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Hybrid mathematical models of cell movement

Roberto Natalini Istituto per le Applicazioni del Calcolo – CNR 11th Meeting on Nonlinear Hyperbolic PDEs and Applications On the occasion of the 60th birthday of Alberto Bressan SISSA, Trieste, June 14th, 2016

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Outline

1

Short biological background on cell movements

2

Continuous models of chemotaxis

3

A hybrid model for morphogenesis in zebrafish

4

A simple model of collective motion under alignment and chemotaxis

5

Cardiac stem cells and the growth of cardiospheres

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

What is chemotaxis ?

Chemotaxis is the movement of cells (bacteria, human cells...) influenced by a chemical substance called chemoattractant. Dictyostellium discoideum (Dicty) Angiogenesis Stem cells Fibroblasts

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Keller-Segel parabolic system

Classical parabolic model [Keller & Segel, 70]:

• ∂tu = ∇ (Du ∇u − χ(u, φ)∇φ) ,

τ∂tφ = Dc ∆φ + f (u, φ).

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

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Keller-Segel parabolic system

Classical parabolic model [Keller & Segel, 70]:

• ∂tu = ∇ (Du ∇u − χ(u, φ)∇φ) ,

τ∂tφ = Dc ∆φ + f (u, φ). Numerous theoretical and numerical results [Horstmann, 03 & 04] :

• 1D : existence of global solutions
• multiD : global existence (small initial mass) vs blow-up of solutions

in finite time (big initial mass).

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

A hyperbolic alternative: the Greenberg-Alt model

Discrete kinetic 1D model      ∂tu+ + γ ∂xu+ = −µ+(φ, φx) u+ + µ−(φ, φx) u−, ∂tu− − γ ∂xu− = µ+(φ, φx) u+ − µ−(φ, φx) u−, ∂tφ = D ∂xxφ + f (u, φ).

• u± is the density of bacteria with speed ±γ
• µ± are the relative turning rates (influenced by chemicals)
• φ is the density of chemicals

... , it is wave equation with relaxation!      ∂tu + ∂xv = 0, ∂tv + γ2 ∂xu = αu − βv, ∂tφ = D ∂xxφ + f (u, φ). with α = γ(µ− − µ+) and β = µ+ + µ−

• u = u+ + u− total density; v = γ(u+ − u−) total flux

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

A hyperbolic alternative: the Greenberg-Alt model

Discrete kinetic 1D model      ∂tu+ + γ ∂xu+ = −µ+(φ, φx) u+ + µ−(φ, φx) u−, ∂tu− − γ ∂xu− = µ+(φ, φx) u+ − µ−(φ, φx) u−, ∂tφ = D ∂xxφ + f (u, φ).

• u± is the density of bacteria with speed ±γ
• µ± are the relative turning rates (influenced by chemicals)
• φ is the density of chemicals

... , it is wave equation with relaxation!      ∂tu + ∂xv = 0, ∂tv + γ2 ∂xu = αu − βv, ∂tφ = D ∂xxφ + f (u, φ). with α = γ(µ− − µ+) and β = µ+ + µ−

• u = u+ + u− total density; v = γ(u+ − u−) total flux

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

A continuous model for the movement of brain stem cells after an ischemic event A Model of Ischemia-Induced Neuroblast Activation in the Adult Subventricular Zone,

• D. Vergni, F. Castiglione, M. Briani, S. Middei, E. Alberdi, K. G. Reymann, R.

Natalini, C. Volont´ e, C. Matute, F. Cavaliere, Plos One 2009.

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Part III A hybrid mathematical model for self-organizing cells in the early development of the zebrafish lateral line A joint work with Ezio Di Costanzo and Luigi Preziosi, J. Math. Bio. 2015

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Lateral line A fundamental sensory system present in fish and amphibians. Large variety of behaviours: detect movement and vibration in the surrounding water; prey and predator detection; school swimming. Neuromasts Main sensory organs of the lateral line, embedded in the body surface in a rosette-shaped pattern: 1–2 sensory hair cells in the centre, surrounded by other support cells (8–12 cells). Neuromasts extend a ciliary bundle into the water, which detect movement in the surrounding environment.

