Biological self-organisation phenomena on weighted networks

Transcription

1 Biological self-organisation phenomena on weighted networks Lucilla Corrias (jointly with F. Camilli, Sapienza Università di Roma) Mathematical Modeling in Biology and Medicine Universidad de Oriente, Santiago de Cuba Lucilla Corrias CIMPA 2016 June 13, / 22

2 Self-organization Self-organization of biological populations is a very commun but highly complex phenomenon that can be observed in very different populations, such as the bacteria Escherichia coli (E.coli) (commonly found in the lower human intestine) (Budrene & Berg, Nature, 1991) Lucilla Corrias CIMPA 2016 June 13, / 22

3 Self-organization Self-organization of biological populations is a very commun but highly complex phenomenon that can be observed in very different populations, such as the slime mold Dictyostelium discoideum Dictyostelium discoideum (commonly found in soil and moist leaf litter) aggregate to form fruiting bodies and then spores able to migrate to found more suitable environments Lucilla Corrias CIMPA 2016 June 13, / 22

4 Self-organization Self-organization of biological populations is a very commun but highly complex phenomenon that can be observed in very different populations, such as Endothelial cells (cells lining the interior surface of the blood vessels in vertebrates) Endothelial cells are able to sprout from existing small vessels and self-assembly into a vascular labyrinth (the early formation of blood vessels). Angiogenesis is a primary process in solid tumor growth. Lucilla Corrias CIMPA 2016 June 13, / 22

5 Self-organization Self-organization of biological populations is a very commun but highly complex phenomenon that can be observed in very different populations, such as Physarum polycephalum (it lives in moist areas, decaying leaves and logs and may be seen without a microscope) In its plasmodium phase (the main vegetative phase) P. polycephalum is a network of protoplasmic tubes with many nuclei. During this stage the organism look for food and the network evolves. Lucilla Corrias CIMPA 2016 June 13, / 22

6 Understanding self-organization The understanding of the self-organization phenomena is a challenging issue for biology and mathematical-biology, in particular for the applications in biology and medicine : think to tumor growth, cells differentiation,... Several processes play a crucial role in self-organisation. One of this is the so called chemotaxis process. Chemotaxis is the phenomenon undergone by single-cell or simple multicellular organisms when they direct their movements toward or away a higher concentration of chemical stimuli. The mouvement is toward the chemical stimulus when it is favorable to the survive of the organisme (positive chemotaxis) and away from the chemical stimulus when it is dangerous (negative chemotaxis) Lucilla Corrias CIMPA 2016 June 13, / 22

7 Understanding self-organization There are several models for chemotaxis. Choosing the macroscopic description of the population, considering uniquely the cell density u 0 and the chemical concentration c 0, and assuming that cells move randomly (brownian motion) and are attracted by the chemical signal (direct mouvement) the chemical signal is produced by the cells themselves the population is conserved a corresponding mathematical model is given by the density and concentration balance equations [Keller & Segel, 1970] where ε 0, plus boundary conditions t u = ( u u c) ε t c = c + u α c ε diff. coeff. of u diff. coeff. of c, α 0 Lucilla Corrias CIMPA 2016 June 13, / 22

8 t u = ( u u c) ε t c = c + u α c There is a huge quantity of literature on the mathematical analysis of the Keller-Segel system. Nevertheless, several challenging issues are still open. Depending on the space dimension, on ε 0 and on the initial mass M = u 0 (x)dx, different phenomena can occur. Roughly speaking : dim 1 : global existence (solutions are global and bounded) dim 2 : threshold phenomenon (solution are global if M < M and blow up if M > M) dim 3 : global existence and blow-up (whatever the initial mass is) Lucilla Corrias CIMPA 2016 June 13, / 22

9 The Physarum polycephalum Physarum polycephalum is a myxomycete. It lives in moist areas, decaying leaves and logs. In its plasmodium phase is a network of protoplasmic tubes with many nuclei. During this stage the organism look for food and the network evolves. The are experimental evidences that P. polycephalum is able to sense the environment (chemotactically) and to recover the shortest path between nourishment sources Lucilla Corrias CIMPA 2016 June 13, / 22

10 The network V := {v 1,..., v n } the finite set of vertices in R N E := {e 1,..., e m } the finite set of edges (curves in R N ) whose endpoints are vertices Γ := (V, E) the finite connected network (between every pair of nodes there exists an edge) each edges is parametrized by two homeomorphisms Π ± j : [0, 1] R which give rise to two oriented arcs e ± j denoted a j to each edge e j E is associated a positive weight κ(e j ) > 0 (the network is not homogeneous) E(v i ) := {j : e j is incident to v i V }, i = 1,..., n Lucilla Corrias CIMPA 2016 June 13, / 22

11 Functional spaces on the network Γ a function u defined on Γ is a collection of m functions (u j ) m j=1 such that u j := u ej u j is intended to be (u Π ± j ) for x e j At the vertices we defined the exterior normal derivative u j n (I (a (u Π ± j )(h) (u Π ± j )(0) j)) = lim h 0 + h u j n (T (a (u Π ± j )(1 + h) (u Π ± j )(1) j)) = lim h 0 h with I (a j ) and T (a j ) the initial and terminal endpoint of a j resp. Γ u(x) dx = m j=1 κ(e j) e j u j (x)dx with 1 u j (x)dx := (u Π ± j )(ξ) dξ e j 0 Lucilla Corrias CIMPA 2016 June 13, / 22

12 Functional spaces on the network Γ C 0 (Γ) := {u = (u j ) m j=1 : u j(v i ) = u k (v i ) if j, k E(v i ), i} For p [1, ) : L p (Γ) := {u = (u j ) m j=1 : u p L p (Γ) := m j=1 κ(e j) u j p L p (e j ) < } L (Γ) := {u = (u j ) m j=1 : u L (Γ) := max 1 j m κ(e j ) u j L (e j )} W 1, (Γ) := {u C 0 (Γ) : u L (Γ)} H r (Γ) := {u C 0 (Γ) : u 2 H r (Γ) := m j=1 κ(e j) u j 2 H r (e j ) < } Lucilla Corrias CIMPA 2016 June 13, / 22

13 The system on the network Γ The system t u = ( u u c) ε t c = c + u α c translates into m systems (one for each edge e j ) t u j = xx u j x (u j x c j ) on (0, ) e j, j = 1,..., m, ε t c j = xx c j + u j α c j on (0, ) e j, j = 1,..., m, u j (0, x) = uj 0 (x), x Γ, j = 1,..., m, c j (0, x) = cj 0 (x), x Γ, j = 1,..., m, What about the boundary conditions? Lucilla Corrias CIMPA 2016 June 13, / 22

14 Boundary and transmission conditions Boundary conditions have to be defined on vertices. But not all vertices are necessarily part of the boundary. In some vertices the fluid just flow in or out. Γ := {v i1,..., v ik } V = {v 1,..., v n }, 0 i k n, with the condition : if E(v l ) = 1 then v l Γ On Γ we consider u j n (t, v i) = c j n (t, v i) = 0, v i Γ The remaining vertices are the transition vertices V T := V \ Γ In the transition vertices we consider the Kirchhoff type conditions (implying the conservation of the total flux) j E(v i ) κ(e j )[ u j n u c j j n ](t, v i) = 0, v i V T Lucilla Corrias CIMPA 2016 June 13, / 22

15 The system on the network Γ t u j = xx u j x (u j x c j ) on (0, ) e j, j = 1,..., m, ε t c j = xx c j + u j α c j on (0, ) e j, j = 1,..., m, u j (0, x) = uj 0 (x), x Γ, j = 1,..., m, c j (0, x) = cj 0 (x), x Γ, j = 1,..., m, u j n (t, v i) = 0, c j n (t, v i) = 0, j E(v i ) κ(e j )[ u j n u j c j n ](t, v i) = 0, t > 0, v i Γ t > 0, v i Γ t > 0, v i V T Lucilla Corrias CIMPA 2016 June 13, / 22

16 The fundamental solution of the heat equation Let G(t, z) = 1 4π t e z2 4 t. The fundamental solution of the heat equation on Γ is ([Roth, 1984]) where L(t, x, y) = H(t, x, y) = δ i, j κ 1 (e i ) G(t, d(x, y)) + L(t, x, y) (k+2) L(x,y) C C k+2 (x,y) κ 1 (e i ) ɛ(c) G(t, d C (x, y)) and C k (x, y) is the set of oriented paths joining x to y, with length k L(x, y) is the length of any geodesic path joining x to y ɛ(c) is the weight of the path C Lucilla Corrias CIMPA 2016 June 13, / 22

17 The fundamental solution of the heat equation H(t, x, y) =δ i, j κ 1 (e i ) G(t, d(x, y)) + (k+2) L(x,y) C C k+2 (x,y) κ 1 (e i ) ɛ(c) G(t, d C (x, y)) δ i,j is the usual Kronecker s delta function d(x, y) = (Π ± (x) (Π ± (y) The first term is the fundamental solution of the heat equation on each edge e j The second term takes into account the instantaneous propagation of the heat along all the possible infinite many paths joining x to y on the network d C (x, y) is the distance between x and y along the path C d C (x, y) = d(x, T (a j1 )) + d(x, I (a jk )) + C 2 Lucilla Corrias CIMPA 2016 June 13, / 22

18 The fundamental solution of the heat equation For all f C 0 (Γ), the function P t f (y) = (H(t) f )(y) := H(t, x, y)f (x)dx, Γ (t, y) (0, ) Γ with P 0 f = f is the unique continuous solution of the initial valued problem t u j = xx u j u j (0) = f j on (0, ) e j, j = 1,..., m on e j, j = 1,..., m u j (t, v i ) = u k (t, v i ), j, k E(v i ), i = 1,..., n, t > 0 κ(e j ) u j n (t, v i) = 0, i = 1,..., n, t > 0 j E(v i ) Lucilla Corrias CIMPA 2016 June 13, / 22

19 The solution on the network u(t, y) = P t u 0 (y) t c(t, y) = e (α/ε)t P (t/ε) c 0 (y) + 1 ε where Theorem (Camilli, C.) 0 P (t s) x (u(s) x c(s))(y)ds P t f (y) := Γ t 0 e (α/ε)(t s) P ((t s)/ε) u(s)(y)ds H(t, x, y)f (x)dx Assume u 0 L (Γ) and c 0 W 1, (Γ). Then, for any T > 0 there exists a unique integral solution of the system with u L ((0, T ); C 0 (Γ)), c L ((0, T ); W 1, (Γ)). Moreover, if the initial data (u 0, c 0 ) are nonnegative, the solution (u, c) is nonnegative. Lucilla Corrias CIMPA 2016 June 13, / 22

20 Proof The ingredients of the proof are : the properties of the fundamental solution H H(t, x, y)dy = 1, (t, x) (0, ) Γ sup x Γ Γ sup H(t, x, ) L 1 (Γ) C 1 x Γ H(t) L (Γ Γ) C 2 (1 + t 1/2 ) y H(t, x, ) L 1 (Γ) + sup y H(t,, y) L 1 (Γ) C 3 (1 + t 1/2 ) y Γ y H(t) L (Γ Γ) C 4 (1 + t 1 ) A fixed point theorem for the local in time existence : u c ψ(u) A continuation argument for the global existence Lucilla Corrias CIMPA 2016 June 13, / 22

21 Conclusions and Perspectives We have shown the well-posedness of the problem This problem seems more challenging than the same problem solved over R : blow-up as t is not excluded Analyse the asymptotic behavior of the solution as t. There is a numerical evidence that the density fill the edges along the shortest path between two sources [Borsche, Göttlich, Klar, Schillen 2014]. Do this model recover really the shortest paths between (say) two vertices? and for which kind of networks? Lucilla Corrias CIMPA 2016 June 13, / 22

22 Thank you for your attention Lucilla Corrias CIMPA 2016 June 13, / 22