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Olofsson P, Chipkin L, Daileda RC, Azevedo RBR. Mutational meltdown in asexual populations doomed to extinction. J Math Biol 2023; 87:88. [PMID: 37994999 DOI: 10.1007/s00285-023-02019-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/10/2023] [Revised: 07/03/2023] [Accepted: 10/20/2023] [Indexed: 11/24/2023]
Abstract
Asexual populations are expected to accumulate deleterious mutations through a process known as Muller's ratchet. Lynch and colleagues proposed that the ratchet eventually results in a vicious cycle of mutation accumulation and population decline that drives populations to extinction. They called this phenomenon mutational meltdown. Here, we analyze mutational meltdown using a multi-type branching process model where, in the presence of mutation, populations are doomed to extinction. We analyse the change in size and composition of the population and the time of extinction under this model.
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Affiliation(s)
- Peter Olofsson
- Department of Mathematics, Trinity University, San Antonio, TX, 78212, USA
- Department of Mathematics, Physics and Chemical Engineering, Jönköping University, 551 11, Jönköping, Sweden
| | - Logan Chipkin
- Department of Biology and Biochemistry, University of Houston, Houston, TX, 77204, USA
| | - Ryan C Daileda
- Department of Mathematics, Trinity University, San Antonio, TX, 78212, USA
| | - Ricardo B R Azevedo
- Department of Biology and Biochemistry, University of Houston, Houston, TX, 77204, USA.
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2
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Laws of Spatially Structured Population Dynamics on a Lattice. PHYSICS 2022. [DOI: 10.3390/physics4030052] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/01/2023]
Abstract
We consider spatial population dynamics on a lattice, following a type of a contact (birth–death) stochastic process. We show that simple mathematical approximations for the density of cells can be obtained in a variety of scenarios. In the case of a homogeneous cell population, we derive the cellular density for a two-dimensional (2D) spatial lattice with an arbitrary number of neighbors, including the von Neumann, Moore, and hexagonal lattice.are not allowed in the abstract, please move them to the main text, please remember to check the order of references in the full text after revision. We then turn our attention to evolutionary dynamics, where mutant cells of different properties can be generated. For disadvantageous mutants, we derive an approximation for the equilibrium density representing the selection–mutation balance. For neutral and advantageous mutants, we show that simple scaling (power) laws for the numbers of mutants in expanding populations hold in 2D and 3D, under both flat (planar) and range population expansion. These models have relevance for studies in ecology and evolutionary biology, as well as biomedical applications including the dynamics of drug-resistant mutants in cancer and bacterial biofilms.
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Castillo D, Lavrentovich MO. Shape of population interfaces as an indicator of mutational instability in coexisting cell populations. Phys Biol 2020; 17:066002. [PMID: 33210619 DOI: 10.1088/1478-3975/abb2dd] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
Abstract
Cellular populations such as avascular tumors and microbial biofilms may 'invade' or grow into surrounding populations. The invading population is often comprised of a heterogeneous mixture of cells with varying growth rates. The population may also exhibit mutational instabilities, such as a heavy deleterious mutation load in a cancerous growth. We study the dynamics of a heterogeneous, mutating population competing with a surrounding homogeneous population, as one might find in a cancerous invasion of healthy tissue. We find that the shape of the population interface serves as an indicator for the evolutionary dynamics within the heterogeneous population. In particular, invasion front undulations become enhanced when the invading population is near a mutational meltdown transition or when the surrounding 'bystander' population is barely able to reinvade the mutating population. We characterize these interface undulations and the effective fitness of the heterogeneous population in one- and two-dimensional systems.
