1 Traveling Fronts Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306
2 When mature Xenopus oocytes (frog eggs) are loaded with a Ca2+ dye such as fura-2 and stimulated by the fusion of sperm, a wave of elevated Ca2+ is produced in the large egg ( 1.2 mm in diameter). This is called a fertilization Ca2+ wave and it is important for triggering the formation of an envelope around the oocyte that prevents the fusion of other sperm (which would result in polyspermy). This process occurs in eggs of other species, from starfish to mammals. Figure 1: Ca2+ wave in a Xenopus oocyte following fertilization. Time goes from top left to bottom right. From Fall et al., 2002. This fertilization wave is an example of a traveling front, where the variable (Ca2+ concentration in this case) goes from a resting low level to a stimulated high level. This can be described as a bistable system, and the front is perturbing the system from the resting state to the co-existing stimulated state. The wave can be observed using a space-time plot, as shown below.
3 Figure 2: Space-time plot of a traveling Ca2+ wave. From Fall et al., 2002. This can be viewed in a different way by plotting the Ca2+ concentration versus time at several values of x, where x is the distance from sperm fusion, as shown in Fig. 3. Figure 3: Time dynamics for several values of the distance from sperm fusion. From Fall et al., 2002. The fertilization wave is described by the partial differential equation c = D 2c + f (c) t (1)
where c is the intracellular Ca 2+ concentration, D is the diffusion coefficient multiplied by the fraction of Ca 2+ that is free, and f(c) is a reaction term that describes the net Ca 2+ flux from the ER into the cytosol. The reaction term is f(c) = [k leak + k IP3 ][c ER c] k SERCA c (2) 4 where the first terms describe influx through ER membrane leak and IP 3 receptors, and the last term is efflux through SERCA pumps. This is an example of a reaction-diffusion system that gives rise to traveling front solutions for some parameter values. Rather than studying this biophysical model, we will focus on simpler reaction-diffusion systems that also produce traveling fronts. Fisher s Equation In a paper published by Fisher in 1937 the spread of an advantageous allele of a gene (call it a ) into a region in which a second allele (call it A ) was initially present was considered. The spread was assumed to be random. If p = Prob[a is present], then 1 p = Prob[A is present]. Using the standard rules of genetics (Hardy-Weinberg genetics), it can be shown that the rate at which the frequency p evolves over time is described by p t = D 2 p + αp(1 p) (3) x2
where D is a diffusion or mixing rate and α is the intensity of selection. This is called Fisher s Equation. We will analyze the system using D = α = 1. Then, p t = 2 p + p(1 p) (4) x2 This PDE has two constant solutions, p 1 = 0 and p 2 = 1. The stability of these solutions can be observed by plotting the reaction term (Fig. 4). Clearly, the equilibrium at 1 is stable, while the equilibrium at 0 is not. f(p) 5 1 p Figure 4: Analysis of the Fisher reaction term shows stability properties of equilibria. In terms of the biology, this means that, ultimately, the probability that an individual has the a allele is 1, and that it has the A allele is 0. This is an asymptotic result, but what about the temporal dynamics? As with the fertilization wave, the spread of the advantageous allele can occur as a traveling front. This can be viewed in the (t, x)-plane (Fig. 5). Here, p = 0 ahead of the wave front, and p = 1 behind the wave front. The
front is assumed to move at a constant velocity c, so the shape of the front in the (t, x)-plane is a line with slope c. This line has a vertical-intercept of z f. The value of p for any other point in this plane depends on whether it lies above or below the front line, and this in turn is determined by whether the line of slope c containing that point has a vertical-intercept z that is greater than or less than z f. That is, if a point (t, x) is on a line of slope c with vertical-intercept z, then the value of p at that point depends entirely on the value of z. Therefore, we look for a traveling front solution, which would be a function depending only on z. Here, x = ct + z (since slope of the line is c and z is the vertical-intercept), so z = x ct. (5) We call this coordinate z the traveling coordinate. The traveling x p=0 z (t,x) front 6 z f p=1 Figure 5: Traveling wave as viewed in the (t, x)-plane. The traveling coordinate z is the verticalintercept of the line with slope c through a point (t, x). t
