Based on the crystal structure of the formin–barbed end complex (Otomo et al., 2005
), we propose a second mode of transition from the closed to the open state, which we will refer to as the screw mode. This mode involves screwlike movement of the FH2 dimer around the barbed end, as illustrated in c. Within the screw mode, each FH2 bridge moves along the short-pitch actin helix. As a result, one bridge undergoes a transition from actins 1 and 2 to actin 1 and opens its post site for binding of a new actin monomer, whereas the second bridge moves from actins 2 and 3 to actins 1 and 2 ( c). The twist direction of the short-pitch actin helix is opposite to that of the long-pitch helix (Holmes et al., 1990
). Therefore, rotation of the FH2 dimer in the screw mode is opposite to that of the stair-stepping mode. Transition from the closed to the open state within the screw mode is coupled to rotation of the FH2 dimer with respect to the bulk of the filament by ~166°, which is equal to the angle between two sequential monomers in the short-pitch actin helix ( c).
For the following treatment, we refer to rotation of the formin dimer with respect to the filament bulk as torsion and define it as positive for the stair-stepping mode and as negative for the screw mode. According to this definition, the torsion angle of one stair-stepping step is SS
≈ 14°, whereas that of the screw step is SCR
≈ −166°. In the case that is relevant for this study, the formin cap is fixed on a membrane or substrate and cannot freely rotate as required by the helical structure of the filament. In this case, the rotation of formin relative to the filament bulk is substituted by the torsional deformation of the actin filament (see supplemental material, available at http://www.jcb.org/cgi/content/full/jcb.200504156/DC1
). This deformation will be referred to as the torsion strain (τ).
The most prominent difference between the two modes is in the opposite directions of FH2 dimer rotation with respect to the actin filament bulk. We propose that, as a result of this feature, torsion strains that are produced by the two modes can mutually compensate, and, hence, processive capping consisting of an optimal combination of the two modes is largely free of the rotation paradox. Indeed, a straightforward consideration shows that torsion strain produced by a sequence of ~12 stair-stepping steps that require the relative actin filament–formin rotation by ~14° each can be compensated for by one screw step that generates rotation in the opposite direction by approximately −166°. Hence, processive capping that consists of repeated sequences of a mean of ~12 stair-stepping steps for each screw step will not lead to a persistent one-directional formin rotation and to the related problems of stress–strain accumulation within growing actin filaments. Stress–strain relaxation by periodic switching between the stair-stepping and screw modes will prevent filament supercoiling. Thus, a combination of the stair-stepping and screw modes could enable an essentially torsion stress–free polymerization of long actin cables in vivo (Sagot et al., 2002
; Yang and Pon, 2002
) and could explain the absence of relative rotation of the formin cap and actin filament, as was observed in vitro (Kovar and Pollard, 2004
To substantiate these conclusions, we performed an elastic energy analysis of the process of actin filament polymerization upon processive capping by a formin dimer (see supplemental material for mathematical details and a more expanded analysis). We considered an elastic model of a system that consists of an actin filament undergoing polymerization upon processive capping by a formin dimer. We assumed that at each step of polymerization, the system can choose between two alternative modes (the stair-stepping and screw mode), contributing 14 and −166°, respectively, to the torsion strain that accumulated as a result of previous steps. The essence of the analysis was to calculate and compare the energies of the two alternatives. Using a deterministic approach, we stated that the mode of the polymerization step undertaken by the system is that of the smaller energy. By performing this analysis step by step, we determined the optimal regime of polymerization and accompanying variations in the torsion strain and elastic energy. A more general probabilistic treatment of processive capping steps is expected to give qualitatively similar results and will be published elsewhere.
