Volume Effects in Chemotaxis

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1 Volume Effects in Chemotaxis Thomas Hillen University of Alberta supported by NSERC with Kevin Painter (Edinburgh), Volume Effects in Chemotaxis p.1/??

2 Eschirichia coli Berg - Lab (Harvard) Volume Effects in Chemotaxis p.2/??

3 Azotobacter vinelandii Page - Lab (Edmonton) Volume Effects in Chemotaxis p.3/??

4 Dictyostelium discoideum Firtel - Lab (US San Diego): Volume Effects in Chemotaxis p.4/??

5 Relevance in ecology oriented movement chemotaxis models as prototype pattern formation and spatial distributions. extinction and coexistence. prey-taxis (with Lee, Lewis). Volume Effects in Chemotaxis p.5/??

6 Outline (1) The Classical Chemotaxis Equations (2) Derivation from a Master Equation (3) The Volume Filling Approach (4) The Quorum Sensing Approach (5) The Finite Sampling Radius Approach (6) Other Approaches (pressure, multi-phase flows) (7) Conclusions (8) Future Research Volume Effects in Chemotaxis p.6/??

7 (1) The Classical Chemotaxis Model u t = D u u χ {u v} v t = D v v + g(u, v) u(t, x): particle density v(t, x): concentration of chemical signal D u : diffusion coefficient, χ chemotactic sensitivity g(u, v) : production and consumption of signal. Volume Effects in Chemotaxis p.7/??

8 References Patlak 1953 Keller + Segel Othmer + Stevens 1997: The ABC of Chemotaxis.... Horstmann 2004: Review. Hillen + Potapov 2004: spikes in 1-D. Volume Effects in Chemotaxis p.8/??

9 Review Dirk Horstmann From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Part I: Jahresbericht der DMV, Vol. 105 (3), , Part II: Jahresbericht der DMV, Vol. 106 (2), 51-69, Volume Effects in Chemotaxis p.9/??

10 (2) Derivation from a Master Equation Random walk description (Othmer-Stevens 1997) _ T T + i i x x i 1 i i+1 u i (t) : Probability to find a particle at x i at time t. T ± i : Transitional probabilities per unit of time for one jump to the right (+) or left (-). Master equation: x du i dt = T + i 1 u i 1 + T i+1 u i+1 (T + i + T i ) u i. Volume Effects in Chemotaxis p.10/??

11 Example: Diffusion Assume the grid size is h and T ± i = α. du i dt = α(u i 1 + u i+1 2u i ) Hence with D u := lim h 0 αh 2 we find for u(t, x): du dt = D 2 u x 2u Volume Effects in Chemotaxis p.11/??

12 Now with Chemotaxis v i : Concentration of a chemical signal. T ± i = α + β(τ(v i±1 ) τ(v i )) τ : sensitivity function, α, β 0 du i dt = α (u i+1 2u i + u i 1 ) β((u i+1 + u i )(τ i+1 τ i ) (u i + u i 1 )(τ i τ i 1 )) Volume Effects in Chemotaxis p.12/??

13 Continuous Limit Limit h 0: u t = D u u xx (uχ(v)v x ) x, D u = lim αh 2, χ(v) = lim 2h 2 β τ(v) v : chemotactic sensitivity. u t = D u u {uχ(v) v} v t = D v v + g(u, v) Volume Effects in Chemotaxis p.13/??

14 Results on Spikes and Finite Time Blow Up u t v t = u χ {u v} = D v v + γu δv x time Volume Effects in Chemotaxis p.14/??

15 Theorem 1 ( ) (Childress, Percus, Jäger, Luckhaus, Nagai, Senba, Yoshida, Herrero, Velazquez, Levine, Sleeman, Gajewski, Zacharias, Biler, Post, Horstmann, Suzuki, Yagi, Potapov, Hillen, Renclawowicz, etc ) ū 0 = u 0 (x)dx 1-D: Spike formation, no blow-up. 2-D: There exists a threshold θ such that ū 0 θ blow-up θ 2 ū 0 < θ boundary blow-up ū 0 < θ 2 no blow-up. n-d: There is a threshold as well (Renclawowicz, Hillen 2005). Volume Effects in Chemotaxis p.15/??

16 So What? Question 1: What does finite time blow-up tell us about the biology? Question 2: What does finite time blow-up tell us about the modeling? Volume Effects in Chemotaxis p.16/??

17 Regularizations (A) Saturation Effects (Neves et al. Biler, Othmer-Stevens) (B) Volume Filling (H, Painter) (C) Quorum Sensing (H, Painter) (D) Finite Sampling Radius (H, Painter, Schmeiser) (E) Pressure (Preziosi et al.) (F) Multi-phase flow (Owen et al.) (G) Cell Kinetics (Boy, Wrzosek) Volume Effects in Chemotaxis p.17/??

