Active cellular protrusion: continuum theories and models |
213 |
M |
Position |
M |
Position |
Fig. 10-3. Monomer concentration vs. position (dots – free monomers; dashes – bound monomers; solid – total monomers).
Special cases of network–membrane interaction: polymerization force, brownian and motor ratchets
In recent years, the concept of “polymerization force” has gained traction as a putative explanation for cellular protrusion. While the concept is often used in a vague manner, it has been formalized on more solid physical grounds in two interesting models: the Brownian ratchet model (see Mogilner and Oster, 1996) and the clamped filament elongation model (Dickinson and Purich, 2002). In essence, these models are special cases of network–membrane interactions in that they rely on the hard-core repulsion between monomers and membrane as an interaction potential. For instance, Eq. 10.14 is identical to that commonly given for the force produced by a Brownian ratchet (see for example Howard, 2001), where δ is taken to be the incremental lengthening of the polymer by addition of a monomer.
Network–network interactions
The basic idea behind a network swelling stress is similar to that of a membrane– network disjoining stress and we shall follow the approach of the previous section. One begins with the assumption that there exists a repulsive force between actin monomers, free or bound. Again, for free (G-actin) subunits, this has no dynamical consequences, as redistribution occurs freely in the cytosol. However, once subunits are sequestered into the cytoskeleton by polymerization, the repulsive force has dynamical consequences because it endows the cytoskeleton with a macroscopic stress. Under these conditions, one can intuitively perceive how the energy of the chemical process of polymerization can be transformed into expansion work.
Fig. 10-3 illustrates this principle. On the left, polymerizing activity is moderate; wherever there is a mild excess of monomers bound in filaments, free monomers are driven out by the repulsive interaction. The end result is that the total concentration of monomers (bound and unbound) varies little.
The amount of variation is determined by the relative magnitudes of the time scale of thermal-driven diffusion of monomer into regions of excess polymer τdiff = l2/D and the time scale of force-driven diffusion out of regions of excess polymer τforce = l2/[D(ψ/ kB T )] (where l is the length scale of the region, D the monomeric diffusion coefficient, and ψ the repulsive potential, and where we have used Einstein’s relation between viscosity and diffusion coefficients).
214 M. Herant and M. Dembo
If, however, polymerization becomes intense – for instance due to uncapping of filaments – it can drive the number of monomers sequestered in filaments above that of the background, and we have the case depicted on the right of Fig. 10-3. The local free monomer concentration goes near zero but cannot be negative, so that the total monomer concentration has a significant bump. A monomer moving around therefore sees a repulsive potential force in the bump that is higher than the baseline away from the bump.
Our formal development here will approximately parallel that of the network– membrane interaction problem in the previous section. We assume a pairwise repulsive potential force (potential φ) between actin monomers either free or part of a filament. The total force exerted on a monomer M is therefore the result of a sum on all other monomers Mi :
FM Mi |
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∂ φM Mi |
− |
ψ n . |
(10.18) |
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i |
= − i ∂ rM Mi |
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Here ψ n is the part of the potential that comes from fixed monomers sequestered in filaments. Generally, this will be the dominant contribution wherever network is highly concentrated, as free monomers will naturally diffuse away and lower the free monomer concentration (see Fig. 10-3). Just like the case of membrane–network interactions, we have a Boltzmann factor,
[Mfree(bump)] |
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= |
exp |
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ψ n (bump) |
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(10.19) |
[Mfree(baseline)] |
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− |
kB T |
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where we have assumed that the network concentration outside of the “bump” is so low as to make ψ n (bump) ψ n (baseline) 0. In this picture, ψ n (bump) is the work of bringing one free monomer from baseline concentration into bump (Fig. 10-3).
Again, we have the chemical reaction
Mfree Mbound, |
(10.20) |
which goes to the right if [Mfree(x)] > [Mcritfree] and to the left if [Mfree(x)] < [Mcritfree] where, as before, [Mcritfree] is the critical free monomeric concentration above which
free ends of polymers are lengthened by monomer addition.
In regions of very high network density, ψ n is large, and by Eq. 10.19 this leads to [Mfree] < [Mcritfree]. This drives Eq. 10.20 to the left (depolymerization) so that one can say that such a region cannot be created by polymerization (although external or con-
tractile forces could compress network above such a threshold). It is therefore clear that the highest network concentration achievable by chemical network polymerization is that for which [Mfree(x)] [Mcritfree], and that therefore, the highest achievable network repulsive stress per monomer is:
ψ n |
= |
kB T ln |
[Mfree(baseline)] |
. |
(10.21) |
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Mcritfree |
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Eq. 10.21 makes it clear that in general, the stress contribution by polymerizing a single monomer into the bump is at most of order 10 kB T . Let VM be the volume of