Mark Chaplain, The SIMBIOS Centre, Division of Mathematics, University of Dundee, Dundee, DD1 4HN

1. Mathematical modelling of cell cytoskeleton biomechanics and cell membrane deformation Mark Chaplain, The SIMBIOS Centre, Division of Mathematics, University of Dundee, Dundee, DD1 4HN SCOTLAND chaplain@maths.dundee.ac.uk http://www.maths.dundee.ac.uk/~chaplain http://www.simbios.ac.uk

2. Collaborative work Dr. Angélique Stéphanou, Dr. Philippe Tracqui, Laboratoire TIMC-IMAG, CNRS UMR 5525, Equipe Dynacell, 38706 La Tronche Cedex France “A mathematical model for the dynamics of large membrane deformations of isolated fibroblasts” Bull. Math. Biol. 66, 1119-1154 (2004)

3. Talk Overview • Biological background • Examples of cell migration • Model derivation • Linear stability analysis • Numerical computations • Application to chemotaxis • Conclusions

4. Schematic cell Figure 1. Schematic representation of the cell which illustrates the phospholipid bilayer, the cell membrane and the interconnected network of F-actin filaments which forms the actin cytoskeleton. The cell body is assumed to be confined in a circular area with radius R0. The cell cortex where remodelling of the actin cytoskeleton mainly occurs corresponds to the area bounded at one side by the outer boundary of the cell body and at the other side by the cell membrane. The width of the cell cortex in any angular direction θ is given by L(θ).

5. Figure 2. Videomicrograph of non-migrating L929 fibroblasts observed with phase contrast microscopy. This videomicrograph shows the most typical morphologies exhibited by this type of cell at their resting state (namely a non-migrating state). Fibroblasts typically present ‘starry’ morphologies involving from 2 to 4 thin membrane extensions which are more often homogeneously distributed around the cell body.

6. Figure 3. Spatio-temporal representations of the cells (cell polarity maps) which illustrates a variety of typical cell morphologies observed experimentally, with cells presenting, respectively, 2, 3 and 4 simultaneous protrusions each; the protrusive directions usually remain located along one axis for significantly long time periods (up to 12 h).

7. Cell migratory response to soluble chemicals: CHEMOTAXIS Aggregation of Dictyostelium amoebae towards a cAMP point source. Movie produced by G. Gerisch, Max-Planck-Institut fur Biochemie, Martinsried, Germany. A single cell moves chemotactically towards a cAMP point source. Movie produced by G. Gerisch, Max-Planck-Institut fur Biochemie, Martinsried, Germany.

8. No ECM with ECM ECM + tenascinEC & Cell migratory response to local tissue environment cues Non-diffusible molecules bound to the extracellular matrix HAPTOTAXIS

9. Chemotaxis TAF Receptor Integrins Haptotaxis Extracellular Matrix The Tissue Response Unit

10. Angiogenesis

11. The Role of Angiogenesis in the Metastatic Cascade

12. Cell migration in wound healing

13. The Individual Cancer Cell “A Nonlinear Dynamical System”

14. Schematic cell Schematic representation of the cell which illustrates the phospholipid bilayer, the cell membrane and the interconnected network of F-actin filaments which forms the actin cytoskeleton. The cell body is assumed to be confined in a circular area with radius R0. The cell cortex where remodelling of the actin cytoskeleton mainly occurs corresponds to the area bounded at one side by the outer boundary of the cell body and at the other side by the cell membrane. The width of the cell cortex in any angular direction θ is given by L(θ).

15. Modelling hypotheses • Sol/gel transition of actin regulated by local • calcium concentration • Actin polymerisation in neighbourhood of • membrane causes protrusion – Brownian ratchet • mechanism • Myosin I + actin = propulsion of filaments towards • membrane • Pressure-driven protrusion

16. Model Variables Stress σ in the cytoskeleton - mechanical properties F-actin concentration a – chemical dynamics of cytoskeleton Membrane deformation L - linked to actin dynamics

17. Stress Equation viscous stress elastic stress contractile stress osmotic stress

19. Actin Equation ac = F-actin concentration differentiating states of polymerization and depolymerization of actin

20. Membrane Deformations • Mechanical forces acting on cell membrane: • friction force between membrane and substrate, ΦV = Φ ∂L/∂t • constant protrusive force P due to hydrostatic pressure within cell • an active force σ(a) which depends on local concentration of actin • a membrane curvature-dependent force τΚL , where τ is a constant characterizing the membrane tension and ΚL is the curvature

