + f2 (x) α1 + α2 1+µ 0.1 −H± (x, p) = − +A 2 2 2 q 1 0.0 c 0 1 2 3 4 (f1 (x) − f2 (x) − (α1 − α2 ) − (1 − µ)A)2 + 4α1 α2 , ± 2 1 − xs 1 + x 2p 1 − x −2p FIG. 1: The mean fitness versus the recombination rate for −c c1 e + c2 e . A = −1 + f1 (x) = x2 , f2 (x) = 3x2 , µ = 3, L = 1000. 2 2 2
Journal Pre-proof 4 R 3.07
the number of mutations. The effective fitness takes the form:
3.06 3.05
f1 (x) + f2 (x) α1 + α2 − (8) 2 q2 1 2 (f1 (x) − f2 (x) − (α1 − α2 )) + 4α1 α2 . ± 2
fˆ(x) =
3.04 3.03 3.02
We see that there is an effective fitness different from the averaged fitness value over two system states f¯(x) = f1 (x)α2 +f2 (x)α1 . Let us consider different choices of the α1 +α2 functions f1 , f2 . The case with the complementary fitness landscapes f2 (x) = −f1 (x) is of particular interest, since the mean of the fitness value turns to zero. For the effective fitness, we derive for α1 = α2 = α:
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3.01 c
3.00 0
1
2
3
4
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FIG. 2: The mean fitness versus the recombination rate for f1 (x) = 2x, f2 (x) = 6x, µ = 3, L = 1000. R 0.5
1p fˆ(x) = −α + (f1 (x) − α)2 + α2 2
0.4 0.3
Epistasis is another important concept in evolutionary biology to discuss in this study. Changes in the genomic sequences can exhibit the synergetic effects: fitness of a collection of the genes differs from the sum of the contribution of different genes to the fitness. For the linear fitness function in the Crow-Kimura model, there is a zero epistasis. In population genetics, the epistasis ε is the measure of fitness interactions between alleles at different loci. For the two-locus measure, ε = f itness(AB)f itness(ab)f itness(Ab)f itness(aB). For our case with many loci, we can define a sign of epistasis as the sign of the second-order derivative of our fitness function, f ′′ (x). It is easy to verify that our definition of epistasis sign coincides with the definition in population genetics. Moreover, we looked at the epistasis of effective fitness altogether with the change of the mean fitness with the recombination. We found that the transition between the landscapes can reduce the degree of negativeness of the landscape; as a result, the recombination reduces the mean fitness, see Fig. 5.
0.0 0.0
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0.2 0.1
c
0.5
1.0
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2.0
2.5
3.0
FIG. 3: The surplus versus the recombination rate for f1 (x) = 1.5x2 , f2 (x) = x2 , µ = 2, L = 1000.
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D. Analytical results for the two-landscape case with same mutation and recombination rates, but different fitnesses
R 1
1
1
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1
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2
3
(x)
f
1.0 0.5 c 4
-1.0
-0.5
0.5
1.0
-0.5
FIG. 4: The mean fitness versus the recombination rate for f1 (x) = x2 , f2 (x) = 3x2 , µ = 1, L = 1000.
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Consider a simpler case when two landscapes have the same mutation and recombination rates µ = 1, C = c, while different fitness functions. Repeating the derivations of the previous subsection, we get: # " r 2 c2 s2 p c − R = max fˆ(x) − µ + µ 1 − x2 1+ 2 4
s
R = fˆ(s),
(9)
(7)
where s = 1 − h 2l L i is the surplus of the population, l is
-2.0 -2.5
FIG. 5: The second derivative f ′′ (x) versus x for the model with α1 = α2 = 1, f1 (x) = 2(1 + x)k , f2 (x) = 2(1 − x)k for the values k = 0.1, 0.33, 0.5 from low to high. The epistasis is defined by the sign of the calculated function. We found that the recombination increases the mean fitness in case of negative epistasis (f ′′ (x) < 0 in the whole interval), otherwise it reduces the mean fitness.
Journal Pre-proof 5 the mean fitness we look det(A(R)) with
To generalize the obtained results, we introduce n different evolutionary states with transitions between them. Each states is characterized by its fitness function fm (x), m = 1, . . . , nP . To define the transition matrix, we use aij and a ˆi ≡ j6=i aij . For the probability distribution over the genetic space and environments, we have now Plm , 0 ≤ l ≤ L, 1 ≤ m ≤ n.
R(t)
=
X
alh Plh
+
Plm
h6=m
(fm (xl ) − (µ + a ˆl )) +
xl = 1 − 2l/L, 0 ≤ l ≤ L.
