Address correspondence to: Naoki Watanabe, Laboratory of Single-Molecule Cell Biology, Tohoku University Graduate School of Life Sciences, Sendai, Miyagi 980-8578, Japan. E-mail: email@example.com
Advances in fluorescence single-molecule-based approaches have offered us a window for direct viewing of biochemical reactions in living cells [Vale,2008; Watanabe,2010]. Cytoskeleton-associated proteins are suitable targets for this approach. Acquisition of images of low-density fluorescently labeled proteins with long exposure time allows the visualization of only cell structure-associated probes as discrete spots [Watanabe and Mitchison,2002]. Using the single-molecule speckle (abbreviated hereafter as SiMS) method [Watanabe,2012], F-actin lifetime distribution [Watanabe and Mitchison,2002] and the actin-binding kinetics of the major actin regulators [Miyoshi et al.,2006; Tsuji et al.,2009] in lamellipodia have been eluciated in a single defined cell system. Figure 1 summarizes the observed kinetics of the major actin-regulatory proteins in the lamellipodia of Xenopus laevis XTC cells [Pudney et al.,1973]. By putting SiMS data together into a mathematical model, this study aims to examine a longstanding question, as to whether the turnover of rapidly disassembling F-actin species can be ascribed to filament treadmilling without the involvement of filament severing.
Actin forms a polarized filament. At steady state in vitro, actin assembles at one end (the barbed end) and depolymerizes at the other end (the pointed end), which is referred to as “treadmilling”. F-actin directs its fast-growing barbed end outward [Small et al.,1978]. In lamellipodia, the actin network is constantly moved to the cell center by the retrograde actin flow [Wang,1985; Forscher and Smith,1988]. These observations led researchers to hypothesize that actin undergoes treadmilling in a manner such that actin polymerizes at the leading edge and depolymerizes at the base of lamellipodia [Small et al.,1995]. These processes are unified in the array treadmilling model in which F-actin moves as a connected array [Borisy and Svitkina,2000].
The growth of the barbed end is ensured by the presence of high amounts of the profilin–actin complex. Profilin-bound G-actin adds onto the free barbed end as fast as free G-actin. The concentration range of profilin in the cells is 10–40 μM [Southwick and Young,1990]. For example, in XTC cells, it is 22 μM [Higashida et al.,2008]. Indeed, the barbed end elongates fast in the cell as evidenced by the speed of processive actin polymerization of a formin homology protein, mDia1 [Higashida et al.,2004]. The estimated rate of elongation of the barbed end of native actin filaments is 66 s−1 in the lamellipodium body [for details, see Figure S5 in Miyoshi et al.,2006]. At the tip of lamellipodia, filament growth is restricted by physical constraint of the leading edge of the cell.
SiMS analysis has also elucidated the kinetics of dynamic capping and uncapping of the barbed end. The dissociation of capping protein (CP) occurs at 0.58 s−1 in lamellipodia [Miyoshi et al.,2006], ≈3 orders of magnitude faster than in vitro [Schafer et al.,1996]. This CP dissociation rate is roughly 20-times faster than the average rate of actin disassembly. Importantly, recombinant EGFP-CP caps the barbed end with a dissociation rate of 0.005 s−1 in vitro [Miyoshi et al.,2006], which is similar to that of the native capping protein [Schafer et al.,1996]. A recent study, which strongly argued that the lamellipod actin turnover is slow [Lai et al.,2008], also detected fast dissociation of CP with a T1/2 of 7.2 s by using fluorescence recovery after photobleaching (FRAP). Thus, the previously prevailing view that CP blocks the growth of the barbed end until the end of the lifetime of its bound filament needs to be revised. None of the other barbed end-interacting proteins, namely, Eps8, VASP, gelsolin [Miyoshi et al.,2006], wild-type mDia1 [Higashida et al.,2008], and AIP1 [Tsuji et al.,2009], shows persistent association with the lamellipod actin network.
