1. Free Boundaries in Biological Aggregation Models

2. Joint Work with Yasmin Dolak-Struss, Vienna / FFG
Biological Aggregation
Joint Work with
Yasmin Dolak-Struss, Vienna / FFG
Christian Schmeiser, Vienna
Marco DiFrancesco, L‘Aquila
Daniela Morale, Milano Vincenzo Capasso, Milano Peter Markowich, Cambridge
Jan Pietschmann, Münster / Cambridge Mary 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. Aggregation Phenomena
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. d X = F ( ) t + ¾ W Microscopic Models F ( X ) = ¡ r V + 1 [ G R ]
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 d X N j = F ( ) t + W F j ( X N ) = r V + 1 k 6 [ G R ]

7. Short-Range Repulsion
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. Taxis (= ordering, greek)
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 equations Othmer-Stevens, ABC‘s of Taxis, Hill-Häder 97, Keller-Segel 73, Erban, Othmer, Maini, .

9. Taxis (= ordering, greek)
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. Chemotaxis @ % + r ¢ ( q ) S ¡ ² =
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) @ t % + r ( q ) S =

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

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

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

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

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

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

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

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

20. Gradient Flow Perspective
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
" = E [ % ] @ t % = r (
" ) @ t % = r ( 1 )
"

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

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

25. Asymptotic Expansion Next order determines interfacial motion
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. ¡ ¹ = Surface Diffusion D ¡ =
Biological Aggregation
Surface Diffusion
We obtain a surface diffusion law with diffusivity and potential
Corresponding energy functional D = 2 @ n S = S 2 [ ]

28. Conservation and Dissipation
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. Conservation and Dissipation
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 perturbation Instability without entropy condition !
Otherwise high-frequency stability, possible low-frequency instability

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

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

33. Low Frequency Instability
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. Cluster Motion in Complicated Geometry
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. Swarming Swarming phenomena arise at the macroscale
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. Nonlinear Fokker-Planck Equations
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. Entropy for Mean-Field Fokker-Planck
Biological Aggregation
Entropy for Mean-Field Fokker-Planck
Entropy functional

42. Mean-Field Fokker-Planck
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. Potential Difficulties
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. Conservation for Nonlinear Fokker-Planck
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. Application to Pattern Formation
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. Application to Pattern Formation
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. Application to Pattern Formation
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. Application to Pattern Formation
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

51. Application to Pattern Formation
Biological Aggregation
Application to Pattern Formation
Long-time behaviour of Fokker-Planck equations:
Convergence to adjacent concentrated solution if initial value is sufficiently close (in the Wasserstein metric)
Asymptotic speed of convergence only determined by local properties of G around zero (estimate for integral operator and ODE in Banach space at the level )

52. Simulation of Pattern Formation
Biological Aggregation
Simulation of Pattern Formation
Gaussian interaction kernel

53. Simulation of Pattern Formation
Biological Aggregation
Simulation of Pattern Formation
Gaussian interaction kernel, rescaled density

54. Refined Asymptotic: Self-Similar Solutions
Biological Aggregation
Refined Asymptotic: Self-Similar Solutions
Let V be convex and with a minimum at 0
Then there exist self-similar solutions of the form
where y is determined from the ODE
and y tends to zero until

55. Simulation of Pattern Formation
Biological Aggregation
Simulation of Pattern Formation
Kernel with finite support, rescaled density

56. Interpretation of Pattern Formation
Biological Aggregation
Interpretation of Pattern Formation
Opinion (consensus) formation models (Hegselmann-Krause, Sznajd-Weron) lead asymptotically exactly to above mean-field equations with zero diffusion (no local repulsion of opinions)
Only few majority opinions survive in the long run
With local interaction more than one majority opinion can be obtained, but typically low number (compare analysis of the number of parties surviving in democracies, BenNaim et al)

57. Local Repulsion: Swarming, Crowding
Biological Aggregation
Local Repulsion: Swarming, Crowding
In biological systems (swarms, crowds, cells) there is a small local repulsion, hence small nonlinear diffusion
Conjecture: stationary solutions are clusters with finite support, close to concentrated measures but with density
Idea of proof: perturbation argument around zero diffusion
How to expand around a Dirac-delta ?

58. Asymptotic Expansion by Optimal Transport
Biological Aggregation
Asymptotic Expansion by Optimal Transport
Closeness to Dirac-delta means small Wasserstein-metric
Hence there exists a „short“ optimal transport (geodesics of Wasserstein metric)
Note: close to concentrated solution, the integral operator behaves like a local operator, analogous to confining potential
Hence, similar stationary states as for nonlinear diffusions with confining potential

59. Asymptotic Expansion by Optimal Transport
Biological Aggregation
Asymptotic Expansion by Optimal Transport
At the level of the pseudo-inverse we simply have
First-order expansion solves
yields density to highest order

60. Asymptotic Expansion by Optimal Transport
Biological Aggregation
Asymptotic Expansion by Optimal Transport
For m = 2 (quadratic nonlinear diffusion, natural two-particle interaction) rigorous analysis via implicit function theorem
Yields existence of stationary solutions for
with support having a diameter of order

61. Biological Aggregation
Large diffusion
In general interplay between the repulsion (diffusion) and attraction (integral operator)
For large diffusion, repulsion becomes too strong, densities decay to zero
Necessary condition for existence of stationary solutions

62. Swarming: Zero vs. Small diffusion
Biological Aggregation
Swarming: Zero vs. Small diffusion

63. Swarming: Zero vs. Small diffusion
Biological Aggregation
Swarming: Zero vs. Small diffusion

64. Swarming: Zero vs. Small diffusion
Biological Aggregation
Swarming: Zero vs. Small diffusion

65. Swarming: Zero vs. Small diffusion
Biological Aggregation
Swarming: Zero vs. Small diffusion

66. Swarming: Zero vs. Small diffusion
Biological Aggregation
Swarming: Zero vs. Small diffusion

67. Numerical Analysis Piecewise constant FE spaces for r and m
Biological Aggregation
Numerical Analysis
Piecewise constant FE spaces for r and m
Raviart-Thomas for J
Numerical Analysis (implicit scheme)
- Well-posed convex programming problem in each time step (Newton method)
- Discrete energy dissipation
- Conserved quantities (mass, center of mass)
- Discrete maximum principle for m
- Error estimates for smooth solutions

68. References www.math.uni-muenster.de/u/burger
Biological Aggregation
References
mb, M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Networks and Heterogeneous Media (2008), to appear. mb, Y.Dolak-Struss, C.Schmeiser, Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions, Commun. Math. Sci. 6 (2008), mb, M. Di Francesco, Y.Dolak-Struss, The Keller-Segel model for chemotaxis: linear vs. nonlinear diffusion, SIAM J. Math. Anal. 38 (2006),
mb, V.Capasso, D. Morale, On an aggregation model with long and short range interactions, Nonlinear Analysis. Real World Application s 8 (2007),