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Attention of biologists in the lateral line to understand: interactions between multiple signaling and the collective migration of cell during morphogenesis; how an organ responds to injury and replaces damaged components; how the genetic defects cause disorders in the nervous system. Recent studies have investigated the zebrafish (Danio rerio) lateral line (Gilmour et al, 2008; Nechiporuk et al, 2008). Zebrafish is an important vertebrate model organism in scientific research for many scientific reasons: regenerative abilities (major organs are visible in 36 hpf); embryos are robust, transparent, easily observable and testable; genome has been fully sequenced. Improvements in the fields of oncology, genetics, stem cell research, and regenerative medicine.

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Experimental observations An initial elongated group (80-100 cells) of mesenchymal cells (primordium), with a trailing region near the head and a leading region towards 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 neuromasts assembly and their detachment. Movie zebrafish (Gilmour et al, 2006).

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Collective migration Two main factors:

1

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

2

the receptor CXCR4b expressed by the primordium itself. Cell-cell and cell-substratum interactions Mechanical forces via filopodia (cadherins, integrins).

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Neuromasts assembly Two main factors:

1

fibroblast growth factors FGF3–FGF10, strongly chemotactic;

2

receptor FGFR. Experimental observations on the FGF activity

1

FGF3 and FGF10 are substantially equivalent (robustness of the system);

2

FGF and FGFR are mutually exclusive.

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

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-shaped structures (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 Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

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

1

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

2

ensure the existence and stability of the rosette structures of the neuromasts, as stationary solutions. Request 2) provides a restriction for some parameters. Hybrid discrete in continuous description discrete on cellular scale (but nonlocal sensing area); continuous at molecular scale.

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

A second order model

Xi(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,

• therwise,

FGF rate in time

• ∂tf

=

diffusion

• D∆f +

production

ξF4(X) −

molecular degradation

• ηf

,

SDF rate in time

• ∂ts

= −

degradation

σsF5(X),

Steady states Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

¨ Xi = αF1 (∇s) + γ(1 − ϕi)F1 (∇f ) + F2( ˙ X) + F3(X) − [µF + (µL − µF)ϕi] ˙ Xi. Alignment effect Cucker and Smale–like flocking term: F2( ˙ X) := 1 ¯ Ni

• j:Xj ∈B(Xi ,R1)\{Xi }

H( ˙ Xj − ˙ Xi), H( ˙ Xj − ˙ Xi) := [βF + (βL − βF)ϕiϕj] R2

1

R2

1 + ||Xj − Xi||2 ( ˙

Xj − ˙ Xi), ¯ Ni := card {j : Xj ∈ B(Xi, R1)} . Unlike the original Cucker-Smale model: flocking term is coupled with other effects (chemotaxis, attraction/repulsion, damping); it acts on a truncated domain.

Alignment effect Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

¨ Xi = αF1 (∇s) + γ(1 − ϕi)F1 (∇f ) + F2( ˙ X) + F3(X) − [µF + (µL − µF)ϕi] ˙ Xi.

Adhesion-repulsion effect F3(X) :=

• j:Xj∈B(Xi,R5)\{Xi}

K(Xj − Xi),

K(Xj − Xi) :=

        

−ωrep

• 1

||Xj − Xi|| − 1 R4

• Xj − Xi

||Xj − Xi|| , if ||Xj − Xi|| ≤ R4; ¯ ωadh (||Xj − Xi|| − R4) Xj − Xi ||Xj − Xi|| , if R4 < ||Xj − Xi|| ≤ R5;

Similar terms can be found in D’Orsogna et al (2006), Cristiani et al (2011), Albi and Pareschi (2013).

Three zone models Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Lateral inhibition mechanism

Leader to Follower transition

We count the number ni of cells in a suitable neighborhood of the i-th cell, with radius of influence R2: Γ(ni) := eni eni + Γ0 − 1 1 + Γ0 , ni := card j : Xj ∈ ˚ B(Xi, R2)\ {Xi} , Sharp difference between a cell in the centre or a cell on the boundary; Fast saturation effect as ni increases (possibility to obtain neuromasts with 8-12 cells).

