Mathematical models for cell movement Part I

Transcription

1 Mathematical models for cell movement Part I FABIO A. C. C. CHALUB Centro de Matemática e Aplicações Fundamentais Universidade de Lisboa Mathematical models for cell movementpart I p.

2 Overview Biological background Keller-Segel model Kinetic models Scaling up and down Mathematical models for cell movementpart I p.

3 Overview Today Biological background Cells move for what? Life and death of Dictyostelium discoideum Choosing a problem: the initiation of the aggregation The Keller-Segel model. Mathematical models for cell movementpart I p.

4 Chemotaxis Figure 1: A neutrophil (white blood cell) chasing a bacterium. Film by Peter Devreotes. Mathematical models for cell movementpart I p.

5 Why do cells move? In general: looking for better places to live. Other reasons are: immunology, embryology and development, aggregation. Mathematical models for cell movementpart I p.

6 One day in the life of Dd Figure 2: Life cycle of Dictyostelium discoideum. Picture made by Florian Seigert and Kees Wiejer (Zoologisches Institut München Ludwig- Maximilians-Universität München). Mathematical models for cell movementpart I p.

7 The initiation of the aggregation The Dictyostelium discoideum moves toward higher concentrations of camp. Video 1 by Peter Devreotes. Video 2 by Peter Devreotes. Mathematical models for cell movementpart I p.

8 The formation of a core Cells of Dd aggregates in a core. Video 1 by Kees Wiejer and Florian Seigert. Mathematical models for cell movementpart I p.

9 The slug phase The aggregate starts to behave like a slug and migrates. Video 1 by Kees Wiejer and Florian Seigert. Video 2 by by Kees Wiejer and Florian Seigert. Video 3 by by Kees Wiejer and Florian Seigert. Mathematical models for cell movementpart I p.

10 The Culmination The slug culminates with a spore on the top. Video 1 by Kees Wiejer and Florian Seigert. Video 2 by Rex Chisolm. Mathematical models for cell movementpart I p. 1

11 Why it is important to study the Dd? Many reasons: Mathematical models for cell movementpart I p. 1

12 Why it is important to study the Dd? Many reasons: Cell motility, Mathematical models for cell movementpart I p. 1

13 Why it is important to study the Dd? Many reasons: Cell motility, Cell communication, Mathematical models for cell movementpart I p. 1

14 Why it is important to study the Dd? Many reasons: Cell motility, Cell communication, Cooperation among non-clonal individuals. Mathematical models for cell movementpart I p. 1

15 Keller-Segel model Initiation of Slime Mold Aggregation Viewed as an Instability, Evelyn Keller and Lee Segel, J. Theor. Biol. (1970) 26, Mathematical models for cell movementpart I p. 1

16 Keller-Segel model Initiation of Slime Mold Aggregation Viewed as an Instability, Evelyn Keller and Lee Segel, J. Theor. Biol. (1970) 26, Variables: Mathematical models for cell movementpart I p. 1

17 Keller-Segel model Initiation of Slime Mold Aggregation Viewed as an Instability, Evelyn Keller and Lee Segel, J. Theor. Biol. (1970) 26, Variables: ρ(x, t) = density of amoebas. Mathematical models for cell movementpart I p. 1

18 Keller-Segel model Initiation of Slime Mold Aggregation Viewed as an Instability, Evelyn Keller and Lee Segel, J. Theor. Biol. (1970) 26, Variables: ρ(x, t) = density of amoebas. S(x, t) = density of camp. Mathematical models for cell movementpart I p. 1

19 Keller-Segel model Initiation of Slime Mold Aggregation Viewed as an Instability, Evelyn Keller and Lee Segel, J. Theor. Biol. (1970) 26, Variables: ρ(x, t) = density of amoebas. S(x, t) = density of camp. η(x, t) = density of phosphodiesterase. Mathematical models for cell movementpart I p. 1

20 Keller-Segel model Initiation of Slime Mold Aggregation Viewed as an Instability, Evelyn Keller and Lee Segel, J. Theor. Biol. (1970) 26, Variables: ρ(x, t) = density of amoebas. S(x, t) = density of camp. η(x, t) = density of phosphodiesterase. c(x,t) = density of a certain instable substance. Mathematical models for cell movementpart I p. 1