7 wave solution (if it exists) has the form p(x, t) = u(z) = u(x ct) (6) where c is the unknown front speed. Our goal is to pick the true front speed, and then find the traveling front solution u(z). Now So the traveling front would satisfy p = du dz t dz dt = cdu dz (7) 2 p x = d2 u 2 dz 2 (8) p(1 p) = u(1 u). (9) c du dz = d2 u + u(1 u). (10) dz2 This can be converted into two first-order ODEs using v = du dz, so du dz = v (11) dv dz = cv u(1 u). (12) ( ( 0 1 The system Eqs. 11, 12 has two steady states, u 1 = and u 0) 2 =. 0) For stability we linearize the system and look at the eigenvalues of the Jacobian matrix, J = ( ) 0 1 2u 1 c. (13) Evaluated at u 1 the eigenvalues are λ = ( c ± c 2 4)/2 and since c > 0 they have negative real parts. They are real when c 2 and
8 complex otherwise. Hence, u 1 = { stable node when c 2 stable spiral when c < 2. (14) At u 2 the eigenvalues are λ = ( c± c 2 + 4)/2. There will be one positive real eigenvalue and one negative real eigenvalue, so u 2 is a saddle point. Hence, u 1 has a 2-dimensional stable manifold and u 2 has 1-dimensional stable and unstable manifolds. v u 2 u Figure 6: The Fisher model with c < 2. A heteroclinic orbit (green) connects the saddle point at u 2 to the stable spiral at u 1. A couple of additional trajectories are shown (black). This is non-physical since it would require some negative values of u. In the case of c < 2 (Fig. 6) the trajectories will have some negative values of u as they spiral in to u 1. This includes the heteroclinic orbit that connects one branch of the u 2 unstable manifold to the stable spiral at u 1. Since u is a probability, this is non-physical. In the case of c 2 (Fig. 7) the trajectories enter u 1 without spiraling, so u can remain positive, and thus provide physical solutions.
9 v u 2 u Figure 7: The Fisher model with c 2. A heteroclinic orbit (green) connects the saddle point at u 2 to the stable node at u 1. This can be done without negative u values. The traveling wave solution starts with the system in a state in which the population is dominated by one allele of the gene and is eventually dominated by the invading allele. In terms of the system of 2 ODEs, this corresponds to the heteroclinic orbit in Fig. 7. In this case, u(z) 1 as z and u(z) 0 as z. A sketch of the heteroclinic orbit as a function of z is shown in Fig. 8. This wave has the shape of a moving u 1 z Figure 8: The Fisher model with c 2. The heteroclinic trajectory is shown as a function of the traveling coordinate z.
front. For populations on the left side of the diagram the invading allele a is dominant (u is near 1); for those on the right the original allele A is dominant (u is near 0). The analysis above was performed in the frame of reference of traveling coordinate z = x ct. In this frame, the wave is stationary. In the frame of reference of a stationary observer there is a traveling front moving from left to right (since c > 0, Fig. 9). One can also view this by looking at time courses at different spatial locations (Fig. 10). u 1 10 t t t 0 1 2 x Figure 9: Traveling front moving from left to right in the stationary frame of reference (t 0 < t 1 < t 2 ). u 1 x 0 x 1 Figure 10: Time courses showing the traveling pulse at two different locations (x 0 < x 1 ). t
Eventually u 1 for any location x and thus the invading allele a dominates the original (and not as fit) allele A in the population. In this model, traveling front solutions exist for any front velocity c 2. That is, there is a continuum of traveling fronts, each with a different front speed. Which one actually occurs depends upon the initial conditions. Spreading Microorganisms In a typical laboratory experiment on yeast, one fills the bottom of a petri dish with a solidified gel called agar, which permits the diffusion of small molecules like glucose. Then, a small number of yeast cells are put on top of the agar. The glucose below provides the necessary nutrients for the yeast. The yeast reproduce and migrate in directions where glucose is available. A 1-dimensional model of yeast spread can be constructed by keeping track of the density of yeast and concentration of glucose: n(x, t) = density of yeast g(x, t) = glucose concentration. As in the last example, we will consider an infinite domain, so x (, ) with no boundaries. Then the yeast PDE is n t = D 2 n n x + Kn(g g 1) (15) 2 11
