Lecture 5: Travelling Waves

Transcription

1 Computational Biology Group (CoBI), D-BSSE, ETHZ Lecture 5: Travelling Waves Prof Dagmar Iber, PhD DPhil MSc Computational Biology 2015

2 26. Oktober / 68 Contents 1 Introduction to Travelling Waves Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation 2 Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis 3 Wave Pinning

3 Introduction to Travelling Waves Computational Biology Group (CoBI), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 5 MSc Oktober / 68

4 26. Oktober / 68 Travelling Wave-like Phenomena in Biology Calcium Waves on Amphibian Eggs

5 26. Oktober / 68 Travelling Wave-like Phenomena in Biology Somitogenesis Movie by Andy Oates

6 26. Oktober / 68 Travelling Wave Reference Frame Travelling Wave travels without change of shape. if u(x, t) represents a travelling wave, the shape of the solution will be the same for all time speed of propagation is a constant, which we denote c. If we look at this wave in a travelling wave frame moving at speed c then this wave will appear stationary.

7 26. Oktober / 68 Travelling Wave in one spatial dimension In one spatial dimension, x, diffusion of a molecule with concentration c(x, t) can be described as du dt = D d 2 u = D u. (1) dx 2 In mathematical terms, u(x, t) is a travelling wave that moves at constant speed c in the positive x-direction, if u(x, t) = u(x ct) = u(z), z = x ct. (2) z is referred to as wave variable.

8 26. Oktober / 68 PDE Set of ODEs can be rewritten as PDE Set of ODEs u(x, t) = u(x ct) = u(z), u t u x = c du dz z = x ct = du dz. (3) Partial differential equations in x and t become sets of ordinary differential equations.

9 26. Oktober / 68 Biological Constraints on PDEs To be physically realistic u(z) has to be bounded for all z and non-negative.

10 26. Oktober / 68 Linear Parabolic PDEs It can be shown that there are no physically realistic travelling wave solutions for reaction-diffusion equations of the form if f (u) is linear. u t = f (u) + Du xx (4) In biological systems f (u) is typically non-linear.

11 Propagating Wave Solutions Computational Biology Group (CoBI), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 5 MSc Oktober / 68

12 Example: Logistic Growth Consider an ODE that describes logistic growth of the variable u, du dt = u(1 u). (5) The general solution with integration constant c is given by u(x, t) = c exp (t) 1 + c exp (t). (6) If we chose as initial conditions u(x, 0) = exp (x) (7) then we obtain u(x, 0) = c 1 + c = 1 c = exp ( x). (8) 1 + exp (x) Computational Biology Group (CoBI), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 5 MSc Oktober / 68

13 26. Oktober / 68 u(x, t) = can be written as c exp (t) 1 + c exp (t) u(x, t) = Using the wave variable z = t x, we then have u(z) = with c = exp ( x) (9) exp (t x) 1 + exp (t x). (10) exp (z) 1 + exp (z). (11)

14 26. Oktober / 68 Travelling Wave Solution u(z) = exp (z) 1 + exp (z) If z = x t = const then the shape does not change, i.e. if dx dt = 1. The shape of the wave thus does not change if one travels with the wave at speed 1. This wave depends on the initial conditions and is highly unstable.

15 26. Oktober / 68 Example: Fisher Equation Consider the Fisher equation u t = ku(1 u) + Du xx. (12) This equation can be non-dimensionalized by using 1/k as timescale and D/k as length scale. The non-dimensionalized Fisher equation reads u t = u(1 u) + u xx. (13)

16 26. Oktober / 68 Transformation to travelling wave coordinates Consider the non-dimensionalized Fisher equation u t = u(1 u) + u xx. (14) We write u(x, t) = u(x ct) = U(z), z = x ct, c 0. (15) and upon substitution into Eqn.(47) we obtain U + cu + U(1 U) = 0 (16)

17 26. Oktober / 68 PDE Set of ODEs U + cu + U(1 U) = 0 can be written as set of ODEs, U = V V = cv U(1 U). (17)

18 26. Oktober / 68 Phase Plane Analysis Determine Nullclines Determine Steady States Determine Stability of Steady States Determine Trajectories and Phase Vectors

19 26. Oktober / 68 Nullclines and Steady States U = V = 0 V = cv U(1 U) = 0. (18) Nullclines: The U-nullcline is given by V = 0 The V-nullcline is given by V = U(1 U) c Steady states: (U,V) = (0,0) and (U, V) = (1,0).