The process of actin disassembly is far less understood than the assembly process. The cofilin/ADF family plays an essential role in cellular F-actin turnover [Hotulainen et al.,2005; Bamburg and Bernstein,2010]. Currently, it remains elusive whether cofilin/ADF functions by accelerating pointed end depolymerization or via its non-catalyzing filament-severing activity [Moriyama and Yahara,1999]. Recent microscopic studies have revealed that cofilin severs the F-actin most efficiently at 10–250 nM [Andrianantoandro and Pollard,2006; Pavlov et al.,2007; Chan et al.,2009]. At higher concentrations, cofilin does sever F-actin, but only transiently. Direct microscopic observation suggests ineffective acceleration of pointed end actin disassembly by cofilin [Andrianantoandro and Pollard,2006]. Pointed end disassembly can more effectively be accelerated by another less abundant [Estornes et al.,2007] isoform ADF (also known as destrin); by ∼22 folds by Arabidopsis thaliana ADF1 [Carlier et al.,1997] and ∼30 folds by human ADF [Maciver,1998]. This question concerning cofilin-catalyzed actin disassembly pathways has long attracted the interest of researchers because severing leads to the formation of newly growing actin barbed ends. In vivo, cofilin inactivation abolishes actin assembly and the increase in barbed ends induced by the epidermal growth factor [Chan et al.,2000] or neuregulin-1β [Nagata-Ohashi et al.,2004] in breast carcinoma cells. In XTC cells, cofilin is required for serum-induced assembly of lamellipod actin [Tsuji et al.,2009]. Photorelease of caged cofilin induces rapid local cell edge extension [Ghosh et al.,2004]. These cofilin-dependent cell edge protrusions have been attributed to barbed end generation by cofilin-catalyzed filament severing. However, cofilin/ADF may also promote lamellipod protrusion by increasing G-actin supply to the cell edge [Kiuchi et al.,2007; Kiuchi et al.,2011]. Thus, the observed cofilin-dependent lamellipod protrusion and free barbed end formation strongly support, but would not perfectly prove, the requirement of filament severing by cofilin/ADF in the remodeling of lamellipodia. Quantitative verification using in vivo molecular kinetics data would further strengthen the requirement of filament severing.
While based on actin FRAP data, a group of researchers conclude that F-actin undergoes treadmilling over the width of the lamellipodia [Lai et al.,2008], fast disintegration of actin photoactivation of fluorescence (PAF) signals in fish keratocytes indicated that actin frequently polymerizes away from the leading edge [Theriot and Mitchison,1991]. Consistent with the latter study, our SiMS analysis revealed that one-third of new F-actin disassembles within 10 s in the lamellipodia [Watanabe and Mitchison,2002] (see recent studies showing similar actin-turnover kinetics measured using improved optics [Smith et al.,2011; Millius et al.,2012; Ryan et al.,2012a]). These PAF and SiMS studies clearly argue against filament treadmilling over the width of lamellipodia. Nonetheless, it remains to be determined whether the turnover of rapidly disassembling F-actin species can be ascribed to the treadmilling of short actin filaments, which is the aim of this study.
Thus far, attempts have been made to model the actin filament turnover at the leading edge [Mogilner and Edelstein-Keshet,2002; Bindschadler et al.,2004; Schaus and Borisy,2008; Ditlev et al.,2009; Stuhrmann et al.,2011; Tania et al.,2011]. These studies, however, estimate several biochemical parameters based on in vitro data in the literature, and in vivo data such as the cofilin-dependent ultrafast CP off-rate [Miyoshi et al.,2006] have not been contemplated because of the lack of consensus on in vivo filament recycling pathways. Given the close relationship between filament-severing activity and other activities including barbed end capping [Carlsson,2006], it would be necessary to quantitatively reevaluate the treadmilling model based on the parameters obtained in vivo.
Here, we describe a method that calculates the lifetime of individual F-actin subunits by the Gamma distribution and its extension under the constraint of filament treadmilling. By incorporating the kinetics of the major filament end regulators such as Arp2/3 complex and CP into the model, our current analysis estimates the rate of pointed end depolymerization required to achieve the observed rapid actin turnover under the constraint of treadmilling. We also discuss a possible alternative actin filament-turnover pathway based on our SiMS data. Our SiMS analysis points to the insufficiency of classical treadmilling filament turnover models, and therefore, it should prompt reinvestigation of complex actin depolymerization processes such as bursting mode of actin disruption [Brieher et al.,2006; Kueh et al.,2008] for better understanding of the actin remodeling mechanism employed by the leading edge of the cell.
