Introduction to Travelling Waves Modeling Examples. Travelling Waves. Dagmar Iber. Computational Biology (CoBI), D-BSSE, ETHZ

Transcription

1 Introduction to Travelling Waves Modeling Examples Travelling Waves Dagmar Iber Computational Biology (CoBI), D-BSSE, ETHZ 1

2 2 Introduction to Travelling Waves Modeling Examples Outline 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 3 Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation Introduction to Travelling Waves

4 4 Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation 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. Reference Frame If we look at this wave in a travelling wave frame moving at speed c then this wave will appear stationary.

5 5 Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation 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.

6 6 Introduction to Travelling Waves Modeling Examples PDE Set of ODEs Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation can be rewritten as 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 therefore become sets of ordinary differential equations.

7 Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation To be physically realistic u(z) has to be bounded for all z and non-negative.

8 8 Introduction to Travelling Waves Modeling Examples Linear Parabolic PDEs Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation 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.

9 9 Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation Propagating Wave Solutions

10 10 Introduction to Travelling Waves Modeling Examples Example: Logistic Growth Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation du dt = u(1 u); u(x, 0) = This equation can be solved to give u(x, t) = c exp (t) 1 + c exp (t) exp (x) (5) (6) To satisfy the initial conditions we further require u(x, 0) = c 1 + c = 1 c = exp ( x) (7) 1 + exp (x)

11 Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation is thus solved by u t = u(1 u); u(x, 0) = u(x, t) = exp (t x) 1 + exp (t x) exp (x) (8) (9) If we set z = t x we thus have u(z) = exp (z) 1 + exp (z) (10)

12 Introduction to Travelling Waves Modeling Examples Travelling Wave Solution Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation u(z) = exp (z) 1 + exp (z) If z = x t = const then the shape does not change, i.e. if dx dt = 1. (11) 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.

13 13 Introduction to Travelling Waves Modeling Examples Example: Fisher Equation Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of 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: u t = u(1 u) + u xx. (13)

14 Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation 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) U + cu + U(1 U) = 0 (16)

15 15 Introduction to Travelling Waves Modeling Examples PDE Set of ODEs Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation U + cu + U(1 U) = 0 can be written as U = V V = cv U(1 U). (17)

16 16 Introduction to Travelling Waves Modeling Examples Phase Plane Analysis Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation Determine Nullclines Determine Steady States Determine Stability of Steady States Determine Trajectories and Phase Vectors

17 Introduction to Travelling Waves Modeling Examples Nullclines and Steady States Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation 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).

18 18 Introduction to Travelling Waves Modeling Examples Stability of Steady States Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation 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)

19 19 Introduction to Travelling Waves Modeling Examples Stability of Steady States Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation 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 stable spiral (2 conjugate complex eigenvalues with negative real part) if c < 2.

20 20 Phase Plane Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation Fisher Equation: U = V V = cv U(1 U). Steady States: 1. (U, V ) = (1, 0): saddle node 2. (U, V ) = (0, 0): stable node if c 2 Phase Plane Trajectory: dv du = cv U(1 U) V

21 Phase Plane Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation 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.

22 22 Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation For travelling waves to exist we require one stable node and one saddle node.

23 23 Initial conditions Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation 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 then we obtain travelling wave solutions with c = c min = 2. For other initial data the solution critically depends on the behaviour of u(x, 0) as x ±.

24 24 Introduction to Travelling Waves Modeling Examples Initial conditions continued Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation Consider the leading edge of the evolving wave where since u is small u 2 << u such that u t u + u xx u(x, 0) = A exp ( αx), α > 0 (22) with solution if i.e. u(x, t) = A exp ( α(x ct)) (23) αcu = u + α 2 u (24) c = 1 α + α (25) The wave speed c depends on the initial conditions α.

25 25 General Equation Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher 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)

26 26 Introduction to Travelling Waves Modeling Examples The minimal travelling wave speed Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation The minimal wave speed of can be determined as c = f (u) α + α (31) by differentiating c with respect to α c min = 2 f (u) (32) dc dα = 1 α 2 f (u) + 1 = 0 α = ± f (u) (33)

27 27 Introduction to Travelling Waves Modeling Examples The travelling wave solution Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation 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)

28 28 Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation 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.

29 29 Introduction to Travelling Waves Modeling Examples Stability of travelling wave Solutions Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation Introduce perturbation around travelling wave solution u(z, t) = u c (z) + wv(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.