20 26. Oktober / 68 Stability of Steady States To determine the stability of the steady states we linearize the set of equations at these steady states (U, V ) and determine the eigenvalues of the Jacobian, J = ( 0 1 (1 2U ) c ). (19) as λ ± = 1 2 ( ) tr(j) ± tr(j) 2 4det(J) (20)

21 26. Oktober / 68 Stability of Steady States λ ± = 1 2 ( ) tr(j) ± tr(j) 2 4det(J) Steady States: 1 (U, V ) = (1, 0): saddle node (1 positive and 1 negative eigenvalue) 2 (U, V ) = (0, 0): stable node (2 real, negative eigenvalues) if c 2 spiral (2 conjugate complex eigenvalues with negative real part) if c < 2.

22 26. Oktober / 68 Phase Plane Fisher Equation: U = V V = cv U(1 U). Steady States: 1. (U, V ) = (1, 0): saddle node 2. (U, V ) = (0, 0): stable stst Phase Plane Trajectory: dv du = cv U(1 U) V

23 26. Oktober / 68 Phase Plane Since U can assume negative values if there is a stable spiral at (U, V ) = (0, 0) we require c 2 for physically realistic travelling wave solutions.

24 26. Oktober / 68 For travelling waves to exist we require one stable node and one saddle node. A key question at this stage is: What kind of initial conditions u(x, 0) will evolve to a travelling wave solution? And, if such solution exists, what is its wave speed c?

25 26. Oktober / 68 Initial conditions & Wave Speed Kolmogoroff et al (1937) proved that if u(x, 0) has compact support, u(x, 0) = u 0 (x) 0, u 0 (x) = { 1 if x x1 0 if x x 2 (21) where x 1 < x 2 and u 0 (x) is continuous in x 1 < x < x 2, then the solution u(x, t) of the Fisher Equation evolves to a travelling wave solutions with c = c min = 2.

26 More general Initial Conditions For other initial data the solution critically depends on the behaviour of u(x, 0) as x ±. To see this, consider the leading edge of the evolving wave where since u is small u 2 << u such that Consider now as initial condition We then have if u t u + u xx. (22) u(x, 0) A exp ( αx); α > 0 as x. (23) u(x, t) = A exp ( α(x ct)) (24) αcu = u + α 2 u c = 1 α + α (25) Computational Biology Group (CoBI), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 5 MSc Oktober / 68

27 26. Oktober / 68 Wave Speed depends on Initial Conditions For initial condition u(x, 0) A exp ( αx) as x the wave speed c = 1 α + α depends on the initial conditions α > 0.

28 26. Oktober / 68 General Equation u t = f (u) + u xx u(x, 0) A exp ( αx), α > 0 (26) can be linearized as u t = f (u)u + u xx u(x, 0) A exp ( αx) (27) with solution if i.e. u(x, t) = A exp ( α(x ct)) (28) αcu = f (u)u + α 2 u (29) c = f (u) α + α (30)

29 26. Oktober / 68 The minimal travelling wave speed The minimal wave speed of can be determined as by differentiating c with respect to α c = f (u) α + α (31) c min = 2 f (u) (32) dc dα = 1 α 2 f (u) + 1 = 0 α = ± f (u) (33)

30 26. Oktober / 68 The travelling wave solution u t = f (u) + u xx u(x, 0) A exp ( αx), α > 0 (34) with f (u) having two zeros u 1, u 2 > u 1 If f (u 1 ) > 0 and f (u 2 ) < 0 wavefront solutions monotonically evolve with u going monotonically from u 1 to u 2 with wave speed c c min = 2 f (u) (35)

31 The flatter the wave, the faster it moves Recall that the V-nullcline is given by V = f (U) c. Moreover, du dz = V = f (U) c. Thus the flatter the wave, the faster it moves. Computational Biology Group (CoBI), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 5 MSc Oktober / 68

32 26. Oktober / 68 Stability of travelling wave Solutions Introduce perturbation around travelling wave solution u(z, t) = u c (z) + ωv(z, t) 0 < w 1 (36) The travelling wave solution u c (z) is stable if If lim v = 0. (37) t lim v = du c t dz. (38) then there are small translations along the x-axis.

33 26. Oktober / 68 Stability of travelling wave Solutions u(z, t) = u c (z) + ωv(z, t) 0 < ω 1 (39) Ansatz: v(z, t) = g(z) exp ( λt) (40) Determine eigenvalues to evaluate long-term behaviour of v(z, t).

