Waves of stationary shape

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NOTE: I had a couple typos in my equations in the original. This is updated and fixed, and hopefully totally correct. Thanks to bioIgnoramous for pointing it out.

Over at Scienceblogs, people are talking about waves. Of course, everyone thinks that waves are in the domain of physics, and people always forget about one of my favorite subjects: waves of advance. Way back in the day, RA Fisher wondered what might happen if genes had to spread not just locally but across space, and he published his findings in a landmark article called The Wave of Advance of Advantageous Genes. This paper was not just important for its contributions to population genetics, but because of fundamental contributions to applied mathematics. As far as I can tell, Fisher and the great mathematician Kolmogorov published similar findings on this same subject in the same year. To that end, these kinds of waves of advance are often referred to as “Fisherian” waves.

What was Fisher’s model, what did he find, and how has it been extended?


Fisher assumed (without much justification) that genes should diffuse through a population, much like dye diffusing through a glass of water. In addition, at each point the dynamics would be affected by natural selection. This led Fisher to writing down a partial differential equation:

Where p is the frequency of the advantageous allele, t is time, and x is space. The parameter sigma^2 is the varaiance in the parent-offspring distance, and s is the selective advantage of one allele over the other. He then assumed that the solution had the form of a wave of stationary shape, and was able to derive a necessary condition on the velocity, namely that the velocity is at least

This showed that even though a gene can sweep through a local population relatively quickly, e.g. ~900 generations to go from a frequency of of .01 to .99 with a selective advantage of 1%, it will take a while for it to spread spatially. That same gene will take an additional 250 generations to fix, say, 50 meters away, assuming parent-offspring variance of 2 meters^2 per unit time. An interesting observation, at least.

But what’s particularly interesting are some extensions of this work. One of the major extensions comes when the “reaction” term is changed to something just a little bit more complex:

Where the p with the triangle on it, called “p-hat” by those in the know, is an internal equilibrium. What kind of biology might be relevant to this situation? Well, that equation is one way of expressing the famous allee effect in ecology, which describes populations who have lowered growth rates both when the population is large (too much competition, for example) and when it is small (it’s hard to find a mate). In evolution, it was thought that that equation could describe the phenomenon of hybrid zones. In another landmark paper, NH Barton described The Dynamics of Hybrid Zones (unfortunately not open access). Now, to be quite honest, I have no idea what Barton is doing in this paper—he’s using methods from physics that I’m not really familiar with. Nonetheless, he proved that if the internal equilibrium is too large, specifically if p-hat is bigger than 1/2, then the wave of advance stops dead in its tracks. It can also be stopped by steep changes in population density, among other things. This result is pretty cool: it seemed to explain why some hybrid zones moved, and why some hybrid zones stayed in their placee.

Unfortuantely, hybrid zones probably AREN’T described by the simple dynamics above. But we can use that simple equation to learn a lot of stuff about them, nonetheless!

23 Comments

  1. Unclear. How low or high population density may stop the advance of advantageous genes?

  2. Absolutely fascinating post! Thanks. So many interesting ideas and insights, and so many things I would never hear about otherwise, all conveniently located in one place. This is such a delightful blog to read.

  3. I should be clear that it’s only in the case where there is an internal equilibrium that the wave can stop because of the population density. And technically it’s due to the gradient in density, not just the actual density.

  4. In the PDE you show first, the term on the LHS has dimensions of inverse time. The two terms on the right are dimensionally incompatible with that and with each other. I’d guess that some sort of diffusivity and rate constant are missing, unless the equation had been nondimensionalised so that t and x are dimensionless already. But if so, it is odd that it is completely free of parameters, even dimensionless ones. 
     
    What’s s?

  5. So it seems that advantageous genes propagate rapidly through fixed populations, but slowly through space. Would I be correct in viewing this as good support for punctuated equilibrium?

  6. “In the PDE you show first, the term on the LHS has dimensions of inverse time. The two terms on the right are dimensionally incompatible with that and with each other. I’d guess that some sort of diffusivity and rate constant are missing, unless the equation had been nondimensionalised so that t and x are dimensionless already. But if so, it is odd that it is completely free of parameters, even dimensionless ones.” 
     
    Ah yeah, there is a diffusivity constant I left off. Technically there should be a “sigma^2/2″ multiplying the partial wrt x, which is the variance in parent-offspring distance per unit time. “s” is the selective advantage of one allele over the other, given in units of 1/time—it can be interpreted as the less fit gene leaves (1-s) as many copies as the more fit gene per unit time. 
     