Lifetime of an Individual F-actin Subunit Assembled at a Given Distance from the Uncapped and Capped Pointed Ends of a Single Filament
We aimed to examine the likelihood of the treadmilling mechanism governing actin turnover by estimating the rate of pointed end depolymerization required under the constraint of treadmilling. Under this assumption, filament severing and the resultant formation of a new filament end are not considered. Consequently, the model assumes that the filaments are exclusively nucleated by the Arp2/3 complex, the major actin nucleator in lamellipodia. Individual actin subunits assemble to the barbed end of Arp2/3 complex-nucleated filaments and subsequently disassemble by depolymerization from the pointed end. Pointed end depolymerization starts after the dissociation of the Arp2/3 complex from the pointed end.
Under these assumptions, the lifetime of each F-actin subunit can be determined by its distance from the pointed end and the presence or absence of the pointed end cap by the Arp2/3 complex at the time of assembly. If a monomer assembles to an N-subunit filament without Arp2/3 complex at the pointed end (Fig. 2A), the probability density function of filament subunit lifetime is given by the Gamma distribution, according to the queuing theory [Ross,1996]:
where t is the lifetime, and θ is the off-rate of actin from the pointed end. For a monomer that attaches to an N-subunit filament with Arp2/3 complex at the pointed end (Fig. 2B), the probability density function is an extension of the Gamma distribution:
where λ is the off-rate of Arp2/3 complex from the pointed end (See Appendix for detail).
This extension of Gamma distribution is applicable for estimating the time required to generate or degrade other multi-subunit structures. For example, at a given filament nucleation rate and an elongation rate, it can estimate the time required to generate newly nucleated filaments of a given length.
Estimation of F-actin Lifetime Distribution in the Entire Lamellipodia Assuming Non-severing Treadmilling
To calculate the lifetime distribution of F-actin in the entire actin network, the length distribution of the filaments needs to be known. Despite extensive electron microscopy studies on the actin network of the leading edge [Small et al.,1995; Svitkina and Borisy,1999; Urban et al.,2010], the precise distribution of filament length is yet to be determined. F-actin is densely packed at ∼1000 μM in lamellipodia [Abraham et al.,1999], which hampers accurate tracing of filament image. It also appears difficult to ultra-structurally distinguish between the capped and uncapped barbed ends, given the high-resolution cryo-electron microscopy image of the capped barbed end [Narita et al.,2006]. We, therefore, estimated the distance of newly assembled F-actin subunits from the pointed end by kinetic simulation.
The parameters are summarized in Table 1. The dissociation rate of the Arp2/3 complex is 0.048 s−1 [Miyoshi et al.,2006]. Although we have attributed the observed fast-paced CP dissociation to frequent filament severing near the barbed end [Miyoshi et al.,2006; Tsuji et al.,2009], the present analysis adopts 0.58 s−1 as the spontaneous rate of dissociation of CP from the barbed end. The actin elongation rate, i.e., 66 s−1, can be estimated on the basis of the speed of processive actin-polymerizing movement of the FH2 mutant of mDia1 [Higashida et al.,2004; Kovar et al.,2006; for detail, see Figure S5 in Miyoshi et al.,2006]. The minimal rate of pointed end depolymerization required under the constraint of treadmilling is the parameter to be identified.
