Home | About | Journals | Submit | Contact Us | Français |

**|**HHS Author Manuscripts**|**PMC2668770

Formats

Article sections

Authors

Related links

Phys Rev Lett. Author manuscript; available in PMC 2009 April 14.

Published in final edited form as:

Published online 2008 September 17.

PMCID: PMC2668770

NIHMSID: NIHMS96624

Richard D. Berlin Center for Cell Analysis and Modeling, Department of Cell Biology, University of Connecticut Health Center, Farmington, Connecticut 06030, USA

The publisher's final edited version of this article is available at Phys Rev Lett

See other articles in PMC that cite the published article.

Bundling of rapidly polymerizing actin filaments underlies dynamics of filopodial protrusions that play an important role in cell migration and cell-cell interaction. Recently, formation of actin bundles has been reconstituted *in vitro* and two scenarios of bundle initiation, involving binding of two filament tips and, alternatively, linking of the tip of one filament to the side of the other, -have been discussed. A first theoretical analysis is presented indicating that the two mechanisms can be distinguished experimentally. While both of them result counter-intuitively in comparable numbers of bundles, these numbers scale differently with the average bundle length. We propose an experiment for determining which of the two mechanisms is involved in the *in vitro* bundle formation.

Filamentous actin constitutes an important component of cell cytoskeleton [1]. Structure and patterns of actin meshwork change dynamically as filaments polymerize, branch, and bundle, which enables living cells to vary their shape and migrate. One such dynamic pattern, used by cells to explore their environment and build adhesive outposts, are filopodia - long, finger-like protrusions that result from formation and growth of bundles of actin filaments [2–4]. The filaments are thought to be held in a bundle by linker proteins such as fascin [5]. Recently, actin bundles have been reconstituted *in vitro* in the presence, or after addition, of fascin [6, 7]. In these experiments, the bundles emerge from a dense quasi-two-dimensional meshwork of actively polymerizing filaments and form star-like structures (Fig 1 a). Importantly, the conditions of the *in vitro* polymerization and bundling are strongly non-equilibrium. This is due to the excess of fascin and Arp2/3, the protein promoting nucleation of new filaments through branching [8]. In addition, the assays do not include any capping proteins or depolymerization factors.

It is generally accepted that the bundle arises from two filaments zipped up by a linker protein [6, 7] but initiation mechanisms are a subject of active research. Two scenarios have been recently proposed [6, 9, 10]. One involves binding of two filament tips into a tip complex, which triggers the zipping if the filaments are at a sufficiently small angle [6, 11–14]. In an alternate scenario, binding of the tip of one filament to the side of the other is thought to be sufficient for the initiation of linking [10], again on a condition of a small angle between the filaments. In this Letter, we analyze how the two mechanisms would affect dynamics of bundling in order to determine if they are experimentally distinguishable. Our results indicate that while, surprisingly, both of them may result in comparable numbers of bundles, these numbers scale differently with the average linear size of the aster. Therefore, the mechanisms can be distinguished by measuring the size dependence of the final number of bundles.

Development of the system of actin filaments and bundles in non-equilibrium assays is analyzed with the aid of a model, which accounts for four essential processes: Arp2/3-mediated nucleation of filaments, polymerization of filaments (both individual and in the bundle), initiation of a bundle from two unbundled filaments, and thickening of a bundle as it absorbs individual filaments [15] (Fig 1 c). Effects of slower processes, such as depolymerization in the absence of depolymerization factors [7, 16] and thermal fluctuations of the filaments [17], can be ignored because a pool of actin monomers in the *in vitro* assays is exhausted quickly due to the facilitated nucleation and rapid polymerization of filaments [7]. Aside from a small number of seed filaments, all new linear filaments nucleate from Arp2/3 complexes at the sides of the existing filaments (Fig 1 c-A) (new filaments do not nucleate from the bundles, possibly because of a dense decoration of those by the linker [5, 6]). Because each nucleated filament has one point of origin and one growing tip, the whole branched system, despite its complexity, can be unambiguously described as a set of linear filaments. To initiate a bundle (Fig 1 c-B) or to link an unbundled linear filament to an existing bundle (Fig 1c-C), the linker must bend the filaments. This is modeled as an ”all-or-none” transition [11, 13, 14] depending on whether the angle between the filaments (or between a filament and a bundle) is below or above a critical value, *α _{c}*, determined by cross-linking strength and elastic properties of the filaments and estimated to be in the range 0.005–0.5 [13, 14].

