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

Mathematical models for cell movement Part II F ABIO A. C. C. C HALUB Centro de Matem´ ¸˜ atica e Aplicac oes Fundamentais Universidade de Lisboa Mathematical models for cell movementPart II – p. 1

Overview Biological background Keller-Segel model Kinetic models Scaling up and down Mathematical models for cell movementPart II – p. 2

Overview – Today The Keller-Segel model. Variations on the same theme. Models with global existence. Kinetic models Mathematical models for cell movementPart II – p. 3

The Keller-Segel Model For x ∈ R 2 , we call the classical Keller-Segel model : ∂ t ρ = ∇ · ( ∇ ρ − χρ ∇ S ) , ∆ S = − ρ , with ρ ( · , 0) = ρ I , with χ = χ 0 = const. Mathematical models for cell movementPart II – p. 4

Keller-Segel Models Corollary. In the two dimensional case, for the classical Keller-Segel model, we have: Mathematical models for cell movementPart II – p. 5

Keller-Segel Models Corollary. In the two dimensional case, for the classical Keller-Segel model, we have: if M < 8 π/χ : global existence of solutions, Mathematical models for cell movementPart II – p. 5

Keller-Segel Models Corollary. In the two dimensional case, for the classical Keller-Segel model, 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 II – p. 5

Keller-Segel Models Consider the following Keller-Segel model (with prevention of overcrowding ) (Hillen-Painter model): ∂ t ρ = ∇ · ( ∇ ρ − χ ( ρ ) ρ ∇ S ) ∆ S = − ρ , where χ ( ρ ) = 0 , ρ ≥ ¯ ρ > 0 . Mathematical models for cell movementPart II – p. 6

Keller-Segel Models Consider the following Keller-Segel model (with prevention of overcrowding ) (Hillen-Painter model): ∂ t ρ = ∇ · ( ∇ ρ − χ ( ρ ) ρ ∇ S ) ∆ S = − ρ , where χ ( ρ ) = 0 , ρ ≥ ¯ ρ > 0 . Theorem. (Hillen, Painter, 2002) Solutions of the HP model exist globally. Mathematical models for cell movementPart II – p. 6

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 II – p. 7

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 II – p. 7

Keller-Segel Models Consider a sensitivity (Velazquez’ model): ρ χ ( ρ ) = χ µ ( ρ ) = 1 + µρ , Mathematical models for cell movementPart II – p. 8

Keller-Segel Models Consider a sensitivity (Velazquez’ model): ρ χ ( ρ ) = χ µ ( ρ ) = 1 + µρ , Theorem. (Velazquez, 2004) The V model has global existence of solutions for any µ > 0 . Mathematical models for cell movementPart II – p. 8

Keller-Segel Models Consider a sensitivity (Velazquez’ model): ρ χ ( ρ ) = χ µ ( ρ ) = 1 + µρ , Theorem. (Velazquez, 2004) The V model has global existence of solutions for any µ > 0 . Mathematical models for cell movementPart II – p. 8

Keller-Segel Models Consider a sensitivity (Velazquez’ model): ρ χ ( ρ ) = χ µ ( ρ ) = 1 + µρ , Theorem. (Velazquez, 2004) The V model has global existence of solutions for any µ > 0 . For t < T , lim µ → 0 ρ µ = ρ 0 . Mathematical models for cell movementPart II – p. 8

Keller-Segel Models Consider a sensitivity (Velazquez’ model): ρ χ ( ρ ) = χ µ ( ρ ) = 1 + µρ , Theorem. (Velazquez, 2004) The V model has global existence of solutions for any µ > 0 . For t < T , lim µ → 0 ρ µ = ρ 0 . This cannot be extended after T because ρ 0 no longer exists ( T is the blow up time). Mathematical models for cell movementPart II – p. 8

Keller-Segel Models Consider a sensitivity (Velazquez’ model): ρ χ ( ρ ) = χ µ ( ρ ) = 1 + µρ , Theorem. (Velazquez, 2004) The V model has global existence of solutions for any µ > 0 . For t < T , lim µ → 0 ρ µ = ρ 0 . This cannot be extended after T because ρ 0 no longer exists ( T is the blow up time). For any µ > 0 , ρ µ exists for any time t . Mathematical models for cell movementPart II – p. 8

Kinetic Models f ( x, v, t ) is the density of cell in space-time point ( x, t ) with velocity v ( phase-space density ). Mathematical models for cell movementPart II – p. 9

Kinetic Models f ( x, v, t ) is the density of cell in space-time point ( x, t ) with velocity v ( phase-space density ). The cell goes in straight line for a certain characteristic time and then changes its direction from v ′ to v (in a space-time point ( x, t ) in the presence of the substance S and cell density ρ ) according to a certain turning kernel T [ S, ρ ]( x, v, v ′ , t ) . Mathematical models for cell movementPart II – p. 9