. To drive the HJE, the ansatz is applied: Plm (t) = P m (x, t) = v m eLu(x,t) . From here, it follows: X
(10)
(11)
h
alh v +
h6=m
al
∂u m v = ∂t
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c ˆm − 1 − (1 − ms) + v m fm (xl ) − a 2 ′ 1 + x 1 − s vm e2u 1+c + (12) 2 2 ′ 1 − x 1+s 1+c . e−2u 2 2 We are looking at Eq. (12) as a system of equations for v m . Then, we define a matrix element:
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∂u c Aii (q) = − + fi (x) − a ˆi − 1 − (1 − ms) + ∂t 2 1−s 1+s 2u′ 1 + x −2u′ 1 − x e 1+c +e 1+c , 2 2 2 2 Aij (q) = aij (13) From these expressions, we get the following consistency condition for the linear system of equations: det(A(q)) = 0.
A¯ii (s) = −R + fi (s) − a ˆi ), ¯ Aij (s) = aij
(16)
and we have an equation
¯l1 m L−l+1 µ Pl−1 1+c + L L L − ¯l1 m l+1 1+c − Pl+1 L L ¯ ¯l1 l1 l l Plm R(t) − cPlm 1− 1− + L L LL X m = (Pl (t)fm (xl )) , m,l
Then another equation, at the maximum point
det(A(s)) = 0.
(17)
Consider the case of transitions between three types. The we get HJE (−q + f1 (x) − a + F )(−q + f2 (x) − 2a + F ) × (−q + f3 (x) − a + F ) − (−q + f1 (x) − a + F )a2 − (−q + f3 (x) − a + F )a2 = (018)
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dPlm (t) Ldt
c Aˆii (R) = −R + fi (x) − a ˆi − 1 − (1 − ms) + 2 r 2 c2 s2 p c 1 − xs 1+ 1 − x2 − −c . 2 4 2 Aˆij (R) = aij (15)
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Transitions between more than two evolutionary regimes
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E.
(14)
For this case, similar to the previous ones, we solve the latter equation for q ≡ ∂u ∂t and derive the HJE. To find
where we denoted ′ 1 + x 1−s c 1+c + F = −1 − (1 − ms) + e2u 2 2 2 ′ 1 − x 1+s 1+c e−2u (19) 2 2
Thus the Hamiltonian has three branches. We find the steady state mean fitness as a maximal solution R of the equation
(Fˆ − R + f1 (x) − a)(Fˆ − R + f2 (x))(Fˆ − R + f3 (x)) −(Fˆ − R + f1 (x) − a)a2 − (Fˆ − R + f3 (x) − a)a2 = (020) where we denoted p c Fˆ = −1 − (1 − ms) + 1 − x2 2
r
c c2 s2 (1 + )2 − (21) 2 4
Looking the maximum point, we get another equation (−R + f1 (s) − a)(−R + f2 (s))(−R + f3 (s)) −(−R + f1 (s) − a)a2 − (−R + f3 (s) − a)a2 = 0 (22) Thus we have Eqs. (22), Eq. (21) and the maximum R condition for Eq. (21) We have two equations, related to third order algebraic equation. Contrary, the two type version of mutator is solved via a system of two equations, (5),(6), related to second order algebraic equation. III.
CONCLUSION
The mutator phenomenon and genome instability belong to the area of modern evolution research, attracting
Journal Pre-proof 6
R = H(x, z, s),
(A.3)
of
B1 + B2 H(x, z, s) = f+ (x) − α+ + 2 s 2 B1 − B2 + α1 α2 . f− − α− + + 2
2 (x) 2 , α+ = α1 +α , α− = Here, we denoted f+ = f1 (x)+f 2 2 α1 −α2 . We get the following equation for the mean fitness 2 R: R = max(min[H(x, z, s)] ) (A.4)
z
x
We have another equation, looking the maximum point in Eq. (A.1), so putting z=1, Bi = fi (x). We complete the latter with Eq. (6) to have two system of equations for two unknown variables x, s. Fig. 6 illustrates the accuracy of our analytical results by Eqs. (A.3, A.4),(6)
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Appendix A: The solution of the general case evolutionary model
and z = e2p . Then, we derive the mean fitness and the Hamiltonian:
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the attention of the wide range of experts. We proposed a new model with recombination and mutator gene based on the classical Crow-Kimura model. We apply a statistical physics approach, analyzing the system’s behaviour for symmetric fitness function and different combinations of evolutionary parameters. Nowadays, the quasispecies models are one of the common in interdisciplinary research in this field, partially due to the existing analytical solutions for them. In this study, we solved the mutator-recombination model using the HJE method developed for the case of the master equation with the one-dimensional transition. The numerical simulations supported our analytical results. In the simplest case (the mutation rate is the same for both wild-types and mutator-types), we have shown that the model can be mapped into the Crow-Kimura model with some effective fitness function. The work is supported by the Russian Science Foundation under grant 19-11-00008.
