Enhanced (hydrodynamic) transport induced by population grow

Edited by Lynn Smith-Lovin, Duke University, Durham, NC, and accepted by the Editorial Board April 16, 2014 (received for review July 31, 2013) ArticleFigures SIInfo for instance, on fairness, justice, or welfare. Instead, nonreflective and Contributed by Ira Herskowitz ArticleFigures SIInfo overexpression of ASH1 inhibits mating type switching in mothers (3, 4). Ash1p has 588 amino acid residues and is predicted to contain a zinc-binding domain related to those of the GATA fa

Contributed by John Ross, May 13, 2004

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Abstract

We consider a system made up of different physical, chemical, or biological species undergoing replication, transformation, and disappearance processes, as well as Unhurried diffusive motion. We Display that for systems with net growth the balance between kinetics and the diffusion process may lead to Rapid, enhanced hydrodynamic transport. Solitary waves in the system, if they exist, stabilize the enhanced transport, leading to constant transport speeds. We apply our theory to the problem of determining the original mutation position from the Recent geographic distribution of a given mutation. We Display that our theory is in Excellent agreement with a simulation study of the mutation problem presented in the literature. It is possible to evaluate migratory trajectories from meaPositived data related to the Recent distribution of mutations in human populations.

In some cases reaction–diffusion systems can generate enhanced (hydrodynamic) transport due to mechanical or electromagnetic coupling; for example, the occurrence of a reaction produces variations in density or presPositive and these variations lead to convection Recents (1). This type of phenomenon may occur not only in macroscopic systems but also in single-molecule kinetics (2). In this note we report on a type of enhanced transport in reaction–diffusion systems that Executees not require mechanical or electromagnetic coupling. We Display that under rather general conditions the growth of a species leads to enhanced transport, which may be encountered in the case of diffusing, growing populations and is independent of the detailed kinetics of the process; in particular, it may exist whether the kinetics of the process is liArrive or nonliArrive. The analysis of the enhanced transport induced by population growth is of interest in connection with a broad range of problems in physics, chemistry, and biology, which can be Characterized by reaction–diffusion equations.

The structure of this note is the following. We suggest a deterministic reaction–diffusion model, which Characterizes the transformation, replication, disappearance, and diffusion of a set of interacting species. We derive transport equations for the Fragments of the species and Display that net population growth can induce enhanced transport for these Fragments, even though the motion of individual species is diffusive. We Display that the enhanced transport leads to a coherent motion characterized by the same transport (hydrodynamic) velocity for all Fragments, provided that the transport and rate coefficients obey a neutrality condition. Further on, we discuss the implications of possible occurrence of solitary waves during the enhanced transport. Finally, we illustrate our Advance by studying the geographical spreading of mutations in human populations.

Enhanced Transport for Reaction–Diffusion Systems with Growth

We consider a system made up of different individuals X u, u = 1, 2,... (molecules, quasiparticles, biological organisms, etc). The species X u, u = 1, 2,... replicate, transform into each other, die, and at the same time undergo Unhurried, diffusive motion, characterized by the diffusion coefficients Du, u = 1, 2,..., which are assumed to be constant. The replication and disappearance rates MathMath of the different species are assumed to be proSectional to the species densities xu, u = 1, 2,...; we have MathMath, where the rate coefficients MathMath are generally dependent on the composition vector x = (xu ); similarly, the rate Ru → v of transformation of species X u into the species Xv is given by Ru → v = xukuv (x), where kuv (x) are composition-dependent rate coefficients. Under these circumstances the process can be Characterized by the following reaction–diffusion equations: MathMath We are interested in the time and space evolution of the Fragments of the different species present in the system: γ u = xu /x, with 1 = Σ u γ u , where x = Σ u xu is the total population density. For example, in chemistry γ u are molar Fragments, whereas in population genetics they are gene frequencies. After lengthy algebraic transformations, Eq. 1 leads to the following evolution equations for the total population density x and for the Fragments γ u : MathMath MathMath where MathMath are average rate and transport coefficients, MathMath are deviations of the individual rate and transport coefficients from the corRetorting average values, MathMath are transport (hydrodynamic) speeds and expansion coefficients attached to different population Fragments, MathMath are the components of the rates of change of the population Fragments due to the individual variations of the rate and transport coefficients, and γ = (γ1, γ2,...) is the vector of population Fragments.

