Warning!!! -- This page contains a java applet and many images, so it may take some time to load. Please be patient. Thanks.


Actin Filament Lengths Affect Actin Network Formation

by Athan Spiros and Leah Keshet

Actin

Actin is the major component of the cytoskeleton. Actin filaments interact with actin crosslinkers to form different types of networks. One such network structure is a gel. The filaments in a gel are equally distributed and show little or no preference in their orientation. Alternatively, filaments may also form bundles. Filaments in bundles bunch together in parallel arrays. These structures not only influence the shape of the cell and the cell's mechanical behaviour, but are also responsible for such actions as cell motility and cell division.

Below is a diagram of a filopod which occurs at the leading edge of a motile cell. Both bundles (seen as spikes) and gels (part of the cell body) are present in such cells.
filopod


Actin filaments are formed from actin monomers which are proteins inside the cell. Filaments are constantly gaining and losing monomers depending on the amount of free monomers present. Approximately, 370 monomers make a filament one micron in length. Actin networks are formed when filaments bind to one another with the aid of actin crosslinkers.

polymerization


Modeling Actin Filament Interactions

When studying overall filament interactions we classify two distinct filament classes: Network (N) filaments (filaments that are bound to other filaments) and Free (F) filaments (filaments that are not bound). The following actions are then modeled:

By letting $ N(x, \theta, t) $ be the concentration of network filaments and $ F(x, \theta, t) $ be the concentation of free filaments with centers at x and orientation $ \theta $ at time t, the filament interactions can be descibed with the following system of integro-PDE's:

\begin{eqnarray*} N_{t}(x, \theta, t) & = & \underbrace{\beta_{1} F K * F + \beta_{2} N K * F}_{ \begin{array}{c} \makebox{filament} \\ \makebox{association} \end{array}} \underbrace{ - \gamma N}_{ \begin{array}{c} \makebox{filament} \\ \makebox{dissociation} \end{array}} \\ F_{t}(x, \theta, t) & = & \overbrace{-\beta_{1} F K * F -\beta_{2} F K * N} \makebox[0.2in]{} \overbrace{ +\gamma N} \\ & & \makebox[.3in]{} \underbrace{+ \mu_{1} F_{\theta \theta}}_{ \begin{array}{c} \makebox{rotational} \\ \makebox{diffusion} \end{array}} \underbrace{ + \mu_{2} F_{x x}}_{ \begin{array}{c} \makebox{translational} \\ \makebox{diffusion} \end{array}} \end{eqnarray*}

where
\[ K*F = \displaystyle \int_{-\pi}^{\pi} \displaystyle \int_{\Omega} K(\theta - \theta', x-x') F(x', \theta') d \theta' dx' \]

The terms like K*F, are convolutions and can be thought of as the expected number of free filaments that are able to interact.

Instability Analysis

By setting the above system to zero, the homogeneous steady state may be found. The homogeneous steady state indicates a gel structure because filaments are spread equally throughout the solution at all angles. The question then is when does the homogeneous steady state become unstable. To determine under what circumstances this happens, perturbations away from the steady state are studied:

\begin{eqnarray*} N & = & n_{0} e^{i k_{1} \theta} e^{i k_{2} x} e^{\lambda t} \, , \\ F & = & f_{0} e^{i k_{1} \theta} e^{i k_{2} x} e^{\lambda t} \, . \end{eqnarray*}

Linear analysis predicts that the steay state is unstable when the following inequality is satisfied:

\[ \mu_{1} k_{1}^{2} + \mu_{2} k_{2}^{2} < \frac{\left( \beta_{1} M \right)^{2}}{\gamma} \hat{K} \left( 1 - \hat{K} \right) \,, \]

where M is a measure of the total amount of actin in the system and $ \hat{K} $ is the Fourier transform of K.

As one can see, if $ \mu_{1} $ is small enough, the instability condition will be satisfied for $ k_{1} $ ; indicating that the orientation of the filaments will be unstable so that filaments will favor some orientation, and thus align. Similarly, if $ \mu_{2} $ is small enough, the instability condition will be satisfied for $ k_{2} $ ; indicating that the position of the filaments will be unstable so that filaments will cluster.

The length of the filament directly affects the diffusion rates. Polymer physics gives estimates for these rates diffusion in a semi-dilute solution. The tranlational rate of diffusion, $ \mu_{2} $ , falls off roughly as 1 over the filament length, while the rotational rate of diffusion, $ \mu_{1} $ , falls off as 1 over the filament length raised to the seventh power. Thus, increasing the filament length leads to the formation of networks other than gels.

rates of diffusion


Simulations

I ran many different simulations under various conditions (by numerically solving the system of integro-PDE's) to see what kind of networks may form. You can see what type of networks form as well as how they form when the filament length is changed by using the following applet:
The above pictures show the concentration of filaments with rays. Long, dark rays indicate big concentrations while short, light rays indicate small concentrations. The beginning of the ray indicates the position of filaments and the angle of the ray indicates the orientation of filaments. Thus each ray indicates the concentration of network filaments at that point in space with that particular orientation (simlilar to angular histograms throughout space).

Key points to recognize:



For more information, please see our technical report or send e-mail to either spiros@iam.ubc.ca or keshet@math.ubc.ca.