Tethered Ratchet Model

  exp / Eq. 3-2

where  · · is the free polymerization velocity and   ·  is the depolymerization velocity.

3.4.2 Tethered Ratchet Model 

The elastic Brownian ratchet model was developed to account for the finding that the velocity of an actin propelled object is invariant with the size of the ob-ject. It is an extension of the classic thermal ratchet [62] model, developed to include this additional aspect. However, another incompatible observation arose, namely the actin filament appeared to be attached at the Listeria mem-brane or bead surface. This was shown in a number of experiments using elec-tron microscopy [6], [64] or Listeria diffusion measurements [42] and (pull-off) force measurements [56]. These observations fit very well with biochemical models on ARP2/3 mediated polymerization, which proposed that ARP2/3 forms a complex with surface bound nucleation promoting factors (NPFs) [22]

to initiate actin polymerization. The problem in the Elastic Brownian Model is that it remains unclear how the filaments can insert monomers and generate force when they are attached to the surface. The “Tethered Ratchet” model pro-posed by Mogilner and Oster [41] solves this problem by assuming that the fi-laments attach to the bacterial surface only transiently.

The model considers two population of filaments: a) attached filaments, that are stretched due to the forward motion of the load and hence resist forward mo-tion by imposing a force ; b) dissociated, growing filaments that are com-pressed and working against the load each with an elastic force , see Figure 3-4 for a graphical illustration.

Figure 3-4 Sketch of the Tethered Brownian Ratchet model. Working filaments (curved) are formed when attached filaments dissociate and with rate  and are capped with rate  . Attached filaments are generated with nucleation rate . Force balance: the polymerization ratchet force, , generated by the working fi-laments is balanced by the force of attachment, , and load force, .

The model consists of three principal equations that describe a) the number of attached/detached filaments at the surface, b) a force balance equation and c) equations that connect the force with the filament dissociation rate. In the fol-lowing these relations will be written down and briefly explained. The complete derivation can be found in the publication by Mogilner and Oster [41].

a) Number of filaments near the surface: The rate of attached filament for-mation is  / n , where n is the nucleation rate of attached fila-ments, the dissociation rate and the number of filaments attached to the sur-face. The working filament formation rate is  / , where is the capping rate of the working filaments, and are the number of the work-ing and attached filaments, respectively. The solutions of these equations are

/ , / . Eq. 3-3

We will see in the following that the number of attached filaments ( ) and detached filaments ( ) depend on the propulsion velocity of the load ( ).

b) Force balance: With the forces exerted by the attached and working fila-ments and , respectively, and  the load force the force balance reads

3.4 Force Generation of Actin Filaments: Microscopic View

25

· · , ^{Eq. 3-4}

c) force-dissociation relation: The dissociation rate of filaments from the bead surface depends on the force acting on the link: pulling on a bond lowers the activation barrier and enhances dissociation. This has been observed expe-rimentally by [55] who showed that the comet tail could be detached from the bead by tearing at it. The actual relation between the dissociation rate and the force depends on the form of the potential associated with the link, but in many cases it can be approximated by an exponential relation [65].

, exp / ^{Eq. 3-5}

where is the characteristic strength of an attachment bond. If the load moves at a velocity , then at a time after an attachment the force applied to the molecular link is   . Thus the force applied to the attachment link is velocity dependent. By a similar argument the dissociation rate is velocity

de-pendent, _, exp / .

Characteristics of the Model: Substituting the force balance Eq. 3-4 into the force-velocity equation obtained from the Elastic Brownian Model (Eq. 3-2) leads to:

  exp / / / ^{Eq. 3-6}

The number of the working filaments and attached filaments are are un-known and depend on the velocity of polymerization, as depend on the veloc-ity (Eq. 3-5) and is connected with and (Eq. 3-3). To overcome this prob-lem a velocity scale is introduced and Eq. 3-6 can be rewritten. Roughly speaking, when the polymerization velocity exceeds a certain velocity

then the filament attachment links are deformed, and the bonds break faster than with their free dissociation rate. Hence decreases and increases in this case.

If then the bonds break with their free polymerization rate. Eq. 3-6 can be rewritten the dimensionless form with / ,  the work done per fila-ment in breaking an attachfila-ment, the dimensionless free polymerization

ve-locity, the dimensionless free depolymerization veve-locity, the work done per working filament on the load:

exp ^{Eq. 3-7}

This equation can be used to analyze the Brownian ratchet model. Here is a function that describes the velocity dependence of the dissociation rate and the attachment force   . It has the following behavior: a) For slow ment,  1, _, ; that is the dissociation rate is equal to the free dissocia-tion rate, and   ; i.e. the attachment force is proportional to the velocity. b) For fast movement 1, _, /ln  ; the dissociation rate increases with the velocity in a sub linear way, and   ln  ; the detachment force in-creases logarithmically with the velocity. With the parameters of ARP2/3 me-diated polymerization shown in appendix 8.1 Eq. 3-7 can be numerically solved.

The result is illustrated in Figure 3-5A. For small velocities Eq. 3-7 is a decreas-ing function because for slow movement the dissociation is constant, whereas the force of attachment that resists the working filaments is proportional to the velocity. For faster movement, the force of attachment increases with velocity more slowly than the dissociation rate. Therefore, Eq. 3-7 is a slowly increasing function of the velocity.

Mogilner and Oster predict that the force-velocity relation for a bead or Lis-teria computed with Eq. 3-7 is biphasic. At small loads, the velocity decreases very fast, while for greater loads the velocity decreases more slowly. The reason is, that for almost zero load ( ), when the object moves in the fast regime, the attachments break quickly and the resistance from the attachment links is small.

For a small increase in load the bacterium is slowed which increases the drag as the filaments stay attached longer. This positive feedback decreases the velocity very quickly as the load grows. At still larger loads the object moves in the slow regime where the attachment links break at their free dissociation rate. In this case a positive feedback is not observed. As a result, the velocity decreases not as strongly for larger loads.

3.4 Force Generation of Actin Filaments: Microscopic View

27

Figure 3-5 A) The right-hand side (RHS) of Eq. 3-7 (magenta) is displayed as the function of the dimensionless velocity, . The left-hand side corresponds to the straight line. The intersection gives the steady-state value of , and shows that ac-tin propelled Listeria move in the “fast” regime (see text) . The non monotonic shape of the right-hand side accounts for the biphasic behavior of the load-velocity curve in B): The force-velocity curve for Listeria. The solid curve corresponds to the parameter values in appendix 8.1. The dashed curve corresponds to a threefold increase in nucleation rate over the solid curve, and illustrates the effect of fila-ment density on the load-velocity behavior. Both graphs were taken from [41].

A B