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