Hybrid mathematical models of cell movement Roberto Natalini - PowerPoint PPT Presentation

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 Short biological background on cell movements 1 Continuous models of chemotaxis 2 A hybrid model for morphogenesis in zebrafish 3 A simple model of collective motion under alignment and chemotaxis 4 Cardiac stem cells and the growth of cardiospheres 5 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]: � ∂ t u = ∇ ( D u ∇ u − χ ( u , φ ) ∇ φ ) , τ∂ t φ = D c ∆ 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]: � ∂ t u = ∇ ( D u ∇ u − χ ( u , φ ) ∇ φ ) , τ∂ t φ = D c ∆ 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 ∂ t u + + γ ∂ x u + = − µ + ( φ, φ x ) u + + µ − ( φ, φ x ) u − ,    ∂ t u − − γ ∂ x u − = µ + ( φ, φ x ) u + − µ − ( φ, φ x ) u − ,   ∂ t φ = D ∂ xx φ + 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!  ∂ t u + ∂ x v = 0 ,  with α = γ ( µ − − µ + )  ∂ t v + γ 2 ∂ x u = α u − β v , and β = µ + + µ −   ∂ t φ = D ∂ xx φ + f ( u , φ ) . • 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 ∂ t u + + γ ∂ x u + = − µ + ( φ, φ x ) u + + µ − ( φ, φ x ) u − ,    ∂ t u − − γ ∂ x u − = µ + ( φ, φ x ) u + − µ − ( φ, φ x ) u − ,   ∂ t φ = D ∂ xx φ + 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!  ∂ t u + ∂ x v = 0 ,  with α = γ ( µ − − µ + )  ∂ t v + γ 2 ∂ x u = α u − β v , and β = µ + + µ −   ∂ t φ = D ∂ xx φ + f ( u , φ ) . • 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: a collective cell migration guided by a haptotactic signal, with 1 constant velocity of about 69 µ m h − 1 ; a process of differentiation in the trailing region that induces a 2 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: chemokine protein SDF-1a ( stromal cell-derived factor-1a ), strongly 1 haptotactic and expressed by the substratum; the receptor CXCR4b expressed by the primordium itself. 2 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: fibroblast growth factors FGF3 – FGF10 , strongly chemotactic; 1 receptor FGFR . 2 Experimental observations on the FGF activity FGF3 and FGF10 are substantially equivalent (robustness of the 1 system); FGF and FGFR are mutually exclusive . 2 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 at the beginning, all cells are leader; 1 leader–follower differentiation (MET 2 transition) produces rosette-shaped structures (proto-neuromasts); neuromasts deposition. 3 Three sufficient conditions for the leader–follower transition a low level of SDF-1a (trailing zone is preferred for transition); 1 a high level of FGF; 2 a lateral inhibition effect ( leader/follower transition favored by a low number of 3 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: describe the collective cell migration, the detachment of the 1 neuromasts, in the physical spatial and temporal scale; ensure the existence and stability of the rosette structures of the 2 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 X i ( t ) : position of the i -th cell; ϕ i ( t ) : switch variable for the i -th cell ( ϕ i = 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 alignment haptotaxis chemotaxis attraction/repulsion  ���� � �� �  � �� � � �� � � �� �  ¨ F 2 ( ˙  X i = α F 1 ( ∇ s ) + γ ( 1 − ϕ i ) F 1 ( ∇ f ) + X ) + F 3 ( X )    damping   � �� �   [ µ F + ( µ L − µ F ) ϕ i ] ˙  − X i ,       �  leader-follower state SDF conc. FGF conc. lateral inhib.   � �� � � �� � � �� � ���� ϕ i = 0 , if δ F 1 ( s ) − [ k F + ( k L − k F ) ϕ i ] F 1 ( h ( f )) + λ Γ( n i ) ≤ 0 , 1 , otherwise ,        production molecular degradation FGF rate in time diffusion   � �� � ���� ���� ����   ∂ t f = D ∆ f + ξ F 4 ( X ) − η f ,        SDF rate in time degradation   � �� � ����  ∂ t s = − σ sF 5 ( X ) , Steady states Roberto Natalini IAC–CNR Hybrid mathematical models of cell movement