martin burger institute for computational and applied mathematics european institute for molecular...

Martin Burger Institute for Computational and Applied Mathematics

European Institute for

Molecular Imaging

Center for

Nonlinear Science CeNoS

Free Boundaries in Biological Aggregation Models

Biological Aggregation 2

13.6.2008Martin Burger

Joint Work withYasmin Dolak-Struss, Vienna / FFGChristian Schmeiser, Vienna

Marco DiFrancesco, L‘Aquila

Daniela Morale, MilanoVincenzo Capasso, MilanoPeter Markowich, CambridgeJan Pietschmann, Münster / CambridgeMary Wolfram, Münster / Linz



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



Aggregation PhenomenaMany 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



IntroductionThese 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



Microscopic ModelsMicroscopic 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

dX Nj = F j (X

N )dt +¾Nj (X N )dW j

t

¡¡

F j (XN ) = ¡ r V(X N

j ) +1N

X

k6=j

[r G(X Nj X N

k ) + r RN (X Nj X N

k )]



Short-Range RepulsionDifferent 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



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, .



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)



ChemotaxisKeller-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(%)r S ¡ ²(q(%) ¡ %q0(%))r %) = 0



Aggregation in ChemotaxisKeller-Segel Model with small diffusion and logistic sensitivity at small time scales: Cluster formation



Coarsening and Cluster MotionKeller-Segel Model with small diffusion and logistic sensitivity at large time scales: Cluster motion



Fast Time ScaleSame scaling as before

Obvious limit for diffusion coefficient to zero



Fast Time Scale Asymptotic – Entropy ConditionLimit for density is a nonlinear (and also nonlocal) conservation law – needs entropy condition

Entropy inequality



Fast Time Scale Asymptotic – MetastabilityPossible stationary solutions of the form

Entropy inequality



Large Time Scale – Cluster MotionAsymptotics for large time by time rescaling

Look for metastable solutions



Similarities to Cahn-Hilliard To understand cluster motion, note similarities to Cahn-Hilliard equation with degenerate diffusivity

Keller-Segel rewritten

"@t%= r ¢(½(1¡ ½)r (" log%¡ " log(1¡ %) ¡ r S[%]))

"@t%= r ¢(½(1¡ ½)r (¡ "¢ %+1"W(%)))



Degenerate (logistic) Diffusivity General Structure

with potential being variation of energy functional

¹ " = E 0"[%]

@t%= r ¢(%(1¡ %)r ¹ ")



Energy functionals Cahn-Hilliard

Keller-Segel

E ²[%] =

Z µ"2jr %j2 +

1"W(%)

¶dx

E ²[%] =

Z µ"F (%)

12%S[%]

¶dx¡

F (%) = %log%+ (1¡ %) log(1¡ %)



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

¹ " = E 0"[%]@t%= r ¢(%(r ¹ ")

@t%= r ¢(%(1¡ %)r ¹ ")



Gradient Flow Perspective Metric gradient flow with an appropriate optimal transport distance

Subject to

d(%1;%2)2 = infZ 1

0

Zu(1¡ u)jvj2 dx dt

@tu + r ¢(u(1¡ u)v) = 0

ujt=0 = %1; ujt=1 = %2



Gradient Flow Perspective Energies are -convex on geodesics for positive

Limiting energies are not -convex

Leads to singular behaviour: 0-1 constraints for density areattained

Interfacial motion apppears



Asymptotic Expansion Asymptotic expansion in interfacial layer (similar to degenerate-diffusivity Cahn-Hilliard)

Tangential variable , signed distance in normal direction



Asymptotic Expansion Leading order determines profile in normal direction

For general quorum sensing model

@»%̂0 =%̂0q(%̂0)

q(%̂0) ¡ %̂0q0(%̂0)@nS0

%̂0 =1

1+ exp(¡ »@nS0)



Asymptotic Expansion Next order determines interfacial motion



Surface diffusion Integration in normal direction and insertion of leading order equation implies

Note: entropy condition crucial for forward surface diffusion



Surface Diffusion We obtain a surface diffusion law with diffusivity

and potential

Corresponding energy functional

D = ¡ 2@n S

¹ = ¡ S2 = ¡ S[ ]

2



Conservation and Dissipation Flow is volume conserving (conservation of cell mass)

Flow has energy dissipation



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)



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



Low Frequency InstabilityPerturbation of flat surface, small density



Low Frequency InstabilityPerturbation of flat surface, smaller density



Low Frequency InstabilityPerturbation of flat surface, large density



Cluster MotionSurface diffusion with violated entropy condition at the end



Cluster Motion in Complicated GeometrySurface diffusion with violated entropy condition at the end



Cluster Motion in 3D



OutlookMethodology 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) ?



OutlookProblems 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 numericallyE.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



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



Nonlinear Fokker-Planck EquationsCoarse-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



Entropy for Mean-Field Fokker-PlanckEntropy functional



Mean-Field Fokker-PlanckMetric gradient flow in manifold of probability measures

with Wasserstein metric (optimal transport theory)



Important Questions- Existence and Uniqueness (follows from -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



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



Spatial Dimension OneIn spatial dimension one, there is a unique optimal transport plan, which can be computed via the pseudo-inverse of the distribution function. Let

Then



Conservation for Nonlinear Fokker-PlanckEquation conserves zero-th and first moment of the density , 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



Application to Pattern FormationBack to the canonical model

Write one-dimensional case in terms of the pseudo-inverse of the distribution function (Lagrangian formulation, z [0,1])



Application to Pattern FormationStart 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



Application to Pattern FormationExistence 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 solutionComplete aggregation !



Application to Pattern FormationUniqueness / 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



Application to Pattern FormationLong-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 )



Simulation of Pattern FormationGaussian interaction kernel



Simulation of Pattern FormationGaussian interaction kernel, rescaled density



Refined Asymptotic: Self-Similar SolutionsLet V be convex and with a minimum at 0Then there exist self-similar solutions of the form

where y is determined from the ODE

and y tends to zero until



Simulation of Pattern FormationKernel with finite support, rescaled density



Interpretation of Pattern FormationOpinion (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)



Local Repulsion: Swarming, CrowdingIn 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 ?



Asymptotic Expansion by Optimal TransportCloseness 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



Asymptotic Expansion by Optimal TransportAt the level of the pseudo-inverse we simply have

First-order expansion solves

yields density to highest order



Asymptotic Expansion by Optimal TransportFor 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



Large diffusionIn 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



Swarming: Zero vs. Small diffusion



Numerical AnalysisPiecewise constant FE spaces for and 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 - Error estimates for smooth solutions



Referencesmb, 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), 1-28.

mb, M. Di Francesco, Y.Dolak-Struss, The Keller-Segel model for chemotaxis: linear vs. nonlinear diffusion, SIAM J. Math. Anal. 38 (2006), 1288-1315.

mb, V.Capasso, D. Morale, On an aggregation model with long and short range interactions, Nonlinear Analysis. Real World Application s 8 (2007), 939-958.

www.math.uni-muenster.de/u/[email protected]

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