Full Model Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Steady states and stability

Particular steady states, biologically relevant, corresponding to the neuromast basic structure. Stationary model                                   

γ(1 − ϕi) W

• B(Xi ,¯

R)

∇f (x)wi(x) dx +

• j:Xj ∈B(Xi ,R5)\{Xi}

K(Xj − Xi) = 0, ϕi =

    

if − kF + (kL − kF)ϕi W

• B(Xi ,¯

R)

f (x) 1 + f (x) wi(x) dx + Γ(ni) ≤ 0, 1

• therwise,

D∗∆f = ηf − ξ

Ntot

• j=1

ϕjχB(Xj ,R3), ∂f ∂n = 0,

• n ∂Ω.

s = 0,

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Definition A N-rosette (N ≥ 2) is a configuration formed by a leader cell surrounded by N follower cells located on the vertices of a regular polygon of N sides centered in the leader cell. Hypotheses

1

the range of lateral inhibition equal to the range of repulsion between cells;

2

the followers are located in the range of the lateral inhibition of the leader;

3

there is no repulsion between adjoining followers if N = 2, 3; there is no repulsion between followers in alternating position if N ≥ 4.

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Proposition 1 There exist N-rosettes if and only if N ≤ 12. Moreover the distance d1, depending on N, can vary in the following ranges: 1 2 sin π

N

≤ d1 R4 ≤ 1, if N = 2, 3, 1 2 sin 2π

N

≤ d1 R4 ≤ 1, if 4 ≤ N ≤ 12. The maximum number of cells is consistent with the experimental observations (Gilmour et al, 2008). Proposition 2 In a N-rosette there are repulsion and lateral inhibition effects between adjoining followers if and only if N ≥ 4. In particular if N ≥ 6 these effects do not depend on d1, and if N = 4, 5 this holds if and only if 1 2 sin 2π

N

≤ d1 R4 < 1 2 sin π

N

, N = 4, 5.

Parameter range Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Numerical test of the steady solution.

Dynamical simulation with initial zero velocity, a FGF concentration given by the stationary equation D∆f = ηf − ξχB(XL,R3), ∂f ∂n = 0,

• n ∂Ω,

and a zero initial concentration of SDF-1a.

Figure : A 8-rosette

5 10 15 20 25 30 35 40 45 50 0.5 1 1.5 2 2.5 3 x 10

−3

Emax,rel Time (h)

Figure : maxi Xi(t) − Xi0 /R

5 10 15 20 25 30 35 40 45 50 1 2 3 4 5 6 7 x 10

−4

Time (h) Vmax (µm h−

1 )

Figure : maxi

• ˙

Xi(t)

• Numerical simulation

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Dynamical model

Parameters estimates Biological literature, stationary model, large parameter sweep.

Data fitting Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Dynamical model

Numerical simulation Tip velocity during migration:

1 2 3 4 5 6 10 20 30 40 50 60 70 80 90 100 Time (h) Tip velocity (µmh−

1 )

Test 2 Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Part IV A hybrid mathematical model of collective motion under alignment and chemotaxis (joint with Ezio Di Costanzo): a short analytical intermezzo

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Cucker-Smale model Behaviour of a flock of birds, general phenomena where autonomous agents 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) 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).

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Cucker-Smale model

      

˙ Vi = β N

N

• j=1

α1

• α2 + Xi − Xj2σ (Vj − Vi),

˙ Xi = Vi, Applications: biological field, collective dynamics of different interacting groups (Szab`

• et al, 2006; Sepulveda et al, 2013; Albi and Pareschi, 2013).

Extensions: repulsion, individuals with preferred directions. Collective motion under alignment and chemotaxis

              

˙ Vi = β N

N

• j=1

1

• 1 + Xi −Xj2

R2

σ (Vj − Vi) + γ∇f (Xi),

˙ Xi = Vi, ∂tf = D∆f + ξ

N

• j=1

χB(Xj ,R) − ηf , (1)

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Existence and uniqueness

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 a constant P with y − y0 ≤ P on [0, T), then there is a ¯ T > T such that 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 Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Numerical simulations

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

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Part V

• E. Di Costanzo, A. Giacomello, E. Messina, R. Natalini, G. Pontrelli, F.

Rossi, R. Smits, M. Twarogowska A discrete in continuous mathematical model of cardiac progenitor cells formation and growth as spheroid clusters (Cardiospheres), preprint, 12/2015

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Cardiac stem cells

Cardiac biopsy-derived progenitor cells, growing in vitro as niche-like microtissue (Cardiospheres), represent a widely adopted platform technology holding great promise to realize a powerful cell therapy system Cardiosphere are spheroid of cellular clusters with a central nucleus of less differentiated elements surrounded by outer layers of cells more committed toward different levels of cardio-vascular differentiation and

• proliferation. (Messina et al. 2004, Chimenti et al. 2011)

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Cardiac stem cells

Cardiac biopsy-derived progenitor cells, growing in vitro as niche-like microtissue (Cardiospheres), represent a widely adopted platform technology holding great promise to realize a powerful cell therapy system Cardiosphere are spheroid of cellular clusters with a central nucleus of less differentiated elements surrounded by outer layers of cells more committed toward different levels of cardio-vascular differentiation and

• proliferation. (Messina et al. 2004, Chimenti et al. 2011)

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

A first model of cardiospheres

Main goals Cell aggregation in spherical layers, with a realistic density Differentiation towards the external layers Differentiation and proliferation are driven by jump processes, submitted to environmental factors Understanding and control the role of the environmental factors (growth factors and nutrients)

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

A first model of cardiospheres

Xi(t) position of the i-eme cell at time t c = c(x, t) concentration of oxigen (nutrients) S = S(x, t) concentation of TGF (chemical signal) ¨ Xi =

• j:Xj∈B(Xi,R2)\{Xi}

K(rij) − µ ˙ Xi + αF(∇S(Xi, t)) ∂tc = ∇ · (Dc∇c) −

N

• i=1

λ(ϕi)cγ+1 k(ϕi) + cγ+1 χB(Xi ,R3) + H ¯ B(c0 − c), ∂tS = ∇ · (DS∇S) +

N

• i=1

ξ(ϕi)χB(Xi ,R3) − ηS with D(x) :=

Dmax 1+ρ¯ A(x), where ¯

A(x) is the fraction of occupied volume near x.

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Differentiation

cells undergoing different states of differentiation: ϕi(t) = 1, 2, 3, d (d=dead) ϕi(t + ∆t) = 1 ϕi(t + ∆t) = 2 ϕ(t + ∆t) = 3 ϕi(t) = 1

• ϕi(t) = 2
• ϕi(t) = 3
• Table : Possible levels of differentiation between time t and t + ∆t

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Differentiation II

Differentiation is given by a non homogeneous Poisson process Γi : inhibition factor for the i-eme cell at time t Γi(t) = 1, if ni ≤ 4, 0, if ni > 4, ni := card{j : Xj ∈ B(Xi, R4)}, i = 1, . . . , N qi(t) := σ(ϕi) · F(S(Xi, t)) · Γi(t) Smax , i = 1, . . . , N, diff. treshold where σ(ϕi) (ϕi = 1, 2) are constants and Smax is the max of TGF-β

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Differentiation III

qi(t) is the intensity of the Poisson process P(ϕi(t + ∆t) − ϕi(t) ≥ 1) ≅ qi(t)∆t, (2) which yields the stochastic differential equation dϕi = dP t qi(s) ds

• ,

i = 1, . . . , N, (3) where P( t

0 qi(s) ds) is the process. A computational approach is

ϕi(t + ∆t) = ϕi(t) + 1, se ¯ qi < qi∆t, 0, altrimenti, i = 1, . . . , N, where ¯ qi is random in [0,1].

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Proliferation

Poisson based on the oxygen dφi = dP t pi(s) ds

• ,

i = 1, . . . , N,

3 6 9 12 15 18 21 0.5 1 1.5 2 2.5 3 Oxygen concentration (%) Proliferation thresholds Cell type 1 Cell type 2 Cell type 3

Figure : Proliferation tresholds based on the cellular type and the oxygen concentration

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

From experiments to simulations

50 100 150 200 50 100 150 200

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Growth at 21% of oxygen concentration

50 100 150 200 50 100 150 200

(a) t=0

50 100 150 200 50 100 150 200

(b) t=24

50 100 150 200 50 100 150 200

(c) t=48

50 100 150 200 50 100 150 200

(d) t=72

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

Growth at 5% of oxygen concentration

50 100 150 200 50 100 150 200

(e) t=0

50 100 150 200 50 100 150 200

(f) t=24

50 100 150 200 50 100 150 200

(g) t=48

50 100 150 200 50 100 150 200 Space (µm) Space (µm) Time = 72 h; N = 172, N1 = 114, N2 = 22, N3 = 0, Nd = 36

(h) t=72

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement

50 100 150 200 250 300 50 100 150 200 250 300 1.295 1.3 1.305 1.31 Space (µm) Concentration of TGF-β Space (µm) 50 100 150 200 250 300 50 100 150 200 250 300 2 3 4 5 6 7 x 10

−13

Space (µm) Concentration of Oxygen Space (µm)

Numerical simulation

Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement