Mathematical models for cell movement Part I F ABIO A. C. C. C - PowerPoint PPT Presentation

Keller-Segel model We write again the system: ∂ρ = −∇ · ( D 1 ∇ ρ ) + ∇ ( D 2 ∇ S ) , ∂t ∂S = − k ( S ) S + ρf ( S, ρ ) + D S ∆ S , ∂t where k ( S ) = η 0 k 2 K/ (1 + KS ) . Mathematical models for cell movementPart I – p. 21

Keller-Segel model We resume the model Amoebas have a random and a chemotactical movement. Mathematical models for cell movementPart I – p. 22

Keller-Segel model We resume the model Amoebas have a random and a chemotactical movement. Chemoattractant S diffuses, is created by the amoebas at rate f and decays at rate k . Mathematical models for cell movementPart I – p. 22

Keller-Segel model From now on, we call the following system the Keller-Segel system ∂ρ = ∇ · ( D ∇ ρ − χρ ∇ S ) , ∂t ∂S = D S ∆ S + ϕ ( ρ, S ) , ∂t where D = D ( S, ρ ) , χ = χ ( S, ρ ) e D S = D S ( S, ρ ) . Mathematical models for cell movementPart I – p. 23

Keller-Segel model From now on, we call the following system the Keller-Segel system ∂ρ = ∇ · ( D ∇ ρ − χρ ∇ S ) , ∂t ∂S = D S ∆ S + ϕ ( ρ, S ) , ∂t where D = D ( S, ρ ) , χ = χ ( S, ρ ) e D S = D S ( S, ρ ) . ϕ ( ρ, S ) describes production and decay of the chemoattractant. Typically ϕ = αρ − βS . Mathematical models for cell movementPart I – p. 23

The Keller-Segel Model ρ =density of cells. S =density of chemo-attractant. Mathematical models for cell movementPart I – p. 24

The Keller-Segel Model ρ =density of cells. S =density of chemo-attractant. ∂ρ = ∇ · ( D ∇ ρ − χρ ∇ S ) , ∂t ∂S = D 0 ∆ S + αρ − βS , ∂t where D = D ( S, ρ ) , χ = χ ( S, ρ ) , D = D ( S, ρ ) e D 0 = D 0 ( S, ρ ) , α > 0 , β ≥ 0 . Mathematical models for cell movementPart I – p. 24

The Keller-Segel Model Some important cases: Mathematical models for cell movementPart I – p. 25

The Keller-Segel Model Some important cases: The classical Keller-Segel model: χ, D, D 0 = const. Mathematical models for cell movementPart I – p. 25

The Keller-Segel Model Some important cases: The classical Keller-Segel model: χ, D, D 0 = const. The no-decay case: β = 0 . Mathematical models for cell movementPart I – p. 25

The Keller-Segel Model Some important cases: The classical Keller-Segel model: χ, D, D 0 = const. The no-decay case: β = 0 . The fast-diffusion limit: elliptic equation for S . Mathematical models for cell movementPart I – p. 25

The Keller-Segel Model Some important cases: The classical Keller-Segel model: χ, D, D 0 = const. The no-decay case: β = 0 . The fast-diffusion limit: elliptic equation for S . With these assumption the equation for S is ∆ S = − ρ . Mathematical models for cell movementPart I – p. 25

The Keller-Segel Model Finally, consider for x ∈ R 2 ∂ t ρ = ∇ · ( ∇ ρ − χρ ∇ S ) , ∆ S = − ρ , with ρ ( · , 0) = ρ I . Mathematical models for cell movementPart I – p. 26

The Keller-Segel Model Properties: Mathematical models for cell movementPart I – p. 27

The Keller-Segel Model Theorem. (Perthame, Dolbeaut, 2004) Consider a solution of the KS system, such that R 2 | x | 2 ρ ( x, t ) dx < ∞ , � 1+ | x | | x − y | ρ ( y, t ) dy ∈ L ∞ ((0 , T ) × R 2 ) . � R 2 Then d � 1 − χM � � R 2 | x | 2 ρ ( x, t ) dx = 4 M . dt 8 π Mathematical models for cell movementPart I – p. 28