    “So it seems that advantageous genes propagate rapidly through fixed populations, but slowly through space. Would I be correct in viewing this as good support for punctuated equilibrium?” 
     
    Hrm, to be honest I’ve never thought about that. I don’t think it would be a support for the process of punctuated equilibrium, but could explain the fossil record in some sense: suppose that many new advantageous genes arise in peripheral populations and have to diffuse into the central population, which may be where we get most of our fossils from. Just some speculation.

  7. It’s perfectly OK to delete my comment above; I was trying to be helpful, not to score points.

  8. Just a note for readers who haven’t done a lot of differential equation modeling, the “internal equilibrium” p-hat doesn’t mean that it is a stable equilibrium. 
     
    In this case, it’s not — if you are already at p-hat, you will continue to stay at that frequency. But if you are perturbed just a bit in either direction, you will go to either p = 0 or p = 1. 
     
    To convince yourself, ignore the spatial term, and just make a simple plot where dp/dt is on the vertical and p is on the horizontal: 
     
    dp/dt = sp(1 – p)(p – phat) 
     
    This has roots at p = 0, p = 1, and p = phat. It’s a cubic equation, and by looking at the leading sign — by multiplying sp * (-p) * p — we see it’s negative, and thus a cubic that goes down-up-down from left to right. 
     
    Between p = 0 and p = phat, the curve (i.e., dp/dt) is negative, meaning that the frequency will shrink. Between p = phat and p = 1, the curve is positive, meaning the frequency will grow. This pushes the frequency away from phat and toward either 0 or 1, depending on which side of phat you started on. 
     
    What would make the internal equilibrium stable? Just change the sign of the cubic so that it goes up-down-up. We need to keep the sp(1 – p) since that’s the logistic growth part of the model, so we’d just re-write the extra term as (phat – p). 
     
    In this case, there is no change in frequency once p gets to 0, phat, or 1, as before. Except now when you plot dp/dt as a function of p, you’ll see that the frequency grows between p = 0 and p = phat, and shrinks between p = phat and p = 1. So, no matter where you start, p is pushed toward phat, where it stays. 
     
    Take-home message — if you haven’t taken an applied math / mathematical modeling course or two, it’s not that hard to pick up, but yields huge dividends.

  9. For those who are unfamiliar with the reaction diffusion formalism…It may be useful to point out that the first equation can be motivated by considering an ordered set of one dimensional compartments along the x-axis. Each compartment has an allele frequency at a given time p(x_i,t_j). The allele frequency in a compartment can change for only one of two reasons:  
     
    1. exogenous inflow/outflow from adjacent cells (diffusion, the partial^2 p/ partial x^2 term) 
     
    2. endogenous rise/fall in allele frequency from within compartment activity (reaction, the sp(1-p) term) 
     
    In the standard chemistry motivated reaction diffusion framework one would also have conservation equations, because usually a given parameter cannot rise in some comparment without others decreasing (e.g. if H_2 0 is going up, then the numbers of H and O molecules are going down).  
     
    In this case that conservation equation is only important if there are more than two alleles at the locus, with each of them being interchangeable for the other. In that case, for n alleles you would have (n-1) variables p_1,…,p_{n-1}, with p_n given by 1-(p_1 + … p_{n-1}). 
     
    Note that a more realistic model might model individuals rather than relying on aggregate frequencies…that would allow time varying populations and closer fidelity to the underlying dynamics of what’s going on (i.e. actually simulating mating events).

  10. PS: the technique agnostic is using there is called a phase space plot.  
     
    At each point in the (p,dp/dt) plane, you draw an arrow corresponding to the direction in which (p,dp/dt) will evolve.  
     
    Especially for two dimensional systems, you can quickly get an intuition for the dynamics of the system, corresponding to paths in phase space.  
     
    There’s some good R and sagemath methods for doing quiver plots of vector fields. 
     
    PSS: if you’ve got a multidimensional problem or want to automate the intuition which agnostic ably explained, you should do a local linearization of the nonlinear equation around each possible equilibrium point to make it a linear ODE of a vector state variable u. In this case state u = (p,dp/dt). Then calculate the eigenvalues of the linearized transition matrix at each point. That will tell you whether it’s a stable equilibrum, unstable, saddle point, etc and has the advantage that a computer can do it.

  11. And for the people who come here for other topics, that model of a gene spreading can be applied to anything that obeys the same growth laws and spatial spreading laws. 
     
    A contagious disease is one example. Say it sort of diffuses around in space because people aren’t locked into place, but wander around their home turf like a bell curve (the mean location is their turf, but they also wander to either side, though less likely the further out you’re talking). And the rise in the percent of people who are infected, over time, also follows the sp(1 – p) law. 
     
