Large time behavior of evolutionary problems in cell biologydinamici5.dm.unibo.it/pdf/natalini.pdf · 5 A simple model of collective motion under alignment and chemotaxis Roberto

Large time behavior of evolutionary problemsin cell biology

Roberto NataliniConsiglio Nazionale delle Ricerche

Istituto per le Applicazioni del Calcolo

Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology

Outline

1 A short introduction to population dynamics: stability vs. instability2 Continuous semilinear models of chemotaxis3 A quasilinear hyperbolic model in vasculogenesis4 A hybrid model for morphogenesis in zebrafish5 A simple model of collective motion under alignment and chemotaxis


Basics in population dynamics

Malthus modelu′ = αu

α > 0, u(t)→∞

α < 0, u(t)→ 0

Logistic modelu′ = αu(1− u)

α > 0, u(t)→ 1

α < 0, u(t)→ 0

→ Lotka-Volterra model


Basics in population dynamics

Malthus modelu′ = αu

α > 0, u(t)→∞

α < 0, u(t)→ 0

Logistic modelu′ = αu(1− u)

α > 0, u(t)→ 1

α < 0, u(t)→ 0

→ Lotka-Volterra model


Pattern formation: spaceKPP-Fischer equation u = u(x .t) with x ∈ (0, L)

ut = D∆u + αu(1− u)

u(0, t) = u(L, t) = 0For D = 0: Logistic u(t)→ 1

For α = 0: Diffusion u(t)→ 0

What happens in general?

If αL2

D < π2 then u(t)→ 0

otherwise αL2

D > π2 then u(t)→ U


Turing instability and pattern formationLet f (0, 0) = g(0, 0) = 0

ut = Du∆u + αf (u, v)

vt = Dv ∆v + αg(u, v)

Assume that for Du = Dv = 0 the point (0, 0) is stable.

Turing 1952: Under some conditions on the linearized system and forDu 6= Dv , the are some unstable modes, which create patterns in thenonlinear case.


What is chemotaxis?

Chemotaxis is the movement of cells (bacteria, human cells...) influencedby a chemical substance called chemoattractant.

Dictyostellium discoideum (Dicty)

Angiogenesis

Stem cells Fibroblasts


Diffusive models of chemotaxis

Patlak 1953; Keller-Segel 1970

∂tu = div (Du∇u − χ(u, φ)∇φ) ,

τ∂tφ = Dc ∆φ+ f (u, φ).

• u is the density of bacteria,• φ is the density of the chemoattractant.

Many analytical and numerical results [Horstmann, 03 & 04]:existence of global solutions vs. finite time blow-up;analysis of the blow-up profile,self-similar solutions, traveling waves...





τ∂tφ = Dc ∆φ+ f (u, φ).







τ∂tφ = Dc ∆φ+ f (u, φ).







τ∂tφ = Dc ∆φ+ f (u, φ).




Hyperbolic models of chemotaxis

Cattaneo-type model (Hillen)∂tu + div(uV ) = 0,∂t(uV ) + γ2∇u = u∇φ− uV ,∂tφ = D ∂xxφ+ f (u, φ).

Euler-type model (Preziosi)

∂tu + div(uV ) = 0,∂t(uV ) + div(uV ⊗ V ) +∇P(u) = u∇φ− uV ,∂tφ = D ∂xxφ+ f (u, φ).


Hyperbolic models of chemotaxis

Cattaneo-type model (Hillen)∂tu + div(uV ) = 0,∂t(uV ) + γ2∇u = u∇φ− uV ,∂tφ = D ∂xxφ+ f (u, φ).

Euler-type model (Preziosi)

∂tu + div(uV ) = 0,∂t(uV ) + div(uV ⊗ V ) +∇P(u) = u∇φ− uV ,∂tφ = D ∂xxφ+ f (u, φ).


Why hyperbolic models?

