Modelli Matematici e Calcolo Scientifico per la Biologia Cellulare Luigi Preziosi Angiogenesis Vasculogenesis Short version of the movie at: www.nature.com/emboj/journal/v22/n8/suppinfo/7595085as1.html Vasculogenesis • What are the mechanisms driving the generation of the patterns? • Why is the size of a successful patchwork nearly constant? • What is the explanation of the transition obtained for low and high densities? n = 50 cells/mm2 n = 100 cells/mm2 n = 200 cells/mm2 n = 400 cells/mm2 Assumptions • Cells move on a surface and do not duplicate • The cell population can be described by a continuous distribution of density n and velocity v • Cells release chemical mediators (c) • Cells are accelerated by gradients of soluble mediators “Original” at and slowed down by friction www.biochemweb.org/neutrophil.shtml • For low densities (early stages) the cell population can be modelled as a fluid of non directly interacting particles showing a certain degree of persistence in their motion • Tightly packed cells respond to compression Original at www.borisylab.northwestern.edu/mov/fig5.mov Checking the hypothesis Mathematical Model D = diffusion coefficient w = attractive strength a = rate of release of soluble mediators = degradation time of soluble mediators b = friction coefficient = (D) 1/2 ~ 0.1-0.2 mm ~ 10-7 cm2/s ~ 103 s ~ 20 min D r = typical dimension of endothelial cells ~ .02 mm D. Ambrosi, A. Gamba, G. Serini, E. Giraudo, L. Preziosi, F. Bussolino, EMBO J., (2004) Mathematical Model p=0 blow-up p = ln n Keller Segel p = convex no blow-up R. Kowalczyk, J. Math. Anal. Appl., (2005) Temporal evolution n = 50 cells/mm2 n = 100 cells/mm2 n = 200 cells/mm2 n = 400 cells/mm2 Phase transition Percolative transition Swiss-cheese transition A. Gamba et al., Phys. Rev. Letters, 90, 118101 (2003) R. Kowalczyk, A. Gamba, L.P. Discr. Cont. Dynam. Sys. B 4 (2004) Nested models in IBMs Sub-cellular level Nested models in IBMs VEGF MOTILITY POLARIZATION TUBULOGENESIS Cellular Potts Model A cell is represented by several nodes 0-nodes represent the outer environment, or the extracellular matrix - Based on the identification of a generalized energy H - Evolution stochastically tries to minimize the energy of the system Cellular Potts Model - A cell is represented by several nodes - Based on the identification of a generalized energy H - Evolution stochastically tries to minimize the energy of the system Cellular Potts Model Choose a node at random Choose a neighbour at random Choose an action Compute the change in energy H H > 0 ? H < 0 ? Accept the action (always) P ( DH ) = e Accept the action with probability decreasing exponentially with H -DH + H 0 T Repeat till all nodes are chosen increase time Sub-Cellular Components in CPM M. Scianna & L.P., Multiscale Model. Simul. 10, 342-382 (2012) CPM for vasculogenesis M. Scianna, L. Munaron & L.P., Progr. Biophys. Molec. Biol. 106, 450-462 (2011) CPM for vasculogenesis M. Scianna, L. Munaron & L.P., Progr. Biophys. Molec. Biol. 106, 450-462 (2011) CPM for vasculogenesis M. Scianna, L. Munaron & L.P., Progr. Biophys. Molec. Biol. 106, 450-462 (2011) CPM for vasculogenesis CPM for angiogenesis Cell-ECM interaction (P. Friedl) Motion in 3D ECM C. Verdier P. Friedl, K. Wolf Rolli et al., PlosOne 5, e8726 (2010) Motion in 3D ECM J. Cell Biol. 201, 1069-1084 (2013) Motion in 3D ECM Collagen concentration = HT1080 migration in rat tail collagen (1.7 mg/ml) in presence of Mmp inhibitor Neutrophil migration in rat tail collagen (1.7 mg/ml) in presence of IL-8 Motion in 3D ECM Heart Sclera MMP secretion Taking into account of the nucleus Giverso & L.P., Biomech. Model. Mechanobiol. 13, 481-502 (2014) Measuring cell traction The classical (direct) problem in continuum mechanics: Given the stress f, find the deformation u of the substratum such that :f→u The inverse problem Set of forces acting on Wc with null resultant and momentum Take the smallest force possible Compute the deformation for a given virtual force Measured deformation Evaluate the computed deformation in the measurement points The inverse problem Cell traction Criterium for invasion Work done by traction > Energy required to squeeze the nucleus No way in Cells with rigid nuclei in microchannel (J. Guch) M. Scianna, L.P., J. Theor. Biol. 317, 394-406 (2013) Cells with deformable nuclei in microchannel (J. Guch) Effect of pore size in ECM M. Scianna, L.P., & K. Wolf, Biosci. Engng. 10, 235-261 (2013) Effect of adhesion in 2D Palecek et al., Nature 385, 537-540 (1997) Effect of deformability Varying fiber elasticity Rigid nucleus Varying nucleus elasticity Elastic nucleus Back to the continuous model Implications for ? It depends on: Pore (and nucleus) size ✔ Nucleus elasticity ✔ Fiber elasticity ✔ Bimodal behaviour (might) ✔ A. Arduino & L.P., Int. J. Nonlinear Mech. (2015) A0 A Back to the continuous model Implications for ? It depends on: Pore (and nucleus) size ✔ Nucleus elasticity ✔ Fiber elasticity ✔ Bimodal behaviour (might) ✔ A. Arduino & L.P., Int. J. Nonlinear Mech. (2015) Back to the continuous model Effect of ECM concentration Effect of nucleus deformability Heterogeneus ECM Invasion of ovary cancer cells C. Giverso, M. Scianna, L.P., N. Lo Buono & A. Funaro Math. Model. Nat. Phenom. 5, 203-223 (2010) A. Chauviere C. Verdier S. Astanin C. Giverso M. Scianna D. Ambrosi A. Tosin G. Vitale V. Peschetola Istituto per la Ricerca e la Cura del Cancro Divisione di Angiogenesi Molecolare F. Bussolino Università di Torino Dipartimento di Scienze della Vita e Biologia dei Sistemi L. Munaron Ospedale Molinette Laboratorio di Immunogenetica A. Funaro
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