34 26. Oktober / 68 Waves in 3D Ansatz: transform to spherical coordinate system x = r cos(ψ) sin(θ) y = r sin(ψ) sin(θ) z = r cos(θ) ψ [0, 2π) the azimuthal angle, and θ [0, π] the polar angle. Spherical coordinate system u(t, x, y, z) = u(t, r cos(ψ) sin(θ), r sin(ψ) sin(θ), r cos(θ))

35 26. Oktober / 68 The diffusion equation then reads where s is the spherical Laplace operator, s = 1 r 2 r (r 2 r ) + 1 r 2 sin(θ) du dt = f (u) + D su. (41) 2 θ (sin(θ) θ ) + 1 r 2 sin 2 (θ) ψ 2. (42)

36 26. Oktober / 68 Axisymmetry θ = 0 The spherical Laplace operator then simplifies to ψ = 0 (43) a = 2 r r r (44)

37 26. Oktober / 68 Transformation to travelling wave coordinates We write u(r, t) = u(r ct) = U(z), z = r ct, c 0. (45) u t u r but 1 r cannot be transformed. = c du dz = du dz (46)

38 26. Oktober / 68 Axisymmetric form of Fisher equation u t = u(1 u) + 1 r u r + u rr (47) does not become an ordinary differential equation in the variable z = r ct. does not possess travelling wavefront solutions in which a wave spreads out with constant speed, because of the u r /r term. wavespeed c(r) is a function of r: increases monotonically with r and reaches c(r) 2 for r large.

39 The calcium-stimulated-calcium release mechanism Computational Biology Group (CoBI), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 5 MSc Oktober / 68

40 Elevated levels of Ca 2+ in the cytoplasm stimulate further release of Ca 2+ from the endoplasmatic reticulum (ER) and sarcoplasmatic reticulum (SR) where available. Computational Biology Group (CoBI), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 5 MSc Oktober / 68

41 26. Oktober / 68 The kinetics for Ca 2+ can therefore be described by the following phenomenological model u = A(u) r(u) + L = f (u) (48) A(u) = k 1u 2 k 2 +u 2 : autocatalytic Ca 2+ accumulation r(u): linear resequestration process L: constant leakage flux

42 26. Oktober / 68 Graphical Analysis u = k 1u 2 k 2 + u 2 k 3u + L = f (u)

43 26. Oktober / 68 Importantly f (u) has three zeros, and in the following we will consider a simplified version, i.e. f (u) = A(u u 1 )(u 2 u)(u u 3 ), u 1 < u 2 < u 3, A > 0.

44 26. Oktober / 68 Since Ca 2+ can diffuse we have to consider a reaction-diffusion equation to adequately describe the Ca 2+ kinetics and we thus have As before we write u = f (u) + D d 2 u dx 2. (49) u(x, t) = u(x ct) = U(z), z = x ct, c 0. (50) and upon substitution into Eqn.(49) DU + cu + f (U) = 0 (51)

45 26. Oktober / 68 DU + cu + f (U) = 0 We need to have U = 0 at U = u 1 and U = u 3. Ansatz: U = α(u u 1 )(U u 3 ) and obtain upon substitution (U u 1 )(U u 3 )[Dα 2 (2U u 1 u 3 ) + cα A(U u 2 )] = 0. We thus require 2Da 2 A = 0 Da 2 (u 1 + u 3 ) cα Au 2 = 0 A α = 2D, c = A 2D (u 1 2u 2 + u 3 ).

46 26. Oktober / 68 As solution we thus obtain U(z) = u 3 + Ku 1 exp (a(u 3 u 1 )z) 1 + K exp (a(u 3 u 1 )z) (52) where K is an integration constant that determines the origin in the z-plane. The sign of c depends on the reaction kinetics, i.e. on u 1, u 2, u 3.

47 Calcium Waves on Amphibian Eggs Models by Cheer et al (1987) and Lane et al (1987) Computational Biology Group (CoBI), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 5 MSc Oktober / 68

48 A number of wave-like events can be observed upon fertilization. There are for example both chemical and mechanical waves which propagate on the surface of many vertebrate eggs. These waves arise from a combination of local reactions and long-range diffusion. Computational Biology Group (CoBI), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 5 MSc Oktober / 68

49 26. Oktober / 68 Waves are on the surface of a sphere, i.e. Ca 2+ wavefront is a ring that is propagating over the surface. We therefore consider diffusion in 3 dimensions and use the spherical coordinate system, i.e. u(t, x, y, z) = u(t, r cos(ψ) sin(θ), r sin(ψ) sin(θ), r cos(θ)) ψ [0, 2π) the azimuthal angle, and θ [0, π] the polar angle.