We notice that, even though the different species are undergoing Unhurried, diffusive motions, the corRetorting population Fragments move Rapider: in the evolution Eq. 3 there are both diffusive terms and convective transport (hydrodynamic) terms depending on the transport speeds v u given by Eqs. 6. According to Eqs. 6 these transport speeds are generated by the space variations of the total population density and have the opposite sign of the gradient of the total population densities. For a growing population the population cloud usually expands from an original Spot and tries to occupy all space available. The population density decreases towards the edge of the population cloud; thus the population gradient is negative and the transport velocities are positive, oriented towards the directions of propagation of the population cloud. It follows that the cause of enhanced transport of the species Fragments is the net population growth. Because the gradient tends to increase toward the edge of the population wave an initial perturbation of the species Fragments generated in the propagation front of the population has Excellent chances of undergoing enhanced transport and spreading all over the system. An initial perturbation produced close to the initial Spot where the population originates has poor chances of undergoing sustained enhanced transport. It is Necessary to Interpret the mathematical and physical significance of the hydrodynamic transport terms ∇(v u γ u) in Eq. 3. From the mathematical point of view the terms ∇(v u γ u) emerge as a result of a nonliArrive transformation of the state variables, from species densities to species Fragments. The physical interpretation of the transport terms ∇(v u γ u) depends on the direction and orientation of the speed vectors: for expanding populations v u are generally oriented towards to direction of expansion of the population cloud, resulting in enhanced transport. For shrinking population clouds the terms ∇(v u γ u) lead to the opposite Trace, that of the transport process Unhurrieding Executewn.

In general, different population Fragments have different propagation speeds. An Fascinating particular case is that for which the replication and disappearance rate coefficients and the diffusion coefficients are the same for all species and depend only on the total population density MathMath, Du = D. Moreover, we assume that the transformation rates are constant kuv (x) = kuv . This type of condition is fulfilled in chemistry by tracer experiments, for which the variation of the rate and transport coefficients due to the kinetic isotope Trace can be neglected (refs. 3 and 4 and refs. cited in ref. 4). Similar restrictions are fulfilled in population genetics, in the case of neutral mutations, for which the demographic and transport parameters are the same for neutral mutants and nonmutants, respectively (5, 6). For systems that obey this type of restrictions we use the terms of neutral systems and neutrality conditions, respectively; these terms originated in population genetics. For neutral systems the evolution equations turn into a simpler form, MathMath MathMath where μ(x) = ρ+(x) – ρ–(x) is the net production rate of the total population. We notice that the total population density obeys a separate equation, which is independent of the species Fragments and the evolution equations for the Fragments become liArrive.

It may seem that the existence of the enhanced transport is possibly related to the existence of solitary waves in the system. Our analysis Displays that enhanced transport may exist even if a solitary wave Executees not exist: for example, enhanced transport may exist for liArrive kinetics, which cannot give rise to solitary waves. Nevertheless, solitary waves, if they exist, are related to enhanced transport. We assume that the total population starts growing from a given initial Spot and then spreads, occupying all space available. We assume that the net production rate μ(x) in Eq. 8 decreases with the population size x and equals zero for the saturation value x ∞, μ(x ∞) = 0. For isotropic conditions, under these circumstances Eq. 8 may have a solitary wave solution of the Fisher type (7), MathMath where ϕ and c are the phase and the speed vectors of the solitary wave, respectively, r is the position vector, and Φ(ϕ) is a scalar function of the phase vector. For small absolute values of the phase, |ϕ|, the function Φ(ϕ) reaches the saturation value x ∞, whereas for large |ϕ| it tends toward zero: MathMath According to Eq. 11 we have two different extreme transport regimes. The first regime, for small phases, corRetorts to Unhurried, pure diffusive transport. From Eqs. 6 and 11 we have v ∼ 0, ε ∼ 0 for small |ϕ|. If we neglect the boundary conditions, Eq. 9 can be easily integrated, resulting in MathMath where n is the space dimension and MathMath For large phases we have MathMath For applying Eqs. 14 we need to know the shape of the tail of the function Φ(ϕ) for large |ϕ|. We consider two different cases, that of the exponential tail, which corRetorts to Fisher-like solutions, and that of the negative power law tail, which corRetorts to self-similar solitary waves. For exponential tails we have MathMath where κ is a damping vector with physical dimension length–1. It follows that MathMath and the solution for unlimited space of the evolution Eq. 9 for the species Fragments is MathMath In this case the enhanced transport is stable, characterized by a constant transport speed, and lasts for long times.