30 30 Introduction to Travelling Waves Modeling Examples Stability of travelling wave Solutions Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation Ansatz: u(z, t) = u c (z) + wv(z, t) 0 < w 1 (39) v(z, t) = g(z) exp ( λt) (40) Determine eigenvalues to evaluate long-term behaviour of v(z, t).

31 31 Waves in 3D Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation transform to spherical coordinate system 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.

32 32 Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation The diffusion equation then reads du dt where s is the spherical Laplace operator, s = 1 r 2 r (r 2 r ) + 1 r 2 sin(θ) = f (u) + D s u. (41) θ (sin(θ) θ ) + 1 r 2 sin 2 (θ) ψ 2. (42) 2

33 33 Axisymmetry Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation θ = 0 The spherical Laplace operator then simplifies to ψ = 0 (43) a = 2 r r r (44)

34 34 Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation 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)

35 35 Introduction to Travelling Waves Modeling Examples Definitions Propagating Wave Solutions Initial conditions Axisymmetric form of Fisher equation 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.

36 36 Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis The calcium-stimulated-calcium release mechanism

37 37 Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis 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.

38 38 Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis 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

39 39 Introduction to Travelling Waves Modeling Examples Graphical Analysis The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis u = k 1u 2 k 2 + u 2 k 3u + L = f (u)

40 40 Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis 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.

41 41 Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis Since Ca 2+ can diffuse we have to consider a reaction-diffusion equation to adequately describe the Ca 2+ kinetics and we thus have u = f (u) + D d 2 u dx 2. (49) As before we write 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)

42 Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis 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 A α = 2D, c = 2D (u 1 2u 2 + u 3 ).

43 43 Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis 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.

44 Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis Calcium Waves on Amphibian Eggs Models by Cheer et al (1987) and Lane et al (1987)

45 45 Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis 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.

46 46 Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis 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.

47 47 Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis The diffusion equation then reads du dt where s is the spherical Laplace operator, = f (u) + D s u. (53) s = 1 r 2 r (r 2 r ) + 1 r 2 sin(θ) 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

48 48 Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis 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)

49 49 Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis 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

50 50 Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis 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 to 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.

51 51 Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis Models for Somitogenesis

52 52 Somitogenesis Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis

53 53 Introduction to Travelling Waves Modeling Examples Clock and Wavefront Model The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis 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),

54 54 Somitogenesis Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models 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

55 55 Introduction to Travelling Waves Modeling Examples Bistability model for Somitogenesis The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models 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

56 56 Thanks!! Introduction to Travelling Waves Modeling Examples The calcium-stimulated-calcium release mechanism Calcium Waves on Amphibian Eggs Models for Somitogenesis Thanks for your attention! Slides for this talk will be available at: Travelling Waves Dagmar Iber Computational Biology (CoBI), D-BSSE, ETHZ

57 Models for Chemotaxis Chemotaxis and Directed Cell Movements Dagmar Iber Computational Biology (CoBI), D-BSSE, ETHZ 1

58 2 Models for Chemotaxis Chemotaxis

59 3 Models for Chemotaxis Outline 1 Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis 2

60 Models for Chemotaxis Chemotaxis Chemotaxis The presence of a gradient in a chemoattractant a(x, t) gives rise to movement of a species (density n(x, t) up the concentration gradient.

61 5 Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Models for Chemotaxis

62 6 Models for Chemotaxis Keller-Segel Model (1971) General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Fluxes in Keller-Segel Model Chemotactic Flux: Diffusional Flux: Total Flux: J C = nχ(a) a J D = D n J = J C + J D Keller-Segel Model (1971) Cells: Chemoatttractant: n t a t = (D n (a) n χ(a)n a) = D a 2 a nδ(a)

63 Models for Chemotaxis Chemotactic Sensitivity χ(a) General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Experiments: The chemotactic effect increases as the chemoattractant concentration a(x, t) decreases. Chemotactic Sensitivity Log Law: χ(a) = χ 0 a Receptor Law: χ(a) = χ 0 k 2 (k + a) 2

64 8 Diffusion D n (a) Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Diffusion Enhancement of Motility: Constant Motility: ( ) ak D n (a) = D 1 + α (a + K ) 2 D n (a) = D

65 9 Models for Chemotaxis Chemoattractant Degradation δ(a) General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Chemoattractant Degradation Typically neglected: δ(a) = 0 Nonlinear: a δ(a) (a + K ) Linear: δ(a) a

66 10 Models for Chemotaxis Interesting qualitative Behaviours General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Interesting qualitative Behaviours 1 Travelling Waves: χ(a) = χ 0 a Aggregation: χ(a) = χ 0 n Work this out as part of your homework!