R 0.5
To solve the original system (1) for different mutation and recombination rates, let us look at Eq. (3). The zero determinant condition gives:
0.4 0.3
q 2 − q(B1 + B2 − α1 − α2 ) + (B1 − α1 )(B2 − α2 ) −α1 α2 = 0 (A.1)
0.2 0.1
al
where we denoted B1 = f1 (x) − c 1−xs + A1 , B2 = f2 − 2 Cµ 1−xs + µA , 2 2 1−x 1 1 − xs 1 + x + c1 z + c2 − 1, 2 2 2 z 1 − xs 1+x B2 = f2 (x) − Cµ + µ( C1 z 2 2 1−x 1 + C2 − 1).(A.2) 2 z
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B1 = f1 (x) − c
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[1] Treffers, H. P., Spinelli, V., & Belser, N. O. A factor (or mutator gene) influencing mutation rates in Escherichia coli. Proceedings of the National Academy of Sciences, 40(11), 1064–1071 (1954). [2] Cox, E. C., & Yanofsky, C. Mutator gene studies in Escherichia coli. Journal of bacteriology 100(1), 390–397 (1969). [3] Cox, E. C. Bacterial mutator genes and the control of spontaneous mutation. Annual review of genetics, 10(1), 135–156 (1976). [4] Pierce, B. L. S. The effect of a bacterial mutator gene upon mutation rates in bacteriophage T4. Genetics, 54, 657–62 (1966). [5] Ives, P. T. The importance of mutation rate genes in evolution. Evolution, 4(3), 236–252(1970). [6] Green, M. M. . The genetics of a mutator
0.0 0
1
2
3
4
C
FIG. 6: The mean fitness versus the recombination rate C for f1 (x) = 3/2x2 , f2 (x) = x2 , µ = 1, c = 1, L = 1000. The smooth line is the result of numerical simulations, solid dots correspond to the analytical result.
[7] [8] [9] [10]
[11] [12]
gene in Drosophila melanogaster. Mutation Research/Fundamental and Molecular Mechanisms of Mutagenesis, 10(4), 353–363 (1970). Von Borstel, R. C., Graham, D. E., La Brot, K. L., Resnick, M. A.: Mutator activity of a X-radiation sensitive yeast. Genetics, 60, 233 (1968). Mohn, G., & Wrgler, F. E. Mutator genes in different species. Human Genetics, 16(1), 49–58 (1972). Nowell, C. P. The clonal evolution of tumor cell populations. Science 194, 2328 (1976). Merlo, L. M. F., Pepper, J. W., Reid B. J. & Maley, C. C. Cancer as an evolutionary and ecological process. Nature Review Cancers 6, 924-935 (2006). Greaves, M.& Maley, C. C. Clonal evolution in cancer. Nature 481, 306-311 (2012). Hanahan, D. & Weinberg, R. A. The hallmarks of cancer.
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(pp. 121-139). Springer (2015). [24] Yakushkina T., DB Saakian, CK Hu, Exact dynamics for a mutator gene model Chinese Journal of Physics 53 (5), 100904-1 (2015). [25] W. J. Ewens, Mathematical Population Genetics (Springer- Verlag, New York, 2004). [26] R. Burger, The Mathematical Theory of Selection, Recombination, and Mutation (Wiley, New York, 2000). [27] Felsenstein J., The evolutionary advantage of recombination. Genetics,78:737-56 (1974). [28] Feldman M. W., Christiansen FB, Brooks L. D. Evolution of recombination in a constant environment. Proc. Natl Acad. Sci. USA 77, 4838-4841.(1980). [29] S. Bonhoeffer et al, Evidence for positive epistasis in HIV1.Science. 306:1547–50 (2004). [30] S. P. Otto and Th. Lenormand, Nat. Rev. Genetics 3, 252 (2002). [31] J. Arjan, G. M. de Visser, and S. F. Elena, Nat. Rev. Genetics 8, 139 (2007). [32] Park JM and Deem MW,Phys. Rev. Lett. 98, 058101 (2007). [33] Baake E and Wagner H, Mutation-selection models solved exactly with methods of statistical mechanics. Genet. Res. Genet. Res. 78, 93(2001). [34] Z. Kirakosyan, D. B. Saakian, and C.-K. Hu. Finite Journal of Statistical Physics, pages 144,149, (2011). [35] Saakian DB, Gazaryan M, Hu CK, The shock waves and positive epistasis in evolution, Phys. Rev. E 90: 022712 (2014)
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al
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Cell 100, 57 (2000) [13] Loeb, L. A., Springgate, C. F.& Battula, N. Errors in DNA replication as a basis of malignant change. Cancer Res. 34, 2311-2321 (1974). [14] Loeb, L. A., Loeb, K. R. & Anderson, J. P. Multiple mutations and cancer. Proc. Natl. Acad. Sci. USA 100, 776–781 (2003). [15] Beckman, R. A.& Loeb, L. A. Efficiency of carcinogenesis with and without a mutator mutation. Proc. Natl. Acad. Sci. USA 103, 1414014145 (2006). [16] Beckman, R. A. Mutator Mutations Enhance Tumorigenic Efficiency across Fitness Landscapes. PLoS One 4, e5860 (2009). [17] Desai, M. M.& Fisher, D. S. 188, 997-1014 (2011). [18] Kussell, E. & Vucelja, M. Non-equilibrium physics and evolution-adaptation, extinction, and ecology: a Key Issues review. Rep. Prog. Phys. 77, 102602 (2014). [19] Kessler, D.& Levine, H. Mutator dynamics on a smooth evolutionary landscape. Phys. Rev. Lett. 80, 2012 (1998). [20] Nagar A. & Jain, K. Exact phase diagram of quasispecies model with a mutator rate modifier. Phys. Rev. Lett. 102, 038101 (2009). [21] Eigen M, McCascill J & Schuster P The Molecular Quasi-Species. Advances in Chemical Physics 75, 149– 263(1989). [22] Saakian DB, T Yakushkina, CK Hu, The rich phase structure of a mutator model Scientific reports 6, 34840 (2016). [23] Saakian, D. B., Hu, C. K. . Mathematical models of quasi-species theory and exact results for the dynamics. In Quasispecies: From Theory to Experimental Systems
PII: DOI: Reference:
S0378-4371(19)31744-3 https://doi.org/10.1016/j.physa.2019.123091 PHYSA 123091
To appear in:
Physica A
Received date : 13 June 2019 Revised date : 9 September 2019 Please cite this article as: D.B. Saakian, E. Vardanyan and T. Yakushkina, Evolutionary model with recombination and randomly changing fitness landscape, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123091. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Journal Pre-proof *Highlights (for review)
. We solved the mutator model with recombination. . For the same muitation rates the model is mapped to the Crow-Kimura (CK) model.
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. We defined the effective fitness.
Journal Pre-proof *Manuscript Click here to view linked References
Evolutionary model with recombination and randomly changing fitness landscape David B. Saakian1,2 ,∗ Edgar Vardanyan3,4, and Tatiana Yakushkina5 1
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3
Laboratory of Applied Physics, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam 2 Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia 4 Yerevan State University, Yerevan, Armenia and 5 National Research University Higher School of Economics, Moscow, Russia (Dated: September 9, 2019)
PACS numbers: 87.23.Kg
I.
INTRODUCTION
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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.
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The role of changeable fitness landscapes in evolutionary processes of various nature has been extensively studied for more than 50 years, and a growing body of literature addresses different aspects of this problem. Fitness variations are often induced by mutator genes, which responsible for mechanisms that maintain genomic stability. Alterations in such genes result in a type with increased spontaneous mutation rate called a “mutator” phenotype. Over decades, experimental research on bacteria, viruses, and simple model organisms extends our understanding of mutator phenomenon. The early studies describe particular mutator genes and their evolutionary parameters, that provided evidence for mutator effect: E. coli [1–3], bacteriophage [4], and even for prokaryotes as drosophila [5, 6] or yeast [7, 8]. However, in many cases, the genes that cause the mutator effect and the following evolutionary path are still to be revealed. One of the areas of application for the mutator dynamics is cancer evolution, which significantly advanced with the studies on clonal evolutionary dynamics [9–11] and genome instability [12]. As most of the cancers are characterized by multiple mutations, the changes in mutator genes have been considered [13, 14] as one of the causes of malignant type development.
∗ Electronic
address: [email protected]
Recombination is one of the key factors of evolution and generally assumed as one of the origins of biological life complexity. Its investigation [25–28, 30–32] is certainly one of main directions of evolution research. It is especially important to clarify the advantages of evolutionary dynamics due to recombination. Since [28], and it is widely accepted [29] that recombination leads to an increase in the mean fitness of the population in the steady-state for negative epistasis, see reviews [30],[31] . The current article considers the synergy of recombination and mutator phenomena. There have been many attempts to mathematically formalize and explain evolutionary dynamics for systems with mutator genes. One of the first approaches to the mutator phenomenon was the phenomenological one [15– 18]. From this perspective, the probability of back mutations is generally assumed to be insignificantly low. This class of models is beneficial for investigation of system dynamics in general fitness case; however, it is not accurate for describing the steady-state solutions. Later, the quasispecies models have been constructed and examined to provide an adequate description of the steady-state of the evolving population. This research direction arose from [19], which introduced a simple Crow-Kimura model with linear fitness function. For this case, the authors provided some analytical estimates as well as numerical analysis. In [20], the solution and the phase structure for the Eigen model with mutator gene and the multiplicative Wrightian fitness function have been obtained. As it has been first found by M. Eigen [21], there are selective
Journal Pre-proof 2 Evolutionary dynamics of the system is shaped by several processes acting together: • mutation with the rate per genome equal to 1 for wild-type and µ for mutator type,
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• recombination with the rate c for wild-type sequences and µC for mutator sequences: at any position in the genome main part, an allele can be replaced by 1 with the probability ¯lτ /L or by −1 with the probability 1 − ¯lτ /L, τ = 1, 2,
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• replication with the fitness f1 (xl ) for normal sequences and f2 (xl ) for the mutator sequences, where xl = 1 − 2l/L stands for the average gene state corresponding to the number of accumulated mutations l, • transitions from the normal state to the mutator state with the rate α1 and back with the rate α2 ,
• limited resource supply: the population size is fixed.