18 (A) Saturation Receptor argument (Othmer, Stevens 1997) R + V k 1 A k 1 R: free receptor, V signal, A occupied receptor. K = R + A = const. quasisteady state: a = Kv v+γ, γ = k 1 k 1. gradient sensing: τ(v) a, χ(v) Kγ (γ + v) 2. Volume Effects in Chemotaxis p.18/??

19 Applications Segel, Jackson (1973): Weber-Fechner law χ(v) = c 1 v. Rivero, Tranquillo, ( Buettner, Lauffenburger (1989): χ(v)v x tanh c 1 v x c 2 +v 2 ). Ford, Lauffenburger (1991): χ(v) = c 1 c 2 +v 2. Höfer, Sherrat, Maini (1995): χ(v) = χ 0v m A m +v m. Othmer, Stevens (1997), Levine Sleeman (1997) Tyson, Lubkin, Murray (1999), etc. Volume Effects in Chemotaxis p.19/??

20 Theoretical Biler (1999): χ(v) v = φ(v) φ(v) is strictly sublinear, if χ(v) > 0, χ(v) 0, as v, vχ(v) increasing. Theorem : n = 2 and strictly sublinear = global existence. n 2, χ(v) = χ 0 /v, χ 0 < 2/n = global existence. Post (1999), Nagai, Senba (1998), Velazquez (2003) etc. Volume Effects in Chemotaxis p.20/??

21 (B) The Volume Filling Approach (w. K. Painter) Increasing chemoattractant concentration A B C Introduce q(u) : probability to find space at a local cell density u Assumption q(u max ) = 0 and q(u) 0 for all 0 u < U max. Standard example: U max = 1, q(u) = 1 u. Volume Effects in Chemotaxis p.21/??

22 The Volume Filling Model T ± i = q(u i±1 ) (α + β(τ(v i±1 ) τ(v i ))) Substitute T ± i into the above master equation and let h 0: u t = (D u (q(u) q (u)u) u q(u)uχ(v) v) v t = D v v + f(u, v) Volume Effects in Chemotaxis p.22/??

23 Complete Picture [1]-[7] [1] Hillen + Painter 2000: 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) Volume Effects in Chemotaxis p.23/??

24 Pattern Formation in 1-D g(u, v) = u v. cell density log 10 t space 30 Volume Effects in Chemotaxis p.24/??

25 Pattern Formation in 2-D ū 0 = 0.5 (top), 0.25 (middle), 0.75 (bottom) Volume Effects in Chemotaxis p.25/??

26 Complete Picture [1]-[7] [1] Hillen + Painter 2000: [2] Painter + Hillen 2002: Derivation from a random walk description, pattern formation, coarsening. [3] D. Wrzosek 2003: Existence of a compact global attractor. [4] D. Wrzosek 2004: Lyapunov function. ω-limit sets are steady states. Volume Effects in Chemotaxis p.26/??

27 Complete Picture [1]-[7] [5] Potapov + Hillen 2004: Bifurcation diagram, metastability, numerical estimates of leading eigenvalues, scaling analysis and pattern interaction. Bifurcation Diagram Volume Effects in Chemotaxis p.27/??

28 Complete Picture [1]-[7] [6] Dolak + Schmeiser 2004: Asymptotic analysis of pattern interaction. [7] Dolak + Hillen 2003: Application to Dictyostelium discoideum and to Salmonella typhimurium. Volume Effects in Chemotaxis p.28/??

29 Application to Dictyostelium discoideum Volume Effects in Chemotaxis p.29/??

30 Application to Salmonella typhimurium Volume Effects in Chemotaxis p.30/??

31 (C) The Quorum Sensing Approach Increasing chemoattractant concentration (b) quorum based approach w i : concentration of quorum sensing molecule. Case 1: Interfering substances T ± i = (α + β(w i )(τ(v i±1 ) τ(v i ))) Case 2: Non-interfering substances T ± i = (α + β(τ(v i±1, w i±1 ) τ(v i, w i )) Volume Effects in Chemotaxis p.31/??

32 Derivation: quorum sensing Case 1: Interfering substances T ± i = (α + β(w i )(τ(v i±1 ) τ(v i ))) u t = D u u xx (β(w)uχ(v)v x ) x v t = D v v xx + g 1 (u, v, w) = D w w xx + g 2 (u, v, w), w t Volume Effects in Chemotaxis p.32/??

33 Derivation: quorum sensing Special case of D w = 0, w equilibrates fast w = w(u) monotonic function Then β(w(u)) = β(u) a volume filling model follows. Volume Effects in Chemotaxis p.33/??

34 Derivation: quorum sensing Case 2: Non-interfering substances T ± i = (α + β(τ(v i±1, w i±1 ) τ(v i, w i )) u t = D u u xx (uχ v v x ) x + (uχ w w x ) x v t = D v v xx + g 1 (u, v, w) = D w w xx + g 2 (u, v, w), w t Volume Effects in Chemotaxis p.34/??