67 11 Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Non-dimensioanl Aggregation Model Cells: Chemoatttractant: Parameters: α, χ, D u, D v > 0 u = D u 2 u α (uχ(v) a) + f (u, v) t v = D v 2 v + g(u, v) t

68 Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Linearization around the steady state The spatially homogenous problem ( ) u u = u = v has steady states (u, v ). ( f (u, v) g(u, v) ) Linearization at the steady state We write u = u + ɛu 1, v = v + ɛv 1, where 0 < ɛ 1, such that Cells: Chemoatttractant: u 1 = D u 2 u 1 α (u χ a) + fu u 1 + fv v 1 t v 1 = D v 2 v 1 + g t uu 1 + gvv 1

69 13 Models for Chemotaxis Linear stability of the steady state General Formulation Example Linear Boundary Value Problem Nonlinear Analysis The stability of the steady states can be determined by studying the long-term behaviour of perturbations of the steady state ( ) u u w = v v ẇ = Jw + D w ( fu f J = v g u g v ) = ( ) ( Du αχu ; D = 0 D v )

70 Separable Solution Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Time-dependent Solution Φ = JΦ Φ(t) exp (λt) where λ represents the eigenvalues of J. SpatialSolution 0 = JW + D W W (x) exp (ikx) where Dk 2 are the eigenvalues of J. k is referred to as wavenumber.

71 15 Models for Chemotaxis Ansatz: Separable Solution General Formulation Example Linear Boundary Value Problem Nonlinear Analysis w(x, t) = Φ(t)W (x) exp (λt + ikx) We can rewrite this as λw = Jw k 2 Dw x Hw = λw H = J k 2 D. ( f H = u D u k 2 fv + αχu k 2 gu gv D v k 2 )

72 16 Eigenvalues of H Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Hw = λw H = J k 2 D. We then have tr(h) = (D u + D v )k 2 + (f u + g v) < 0 det (H) = h(k 2 ) = D u D v k 4 (D u g v + D v f u + αu χ g u)k 2 +f u g v f v g u

73 17 Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Stability to temporal perturbations, k = 0 To obtain Re(λ(k 2 = 0)) < 0 we require tr(j) = (f u + g v) < 0 det (J) = h(k 2 = 0) = f u g v f v g u > 0

74 18 Dispersion Relation Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis For patterns to emerge we require det (H) = h(k 2 ) = D u D v k 4 (D u gv + D v fu + αu χ gu)k 2 +fu gv fv gu < 0 We thus want h min < 0. The critical case occurs for h c (k 2 ) = D u D v k 4 (D u gv + D v fu + αu χ gu)k 2 +fu gv fv gu = 0 and thus k 2 c = D ug v + D v f u + αu χ g u 2D u D v

75 19 Dispersion Relation Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Replacing in k 2 c = D ug v + D v f u + αu χ g u 2D u D v h c (k 2 ) = D u D v k 4 (D u g v + D v f u + αu χ g u)k 2 +f u g v f v g u = 0 yields α = (D ugv + D v fu + 2 D u D v (fu gv fv gu)) u χ gu kc 2 fu gv fv gu = D u D v

76 20 Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Example: your HOMEWORK

77 21 Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Example: Aggregation of Dictyostelium discoideum Model for the aggregation of the amoebae state of the slime mold Dictyostelium discoideum. The population n(x, t) secretes a chemical attractant, cyclic-amp, a(x, t), that attracts the amoebae. Cells: Chemoatttractant: n t a t Parameters: h, δ, ξ, D n, D a > 0 = D n 2 n ξ (n a) = D a 2 a + hn δa

78 22 Dispersion Relation Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis For patterns to emerge we require det (H) = h(k 2 ) = D n k 2 (1 + D a k 2 ) ξn k 2 < 0 We thus want h min < 0. The critical case occurs for and thus h c (k 2 ) = D n k 2 (1 + D a k 2 ) ξn k 2 = 0 k 2 c = ξn D n D n D a In the infinite domain we thus only require k 2 c > 0, i.e. ξn > D n for pattern to emerge.

79 23 Finite Domain Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis In the finite domain [0, 1] solutions are with zero flux boundary conditions w exp (λt) cos (kx), k = nπ We thus require k 2 c = χn D n D n D a > π 2 for n = n to be unstable. The critical wavelength is the first to go unstable, namely k 1 = π.

80 24 Models for Chemotaxis Dimensional Conditions General Formulation Example Linear Boundary Value Problem Nonlinear Analysis In the finite domain [0, L] solutions are with zero flux boundary conditions w exp (λt) cos (kx), k = nπ L We thus require k 2 c = χhδ2 n δd n D n D a > π2 L 2 for n = n to be unstable. The critical wavelength is the first to go unstable, namely k 1 = π/l.