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and non-selective phases in the model, similar to ferromagnetic and paramagnetic phases in the Ising model. From our perspective, the investigation of a new (mutator) phase [20] was particularly important. In our previous paper [22], one can find a more detailed discussion of the different approaches to mathematical modelling of mutator phenomena. Let us focus on the quasispecies models by Crow and Kimura and by Eigen, which were successfully applied to many problems in microbiological evolution [23]. From the classical viewpoint, this method considers the evolution with constant mutation rates on the static fitness landscape. Most of the studies use symmetric fitness landscapes, where fitness is a function of the total number of accumulated mutations compare to the reference genetic state. However, the version of the model with the multiple reference sequences is also solvable. Both CrowKimura and Eigen models can be solved in the dynamics with O(1/L) accuracy (L is the genome length) [23]. In the statics, the finite genome size corrections have derived as well. Here, we demonstrate a smooth large L limit ( which is discussed below together with the illustration on Figs. 1-3). In [22], we solved the mutator model in the bulk approximation calculating the mean fitness, attending mainly the uni-directional transitions from normal allele to mutator allele. The dynamics of the model has been obtained in [24], looking at the symmetric transition rates between the normal allele to mutator. In the current article, we examine the role of epistasis for recombination. While we look at the recombination phenomenon in the mutator-like model (the mutator allele brings to the change of fitness landscape), our results are very relevant for the understanding of the recombination phenomenon in standard evolution models. First, we obtain an effective fitness landscape for our mutator model, then look at the standard evolutionary model with such effective fitness landscape. From this approach, we derive an unusual and counter-intuitive result.
The system dynamics is described by master equations for the probabilities Pl and Ql : the total frequency of all wild-type sequences with l mutations (normal mutation rate) and all mutator type sequences with l mutations (increased mutation rate µ) correspondingly. In these two cases, we have different in general recombination rates c and µC.
II.
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dPl (t) = Ldt
SOLUTION OF MUTATOR GENE MODEL WITH RECOMBINATION A.
dQl (t) = Ldt
The mathematical model
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We consider a simplified evolution model with two types of biallelic genes and symmetric fitness landscape. The genome includes a mutator gene, having a normal and mutator allele, and the main part with L genes. Each gene in a sequence is encoded by ±1: +1 stands for the wild-type allele for the gene, −1 — for the mutant allele. Thus, the genetic space consists of 2L+1 different sequences, which can be divided into Hamming classes according to the number of accumulated mutations in the main part. Depending on the mutator gene, the genetic sequence can be in one of two states: the wild-type state with the mean number of mutations ¯l1 ; and the mutator state with the mean number of mutations ¯l2 .
R(t) =
α2 Ql + Pl (f1 (xl ) − (1 + α1 )) + ¯l1 L−l+1 1+c + Pl−1 L L L − ¯l1 l+1 1+c − Pl+1 L L ¯ ¯l1 l1 l l 1− cPl 1− + − Pl R(t), L L LL α1 Pl + Ql (f2 (xl ) − (µ + α2 )) + ¯ L−l+1 l2 µ Ql−1 1+C + L L L − l¯2 l+1 1+C − Ql+1 L L ¯ l¯2 l2 l l 1− CQl 1− + − Ql R(t), L L LL X (Pl (t)f1 (xl ) + Ql (t)f2 (xl )) , l
¯l1 =
X l
Pl l, ¯l2 =
X
Ql l,
l
xl = 1 − 2l/L, 0 ≤ l ≤ L. We can think of this model as a combination of the basic mutator model [22] and standard recombination model
(1)
Journal Pre-proof 3
b. Different mutation rates, but same recombination rates µ 6= 1, c = C. c. Same mutation and recombinations rates µ = 1, c = C. d. A generalization of Eq. (1) for several fitness landscapes with µ = 1, c = C. To solve Eq. (1), we use the following ansatz [22]: Pl = P (x, t) = v1 (x)eLu(x,t) , (2)
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where x is obtained from xl = 1 − 2l/L in a large genome limit. Applying the expressions (2) to the system (1), we derive: 1 − xs v2 α2 + v1 −u′t + f1 (x) − α1 − c + 2 1 + x 2u′x 1 − x −2u′x c1 e + c2 e −1 = 0, 2 2 1 − xs v1 α1 + v2 −u′t + f2 (x) − α2 − Cµ + 2 ′ ′ 1−x 1+x C1 e2ux + C2 e−2ux − 1 = 0. (3) µ 2 2 , c2 = 1 + c 1+s , Here, we denoted c1 = 1 + c 1−s 2 2 1+s C1 = 1 + C 1−s , C = 1 + C . While deriving 2 2 2 Eq. (3), we dropped the terms of the order O(1/L). We can consider Eq. (3) as a system of linear equations with respect to the variables v1 , v2 . The consistency of a system of linear equations is guaranteed by a zero determinant det A = 0 condition, where we denoted the matrix on the right-hand side of Eq. (3) as A. By solving detA(q) = 0 for q ≡ ut , we obtain a Hamilton–Jacobi equation (HJE). In Appendix A, we derive the HJE for the general case. Here, we provide the solution for the case C = c: u′t
Having the HJE derived, we can calculate the mean fitness using the minimax principle [23]. To apply it, we take the minimum of the Hamiltonian with respect to p = u′x and then the maximum with respect to x. The latter can be derided following the idea of the study [33] that the fitness equals to the value of the fitness function at the surplus point. Thus, we get the expressions for the mean fitness value in the steady-state: f1 (x) + f2 (x) α1 + α2 1+µ R = max − +A 2 2 2 q 1 2 ± (f1 (x) − f2 (x) − (α1 − α2 ) − (1 − µ)A) + 4α1 α2 , 2 x r p 2 2 2 c s 1 − xs c A = 1 − x2 − −c . (5) 1+ 2 4 2 Then, taking into account p = 0 at the point x = s, we obtain from Eq. (4) an additional equation:
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Ql = Q(x, t) = v2 (x)eLu(x,t) ,
Analytical results for the two-landscape case with different mutation rates
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a. Different mutation and recombination rates µ 6= 1, C 6= c (see Appendix A).