21 Keller-Segel model Initiation of Slime Mold Aggregation Viewed as an Instability, Evelyn Keller and Lee Segel, J. Theor. Biol. (1970) 26, Variables: ρ(x, t) = density of amoebas. S(x, t) = density of camp. η(x, t) = density of phosphodiesterase. c(x,t) = density of a certain instable substance. We assume n = 2. Mathematical models for cell movementpart I p. 1

22 Keller-Segel model We shall implement: Mathematical models for cell movementpart I p. 1

23 Keller-Segel model We shall implement: camp is produced by the amoebas at a rate f(s,ρ) per amoeba. Mathematical models for cell movementpart I p. 1

24 Keller-Segel model We shall implement: camp is produced by the amoebas at a rate f(s,ρ) per amoeba. camp is degraded by an extra-cellular enzyme (phosphodiesterase) at rate g(s,η) by amoeba. Mathematical models for cell movementpart I p. 1

25 Keller-Segel model We shall implement: camp is produced by the amoebas at a rate f(s,ρ) per amoeba. camp is degraded by an extra-cellular enzyme (phosphodiesterase) at rate g(s,η) by amoeba. camp and phosphodiesterase react making a new unstable compound C that decays immediately in phosphodiesterase and certain degenerated product. S + η C η + product. Mathematical models for cell movementpart I p. 1

26 Keller-Segel model camp, phosphodiesterase and the compound diffuse according to Fick s law. Mathematical models for cell movementpart I p. 1

27 Keller-Segel model camp, phosphodiesterase and the compound diffuse according to Fick s law. J = current, φ = flux, J = D φ. Mathematical models for cell movementpart I p. 1

28 Keller-Segel model camp, phosphodiesterase and the compound diffuse according to Fick s law. J = current, φ = flux, J = D φ. Amoebas concentration varies due to random diffusion and chemotaxis, in the positive direction of camp s gradient. Mathematical models for cell movementpart I p. 1

29 Keller-Segel model For any substance with density a i, we have a flux J i and a creation/destruction term Q i such that a i t = Qi J i. Mathematical models for cell movementpart I p. 1

30 Keller-Segel model For any substance with density a i, we have a flux J i and a creation/destruction term Q i such that a i t = Qi J i. Mass conservation implies that Q ρ = 0. Mathematical models for cell movementpart I p. 1

31 Keller-Segel model In order to obtain Q S, Q η and Q C we shall consider the production by the amoebas and the chemical reaction: Q S = k 1 Sη + k 1 c + ρf(s,ρ), Q η = k 1 Sη + (k 1 + k 2 )c + ρg(s,η), Q c = k 1 Sη (k 1 + k 2 )c. Mathematical models for cell movementpart I p. 1

32 Keller-Segel model Fluxes are given by the following expressions: J ρ = D 1 ρ + D 2 S, J S = D S S, J η = D η η, J c = D c c. Mathematical models for cell movementpart I p. 1

33 Keller-Segel model Fluxes are given by the following expressions: J ρ = D 1 ρ + D 2 S, J S = D S S, J η = D η η, J c = D c c. Diffusion coefficients are: D i = D i (ρ,s,η,c). Mathematical models for cell movementpart I p. 1

34 Keller-Segel model Finally, we write the full Keller-Segel system: Mathematical models for cell movementpart I p. 1

35 Keller-Segel model Finally, we write the full Keller-Segel system: ρ t S t η t c t = (D 1 ρ) (D 2 S), = k 1 Sη + k 1 c + ρf(s,ρ) + D S S, = k 1 Sη + (k 1 + k 2 )c + ρg(s,η) + D η η, = k 1 Sη (k 1 + k 2 )c + D c c. Mathematical models for cell movementpart I p. 1

36 Keller-Segel model Now, we simplify the model. Mathematical models for cell movementpart I p. 1

37 Keller-Segel model Now, we simplify the model. We introduce two biologically reasonable assumptions: Mathematical models for cell movementpart I p. 1

38 Keller-Segel model Now, we simplify the model. We introduce two biologically reasonable assumptions: The substance C is in chemical equilibrium: k 1 Sη (k 1 + k 2 )c = 0. (Haldane s assumption) Mathematical models for cell movementpart I p. 1