The Keller-Segel Model Proof: − 1 � S ( x, t ) = R 2 log | x − y | ρ ( y, t ) dy , 2 π Mathematical models for cell movementPart I – p. 29

The Keller-Segel Model Proof: − 1 � S ( x, t ) = R 2 log | x − y | ρ ( y, t ) dy , 2 π − 1 � x − y ∇ S ( x, t ) = | x − y | 2 ρ ( y, t ) dy . 2 π R 2 Mathematical models for cell movementPart I – p. 29

The Keller-Segel Model Proof: − 1 � S ( x, t ) = R 2 log | x − y | ρ ( y, t ) dy , 2 π − 1 � x − y ∇ S ( x, t ) = | x − y | 2 ρ ( y, t ) dy . 2 π R 2 R 2 ρ ( x ) ρ ( y ) x · ( x − y ) R 2 ρ ( x ) ρ ( y ) y · ( x − y ) � � | x − y | 2 dydx = − | x − y | 2 dydx = Mathematical models for cell movementPart I – p. 29

The Keller-Segel Model Proof: − 1 � S ( x, t ) = R 2 log | x − y | ρ ( y, t ) dy , 2 π − 1 � x − y ∇ S ( x, t ) = | x − y | 2 ρ ( y, t ) dy . 2 π R 2 R 2 ρ ( x ) ρ ( y ) x · ( x − y ) R 2 ρ ( x ) ρ ( y ) y · ( x − y ) � � | x − y | 2 dydx = − | x − y | 2 dydx = R 2 ρ ( x ) ρ ( y ) dxdy = M 2 1 R 2 ρ ( x ) ρ ( y )( x − y ) · ( x − y ) dydx = 1 � � 2 . | x − y | 2 2 2 Mathematical models for cell movementPart I – p. 29

The Keller-Segel Model 1 d 1 � � R 2 | x | 2 ρ ( x ) dx R 2 | x | 2 ∇ · ( ∇ ρ − χρ ∇ S ) dx = 2 dt 2 Mathematical models for cell movementPart I – p. 30

The Keller-Segel Model 1 d 1 � � R 2 | x | 2 ρ ( x ) dx R 2 | x | 2 ∇ · ( ∇ ρ − χρ ∇ S ) dx = 2 dt 2 � = − R 2 x ( ∇ ρ − χρ ∇ S ) dx Mathematical models for cell movementPart I – p. 30

The Keller-Segel Model 1 d 1 � � R 2 | x | 2 ρ ( x ) dx R 2 | x | 2 ∇ · ( ∇ ρ − χρ ∇ S ) dx = 2 dt 2 � = − R 2 x ( ∇ ρ − χρ ∇ S ) dx � χ �� R 2 × R 2 ρ ( x ) ρ ( y ) x · ( x − y ) = R 2 2 ρdx − | x − y | 2 dydx 2 π Mathematical models for cell movementPart I – p. 30

The Keller-Segel Model 1 d 1 � � R 2 | x | 2 ρ ( x ) dx R 2 | x | 2 ∇ · ( ∇ ρ − χρ ∇ S ) dx = 2 dt 2 � = − R 2 x ( ∇ ρ − χρ ∇ S ) dx � χ �� R 2 × R 2 ρ ( x ) ρ ( y ) x · ( x − y ) = R 2 2 ρdx − | x − y | 2 dydx 2 π = Mathematical models for cell movementPart I – p. 30

The Keller-Segel Model 1 d 1 � � R 2 | x | 2 ρ ( x ) dx R 2 | x | 2 ∇ · ( ∇ ρ − χρ ∇ S ) dx = 2 dt 2 � = − R 2 x ( ∇ ρ − χρ ∇ S ) dx � χ �� R 2 × R 2 ρ ( x ) ρ ( y ) x · ( x − y ) = R 2 2 ρdx − | x − y | 2 dydx 2 π M 2 2 M − χ = 2 π 2 Mathematical models for cell movementPart I – p. 30

The Keller-Segel Model 1 d 1 � � R 2 | x | 2 ρ ( x ) dx R 2 | x | 2 ∇ · ( ∇ ρ − χρ ∇ S ) dx = 2 dt 2 � = − R 2 x ( ∇ ρ − χρ ∇ S ) dx � χ �� R 2 × R 2 ρ ( x ) ρ ( y ) x · ( x − y ) = R 2 2 ρdx − | x − y | 2 dydx 2 π M 2 � � 2 M − χ 1 − χM = = 2 M . 2 π 2 8 π Mathematical models for cell movementPart I – p. 30