B.
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[32], where the set of terms in the equations (1) is examined. The parameters of this model (1) define the balance between replication, mutation, and recombination, and, thus, the evolutionary behaviour of the system. We will consider the following scenarios for the model:
f1 (s) + f2 (s) α1 + α2 R(s) = − (6) 2 2 q 1 2 (f1 (s) − f2 (s) − (α1 − α2 )) + 4α1 α2 , + 2
where s = 1 − h 2l L i is the surplus of the population, l is the number of mutations. Thus, we get a system of two equations for the variables x and s. However, for the considered case, we can not introduce the effective fitness. C. Numerical results for the two-landscape case with different mutation and recombination rates and different fitnesses
Figures 1–3 describe the case referred as (a) in our list, where µ 6= 1, C 6= c, and the fitness functions for mutator and normal types are different. The numerics confirms the validity of our results for mean fitness and surplus. R 0.5 0.4 0.3
+ H± (x, p) = 0, (4) 0.2 f1 (x) + f2 (x) α1 + α2 1+µ 0.1 −H± (x, p) = − +A 2 2 2 q 1 0.0 c 0 1 2 3 4 (f1 (x) − f2 (x) − (α1 − α2 ) − (1 − µ)A)2 + 4α1 α2 , ± 2 1 − xs 1 + x 2p 1 − x −2p FIG. 1: The mean fitness versus the recombination rate for −c c1 e + c2 e . A = −1 + f1 (x) = x2 , f2 (x) = 3x2 , µ = 3, L = 1000. 2 2 2
Journal Pre-proof 4 R 3.07
the number of mutations. The effective fitness takes the form:
3.06 3.05
f1 (x) + f2 (x) α1 + α2 − (8) 2 q2 1 2 (f1 (x) − f2 (x) − (α1 − α2 )) + 4α1 α2 . ± 2
fˆ(x) =
3.04 3.03 3.02
We see that there is an effective fitness different from the averaged fitness value over two system states f¯(x) = f1 (x)α2 +f2 (x)α1 . Let us consider different choices of the α1 +α2 functions f1 , f2 . The case with the complementary fitness landscapes f2 (x) = −f1 (x) is of particular interest, since the mean of the fitness value turns to zero. For the effective fitness, we derive for α1 = α2 = α:
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3.01 c
3.00 0
1
2
3
4
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FIG. 2: The mean fitness versus the recombination rate for f1 (x) = 2x, f2 (x) = 6x, µ = 3, L = 1000. R 0.5
1p fˆ(x) = −α + (f1 (x) − α)2 + α2 2
0.4 0.3
Epistasis is another important concept in evolutionary biology to discuss in this study. Changes in the genomic sequences can exhibit the synergetic effects: fitness of a collection of the genes differs from the sum of the contribution of different genes to the fitness. For the linear fitness function in the Crow-Kimura model, there is a zero epistasis. In population genetics, the epistasis ε is the measure of fitness interactions between alleles at different loci. For the two-locus measure, ε = f itness(AB)f itness(ab)f itness(Ab)f itness(aB). For our case with many loci, we can define a sign of epistasis as the sign of the second-order derivative of our fitness function, f ′′ (x). It is easy to verify that our definition of epistasis sign coincides with the definition in population genetics. Moreover, we looked at the epistasis of effective fitness altogether with the change of the mean fitness with the recombination. We found that the transition between the landscapes can reduce the degree of negativeness of the landscape; as a result, the recombination reduces the mean fitness, see Fig. 5.