Table 1. A Parameters used in the Simulation
aThe rate of actin elongation was estimated from the speed of processive actin-polymerizing movement of the FH2 domain mutant of mDia1, namely, mDia1F2 (a.a. 752–1182) in XTC cells [Miyoshi et al.,2006]. mDia1F2 does not affect F-actin distribution and dynamics [Watanabe et al.,1999; Higashida et al.,2004]. In the previous study [Miyoshi et al.,2006], the average speed of mDia1F2 was found to be 0.081 and 0.064 μm/s in the lamella and lamellipodia, respectively (n = 12 cells). In the presence of the profilin:ATP-actin complex in vitro, the mDia1FH2-bound barbed end elongates at a rate 3.2/9.1-times slower than the native barbed end [Kovar et al.,2006]. Based on these results, we estimate the elongation rate of the free barbed end in the lamellipodia of XTC cells to be 66 subunits·s−1. A previous mathematical study demonstrated that the concentration of G-actin in the form of profilin:actin may decrease ∼1.5 times from the rear end to the front end of the lamellipodium [Novak et al.,2008]. However, Novak et al. chose keratocytes as a model and adopted an extremely fast retrograde actin flow rate of 0.3 μm/s. In our cell system, the retrograde actin flow rate is 0.02∼0.05 μm/s in lamellipodia. Consequently, G-actin concentration is estimated to decrease only by 10–20% [Smith et al.2013]. We therefore adopted a single barbed end growth rate for simplicity in this study.
bDisassembly of the barbed end and elongation of the pointed end were omitted because the most abundant source of polymerization-competent G-actin, the profilin–actin complex, does not assemble to the pointed end [Pollard et al.,2000]; its abundance could prevent actin dissociation at the barbed end, which occurs after ATP hydrolysis and γ-phosphate release [Fujiwara et al.,2007].
Geometric constraints at the leading edge were introduced by allocating a maximal elongation limit, d/cosθ, to individual filaments on the basis of the data regarding the appearance of the Arp2/3 complex speckle [Miyoshi et al.,2006] and the distribution of filament angle [Maly and Borisy,2001] (Fig. 3A). We tallied the distance from the pointed end for newly assembled individual actin subunits onto each filament (Fig. 3B). After estimating the position distribution of newly assembled actin in the filaments, the lifetime distribution of the whole filament network was calculated using the above-described Gamma distribution. Extrapolation of this computation (Fig. 3) can also yield the entire lifetime distribution of F-actin without calculation by the above-described Gamma distribution. However, we followed a two-step calculation method because it could substantially reduce the computation time required, especially under conditions where the pointed end disassembly rate is low.
The calculated lifetime distribution was compared with that obtained from the F-actin lifetime data of actin speckles that are >0.5 μm away from the cell edge [Watanabe and Mitchison,2002], because actin in contact with the cell edge might assemble following a different mechanism [Mogilner and Oster,2003]. Indeed, actin speckles had slightly prolonged lifetime in the tip region of lamellipodia [Watanabe and Mitchison,2002]. Because majority of actin assembly occurs in the body of the lamellipodia [Watanabe and Mitchison,2002], it is appropriate to use this selected F-actin lifetime data to test the validity of the treadmilling model.
Treadmilling Requires >100-fold Acceleration of Pointed End Disassembly to Achieve the Observed F-actin Lifetime Distribution
F-actin lifetime distributions with various pointed end disassembly rates and the comparison with the observed data are shown in Fig. 4. F-actin lifetime shortens as the pointed end depolymerization rate increases. At 8.1 s−1, F-actin lifetime distributes widely over 10∼100 s. At a rate faster than 27 s−1, which corresponds to ∼100-fold acceleration of the in vitro ADP-actin dissociation rate at the pointed end [Pollard et al.,2000; Fujiwara et al.,2007], F-actin lifetime distribution was estimated to become comparable to, but still slightly slower than, the in vivo data. In vitro, the pointed end disassembly rate can be accelerated ∼22 folds by Arabidopsis thaliana ADF1 [Carlier et al.,1997] and ∼30 fold by human ADF [Maciver,1998]. Acceleration by 30 folds corresponds to 8.1 s−1. These results suggest that the treadmilling mechanism with the accelerated pointed depolymerization rate achieved by cofilin/ADF alone does not account for the actin-turnover kinetics observed in the lamellipodia of XTC cells.
Combination of 30-fold Acceleration of Pointed End Depolymerization with a Barbed End-capping Rate of 6.0 s−1 May Achieve the Observed Actin Turnover
We also investigated another possible compensatory mechanism, an excessive barbed end-capping activity. If high amounts of barbed end cappers exist, the ratio of free barbed ends and pointed ends reduces. This mechanism, referred to as funneling, could help balance slow-paced pointed end depolymerization with fast-paced barbed end elongation.
