Enhanced (hydrodynamic) transport induced by population growth in reaction–diffusion systems with application to population genetics

Vlad, Marcel Ovidiu; Cavalli-Sforza, L. L.; Ross, John

doi:10.1073/pnas.0403419101

Cited by 28 publications

(37 citation statements)

References 10 publications

(12 reference statements)

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“…Under this classical KPP model, any fraction except that located at the head of the front vanishes progressively; this shows that the "enhanced transport" of neutral fractions described by Vlad et. al (28) for similar equations is only a transient phenomenon.…”

Section: Numerical Computationsmentioning

confidence: 96%

Allee effect promotes diversity in traveling waves of colonization

Roques

Garnier

Hamel

et al. 2012

Proc. Natl. Acad. Sci. U.S.A.

168

225

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Most mathematical studies on expanding populations have focused on the rate of range expansion of a population. However, the genetic consequences of population expansion remain an understudied body of theory. Describing an expanding population as a traveling wave solution derived from a classical reaction-diffusion model, we analyze the spatio-temporal evolution of its genetic structure. We show that the presence of an Allee effect (i.e., a lower per capita growth rate at low densities) drastically modifies genetic diversity, both in the colonization front and behind it. With an Allee effect (i.e., pushed colonization waves), all of the genetic diversity of a population is conserved in the colonization front. In the absence of an Allee effect (i.e., pulled waves), only the furthest forward members of the initial population persist in the colonization front, indicating a strong erosion of the diversity in this population. These results counteract commonly held notions that the Allee effect generally has adverse consequences. Our study contributes new knowledge to the surfing phenomenon in continuous models without random genetic drift. It also provides insight into the dynamics of traveling wave solutions and leads to a new interpretation of the mathematical notions of pulled and pushed waves. R apid increases in the number of biological invasions by alien organisms (1) and the movement of species in response to their climatic niches shifting as a result of climate change have caused a growing number of empirical and observational studies to address the phenomenon of range expansion. Numerous mathematical approaches and simulations have been developed to analyze the processes of these expansions (2, 3). Most results focus on the rate of range expansion (4), and the genetic consequences of range expansion have received little attention from mathematicians and modelers (5). However, range expansions are known to have an important effect on genetic diversity (6, 7) and generally lead to a loss of genetic diversity along the expansion axis due to successive founder effects (8). Simulation studies have already investigated the role of the geometry of the invaded territory (9-11), the importance of long-distance dispersal and the shape of the dispersal kernel (12-14), the effects of local demography (15), or existence of a juvenile stage (13). Further research is needed to obtain mathematical results supporting these empirical and simulation studies, as such results could determine the causes of diversity loss and the factors capable of increasing or reducing it.In a simulation study using a stepping-stone model with a lattice structure, Edmonds et. al (16) analyzed the fate of a neutral mutation present in the leading edge of an expanding population. Although in most cases the mutation remains at a low frequency in its original position, in some cases the mutation increases in frequency and propagates among the leading edge. This phenomenon is described as "surfing" (15). Surfing is caused by the strong genetic drift ...

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Section: Numerical Computationsmentioning

confidence: 96%

Allee effect promotes diversity in traveling waves of colonization

Roques

Garnier

Hamel

et al. 2012

Proc. Natl. Acad. Sci. U.S.A.

168

225

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“…Even deleterious alleles may reach a high frequency because of surfing (Klopfstein et al 2006;Travis et al 2007;Excoffier and Ray 2008;Hallatschek and Nelson 2010;Hallatschek 2011). In addition to these simulation-based studies, the analytic theory of surfing in terms of reaction-diffusion equations has been developed (Vlad et al 2004;Hallatschek and Nelson 2008;Hallatschek 2011). The serial-founder model has also been investigated by coalescent approaches (Austerlitz et al 1997;DeGiorgio et al 2011) and the theory has been used to date the time of the onset of human expansions from Africa (Liu et al 2006).…”

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confidence: 99%

Serial Founder Effects During Range Expansion: A Spatial Analog of Genetic Drift

2012

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Range expansions cause a series of founder events. We show that, in a one-dimensional habitat, these founder events are the spatial analog of genetic drift in a randomly mating population. The spatial series of allele frequencies created by successive founder events is equivalent to the time series of allele frequencies in a population of effective size k e , the effective number of founders. We derive an expression for k e in a discrete-population model that allows for local population growth and migration among established populations. If there is selection, the net effect is determined approximately by the product of the selection coefficients and the number of generations between successive founding events. We use the model of a single population to compute analytically several quantities for an allele present in the source population: (i) the probability that it survives the series of colonization events, (ii) the probability that it reaches a specified threshold frequency in the last population, and (iii) the mean and variance of the frequencies in each population. We show that the analytic theory provides a good approximation to simulation results. A consequence of our approximation is that the average heterozygosity of neutral alleles decreases by a factor of 1 -1/(2k e ) in each new population. Therefore, the population genetic consequences of surfing can be predicted approximately by the effective number of founders and the effective selection coefficients, even in the presence of migration among populations. We also show that our analytic results are applicable to a model of range expansion in a continuously distributed population.R ANGE expansion through a series of colonization events can produce geographic patterns in allele frequencies that are quite different from what is expected in equilibrium populations. One consequence of range expansion is the steady reduction of heterozygosity with increasing distance from the ancestral population (Austerlitz et al. 1997;DeGiorgio et al. 2011), a pattern well documented in human populations (Prugnolle et al. 2005;Ramachandran et al. 2005;Handley et al. 2007;Li et al. 2008;DeGiorgio et al. 2009;Deshpande et al. 2009). Another consequence is that some alleles may reach a high frequency because of repeated founder events (Edmonds et al. 2004), a process called genetic surfing (Excoffier et al. 2009). Even deleterious alleles may reach a high frequency because of surfing (Klopfstein et al. 2006;Travis et al. 2007;Excoffier and Ray 2008;Hallatschek and Nelson 2010;Hallatschek 2011). In addition to these simulation-based studies, the analytic theory of surfing in terms of reaction-diffusion equations has been developed (Vlad et al. 2004;Hallatschek and Nelson 2008;Hallatschek 2011). The serial-founder model has also been investigated by coalescent approaches (Austerlitz et al. 1997;DeGiorgio et al. 2011) and the theory has been used to date the time of the onset of human expansions from Africa (Liu et al. 2006).In this article, we show that the effect o...

