free boundaries in biological aggregation models
DESCRIPTION
Free Boundaries in Biological Aggregation Models. Joint Work with. Yasmin Dolak-Struss , Vienna / FFG Christian Schmeiser , Vienna Marco DiFrancesco , L‘Aquila Daniela Morale , Milano Vincenzo Capasso , Milano Peter Markowich , Cambridge - PowerPoint PPT PresentationTRANSCRIPT
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
Biological Aggregation 3
13.6.2008Martin Burger
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
Biological Aggregation 4
13.6.2008Martin Burger
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
Biological Aggregation 5
13.6.2008Martin Burger
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
Biological Aggregation 6
13.6.2008Martin Burger
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 )]
Biological Aggregation 7
13.6.2008Martin Burger
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
Biological Aggregation 8
13.6.2008Martin Burger
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, .
Biological Aggregation 9
13.6.2008Martin Burger
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)
Biological Aggregation 10
13.6.2008Martin Burger
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
Biological Aggregation 11
13.6.2008Martin Burger
Aggregation in ChemotaxisKeller-Segel Model with small diffusion and logistic sensitivity at small time scales: Cluster formation
Biological Aggregation 12
13.6.2008Martin Burger
Coarsening and Cluster MotionKeller-Segel Model with small diffusion and logistic sensitivity at large time scales: Cluster motion
Biological Aggregation 13
13.6.2008Martin Burger
Fast Time ScaleSame scaling as before
Obvious limit for diffusion coefficient to zero
Biological Aggregation 14
13.6.2008Martin Burger
Fast Time Scale Asymptotic – Entropy ConditionLimit for density is a nonlinear (and also nonlocal) conservation law – needs entropy condition
Entropy inequality
Biological Aggregation 15
13.6.2008Martin Burger
Fast Time Scale Asymptotic – MetastabilityPossible stationary solutions of the form
Entropy inequality
Biological Aggregation 16
13.6.2008Martin Burger
Large Time Scale – Cluster MotionAsymptotics for large time by time rescaling
Look for metastable solutions
Biological Aggregation 17
13.6.2008Martin Burger
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(%)))
Biological Aggregation 18
13.6.2008Martin Burger
Degenerate (logistic) Diffusivity General Structure
with potential being variation of energy functional
¹ " = E 0"[%]
@t%= r ¢(%(1¡ %)r ¹ ")
Biological Aggregation 19
13.6.2008Martin Burger
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¡ %)
Biological Aggregation 20
13.6.2008Martin Burger
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 ")
Biological Aggregation 21
13.6.2008Martin Burger
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
Biological Aggregation 22
13.6.2008Martin Burger
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
Biological Aggregation 23
13.6.2008Martin Burger
Asymptotic Expansion Asymptotic expansion in interfacial layer (similar to degenerate-diffusivity Cahn-Hilliard)
Tangential variable , signed distance in normal direction
Biological Aggregation 24
13.6.2008Martin Burger
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)
Biological Aggregation 25
13.6.2008Martin Burger
Asymptotic Expansion Next order determines interfacial motion
Biological Aggregation 26
13.6.2008Martin Burger
Surface diffusion Integration in normal direction and insertion of leading order equation implies
Note: entropy condition crucial for forward surface diffusion
Biological Aggregation 27
13.6.2008Martin Burger
Surface Diffusion We obtain a surface diffusion law with diffusivity
and potential
Corresponding energy functional
D = ¡ 2@n S
¹ = ¡ S2 = ¡ S[ ]
2
Biological Aggregation 28
13.6.2008Martin Burger
Conservation and Dissipation Flow is volume conserving (conservation of cell mass)
Flow has energy dissipation
Biological Aggregation 29
13.6.2008Martin Burger
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)
Biological Aggregation 30
13.6.2008Martin Burger
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
Biological Aggregation 31
13.6.2008Martin Burger
Low Frequency InstabilityPerturbation of flat surface, small density
Biological Aggregation 32
13.6.2008Martin Burger
Low Frequency InstabilityPerturbation of flat surface, smaller density
Biological Aggregation 33
13.6.2008Martin Burger
Low Frequency InstabilityPerturbation of flat surface, large density
Biological Aggregation 34
13.6.2008Martin Burger
Cluster MotionSurface diffusion with violated entropy condition at the end
Biological Aggregation 35
13.6.2008Martin Burger
Cluster Motion in Complicated GeometrySurface diffusion with violated entropy condition at the end
Biological Aggregation 36
13.6.2008Martin Burger
Cluster Motion in 3D
Biological Aggregation 37
13.6.2008Martin Burger
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) ?
Biological Aggregation 38
13.6.2008Martin Burger
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
Biological Aggregation 39
13.6.2008Martin Burger
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
Biological Aggregation 40
13.6.2008Martin Burger
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
Biological Aggregation 41
13.6.2008Martin Burger
Entropy for Mean-Field Fokker-PlanckEntropy functional
Biological Aggregation 42
13.6.2008Martin Burger
Mean-Field Fokker-PlanckMetric gradient flow in manifold of probability measures
with Wasserstein metric (optimal transport theory)
Biological Aggregation 43
13.6.2008Martin Burger
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
Biological Aggregation 44
13.6.2008Martin Burger
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
Biological Aggregation 45
13.6.2008Martin Burger
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
Biological Aggregation 46
13.6.2008Martin Burger
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
Biological Aggregation 47
13.6.2008Martin Burger
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])
Biological Aggregation 48
13.6.2008Martin Burger
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
Biological Aggregation 49
13.6.2008Martin Burger
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 !
Biological Aggregation 50
13.6.2008Martin Burger
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
Biological Aggregation 51
13.6.2008Martin Burger
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 )
Biological Aggregation 52
13.6.2008Martin Burger
Simulation of Pattern FormationGaussian interaction kernel
Biological Aggregation 53
13.6.2008Martin Burger
Simulation of Pattern FormationGaussian interaction kernel, rescaled density
Biological Aggregation 54
13.6.2008Martin Burger
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
Biological Aggregation 55
13.6.2008Martin Burger
Simulation of Pattern FormationKernel with finite support, rescaled density
Biological Aggregation 56
13.6.2008Martin Burger
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)
Biological Aggregation 57
13.6.2008Martin Burger
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 ?
Biological Aggregation 58
13.6.2008Martin Burger
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
Biological Aggregation 59
13.6.2008Martin Burger
Asymptotic Expansion by Optimal TransportAt the level of the pseudo-inverse we simply have
First-order expansion solves
yields density to highest order
Biological Aggregation 60
13.6.2008Martin Burger
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
Biological Aggregation 61
13.6.2008Martin Burger
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
Biological Aggregation 62
13.6.2008Martin Burger
Swarming: Zero vs. Small diffusion
Biological Aggregation 63
13.6.2008Martin Burger
Swarming: Zero vs. Small diffusion
Biological Aggregation 64
13.6.2008Martin Burger
Swarming: Zero vs. Small diffusion
Biological Aggregation 65
13.6.2008Martin Burger
Swarming: Zero vs. Small diffusion
Biological Aggregation 66
13.6.2008Martin Burger
Swarming: Zero vs. Small diffusion
Biological Aggregation 67
13.6.2008Martin Burger
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
Biological Aggregation 68
13.6.2008Martin Burger
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]