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

Ocean University of China Outline Background Phase field model Fluid phase field systems Multi-component vesicles Vesicle adhesion Cell oscillations Ocean University of China

Ocean University of China 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) Ocean University of China

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  Ocean University of China

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 Ocean University of China

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 Ocean University of China

Ocean University of China 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. Ocean University of China

Ocean University of China 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 Ocean University of China

Ocean University of China 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. Ocean University of China

Elastic Bending Energy in Phase Field Model For , how to formulate our problem: min subject to volume/surface constraints. Ocean University of China

Ocean University of China Phase Field Model Inside volume of  Surface area of  Phase Field Model: Min with constraints: Ocean University of China

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. Ocean University of China

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. Ocean University of China

Ocean University of China Equilibrium Shapes 3d view of some equilibrium configurations Ocean University of China

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 Ocean University of China

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 Ocean University of China

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 Ocean University of China

Ocean University of China Line Tension Energy Surface area of  Length of  minimizing L(, ) => if  ? ? Ocean University of China

Ocean University of China 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 Ocean University of China

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 Ocean University of China

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. Ocean University of China

Ocean University of China 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 Ocean University of China

Ocean University of China Cell Oscillation These cell oscillations are driven by actin and myosin dynamics. Ocean University of China

Cell cortex: Stress due to myosin motors Ocean University of China

A mechanism for the oscillation Actin Myosin Ocean University of China [E. Paluch, M. Piel, J. Prost, M. Bornens, C. Sykes, Biophys. J., 89:724-733]

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. Ocean University of China

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 . Ocean University of China

Ocean University of China 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. Ocean University of China

Ocean University of China 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 Ocean University of China

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: Ocean University of China

Ocean University of China 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: Ocean University of China

Phase Field Formulations Surface tension: where Elastic bending energy: Line tension energy with Osmotic potential energy with Ocean University of China

Phase Field Formulations Perpendicular of  and ? : Tanh profile preserving of  : Total energy: System with gradient flow: Ocean University of China

Ocean University of China 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 . Ocean University of China

Ocean University of China Numerical Results Ocean University of China

Ocean University of China Numerical Results Ocean University of China

Ocean University of China Cell Blebbing Instead of the breakage of the actin gel, the cell blebbing is due to the detaching of the actin gel. Ocean University of China

Ocean University of China Phase Field Functions Ocean University of China

Ocean University of China Energy formulations Surface energy Surfance area Ocean University of China

Ocean University of China Total Energy Elastic Bending Energy Total energy Ocean University of China

Acto-Myosin Dynamics and Lipid Transfer Ocean University of China

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 Ocean University of China

Ocean University of China Moving of the cell The moving distance of the cell mass center according to time Ocean University of China

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 Ocean University of China

Ocean University of China 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. Ocean University of China