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“…An application that is of interest both in chemical kinetics and genetics is the enhanced transport in reaction-diffusion systems induced by population growth (14,15). Although this type of enhanced transport may occur both in chemistry and genetics, here we discuss it in connection with the genetic applications.…”

Section: Implications In Genetics and Chemical Kineticsmentioning

confidence: 99%

“…However, if the mutation occurs at a position close to the propagation front of the total population, then the transport of the mutation is enhanced and has a diffusive (slow) component as well as a hydrodynamic (fast) component. Considering a single locus with many alleles and isotropic diffusion, the hydrodynamic speed v u of the subpopulation carrying the allele u is given by v u ϭ 2D u ٌ x ln n(x, t), where D u is the diffusion coefficient of the subpopulation u and n(x, t) is the total population density (15). For anisotropic diffusion characterized by a diffusion tensor ‫ބ‬ u the hydrodynamic speed is given by the more general relation…”

Section: Implications In Genetics and Chemical Kineticsmentioning

confidence: 99%

“…For example, a recent research technique uses certain chain reaction for amplifying chemical signals in gene expression and sequencing studies (13); the contribution of response signals of chains of different lengths may be expressed in terms of a Fisher's theorem. Similar approaches may be useful for the study of pattern formation in reaction-diffusion systems as well as geographical population genetics (14,15). We eventually came up with two sets of generalized Fisher equations that can be applied to various physical, chemical, and biological problems, involving time-and͞or space-dependent density fields.…”

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confidence: 99%

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Fisher's theorems for multivariable, time- and space-dependent systems, with applications in population genetics and chemical kinetics

Vlad

Szedlacsek

Pourmand

et al. 2005

Proc. Natl. Acad. Sci. U.S.A.

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We study different physical, chemical, or biological processes involving replication, transformation, and disappearance processes, as well as transport processes, and assume that the time and space dependence of the species densities are known. We derive two types of Fisher equations. The first type relates the average value of the time derivative of the relative time-specific rates of growth of the different species to the variance of the relative, time-specific rates of growth. A second type relates the average value of the gradient or the divergence of the relative, space-specific rates of growth to the space correlation matrix of the relative, space-specific rates of growth. These Fisher equations are exact results, which are independent of the detailed kinetics of the process: they are valid whether the evolution equations are linear or nonlinear, local or nonlocal in space and͞or time and can be applied for the study of a large class of physical, chemical, and biological systems described in terms of time-and͞or spacedependent density fields. We examine the implications of our generalized Fisher relations in population genetics, biochemistry, and chemical kinetics (reaction-diffusion systems). We show that there is a connection between the enhanced (hydrodynamic) transport of mutations induced by population growth and space-specific rate vectors: the velocity of enhanced transport is proportional to the product of the diffusion coefficient of the species and the space rate vector; this relation is similar to a fluctuation-dissipation relation in statistical mechanics. F isher's fundamental theorem of natural selection is one of the basic laws of population genetics. In 1930, Fisher showed that for single-locus genetic systems with pure selection and constant selection coefficients, the rate of variation of the average population fitness equals the genetic variance of the fitness (1). Because the variance is nonnegative, it follows that for systems with pure selection and constant rate coefficients, the average fitness always increases in time, a result that is compatible with general ideas of biological evolution. Fisher claimed that this law should hold the same position among biological sciences as the second law of thermodynamics in physical sciences (1). Many generalizations of Fisher's theorem have been suggested in the literature. A first generalization was introduced by Fisher himself, by taking into account the effects of overpopulation and the effect of deterioration of the environment (1). Further generalizations were introduced by Wright (2), Kimura (3), Price (4), Frank (5) and Ewens (6), Lessard (7), Crow (8), Nagylaki (9), and Edwards (10). A recent article in PNAS (11) reports on a general f luctuationdissipation relation for biological systems, which is qualitatively similar to the Fisher equation. Fisher's theorem also was used for the analysis of economic competition (12).Our research on Fisher's theorem was stimulated by a number of different problems in chemistry and biology. We noticed...

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Enhanced (hydrodynamic) transport induced by population growth in reaction–diffusion systems with application to population genetics

Cited by 28 publications

References 10 publications

Allee effect promotes diversity in traveling waves of colonization

Allee effect promotes diversity in traveling waves of colonization

Serial Founder Effects During Range Expansion: A Spatial Analog of Genetic Drift

Fisher's theorems for multivariable, time- and space-dependent systems, with applications in population genetics and chemical kinetics

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