0.0 0.0
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0.2 0.1
c
0.5
1.0
1.5
2.0
2.5
3.0
FIG. 3: The surplus versus the recombination rate for f1 (x) = 1.5x2 , f2 (x) = x2 , µ = 2, L = 1000.
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D. Analytical results for the two-landscape case with same mutation and recombination rates, but different fitnesses
R 1
1
1
1.46 0
1
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2
3
(x)
f
1.0 0.5 c 4
-1.0
-0.5
0.5
1.0
-0.5
FIG. 4: The mean fitness versus the recombination rate for f1 (x) = x2 , f2 (x) = 3x2 , µ = 1, L = 1000.
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-1.0 -1.5
Consider a simpler case when two landscapes have the same mutation and recombination rates µ = 1, C = c, while different fitness functions. Repeating the derivations of the previous subsection, we get: # " r 2 c2 s2 p c − R = max fˆ(x) − µ + µ 1 − x2 1+ 2 4
s
R = fˆ(s),
(9)
(7)
where s = 1 − h 2l L i is the surplus of the population, l is
-2.0 -2.5
FIG. 5: The second derivative f ′′ (x) versus x for the model with α1 = α2 = 1, f1 (x) = 2(1 + x)k , f2 (x) = 2(1 − x)k for the values k = 0.1, 0.33, 0.5 from low to high. The epistasis is defined by the sign of the calculated function. We found that the recombination increases the mean fitness in case of negative epistasis (f ′′ (x) < 0 in the whole interval), otherwise it reduces the mean fitness.
Journal Pre-proof 5 the mean fitness we look det(A(R)) with
To generalize the obtained results, we introduce n different evolutionary states with transitions between them. Each states is characterized by its fitness function fm (x), m = 1, . . . , nP . To define the transition matrix, we use aij and a ˆi ≡ j6=i aij . For the probability distribution over the genetic space and environments, we have now Plm , 0 ≤ l ≤ L, 1 ≤ m ≤ n.
R(t)
=
X
alh Plh
+
Plm
h6=m
(fm (xl ) − (µ + a ˆl )) +
xl = 1 − 2l/L, 0 ≤ l ≤ L.
. To drive the HJE, the ansatz is applied: Plm (t) = P m (x, t) = v m eLu(x,t) . From here, it follows: X
(10)
(11)
h
alh v +
h6=m
al
∂u m v = ∂t
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c ˆm − 1 − (1 − ms) + v m fm (xl ) − a 2 ′ 1 + x 1 − s vm e2u 1+c + (12) 2 2 ′ 1 − x 1+s 1+c . e−2u 2 2 We are looking at Eq. (12) as a system of equations for v m . Then, we define a matrix element:
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∂u c Aii (q) = − + fi (x) − a ˆi − 1 − (1 − ms) + ∂t 2 1−s 1+s 2u′ 1 + x −2u′ 1 − x e 1+c +e 1+c , 2 2 2 2 Aij (q) = aij (13) From these expressions, we get the following consistency condition for the linear system of equations: det(A(q)) = 0.
A¯ii (s) = −R + fi (s) − a ˆi ), ¯ Aij (s) = aij
(16)
and we have an equation
¯l1 m L−l+1 µ Pl−1 1+c + L L L − ¯l1 m l+1 1+c − Pl+1 L L ¯ ¯l1 l1 l l Plm R(t) − cPlm 1− 1− + L L LL X m = (Pl (t)fm (xl )) , m,l
Then another equation, at the maximum point
det(A(s)) = 0.
(17)
Consider the case of transitions between three types. The we get HJE (−q + f1 (x) − a + F )(−q + f2 (x) − 2a + F ) × (−q + f3 (x) − a + F ) − (−q + f1 (x) − a + F )a2 − (−q + f3 (x) − a + F )a2 = (018)
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dPlm (t) Ldt
c Aˆii (R) = −R + fi (x) − a ˆi − 1 − (1 − ms) + 2 r 2 c2 s2 p c 1 − xs 1+ 1 − x2 − −c . 2 4 2 Aˆij (R) = aij (15)
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Transitions between more than two evolutionary regimes
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E.
(14)
For this case, similar to the previous ones, we solve the latter equation for q ≡ ∂u ∂t and derive the HJE. To find
where we denoted ′ 1 + x 1−s c 1+c + F = −1 − (1 − ms) + e2u 2 2 2 ′ 1 − x 1+s 1+c e−2u (19) 2 2
Thus the Hamiltonian has three branches. We find the steady state mean fitness as a maximal solution R of the equation
(Fˆ − R + f1 (x) − a)(Fˆ − R + f2 (x))(Fˆ − R + f3 (x)) −(Fˆ − R + f1 (x) − a)a2 − (Fˆ − R + f3 (x) − a)a2 = (020) where we denoted p c Fˆ = −1 − (1 − ms) + 1 − x2 2
r
c c2 s2 (1 + )2 − (21) 2 4
Looking the maximum point, we get another equation (−R + f1 (s) − a)(−R + f2 (s))(−R + f3 (s)) −(−R + f1 (s) − a)a2 − (−R + f3 (s) − a)a2 = 0 (22) Thus we have Eqs. (22), Eq. (21) and the maximum R condition for Eq. (21) We have two equations, related to third order algebraic equation. Contrary, the two type version of mutator is solved via a system of two equations, (5),(6), related to second order algebraic equation. III.
CONCLUSION
The mutator phenomenon and genome instability belong to the area of modern evolution research, attracting
Journal Pre-proof 6
R = H(x, z, s),
(A.3)
of
B1 + B2 H(x, z, s) = f+ (x) − α+ + 2 s 2 B1 − B2 + α1 α2 . f− − α− + + 2
2 (x) 2 , α+ = α1 +α , α− = Here, we denoted f+ = f1 (x)+f 2 2 α1 −α2 . We get the following equation for the mean fitness 2 R: R = max(min[H(x, z, s)] ) (A.4)
z
x
We have another equation, looking the maximum point in Eq. (A.1), so putting z=1, Bi = fi (x). We complete the latter with Eq. (6) to have two system of equations for two unknown variables x, s. Fig. 6 illustrates the accuracy of our analytical results by Eqs. (A.3, A.4),(6)
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Appendix A: The solution of the general case evolutionary model
and z = e2p . Then, we derive the mean fitness and the Hamiltonian:
p ro
the attention of the wide range of experts. We proposed a new model with recombination and mutator gene based on the classical Crow-Kimura model. We apply a statistical physics approach, analyzing the system’s behaviour for symmetric fitness function and different combinations of evolutionary parameters. Nowadays, the quasispecies models are one of the common in interdisciplinary research in this field, partially due to the existing analytical solutions for them. In this study, we solved the mutator-recombination model using the HJE method developed for the case of the master equation with the one-dimensional transition. The numerical simulations supported our analytical results. In the simplest case (the mutation rate is the same for both wild-types and mutator-types), we have shown that the model can be mapped into the Crow-Kimura model with some effective fitness function. The work is supported by the Russian Science Foundation under grant 19-11-00008.
R 0.5
To solve the original system (1) for different mutation and recombination rates, let us look at Eq. (3). The zero determinant condition gives:
0.4 0.3
q 2 − q(B1 + B2 − α1 − α2 ) + (B1 − α1 )(B2 − α2 ) −α1 α2 = 0 (A.1)
0.2 0.1
al
where we denoted B1 = f1 (x) − c 1−xs + A1 , B2 = f2 − 2 Cµ 1−xs + µA , 2 2 1−x 1 1 − xs 1 + x + c1 z + c2 − 1, 2 2 2 z 1 − xs 1+x B2 = f2 (x) − Cµ + µ( C1 z 2 2 1−x 1 + C2 − 1).(A.2) 2 z
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B1 = f1 (x) − c
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[1] Treffers, H. P., Spinelli, V., & Belser, N. O. A factor (or mutator gene) influencing mutation rates in Escherichia coli. Proceedings of the National Academy of Sciences, 40(11), 1064–1071 (1954). [2] Cox, E. C., & Yanofsky, C. Mutator gene studies in Escherichia coli. Journal of bacteriology 100(1), 390–397 (1969). [3] Cox, E. C. Bacterial mutator genes and the control of spontaneous mutation. Annual review of genetics, 10(1), 135–156 (1976). [4] Pierce, B. L. S. The effect of a bacterial mutator gene upon mutation rates in bacteriophage T4. Genetics, 54, 657–62 (1966). [5] Ives, P. T. The importance of mutation rate genes in evolution. Evolution, 4(3), 236–252(1970). [6] Green, M. M. . The genetics of a mutator
0.0 0
1
2
3
4
C
FIG. 6: The mean fitness versus the recombination rate C for f1 (x) = 3/2x2 , f2 (x) = x2 , µ = 1, c = 1, L = 1000. The smooth line is the result of numerical simulations, solid dots correspond to the analytical result.
[7] [8] [9] [10]
[11] [12]
gene in Drosophila melanogaster. Mutation Research/Fundamental and Molecular Mechanisms of Mutagenesis, 10(4), 353–363 (1970). Von Borstel, R. C., Graham, D. E., La Brot, K. L., Resnick, M. A.: Mutator activity of a X-radiation sensitive yeast. Genetics, 60, 233 (1968). Mohn, G., & Wrgler, F. E. Mutator genes in different species. Human Genetics, 16(1), 49–58 (1972). Nowell, C. P. The clonal evolution of tumor cell populations. Science 194, 2328 (1976). Merlo, L. M. F., Pepper, J. W., Reid B. J. & Maley, C. C. Cancer as an evolutionary and ecological process. Nature Review Cancers 6, 924-935 (2006). Greaves, M.& Maley, C. C. Clonal evolution in cancer. Nature 481, 306-311 (2012). Hanahan, D. & Weinberg, R. A. The hallmarks of cancer.
Journal Pre-proof 7
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