39 Keller-Segel model Now, we simplify the model. We introduce two biologically reasonable assumptions: The substance C is in chemical equilibrium: (Haldane s assumption) k 1 Sη (k 1 + k 2 )c = 0. Enzyme concentration (in both free and bound forms) is constant η + c = η 0. Mathematical models for cell movementpart I p. 1

40 Keller-Segel model Solving the system, we find η = η Kρ, K = k 1 k 1 + k 2. Mathematical models for cell movementpart I p. 2

41 Keller-Segel model We write again the system: ρ t S t = (D 1 ρ) + (D 2 S), = k(s)s + ρf(s,ρ) + D S S, where k(s) = η 0 k 2 K/(1 + KS). Mathematical models for cell movementpart I p. 2

42 Keller-Segel model We resume the model Amoebas have a random and a chemotactical movement. Mathematical models for cell movementpart I p. 2

43 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. 2

44 Keller-Segel model From now on, we call the following system the Keller-Segel system ρ t S t = (D ρ χρ S), = D S S + ϕ(ρ,s), where D = D(S,ρ), χ = χ(s,ρ) e D S = D S (S,ρ). Mathematical models for cell movementpart I p. 2

45 Keller-Segel model From now on, we call the following system the Keller-Segel system ρ t S t = (D ρ χρ S), = D S S + ϕ(ρ,s), 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. 2

46 The Keller-Segel Model ρ=density of cells. S=density of chemo-attractant. Mathematical models for cell movementpart I p. 2

47 The Keller-Segel Model ρ=density of cells. S=density of chemo-attractant. ρ t S t = (D ρ χρ S), = D 0 S + αρ βs, 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. 2

48 The Keller-Segel Model Some important cases: Mathematical models for cell movementpart I p. 2

49 The Keller-Segel Model Some important cases: The classical Keller-Segel model: χ,d,d 0 = const. Mathematical models for cell movementpart I p. 2

50 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. 2

51 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. 2

52 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. 2

53 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. 2

54 The Keller-Segel Model Properties: Mathematical models for cell movementpart I p. 2

55 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 <, R 2 1+ x x y ρ(y,t)dy L ((0,T) R 2 ). Then d dt R 2 x 2 ρ(x,t)dx = 4M ( 1 χm 8π ). Mathematical models for cell movementpart I p. 2

56 The Keller-Segel Model Proof: S(x,t) = 1 2π R 2 log x y ρ(y,t)dy, Mathematical models for cell movementpart I p. 2

57 The Keller-Segel Model Proof: S(x,t) = 1 2π S(x,t) = 1 2π log x y ρ(y,t)dy, R 2 x y x y 2ρ(y,t)dy. R 2 Mathematical models for cell movementpart I p. 2

58 The Keller-Segel Model Proof: R 2 ρ(x)ρ(y) S(x,t) = 1 2π S(x,t) = 1 2π log x y ρ(y,t)dy, R 2 x y x y 2ρ(y,t)dy. R 2 x (x y) x y dydx = y (x y) ρ(x)ρ(y) 2 R x y dydx = 2 2 Mathematical models for cell movementpart I p. 2

59 The Keller-Segel Model Proof: 1 2 ρ(x)ρ(y) R 2 R 2 ρ(x)ρ(y) S(x,t) = 1 2π S(x,t) = 1 2π log x y ρ(y,t)dy, R 2 x y x y 2ρ(y,t)dy. R 2 x (x y) x y dydx = y (x y) ρ(x)ρ(y) 2 R x y dydx = 2 2 (x y) (x y) x y 2 dydx = 1 2 R2 ρ(x)ρ(y)dxdy = M2 2. Mathematical models for cell movementpart I p. 2

60 The Keller-Segel Model 1 2 d dt R 2 x 2 ρ(x)dx = 1 2 R 2 x 2 ( ρ χρ S) dx Mathematical models for cell movementpart I p. 3

61 The Keller-Segel Model 1 2 d dt x 2 ρ(x)dx = 1 x 2 ( ρ χρ S) dx R 2 2 R 2 = x ( ρ χρ S) dx R 2 Mathematical models for cell movementpart I p. 3