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Affiliation(s)
- Daniel Castillo
- Department of Physics & Astronomy, University of Tennessee, Knoxville, Tennessee 37996, United States of America
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Wodarz D, Komarova NL. Mutant Evolution in Spatially Structured and Fragmented Expanding Populations. Genetics 2020; 216:191-203. [PMID: 32661138 PMCID: PMC7463292 DOI: 10.1534/genetics.120.303422] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/17/2020] [Accepted: 06/23/2020] [Indexed: 11/18/2022] Open
Abstract
Mutant evolution in spatially structured systems is important for a range of biological systems, but aspects of it still require further elucidation. Adding to previous work, we provide a simple derivation of growth laws that characterize the number of mutants of different relative fitness in expanding populations in spatial models of different dimensionalities. These laws are universal and independent of "microscopic" modeling details. We further study the accumulation of mutants and find that, with advantageous and neutral mutants, more of them are present in spatially structured, compared to well-mixed colonies of the same size. The behavior of disadvantageous mutants is subtle: if they are disadvantageous through a reduction in division rates, the result is the same, and it is the opposite if the disadvantage is due to a death rate increase. Finally, we show that in all cases, the same results are observed in fragmented, nonspatial patch models. This suggests that the patterns observed are the consequence of population fragmentation, and not spatial restrictions per se We provide an intuitive explanation for the complex dependence of disadvantageous mutant evolution on spatial restriction, which relies on desynchronized dynamics in different locations/patches, and plays out differently depending on whether the disadvantage is due to a lower division rate or a higher death rate. Implications for specific biological systems, such as the evolution of drug-resistant cell mutants in cancer or bacterial biofilms, are discussed.
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Affiliation(s)
- Dominik Wodarz
- Department of Population Health and Disease Prevention, Program in Public Health, Susan and Henry Samueli College of Health Sciences, University of California Irvine, California 92697
- Department of Mathematics, University of California Irvine, California 92697
| | - Natalia L Komarova
- Department of Mathematics, University of California Irvine, California 92697
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Abstract
Evolution can potentially rescue populations from being driven extinct by biological invasions, but predictions for this occurrence are generally lacking. Here I derive theoretical predictions for evolutionary rescue of a resident population experiencing invasion from an introduced competitor that spreads over its introduced range as a traveling spatial wave that displaces residents. I compare the likelihood of evolutionary rescue from invasion for two modes of competition: exploitation and interference competition. I find that, all else equal, evolutionary rescue is less effective at preventing extinction caused by interference-driven invasions than by exploitation-driven invasions. Rescue from interference-driven invasions is, surprisingly, independent of invader dispersal rate or the speed of invasion and is more weakly dependent on range size than in the exploitation-driven case. In contrast, rescue from exploitation-driven invasions strongly depends on range size and is less likely during fast invasions. The results presented here have potential applications for conserving endemic species from nonnative invaders and for ensuring extinction of pests using intentionally introduced biocontrol agents.
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Gralka M, Hallatschek O. Environmental heterogeneity can tip the population genetics of range expansions. eLife 2019; 8:e44359. [PMID: 30977724 PMCID: PMC6513619 DOI: 10.7554/elife.44359] [Citation(s) in RCA: 33] [Impact Index Per Article: 6.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2018] [Accepted: 04/11/2019] [Indexed: 12/12/2022] Open
Abstract
The population genetics of most range expansions is thought to be shaped by the competition between Darwinian selection and random genetic drift at the range margins. Here, we show that the evolutionary dynamics during range expansions is highly sensitive to additional fluctuations induced by environmental heterogeneities. Tracking mutant clones with a tunable fitness effect in bacterial colonies grown on randomly patterned surfaces we found that environmental heterogeneity can dramatically reduce the efficacy of selection. Time-lapse microscopy and computer simulations suggest that this effect arises generically from a local 'pinning' of the expansion front, whereby stretches of the front are slowed down on a length scale that depends on the structure of the environmental heterogeneity. This pinning focuses the range expansion into a small number of 'lucky' individuals with access to expansion paths, altering the neutral evolutionary dynamics and increasing the importance of chance relative to selection.
