Some models of cell movement Beno t Perthame OUTLINE OF THE - PowerPoint PPT Presentation

Some models of cell movement Beno ˆ ıt Perthame

OUTLINE OF THE LECTURE I. Why study bacterial colonies growth ? II. Macroscopic models (Keller-Segel) III. The hyperbolic Keller-Segel models IV. Proof through the kinetic formulation V. Movement at a microscopic scale (kinetic models)

WHY .

WHY Biologist can now access to • Individual cell motion • Molecular content in some proteins • They act on the genes controlling these proteins But the global effects are still to explain : nutrients, chemoattraction, chemorepulsion, response to light, effectivity of propulsion, effects of surfactants, cell-to-cell interactions and exchanges, metabolic control loops...

WHY Examples of application fields • Ecology : bioreactors, biofilms • Health : biofilms, cancer therapy .

MACROSCOPIC MODELS MIMURA’s model  � � µ n ∂ ∂t n ( t, x ) − d 1 ∆ n = r n S − ,   ( n 0 + n )( S 0 + S )          ∂ ∂t S ( t, x ) − d 2 ∆ S = − r nS,        µ n ∂  ∂t f ( t, x ) = r n    ( n 0 + n )( S 0 + S ) The dynamics is driven by the source terms, i.e., by bacterial growth.

MACROSCOPIC MODELS

CHEMOTAXIS : Keller-Segel model The mathematical modelling of cell movement goes back to Patlak (1953), E. Keller and L. Segel (70’s) n ( t, x ) = density of cells at time t and position x, c ( t, x ) = concentration of chemoattractant , In a collective motion, the chemoattractant is emited by the cells that react according to biased random walk. � ∂ x ∈ R d , ∂t n ( t, x ) − ∆ n ( t, x ) + div( nχ ∇ c ) = 0 , − ∆ c ( t, x ) = n ( t, x ) , The parameter χ is the sensitivity of cells to the chemoattractant.

CHEMOTAXIS : Keller-Segel model  ∂ x ∈ R d , ∂t n ( t, x ) − ∆ n ( t, x ) + div( nχ ∇ c ) = 0 ,     − ∆ c ( t, x ) = n ( t, x ) , This model, although very simple, exhibits a deep mathematical structure and mostly only dimension 2 is understood, especially ”chemotactic collapse”. This is the reason why it has attracted a number of mathematicians J¨ ager-Luckhaus, Biler et al , Herrero- Velazquez, Suzuki-Nagai, Brenner et al , Lauren¸ cot, Corrias.

CHEMOTAXIS : Keller-Segel model Theorem (dimensions d ≥ 2) - (method of Sobolev inequalities) (i) for � n 0 � L d/ 2 ( R d ) small, then there are global weak solutions, (ii) these small solutions gain L p regularity, (iii) � n ( t ) � L ∞ ( R d ) → 0 with the rate of the heat equation, � � | x | 2 n 0 � ( d − 2) < C � n 0 � d (iii) for L 1 ( R d ) with C small, there is blow-up in a finite time T ∗ .

CHEMOTAXIS : Keller-Segel model The existence proof relies on J¨ ager-Luckhaus argument � n ( t, x ) p � |∇ n p/ 2 | 2 + � p ∇ n p − 1 n χ ∇ c d = − 4 dt p � �� � χ � ∇ n p ·∇ c = − χ � n ∆ c � � − 4 |∇ n p/ 2 | 2 n p +1 = + χ p � �� � � �� � hyperbolic effect parabolic dissipation Using Gagliardo-Nirenberg-Sobolev ineq. on the quantity u ( x ) = n p/ 2 , we obtain � n p +1 ≤ C gns ( d, p ) �∇ n p/ 2 � 2 L 2 � n � d L 2

CHEMOTAXIS : Keller-Segel model In dimension 2, for Keller and Segel model : � ∂ x ∈ R 2 , ∂t n ( t, x ) − ∆ n ( t, x ) + div( nχ ∇ c ) = 0 , − ∆ c ( t, x ) = n ( t, x ) , Theorem (d=2) (Method of energy) (Blanchet, Dolbeault, BP) for � n 0 � L 1 ( R 2 ) < 8 π (i) χ , there are smooth solutions, (ii) for � n 0 � L 1 ( R 2 ) > 8 π χ , there is creation of a singular measure (blow-up) in finite time. (iii) For radially symmetric solutions, blow-up means n ( t ) ≈ 8 π χ δ ( x = 0) + Rem.

