Modeling of Actomyosin Driven Cell Oscillations

1. Modeling of Actomyosin Driven Cell Oscillations Xiaoqiang Wang Florida State Univ. KITPC - Membrane Biophysics

2. Outline • Background • Facts determine vesicle shape • A mechanism for the oscillation • Mathematical model and Phase field formulations • Numerical experiment • Future work and conclusion KITPC - Membrane Biophysics

3. Membranes • Cellular membranes are composed mostly of lipids. • Lipid has one polar (hydrophilic) head and one or more hydrophobic tails. KITPC - Membrane Biophysics

4. Vesicle membranes • Lipids form a bilayer structure which is a basic building block for all bio-membranes. • Membranes are fluid-like: lipids have rapid lateral movement and slowly flip-flop movement. KITPC - Membrane Biophysics

5. Cell Oscillation These cell oscillations are driven by actin and myosin dynamics. KITPC - Membrane Biophysics

6. Fragments of L929 fibroblasts Cytoplast Centrifugation after microfilaments and microtubules depolymerization Fragments L929 fibroblasts Nucleus [E. Paluch, M. Piel, J. Prost, M. Bornens, C. Sykes, Biophys. J., 89:724-733] KITPC - Membrane Biophysics

7. Facts determine vesicle shape • Elastic bending energy, measured by the bending curvatures of the surface. • Osmotic pressure • Surface tension • Two components Line tension energy • Dynamics inside the cell membrane (actin filaments, microtubules). KITPC - Membrane Biophysics

8. Elastic bending energy • The Elastic Bending Energy is determined by the surface curvatures: k: bending rigidity C0: spontaneous curvature Helfrich W., Z. Naturforsch, 1973 Lipid bilayer builds  KITPC - Membrane Biophysics

9. Osmotic Potential Energy • Osmotic pressure depends on the salty density difference between inside and outside of the membrane. The osmotic pressure is proportional to density difference between inside and outside. • In the case with zero outside density, the osmotic pressure inverse proportional to the inside volume i.e. • We formulate the osmotic potential energy by KITPC - Membrane Biophysics

10. Surface Tension and Line Tension • The oscillating cell membranes can be divided into two components, with different actin and myosin concentrations. • Besides the Elastic Bending Energy, different components has different surface tension, which can be formulated by • Line tension energy involves between the two components. It can be formulated by or KITPC - Membrane Biophysics

11. Dynamic characterization of actin during the oscillation KITPC - Membrane Biophysics

12. Dynamic characterization of myosin II during the oscillation KITPC - Membrane Biophysics

13. Cell cortex: Stress due to myosin motors KITPC - Membrane Biophysics

14. Actin Myosin A mechanism for the oscillation KITPC - Membrane Biophysics [E. Paluch, M. Piel, J. Prost, M. Bornens, C. Sykes, Biophys. J., 89:724-733]

15. Total Energy • All together with the elastic bending energy, surface tension, line tension and osmotic potential 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. KITPC - Membrane Biophysics

16. Lipids Transfer • The total number of the lipid molecules is fixed and the lipids may move from one component to the other. • Suppose the nature 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 KITPC - Membrane Biophysics

17. 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: KITPC - Membrane Biophysics

18. 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: KITPC - Membrane Biophysics

19. =0 >0 <0 Membrane Phase Field Function • Introduce a phase function , defined on a computational domain , to label the inside/outside of the vesicle • Membrane : the level set KITPC - Membrane Biophysics

20.  1 =1 -1  =-1 Membrane Phase Field Function • Ideal phase field function • d: distance function • +1 inside, -1 outside • Sharp interface as ! 0 KITPC - Membrane Biophysics

21. Elastic Bending Energy in Phase Field Model • Taking • On the other hand, minimizing => KITPC - Membrane Biophysics

22. =1 =1 =-1 =-1 Component Phase Field Function • phase field function • d: distance function • +1 one component, -1 another • bending rigidity  is a function of  KITPC - Membrane Biophysics

23. Phase Field Formulations • Surface tension: where • Elastic bending energy: where • Line tension energy with • Osmotic potential energy with KITPC - Membrane Biophysics

24. Phase Field Formulations • Perpendicular of  and ? : • Tanh profile preserving of  : • Total energy: • System with gradient flow: KITPC - Membrane Biophysics

25. 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 . KITPC - Membrane Biophysics

26. Numerical Results KITPC - Membrane Biophysics

27. Numerical Results KITPC - Membrane Biophysics

28. Future work • Simulations of the breakage • More numerical simulations for examining the effect of osmotic pressure, spontaneous curvature, line tension, etc. • Coupling with fluid • Reaction diffusion of actin and myosin • Improvement of our phase field formulations • More rigorous theoretical analysis of our models KITPC - Membrane Biophysics

29. Summary • We proposed a model together with the phase field simulation to explain the oscillation of cell membrane. • Some preliminary analysis and numerical simulations have been carried out and compared with experiment findings. • The simulation results illustrate how the cell membrane interact with the interior actin dynamics, the competition of the surface tension, bending stiffness and the interfacial line tension. • More studies are underway… KITPC - Membrane Biophysics