The Keller-Segel Model Theorem. (Perthame, Dolbeaut, 2004) Consider the KS system such that M < 8 π/χ , ρ I ∈ L 1 ( R 2 , (1 + | x | 2 ) dx ) . Then, the system has a weak solution such that (1 + | x | 2 + log ρ ) ρ ∈ L ∞ loc ( R + , L 1 ( R 2 )) , � ∞ R 2 ρ |∇ log ρ − χ ∇ S | 2 dx dt < ∞ , � 0 ρ, ∇√ ρ ∈ L 2 ([0 , T ] × R 2 ) , ∀ T > 0 . Mathematical models for cell movementPart I – p. 31

The Keller-Segel Model Proof: The proof consists in three steps: Mathematical models for cell movementPart I – p. 32

The Keller-Segel Model Proof: The proof consists in three steps: Regularization of solutions, Mathematical models for cell movementPart I – p. 32

The Keller-Segel Model Proof: The proof consists in three steps: Regularization of solutions, Estimations, Mathematical models for cell movementPart I – p. 32

The Keller-Segel Model Proof: The proof consists in three steps: Regularization of solutions, Estimations, Limits. Mathematical models for cell movementPart I – p. 32

The Keller-Segel Model Step 1: (Regularization) Consider � − 1 2 π log | z | if | z | > ε , K ε ( z ) = − 1 2 π log ε if | z | ≤ ε . Mathematical models for cell movementPart I – p. 33

The Keller-Segel Model Step 1: (Regularization) Consider � − 1 2 π log | z | if | z | > ε , K ε ( z ) = − 1 2 π log ε if | z | ≤ ε . This means that, from S ( x, t ) = − 1 � R 2 log | x − y | ρ ( y, t ) dy 2 π we change to � S ε ( x, t ) = R 2 K ε ( x − y ) ρ ( y, t ) dy = ( K ε ∗ ρ ε ) ( x, t ) . Mathematical models for cell movementPart I – p. 33

The Keller-Segel Model Step 2: (Estimations) Equation: ∂ t ρ ε = ∇ · ( ∇ ρ ε − χρ ε ∇ ( K ε ∗ ρ ε )) . d dt Mathematical models for cell movementPart I – p. 34

The Keller-Segel Model Step 2: (Estimations) Equation: ∂ t ρ ε = ∇ · ( ∇ ρ ε − χρ ε ∇ ( K ε ∗ ρ ε )) . Then: d � 4 M − χ � R 2 | x | 2 ρ ε ( x, t ) dx = ρ ε ( x, t ) ρ ε ( y, t ) dxdy dt 2 π | y − x | >ε Mathematical models for cell movementPart I – p. 34

The Keller-Segel Model Step 2: (Estimations) Equation: ∂ t ρ ε = ∇ · ( ∇ ρ ε − χρ ε ∇ ( K ε ∗ ρ ε )) . Then: d � 4 M − χ � R 2 | x | 2 ρ ε ( x, t ) dx = ρ ε ( x, t ) ρ ε ( y, t ) dxdy dt 2 π | y − x | >ε ≤ 4 M , Mathematical models for cell movementPart I – p. 34

The Keller-Segel Model Step 2: (Estimations) Equation: ∂ t ρ ε = ∇ · ( ∇ ρ ε − χρ ε ∇ ( K ε ∗ ρ ε )) . Then: d � 4 M − χ � R 2 | x | 2 ρ ε ( x, t ) dx = ρ ε ( x, t ) ρ ε ( y, t ) dxdy dt 2 π | y − x | >ε ≤ 4 M , We prove similar ε -independent bounds and take the limit ε → 0 . Mathematical models for cell movementPart I – p. 34

The Keller-Segel Model Corollary. In the two dimensional case, we have: Mathematical models for cell movementPart I – p. 35

The Keller-Segel Model Corollary. In the two dimensional case, we have: if M < 8 π/χ : global existence of solutions, Mathematical models for cell movementPart I – p. 35