CP contributes most of the actin-capping activity in cell lysates [DiNubile et al.,1995; Hug et al.,1995], and plays a critical role in the regulation of lamellipodium morphology [Rogers et al.,2003]. Indeed, 90% of the barbed end-capping activity in the neutrophil lysate derives from CP [DiNubile et al.,1995]. The kinetics of CP presumably represents most of the capping activities in lamellipodia. Other proteins that may potentially interfere with barbed end elongation include gelsolin, Eps8 [Disanza et al.,2004], and AIP1 [Okada et al.,2002, 2006; Ono,2003]. Eps8 and VASP show fast-paced dissociation kinetics from the actin network, whereas gelsolin shows a highly rare association with the lamellipod actin network [Miyoshi et al.,2006]. VASP enhances the elongation of its associated barbed end [Breitsprecher et al.,2008; Hansen and Mullins,2010]. These molecules thus do not appear to cap high amounts of barbed ends. AIP1 dissociates from the lamellipod actin network at 1.1 s−1, which is faster than CP, and the AIP1 is present at ∼1.6 μM in lamellipodia. Free CP is present at 0.4∼2 μM in the lysates obtained from Dictyostelium cells [Hug et al.,1995] and neutrophils [DiNubile et al.,1995], and CP dissociates from the actin network at 0.58 s−1 [Tsuji et al.,2009]. Thus, albeit less prominent than CP, AIP1 may also quantitatively contribute to the barbed end capping activities in lamellipodia. We therefore examined the effect of increased barbed end-capping rates considering the participation of multiple cappers.
We calculated F-actin lifetime distribution with a fixed accelerated pointed-end dissociation rate of 8.1 s−1 (30-fold acceleration) and various barbed end-capping rates. The capping rate of 6.0 s−1 turned out to be the minimal rate to achieve the in vivo F-actin lifetime distribution (Fig. 5). This capping rate exceeds the rate of the capping activities reported in the cytoplasm [DiNubile et al.,1995; Hug et al.,1995]. The diffusion constant of proteins is roughly one-order smaller in the crowded cytoplasm than in solution. Therefore, free barbed end cappers would be required at several-to-several-tens micromolar concentrations to achieve this fast-paced capping rate. Our analysis suggests that a part of F-actin turnover is driven by processes other than filament treadmilling.
Conclusions and Perspectives
The present study has quantitatively inspected the filament treadmilling mechanism and has shown its insufficiency for solely facilitating the fast actin turnover observed in the lamellipodia. Our calculation indicates that >100-fold acceleration of pointed end depolymerization would be required if treadmilling is the sole actin turnover mechanism in lamellipodia. If pointed end depolymerization is accelerated 30 folds, which is the maximal acceleration achieved by cofilin/ADF in vitro, the barbed end capping rate needs to be accelerated to exceptionally high values of >6.0 s−1. These estimates provide evidence for the involvement of non-treadmilling mechanisms in the in vivo actin turnover cycle.
What alternative processes could facilitate fast actin turnover? Recent work has discovered a highly active tri-component actin disassembling system consisting of cofilin, AIP1, and coronin [Brieher et al.,2006]. This system facilitates actin disassembly from both the barbed and pointed ends by abruptly removing filaments of a mean size of 260 subunits [Kueh et al.,2008]. Although it remains unclear whether the mode of such abrupt actin disruption by this system differs from the filament severing mechanism by cofilin, the residual filaments appear to be transiently capped by AIP1 [Kueh et al.,2008]. These findings are consistent with previous studies showing that AIP1 specifically recognizes and binds the barbed ends generated by cofilin-induced filament severing [Okada et al.,2002; Ono,2003].
