Free Boundaries in Biological Aggregation Models

1. Free Boundaries in Biological Aggregation Models

2. Biological Aggregation Joint Work with Yasmin Dolak-Struss, Vienna / FFG Christian Schmeiser, Vienna Marco DiFrancesco, L‘Aquila Daniela Morale, MilanoVincenzo Capasso, MilanoPeter Markowich, Cambridge Jan Pietschmann, Münster / CambridgeMary Wolfram, Münster / Linz

3. Biological Aggregation Why FBP in Biomedicine ? „Biology works at very specific conditions, selected by evolution. This leads always to some small parameters, hence singular perturbations and asymptotic expansions are very appropriate“ Bob Eisenberg, Dep. of Physiology, Rush Medical University, Chicago In many cases such asymptotics can be used to describe moving boundaries

4. Biological Aggregation Aggregation Phenomena Many herding models can be derived from microscopic models for individual agents, using similar paradigms as statistical physics: • Ions at subcellular levels (channels) • Cell aggregation (chemotaxis) • Swarming / Herding / Schooling / Flocking (birds, fish, insect colonies, human crowds in evacuation) • Opinion formation • Volatility clustering, price herding on markets

5. Biological Aggregation Introduction These processes can be modelled as stochastic systems at the microscopic level Examples are jump processes, random walks, forced Brownian motions, molecular dynamics, Boltzmann equations With appropriate scaling, they all lead to nonlinear Fokker-Planck- type equations as macroscopic limits

6. ¡ ¡ j N N N N ( ) ( ) d d d X X X F W t + N N ¾ ( ) ( ) = X F r V X j ¡ + j j t = j j 1 X N N N N [ ( ) ( ) ] r G X X r R X X + N k k j j N k 6 j = Biological Aggregation Microscopic Models Microscopic models can be derived in terms of SDEs, Langevin equations for particle position (biology always overdamped) Interaction kernels are mainly determined by long-range attraction – kernel with maximum at zero

7. Biological Aggregation Short-Range Repulsion Different paradigms for modelling short-range repulsion • Smooth finite force (like scaled Gaussian): Swarming / Chemotaxis • Smooth force with singularity (Lennard-Jones): Ions • Nonsmooth infinite force (hard-core): Ions, cells • With appropriate scaling all lead to nonlinear diffusion and / or modified mobilities • Cf. Talks of Fasano, King, Calvez

8. Biological Aggregation Taxis (= ordering, greek) • Taxis phenomena arise in various biological processes, typically in cell motion: chemotaxis, haptotaxis, galvanotaxis, phototaxis, gravitaxis, … • Various mathematical models at different scales. Often microscopic random walk models upscaled to macroscopic continuum equationsOthmer-Stevens, ABC‘s of Taxis, Hill-Häder 97, Keller-Segel 73, Erban, Othmer, Maini, .

9. Biological Aggregation Taxis (= ordering, greek) • Taxis includes a long range aggregation and leads to formation of clusters • Original models do not take into account finite size of cells, result can be blow-up of density • Recently modified models have been derived avoiding overcrowding and blow-up (quorum sensing)

10. 0 ( ( ) ( ( ) ( ) ) ) @ r r S r 0 + ¡ ¡ ¢ % % q % ² q % % q % % = t Biological Aggregation Chemotaxis Keller-Segel Model with small diffusion and logistic sensitivity • Sensitivity function for quorum sensing derived by Painter and Hillen 2003 from microscopic model: • q needed to be concave (logistic is extreme one)

11. Biological Aggregation Aggregation in Chemotaxis Keller-Segel Model with small diffusion and logistic sensitivity at small time scales: Cluster formation

12. Biological Aggregation Coarsening and Cluster Motion Keller-Segel Model with small diffusion and logistic sensitivity at large time scales: Cluster motion

13. Biological Aggregation Fast Time Scale Same scaling as before • Obvious limit for diffusion coefficient e to zero

14. Biological Aggregation Fast Time Scale Asymptotic – Entropy Condition Limit for density is a nonlinear (and also nonlocal) conservation law – needs entropy condition • Entropy inequality

15. Biological Aggregation Fast Time Scale Asymptotic – Metastability Possible stationary solutions of the form • Entropy inequality

