1. Xiaoqiang Wang Florida State University
Phase Field Modeling of Actomyosin Driven Cell Oscillations and Cell Blebing
Xiaoqiang Wang
Florida State University
Ocean University of China

2. Ocean University of China
Outline
Background
Phase field model
Fluid phase field systems
Multi-component vesicles
Vesicle adhesion
Cell oscillations
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Our goal
Goal: to study, together with bio-scientists, the deformation/interaction of vesicle membranes under external (flow / electric / magnetic) fields.
Models:
Atomistic: ab initio, MD
Coarse-grained: MC, Fluid mosaic, effective particle
Continuum mechanics: bending elasticity model (our starting point)
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4. Bending elasticity model
Basic hypothesis: the observed (static) vesicle shape is a surface  that minimizes the Elastic Bending Energy subject to fixed volume/area constraints.
The Elastic Bending Energy is mainly determined by the curvature of the membrane surface .
Lipid bilayer builds 
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5. Elastic Bending Energy
Helfrich:
a: surface tension
b, c: bending rigidities
c0: spontaneous curvature
describing the inside-outside asymmetry
H: mean curvature K: Gaussian curvature.
Special case with the spontaneous curvature:
Helfrich W., Z. Naturforsch, 1973
(may depend on the local heterogeneous concentration of the species such as protein molecules in blood cells)
For simplicity, we only study a special case with the effect of spontaneous curvature.
k: bending rigidity
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6. Elastic Bending Energy
Our problem:
Minimize E subjects to fixed volume/area.
Isotropic case without spontaneous curvature:
is related to the Willmore’s problem:
min of for tori with fixed area
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Solution techniques
Numerical simulations employing meshes on the interface that undergo deformation.
Interface tracking
Hard to deal with topological events
Our approach:
phase field model.
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Phase Field Model
Introduce a phase function , defined on a computational domain , to label the inside/outside of the vesicle
Membrane : the level set
=0
>0
<0
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Phase Field Function
Ideal phase field function
d: distance function
+1 inside, -1 outside
Sharp interface as  ! 0
=1
=-1 -1 1 
Diffusive interface model.
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10. Elastic Bending Energy in Phase Field Model
For , how to formulate our problem:
min subject to volume/surface
constraints.
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Phase Field Model
Inside volume of 
Surface area of 
Phase Field Model: Min with constraints:
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12. Euler-Lagrange Equation
The Euler-Lagrange equation for this problem reads:
Let
We have
We proved the uniformly bounds for the Lagrange multipliers   0 .
To solve this minimization phase field model, we can use Euler- Lagrange approach with the E-L multipliers.
And the equation has the following paritcular system with lamda1 and lambda2 these multipliers.
We also have obtained some analytic result. For example, we proved that the L multipliers are uni. Boun. As eps goes to zero.
And this is also very important to justify the sharp interface limit.
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13. Penalty Constraints
Apply a penalty formulation for constrained minimization.
M1, M2: penalty coefficients
With large M1, M2 :
Min W() with A() =  and B() = 
Min WM()
Solve via a gradient flow:
The computation is acturally more covenient to apply the penalty formulation for a constrained minimization.
We add the error of the volume and the error of the surface area, square them, multiply them by large coefficients M1 and M2. So we only need to minimize the modified elastic bending energy W_M.
To solve this minimizing problem, we can use a gradient flow approach, which gives us a nonlinear evolution equation.
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Equilibrium Shapes
3d view of some equilibrium configurations
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15. Multi-component Vesicles
More complex equilibrium shapes for vesicles with multiple lipid components:
Naturelly, there are different kinds of lipid moleculus. And vesicles may be composed by multiple lipid components.
This is an experiment published in nature, in which they have two different type of molecules, which form very complex patterns.
T.~Baumgart, S.~ Hess and W.~ Webb, Nature 2003
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16. Two-component Vesicles
Total energy involves elastic bending energy and line tension energy
Line tension energy involves between every two components or at the boundary of the open membrane
Our problem:
Minimize E subjects to fixed volume/area.
In the model of multi-component vesicles, the total energy ……,
Line tension happens between the interface of two components.
To formulate this problem, we introduce another pahse field function
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17. Component Phase Field Function
d: distance function
+1 one component, -1 another
bending rigidity  is a function of 
=1
=1
=-1
Here we start from an ideal phase field function, ….
=-1
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Line Tension Energy
Surface area of 
Length of 
minimizing L(, ) => if  ? ?
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Phase field model
Define the total energy by a phase field function pair (, )
Phase field model:
Minimize E(, ) with some constraints and regularization terms on total volume, and surface areas of individual components
In the model of multi-component vesicles, the total energy ……,
Line tension happens between the interface of two components.
To formulate this problem, we introduce another pahse field function
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20. Constraints and Regularization terms
Inside volume of 
Surface area of 
Perpendicular of  and ?