35 Quorum sensing, case 1, in 2-D Ring-shaped patterns: Chi = 2.0 Chi = 5.0 Chi= Volume Effects in Chemotaxis p.35/??

36 Quorum sensing in 1-D Quorum sensing model (case 2, attraction-repulsion) Hopf bifurcations possible. time space Volume Effects in Chemotaxis p.36/??

37 Analysis of the non-interfering model (w. J. Renclawowicz) u t = (D u u χ v u v + χ w u w) v t = D v v + αu βv w t = D w w + γu δw Some rescaling: u t = ( u u v + u w) v t = D v v + u βv w t = D w w + γu δw Volume Effects in Chemotaxis p.37/??

38 Concentration difference, z Introduce z := v w Special case: D v = D w = D, β = δ: u t = u (u z) z t = D z + (1 γ)u βz repulsive case: γ > 1, attractive case: γ < 1. Volume Effects in Chemotaxis p.38/??

39 Results on the Quorum Sensing Model (w. J. Renclawowicz) For attractive and repulsive case: Local existence. u 0 0, z 0 0, = u(t) 0, z(t) 0. In 1-D, global existence. Norm-estimates of Hillen + Potapov n-d, n 3: Assume p > n/2 + 1 and u 0 p, z 0 p are small. Then solutions exist globally in time. New proof based on L p estimates. Volume Effects in Chemotaxis p.39/??

40 More Results For the attractive case: 2-D: Lyapunov function. Modify Gajewsky + Zacharias 1998 and Biler 1998 for z <> 0. Consequence: Global existence below a threshold u 0 < 4Dθ. 2-D: Existence of blow-up solutions. apply Herrero-Velazquez 1997 Volume Effects in Chemotaxis p.40/??

41 Open Questions for Quorum Sensing Global existence for the repulsive case in n-d. Hopf bifurcation and periodic solutions. Volume Effects in Chemotaxis p.41/??

42 (D) Finite Sampling Radius (with Painter and Schmeiser) Othmer, H (2002): Nonlocal gradient: n v(x, t) := σv(x + Rσ, t) dσ. S n 1 R S n 1 Sampling radius R. (i) If v =const. then v = 0. (ii) For each (x, t) we have lim R 0 v = v. Volume Effects in Chemotaxis p.42/??

43 Modified Model u t = u χ { u } v v t = D v v + γu δv Theorem : In any dimension: if R > 0 = global existence. Volume Effects in Chemotaxis p.43/??

44 Simulations Volume Effects in Chemotaxis p.44/??

45 Other Approaches Volume Effects in Chemotaxis p.45/??

46 (E) Preziosi et al mass and momentum conservation: u t + ( γ)u = 0 γ t + ( γ)γ + p(u) = µ v γ 0 = D v + αu βv p(u): pressure γ(x, t): velocity. Aggregation and mesh formation, application to vasculature. Volume Effects in Chemotaxis p.46/??

47 Relation to Volume Filling Given q(u), then p(u) = q(u) + q(u) u Given p(u), then q(u) = u p (u) u Volume Effects in Chemotaxis p.47/??

48 (F) Owen, Byrne 2005 Two phase flow: cells u, water 1 u. Chemotaxis equation: u t = (µ(u, v) u χ(u, v) v) µ(u, v) = χ(u, v) = u(1 u)2 k(u) u(1 u)2 k(u) (uλ(u, v)) u Λ(u, v) v Λ(u, v): chemotactic stress k(u): interphase drag strength. Volume Effects in Chemotaxis p.48/??

49 Conclusions on Blow-up Blow-up describes the onset of aggregation and the underlying instability very well. Blow-up models are singular limit cases of more realistic models. More realistic models include volume effects. Volume Effects in Chemotaxis p.49/??

50 Conclusions on Volume Effects Volume Filling Relatively easy to include into a model. Leads to global existence and pattern formation. Attractors and bifurcations are understood. Quorum Sensing Can show all phenomena, blow-up, spikes, or global patterns. Open questions: Global existence for the repulsive case, and Hopf bifurcation. Finite Sampling Radius The finite sampling radius immediately regularizes the problem. Open question: 3-D? Open question: Does the non local gradient have additional regularity properties? Volume Effects in Chemotaxis p.50/??

51 (8) Future Research Chemotaxis of D. discoideum, S. typhimurium, E. col i, etc. on the population level are well understood and well modeled. Current research focusses on the individual movement behavior. Other applications, like endothelial cells. angiogenesis, tumor growth, cancer therapies. wound healing. development, pigmentation patterns. in ecology (A. vinelandii). Volume Effects in Chemotaxis p.51/??

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