16. Biological Aggregation Large Time Scale – Cluster Motion Asymptotics for large time by time rescaling • Look for metastable solutions

17. ( ( ) ( ( ) [ ] ) ) @ l l r r r S 1 1 ¡ ¡ ¡ ¡ ¢ " % ½ ½ " o g % " o g % % = t 1 ( ( ) ( ( ) ) ) @ r r ¢ W 1 ¡ ¡ + ¢ " % ½ ½ " % % = t " Biological Aggregation Similarities to Cahn-Hilliard To understand cluster motion, note similarities to Cahn-Hilliard equation with degenerate diffusivity • Keller-Segel rewritten

18. 0 ( ( [ ] ) ) " @ r E r 1 ¡ ¢ % ¹ % % % ¹ = = t " " Biological Aggregation Degenerate (logistic) Diffusivity General Structure • with potential being variation of energy functional

19. µ µ ¶ ¶ Z Z 1 1 " ( ) ( ) ( ) l l F 1 1 + ¡ ¡ ¡ 2 % % o g % % o g % [ [ ] ] j ( j ) [ ( ] ) = d d E E F r W S + % % " % % % % % x x = = ² ² 2 2 " Biological Aggregation Energy functionals Cahn-Hilliard • Keller-Segel

20. 0 ( ( ( [ ( ] ) ) ) " " @ @ r r E r r 1 ¡ ¢ ¢ % % ¹ % % % % ¹ ¹ = = = t t " " Biological Aggregation Gradient Flow Perspective Compare to recently explored gradient flows in the Wasserstein metric on manifold of probability measures • Now even smaller manifold, measures with density bounded by 1

21. 1 Z Z j ( ( j ) ) @ r 1 0 + ¡ ¢ u u % u u u v % 2 2 = = = 0 1 1 2 ( ) ( ) j j t t t d f d d i 1 t ¡ = = ; % % n u u v x = 1 2 ; 0 Biological Aggregation Gradient Flow Perspective Metric gradient flow with an appropriate optimal transport distance • Subject to

22. Biological Aggregation Gradient Flow Perspective Energies are l-convex on geodesics for positive e Limiting energies are not l-convex Leads to singular behaviour: 0-1 constraints for density are attained Interfacial motion apppears

23. Biological Aggregation Asymptotic Expansion Asymptotic expansion in interfacial layer (similar to degenerate-diffusivity Cahn-Hilliard) • Tangential variable s, signed distance in normal direction ex

24. ( 1 ) ^ ^ % q % 0 0 ^ @ ^ @ S % = % 0 = » 0 0 ( ) n » @ S ( ) ( ) 1 ^ ^ ^ + ¡ 0 ¡ e x p q % % q % 0 0 0 0 n Biological Aggregation Asymptotic Expansion Leading order determines profile in normal direction • For general quorum sensing model

25. Biological Aggregation Asymptotic Expansion Next order determines interfacial motion

26. Biological Aggregation Surface diffusion Integration in normal direction and insertion of leading order equation implies • Note: entropy condition crucial for forward surface diffusion

27. 2 D ¡ [ ] S ­ = S @ S ¡ ¡ ¹ = = n 2 2 Biological Aggregation Surface Diffusion • We obtain a surface diffusion law with diffusivityand potential • Corresponding energy functional

28. Biological Aggregation Conservation and Dissipation • Flow is volume conserving (conservation of cell mass) • Flow has energy dissipation

29. Biological Aggregation Stationary Solutions • Stationary solutions can be computed in special situations, • e.g. quasi-one dimensional solutions (flat surfaces) • Stability would naturally be done in terms of a linear stability analysis. Perform linear stability with respect to the free boundary – shape sensitivity analysis • Stationary solutions are critical points of the energy functional(subject to volume constraint)

30. Biological Aggregation Conservation and Dissipation • Stability of stationary solutions can be studied based on second (shape) variations of the energy functional • Stability condition for normal perturbationInstability without entropy condition ! • Otherwise high-frequency stability, possible low-frequency instability

31. Biological Aggregation Low Frequency Instability Perturbation of flat surface, small density

32. Biological Aggregation Low Frequency Instability Perturbation of flat surface, smaller density

33. Biological Aggregation Low Frequency Instability Perturbation of flat surface, large density

34. Biological Aggregation Cluster Motion Surface diffusion with violated entropy condition at the end

35. Biological Aggregation Cluster Motion in Complicated Geometry Surface diffusion with violated entropy condition at the end

36. Biological Aggregation Cluster Motion in 3D

37. Biological Aggregation Outlook • Methodology can be carried over to situations with small diffusivity and a driving potential • Always leads to generalized surface diffusion law • Next (still open) step: • Problems with multiple species • E.g. solutions or channels with several ion types – where / how are the clusters (attraction only among differently charged ions) ?

