Pattern Formation in Chemotaxis

Pattern Formation in Chemotaxis Thomas Hillen University of Alberta, Edmonton (with K. Painter, Heriot-Watt) Pattern Formation in Chemotaxis p.1/28

Outline (1) The Minimal Model (M1) Pattern Formation in Chemotaxis p.2/28

Outline (1) The Minimal Model (M1) (2) The Volume Filling Model (M3a) Pattern Formation in Chemotaxis p.2/28

Outline (1) The Minimal Model (M1) (2) The Volume Filling Model (M3a) (3) Regularizations (M2)-(M8) Pattern Formation in Chemotaxis p.2/28

Outline (1) The Minimal Model (M1) (2) The Volume Filling Model (M3a) (3) Regularizations (M2)-(M8) (4) Pattern Formation Pattern Formation in Chemotaxis p.2/28

Outline (1) The Minimal Model (M1) (2) The Volume Filling Model (M3a) (3) Regularizations (M2)-(M8) (4) Pattern Formation (5) Open Questions and Conclusions Pattern Formation in Chemotaxis p.2/28

(1) Chemotaxis Equation Patlak 53 Keller and Segel, 7 Minimal model u t = (D u χu v), v t = 2 v + u v. (M1) u(x, t): cell density v(x, t): chemical signal Pattern Formation in Chemotaxis p.3/28

Horstmann s review 3 [D. Horstmann, Jahresberichte DMV, 15, 13-165, 23.] model derivations steady states blow-up general mathematical properties self similar solutions, traveling waves Pattern Formation in Chemotaxis p.4/28

(M1) shows Finite Time Blow Up 2 18 16 14 12 1 8 6 4 2.2.4.6 x.8 1 2 4 1 8 6 time Pattern Formation in Chemotaxis p.5/28

Theorem 1 (1992-27) (Childress, Percus, Jäger, Luckhaus, Nagai, Senba, Yoshida, Herrero, Velazquez, Levine, Sleeman, Gajewski, Zacharias, Biler, Post, Horstmann, Suzuki, Yagi, Potapov, Hillen, Perthame, etc ) ū = u (x)dx 1-D: Spike formation, no blow-up. 2-D: quantized blow-up (Suzuki 5, Hortsmann 3) boundary blow-up takes mass 4π χ interior blow-up takes mass 8π χ. If u I 1 < 4π χ then global existence. Pattern Formation in Chemotaxis p.6/28

(2) The Volume Filling Approach (M3a) (w. K. Painter) Increasing chemoattractant concentration A B C Pattern Formation in Chemotaxis p.7/28

(2) The Volume Filling Approach (M3a) (w. K. Painter) Increasing chemoattractant concentration A B C q(u) : probability to find space at a local cell density u Pattern Formation in Chemotaxis p.7/28

(2) The Volume Filling Approach (M3a) (w. K. Painter) Increasing chemoattractant concentration A B C q(u) : probability to find space at a local cell density u Assumption q(u max ) = and q(u) for all u < U max. Pattern Formation in Chemotaxis p.7/28

The Volume Filling Model We use a Master equation approach to derive: u t = (D(q(u) q (u)u) u q(u)uχ(v) v) v t = v + u v Pattern Formation in Chemotaxis p.8/28

Pattern Analysis [1]-[7] [1] Hillen + Painter 2: First mention of the volume filling model; proof of global existence for special cases; numerical pattern formation. Pattern Formation in Chemotaxis p.9/28

Pattern Analysis [1]-[7] [1] Hillen + Painter 2: First mention of the volume filling model; proof of global existence for special cases; numerical pattern formation. If the domain is large enough we obtain non trivial steady states. u(x) τ (x) Pattern Formation in Chemotaxis p.9/28

Pattern Formation in 1-D cell density 1.5 9 6 log 1 t 3 2 1 space 3 Pattern Formation in Chemotaxis p.1/28

Pattern Formation in 2-D ū =.5 (top),.25 (middle),.75 (bottom) Pattern Formation in Chemotaxis p.11/28

Pattern Analysis [1]-[7] [1] Hillen + Painter 2: [2] Painter + Hillen 22: Derivation from a random walk description, pattern formation, coarsening. Pattern Formation in Chemotaxis p.12/28

Pattern Analysis [1]-[7] [1] Hillen + Painter 2: [2] Painter + Hillen 22: Derivation from a random walk description, pattern formation, coarsening. [3] D. Wrzosek 23: Existence of a compact global attractor. Pattern Formation in Chemotaxis p.12/28

Pattern Analysis [1]-[7] [5] Potapov + Hillen 24: Bifurcation diagram, metastability, numerical estimates of leading eigenvalues, scaling analysis and pattern interaction. Pattern Formation in Chemotaxis p.13/28

Pattern Analysis [1]-[7] [6] Dolak + Schmeiser 24: Asymptotic analysis of pattern interaction. Pattern Formation in Chemotaxis p.14/28

Pattern Analysis [1]-[7] [6] Dolak + Schmeiser 24: Asymptotic analysis of pattern interaction. [7] Dolak + Hillen 23: Application to Dictyostelium discoideum and to Salmonella typhimurium. Pattern Formation in Chemotaxis p.14/28

(3) Regularizations Signal dependent sensitivity (M2a), (M2b) Density dependent sensitivity (M3a), (M3b) Non-local gradient (M4) Non-linear diffusion (M5) Non-linear signal kinetics (M6) Non-linear gradient (M7) Cell kinetics (M8) Pattern Formation in Chemotaxis p.15/28

Receptor Model (M2a) u t = Othmer, Stevens 97 ( D u v t = 2 v + u v ) χu (1 + αv) v 2 Segel 76 77, Ford et al 91, Tyson et al 99, Levine and Sleeman 97 Pattern Formation in Chemotaxis p.16/28

Logistic model (M2b) u t = ( D u χu 1 + β ) v + β v v t = 2 v + u v Keller, Segel 71, Dahlquist et al 72, Nanjundiah 73 see review in Horstmann 3 Pattern Formation in Chemotaxis p.17/28

Volume-Filling (M3a) u t = ( ( D u χu 1 u ) ) v γ H, Painter 1 2 v t = 2 v + u v, Wrzosek 4, Dolak, Schmeiser 5 Pattern Formation in Chemotaxis p.18/28

Velazquez model (M3b): Velazquez 4 u t = ( D u χ u ) 1 + ɛu v v t = 2 v + u v, Pattern Formation in Chemotaxis p.19/28

Non-Local Model (M4) u t = ( D u χu ) ρ v v t = 2 v + u v Non-local gradient: ρ v(x, t) = Othmer, H 2 n S n 1 ρ H, Painter, Schmeiser 6 S n 1 σv(x + ρσ, t) dσ, Pattern Formation in Chemotaxis p.2/28

Nonlinear Diffusion (M5) u t = (Du n u χu v) v t = 2 v + u v, Kowalczyk 5 Eberl 1, biofilm Pattern Formation in Chemotaxis p.21/28

Nonlinear Signal Kinetics (M6) u t = (D u χu v) v t = 2 v + u 1 + φu v, Höfer et al 95, Woodward et al 95 Myerscough et al 98, Post 99. Pattern Formation in Chemotaxis p.22/28

Nonlinear Gradient (M7) u t = (D u χuf c ( v)) v t = 2 v + u v, F c ( v) = 1 c ( ( ) ( )) cvx1 cvxn tanh,..., tanh. 1 + c 1 + c Rivero et al 89 Pattern Formation in Chemotaxis p.23/28

Cell Kinetics (M8) u t = (D u χu v) + ru(1 u) v t = 2 v + u v Tyson 99, Woodward 95, Chaplain, Stuart 93 Osaki et al 2, Wrzosek 4, Pattern Formation in Chemotaxis p.24/28

(4) Pattern Formation Pattern Formation in Chemotaxis p.25/28

2-D Aggregations 15 1 5 (M1) t =.93 1.5.5 1 1 (M2a) α =.5 5 1.5.5 1 15 1 5 (M2b) β =. 1.5.5 1 (M3a) γ = 3. 3 2 1 1.5.5 1 (M3b) ε = 1. 2 1 1.5.5 1 (M4) ρ =.25 3 2 1 (M5) n = 1. 2 1 (M6) φ = 1. 2 1 1 (M7) c = 3. 5 (M8) r =.25 4 2 1.5.5 1 1.5.5 1 1.5.5 1 1.5.5 1 1.5.5 1 (M1) (a) Cell density profiles 1 4 1 3 1 2 1 1 (M2b) (M2a) 1 3 1 2 1 1 (M5) (M4) (M3b) 1 2 1 1 (M8) (M7) (M6) 2 4 6 (M3a) 2 4 6 8 1 2 3 4 5 (b) Peak cell density evolution Pattern Formation in Chemotaxis p.26/28

(5) Open Questions Global existence for (M2b) for n-d, n 3 and β > Global existence for (M2b) for β =. Possibly threshold and blow-up? Prove existence of a global attractor for (M2a), (M2b), (M3b), (M4), (M5), (M6), (M7) analysis of merging dynamics (so far only Dolak et al.) analysis of merging-emerging dynamics for (M8) Pattern Formation in Chemotaxis p.27/28

Conclusions List of models not complete Pattern Formation in Chemotaxis p.28/28

Conclusions List of models not complete A combination of effects might be needed for applications Pattern Formation in Chemotaxis p.28/28

Conclusions List of models not complete A combination of effects might be needed for applications All models show coarsening dynamics Pattern Formation in Chemotaxis p.28/28

Conclusions List of models not complete A combination of effects might be needed for applications All models show coarsening dynamics Thanks to K. Painter, and to NSERC Pattern Formation in Chemotaxis p.28/28