Tanh profile preserving of 
Phase Field Model:
Min with constraints and regularizations
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21. Gradient Flow with Penalty Formulation
Penalty Formulations for constraints and regularization terms
with large penalty coefficients Mi.
Gradient Flow
Equilibrium shapes reach as
And this evolution equation is written in the following particular form.
We can show that as t-> infinity, the equilibrium shapes will be the energy minimizers. Because the energy decreases monotonly in time.
As M goes to infty, the minimizer of E_m phi convergents to phi star, the minimizer of E \phi.
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Numerical Schemes
We have developed a number of discretization schemes for the equilibrium systems
Spatial discretization:
Finite Difference and Fourier Spectral.
Time discretization: Exponential Time Differential Rounge-Kutta
Strictly preserving the Energy Law
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Cell Oscillation
These cell oscillations are driven by actin and myosin dynamics.
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24. Cell cortex: Stress due to myosin motors
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25. A mechanism for the oscillation
Actin
Myosin
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[E. Paluch, M. Piel, J. Prost, M. Bornens, C. Sykes, Biophys. J., 89: ]

26. A Model of the Oscillation
Two components with different
density values of actin and
myosin II, and therefore different
surface tension;
Interior osmotic pressure;
Line tension of the ring between
two components;
Bending rigidity of the lipid cell membrane;
Moving of lipid molecules from one component to the other: shrinking and bulging surfaces.
Polymerization and depolymerization of actin, combination of myosin II with actin; diffusion of actin and myosin II.
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27. Osmotic Potential Energy and Line Tension Energy
Osmotic pressure depends on the salty density difference between inside and outside of the membrane. In the case with zero outside density, the osmotic pressure is inverse proportional to the inside volume i.e.
We formulate the osmotic potential energy by
Line tension energy results from the myosin ring  between two components. It can be formulated by or
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Total Energy
All together with the elastic bending energy of the lipid membrane, we have the total energy
where i are the surface tension coefficients, i are the bending rigidities, Ci are the spontaneous curvatures.
The surface tension and spontaneous curvatures are depending on the density of myosin II of each component, and we set
where yi are the density values of myosin.
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Lipids Transfer
The total number of the lipid molecules is fixed which results a constant surface area and the lipids may move from one component to the other.
Suppose the area of two components are A1(t) and A2(t), written by the penalty formulation to the total energy:
The interior surface tension / pressure is proportional to the Lagrange terms, i.e.
The lipid moving rate from one component to the other is assumed to be proportional to the pressure difference, i.e.
And we have
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30. Polymerization and Diffusion of Actin
The concentrations of actin and myosin II are different on different membrane components.
Actin polymerization occurs at the surface ends whereas depolymerization occurs at the pointed ends. The growth velocity of the actin gel:
where and are the rate constants at two ends, is the concentration of G-actin available for polymerization.
And we have the mass conservation:
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Diffusion of Myosin II
Myosin II is combined with actin, it disassembles to the solvent as the depolymerization of actin filaments. On the other hand, it attaches to the filaments at any position.
where is the attaching rate of myosin and is the depolymerization of actin filaments, , and are the concentration of myosin II in solvent, component 1 and component 2.
Also the mass conservation:
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32. Phase Field Formulations
Surface tension:
where
Elastic bending energy:
Line tension energy with
Osmotic potential energy with
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33. Phase Field Formulations
Perpendicular of  and ? :
Tanh profile preserving of  :
Total energy:
System with gradient flow:
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Numerical Schemes
Axis-symmetric or truly 3D configurations.
Spatial discretization:
Finite Difference and Fourier Spectral.
Time discretization: Explicit Forward Euler / Implicit Schemes
Time step size is adjusted to ensure the gradient flow part:
Update area A1(t), A2(t), actin concentrations m0, h1, h2, myosin II concentrations y0, y1, y2 every time step after the gradient flow of  and .
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Numerical Results
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Numerical Results
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Cell Blebbing
Instead of the breakage of the actin gel, the cell blebbing is due to the detaching of the actin gel.
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Phase Field Functions
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Energy formulations
Surface energy
Surfance area
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Total Energy
Elastic Bending Energy
Total energy
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41. Acto-Myosin Dynamics and Lipid Transfer
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42. Numerical Experiments
Cell blebbing with lower solution potential inside leads to cell migration. From left to right, first row to second row, phase-field image at time t=0, 0.1, 0.2, 0.3, 0.375, 0.45, 0.525, 0.575
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Moving of the cell
The moving distance of the cell mass center according to time
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44. Comparison with laboratory observations
First row: the image of cell blebbing of zebrafish primordial germ cell from [H. Blaser, and et. 2006]
Second row: our numerical results
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Summary
We discussed the aoto-myosin dynamics and the mathematical model for the acto-myosin driven cell oscillations and cell blebbing.
We give the phase field formulations and described the lipid transferring from one part to the other.
Numerical simulations are compared with laboratory observations.
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