Vessel formation (left) and numerical simulations by Preziosi et al. (right)


The 1D case (Greenberg-Alt)

∂tu + ∂x v = 0,∂tv + γ2 ∂x u = (χ(φ)∂xφ)u − v ,∂tφ = D ∂xxφ+ au − bφ,

x ∈ [0, L], No-flux conditions:

v(0) = v(L) = ∂xφ(0) = ∂xφ(L) = 0(= ∂x u(0) = ∂x u(L)).

Conserved quantities

Mass∫

[0,L]

u(x , t) dx =

∫[0,L]

u(x , 0) dx

Symmetry (u, v , φ)(L− x , t) = (u,−v , φ)(x , t)


The 1D case (Greenberg-Alt)

∂tu + ∂x v = 0,∂tv + γ2 ∂x u = (χ(φ)∂xφ)u − v ,∂tφ = D ∂xxφ+ au − bφ,

x ∈ [0, L], No-flux conditions:

v(0) = v(L) = ∂xφ(0) = ∂xφ(L) = 0(= ∂x u(0) = ∂x u(L)).

Conserved quantities

Mass∫

[0,L]

u(x , t) dx =

∫[0,L]

u(x , 0) dx

Symmetry (u, v , φ)(L− x , t) = (u,−v , φ)(x , t)


The Neumann problem

Joint with Guarguaglini, Mascia, Ribot, DCDS-B 2009Let (U, 0,Φ) be a constant steady state of the Cattaneo model(U + u, v ,Φ + φ) perturbed solutionThe perturbation w = (u, v , φ) satisfies∂tu + ∂x v = 0,

∂tv + γ2 ∂x u − χU ∂xφ+ β v= F1(φ, ∂xφ) + F2(φ, ∂xφ) u + F3(φ, ∂xφ) v ,

∂tφ− D ∂xxφ+ bφ− a u = F4(u, φ),

v = ∂xφ = 0,for x = 0, L

where

F1(φ, ψ) := U(

g(Φ + φ, ψ)− χψ)

= O(|(φ, ψ)|2),

F2(φ, ψ) := g(Φ + φ, ψ) = O(|(φ, ψ)|),F3(φ, ψ) := β − h(Φ + φ, ψ) = O(|(φ, ψ)|),

|(φ, ψ)| → 0

F4(u, φ) := f (U + u,Φ + φ)− a u + b φ = O(|(u, φ)|2) |(u, φ)| → 0


The Neumann problem

Joint with Guarguaglini, Mascia, Ribot, DCDS-B 2009Let (U, 0,Φ) be a constant steady state of the Cattaneo model(U + u, v ,Φ + φ) perturbed solutionThe perturbation w = (u, v , φ) satisfies∂tu + ∂x v = 0,

∂tv + γ2 ∂x u − χU ∂xφ+ β v= F1(φ, ∂xφ) + F2(φ, ∂xφ) u + F3(φ, ∂xφ) v ,

∂tφ− D ∂xxφ+ bφ− a u = F4(u, φ),

v = ∂xφ = 0,for x = 0, L

where

F1(φ, ψ) := U(

g(Φ + φ, ψ)− χψ)

= O(|(φ, ψ)|2),

F2(φ, ψ) := g(Φ + φ, ψ) = O(|(φ, ψ)|),F3(φ, ψ) := β − h(Φ + φ, ψ) = O(|(φ, ψ)|),

|(φ, ψ)| → 0

F4(u, φ) := f (U + u,Φ + φ)− a u + b φ = O(|(u, φ)|2) |(u, φ)| → 0


Global existence for the Neumann case

Theorem

Under the previous assumptions, let (U, 0,Φ) be a constant steady statesuch that

χ = ∂ψg(Φ, 0) > 0, β = h(Φ, 0) > 0, ∂φf (U,Φ) = −b < 0 < a = ∂uf (U,Φ)

Assume the stability condition

U <γ2

χ a

(b +

Dπ2

L2

)Let w0 = (u0, v0, φ0) ∈ H1 the perturbation (with zero mass for u0),and w the corresponding solution. Then there exists ε0 > 0 such that,if ‖w0‖H1 ≤ ε0, then

‖w‖H1 (t) ≤ C ‖w0‖H1 e−θ t . ∀ t > 0.


Global existence for the Neumann case

Theorem

Under the previous assumptions, let (U, 0,Φ) be a constant steady statesuch that

χ = ∂ψg(Φ, 0) > 0, β = h(Φ, 0) > 0, ∂φf (U,Φ) = −b < 0 < a = ∂uf (U,Φ)

Assume the stability condition

U <γ2

χ a

(b +

Dπ2

L2

)Let w0 = (u0, v0, φ0) ∈ H1 the perturbation (with zero mass for u0),and w the corresponding solution. Then there exists ε0 > 0 such that,if ‖w0‖H1 ≤ ε0, then

‖w‖H1 (t) ≤ C ‖w0‖H1 e−θ t . ∀ t > 0.


Stationary solutions

From ∂tu + ∂x v = 0, we have v = 0.Constant solutions

(u, v , φ) =(

U, 0, ab U

), U ≥ 0.

Nonconstant solutions

γ2 ∂x u − u∂xφ = 0,D ∂xxφ+ au − bφ = 0,∂xφ(0) = ∂xφ(L) = 0.




(u, v , φ) =(

U, 0, ab U

), U ≥ 0.






(u, v , φ) =(

U, 0, ab U

), U ≥ 0.




Bifurcation diagram (for the Neumann Pb.)










Comparison between Hyperbolic and Parabolic modelsThe constant case


The Quasilinear model

[Ambrosi, Gamba, Preziosi et al., 03]∂tρ+ ∂x (ρu) = 0∂t(ρu) + ∂x (ρu2) + ∂x P(ρ) = χρ∂xφ− αρu,τ∂tφ = D ∂xxφ+ aρ− bφ,

x ∈ [0, L], with ”no-flux” conditionsStability in 1D and even multi-D of constant stationary states forsmall perturbations [Di Russo, Sepe, 13]Solutions with vacuum zones!


Stationary Solutions

[R.N., M. Ribot, M. Twarogowska]On [0, L], fix a mass M

∂tρ+ ∂x (ρu) = 0

∂t(ρu) + ∂x (ρu2) + ∂x P(ρ) = χρ∂xφ− αρu,

τ∂tφ = D ∂xxφ+ aρ− bφ.

+ Boundary conditions ⇒ ρu = 0




∂tρ+ ∂x (ρu) = 0







∂tρ+ ∂x (ρu) = 0




∂x P(ρ) = χρ∂xφ,

0 = D ∂xxφ+ aρ− bφ.




∂tρ+ ∂x (ρu) = 0




κγργ−1∂xρ = χρ∂xφ,

0 = D ∂xxφ+ aρ− bφ,

with P(ρ) = κργ .


Stationary solutions, again


0 = D ∂xxφ+ aρ− bφ.+ ∂xρ|0,L = ∂xφ|0,L = 0.

Constants: (ρ, φ) =

(ML ,

aMbL

)Otherwise: If γ ≥ 2, we have ρ = 0 or κγργ−2∂xρ = χ∂xφ.For γ = 2, explicit solutions

ρ = 0 et D ∂xxφ− bφ = 0.

ρ =χ

2κφ+ K et D ∂xxφ+(aχ

2κ − b)φ+ aK = 0.

⇒ Explicit solutions (for ω =1D

(aχ2κ − b

)> 0 ).


Stationary solutions, again


0 = D ∂xxφ+ aρ− bφ.+ ∂xρ|0,L = ∂xφ|0,L = 0.

Constants: (ρ, φ) =

(ML ,

aMbL

)Otherwise: If γ ≥ 2, we have ρ = 0 or κγργ−2∂xρ = χ∂xφ.For γ = 2, explicit solutions

ρ = 0 et D ∂xxφ− bφ = 0.

ρ =χ

2κφ+ K et D ∂xxφ+(aχ

2κ − b)φ+ aK = 0.

⇒ Explicit solutions (for ω =1D

(aχ2κ − b

)> 0 ).


A zoological classification of stationary solutions for(γ = 2)

Density ρ.

With κ = 1, χ =10, D = 0.1, a =20, b = 10, L = 1 etM = 10.


Numerical test 1, with respect to the size of the domain Land the friction coefficient α, for P(ρ) = κρ2

L\α 5 50 200

1

7

30


Numerical test 3: hyperbolic–parabolic comparison

t = 0.1 t = 0.5 t = 3

t = 20 - 300 t = 331 t = 333κ = 1, χ = 10, D = 0.1, a = 20, b = 10, L = 1 et γ = 3.Strong difference in the asymptotic behavior between the hyperbolic andthe parabolic cases


Numerical test 3: hyperbolic–parabolic comparison

t = 0.1 t = 0.5 t = 3

t = 20 - 300 t = 331 t = 333κ = 1, χ = 10, D = 0.1, a = 20, b = 10, L = 1 et γ = 3.Strong difference in the asymptotic behavior between the hyperbolic andthe parabolic cases


A hybrid mathematical model for self-organizing cells inzebrafish

A joint work with Ezio Di Costanzo and Luigi Preziosi, J. Math. Bio.2015


Lateral lineA fundamental sensory system present in fish and amphibians.

Large variety of behaviours:detect movement and vibration inthe surrounding water;prey and predator detection;school swimming.

NeuromastsMain sensory organs of the lateral line, embedded in the body surface in arosette-shaped pattern: 1–2 sensory hair cells in the centre, surrounded by othersupport cells (8–12 cells).Neuromasts extend a ciliary bundle into the water, which detect movement in thesurrounding environment.


Experimental observationsAn initial elongated group (80-100 cells) of mesenchymal cells(primordium), with a trailing region near the head and a leading regiontowards the future tail of the embryo.

Two primary mechanisms in the morphogenesis process:1 a collective cell migration guided by a haptotactic signal, with

constant velocity of about 69 µm h−1;2 a process of differentiation in the trailing region that induces a

mesenchymal–epithelial transition and causes the neuromastsassembly and their detachment.

Movie zebrafish (Gilmour et al, 2006).Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology

Collective migrationTwo main factors:

1 chemokine protein SDF-1a (stromal cell-derived factor-1a), stronglyhaptotactic and expressed by the substratum;

2 the receptor CXCR4b expressed by the primordium itself.

Neuromasts assemblyTwo main factors:

1 fibroblast growth factors FGF3–FGF10, strongly chemotactic;2 receptor FGFR.

Experimental observations on the FGF activity1 FGF3 and FGF10 are substantially equivalent (robustness of the

system);2 FGF and FGFR are mutually exclusive.


Leader to follower differentiation

leader mesenchymal cells: produce FGF but the receptor FGFR is not activated;follower epithelial cells: activate FGFR, but do not produce FGF.

Cyclic mechanism

1 at the beginning, all cells are leader;2 leader–follower differentiation (MET

transition) produces rosette-shapedstructures (proto-neuromasts);

3 neuromasts deposition.

Three sufficient conditions for the leader–follower transition

1 a low level of SDF-1a (trailing zone is preferred for transition);2 a high level of FGF;3 a lateral inhibition effect (leader/follower transition favored by a low number of

neighboring cells).

return


The mathematical modelOur aim is to obtain a minimal mathematical model which is able to:

1 describe the collective cell migration, the detachment of theneuromasts, in the physical spatial and temporal scale;

2 ensure the existence and stability of the rosette structures of theneuromasts, as stationary solutions.

Request 2) provides a restriction for some parameters.

Hybrid discrete in continuous descriptiondiscrete on cellular scale (but nonlocal sensing area);continuous at molecular scale.