The Keller-Segel Model Corollary. In the two dimensional case, we have: if M < 8 π/χ : global existence of solutions, if M > 8 π/χ : finite-time-blow-up. Mathematical models for cell movementPart I – p. 35

The Keller-Segel Model Corollary. In the two dimensional case, we have: if M < 8 π/χ : global existence of solutions, if M > 8 π/χ : finite-time-blow-up. This is in agreement with the fact that aggregation occurs only if the initial density of Dd is above certain threshold. Mathematical models for cell movementPart I – p. 35

The Keller-Segel Model Corollary. In the two dimensional case, we have: if M < 8 π/χ : global existence of solutions, if M > 8 π/χ : finite-time-blow-up. This is in agreement with the fact that aggregation occurs only if the initial density of Dd is above certain threshold. What happens if M = 8 π/χ ? Mathematical models for cell movementPart I – p. 35

The Keller-Segel Model Suppose general dimension n . Mathematical models for cell movementPart I – p. 36

The Keller-Segel Model Suppose general dimension n . Theorem. (Corrias, Perthame, Zaag, 2004) There exists a constant K such that || ρ I || L n/ 2 ( R n ) ≤ K then the KS system has a global in time weak solution such that || ρ I || L p ( R n ) , max { 1 , n 2 − 1 } ≤ p ≤ n || ρ ( · , t ) || L p ( R n ) ≤ 2 , C ( t, K, || ρ I || L p ( R n ) ) , n || ρ ( · , t ) || L p ( R n ) ≤ 2 < p ≤ ∞ . Mathematical models for cell movementPart I – p. 36

The Keller-Segel Model Theorem. (Corrias, Perthame, Zaag, 2004) Suppose that n ≥ 3 and | x | 2 � 2 ρ I ( x ) dx ≤ CM n/ ( n − 2) , R n and assume that M ≥ M 0 for some M 0 > 0 . Then the KS system has no global solution with fast decay. Mathematical models for cell movementPart I – p. 37

Keller-Segel Models Consider the following Keller-Segel model (with prevention of overcrowding ) (Hillen-Painter model): ∂ t ρ = ∇ · ( ∇ ρ − χβ ( ρ ) ρ ∇ S ) ∆ S = − ρ , where β ( ρ ) = 0 , ρ ≥ ¯ ρ > 0 . Mathematical models for cell movementPart I – p. 38

Keller-Segel Models Consider the following Keller-Segel model (with prevention of overcrowding ) (Hillen-Painter model): ∂ t ρ = ∇ · ( ∇ ρ − χβ ( ρ ) ρ ∇ S ) ∆ S = − ρ , where β ( ρ ) = 0 , ρ ≥ ¯ ρ > 0 . Theorem. (Hillen, Painter, 2002) Solutions of the HP model exist globally. Mathematical models for cell movementPart I – p. 38

Keller-Segel Models Proof: Consider: J + = { x | ρ ( x, t ) > ¯ ρ } , J 0 = { x | ρ ( x, t ) = ¯ ρ } , J − = { x | ρ ( x, t ) < ¯ ρ } , Mathematical models for cell movementPart I – p. 39

Keller-Segel Models Proof: Consider: J + = { x | ρ ( x, t ) > ¯ ρ } , J 0 = { x | ρ ( x, t ) = ¯ ρ } , J − = { x | ρ ( x, t ) < ¯ ρ } , and define � ρ ( x, t ) − ¯ ρ if x ∈ J + , ρ + ( x, t ) = 0 otherwise. Mathematical models for cell movementPart I – p. 39

Keller-Segel Models 1 d �� � � � dt || ρ + ( · , t ) || 2 ρ + ρ + = + + L 2 ( R n ) t 2 J + J 0 J − Mathematical models for cell movementPart I – p. 40

Keller-Segel Models 1 d �� � � � dt || ρ + ( · , t ) || 2 ρ + ρ + = + + L 2 ( R n ) t 2 J + J 0 J − � = ( ρ − ¯ ρ ) ρ t J + Mathematical models for cell movementPart I – p. 40