50 26. Oktober / 68 The diffusion equation then reads where s is the spherical Laplace operator, s = 1 r 2 r (r 2 r ) + 1 r 2 sin(θ) du dt = f (u) + D su. (53) θ (sin(θ) θ ) + 1 r 2 sin 2 (θ) ψ 2. (54) Given the inherent symmetries the spherical Laplace operator simplifies to 1 s = r 2 sin(θ) θ (sin(θ) θ ) = 1 ( 2 r 2 θ 2 + cot(θ) ). (55) θ 2

51 26. Oktober / 68 Cheer et al (1987) use the phenomenological description f (u) = A(u u 1 )(u u 2 )(u u 3 ) where A is a positive parameter to describe the excitable kinetics. At each fixed θ we obtain a wavefront solution of the form u(θ, t) = U(z), z = Rθ ct. (56)

52 26. Oktober / 68 We therefore have DU + (c + D R cot (θ))u + f (U) = 0 (57) This equation can be solved in the same way as Eqn.(51) as long as we set θ = const. We then obtain for the wave speed in analogy to Eqn.(52) AD c = 2 (u 1 2u 2 + u 3 ) D cot θ. (58) R

53 26. Oktober / 68 c = AD 2 (u 1 2u 2 + u 3 ) D cot θ. R The wave speed thus increases as the wave moves from the animal pole (θ = 0) to the vegetal pole (θ = π). In reality calcium waves slow down as they move towards the vegetal pole. There must therefore be important cortical properties that have been neglected by the model and which prevent the speeding up tendencies for propagating waves on the surface of a sphere.

54 Models for Somitogenesis Computational Biology Group (CoBI), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 5 MSc Oktober / 68

55 Somitogenesis Computational Biology Group (CoBI), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 5 MSc Oktober / 68

56 26. Oktober / 68 Clock and Wavefront Model Cooke, J., & Zeeman, E. C. (1976). A clock and wavefront model for control of the number of repeated structures during animal morphogenesis. Journal of theoretical biology, 58(2),

57 26. Oktober / 68 Somitogenesis Goldbeter, A., Gonze, D.,& Pourquie, O. (2007). Sharp developmental thresholds defined through bistability by antagonistic gradients of retinoic acid and FGF signaling. Developmental dynamics, 236(6), 1495Ð1508

58 26. Oktober / 68 Bistability model for Somitogenesis Goldbeter, A., Gonze, D.,& Pourquie, O. (2007). Sharp developmental thresholds defined through bistability by antagonistic gradients of retinoic acid and FGF signaling. Developmental dynamics, 236(6), 1495Ð1508

59 Wave Pinning Computational Biology Group (CoBI), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 5 MSc Oktober / 68

60 26. Oktober / 68 Wave Pinning Wave Pinning If a travelling wave eventually halts, the phenomenon is referred to as wave pinning.

61 Example: Cell Polarization Membrane translocation cycle of a typical Rho family GTPase. The inactive cytosolic form (B) diffuses much faster than the active membrane-bound form (A) and is approximately uniformly distributed. Mori et al. Biophys J 94, , (2008). Computational Biology Group (CoBI), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 5 MSc Oktober / 68

62 26. Oktober / 68 Conserved System a t = D 2 a a + f (a, b) x 2 b t = D 2 a b f (a, b) x 2 with Db D a and f (a, b) leading to bistability (at least two stable steady states (a, b ), (a +, b + )), e.g. f (a, b) = b ( k 0 + ) γa2 K 2 + a 2 δa For details, see Mori, Y., Jilkine, A. & Edelstein-Keshet, L. Wave-pinning and cell polarity from a bistable reaction-diffusion system. Biophys J 94, , doi: /biophysj (2008).

63 Single Variable Case (fixed b) a t = D 2 a a + f (a, b) x 2 with f (a, b) leading to bistability (at least two stable steady states (a, a + ). Suppose a develops a propagating front on 0 < x < L, sufficiently far from the boundaries. Then we can approximate a region of validity by < z <. Let A(z) = A(x ct) = a(x, t). A + ca + f (a, b) = 0 Computational Biology Group (CoBI), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 5 MSc Oktober / 68

64 Wave Speed A + ca + f (a, b) = 0 Multiply by A and integrate from to A (A + ca + f (a, b))dz = 0 Since A (± ) = 0 (flat part of wave) and A( ) = a and A( ) = a +, this integrates to give c [A ] 2 dz = f (a, b)a dz = a+ a f (a, b)da a+ a f (a, b)da c = ( ) 2 A z dz Computational Biology Group (CoBI), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 5 MSc Oktober / 68

65 26. Oktober / 68 Wave Speed Wave Speed c(b) = a+ a f (a, b)da ( ) 2 A z dz The denominator is always positive, while the numerator changes sign dependent on b.

66 26. Oktober / 68 Conditions for Wave Pinning At least two stable steady states for b min < b < b max a+ a f (a, b)da changes sign at b = b c c(b) = a+ a f (a, b)da ( ) 2 A z dz Mori, Y., Jilkine, A. & Edelstein-Keshet, L. Biophys J 94, , (2008). Pattern formation much faster than by diffusion (Turing Pattern)!!

67 26. Oktober / 68 Example for Wave Pinning Dashed line: initial conditions Mori, Y., Jilkine, A. & Edelstein-Keshet, L. Biophys J 94, , (2008).

68 26. Oktober / 68 Thanks!! Thanks for your attention! Slides for this talk will be available at: