SoSe 2015
22/05/2015
Prof. Dr. E. Frey
Marianne Bauer, Raphaele Geßele,
Florian Gartner und Jonas Denk
Ludwig
Maximilians
Universität
Lehrstuhl für Statistische Physik
Biologische Physik & Weiche Materie
Arnold-Sommerfeld-Zentrum für Theoretische Physik
Department für Physik
Nonlinear Dynamics
(Prof. E. Frey)
Problem set 6
Problem 1
Propagating Signals in Excitable Media
Consider reaction diffusion kinetics of the form
∂t u = ε−1 f (u, v) + ∇2 u
∂t v = g(u, v) + D∇2 v,
for two component concentrations u and v, with different diffusion constants rescaled to D = Dv /Du .
The parameter ε gives the ratio of the reaction time scales. As ε is taken to be small, changes in u
based on the kinetics f (u, v) will be fast compared with those of v.
As a particular example for the reaction kinetics let us consider the following “piecewise linear” form:
(
−u − v
for u < a
f (u, v) =
,
1 − u − v for u > a
g(u, v) = u − bv,
allowing many calculations to be done explicitly.
a) Sketch the u and v nullclines for a = 1/4 and for (i) b = 0.2 and (ii) b = 1. For what range of b
is the reaction kinetics monostable, for what range bistable?
b) Stationary front: The stationary case, defined by the vanishing of viscous friction and equal
potentials for both states shall be considered now. Use the ’rolling ball’ analogy for illustration.
Determine the corresponding potential V for f . Calculate the value of vf (a) (f for front) for
which the front connecting the small u and large u portions of the u nullcline is stationary (i.e.
vf (a) = v ? gives the fixed points at which both potentials are maximal.).
c) Consider for the reaction kinetics with a = 1/4 and with constant inhibitor concentrations
v = vf = 0. There must be a (non-stationary) front connecting the rest state (u = 0, v = 0) and
(u = 1, v = 0). Plot the effective potential V (u) for the “rolling ball” analogy.
d) Moving front: Calculate (as a function of the parameters a and v = vf ) the speed of the front
connecting the rest state (u = 0, v = vf ) with the excited state at this value of v, i.e. (u = 1,
v = vf ). Verify that this front speed is zero for a = 1/2 and vf = 0. (Hint: You may use the
same approximation
that we applied in the lecture for the Nagumo model (see lecture notes, eq.
R +∞ 02
(16)), namely −∞ u (z)dz ≈ 1 )
e) Pulse state: Find the propagating speed c and calculate and plot u(x − ct) and v(x − ct) for the
excitation pulse propagating in the rest state of the reaction diffusion system with a = 1/4 and
b = 1/5.
f) Dispersion relation: Calculate and plot expressions for the dispersion relation C(T ) for the scaled
speed C = ε1/2 c as a function of the temporal period T for waves propagating in the reactiondiffusion system with a = 1/4 and b = 1/5. Do not worry about the breakdown of this scaling
that occurs for small C.
Problem 2
Wave-Pinning and Cell Polarity from a bistable Reaction-Diffusion System
Polarization of the cytoskeleton is a fundamental attribute of cells that is essential for cell locomotion, morphogenesis and division. Different mechanisms are known by which
cells polarize, i.e. break their anterior-posterior-symmetry in response to external signals. Here, we study the so-called mechanism of ”wave-pinning”, where a propagating wave-front
inside the cell is brought to a standstill and the different concentration levels of the corresponding protein before and behind the front, respectively, are thereby ”frozen”.
Consider a signalling protein, that can be in one of two different states, either bound to the cell membrane (active form A)
or solved in the cytosol (inactive form B). We denote the concentrations of A and B, respectively as a and b and the rate
constants of activation (binding to the membrane) and inactivation (unbinding) as kba and kab (rate per molecule), respectively.
Suppose kba and kab are given by
kba = k0 +
γa2
,
K 2 + a2
kab = δ,
(1)
where k0 , γ, K, δ are positive constants, so that in the well-mixed case the concentration of A behaves
like
da
γa2
= b k0 + 2
− δ a =: f (a, b)
(2)
dt
K + a2
a) Give a physical interpretation of the functional form of the rates kba and kab . What is the
meaning of each of the constants k0 , γ, K, δ and of the Hill coefficient n = 2 in kba . What does
the evolution equation for b look like if the total protein concentration a + b is conserved?
Diffusion on the membrane is in general significantly slower than in the cytosol, Da Db ,
such that we can assume that B is uniformly distributed over the cytosol on the time scale of
the dynamics of A. Very generally, what is the necessary condition on the reactive part so that
travelling wave fronts in the concentration of A can emerge if we add diffusion to the system?
Argue why this condition can indeed be fulfilled by the reaction term in (2) (depending on the
parameters). Would this also hold if we chose a Hill coefficient of n = 1 in kba ? Justify your
answer.
We now consider a one-dimensional spatial version of the system on a line (0 ≤ x ≤ L; see
figure) and add diffusion to the above reaction kinetics. The reaction-diffusion equation for a
then reads:
∂a
∂2a
= Da 2 + f (a, b) ;
∂t
∂x
(3)
and similar for b with the respective reaction term and the diffusion constant Db . Furthermore
∂a
∂b
we choose the boundary conditions ∂x
|x=0,L = ∂x
|x=0,L = 0.
b) In order to understand the mechanism of wave-pinning, again regard the uniform concentration
b of B as a parameter in equation (3). Use the rolling-ball-analogy to show that the b-dependent
speed of the travelling wave front is given by (assume an infinitely extended system (−∞ < x <
∞))
R a+
a f (a, b)da
,
(4)
c(b) = R ∞ −
2
−∞ (∂a(ξ)/∂ξ) dξ
where ξ := x − ct and a− (b) < a+ (b) are the two stable fixed points of the reaction term (that
we assume to exist for b within a certain interval, bmin < b < bmax ).
Suppose that initially a is at its lower steady state (a(x, 0) = a− (b)) and b is large enough such
that c(b) > 0 at t = 0. Suppose a local stimulus near the left edge of the cell at x = 0 triggers a
local increase to the level a = a+ via bistability and a wave front starts to propagate to the right
(see figure, bottom part). Describe in detail how the propagation of the wave front affects the
parameter b and thereby its own speed c(b) (use the reaction term (2) for your argumentation).
How and under which conditions does wave-pinning occur, meaning that the wave comes to a
standstill anywhere between 0 and L?
c) Download the file WavePinning.py and solve the system of PDEs, Eq. (3), numerically. Discuss
the discrete version of the Laplacian and the ’no-flux’ boundary conditions implemented in
the code. Use the parameters: k0 = 0.067, γ = 1., K = 1., δ = 1., with homogeneous initial
conditions b(t = 0) = 2. and a(t = 0) = a− (b(t = 0)) in a system of size L = 10. As local
stimulus near the edge x = 0 choose a step function
(
)
S
for t < t1 and x < 1
fs (x, t) =
,
(5)
0
else
with S = 0.1 and t1 smaller than your total simulation time. How does the stimulated system
evolve if you choose Da = Db = 0? Now choose Da = 0.1, Db = 10.; how does this change the
steady state of your system? Compare your results with your expectations from b).
d) Summarise the main conditions that must be fulfilled such that polarization via wave-pinning
can occur for an arbitrary reaction term g(a, b).
For details please refer to Mori, Y., A. Jilkine, and L. Edelstein-Keshet, Wave-pinning and
cell polarity from a bistable reaction-diffusion system, Biophys. J., 94:3684-3697 (2008).
For an interesting application of the wave-pinning mechanism also see Stability of Localized
Wave Fronts in Bistable Systems, S. Rulands, B. Kln̈der, and E. Frey, Phys. Rev. Lett. 110,
038102 (2013).
Due date: 29/05/2015, 13pm in the marked box close to room A348/349.