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Affiliation(s)
- Matti Gralka
- Department of PhysicsUniversity of California, BerkeleyBerkeleyUnited States
| | - Oskar Hallatschek
- Department of PhysicsUniversity of California, BerkeleyBerkeleyUnited States
- Department of Integrative BiologyUniversity of California, BerkeleyBerkeleyUnited States
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Kayser J, Schreck CF, Gralka M, Fusco D, Hallatschek O. Collective motion conceals fitness differences in crowded cellular populations. Nat Ecol Evol 2018; 3:125-134. [PMID: 30510177 PMCID: PMC6309230 DOI: 10.1038/s41559-018-0734-9] [Citation(s) in RCA: 27] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2018] [Accepted: 10/23/2018] [Indexed: 12/15/2022]
Abstract
Many cellular populations are tightly-packed, such as microbial colonies and biofilms, or tissues and tumors in multicellular organisms. Movement of one cell in those crowded assemblages requires motion of others, so that cell displacements are correlated over many cell diameters. Whenever movement is important for survival or growth, these correlated rearrangements could couple the evolutionary fate of different lineages. Yet, little is known about the interplay between mechanical forces and evolution in dense cellular populations. Here, by tracking slower-growing clones at the expanding edge of yeast colonies, we show that the collective motion of cells prevents costly mutations from being weeded out rapidly. Joint pushing by neighboring cells generates correlated movements that suppress the differential displacements required for selection to act. This mechanical screening of fitness differences allows slower-growing mutants to leave more descendants than expected under non-mechanical models, thereby increasing their chance for evolutionary rescue. Our work suggests that, in crowded populations, cells cooperate with surrounding neighbors through inevitable mechanical interactions. This effect has to be considered when predicting evolutionary outcomes, such as the emergence of drug resistance or cancer evolution.
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Affiliation(s)
- Jona Kayser
- Department of Physics, University of California, Berkeley, Berkeley, CA, USA.,Department of Integrative Biology, University of California, Berkeley, Berkeley, CA, USA
| | - Carl F Schreck
- Department of Physics, University of California, Berkeley, Berkeley, CA, USA.,Department of Integrative Biology, University of California, Berkeley, Berkeley, CA, USA
| | - Matti Gralka
- Department of Physics, University of California, Berkeley, Berkeley, CA, USA
| | - Diana Fusco
- Department of Physics, University of California, Berkeley, Berkeley, CA, USA.,Department of Integrative Biology, University of California, Berkeley, Berkeley, CA, USA
| | - Oskar Hallatschek
- Department of Physics, University of California, Berkeley, Berkeley, CA, USA. .,Department of Integrative Biology, University of California, Berkeley, Berkeley, CA, USA.
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Spatially Constrained Growth Enhances Conversional Meltdown. Biophys J 2017; 110:2800-2808. [PMID: 27332138 DOI: 10.1016/j.bpj.2016.05.024] [Citation(s) in RCA: 22] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/28/2016] [Revised: 04/28/2016] [Accepted: 05/16/2016] [Indexed: 11/23/2022] Open
Abstract
Cells that mutate or commit to a specialized function (differentiate) often undergo conversions that are effectively irreversible. Slowed growth of converted cells can act as a form of selection, balancing unidirectional conversion to maintain both cell types at a steady-state ratio. However, when one-way conversion is insufficiently counterbalanced by selection, the original cell type will ultimately be lost, often with negative impacts on the population's overall fitness. The critical balance between selection and conversion needed for preservation of unconverted cells and the steady-state ratio between cell types depends on the spatial circumstances under which cells proliferate. We present experimental data on a yeast strain engineered to undergo irreversible conversion: this synthetic system permits cell-type-specific fluorescent labeling and exogenous variation of the relative growth and conversion rates. We find that populations confined to grow on a flat agar surface are more susceptible than their well-mixed counterparts to fitness loss via a conversion-induced "meltdown." We then present analytical predictions for growth in several biologically relevant geometries-well-mixed liquid media, radially expanding two-dimensional colonies, and linear fronts in two dimensions-by employing analogies to the directed-percolation transition from nonequilibrium statistical physics. These simplified theories are consistent with the experimental results.
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De Luca A, Le Doussal P. Mutually avoiding paths in random media and largest eigenvalues of random matrices. Phys Rev E 2017; 95:030103. [PMID: 28415280 DOI: 10.1103/physreve.95.030103] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/22/2016] [Indexed: 06/07/2023]
Abstract
Recently, it was shown that the probability distribution function (PDF) of the free energy of a single continuum directed polymer (DP) in a random potential, equivalently to the height of a growing interface described by the Kardar-Parisi-Zhang (KPZ) equation, converges at large scale to the Tracy-Widom distribution. The latter describes the fluctuations of the largest eigenvalue of a random matrix, drawn from the Gaussian unitary ensemble (GUE), and the result holds for a DP with fixed end points, i.e., for the KPZ equation with droplet initial conditions. A more general conjecture can be put forward, relating the free energies of N>1 noncrossing continuum DP in a random potential, to the sum of the Nth largest eigenvalues of the GUE. Here, using replica methods, we provide an important test of this conjecture by calculating exactly the right tails of both PDFs and showing that they coincide for arbitrary N.
