Mechanical model of a cell growth
Continuum mechanics, bibliography
Aleš Janka, Florence Yerly
Dept. of Mathematics and Dept. of Biology
group of Profs. J-P. Gabriel, Ch. Mazza (maths) and D. Reinhardt (biology)
Université de Fribourg
SystemsX meeting Bern, Dec 5, 2008
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
Cell structure (plant cell)
(source: Wikipedia ”plant cell”)
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
Mechanical model: balls filled with water
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
Mechanical model: different time-scales
growth of cell walls (time-scale: slow)
growth of cell interior / cell divisions (time-scale: slow)
semi-permeable wall – osmosis (time-scale: slow)
elastic response of the continuum (time-scale: fast)
incompressible cell-interior / cytoplasm (time-scale: fast)
⇒ Need to distinguish slow and fast mechanical model
fast mechanical model: water-filled cell-structure,
impermeable walls, elasto-plastic membrane + hydrostatic
pressure (incompressibility)
slow mechanical model: cell growth, osmosis (transport of
matter), cell divisions + responses of the fast model!
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
Fast model: pre-deformed elastic cell walls – elastic shell
Deformation (strain tensor)
∂uj
1 ∂ui
εij =
+
i, j ∈ {x, y , z}
2 ∂uj
∂ui
u(x)
Constitutive law (material properties)
τk` = Eijkl (εij + εij )
k, ` ∈ {x, y , z}
ε˙ij = G [εij + εij ]+
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
Fast model: pre-deformed elastic cell walls – elastic shell
Deformation (strain tensor)
∂uj
1 ∂ui
εij =
+
i, j ∈ {x, y , z}
2 ∂uj
∂ui
u(x)
u(x)
Constitutive law (material properties)
τk` = Eijkl (εij + εij )
k, ` ∈ {x, y , z}
ε˙ij = G [εij + εij ]+
u(x)
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
Fast model: pre-deformed elastic cell walls – elastic shell
Deformation (strain tensor)
∂uj
1 ∂ui
εij =
+
i, j ∈ {x, y , z}
2 ∂uj
∂ui
u(x,y)
τyy(u)
Constitutive law (material properties)
k, ` ∈ {x, y , z}
ε˙ij = G [εij + εij ]+
F
y
x
Aleš Janka, Florence Yerly
τyy(u)
u(x,y)
F
Mechanical model of a cell growth
εyy (u)
=
τk` = Eijkl (εij + εij )
du
dy
Growth: viscoplastic model – time-dependent plasticity
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
Growth: viscoplastic model – one cell
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
Growth: viscoplastic model – time-dependent plasticity
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
Growth: viscoplastic model – time-dependent plasticity
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
Fast model: turgor pressure vs. mass conservation
Turgor pressure
Hydrostatic pressure + osmosis.
Depends on the state of each cell.
Changes by deformation??
Mass conservation ??
Can wall permeability be neglected for
fast processes?? If yes, then:
∂ρ
+ div(ρu̇(x)) = 0
∂t
u(x)
p
Energy conservation
Z 1
E (u) =
τij εij − f · u dx → min
2
u(x)
Ω
+ constraint
∂ρ
+ div(ρu̇) = 0.
∂t
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
Fast model: turgor pressure vs. mass conservation
Turgor pressure
Hydrostatic pressure + osmosis.
Depends on the state of each cell.
Changes by deformation??
Mass conservation ??
Can wall permeability be neglected for
fast processes?? If yes, then:
∂ρ
+ div(ρu̇(x)) = 0
∂t
u(x)
p
Energy conservation
Z 1
E (u) =
τij εij − f · u dx → min
2
u(x)
Ω
+ constraint
∂ρ
+ div(ρu̇) = 0.
∂t
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
Fast model: turgor pressure vs. mass conservation
Turgor pressure
Hydrostatic pressure + osmosis.
Depends on the state of each cell.
Changes by deformation??
Mass conservation ??
Can wall permeability be neglected for
fast processes?? If yes, then:
∂ρ
+ div(ρu̇(x)) = 0
∂t
u(x)
mass
Energy conservation
Z 1
E (u) =
τij εij − f · u dx → min
2
u(x)
Ω
+ constraint
∂ρ
+ div(ρu̇) = 0.
∂t
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
Fast model: turgor pressure vs. mass conservation
Turgor pressure
Hydrostatic pressure + osmosis.
Depends on the state of each cell.
Changes by deformation??
