Analysis of actin dynamics at the leading edge of crawling cells: implications for the shape of keratocyte lamellipodia

Abstract

Leading edge protrusion is one of the critical events in the cell motility cycle and it is believed to be driven by the assembly of the actin network. The concept of dendritic nucleation of actin filaments provides a basis for understanding the organization and dynamics of the actin network at the molecular level. At a larger scale, the dynamic geometry of the cell edge has been described in terms of the graded radial extension model, but this level of description has not yet been linked to the molecular dynamics. Here, we measure the graded distribution of actin filament density along the leading edge of fish epidermal keratocytes. We develop a mathematical model relating dendritic nucleation to the long-range actin distribution and the shape of the leading edge. In this model, a steady-state graded actin distribution evolves as a result of branching, growth and capping of actin filaments in a finite area of the leading edge. We model the shape of the leading edge as a product of the extension of the actin network, which depends on actin filament density. The feedback between the actin density and edge shape in the model results in a cell shape and an actin distribution similar to those experimentally observed. Thus, we explain the stability of the keratocyte shape in terms of the self-organization of the branching actin network.

Appendix: asymptotic analysis of the model equations

Scaling and non-dimensionalization

Introducing the non-dimensionalized variables x/L, γt and p/(β/γ), we can write the model equations in the case of the steady locomotion in the form:

$$ {{\partial p^ \pm } \over {\partial t}} = \mp \in {\partial \over {\partial x}}\left( {v^ \pm p^ \pm } \right) + {{p^ \mp } \over p} - p^ \pm ,\;\;\;\;\;p = p^ + + p^ - ,\;\;\;\;\;[{\rm{local}}\;{\rm{model]}} $$

(12)

$$ {{\partial p^ \pm } \over {\partial t}} = \mp \in {\partial \over {\partial x}}\left( {v^ \pm p^ \pm } \right) + {{p^ \mp } \over {\bar P}} - p^ \pm ,\;\;\;\;\;\bar P = \int_{ - 1}^1 {p\,{\rm{d}}x} \;\;\;\;\;[{\rm{global}}\;{\rm{model]}} $$

(13)

$$ v^ \pm \left( x \right) = {1 \over {1.42 \mp f'\left( x \right)}},\;\;\;\;\; \in = {V \over {L\gamma }},\;\;\;\;\;p^ + \left( { - 1} \right) = 0,\;\;\;\;\;p^ - \left( 1 \right) = 0 $$

(14)

Here, for simplicity, we keep the same notations for non-dimensional variables. In what follows, we investigate the steady-state solutions of these equations, i.e. when the time derivative is equal to zero. We also concentrate on the case of the flat leading edge, f"(x)=0. In this last case, without loss of generality we assume that v ^±=1, which can be justified by re-scaling the protrusion velocity (f"(x)=0).

Slow rate of capping

In this case (γ≪V/L, ε≫1), the scale of the actin density is defined by the ratio of the branching rate β over the effective rate of fiber disappearance due to the lateral flow, V/L. In the non-dimensional system this means that the scale of actin density is ~1/ε. Substituting this new scale into Eqs. (12) and (13) and dropping the negligible capping term (the third term on the right-hand side), we obtain the following steady-state equations:

$$ - {{{\rm{d}}p^ \pm } \over {{\rm{d}}x}} + {{p^ \mp } \over {\tilde p}} = 0,\;\;\;\;\;\tilde p = \left\{ {\matrix{ {\bar P,\;\;\;\;\;{\rm{global}}\;{\rm{model}}} \cr {p,\;\;\;\;\;{\rm{local}}\;{\rm{model}}} \cr } } \right. $$

(15)

Let us introduce the function s=p ⁺−p ^–. Adding and subtracting Eqs. (15), we obtain the equations:

$$ - {{{\rm{d}}s} \over {{\rm{d}}x}} + 1 = 0,\;\;\;\;\;{{{\rm{d}}p} \over {{\rm{d}}x}} + {s \over p} = 0 $$

