When the cell shape is deformable, the active currents and tensions of cortical actin can induce interesting shape instabilities. Towards the end of the thesis, we propose an active mechanism of plasma membrane deformation due to active tensions that arise in cortical actin.
Active Matter
In this thesis, we provide a theoretical study of the interplay between cell shape and cortical actin dynamics and patterning. The nucleus is shown in green, microtubules in yellow and actin filaments in purple (Photo courtesy: www.cellimagelibrary.org/images/240), (b) Schematic diagram of the cell showing the nucleus and cytoskeleton.
Cell Cortex as an Active Fluid
Active Stresses and Currents in the Cortical Actin
Motor-driven motility: Action of myosin-II motors on the actin filaments can also lead to the relative movement of filaments. Advection of the actomyosin filaments can therefore be due to (a) the velocity associated with the medium that has contributions from the active tension and, (b) relative velocity due to self-propulsion of actin filaments derived from the treadmill and motor-driven motility.
Mechanisms of Dissipation : Viscosity and Friction
Gradients of the active stresses generate flow in the medium described by the hydrodynamic velocity v. The dense cortical actin network creates large friction, leading to small values of the sieve length.
Theoretical Framework : The Active Hydrodynamic Approach
Hydrodynamic Description
The coarse-grained concentration c(r, t) of actomyosin filaments is thus associated with a conservation law and is defined as. We now discuss some examples where cellular shape is important in determining the patterning of actomyosin filaments.
Patterning of Cortical Actin : Substrate Geometry
When the cellular shape is deformable, the dynamics of the shape changes of the plasma membrane are coupled to the active cortical actin. This intimate interplay between actomyosin contractility and shape changes of the plasma membrane gives rise to self-organized waves and ruffles.
Migrating and Spreading Cells
Lateral Membrane Waves and Ruffles
Some common actomyosin structures and the corresponding membrane shape at the leading edge are shown in Fig.1.9[15]. This layer exhibits undulations in membrane shape as well as actomyosin density.
Endocytosis
Active Membranes
Theme of the Thesis
In addition to the actomyosin rings, actomyosin filaments also form cables and junctions in wild-type fission yeast [1], [2] (Fig. 2.1). When the shape of the fission yeast cell is made spherical, actomyosin rings do not maintain their position stably and slide towards the poles [3].
General Framework : Curvature Orientation Coupling
Another slow variable, as described in the first chapter, is the polar orientation of actin filaments, denoted by n. We are working in the limit of high friction, so the momentum density does not correspond to the conservation law.
Patterns on the Cylindrical Cell
As mentioned in the previous section, the coefficients Λ lead to the anisotropic advective current by renormalizing v0 to vi =v0+ ΛRi.
Parameters and Units
To obtain the values of parameters in real units, the number used in the simulation must be multiplied by its appropriate dimension in units of `, τ and n. The values of other parameters chosen for the numerical integration are given in the table .
Spinodal Instability of Homogeneous phases
- Low Mean Concentration of the Actin Filaments
- High Mean Concentration of the Actin Filaments
- Numerical Integration of the Dynamical Equations
- Axi-symmetric Solutions
- Coarsening and Ring Merger
Stationary rings or cables are different from the moving rings or cables in the steady state arrangement of the orientation vector. The steady state patterns are obtained from numerical integration of the dynamic PDEs as before. The speed of the moving rings or cables is set by vθ,z to lowest order.
Nucleation and Growth
Assuming that the concentration is uniform at c0, this nucleation leads to an increase in area A. The aster size is completely fixed by A, while the dimensions of the ring or cable segments are fixed by A and the width. Ring/cable segment: − The width of the ring (cable) segment is determined by the anisotropic Pecl´et length vD. For smaller values of R (below buckling in (a)), the ring segment completely encircles the cylinder, its width now set by the A/2πRand "energy" functional.
Helices on the Cylinder
We demonstrate that the inclusion of such dependence will lead to the slope of segments differing from ψ = 0 (cable segment) and ψ = π2 (ring segment) where ψ is the angle between the segment and the long axis of the cylinder . The width of the tilted segment w(ψ) is determined by a balance of the net current J·mˆ⊥, where mˆ⊥ is a unit vector perpendicular to the boundary. Cylindrical shape of the cell requires the parameters to be a function of cos 2ψ due to ψ →π−ψ →π+ψ → −ψ symmetry.
Summary and Future Directions
We will be interested in classifying the steady-state patterns of acto-myosin filaments on these curved surfaces. Before proceeding with the analysis of steady-state models of acto-myosin filaments in different geometries, we present the general dynamical equations for the concentration and polar orientation of filaments proposed in the last chapter. As stated earlier in this chapter, we focus primarily on axisymmetric patterns of active polar filaments on the surface of the sphere.
Rings on Sphere
Rings as Unstable Solutions
As a result of the anisotropy induced by the curvature-orientation coupling, we obtained rings, cables and asters as steady state patterns on the surface of a cylindrical cell. On the other hand, if the ring is formed slightly away from the equator, it slides towards the nearest pole (Fig. After obtaining the rings as unstable solution on the surface of the sphere, we now follow the dynamics of the ring as it moves on the friction surface substrate of spherical geometry.
Dynamics of Acto-myosin Rings
The center of the ring (green dotted line) has angular coordinate θ0(t) and slides in the direction indicated in the figure. b) An enlarged version of the same ring, showing the unit (facing outwards). Angular velocity of the ring = Mobility×Force (3.12) In the case of a ring perfectly placed on the equator, the total currents at both edges are the same and balance each other. However, when the ring is moved slightly away from the equator, the total currents at the two edges are no longer the same and a net force acts on the ring, resulting in the ring shifting poleward.