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Affiliation(s)
- Andrea De Luca
- Laboratoire de Physique Théorique et Modèles Statistiques (UMR CNRS 8626), Université Paris-Sud, Orsay, France
| | - Pierre Le Doussal
- Laboratoire de Physique Théorique de l'ENS, CNRS & Ecole Normale Supérieure de Paris, Paris, France
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De Luca A, Le Doussal P. Crossing probability for directed polymers in random media. II. Exact tail of the distribution. Phys Rev E 2016; 93:032118. [PMID: 27078303 DOI: 10.1103/physreve.93.032118] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/25/2015] [Indexed: 06/05/2023]
Abstract
We study the probability p ≡ p(η)(t) that two directed polymers in a given random potential η and with fixed and nearby endpoints do not cross until time t. This probability is itself a random variable (over samples η), which, as we show, acquires a very broad probability distribution at large time. In particular, the moments of p are found to be dominated by atypical samples where p is of order unity. Building on a formula established by us in a previous work using nested Bethe ansatz and Macdonald process methods, we obtain analytically the leading large time behavior of all moments p(m) ≃ γ(m)/t. From this, we extract the exact tail ∼ρ(p)/t of the probability distribution of the noncrossing probability at large time. The exact formula is compared to numerical simulations, with excellent agreement.
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Affiliation(s)
- Andrea De Luca
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Pierre Le Doussal
- Laboratoire de Physique Théorique de l'ENS, CNRS and Ecole Normale Supérieure de Paris, Paris, France
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De Luca A, Le Doussal P. Crossing probability for directed polymers in random media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:040102. [PMID: 26565146 DOI: 10.1103/physreve.92.040102] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/06/2015] [Indexed: 06/05/2023]
Abstract
We study the probability that two directed polymers in the same random potential do not intersect. We use the replica method to map the problem onto the attractive Lieb-Liniger model with generalized statistics between particles. Employing both the nested Bethe ansatz and known formula from MacDonald processes, we obtain analytical expressions for the first few moments of this probability and compare them to a numerical simulation of a discrete model at high temperature. From these observations, several large time properties of the noncrossing probabilities are conjectured. Extensions of our formalism to more general observables are discussed.
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Affiliation(s)
- Andrea De Luca
- Laboratoire de Physique Théorique et Modèles Statistiques (UMR CNRS 8626), Université Paris-Sud, Orsay, France
| | - Pierre Le Doussal
- Laboratoire de Physique Théorique de l'ENS, CNRS and Ecole Normale Supérieure de Paris, Paris, France
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Ralph PL, Coop G. The Role of Standing Variation in Geographic Convergent Adaptation. Am Nat 2015; 186 Suppl 1:S5-23. [PMID: 26656217 DOI: 10.1086/682948] [Citation(s) in RCA: 38] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023]
Abstract
The extent to which populations experiencing shared selective pressures adapt through a shared genetic response is relevant to many questions in evolutionary biology. In this article, we explore how standing genetic variation contributes to convergent genetic responses in a geographically spread population. Geographically limited dispersal slows the spread of each selected allele, hence allowing other alleles to spread before any one comes to dominate the population. When selectively equivalent alleles meet, their progress is substantially slowed, dividing the species range into a random tessellation, which can be well understood by analogy to a Poisson process model of crystallization. In this framework, we derive the geographic scale over which an allele dominates and the proportion of adaptive alleles that arise from standing variation. Finally, we explore how negative pleiotropic effects of alleles can bias the subset of alleles that contribute to the species' adaptive response. We apply the results to the malaria-resistance glucose-6-phosphate dehydrogenase-deficiency alleles, where the large mutational target size makes it a likely candidate for adaptation from deleterious standing variation. Our results suggest that convergent adaptation may be common. Therefore, caution must be exercised when arguing that strongly geographically restricted alleles are the outcome of local adaptation. We close by discussing the implications of these results for ideas of species coherence and the nature of divergence between species.
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Affiliation(s)
- Peter L Ralph
- Computational Biology and Bioinformatics, University of Southern California, Los Angeles, California 90089
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