CHEMOTAXIS : dimension 2 Existence part is based on the energy �� � � � � ∇√ n − χ ∇ c d R 2 n log n dx − χ 2 dx . � � � � R 2 n c dx = − � dt 2 R 2 and limit Hardy-Littlewood-Sobolev inequality (Beckner, Carlen-Loss, 96) � � � R 2 f log f dx + 2 R 2 × R 2 f ( x ) f ( y ) log | x − y | dx dy ≥ M (1 + log π + log M ) . M Notice that in d = 2 we have � c ( t, x ) = 1 − ∆ c = n, n ( t, y ) log | x − y | dy 2 π � n ∈ L 1 log = ⇒ nc < ∞ .

. From A. Marrocco (INRIA, BANG)

Hyperbolic Keller-Segel model Why a need for hyperblic models • We see front motion • The parabolic scale does not explain all the phenomena • Experiments access to finer scales

Hyperbolic Keller-Segel model The hyperbolic Keller-Segel system (Dolak, Schmeiser) � �  ∂ x ∈ R d , t ≥ 0 , ∂t n ( t, x ) + div n(1 − n) ∇ c = 0 ,      − ∆ c + c = n,    n 0 ∈ L 1 ( R d ) .  n ( t, x ) = n 0 ( x ) , 0 ≤ n 0 ( x ) ≤ 1 ,  Interpretation -) n ( t, x ) = bacterial density , -) c ( t, x ) = chemical signalling (chemoattraction), -) n (1 − n ) represents quorum sensing, -) random motion of bacterials is neglected (but exists)

Hyperbolic Keller-Segel model : applications By V. Calvez, B. Desjardins on multiple sclerosis

Hyperbolic Keller-Segel model  � � ∂ x ∈ R d , t ≥ 0 , ∂t n ( t, x ) + div n(1 − n) ∇ c = 0 ,      − ∆ c + c = n,    n 0 ∈ L 1 ( R d ) .  n ( t, x ) = n 0 ( x ) , 0 ≤ n 0 ( x ) ≤ 1 ,  Difficulties. All the properties of Scalar Consevation Laws are lost -) TV property is wrong (except in dimension d = 1), -) Contraction is wrong, -) Regularizing effects are wrong (except in dimension d = 1), -) Good news : A priori estimate 0 ≤ n ( t, x ) ≤ 1.

Hyperbolic Keller-Segel model  � � ∂ x ∈ R d , t ≥ 0 , ∂t n ( t, x ) + div n(1 − n) ∇ c = 0 ,      − ∆ c + c = n,    n 0 ∈ L 1 ( R d ) .  n ( t, x ) = n 0 ( x ) , 0 ≤ n 0 ( x ) ≤ 1 ,  Difficulties. All the properties of Scalar Consevation Laws are lost -) TV property is wrong (except in dimension d = 1), -) Contraction is wrong, -) Regularizing effects are wrong (except in dimension d = 1), -) Good news : A priori estimate 0 ≤ n ( t, x ) ≤ 1.

Hyperbolic Keller-Segel model � �  ∂ x ∈ R d , t ≥ 0 , ∂t n ( t, x ) + div n(1 − n) ∇ c = 0 ,   − ∆ c + c = n. Theorem (A.-L. Dalibar, B. P.) There exist a solution n ∈ L ∞ � � R + ; L 1 ∩ L ∞ ( R d ) in the weak sense. It is the strong limit of the same eq. with a small diffusion. � �  ∂ x ∈ R d , t ≥ 0 , ∂t n ε ( t, x ) + div n ε (1 − n ε ) ∇ c ε = ε ∆ n ε ,   − ∆ c ε + c ε = n ε .