12 where D n 2 n x 2 reflects random diffusion-like migration of yeast and Kn(g g 1 ) reflects yeast proliferation. Cells are assumed to increase proportionally to the amount of glucose in excess of g 1. The glucose PDE is g t = D 2 g g x ρkn(g g 1). (16) 2 The D g 2 g x 2 term represents glucose diffusion. The ρkn(g g 1 ) term represents glucose depletion by yeast at a rate of ρ units of glucose per new cell made. To simplify the analysis we omit the yeast diffusion term, which should be much slower than glucose diffusion. That is, the spread of yeast will be due entirely to the diffusion of glucose. Also, we define ĝ(x, t) g(x, t) g 1. Then, n t ĝ t = Knĝ (17) 2 ĝ = D g ρknĝ. (18) x2 We now seek traveling front solutions, using N(z) = n(x, t) (19) G(z) = ĝ(x, t) (20) where z = x ct. Inserting N and G into the differential equations, we
13 obtain the following ordinary differential equations: c dn dz c dg dz = kng (21) = D g d 2 G ρkng. (22) dz2 This could be converted into a set of three first-order ODEs, but then phase plane analysis would not be possible. Instead, multiply the first equation by ρ and add it to the second to obtain Integrate this from to z, cρ dn dz cdg dz = D d 2 G g. (23) dz 2 or cρ z dn ds ds c z dg z ds ds = D g d 2 G ds (24) ds2 cρn(z)+cρn( ) cg(z)+cg( ) = D g dg dz D g dg ( ) (25) dz What does z mean? It means that ct x, or that the location x is far behind the wave front. At this location, one expects the yeast to have reached some carrying capacity N o and the glucose to be at a level at which yeast exhibit neither net proliferation nor death. That is, G = 0 since then g = g 1. Also, one would not expect much change in G as z is
14 increased a small amount. Altogether, so Eq. 25 becomes N( ) = N o (26) G( ) = 0 (27) dg ( ) = 0 dz (28) cρn(z) + cρn o cg(z) = D g dg dz. (29) We thus have two first-order ODEs, dg dz = c G cρ N + cρ N o D g D g D g (30) dn dz = k NG. c (31) Any traveling front solutions must be heteroclinic orbits of these equations. The traveling front equations have two equilibria, (0, N o ) and (ρn o, 0). The first has yeast at its carrying capacity and glucose g = g 1 (so that G = 0). The second has no yeast and glucose at g = ρn o + g 1 (so G = ρn o ). The first equilibrium is a saddle point while the second is a stable node. The heteroclinic orbit is drawn in green in Fig. 11, connecting the saddle point to the stable node. It is sketched in Fig. 12 as a function of the traveling coordinate. So this example again shows how a traveling front can come about, in this case for the population of yeast. Since the traveling
15 N No ρno G Figure 11: Phase portrait for the density of yeast N and glucose G. The heteroclinic orbit is drawn in green. coordinate is z = x ct and c > 0, the front is moving to the right when viewed as N vs. x at different points in time. Hence, at any location x, the glucose level will eventually be g = g 1 and the yeast number will be at the carrying capacity N = N o. N G Figure 12: Traveling front of cell number (N) and glucose concentration (G) as a function of the traveling coordinate z. z In this example, the only constraint on the front speed c is that it be positive so that N moves to its stable carrying capacity as t. In both examples of traveling fronts the system was described in terms
of planar ODEs. In the second example, we were lucky; the reduction to two dimensions was the result of a reduction of order. Other PDEs and systems of PDEs can have traveling front solutions, but the analysis is harder since the system of ODEs that must be analyzed has more than two dimensions. 16
17 References Mathematical Models in Biology, L. Edelstein-Keshet, SIAM Classics in Applied Mathematics 46, 2005.