21. Membrane Equation where L = L(θ) denotes the radial extension of the cell cortex

22. System of Equations

23. Curvature

24. Model Simplification • Main modelling assumption: • 1D approximation • actin dynamics are restricted to a circle of radius r* • no displacement of actin in radial direction, no contraction of cytogel in this direction

30. Membrane and actin dynamics are coupled by means of the following equation describing the conservation of actin:

31. Simplified Equations

32. Linear Stability Analysis Linear stability analysis is carried out in order to determine the conditions required for the model parameters to generate self-sustained oscillations of the membrane – destabilization of uniform steady-state through a Hopf bifurcation

33. Steady State

34. Dispersion Equation The dispersion equation found from the solution of det(A) = 0, is given by:

35. Figure 4. Conditions required to satisfy the Routh–Hurwitz criteria.

36. Static Membrane Deformations Replace retraction force σ(a) by γ(θ): α and m control amplitude of deformation and mode of deformation respectively

37. Figure 5. Potential cell morphologies obtained for various modes of deformation (m = 1, 2, 3, 4, 5, 10) of the function γ (θ) representing a spatial modulation of the F-actin filament stiffness. In each graph, the dotted curve represents the initial cell shape [circular shape L(θ, t = 0) = L0] and the most external curve the analytical solution of the equation for the membrane deformations [equation (11)] taken for τ = 0 (passive membrane). The two internal curves correspond to the numerical solutions of that same equation for two different values of the membrane stiffness coefficient, namely τ = 0.05 and τ = 0.1.

38. Dynamic Membrane Deformations Numerical computation of equations: Crank-Nicholson finite differences, relaxation scheme; periodic boundary conditions; initial conditions random perturbations of F-actin concentration around homogeneous steady-state in circular morphology.

39. Numerical Simulation Results L(θ,t) a(θ,t) Figure 6. Simulation results of the spatio-temporal evolution of the cell membrane deformations (left side), together with the corresponding actin distributions (right side). rotating wave of deformation;

40. a(θ,t) L(θ,t) symmetrical pulsation asymmetrical (or alternating) pulsation

41. Figure 7. Simulated cell membrane deformations (asymptotic state associated with the top graph of Fig. 6). Snapshots are taken every 200 iterations (_t = 0.2). The counterclockwise wave of deformation has a periodicity of about 2.8 normalized time units (sequence to be read from top to bottom).

42. Figure 8. Simulated cell membrane deformations (asymptotical state associated with the middle graph of Fig. 6). Snapshots are taken every 200 iterations (_t = 0.2). The pulsation of the cell deformation has a periodicity of about 2.2 normalized time units (sequence to be read from top to bottom).

43. Figure 9. Simulated cell membrane deformations (asymptotic state associated with the bottom graph of Fig. 6). Snapshots are taken every 200 iterations (_t = 0.2). The alternating pulsation of the cell deformation has a periodicity of about 2.8 normalized time units (sequence to be read from top to bottom).

44. Figure 10. Videomicroscopy sequence of a L929 pulsating fibroblast. The time interval between two consecutive pictures is about 2 min (sequence to be read from top to bottom).

46. Figure 11. Simultaneous plots of actin distribution and corresponding membrane deformations in upper graphs. In the lower rectangular graphs, the associated tangential displacements of actin are displayed. These four graphs correspond to the snapshots 1, 3, 4 and 5 of the sequence of Fig. 7 associated with the normalized times 5, 5.4, 5.6 and 5.8 respectively.

47. Figure 12. Upper graph: evolution over time of the cell membrane deformations and of the associated actin distribution (higher amplitude curve) in normalized units, for a given protrusive direction. Lower graph: simultaneous evolution of the membrane deformations for two protrusive directions at 45◦ from each other.

48. Figure 13. Schematic representation of a migrating cell exhibiting a characteristic domelike shape where the thickest part represents the cell body. From the mechanical point of view, intercalation of molecules in the membrane is responsible for cell morphological instabilities.

49. Figure 14. Schematic diagram exhibiting the two-step mechanism of migration, with first the membrane extension along the migration direction and second the cell body translocation, i.e., the displacement of the cell body at the new position of the cell geometrical barycentre. This second step occurs when the adhesion force becomes able to overcome the tension force of the actomyosin fibres in the cortex.

50. modification of membrane tension coefficient τ in presence of a chemoattractant, concentration C where Λ(C) is a function which characterizes the sensitivity of the cell to the extracellular factor