Mass conservation ??
Can wall permeability be neglected for
fast processes?? If yes, then:
∂ρ
+ div(ρu̇(x)) = 0
∂t
u(x)
mass
Energy conservation
Z 1
E (u) =
τij εij − f · u dx → min
2
u(x)
Ω
+ constraint
∂ρ
+ div(ρu̇) = 0.
∂t
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
Fast model: turgor pressure vs. mass conservation
Turgor pressure
Hydrostatic pressure + osmosis.
Depends on the state of each cell.
Changes by deformation??
Mass conservation ??
Can wall permeability be neglected for
fast processes?? If yes, then:
∂ρ
+ div(ρu̇(x)) = 0
∂t
u(x)
mass
Energy conservation
Z 1
E (u) =
τij εij − f · u dx → min
2
u(x)
Ω
+ constraint
∂ρ
+ div(ρu̇) = 0.
∂t
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
Fast model: turgor pressure vs. mass conservation
Turgor pressure
Hydrostatic pressure + osmosis.
Depends on the state of each cell.
Changes by deformation??
Mass conservation ??
Can wall permeability be neglected for
fast processes?? If yes, then:
∂ρ
+ div(ρu̇(x)) = 0
∂t
Energy conservation
Z 1
E (u) =
τij εij − f · u dx → min
2
u(x)
Ω
+ constraint
Force equilibrium, mass conservation
∇j τij (u) + ∇i p = fi
∂ρ
− div(ρu̇) =
∂t
force f = volumic forces
(gravity) and acceleration
static pressure = Lagrange
multiplier for mass
conservation.
osmosis modelled through
∂ρ
∂t = f (p, C ), fcn. of static
pressure, concentrations, etc..
∂ρ
+ div(ρu̇) = 0.
∂t
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
Applicability of the model
(source: Wikipedia)
continuum mechanics ⇒ not applicable for plasmolyzed cells!
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
More issues
Mechanism for determining the shape of the cells (mechanics
or genetics?)
Rule for division of cells
Max. number of cells (3D) treatable in a reasonable time:
10.000–100.000 ⇒ interest of homogenization??
Is it necessary to follow each cell (expensive complex models,
difficulty to measure experimentally), or a coarse-scale
continuum model would suffice (cells are modeled only
through local material properties)?
Aleš Janka, Florence Yerly
Mechanical model of a cell growth
Bibliography
Florence Yerly
SystemsX meeting Bern, Dec 5, 2008
Florence Yerly
Mechanical model of a cell growth
Virtual plant project: INRIA-CIRAD Montpellier, France
Team : S. Stoma, J. Chopard, C. Godin, J. Traas
Studies on the shoot apical meristem functioning and
development
Studies on the genetical processes involved
Tools : mathematics and computational methods
See Stoma S. et al., 2007 (shape development of the meristem,
mass-spring) and Stoma S. et al., 2008 (auxin transport)
Florence Yerly
Mechanical model of a cell growth
Mechanics of tip growth morphogenesis
Team : J. Dumais and collaborators, Harvard University
Anisotopic-viscoplastic model (see J. Dumais et al., 2006)
Plant cell = thin pressurized shell
Connection between wall stresses, wall strains and cell
geometry
Comparison with rubber balloons (see R. Bernal et al., 2007)
Florence Yerly
Mechanical model of a cell growth
Biological studies of material properties and of cell pressure
L. Zonia and T. Munnik, 2007
New wall materials + osmotic pressure ⇒ growth
Hydrodynamic flow : importance for the cell shape and
structure
Hydrostatic pressure surges = general mechanism of growth ?
A. Boudaoud, 2003
Growth of walled cells = plastic deformation
cells = thin shells with 2 modes of defomation : streching and
bending
turgor pressure → strain on the wall
Florence Yerly
Mechanical model of a cell growth
Growth theory in general continuum mechanics
A. DiCarlo and S. Quiligotti, 2002 and A. DiCarlo, 2005
Modelling bulk growth with continuum physics
Constitutive theory
J.F. Ganghoffer, 2005
Growth = addition of material
Study of the variation of the reference configuration
Florence Yerly
Mechanical model of a cell growth
Mechanics of cell wall
D.J Cosgrove, 2005
Biological mechanisms of cell wall growth
L.S Yellapragada et al., 2008
Fluid-solid interaction
Fluid region ⇒ strain on the wall
Florence Yerly
Mechanical model of a cell growth