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Suvorov V, SaAkian DB, Lynch M. A solution of the Crow-Kimura evolution model on fluctuating fitness landscape. EUROPHYSICS LETTERS 2023; 142:57003. [PMID: 39175800 PMCID: PMC11340687 DOI: 10.1209/0295-5075/acd65b] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 08/24/2024]
Abstract
The article discusses the Crow-Kimura model in the context of random transitions between different fitness landscapes. The duration of epochs, during which the fitness landscape is constant over time, is modeled by an exponential distribution. To obtain an exact solution, a system of functional equations is required. However, to approximate the model, we consider the cases of slow or fast transitions and calculate the first-order corrections using either the transition rate or its inverse. Specifically, we focus on the case of slow transitions and find that the average fitness is equal to the average fitness for evolution on static fitness landscapes, but with the addition of a load term. We also investigate the model for a small number of genes and identify the exact transition points to the transient phase.
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Affiliation(s)
| | - David B. SaAkian
- A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation - 2 Alikhanian Brothers St., Yerevan 375036, Armenia
| | - Michael Lynch
- Biodesign Center for Mechanisms of Evolution, Arizona State University - Tempe, AZ, 85287, USA
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Saakian DB, Cheong KH. Weak mixed phase in the mutator model. Phys Rev E 2021; 103:032113. [PMID: 33862733 DOI: 10.1103/physreve.103.032113] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2020] [Accepted: 02/10/2021] [Indexed: 11/07/2022]
Abstract
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.
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Affiliation(s)
- David B Saakian
- Laboratory of Applied Physics, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam.,Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam
| | - Kang Hao Cheong
- Science, Mathematics and Technology Cluster, Singapore University of Technology and Design, 8 Somapah Road, S487372 Singapore
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Saakian DB, Yakushkina T, Koonin EV. Allele fixation probability in a Moran model with fluctuating fitness landscapes. Phys Rev E 2019; 99:022407. [PMID: 30934266 DOI: 10.1103/physreve.99.022407] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/26/2018] [Indexed: 06/09/2023]
Abstract
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.
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Affiliation(s)
- David B Saakian
- Laboratory of Applied Physics, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
- Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam
| | - Tatiana Yakushkina
- National Research University Higher School of Economics, Moscow 101000, Russia
| | - Eugene V Koonin
- National Center for Biotechnology Information, National Library of Medicine, National Institutes of Health, Bethesda, MD 20894, USA
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Huang GR, Saakian DB, Hu CK. Accurate analytic solution of chemical master equations for gene regulation networks in a single cell. Phys Rev E 2018; 97:012412. [PMID: 29448337 DOI: 10.1103/physreve.97.012412] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/30/2017] [Indexed: 12/21/2022]
Abstract
Studying gene regulation networks in a single cell is an important, interesting, and hot research topic of molecular biology. Such process can be described by chemical master equations (CMEs). We propose a Hamilton-Jacobi equation method with finite-size corrections to solve such CMEs accurately at the intermediate region of switching, where switching rate is comparable to fast protein production rate. We applied this approach to a model of self-regulating proteins [H. Ge et al., Phys. Rev. Lett. 114, 078101 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.078101] and found that as a parameter related to inducer concentration increases the probability of protein production changes from unimodal to bimodal, then to unimodal, consistent with phenotype switching observed in a single cell.
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Affiliation(s)
- Guan-Rong Huang
- Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan
| | - David B Saakian
- Theoretical Physics Research Group, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam.,Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam.,Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan
| | - Chin-Kun Hu
- Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan.,Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan.,Department of Systems Science, University of Shanghai for Science and Technology, Shanghai 200093, China.,Department of Physics, National Dong Hwa University, Hualien 97401, Taiwan
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Saakian DB, Bratus AS, Hu CK. Crossing fitness canyons by a finite population. Phys Rev E 2017; 95:062405. [PMID: 28709354 DOI: 10.1103/physreve.95.062405] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/08/2016] [Indexed: 11/07/2022]
Abstract
We consider the Wright-Fisher model of the finite population evolution on a fitness landscape defined in the sequence space by a path of nearly neutral mutations. We study a specific structure of the fitness landscape: One of the intermediate mutations on the mutation path results in either a large fitness value (climbing up a fitness hill) or a low fitness value (crossing a fitness canyon), the rest of the mutations besides the last one are neutral, and the last sequence has much higher fitness than any intermediate sequence. We derive analytical formulas for the first arrival time of the mutant with two point mutations. For the first arrival problem for the further mutants in the case of canyon crossing, we analytically deduce how the mean first arrival time scales with the population size and fitness difference. The location of the canyon on the path of sequences has a crucial role. If the canyon is at the beginning of the path, then it significantly prolongs the first arrival time; otherwise it just slightly changes it. Furthermore, the fitness hill at the beginning of the path strongly prolongs the arrival time period; however, the hill located near the end of the path shortens it. We optimize the first arrival time by applying a nonzero selection to the intermediate sequences. We extend our results and provide a scaling for the valley crossing time via the depth of the canyon and population size in the case of a fitness canyon at the first position. Our approach is useful for understanding some complex evolution systems, e.g., the evolution of cancer.
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Affiliation(s)
- David B Saakian
- Theoretical Physics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam.,Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam.,Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan
| | - Alexander S Bratus
- Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow 119992, Russia
| | - Chin-Kun Hu
- Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan.,National Center for Theoretical Sciences, National Tsing Hua University, Hsinchu 30013, Taiwan.,Business School, University of Shanghai for Science and Technology, Shanghai 200093, China
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Saakian DB, Hu CK. Solution of classical evolutionary models in the limit when the diffusion approximation breaks down. Phys Rev E 2016; 94:042422. [PMID: 27841654 DOI: 10.1103/physreve.94.042422] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/16/2016] [Indexed: 06/06/2023]
Abstract
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.
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Affiliation(s)
- David B Saakian
- Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan
- A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, 2 Alikhanian Brothers St., Yerevan 375036, Armenia
| | - Chin-Kun Hu
- Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan
- National Center for Theoretical Sciences, National Tsing Hua University, Hsinchu 30013, Taiwan
- Business School, University of Shanghai for Science and Technology, Shanghai 200093, China
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