On the other hand, our SiMS analysis revealed that in lamellipodia, AIP1 rapidly dissociates from the actin network at the rate of 1.1 s−1, and the overall frequency of the actin association/dissociation events of AIP1 is 1.8 μM·s−1. Notably, this frequency is about 15 times the actin nucleating frequency of the Arp2/3 complex in lamellipodia [Tsuji et al.,2009]. Therefore, although quite a lot remains to be elucidated about the in vivo disassembly of the filaments, i.e., about severing, long-range abrupt disruption, or filament-end depolymerization, the actin reorganization process involving cofilin-associated filament disassembly and the resultant formation of new barbed ends, which are recognized by AIP1, occurs far more frequently than actin nucleation by Arp2/3 complex, which is also a critical regulator for the lamellipod actin network [Rogers et al.,2003].
In addition to the difference in the frequency of Arp2/3 complex- and AIP1-associated events, our previous findings related to fast-paced actin dissociation of CP and AIP1, which are both sensitive to filament stabilization by jasplakinolide [Miyoshi et al.,2006; Tsuji et al.,2009], support two notions. First, a part of F-actin may dissociate from the actin network as actin oligomers. Actin remodeling may involve dynamic dissociation and re-annealing of actin filaments and short oligomers. Second, this type of filament disruption/severing most frequently occurs near the barbed end. We previously proposed the frequent filament severing-annealing hypothesis for the mechanism underlying the fast, actin turnover-dependent CP protein dissociation [Miyoshi et al.,2006]. However, the frequency of actin association of AIP1 [Tsuji et al.,2009] falls short of the filament-severing frequency predicted by the frequent filament severing-annealing hypothesis, if we assume that filament severing occurs evenly over the length of F-actin. We therefore postulate that the barbed end portion of F-actin might be more susceptible to the filament-disruption activities involving cofilin and AIP1, rather than the other portions. This notion is partly consistent with the preferred actin-disruption sites by the cofilin/AIP1/coronin system [Kueh et al.,2008]. In vivo, the pointed end of F-actin appears to be more stable than the barbed end side as the Arp2/3 complex dissociates with slower kinetics. Taken together, it is tempting to speculate that F-actin in vivo might undergo rapid disassembly and polymerization cycles near the barbed end, which is similar to the dynamic instability of microtubules (Fig. 6) [Watanabe,2010]. Further extensive studies, both in vitro and in vivo, are required to prove this intriguing possibility.
Apart from the biochemical kinetics-based modeling of the F-actin turnover [Mogilner and Edelstein-Keshet,2002; Bindschadler et al.,2004; Schaus and Borisy,2008; Ditlev et al.,2009; Stuhrmann et al.,2011; Tania et al.,2011], cell protusion dynamics has extensively been studied by coarse grained modeling [reviewed in Ryan et al.,2012b]. It still appears to be a challenge to combine these two approaches because of the lack of the precise knowledge about where cofilin/AIP1/coronin-catalyzed filament severing/disruption occurs in the actin filament (i.e. near the barbed end, near the pointed end or both) and how this reaction might contribute to the free barbed end formation. In our recent work [Smith et al.,2013], a possible involvement of slowly diffusing actin oligomers in actin recycling was noted as a mechanism that could reconcile the apparent discrepancy between the slow recovery kinetics of actin FRAP [Lai et al.,2008] and the fast actin disassembly in SiMS analysis [Watanabe and Mitchison,2002]. Further elucidation of the actin recycling processes would be a crucial aspect to understand how actin remodeling signals propagate within the actin array to drive cell edge protrusion regulating pathophysiological conditions such as immune cell migration, cancer invasion, and neuronal process outgrowth.
Materials and Methods
Filament Lifetime Distribution in the Entire Network
We first simulated the elongation of a large number of filaments to predict the distance distribution of the newly assembled subunits from the pointed end. Our model concerns the lifetime of the actin subunits assembled in the lamellipodium body, which comprise majority of polymerization events [Watanabe and Mitchison,2002], but not those assembled to the barbed end in contact with the plasma membrane of the leading edge. We assigned the maximal distance of elongation to each filament, according to the distribution of the distance of Arp2/3 complex speckle appearance from the cell edge [Miyoshi et al.,2006] and the filament angle distribution by an EM study [Maly and Borisy,2001]. Computation was carried out as follows:
If the barbed end is not capped, an actin subunit is added with a given probability.