(16)

for the local model. The solutions of these equations can be found easily by first solving the first equation of (16), and then the second one, and by using the conditions that s(x) is an odd function, p(x) is an even function, and p(1)=s(1):

$$ s = x,\;\;\;\;\;p = \sqrt {2 - x^2 } $$

(17)

The dimensional form of these solutions is given by Eq. (8). For the global model, adding and subtracting Eqs. (15), we obtain:

$$ - {{{\rm{d}}s} \over {{\rm{d}}x}} + {p \over {\bar P}} = 0,\;\;\;\;\;{{{\rm{d}}p} \over {{\rm{d}}x}} + {s \over {\bar P}} = 0 $$

(18)

Differentiating these equations a second time, we find:

$$ {{{\rm{d}}^2 \left( {s,p} \right)} \over {{\rm{d}}x^2 }} + {{\left( {s,p} \right)} \over {\bar P^2 }} = 0 $$

(19)

Using symmetry properties of functions s(x) and p(x), we have:

$$ p = A\cos \left( {{\pi \over {\bar P}}} \right),\;\;\;\;\;s = A\sin \left( {{x \over {\bar P}}} \right) $$

(20)

Using the condition p(1)=s(1), we find that 1/P=π/4. Finally, integrating p(x) from −1 to 1, we obtain the value of A:

$$ \bar P = \int_{ - 1}^1 {p\left( x \right){\rm{d}}x} = 2A\bar P\sin (1/\bar P),\;\;\;\;\;A = 1/\sqrt 2 $$

(21)

The dimensional form of these solutions is given by Eq. (9).

Fast rate of capping

In this case (γ≫V/L, ε≪1), singular perturbation theory can be used to investigate the local model (Eq. 12). Simple analysis shows that away from two boundary layers of width O(ε) at the spatial domain's edges, the constant "outer" solution, p ^±(x)=0.5, p(x)=1, gives the leading order approximation. Near the boundaries, re-scaling the spatial variable, x→εx, leads to a non-linear system, which cannot be solved analytically. A qualitative analysis shows that the corresponding "inner" solution for p(x) decreases from p=1 inside the spatial interval to p=O(ε) at the boundaries.

Unlike the local model, the global model (Eq. 13) is not characterized by the boundary layers and has an approximate analytical solution in this limit. Adding and subtracting Eqs. (13), we find:

$$ \in {{{\rm{d}}s} \over {{\rm{d}}x}} + \left( {1 - {1 \over {\bar P}}} \right)p = 0,\;\;\;\;\; \in {{{\rm{d}}p} \over {{\rm{d}}x}} + \left( {1 + {1 \over {\bar P}}} \right)s = 0 $$

(22)

Differentiating these equations a second time, we derive:

$$ {{{\rm{d}}^2 \left( {s,p} \right)} \over {{\rm{d}}x^2 }} = - {{\kappa ^2 } \over {\varepsilon ^2 }}\left( {s,p} \right),\;\;\;\;\;\kappa ^2 = - \left( {1 - {1 \over {\bar P}}} \right)\left( {1 + {1 \over {\bar P}}} \right) $$

(23)

Solving these equations and using symmetry properties of functions s(x) and p(x) gives:

$$ p = A\cos \left( {{{\kappa x} \over \varepsilon }} \right),\;\;\;\;\;s = B\sin \left( {{{\kappa x} \over \varepsilon }} \right) $$

(24)

Substituting these solutions into Eq. (22) and using the condition p(1)=s(1), we find that $\bar P \approx 1 - (\pi ^2 \varepsilon ^2 /8)$, κ≈(πε/2), A≈π/4 and B≈π²ε/16. Thus, the non-dimensional approximate solution of the local model in this limit is p=(π/4)cos(πx/2). The dimensional form of the solutions of the global and local models in the fast capping rates limit are given by Eq. (10).