    You’ve heard of logistic growth — exponential growth, but constrained above by a carrying capacity. That’s what the sp(1 – p) term says — one person infects a number of others, who infect a number of others, etc., until pretty soon the carrying capacity of infecteds — 100% of the group — has been infected. Just like one rabbit gives birth to a number of rabbits, and each of them to a number of rabbits, etc., until the carrying capacity is reached. 
     
    Culture works the same way. A technology may diffuse around in space because its users / owners wander around as above. And a possessor of the technology can come into contact with someone who doesn’t know about it, and with some probability transmit the technology to the latter. So you could see “waves of advance” of some useful technology. 
     
    And Allee Effects can be built into that as well — with too few people using the technology, its growth rate is slowed, and likewise if there are too many people using it. 
     
    Anyway, just some further arm-twisting for everyone to learn a little applied math — so that the study of culture doesn’t languish in verbal-only idealizations.

  12. Also, the “space” referred to doesn’t even have to be geographical space. It could refer to the “space” of ages (how some trend spreads from one age-group to another, say), some kind of social “space” (how a technology spreads from high-status turf to low-status turf — or vice-versa), or any other way you could think of arranging the population in a space-like way.

  13. Especially for two dimensional systems, you can quickly get an intuition for the dynamics of the system, corresponding to paths in phase space. 
     
    http://www.math.psu.edu/melvin/phase/newphase.html

  14. Some of these comments illustrate why mathematical modelling can be so fascinating. One almost has to remind oneself that it’s the physics, or biology, that we’re trying to understand, not the math model.

  15. that we’re trying to understand, not the math model. 
     
    point taken. but the precision and clarity of the discussion is certainly refreshing, no? though i suppose economics shows that precision, clarity and elegance of model shouldn’t confuse one into thinking that that banishes uncertainty of one’s understanding of the real world.

  16. I haven’t studied Fisher’s paper, but I assume he distinguishes between a local population which is assumed to be panmictic, i.e. breeding randomly throughout the locality, and a larger population in which the probability of interbreeding between two individuals is some function of distance. Obviously in the panmictic case the only factor influencing the rate of increase of an advantageous gene is its selective advantage, because that is built into the assumptions, whereas in the larger population we also have to allow for rates of interbreeding and migration. I can’t see any particular relevance to the theory of punctuated equilibrium in all this. I think the punk-eek theorists generally assume that punctuation occurs in a geographically isolated population, and forms a new species (reproductively isolated) before spreading out. This is not Fisher’s model, which assumes a continuous population without barriers.

  17. I am familiar with logistic growth (S curve) where the rate of increase is related to the distance to the level of saturation. How is that related to differential reproduction and the velocity of diffusion of advantageous characters? An empty space will stop diffusion, agreed, but a sparely populated space will not.

  18. One almost has to remind oneself that it’s the physics, or biology, that we’re trying to understand, not the math model. 
     
    Greg and Henry apply the Fisher-Kolmogorov equation in their new book. Basically, they point out that strong selection, even if uniform in sign over large geographic areas, will not necessarily result in fixation of the favored allele in a given time interval despite the elementary equations saying that fixation will happen. That’s because the elementary equations only track the frequency of an allele over time and ignore its distribution over space. 
     
    So if the assumptions underlying Fisher’s wave of advance (e.g., lad mates with a lass from the next village over) are accepted as a baseline, then an exceptionally rapid fixation of an allele over a large geographic extent requires special dispersive mechanisms (war, migration, etc.).

  19. I had a look at Fisher’s paper, but it is too advanced for me. There is no reference to mating, so I think it applies to either sexual or asexual populations, provided there is some ‘diffusion’ of offspring away from their parents. 
     
    I also looked at the abstract of Barton’s paper. He seems to be dealing with cases where populations meet and hybrids between them have a selective disadvantage. In this case a gene that might be advantageous in both populations may be stopped from spreading if the hybrid disadvantage is great enough.

  20. David B, 
     
    Then we are not talking about one population, but two dissimilar populations that produce hybrids when they mix.

  21. “I am familiar with logistic growth (S curve) where the rate of increase is related to the distance to the level of saturation. How is that related to differential reproduction and the velocity of diffusion of advantageous characters? An empty space will stop diffusion, agreed, but a sparely populated space will not.” 
     