To be specific and make quantitative estimations, we used parameters corresponding to the experimental design of Kovar and Pollard (2004)
. The system consists of a growing actin filament, whose barbed end is capped by a formin molecule that is attached to a substrate. When the filament reaches a length of ~1 μm, its pointed end is also fastened to the substrate through an N
-ethylmaleimide (NEM)–treated myosin II molecule. After the pointed end is immobilized, free rotation of the filament is prevented, and further polymerization gives rise to torsion stresses within the system. In general, these stresses are shared between the actin filament on one hand and formin and NEM-myosin on the other. To estimate the maximum possible torsion stresses, we assumed formin, NEM-myosin, and their links to the substrate to be much more rigid than the actin filament, whose rigidity is characterized by two elastic moduli: the torsion modulus C
≈ 8 × 10−26
J × m (Tsuda et al., 1996
) and the bending modulus K
≈ 3.6 × 10−26
J × m (Gittes et al., 1993
; Isambert et al., 1995
). A general case of stress distribution is considered in the supplemental material. Although performed for a particular experimental design, the results of the analysis also apply qualitatively to in vivo actin–formin systems such as intracellular actin cables (Yang and Pon, 2002
; Kobielak et al., 2004
Variations of the torsion strain and elastic energy within the optimal regime of processive capping that was determined by our analysis (see supplemental material) are presented in . We found that immediately after the pointed end is immobilized, torsion stresses start to develop within the system, and the first steps of polymerization proceed in the stair-stepping mode, resulting in accumulation of a positive torsion strain. When the strain reaches the level of ~83°, which corresponds to six stair-stepping steps, the screw mode becomes energetically more favorable. The following step is performed in the screw mode, which relaxes the previously accumulated strain and induces a negative strain of approximately −83°. Further processive capping consists of repeating sequences of ~12 stair-stepping steps followed by one screw step. Within each sequence, the torsion strain varies between approximately −83° and ~83° ( a). Periodic relaxation of the deformation, which does not allow the absolute value of the torsion strain to exceed ~83°, prevents supercoiling. Indeed, the torsion strain that must be reached in order to generate supercoiling can be determined through the torsion (C
) and bending (K
) moduli of the actin filament, according to the relationship τ*
, where α = 8.98 is a numeric constant resulting from elastic analysis (Landau and Lifshitz, 1959
). By using the aforementioned values of elastic moduli, we obtain that τ*
≈ 230°. Therefore, within the suggested regime of processive capping, the torsion strain always remains smaller than the critical value τ*
, and supercoiling is not expected, which is in agreement with the experimental observation.
Figure 2. The optimal regime of processive capping, consisting of repeated sequences of ~12 stair-stepping steps followed by one screw step. (a) The torsion strain as a function of the number of polymerization steps after the beginning of elastic stress (more ...)
The torsion elastic energy that accumulates within the system changes periodically, and the amplitude of this variation decreases slowly ( b). For the parameters that we have used, the maximum value of elastic energy is ~20 kB
T (where kB
T ≈ 0.6 kcal/M is the product of the Boltzmann constant and absolute temperature). Such a torsion energy is close to the energy of protein–protein interaction in the actin–formin system (Kozlov and Bershadsky, 2004
) and, hence, appears to be feasible. At the same time, this torsion energy is high enough that it could significantly influence the effective actin–FH2-binding energy underlying the kinetics of stair-stepping and screw modes of processive capping.
The two proposed modes of processive capping differ substantially in terms of the intermediate conformations that the system has to pass on its way from the closed to the open state. These largely unknown factors may determine the relative kinetics of the two modes, whose detailed analysis will be performed elsewhere. In this study, we briefly discuss the most important related issues.
The screw step can be divided into substages. At the first substage, the torsion strain that accumulated during the sequence of stair-stepping steps relaxes to zero, and the system reaches an unstressed state of vanishing elastic energy. In this intermediate state, the filament barbed end is turned with respect to formin in such a way that terminal actin subunits cannot bind to the formin dimer. To reassociate with the FH2 dimer and absorb a new actin monomer, the barbed end has to rotate with respect to formin by another ~83°, which constitutes the next substage of the screw step. This rotation, which is followed by binding, results in accumulation of the torsion elastic energy but allows for release of the actin–formin- and actin–actin-binding energies. The resulting overall energy balance of the screw step is negative, meaning that it is favorable energetically. At the same time, before binding energy is released, the system has to overcome an energy barrier that is produced by the accumulating torsion strain. Note that the energy barrier of the same origin also exists at a stair-stepping step of processive capping. Indeed, every stair-stepping step is accompanied by an ~14° torsion strain and accumulation of the corresponding elastic energy. These elastic energy barriers may contribute to or even determine the kinetics of screw and stair-stepping modes. For the stair-stepping mode, the torsion strain and, hence, the elastic energy barrier are smaller; thus, the related rate limitation should be less significant.