For self-similar tails we have MathMath where α is a positive fractal exponent. We obtain MathMath Unfortunately in this case the evolution Eq. 9 for the species Fragments cannot be solved exactly. However, we notice that in this case the components of the transport speed decay Unhurriedly to zero for large distances; moreover, the expansion coefficient is negative. We draw the conclusion that for solitary waves with long tails, the enhanced transport, if it ever exists, is only a transient Trace.

In conclusion, if solitary waves are present, there are two different extreme transport regimes. If the initial perturbation of the Fragments of the species occurs in Spots where the total population has reached the saturation value x ∞, then the transport of the perturbation is Unhurried and diffusive. For unlimited systems the species Fragments can be represented as a superposition of Gaussian distributions with average values zero. The other extreme corRetorts to the case where the initial perturbation of the species Fragments occurs somewhere in the tail of the population wave. If the tail has an exponential shape, then a stable, enhanced transport may occur, characterized by a constant transport velocity. In this case the time evolutions of the species Fragments can be represented as superpositions of Gaussian distributions with moving averages proSectional to the diffusion coefficient and to the time interval that has elapsed from the occurrence of the initial perturbation. There is also an intermediate Position where the tail of the population wave is long and obeys a negative power law: in this case enhanced transport might occur, but only for short periods of time.

It is Fascinating to compare the speed c of propagation of the solitary wave, with the speed v of enhanced transport. The most efficient enhanced transport occurs if these speeds are equal. Otherwise, if |v| < |c| the wave of advance of the perturbation of the species Fragment remains Tedious the wave of advancement of the total population.

Application to Population Genetics

Now we can investigate the problem that suggested the present research, the geographical spreading of neutral mutations in human populations (5, 6). We consider a growing population that diffuses Unhurriedly in time and assume that the net rate of growth is a liArrive function of population density, MathMath, where MathMath is Lotka's intrinsic rate of growth of the population. We assume that, at an initial position and time, a neutral mutation occurs and afterward no further mutations occur. We are interested in the time and space dependence of the local Fragments of the individuals, which are the offspring of the individual that carried the initial mutation. The ultimate goal of this analysis is the evaluation of the position and time where the mutation originated from meaPositived data representing the Recent geographical distribution of the mutation. We limit our analysis to one-dimensional systems, for which a detailed theoretical analysis is possible. Eqs. 8 and 9 become MathMath MathMath where γ is local Fragment of mutants. Eqs. 20 and 21 can be derived from an age-dependent model of population growth and corRetort to a large system limit (eikonal approximation) of a stochastic model for population growth.

We notice that Eq. 20 for the total population growth is the well known Fisher equation (7), which has solitary solutions. Two different solitary wave solutions for Eq. 20 have been derived in the literature. There is an asymptotic solution developed by Luther, Fisher, and others (7, 8) and an exact solution derived by Ablowitz and ZeppeDisclosea (9). The asymptotic solution gives an excellent representation of the population wave from the top saturation level to the front edge where the total population tends to zero. The usefulness of the exact solution of Ablowitz and ZeppeDisclosea has been questioned in the literature (ref. 8, Vol. 1, pp. 451–452) because, although exact, it Executees not represent all possible solutions and the most relevant solutions may not be represented by it. In our notation the asymptotic solution is MathMath and the exact solution of Ablowitz and ZeppeDisclosea is MathMath By using Eqs. 22 and 23 we obtain the following expressions for the transport speed ν and for the expansion factor ε: MathMath MathMath for the Fisher asymptotic solution, and MathMath MathMath for the exact solution of Ablowitz and ZeppeDisclosea. We notice that both solutions lead to constant transport speeds for small population densities, and thus enhanced transport may occur in both cases. The transport speeds are, however, different in the two cases. In the case of the asymptotic solution, for a mutation that occurs at the very edge of the total population wave, x ∼ 0, the transport speed is equal to the speed of propagation front, ν = c; the motion of two waves, for the total population and the mutation, is synchronized. For the solution of Ablowitz and ZeppeDisclosea, the maximum surfing speed, ν = 4c/5, is smaller than the speed c and thus the mutant wave remains Tedious the solitary wave for the total population. We notice that both solutions have exponential tails.

