1. 9/23/2015 KITPC - Membrane Biophysics 1 Modeling of Actomyosin Driven Cell Oscillations Xiaoqiang Wang Florida State Univ.

2. 9/23/2015 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

3. 9/23/2015 KITPC - Membrane Biophysics 3 Membranes Cellular membranes are composed mostly of lipids. Lipid has one polar (hydrophilic) head and one or more hydrophobic tails.

4. 9/23/2015 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.

5. 9/23/2015 KITPC - Membrane Biophysics 5 Cell Oscillation These cell oscillations are driven by actin and myosin dynamics.

6. 9/23/2015 KITPC - Membrane Biophysics 6 Fragments of L929 fibroblasts L929 fibroblasts Centrifugation after microfilaments and microtubules depolymerization Cytoplast Fragments Nucleus [E. Paluch, M. Piel, J. Prost, M. Bornens, C. Sykes, Biophys. J., 89:724-733]

7. 9/23/2015 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 Line tension energy Dynamics inside the cell membrane (actin filaments, microtubules).

8. 9/23/2015 KITPC - Membrane Biophysics 8 Elastic bending energy The Elastic Bending Energy is determined by the surface curvatures: Lipid bilayer builds  k: bending rigidity C 0 : spontaneous curvature Helfrich W., Z. Naturforsch, 1973

9. 9/23/2015 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

10. 9/23/2015 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 or

11. 9/23/2015 KITPC - Membrane Biophysics 11 Dynamic characterization of actin during the oscillation

12. 9/23/2015 KITPC - Membrane Biophysics 12 Dynamic characterization of myosin II during the oscillation

13. 9/23/2015 KITPC - Membrane Biophysics 13 Cell cortex: Stress due to myosin motors

14. 9/23/2015 KITPC - Membrane Biophysics 14 A mechanism for the oscillation [E. Paluch, M. Piel, J. Prost, M. Bornens, C. Sykes, Biophys. J., 89:724-733] Actin Myosin

15. 9/23/2015 KITPC - Membrane Biophysics 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, C i 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 y i are the density values of myosin.

16. 9/23/2015 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 A 1 (t) and A 2 (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

17. 9/23/2015 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:

18. 9/23/2015 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:

19. 9/23/2015 KITPC - Membrane Biophysics 19 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  =0  >0  <0

20. 9/23/2015 KITPC - Membrane Biophysics 20 Membrane Phase Field Function Ideal phase field function d: distance function +1 inside, -1 outside +1 inside, -1 outside Sharp interface as  ! 0  1  =1  =-1 

21. 9/23/2015 KITPC - Membrane Biophysics 21 Elastic Bending Energy in Phase Field Model Taking On the other hand, minimizing =>

22. 9/23/2015 KITPC - Membrane Biophysics 22 Component Phase Field Function phase field function d: distance function +1 one component, -1 another +1 one component, -1 another bending rigidity  is a function of  bending rigidity  is a function of   =1  =-1 11  -1

23. 9/23/2015 KITPC - Membrane Biophysics 23 Phase Field Formulations Surface tension: where Elastic bending energy: where Line tension energy with Osmotic potential energy with

24. 9/23/2015 KITPC - Membrane Biophysics 24 Phase Field Formulations Perpendicular of  and  ? : Tanh profile preserving of  : Total energy: System with gradient flow:

25. 9/23/2015 KITPC - Membrane Biophysics 25 Numerical Schemes Axis-symmetric or truly 3D configurations. Spatial discretization: Finite Difference and Fourier Spectral. 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 A 1 (t), A 2 (t), actin concentrations m 0, h 1, h 2, myosin II concentrations y 0, y 1, y 2 every time step after the gradient flow of  and .

26. 9/23/2015 KITPC - Membrane Biophysics 26 Numerical Results

27. 9/23/2015 KITPC - Membrane Biophysics 27 Numerical Results

28. 9/23/2015 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

29. 9/23/2015 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…