38. Biological Aggregation Outlook • Problems with reaction terms – different scaling limits possible • (Allen-Cahn or Cahn-Hilliard type): mixed evolution laws • Multiscale issues: complicated 1D problems in normal direction to be solved numerically • E.g. electrical potentials in the human heart – expansion of cardiac bidomain model to derive description of excitation wavefronts cf. Colli-Franzone et al, Nielsen et al, Plank et al, Trayanova et al

39. Biological Aggregation Swarming • Swarming phenomena arise at the macroscale • Animals (birds, fish ..) try to follow their swarm (attractive force) but to keep a local distance (repulsion) • Similar models for consensus formation, but without repulsion

40. Biological Aggregation Nonlinear Fokker-Planck Equations Coarse-graining to PDE-models similar to statistical physics (Boltzmann /Vlasov-type, Mean-Field Fokker Planck) Canonical mean-field equation includes short-range repulsion (nonlinear diffusion) and long-range attraction (interaction kernel G) Capasso-Morale-Ölschläger 04 Interaction of these two effects leads to interesting pattern formationMogilner-Edelstein Keshet 99, Bertozzi et al 03-06

41. Biological Aggregation Entropy for Mean-Field Fokker-Planck • Entropy functional

42. Biological Aggregation Mean-Field Fokker-Planck • Metric gradient flow in manifold of probability measures • with Wasserstein metric (optimal transport theory)

43. Biological Aggregation Important Questions • Existence and Uniqueness (follows from l-convexity of the entropy along geodesics) • Finite speed of propagation: from estimate in the -Wasserstein-metric • Numerical solution: by variational scheme derived from gradient flow structure- Long-time behaviour / pattern formation: difficult due to missing convexity of the entropy

44. Biological Aggregation Potential Difficulties • - convection-dominant for steep potentials • - Nonlocal / nonlinear interaction terms • - degenerate diffusion • - possibly no maximum principle • - bad nonlinearity for optimization / inverse problems • For analysis and robust simulation, look for dissipative formulation

45. Biological Aggregation Spatial Dimension One • In spatial dimension one, there is a unique optimal transport plan, which can be computed via the pseudo-inverse of the distribution function. Let • Then

46. Biological Aggregation Conservation for Nonlinear Fokker-Planck • Equation conserves zero-th and first moment of the density r, • i.e. mass and center of mass (in any dimension if V = 0). In 1D, center of mass becomes in terms of the pseudo-inverse • Finite speed of propagation: by estimate of Wasserstein metric for p to infinity, since

47. Biological Aggregation Application to Pattern Formation • Back to the canonical model • Write one-dimensional case in terms of the pseudo-inverse of the distribution function (Lagrangian formulation, z  [0,1])

48. Biological Aggregation Application to Pattern Formation • Start with pure aggregation model (a = b = 0) • Conjecture: aggregation to concentrated measures (linear combination of Dirac deltas) in the large-time limit • To which, how, and how fast ? • General theory for aggregation kernel G – symmetric with maximum at zero (aggregation most attractive) • No global concavity (decay to zero), only locally concave at 0

49. Biological Aggregation Application to Pattern Formation • Existence of stationary states: let then • is a stationary solution (v corresponds to the pseudo-inverse) • For V = 0 the concentrated measure at the center of mass is a stationary solution • Complete aggregation !

50. Biological Aggregation Application to Pattern Formation • Uniqueness / Non-Uniqueness of stationary states (V=0) • If G has global support, then concentration at center of mass is the unique stationary state • - If G has finite support there is an infinite number of stationary states. Combination of concentrated measures with distance larger than the interaction range is always a stationary solution