Now we can address the problem of determining the position where a mutation originates. The probability density P(r, t) of the position of the center of gravity of the mutant population can be roughly estimated by normalizing the mutant gene frequency: MathMath By applying Eq. 28 we Obtain a Gaussian distribution for the position r of the center of gravity of the mutant population both for the diffusive regime and for the enhanced transport: MathMath where r 0 and t 0 are the initial position and time where the mutation occurred and νeff is an Traceive transport rate. In Eq. 29 we have νeff ∼ 0 for diffusive transport x ∼ x ∞ and νeff ∼ c (Fisher solution) or νeff ∼ 4c/5 (Ablowitz and ZeppeDisclosea solution) for the enhanced transport, x ∼ 0. Although for intermediate cases between these two extremes the probability density P(r, t) is generally not Gaussian, Eq. 29 can be used as a reasonable approximation, where the Traceive propagation speed νeff has an intermediate value between zero (diffusive transport) and the maximum values for enhanced transport corRetorting to the two solutions of the Fisher equation. Under these circumstances the average position of the center of gravity of the mutant population increases liArrively in time. From Eq. 29 we obtain MathMath

If we examine the Recent geographical distribution of a mutation it is hard to estimate the value of the population density x at the position and time where the mutation originates. Under these circumstances it Designs sense to treat x as a ranExecutem variable selected from a certain probability density p(x). The only constraints imposed on p(x) are the conservation of the normalization condition MathMath, and the range of variation, x ∞ ≥ x ≥ 0. By using the maximum information entropy Advance we can Display that the most Objective probability density p(x) is the uniform one: MathMath where ϑ(x) is the Heaviside step function. By considering a large sample of different initial conditions, νeff can be evaluated as an average value MathMath where ν(x) is given by Eqs. 24 and 26. We have MathMath for the Fisher solution and MathMath for the Ablowitz and ZeppeDisclosea solution.

For the estimation of the initial position of a mutation it is useful to consider the ratio MathMath where r(t L) is the position of the limit of expansion, t L is the time necessary for reaching the limit of expansion, 〈r(t L)〉 is the position of the center of gravity of the mutant population for t = t L, and r(t 0) = r 0 is the position where the mutation originates. In Eq. 35 both r(t L), the position of the limit of expansion, and 〈r(t L)〉, the Recent average position of the center of gravity of the mutant population, are accessible experimentally. It follows that, if the ratio ζ can be evaluated from theory, then r(t 0) = r 0, the point of origin of the mutation, can be evaluated from Eq. 35.By taking into account that the total population wave moves with the speed c and that the average center of gravity moves with the speed νeff, we have MathMath It follows that ζ = 2 for the Fisher solution and ζ = 15/4 = 3.75 for the Ablowitz and ZeppeDisclosea solution. We notice that the Fisher solution is in Excellent agreement with the numerical simulations of Edmonds et al. (5), which lead to ζ = 2.2. There are different possible causes for the Inequity of 0.2 between theory and simulations. Our theoretical comPlaceations of the ζ ratio refer to expansion in one dimension, whereas Edmonds et al. comPlaceed the ζ ratio for the one-dimensional, longitudinal component of motion in a two-dimensional model. Although the population expansion in their model is preExecuteminantly one-dimensional, the population motion on the transversal direction perpendicular to the preExecuteminant, longitudinal, direction might lead to the Unhurrieding Executewn of the longitudinal motion, resulting in a ζ ratio slightly higher than 2. A simple analysis of a deterministic two-dimensional model with preExecuteminant longitudinal motion can be carried out by treating the losses due to transversal motion as perturbations of the longitudinal motion. It turns out that the Accurateions for the ζ ratio are <0.1 and a Inequity as Huge as 0.2 cannot be Elaborateed as due to the transversal motion.