    If you look at the reaction term in the Fisher’s equation, p(1-p), you can see that that is a logistic growth type of term. In this case, troughs in population density CANNOT stop the wave of advance. However, in the “bistable” version, p(1-p)(p-phat), the trough can stop the wave. The intuition for this is that if there is too steep a change in population density, then there won’t be enough genes diffusing into the area to overcome the unstable point (recall what Agnostic explained about p < phat implying that the frequency goes down to zero), and hence it will be stopped dead in its tracks.

  22. I am familiar with logistic growth (S curve) where the rate of increase is related to the distance to the level of saturation. How is that related to differential reproduction and the velocity of diffusion of advantageous characters? 
     
    The logistic doesn’t have to do with diffusion really. The diffusion term says how the allele frequency changes across space — does it move like a bell curve, where it stays put on average but wanders in both directions with equal probability, although with smaller probability as the distance from home increases? Does it preferentially move in one direction and away from another, like cells following a gradient toward a signal? Etc. 
     
    The logistic says how the allele frequency changes over time, ignoring space. I won’t derive it, but you can pick up Gillespie’s Population Genetics: A Concise Guide (pp. 61 – 62), or any other pop gen book, to see that the change in frequency of a selected allele is: 
     
    delta p = p*q * [p(w11 - w12) + q(w12 - w22)] / wbar 
     
    Where p = allele frequency of the selected one, q is frequency of the other, w11 is fitness of the selected homozygote, w12 the fitness of the heterozygote, w22 the fitness of the non-selected homozygote, and wbar is mean population fitness = p^2 * w11 + 2pq * w12 + q^2 * w22. 
     
    Since we only have two alleles in the model, q = 1-p. So we can re-write delta p as: 
     
    delta p = [[p(w11 - w12) + q(w12 - w22)] / wbar] * p*(1 – p) 
     
    Anytime you see delta p or dp/dt written as p(1 – p) or p(1 – p/k), with some chunk multiplied in front, that should trigger the association with logistic growth. 
     
    Pursuing this hunch, let’s re-write the above as a simple logistic equation: 
     
    delta p = s * p * (1 – p/K) 
     
    K is the carrying capacity — which, when we’re talking about frequencies or probabilities, is 1. So it drops out, and we just have s * p * (1 – p). 
     
    That must mean, by analogy, that that big ugly chunk in the delta p equation is like the intrinsic growth rate in logistic population growth. Looking just at that chunk: 
     
    s = [p(w11 - w12) + q(w12 - w22)] / wbar 
     
    Forget the wbar in the denominator — it’s just there to normalize the absolute fitnesses, and put them on the scale of “compared to mean population fitness.” That way we can compare cases where fitness is measured in any variety of units. 
     
    Two alleles go into the offspring. You’re the selected allele, and someone else is the non-selected allele. And there are two types of alleles that you could be paired up with — the selected and non-selected. We want to see how well you would fair, compared to the alternative choice, no matter who you are paired with. 
     
    Case 1: the partner is a selected allele. If you are chosen, you form a selected / selected homozygote, whereas if the other is chosen, they form a heterozygote. The advantage that your offspring enjoys over the other offspring is w11 – w12. 
     
    Case 2: the partner is a non-selected allele. If you are chosen, you form a heterozygote, whereas if the other is chosen, they form a non-selected / non-selected homozygote. The advantage that your offspring enjoys over the other offspring is w12 – w22. 
     
    With random mating, case 1 happens with probability p — you’re drawing a partner at random, and p is the frequency of the selected allele. Likewise, case 2 happens with probability q = 1 – p. 
     
    Thus, the weighted average advantage that your offspring would enjoy, compared to the offspring of the non-selected allele, is p(w11 – w12) + q(w12 – w22). Normalize this advantage by comparing it to (dividing it by) mean pop fitness, and you get the formula for s above. 
     
    So, the analogy between logistic population growth and natural selection is done. K, the carrying capacity, is 100%, so it drops out. And the big ugly mess aside from p(1 – p) represents the selective advantage, which is like the intrinsic growth rate. 
     
    The key difference is that, for natural selection, s is not a parameter — it is defined in terms of the variable p, so s changes as p changes. In simple logistic growth models, you usually just treat r as a parameter, although there’s no reason you couldn’t make it a function of the variable N too. 
     
    But if you want to simplify things, you can treat s as a parameter, just like with your garden-variety logistic growth equation, as Fisher is doing in this case. I.e., during the bout of selection, there’s some constant selective advantage of the allele.

  23. Genes can literally travel at the speed of light since they are only information. It may take generations for humanoids to reach Alpha Centuri but their genes will only be 4.3 years out of date. In the same way, the software of planetary probes is updated en route.

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