Upon the addition of an actin subunit, its distance from the pointed end and the presence or absence of Arp2/3 complex in the filament are noted and recorded.
If Arp2/3 complex is absent in the filament, the pointed end is shortened by one subunit with a given probability.
If Arp2/3 complex is present, the Arp2/3 complex is removed with a given probability.
If the barbed end is not capped, CP is added with a given probability.
If the barbed end is capped, CP is removed with a given probability.
The filament was eliminated either when all the actin subunits are lost or when the barbed end reached its own maximal elongation distance.
The steps from i to vii are looped.
When the loop is complete, the distribution of the distance of the new actin subunits from both Arp2/3 complex-associated and Arp2/3 complex-free pointed ends is recorded. Then, the lifetime distribution of the whole filament network is calculated using the Gamma distribution described in the Appendix and Fig. 2.
We thank Ichiro Fujii for advice on the extension of the Gamma distribution and Dimitrios Vavylonis for the helpful discussion. This work was supported by the Cabinet Office, Government of Japan, through the Funding Program for Next Generation World-Leading Researchers (LS013) and by grants from the Human Frontier Science Program (RGP0061/2009-C) and the Takeda Science Foundation.
Determination of F-actin Subunit Lifetime in the Filament Bound to the Arp2/3 Complex
In the non-severing treadmilling model, for a particular actin monomer in a filament, the time span between its assembly and dissociation from the pointed end can be determined by the existence of Arp2/3 complex and by the number of actin monomers between the monomer and the pointed end. We define the time span between binding to the barbed end and releasing from the pointed end as the lifetime of an actin monomer. Under non-severing condition, the lifetime of an actin monomer, which binds to an N-subunit filament without Arp2/3 complex, can be calculated by the probability density function of Gamma distribution according to the queuing theory [Ross,1996],
where t is the lifetime, and θ is the actin off-rate from the pointed end. The lifetime of an F-actin subunit, which binds to an N-subunit filament harboring Arp2/3 complex at the pointed end, can also be calculated. Here, we define T0 as the time span before the Arp2/3 complex leaves the pointed end; Tk as the time span between the release of (k − 1)-th actin and the release of k-th actin; and Sn as the summation of T1, T2, …, Tn. The probability that the Arp2/3 complex and the n actin subunits disassemble from the pointed-end before time t is,
where λ is the Arp2/3 complex dissociation rate and λ < θ. The probability density function of “lifetime” is the differential of P [T0 + Sn ≤ t],
The R platform (http://www.r-project.org) provides the cumulative probability function of the Gamma distribution, called “pgamma.” With this function, the above equation can be calculated as,
Estimation of the Capping Protein On-rate
At the equilibrium, we have
where [B], [C], [C]0, and [BC] are the concentrations of the free barbed ends, free capping protein (CP), total CP, and CP-capped barbed ends. kon and koff are the on- and off-rates of CP. From the above two equations, we have,
The concentration of CP, [C]0, is estimated to be 1 μM [DiNubile et al.,1995; Hug et al.,1995]. The on-rate of CP to the barbed end is 4 μM−1·s−1 [Schafer et al.,1996]. The off-rate is 0.58 s−1 [Miyoshi et al.,2006]. Thus, the on-rate of CP is determined by the free barbed-end concentration. In lamellipodia, 3% of F-actin is replaced every second [Watanabe and Mitchison,2002]. The concentration of free barbed end, [B], can be estimated by,
where [F]0 is the F-actin concentration and Bon is the elongation rate of barbed ends. The concentration of F-actin is 1000 μM [Abraham et al.,1999], and Bon is 66 s−1 in the lamellipodia (Table 1). [B] is then calculated to be 0.45 μM, which is in agreement with the measured free barbed end concentration (0.99 μM) in the lamellipodia of permeabilized XTC cells by using recombinant EGFP-capping protein as a probe [Miyoshi et al.,2006]. This concentration should be larger than the former estimated value because our measurement included both freely growing barbed ends and those in contact with the plasma membrane. The overall capping protein on-rate kon[C] is estimated to be 0.97 s−1 and has been adopted in the primary analysis (Figs. 3 and 4).