Because of the abundance of the linker protein [7, 18], the initiation of a bundle and the absorption of filaments by existing bundles are approximated as collision-controlled, so that the problem involves only two time scales determined by the initial frequency of polymerization events and the frequency of nucleation. The initial polymerization frequency is
${\tau}_{p}^{-1}={k}_{p}{C}_{0}$ where *k _{p}* is the rate constant for binding of a monomer to a filament plus end and

Formally, a state *S* of the system is defined by the number of monomers *Y* [23], the set of unbundled linear filaments, {*f _{i}*:

In our analysis, we utilize both detailed spatial simulations and mean-field approaches. The detailed dynamics are described by the probability *P* (*S*, *t*|*S*_{0}, 0) for the system to be in a state *S* at time *t*, given an initial state *S*_{0} at *t* = 0. The governing equation [26],

$${t}_{P}$$

(1)

is solved numerically using Kinetic Monte-Carlo techniques (a snapshot of a typical realization is shown in Fig 1 b). The transition rates *W _{S}*

Insight into behavior of the system can also be gained from a non-spatial mean-field approximation formulated in terms of averages, such as *B* = Σ* _{S} B* (

$$\tau =\sqrt{2{\tau}_{p}{\tau}_{n}},\phantom{\rule{0.38889em}{0ex}}{L}_{\infty}=\delta \sqrt{\frac{2{\tau}_{n}}{{\tau}_{p}}},\phantom{\rule{0.38889em}{0ex}}{E}_{\infty}={Y}_{0}\sqrt{\frac{2{\tau}_{p}}{{\tau}_{n}}}.$$

(2)

(here and below, the angular brackets denoting ensemble averaging are omitted for brevity). The results (2), obtained in the limit *X*_{0}/*E*_{∞} 1, agree with the solutions of Eq (1) [25]. Also, with the estimates of *τ _{p}* and

An upper bound for the number of bundles *B* can be obtained by ignoring incorporation of individual filaments into existing bundles. For the ”tip-tip” mechanism, the upper mean-field estimate of the final number of bundles is [25]:

$${B}_{\infty}^{\text{tip}-\text{tip}}=\frac{8}{{\pi}^{3}}\frac{{\alpha}_{c}\delta}{h}{Y}_{0}^{2}{\left(\frac{{\tau}_{p}}{{\tau}_{n}}\right)}^{3/2}.$$

(3)

where the aster thickness *h* is on the order of several *δ*. Indeed,
${B}_{\infty}^{\text{tip}-\text{tip}}$ can be viewed as the total number of ”favorable” tip-tip collisions over the time *τ*:
${B}_{\infty}^{\text{tip}-\text{tip}}\approx \tau {R}_{\text{coll}}$ where the collision frequency *R*_{coll} is expressed in terms of the average relative velocity of the tips *υ _{rel}* and the average tip density

Intuitively, the ”tip-side” mechanism should result in a much larger number of bundles because of higher frequency of collisions. Indeed, the mean-field collision rate for this mechanism is *R*_{coll} *X _{M}*, where

$${B}_{\infty}^{\text{tip}-\text{side}}=\frac{{\alpha}_{c}^{}{\pi}^{2}h}{{Y}_{0}^{2}}$$

(4)

with ${\alpha}_{c}^{}$, accurately approximates the solution of Eq (1) for the case of the tip-side binding.

Comparing Eqs (3) and (4) with the account of (2) leads to an important observation: the final numbers of bundles in the two mechanisms scale differently with the final linear size of the aster:
${B}_{\infty}^{\text{tip}-\text{tip}}{({L}_{\infty})}^{-3}$ and
${B}_{\infty}^{\text{tip}-\text{side}}{({L}_{\infty})}^{-2}$. We therefore conclude that, while
${B}_{\infty}^{\text{tip}-\text{tip}}$ and
${B}_{\infty}^{\text{tip}-\text{side}}$ may fall in the same range due to possible variations in parameter values, their dependence on the final linear size of the aster is described by a power law with different exponents (Fig 4), so the two mechanisms can be distinguished experimentally. For this, we propose to perform a series of the *in vitro* experiments, as described in [6, 7], with a same initial concentration of actin monomers, and after the polymerization is over, measure the number of bundles as a function of the linear size of the aster. The data are expected to be less noisy if the experiments are conducted with a fixed saturating concentration of fascin and with varying saturating amounts of Arp2/3. Our theory predicts that in these conditions, the measured dependence will be described by the power law, *B*_{∞} (*L*_{∞})^{−}* ^{β}*. The data yielding