62 The Keller-Segel Model 1 2 d dt x 2 ρ(x)dx = 1 x 2 ( ρ χρ S) dx R 2 2 R 2 = x ( ρ χρ S) dx R 2 = 2ρdx χ (x y) ρ(x)ρ(y)x R 2π 2 R 2 R x y 2dydx 2 Mathematical models for cell movementpart I p. 3

63 The Keller-Segel Model 1 2 d dt x 2 ρ(x)dx = 1 x 2 ( ρ χρ S) dx R 2 2 R 2 = x ( ρ χρ S) dx R 2 = 2ρdx χ (x y) ρ(x)ρ(y)x R 2π 2 R 2 R x y 2dydx 2 = Mathematical models for cell movementpart I p. 3

64 The Keller-Segel Model 1 2 d dt x 2 ρ(x)dx = 1 x 2 ( ρ χρ S) dx R 2 2 R 2 = x ( ρ χρ S) dx R 2 = 2ρdx χ (x y) ρ(x)ρ(y)x R 2π 2 R 2 R x y 2dydx 2 = 2M χ 2π M 2 2 Mathematical models for cell movementpart I p. 3

65 The Keller-Segel Model 1 2 d dt x 2 ρ(x)dx = 1 x 2 ( ρ χρ S) dx R 2 2 R 2 = x ( ρ χρ S) dx R 2 = 2ρdx χ (x y) ρ(x)ρ(y)x R 2π 2 R 2 R x y 2dydx 2 = 2M χ ( M 2 2π 2 = 2M 1 χm ). 8π Mathematical models for cell movementpart I p. 3

66 The Keller-Segel Model Theorem. (Perthame, Dolbeaut, 2004) Consider the KS system such that M < 8π/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 )), 0 R ρ log ρ χ S 2 dxdt <, 2 ρ, ρ L 2 ([0,T] R 2 ), T > 0. Mathematical models for cell movementpart I p. 3

67 The Keller-Segel Model Proof: The proof consists in three steps: Mathematical models for cell movementpart I p. 3

68 The Keller-Segel Model Proof: The proof consists in three steps: Regularization of solutions, Mathematical models for cell movementpart I p. 3

69 The Keller-Segel Model Proof: The proof consists in three steps: Regularization of solutions, Estimations, Mathematical models for cell movementpart I p. 3

70 The Keller-Segel Model Proof: The proof consists in three steps: Regularization of solutions, Estimations, Limits. Mathematical models for cell movementpart I p. 3

71 The Keller-Segel Model Step 1: (Regularization) Consider K ε (z) = { 1 2π 1 2π log z if z > ε, log ε if z ε. Mathematical models for cell movementpart I p. 3

72 The Keller-Segel Model Step 1: (Regularization) Consider { 1 2π K ε (z) = This means that, from S(x,t) = 1 2π 1 2π log z if z > ε, log ε if z ε. R 2 log x y ρ(y,t)dy we change to S ε (x,t) = K ε (x y)ρ(y,t)dy = (K ε ρ ε ) (x,t). R 2 Mathematical models for cell movementpart I p. 3

73 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. 3

74 The Keller-Segel Model Step 2: (Estimations) Equation: t ρ ε = ( ρ ε χρ ε (K ε ρ ε )). Then: d dt R 2 x 2 ρ ε (x,t)dx = 4M χ 2π y x >ε ρ ε (x,t)ρ ε (y,t)dxdy Mathematical models for cell movementpart I p. 3

75 The Keller-Segel Model Step 2: (Estimations) Equation: t ρ ε = ( ρ ε χρ ε (K ε ρ ε )). Then: d dt x 2 ρ ε (x,t)dx = 4M χ R 2π 2 4M, y x >ε ρ ε (x,t)ρ ε (y,t)dxdy Mathematical models for cell movementpart I p. 3

76 The Keller-Segel Model Step 2: (Estimations) Equation: Then: d dt t ρ ε = ( ρ ε χρ ε (K ε ρ ε )). x 2 ρ ε (x,t)dx = 4M χ R 2π 2 4M, y x >ε ρ ε (x,t)ρ ε (y,t)dxdy We prove similar ε-independent bounds and take the limit ε 0. Mathematical models for cell movementpart I p. 3

77 The Keller-Segel Model Corollary. In the two dimensional case, we have: Mathematical models for cell movementpart I p. 3