Kinetic Models f ( x, v, t ) is the density of cell in space-time point ( x, t ) with velocity v ( phase-space density ). The cell goes in straight line for a certain characteristic time and then changes its direction from v ′ to v (in a space-time point ( x, t ) in the presence of the substance S and cell density ρ ) according to a certain turning kernel T [ S, ρ ]( x, v, v ′ , t ) . The set of all possible velocities is given by a compact, spherically symmetric set V . Mathematical models for cell movementPart II – p. 9

Kinetic Models We define an equilibrium distribution F = F ( v ) : � � F > 0 , Fdv = 1 , vFdv = 0 , V V ⇒ T [ S 0 , ρ ]( x, v, v ′ , t ) F ( v ′ ) = T [ S 0 , ρ ]( x, v ′ , v, t ) F ( v ) . if S = S 0 = Mathematical models for cell movementPart II – p. 10

Kinetic Models We define an equilibrium distribution F = F ( v ) : � � F > 0 , Fdv = 1 , vFdv = 0 , V V ⇒ T [ S 0 , ρ ]( x, v, v ′ , t ) F ( v ′ ) = T [ S 0 , ρ ]( x, v ′ , v, t ) F ( v ) . if S = S 0 = Two possible turning kernels: T [ S, ρ ]( x, v, v ′ , t ) = λ ( S, ρ )( x, t ) F ( v ) + a ( S, ρ ) F ( v ) v · ∇ S ( x, t ) , T [ S, ρ ]( x, v, v ′ , t ) = ψ ( S ( x + vt, t ) − S ( x, t )) F ( v ) . Mathematical models for cell movementPart II – p. 10

Kinetic Models ∂ t f ( x, v, t ) + v · ∇ f ( x, v, t ) = � ( T [ S, ρ ]( x, v, v ′ , t ) f ( x, v ′ , t ) − T [ S, ρ ]( x, v ′ , v, t ) f ( x, v, t )) dv ′ . V Mathematical models for cell movementPart II – p. 11

Kinetic Models Notation f = f ( x, v, t ) , f ′ = f ( x, v ′ , t ) , T [ S, ρ ] = T [ S, ρ ]( x, v, v ′ , t ) , T ∗ [ S, ρ ] = T [ S, ρ ]( x, v ′ , v, t ) . Mathematical models for cell movementPart II – p. 12

Kinetic Models Notation f = f ( x, v, t ) , f ′ = f ( x, v ′ , t ) , T [ S, ρ ] = T [ S, ρ ]( x, v, v ′ , t ) , T ∗ [ S, ρ ] = T [ S, ρ ]( x, v ′ , v, t ) . Equation � ( T [ S, ρ ] f ′ − T ∗ [ S, ρ ] f ) dv ′ . ∂ t f + v · ∇ f = V Mathematical models for cell movementPart II – p. 12

Kinetic Model This is an example of a Boltzmann-type integro-differential equation (kinetic model). Mathematical models for cell movementPart II – p. 13

Kinetic Model This is an example of a Boltzmann-type integro-differential equation (kinetic model). The “macroscopic” density ρ is related to the “microscopic” density f by � ρ ( x, t ) = f ( x, v, t ) dv . V Mathematical models for cell movementPart II – p. 13

Kinetic Model This is an example of a Boltzmann-type integro-differential equation (kinetic model). The “macroscopic” density ρ is related to the “microscopic” density f by � ρ ( x, t ) = f ( x, v, t ) dv . V We should consider also an equation for S : ∂ t S = D 0 ∆ S + ϕ ( S, ρ ) . Mathematical models for cell movementPart II – p. 13

Kinetic Model This is an example of a Boltzmann-type integro-differential equation (kinetic model). The “macroscopic” density ρ is related to the “microscopic” density f by � ρ ( x, t ) = f ( x, v, t ) dv . V We should consider also an equation for S : ∂ t S = D 0 ∆ S + ϕ ( S, ρ ) . Mathematical models for cell movementPart II – p. 13

Kinetic Models Theorem. (C., Markowich, Perthame, Schmeiser, 2004; Hwang, Kang, Stevens, 2005) If ψ ( y ) ≤ Ay + B then solutions of the kinetic model exist globally. Mathematical models for cell movementPart II – p. 14

Kinetic Models Theorem. (C., Markowich, Perthame, Schmeiser, 2004; Hwang, Kang, Stevens, 2005) If ψ ( y ) ≤ Ay + B then solutions of the kinetic model exist globally. Proof: (Let us suppose n = 3 , the case n = 2 is technically more complicated but similar.) Mathematical models for cell movementPart II – p. 14

Kinetic Models Theorem. (C., Markowich, Perthame, Schmeiser, 2004; Hwang, Kang, Stevens, 2005) If ψ ( y ) ≤ Ay + B then solutions of the kinetic model exist globally. Proof: (Let us suppose n = 3 , the case n = 2 is technically more complicated but similar.) We divide S ( x, t ) = 1 1 � | x − y | ρ ( y, t ) dy , 4 π R 3 in S = S S + S L , where 1 1 S S = S L = 4 π | · | I {| x | < 1 } ∗ ρ , 4 π | · | I {| x |≥ 1 } ∗ ρ . Mathematical models for cell movementPart II – p. 14