In summary, we have analyzed the dynamics of bundling of actin filaments for two different mechanisms of bundle initiation. In one, binding of the tips of neighboring filaments initiates linking of the filaments into a bundle. In an alternative scenario, the initiation of a bundle is brought about by linking the tip of one filament to the side of the other. Our analysis indicates that while both mechanisms may result in comparable numbers of bundles, the dependence of the final number of bundles on the final linear size of the aster is described by a power law with different exponents: *B*_{∞} (*L*_{∞})^{−}* ^{β}* with

We acknowledge helpful conversations with Gary Borisy, Vladimir Rodionov, Leslie Loew, and Thomas Pollard. The work is supported by National Institutes of Health through grants 1U54-RR022232, P41-RR13186, and 1U54-GM64346-01.

PACS numbers: 87.16.Ka, 87.16.Ac, 87.16.-b

1. Alberts B, et al. Molecular Biology of the Cell. 3 Garland Publishers; New York: 1994.

2. Jacinto A, Wolpert L. Curr Biol. 2001;11:R634. [PubMed]

3. Ridley AJ, et al. Science. 2003;302:1704. [PubMed]

4. Rorth P. Cell. 2003;112:595. [PubMed]

5. Svitkina TM, et al. J Cell Biol. 2003;160:409. [PMC free article] [PubMed]

6. Vignjevic D, et al. J Cell Biol. 2003;160:951. [PMC free article] [PubMed]

7. Haviv L, et al. Proc Natl Acad Sci USA. 2006;103:4906. [PubMed]

8. Pollard TD. Annu Rev Biophys Biomol Struct. 2007;36:451. [PubMed]

9. Vignjevic D, et al. Methods in Enzymol. 2006;406:727. [PubMed]

10. Stewman SF, Dinner AR. Phys Rev E. 2007;76:016103. [PubMed]

11. Kierfeld J, Kühne T, Lipowsky R. Phys Rev Lett. 2005;95:038102. [PubMed]

12. Kierfeld J, et al. J Comput Theor Nanosci. 2006;3:898.

13. Yang L, et al. Bioph J. 2006;90:4295. [PubMed]

14. Yu X, Carlsson AE. Biophys J. 2004;87:3679. [PubMed]

15. Merging of bundles, another possible process, is ignored because these events are rare, with little effect on the overall dynamics.

16. Pollard TD, et al. Annu Rev Biophys Biomol Struct. 2000;29:545. [PubMed]

17. Kuhn JR, Pollard TD. Biophys J. 2005;88:1387. [PubMed]

18. Yamakita Y, et al. J Biol Chem. 1996;271:12632. [PubMed]

19. Pollard TD. J Cell Biol. 1986;103:2747. [PMC free article] [PubMed]

20. Mullins RD, et al. Proc Natl Acad Sci USA. 1998;95:6181. [PubMed]

21. Pantaloni D, et al. Nature Cell Biol. 2000;2:385. [PubMed]

22. Mogilner MA, Edelstein-Keshet L. Biophys J. 2002;83:1237. [PubMed]

23. Diffusion of monomers is fast on a spatial scale of the aster [24].

24. Abraham VC, et al. Biophys J. 1999;77:1721. [PubMed]

25. See EPAPS Document No……… for an appendix containing a detailed description of our model. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html.

26. Gardiner CW. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer-Verlag; 2004.

27. Tseng Y, et al. J Mol Biol. 2001;310:351. [PubMed]

PubMed Central Canada is a service of the Canadian Institutes of Health Research (CIHR) working in partnership with the National Research Council's national science library in cooperation with the National Center for Biotechnology Information at the U.S. National Library of Medicine(NCBI/NLM). It includes content provided to the PubMed Central International archive by participating publishers. |