78 The Keller-Segel Model Corollary. In the two dimensional case, we have: if M < 8π/χ: global existence of solutions, Mathematical models for cell movementpart I p. 3

79 The Keller-Segel Model Corollary. In the two dimensional case, we have: if M < 8π/χ: global existence of solutions, if M > 8π/χ: finite-time-blow-up. Mathematical models for cell movementpart I p. 3

80 The Keller-Segel Model Corollary. In the two dimensional case, we have: if M < 8π/χ: global existence of solutions, if M > 8π/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. 3

81 The Keller-Segel Model Corollary. In the two dimensional case, we have: if M < 8π/χ: global existence of solutions, if M > 8π/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 = 8π/χ? Mathematical models for cell movementpart I p. 3

82 The Keller-Segel Model Suppose general dimension n. Mathematical models for cell movementpart I p. 3

83 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(,t) L p (R n ) ρ I L p (R n ), max{1, n 2 1} p n 2, ρ(,t) L p (R n ) C(t,K, ρ I L p (R n )), n 2 < p. Mathematical models for cell movementpart I p. 3

84 The Keller-Segel Model Theorem. (Corrias, Perthame, Zaag, 2004) Suppose that n 3 and and assume that Rn x 2 2 ρi (x)dx CM n/(n 2), 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. 3

85 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. 3

86 Keller-Segel Models Consider the following Keller-Segel model (with prevention of overcrowding) (Hillen-Painter model): where t ρ = ( ρ χβ(ρ)ρ S) S = ρ, β(ρ) = 0, ρ ρ > 0. Theorem. (Hillen, Painter, 2002) Solutions of the HP model exist globally. Mathematical models for cell movementpart I p. 3

87 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. 3

88 Keller-Segel Models Proof: Consider: and define J + = {x ρ(x,t) > ρ}, J 0 = {x ρ(x,t) = ρ}, J = {x ρ(x,t) < ρ}, ρ + (x,t) = { ρ(x,t) ρ if x J+, 0 otherwise. Mathematical models for cell movementpart I p. 3

89 Keller-Segel Models 1 2 d dt ρ+ (,t) 2 L 2 (R n ) = ( J + + J 0 + J ) ρ + ρ + t Mathematical models for cell movementpart I p. 4

90 Keller-Segel Models 1 2 d dt ρ+ (,t) 2 L 2 (R n ) = = ( + + J + J 0 J + (ρ ρ)ρ t J ) ρ + ρ + t Mathematical models for cell movementpart I p. 4

91 Keller-Segel Models 1 2 d dt ρ+ (,t) 2 L 2 (R n ) = = = ( ) ρ + ρ + t J + + J 0 + J (ρ ρ)ρ t J + (ρ ρ) ( ρ χβ(ρ)ρ S), J + Mathematical models for cell movementpart I p. 4

92 Keller-Segel Models 1 2 d dt ρ+ (,t) 2 L 2 (R n ) = ( ) + + ρ + ρ + t 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. 4

93 Keller-Segel Models + ( ) 1 d 2 dt ρ+ (,t) 2 L 2 (R n ) = + + ρ + ρ + t 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. 4

94 Keller-Segel Models + ( ) 1 d 2 dt ρ+ (,t) 2 L 2 (R n ) = + + ρ + ρ + t 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. 4

95 Keller-Segel Models If ρ I L (R n ) ρ, then ρ + (, 0) = 0, Mathematical models for cell movementpart I p. 4

96 Keller-Segel Models If ρ I L (R n ) ρ, then ρ + (, 0) = 0, ρ + (,t) = 0, Mathematical models for cell movementpart I p. 4

97 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. 4

98 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. 4

99 Keller-Segel Models Define the non-local gradient R f(x,t) = 1 ω n 1 R n 1 S n 1 f(x + yr)dy. Mathematical models for cell movementpart I p. 4

100 Keller-Segel Models Define the non-local gradient R f(x,t) = 1 ω n 1 R n 1 S n 1 f(x + yr)dy. Then the Hillen-Schmeiser-Painter model ( t ρ = ρ χρ ) R S has global existence of solutions., Mathematical models for cell movementpart I p. 4

101 Keller-Segel Models Consider a sensitivity (Velazquez model): β(ρ) = β µ (ρ) = ρ 1 + µρ, Mathematical models for cell movementpart I p. 4

102 Keller-Segel Models Consider a sensitivity (Velazquez model): or more generally: β(ρ) = β µ (ρ) = β µ (ρ) = 1 µ Q(µρ), ρ 1 + µρ, Q(y) y αy 2, y, lim Q(y) <. y Mathematical models for cell movementpart I p. 4

103 Keller-Segel Models Theorem. (Velazquez, 2004) The V model has global existence of solutions for any µ > 0. Mathematical models for cell movementpart I p. 4

104 Keller-Segel Models Theorem. (Velazquez, 2004) The V model has global existence of solutions for any µ > 0. The Velazquez model reproduces the classical KS in the limit µ 0. This allows an extension of the Keller-Segel model after blow up time. Mathematical models for cell movementpart I p. 4

105 Keller-Segel Models Theorem. (Velazquez, 2004) The V model has global existence of solutions for any µ > 0. The Velazquez model reproduces the classical KS in the limit µ 0. This allows an extension of the Keller-Segel model after blow up time. More precisely: Mathematical models for cell movementpart I p. 4

106 Keller-Segel Models Theorem. (Velazquez, 2004) The V model has global existence of solutions for any µ > 0. The Velazquez model reproduces the classical KS in the limit µ 0. This allows an extension of the Keller-Segel model after blow up time. More precisely: For t < T, lim µ 0 ρ µ = ρ 0. Mathematical models for cell movementpart I p. 4

107 Keller-Segel Models Theorem. (Velazquez, 2004) The V model has global existence of solutions for any µ > 0. The Velazquez model reproduces the classical KS in the limit µ 0. This allows an extension of the Keller-Segel model after blow up time. More precisely: 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 I p. 4

108 Keller-Segel Models Theorem. (Velazquez, 2004) The V model has global existence of solutions for any µ > 0. The Velazquez model reproduces the classical KS in the limit µ 0. This allows an extension of the Keller-Segel model after blow up time. More precisely: 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 I p. 4

109 Keller-Segel Models For any µ > 0 the aggregation region is of order µ. We consider each of the N aggregates as a point particles, x j, j = 1,,N, and consider a regular remainder which is important far from these points: ρ(x,t) N j=1 M j (t)δ(x x j (t)) + u reg (x,t). Mathematical models for cell movementpart I p. 4

110 Keller Segel Models The chemoattractant concentration is given by S(x,t) 1 2π N j=1 M j (t) log( x x j (t) ) + S reg (x,t), where S reg = 1 2π R 2 log( x y )u reg (y,t)dy. Mathematical models for cell movementpart I p. 4

111 Keller-Segel Models Theorem. (Velazquez, 2004) Consider a solution of the V model. If we impose the previous Ansatz, then ρ reg t = ρ reg + 1 2π N j=1 M j (t) (x x j(t)) x x j (t) 2 ρ reg (ρ reg S reg ), S reg (x,t) = 1 log( x y )ρ reg (y,t)dy, 2π R 2 ẋ i (t) = Γ(M(t))A i (t), Mathematical models for cell movementpart I p. 4

112 Keller-Segel Models A i (t) = dm i (t) dt N j=1,j i M j (t) 2π + S reg (x i (t),t), = u reg M i (t), where Γ(M) is a function of M > 8π. (x i (t) x j (t)) x i (t) x j (t) 2 Mathematical models for cell movementpart I p. 4

113 Keller-Segel Models After certain time, mass will be only in the aggregates, that will interact as ẋ i (t) = Γ(M i(t))m j (t) 2π ẋ j (t) = Γ(M j(t))m i (t) 2π (x i(t) x j (t)) x i (t) x j (t) 2, (x j(t) x i (t)) x j (t) x i (t) 2 and after certain time they will coalesce in a single aggregate. Mathematical models for cell movementpart I p. 4

114 Keller-Segel Models the function Γ(M) has the following properties: 0 < Γ(M) < 1, lim M 8π Γ(M) = 1, lim M Γ(M) = 0. Mathematical models for cell movementpart I p. 5

115 The Keller-Segel Model Is the extension of the Keller-Segel model made by Velazquez general? Mathematical models for cell movementpart I p. 5