Another origin of kinetic limitations of the two modes of processive capping is determined by the energy barriers that are related to transient detachment of formin bridge elements from the filament barbed end. The stair-stepping mode assumes that in the course of transition from the closed to the open state, only one FH2 bridge detaches from the barbed end, whereas the second bridge remains bound and does not move ( b). Thus, the stair step requires the dissociation of two actin–FH2 interaction sites (Otomo et al., 2005
). The screw mode most probably requires simultaneous detachment of the two bridges from the four actin-binding sites in the closed state, because the tethers connecting the two bridges within an FH2 dimer appear to be not long enough to allow for sequential detachment of the bridges. This is followed by slipping of the two bridges around the filament toward the new position in the open state, where they are again trapped in the corresponding binding sites. The decision of the system to adopt one mode over the other must be strongly influenced by strain. In the absence of strain, the release of only a single bridge element should be significantly more probable than the simultaneous release of both bridge elements. Thus, under stress-free conditions (e.g., when the system is free in solution), stair stepping should be the more probable mode of procession. However, in the presence of torsion strain, binding interactions will be effectively weakened, which enhances the simultaneous dissociation of both bridges and facilitates the screw mode. Under positive torsion strain, the screw mode relieves strain and, therefore, is favored. Under negative torsion strain, stair stepping is favored for the same reason.
The existence of energy barriers, which can limit kinetics of the two modes of processive capping, results from consideration of the minimal system that consists of the barbed end and FH2 dimer. It is possible that overcoming these energy barriers and the acceleration of processive capping are caused by the active participation of FH1 and profilin (Romero et al., 2004
). Indeed, the FH1–FH2 complex in the presence of profilin has been suggested to act as a processive motor, as it increases the barbed end polymerization rate (Romero et al., 2004
Detachment in all four binding sites means that within the screw mode, the formin dimer transiently loses its specific interactions with the barbed end until it reaches three actin-binding sites in the open state. It is important to emphasize that this detachment does not necessarily mean complete separation of formin from the actin filament. Indeed, according to the structure of the formin–actin complex (Otomo et al., 2005
) and the present model, the terminal subunits of the barbed end penetrate the formin ring in both the open and closed state of the system (). Thus, complete separation of formin from the barbed end requires, in addition to the bond detachment, axial translation of the formin ring off barbed end terminal subunits. In addition, if rotation occurs rapidly, the formin could remain associated with the filament through essentially nonspecific contacts during the screw transition. In experimentally relevant situations, the complete formin–actin separation was not expected, which is in agreement with observations of a low formin–barbed end dissociation rate (Zigmond et al., 2003
; Kovar and Pollard, 2004
). Indeed, as previously mentioned in this section, in situations in which formin is not immobilized on any substrate or membrane (Zigmond et al., 2003
), the screw mode and, hence, the related simultaneous unbinding of two formin bridges from the barbed end are not expected. In the case of formin that is fixed on the substrate (Kovar and Pollard, 2004
), actin polymerization produces a force pushing the barbed end against the formin cap. This force keeps barbed end terminal subunits within the formin ring, and separation between the latter is sterically prevented. In addition, electrostatic interaction between the highly positively charged post site and linker of FH2 and highly negatively charged actin groups should provide long-range interprotein attraction. This could help retain formin on the barbed end during rotational movement in the screw mode and lead it into the correct position that corresponds to the new bound state in the open configuration.
Another possibility could be that, as a result of the release of four bonds, the FH2 ring slides along the filament toward the filament bulk and binds behind the barbed end. This would expose an uncapped barbed end to actin monomer solution and allow for unconstrained polymerization. Such a scenario is prevented by distortion of the filament structure by the FH2 ring (Otomo et al., 2005
), which is minimized when the ring is located at the very end of the filament. An effective force that is generated by elastic stresses retains the FH2 ring at the barbed end and prevents a free filament growth.
To summarize, we suggest that torsion elastic stresses determine the optimal regime of actin polymerization upon processive capping by a fastened FH2 dimer. This regime consists, on average, of repeating sequences of 12 stair-stepping steps followed by one screw step. Our model explains the unique properties of formin that enable it to generate arrays of actin filaments whose ends are firmly attached to other cellular structures.