81 25 Models for Chemotaxis Minimal Domain Size General Formulation Example Linear Boundary Value Problem Nonlinear Analysis In the finite domain [0, L] solutions are with zero flux boundary conditions, domain size L must meet the following condition L 2 > π 2 D n D a χhδ 2 n δd n Note that higher expression rate h facilitates the emergence of pattern.

82 26 Models for Chemotaxis Conditions for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis χ measures aggregation, D a, D n dispersion. For pattern to emerge aggregation has to defeat dispersion. Minimal domain size Higher chemoattractant production facilitates pattern formation.

83 27 Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Slime mold solving a maze in the lab

84 28 Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Linear Boundary Value Problem

85 29 Models for Chemotaxis Linear Boundary Value Problem General Formulation Example Linear Boundary Value Problem Nonlinear Analysis The previous analysis showed only whether pattern formation is possible, but NOT which pattern will emerge. The pattern types which are possible depend on the number of different wavevectors k allowed by the boundary conditions.

86 30 Wavevectors Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Consider a rectangular domain S with periodic boundary conditions. The spatial eigenvalue problem with periodic boundary conditions is then 2 ψ + k 2 ψ = 0. The possible eigensolutions of the partial differential equations are ψ = A cos (k n x) + B sin (k n x) where the kn 2 are allowable eigenvectors, ( ) ( ) ( 0 2nπ/lx 2pπ/lx k 1 =, k 2mπ/l 2 =, k y 0 3 = 2qπ/l y ) where m, n, p, and q are all integers.

87 31 Wavevectors Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Consider a rectangular domain S with periodic boundary conditions. The spatial eigenvalue problem with periodic boundary conditions is then 2 ψ + k 2 ψ = 0. The number of such solution vectors which are admitted depends on the relationship between the lengths l x and l y.

88 32 Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Suppose kc 2 = ( ) 2Mπ 2 l and that lx = l y = l. Then for M = 1 there are two possible solution vectors ( ) ( ) 0 2π/l k 1 =, k 2π/l 2 = 0 If M = 5 then there are four such possible solution vectors ( ) ( ) 0 2 5π/lx k 1 =, k 2 5π/l 2 =, y 0 ( ) ( ) 2 3π/lx 2 4π/lx k 3 =, k 2 4π/l 4 = y 2 3π/l y

89 33 Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Nonlinear Analysis

90 34 Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Recall: Non-dimensioanl Aggregation Model Cells: Chemoatttractant: Parameters: α, χ, D u, D v > 0 u = D u 2 u α (uχ(v) a) + f (u, v) t v = D v 2 v + g(u, v) t

91 35 Models for Chemotaxis Multi-scale Asymptotic Analysis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Determine small perturbation solutions that are valid for all time (Tyson, 1996). u = u + ɛu 1 + ɛ 2 u 2 + ɛ 3 u v = v + ɛv 1 + ɛ 2 v 2 + ɛ 3 v where (u, v ) is the spatially homogenous steady state.

92 36 Models for Chemotaxis Multi-scale Asymptotic Analysis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Scale time as T = ˆωt, ˆω = ɛω 1 + ɛ 2 ω where ω i have to be determined.

93 37 Substitutions Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis u = ˆω û t T 2 u = 2 û α (uχ(v) a) =... f (u, v) = f +... Linear, quadratic, cubic, and so on terms in û.

94 38 Rewrite Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis ˆω u T = (D 2 + J ) u + Q ( u) + C ( u) + ( ) û u = ˆv The quantities J(û), Q(û), C(û) represent the linear, quadratic, cubic terms in û. J(û) and D were determined in the linear analysis.

95 Models for Chemotaxis Parameter Perturbations General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Consider the parameter sets that yield λ = 0 and study the impact of perturbations in a given parameter, that we will denote a. We then have λ(a) = λ(a c ) + λ a a c +... = λ a a c +... We take the perturbation to be such that kc 2 a = 0 ac so that the perturbation effect is restricted to a change in the growth rate λ. 39

96 40 Perturbation Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Perturbation of the steady state model about the critical set. If change in a c makes Re(λ(a)) positive, the pattern corresponding to k 2 c is predicted to grow according to linear theory. If this growth is sufficiently slow, we can predict whether or not it will develop into a temporally stable pattern and what the characteristics of the pattern will be such as spots or stripes.