A second order modelXi (t) : position of the i-th cell;ϕi (t) : switch variable for the i-th cell (ϕi = 0, 1 resp. follower-leader);f (x, t) : concentration of FGF (equivalent FGF3 and FGF10);s(x, t) : concentration of SDF-1a;

acceleration i-th cell︷︸︸︷Xi =

haptotaxis︷︸︸︷αF1 (∇s) +

chemotaxis︷︸︸︷γ(1− ϕi )F1 (∇f ) +

alignment︷︸︸︷F2(X) +

attraction/repulsion︷︸︸︷F3(X)

−

damping︷︸︸︷[µF + (µL − µF)ϕi ] Xi ,

leader-follower state︷︸︸︷ϕi =

0, if

SDF conc.︷︸︸︷δF1(s) −

FGF conc.︷︸︸︷[kF + (kL − kF)ϕi ] F1(h(f )) +

lateral inhib.︷︸︸︷λΓ(ni ) ≤ 0,

1, otherwise,

FGF rate in time︷︸︸︷∂t f =

diffusion︷︸︸︷D∆f +

production︷︸︸︷ξF4(X) −

molecular degradation︷︸︸︷ηf ,

SDF rate in time︷︸︸︷∂ts = −

degradation︷︸︸︷σsF5(X) ,


Numerical test of the steady solution.Dynamical simulation with initial zero velocity, a FGF concentrationgiven by the stationary equation

D∆f = ηf − ξχB(XL,R3),∂f∂n = 0, on ∂Ω,

and a zero initial concentration of SDF-1a.

Figure : A 8-rosette

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3x 10

−3

Emax,rel

Time (h)

Figure :maxi ‖Xi(t)− Xi0‖ /R

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7x 10

−4

Time (h)

Vmax(µm

h−1)

Figure : maxi∥∥Xi(t)

∥∥Numerical simulation


Dynamical model

Numerical simulation


The Cucker-Smale modelBehaviour of a flock of birds, general phenomena where autonomousagents reach a consensus. Vi =

β

N

N∑j=1

α1

(α2 + ||Xi − Xj ||2)σ(Vj − Vi );

Xi = Vi ,

i = 1, . . . ,N.

σ: rate of decay of the influence between agents.Cucker and Smale (2007), Ha et al (2009), see also J. A. Carrillo, et al(2010) proved that:

if 0 ≤ σ ≤ 1/2 there is unconditional flocking (Movie with σ = 1/2);if σ > 1/2 there is conditional flocking (Movie with σ = 2).


Cucker-Smale modelVi =

β

N

N∑j=1

α1(α2 + ‖Xi − Xj‖2)σ (Vj − Vi ),

Xi = Vi ,

Applications: biological field, collective dynamics of different interacting groups(Szabo et al, 2006; Sepulveda et al, 2013; Albi and Pareschi, 2013).Extensions: repulsion, individuals with preferred directions.

A simple Collective motion under alignment and chemotaxis

Vi =β

N

N∑j=1

1(1 +‖Xi −Xj‖2

R2

)σ (Vj − Vi ) + γ∇f (Xi ),

Xi = Vi ,

∂t f = D∆f + ξ

N∑j=1

χB(Xj ,R) − ηf ,

(1)


Existence and uniqueness

Joint with E. Di Costanzo 2016

Theorem 1 (Local existence and uniqueness)System (1) has a unique solution on [0,T ], where T is suitably estimated.

Proposition 1 (Continuation of solutions)Let y(t) be a solution of system (1) on a interval [0,T ), if there is aconstant P with ‖y− y0‖ ≤ P on [0,T ), then there is a T > T suchthat y(t) can be continued to [0, T ].

Theorem 2 (Global existence and uniqueness)System (1) has a unique solution in all [0,+∞). go to


Numerical simulations: asymptotic behavior

Simulation 1. Absence of chemotaxis (γ = 0).Simulation 2. Strong flocking state.Simulation 3. Absence of alignment (β = 0).


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Large time behavior of evolutionary problems in cell biologydinamici5.dm.unibo.it/pdf/natalini.pdf · 5 A simple model of collective motion under alignment and chemotaxis Roberto