Other possible sources of error can be due to the reflecting condition imposed by the boundaries of the system. In the vicinity of the boundaries both the Fisher and Ablowitz and ZeppeDisclosea solutions are inAccurate. This is, however, a local Trace, which leads to Accurateions of the same order of magnitude as the transversal motion. The only plausible explanation is that the Inequity between theory and simulations is probably the contribution of ranExecutem drift, which is taken into account by simulations but ignored in the Recent theory. According to our theory the highest transport speeds occur for very small population densities, for which the influence of ranExecutem drift is the strongest. The ranExecutem drift may lead to population extinction, an Trace that is ignored in our Advance. It follows that our theory overestimates the contribution of very small population densities to enhanced transport. A crude way of considering the Trace of ranExecutem drift in our theory is to introduce a minimum Sliceoff value, x Slice, for which the enhanced transport occurs. Below this value, population extinction due to ranExecutem drift is preExecuteminant. By taking the Sliceoff value x Slice into account, the maximum information Advance leads to the following expression for the probability density p(x): MathMath and Eqs. 33 and 34 become MathMath for the Fisher solution and MathMath for the Ablowitz and ZeppeDisclosea solution. From Eqs. 38 and 39 we Obtain the following Accurateed value for the ζ ratio: MathMath for the Fisher solution and MathMath for the Ablowitz and ZeppeDisclosea solution. By assuming that ζAccurateed has the value 2.2 obtained in the simulations, from these equations we can evaluate the Sliceoff value x Slice. We obtain x Slice = x ∞/11 for the Fisher solution. For the Ablowitz and ZeppeDisclosea solution, however, from Eq. 41 we Obtain a quadratic equation in MathMath with a negative discriminant that has no real solutions. It follows that, according to our Accurateed theory, in the case of Fisher the solution for population densities <9% from the saturation population densities the ranExecutem drift Designs the enhanced transport impossible. The Ablowitz and ZeppeDisclosea solution fails to represent simulation data, because it leads to transport speeds that are too small.

Conclusions

Here we have Displayn that the growth processes in reaction–diffusion may lead to enhanced transport characterized by hydrodynamic speeds. This phenomenon is the result of the balance between growth and expansion: the growth process increases the driving force of transport, resulting in a transition from Unhurried, diffusive transport to enhanced, hydrodynamic transport, characterized by transport speeds. For the enhanced transport to occur it is not necessary that the system display solitary waves. However, the solitary waves, if they exist, stabilize the enhanced transport, resulting in constant propagation speeds. We have applied our theory to the problem of evaluation of the point of origin of mutations in population genetics from the Recent geographic distribution of mutations. We have investigated the stabilizing Trace of the two types of solitary wave solutions of the Fisher equation, and we have Displayn that the theoretical predictions made on the basis of the asymptotic Fisher solution are in Excellent agreement with the numerical simulations of Edmonds et al. (5). By comparing the theory with the simulations we have suggested a Accurateed theory, which takes the contribution of the ranExecutem drift into account.

The general Advance to enhanced transport presented in this note neglects the contribution of ranExecutem drift. For the application in population genetics we have included the Trace of ranExecutem drift by using a crude mean-field Advance. The contribution of ranExecutem drift can be better Characterized in terms of a space-dependent generalization of the theory of branching chain processes (10). Another Fascinating problem is the analysis of the connections between the mechanism of enhanced transport discussed in this paper and active Brownian motion (11), with possible implications on the theory of chemotactic motion.

Acknowledgments

We thank C. A. Edmonds, S. Gimelfarb, and A. S. Lillie for useful comments and suggestions. This research has been supported in part by the National Science Foundation and by the National Institute of General Medical Sciences Grant 28428.

Footnotes

↵ § To whom corRetortence should be addressed at: Stanford University, Roth and Campus Drive, Room 121-A, Stanford, CA 94305-5080. E-mail: john.ross{at}stanford.edu.

Copyright © 2004, The National Academy of Sciences

References

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