97 Expansion Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Consider an expansion of the general parameter a and substitute to get a = a c + ɛa 1 + ɛ 2 a 2 + ɛ 3 a ˆω ˆ u T = (D c 2 + J c ) û + Q c ( û) + C c ( û) + â(d c a 2 + J c a ) û + âq c a ( û) + higher-order terms Drop c for better readability in the following.

98 42 O(ɛ) Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis The O(ɛ) equations ( ) ( 0 Du = 2 + fu fv + αχu k 2 0 gu gv + D v 2 ) ( u1 u 2 ) ( u1 = L u 2 ) can be solved by ( u1 u 2 ) = N V 1 l A l k l 2 = kc 2 l l=1 where A l = a l (T ) exp (i k l x) + ā l (T ) exp ( i k l x) is the sinusoidal part of the solution.

99 43 Linear Solution Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Substituting into ( 0 0 ( u1 u 2 ) = N V 1 l A l l=1 ) ( Du = 2 + fu fv + αχu k 2 ) ( u1 gu gv + D v 2 u 2 ) ( u1 = L u 2 ) yields the normalized unity vector V 1 l = ( V 1l1 V 1 l2 ) = 1 (D v k l 2 gv) 2 + (gu) 2 ( Dv k l 2 g v g u )

100 44 Amplitude Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis The solution amplitude is a l (T ). We require the O(ɛ 2 ) equations to solve for the amplitudes a l (T ) and ā l (T ). Constraints on a l (T ) and ā l (T ) are obtained via so-called secular terms that arise in higher-order terms that have undifferentiated terms on the RHS.

101 45 Secular Terms Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Consider the simple equation u + u = ɛu where denotes differentiation with respect to x. Further u(0) = 1, u (0) = 0. We write u = u 0 + ɛu with u i all O(1). At O(1): u 0 = cos(x). At O(ɛ): u 1 + u 1 = sin(x), u 1 (0) = u 1 (0) = 0 u 1 (x) = 0.5(sin (x) x cos (x)) But u 1 is not O(1) for all x because of the xcos(x) term: it is the secular term.

102 46 Amplitude Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Secular Terms are an artifact of the perturbation method and need to be removed. The O(ɛ 2 ) equation do not produce secular terms and we thus require the O(ɛ 3 ) equation. Finding these equations is complicated and detailed and involves Landau equations.

103 47 Models for Chemotaxis Amplitude for 2 modes General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Suppose the number of modes of the solution is N = 2. Then Tyson (1996) showed that the amplitude, or Landau, equations are d a 1 2 dt = a 1 2 (X A a X B a 2 2 ) + Y a 1 2 d a 2 2 dt = a 2 2 (X B a X A a 2 2 ) + Y a 2 2 where the X A, X B, and Y are complicated functions of the parameters of the original system.

104 48 Models for Chemotaxis Steady State Solutions General Formulation Example Linear Boundary Value Problem Nonlinear Analysis (1) a 1 2 = 0 a 2 2 = 0 (2) a 1 2 = 0 a 2 2 = Y X A (3) a 1 2 = Y X A a 2 2 = 0 (4) a 1 2 = Y X A + X B a 2 2 = Y X A + X B (1) zero amplitude pattern, or no pattern at all. (2,3) stripes (amplitude only in one direction) (4) spots (amplitude in both directions)

105 49 Stability Condition Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis

106 50 Pattern Prediction Models for Chemotaxis General Formulation Example Linear Boundary Value Problem Nonlinear Analysis Tyson (1996)

107 51 Models for Chemotaxis

108 52 Models for Chemotaxis How to sense a gradient?

109 53 Models for Chemotaxis Dynamic Range through Adaptation E. coli can swim up a gradient, sensing the attractant concentration over at least five orders of magnitude Adaptation over at least five orders of magnitude Bacteria can detect a change in occupancy of their aspartate receptors of %, corresponding to the binding of one or two molecules per cell

110 54 Models for Chemotaxis Adaptation through integral Feedback Barkai Leibler (1997) Nature, 387, 913ff Yi et al (2000) PNAS, 97, 4649ff Ma et al (2009) Cell, 138, Lecture 5 in Lecture Course on Mathematical Modelling in Systems Biology

111 55 Models for Chemotaxis Literature Murray: Mathematical Biology, Volume 1 Murray: Mathematical Biology, Volume 2, Chapter 5: Bacterial Patterns and Chemotaxis

112 56 Models for Chemotaxis Thanks!! Thanks for your attention! Slides for this talk will be available at: Chemotaxis and Directed Cell Movements Dagmar Iber Computational Biology (CoBI), D-BSSE, ETHZ