Hyperbolic Keller-Segel model Related to a problem coming from oil recovery  � � ∂ x ∈ R d , t ≥ 0 , ∂t n ( t, x ) + div n (1 − n ) u = 0 ,      u = K. ∇ p,     div u = 0 ,  which is still open.

Hyperbolic Keller-Segel model Idea of the proof It is based on the kinetic formulation. In the present case, with A ( n ) = n (1 − n ) , a = A ′ , it is  ∂χ ( ξ ; n ) + a ( ξ ) ∇ y c · ∇ y χ ( ξ ; n ) + ( ξ − c ) A ( ξ ) ∂χ ( ξ ; n ) = ∂m ∂ξ ,   ∂t ∂ξ    m ( t, x, ξ ) a nonnegative measure ,     D 2 c ∈ L p ([0 , T ] × R d ) ,  1 < p < ∞ ,  +1 for 0 ≤ ξ ≤ u,    χ ( ξ, u ) = − 1 for u ≤ ξ ≤ 0 ,   0 otherwise . 

Hyperbolic Keller-Segel model With a small diffusion, the function χ ( ξ ; n ε ) satisfies a similar kinetic equation. Then one can pass to the weak limit and the problem comes from the ’nonlinear’ term in the kinetic formulation ∂χ ( ξ ; n ) +( ξ − c ) A ( ξ ) ∂χ ( ξ ; n ) = ∂m + a ( ξ ) ∇ y c · ∇ y χ ( ξ ; n ) ∂ξ , ∂t ∂ξ � �� � =div[ ∇ y c χ ( ξ ; n )] − ∆ c χ ( ξ ; n ) One obtains ∂ t f + a ( ξ ) ∇ y c · ∇ y f + a ( ξ )( ρ − nf ) + ( ξ − c ) A ( ξ ) ∂ ξ f = ∂ ξ m.

Recalling the standard case ∂ x ∈ R d , t ≥ 0 , ∂tn ( t, x ) + div A ( n ) = 0 , for entropy solutions ∂ t χ ( ξ ; n ) + a ( ξ ) ∇ y χ ( ξ ; n ) = ∂ ξ m, m ≥ 0 . because for S convex � � � S ′ ( ξ ) χ ( ξ ; n ) dξ + div S ′ ( ξ ) a ( ξ ) χ ( ξ ; n ) dξ = S ′ ( ξ ) ∂ ξ mdξ. ∂ t ⇒ ∂ � � + div η S ( n ) ≤ 0 , x ∈ R d , t ≥ 0 , ⇐ ∂tS n ( t, x )

Recalling the standard case Uniqueness follows in three steps 1st step. Convolution ∂ t χ ( ξ ; n ) ∗ ( t,x ) ω ε + a ( ξ ) ∇ y χ ( ξ ; n ) ∗ ( t,x ) ω ε = ∂ ξ m ∗ ( t,x ) ω ε , 2nd step. L 2 linear uniqueness ∂ t | χ ( ξ ; n 1 ) ε − χ ( ξ ; n 2 ) ε | 2 + a ( ξ ) ∇ y | χ ( ξ ; n 1 ) ε − χ ( ξ ; n 2 ) ε | 2 � � χ ( ξ ; n 1 ) ε − χ ( ξ ; n 2 ) ε ∂ ξ ( m 1 ε − m 2 = 2 ε ) � � � � � | χ ( ξ ; n 1 ) ε − χ ( ξ ; n 2 ) ε | 2 dxdξ = 2 δ ( ξ = n 1 ) ε − δ ( ξ = n 2 ) ε m 1 ε − m 2 ∂ t ε 3rd step. Limit as ε → 0 � d | χ ( ξ ; n 1 ) − χ ( ξ ; n 2 ) | 2 dxdξ = 0+ ≤ 0 + 0+ ≤ 0 dt