Keller-Segel Models 1 d �� � � � dt || ρ + ( · , t ) || 2 ρ + ρ + = + + L 2 ( R n ) t 2 J + J 0 J − � = ( ρ − ¯ ρ ) ρ t J + � = ( ρ − ¯ ρ ) ∇ · ( ∇ ρ − χβ ( ρ ) ρ ∇ S ) , J + Mathematical models for cell movementPart I – p. 40

Keller-Segel Models 1 d �� � � � dt || ρ + ( · , t ) || 2 ρ + ρ + = + + L 2 ( R n ) t 2 J + J 0 J − � = ( ρ − ¯ ρ ) ρ t J + � = ( ρ − ¯ ρ ) ∇ · ( ∇ ρ − χβ ( ρ ) ρ ∇ S ) , J + � � |∇ ρ | 2 + = − ( ρ − ¯ ρ ) ∇ ρ · ν J + ∂ J + Mathematical models for cell movementPart I – p. 40

Keller-Segel Models 1 d �� � � � dt || ρ + ( · , t ) || 2 ρ + ρ + = + + L 2 ( R n ) t 2 J + J 0 J − � = ( ρ − ¯ ρ ) ρ t J + � = ( ρ − ¯ ρ ) ∇ · ( ∇ ρ − χβ ( ρ ) ρ ∇ S ) , J + � � |∇ ρ | 2 + = − ( ρ − ¯ ρ ) ∇ ρ · ν J + ∂ J + � � + ( ∇ ρ ) χρβ ( ρ ) ∇ S − ( ρ − ¯ ρ ) χρβ ( ρ ) ∇ S · ν J + ∂ J + Mathematical models for cell movementPart I – p. 40

Keller-Segel Models 1 d �� � � � dt || ρ + ( · , t ) || 2 ρ + ρ + = + + L 2 ( R n ) t 2 J + J 0 J − � = ( ρ − ¯ ρ ) ρ t J + � = ( ρ − ¯ ρ ) ∇ · ( ∇ ρ − χβ ( ρ ) ρ ∇ S ) , J + � � |∇ ρ | 2 + = − ( ρ − ¯ ρ ) ∇ ρ · ν J + ∂ J + � � + ( ∇ ρ ) χρβ ( ρ ) ∇ S − ( ρ − ¯ ρ ) χρβ ( ρ ) ∇ S · ν J + ∂ J + � |∇ ρ | 2 ≤ 0 . = − J + Mathematical models for cell movementPart I – p. 40

Keller-Segel Models If || ρ I || L ∞ ( R n ) ≤ ¯ ρ , then ρ + ( · , 0) = 0 , Mathematical models for cell movementPart I – p. 41

Keller-Segel Models If || ρ I || L ∞ ( R n ) ≤ ¯ ρ , then ρ + ( · , 0) = 0 , ρ + ( · , t ) = 0 , Mathematical models for cell movementPart I – p. 41

Keller-Segel Models If || ρ I || L ∞ ( R n ) ≤ ¯ ρ , then ρ + ( · , 0) = 0 , ρ + ( · , t ) = 0 , || ρ ( · , t ) || L ∞ ( R n ) ≤ ¯ ρ . Mathematical models for cell movementPart I – p. 41

Keller-Segel Models If || ρ I || L ∞ ( R n ) ≤ ¯ ρ , then ρ + ( · , 0) = 0 , ρ + ( · , t ) = 0 , || ρ ( · , t ) || L ∞ ( R n ) ≤ ¯ ρ . If || ρ I || L ∞ ( R n ) > ¯ ρ , then in a neighbourhood of a point x such that ρ ( x, t ) > ¯ ρ , the equation is ∂ t ρ = ∆ ρ and the maximum principle holds. Mathematical models for cell movementPart I – p. 41

Keller-Segel Models Define the non-local gradient 1 � ◦ ∇ R f ( x, t ) = S n − 1 f ( x + yR ) dy . ω n − 1 R n − 1 Mathematical models for cell movementPart I – p. 42

Keller-Segel Models Define the non-local gradient 1 � ◦ ∇ R f ( x, t ) = S n − 1 f ( x + yR ) dy . ω n − 1 R n − 1 Then the Hillen-Schmeiser-Painter model ◦ � � ∂ t ρ = ∇ · ∇ ρ − χρ ∇ R S , has global existence of solutions. Mathematical models for cell movementPart I – p. 42