Asters on Sphere
Mechanical Model
Here we only report the force density characteristics fr that we use to write (and later verify) a simple mechanical model for the ring dynamics. The direction of the contractile force density due to the actomyosin ring is shown in the figure. In the next chapter we compare this expression with the experimental trajectories of the actomyosin rings on spherical Fission yeast cells.
Other Geometries
Cone
Using similar assumptions about the nature of contractile force density, we now write down the dynamics of a ring placed on the surface of a cone (Figure 3.13). ) will read.
Saddle-shaped Geometry
Composite Shapes : Examples and Predictions
Endocytic Buds
When the ring reaches the neck, it will stabilize and may lead to tightening as a result of the active contractile force created in the ring.
Shape of the Dividing Cells
In the previous chapter, we found that cell shape plays an important role in determining the stability of steady-state models of active polar filaments. For successful cell division, precise spatiotemporal regulation of the positioning, assembly and contraction of actomyosin rings is required. Continuing with this, we investigate the role of cell shape and active mechanical stresses in the positioning and stable contraction of acto-myosin rings assembled in fission tip cells [3].
Stability of the Acto-myosin Rings in Fission Yeast Cells
In mid1p protein-defective cells, acto-myosin rings, if formed at the hemispherical end caps, slide toward the poles (Fig. due to the local spherical geometry, they were not stable in position and slipped. The driving force for ring contraction has been identified to be acto-myosin contractility [3], [4].
The Mechanical Model
Predictions and Comparison with the Experiments
This can be solved to give the time dependence of the angular position of the ring. Ring angular position versus time curves for 15 cells without this time shift are shown in Figs. These images show that the fluorescence intensity of Myosin II per unit length of the ring and.
Discussions and Future Directions
We verify several predictions of this model, in particular the scaling of the ring slip time with cell radius. If the difference between the two is significant, it will imply that the septation itself affects the recruitment and turnover of the components of the actomyosin ring. We begin by citing some examples of the actomyosin and membrane waves and ruffles observed in experiments.
Existing Theoretical models
Curvature-based Models
These models use the curvature-sensing properties of proteins that regulate actin dynamics at the plasma membrane (eg, actin nucleators or actin polymerization proteins) and membrane proteins [8], [9]. This creates a local indentation in the membrane that causes segregation of the activators away from the drop zone. Similarly, in [9], density fluctuations and thermal diffusion of curved actin polymerization activators together with membrane shape dynamics lead to waves, while the activators themselves are activated in response to an external stimulus.
Reaction-Diffusion based Models
The undulatory response in these models is crucially dependent on the sign of spontaneous curvature, which is either favored or induced by activator proteins. To understand the spontaneous formation of actin and membrane waves and membrane instability, we now propose a coarse-grained model for the coupled shape dynamics of plasma membrane and cortical actin for spreading and motile cells. As explained in previous chapters, the form of the dynamical equations is determined by generic symmetry arguments.
Theoretical Framework : Coupled dynamics of Plasma Membrane and Corti-
Symmetries and the Dynamical Equations
- Dynamics of Filament Concentration
- Dynamics of Filament Orientation
- Dynamics of Membrane Shape
The pressure of actin nucleators on the inside of the membrane gives rise to the formation of cortical actin, leading to a well-defined 'inside' and 'outside' of the membrane. The dynamic equations should be invariant under the simultaneous rotations of the local membrane normal N and the three-dimensional polar orientation vector P (Fig. Active deformation of an active composite membrane 86. b) Schematic diagram of an exploded part of the leading edge of the membrane, showing the symmetry of the system during simultaneous rotation of the membrane normal.
Discussion of Coupling to the Membrane Slope
The parameters µ1,2,3 are phenomenological parameters that appear in the dynamic equations as the membrane slope coefficients. At first glance, the dependence on the membrane slope via terms such as p · ∇h seems to break rotational invariance. The common point in all these examples is the broken rotational invariance due to the presence of an external direction, which in our case is caused by actin-based cell propulsion on a substrate.
Mechanisms of Instabilities
Coupling between handc: Coupling between height and concentration of horizontal actin filaments occurs via an active current proportional to the height gradient and the spontaneous curvature terms included in the parameters µ1 and κ1, respectively. The fact that the current for actin filaments is proportional to the height gradients introduces interesting new and lower instabilities and can lead to membrane instabilities or waves of actomyosin concentration, as well as to membrane height with increasing amplitudes.
Active Membrane
Instabilities of the Plasma Membrane and Cortical Actin
Shape and Polar Order
- Low Mean Concentration
- High Mean Concentration
This instability is expected to drive the membrane to form tubules separated by a distance given by the inverse of qmax−1 and is labeled in the phase diagram shown in Fig. This instability is expected to drive the membrane to develop ridges parallel to the direction of ordering. Growth rates when fastest growing wave vector is (a) perpendicular and (b) parallel to the direction of polar order.
Phase Diagram
Shape and Acto-myosin Concentration
The phase boundaries are obtained from linear stability analysis as explained in the text, and the corresponding phases are predicted from the nature of dispersion relations. In the former case, the instability competes with the diffusion coefficient to build, while in the latter case, instability competes with the surface tension Σ. These instabilities rise as they grow and suggest traveling waves of the actomyosin concentration and membrane height, a situation realized in various experiments described at the beginning of this chapter.
Current and Future Directions
- Direct Numerical Simulations
- Travelling Actin Waves
- Membrane Buckling Under Horizontal Stress
- Active Elastomeric Membrane
This treatment is valid for time scales larger than the time scale of cortical actin remodeling (typically on the order of 10s). Using the above expressions for the differential operators, we obtain the form of the dynamic equation for the concentration as Of the three modes, one has a nonzero term of zeroth order and does not become unstable.