Mathematical Models and Simulations of Cellular Processes Based on Actin Filaments

Actin filaments help to maintain the physical integrity of cells and participate in many processes that produce cellular movements. Studies of the processes that depend on actin filaments have progressed to the point where mathematical models and computer simulations are an essential part of the experimental toolkit. These quantitative models integrate knowledge about the structures of the key proteins and the rate and equilibrium constants for the reactions for comparison with a growing body of quantitative measurements of dynamic processes in live cells. Models and simulations are essential because it is impossible to appreciate by intuition alone the properties that emerge from a network of coupled reactions, particularly when the system contains many components, and force is one of the parameters. Actin filaments help to maintain the physical integrity of cells and participate in many processes that produce cellular movements. Studies of the processes that depend on actin filaments have progressed to the point where mathematical models and computer simulations are an essential part of the experimental toolkit. These quantitative models integrate knowledge about the structures of the key proteins and the rate and equilibrium constants for the reactions for comparison with a growing body of quantitative measurements of dynamic processes in live cells. Models and simulations are essential because it is impossible to appreciate by intuition alone the properties that emerge from a network of coupled reactions, particularly when the system contains many components, and force is one of the parameters. We use a few examples to illustrate how mathematical models advance the understanding of the actin system from side chain motions of proteins to the behavior of whole cells. Readers will find references to experimental work in the papers cited. Diverse methods (supplemental data) are required given the range of complexity (single proteins to cells), dimensions (10–9–10–4 m), and time (10–12–102 s). Internal Motions of Actin Monomers—Actin consists of four subdomains surrounding a cleft that binds ATP or ADP (Fig. 1). In molecular dynamics (MD) 2The abbreviations used are: MD, molecular dynamics; pN, piconewton(s); FH, formin homology; ADF, actin-depolymerizing factor. simulations, the DNase loop in subdomain 2 is the most flexible part of the protein with a weak tendency to form a π -oran α-helix (1Zheng X. Diraviyam K. Sept D. Biophys. J. 2007; 93: 1277-1283Abstract Full Text Full Text PDF PubMed Scopus (66) Google Scholar, 2Dalhaimer P. Pollard T.D. Nolen B. J. Mol. Biol. 2008; 376: 166-183Crossref PubMed Scopus (46) Google Scholar), so crystal contacts may stabilize the helix when it is present. The nucleotide-binding cleft of actin remains closed in MD simulations with bound ATP or ADP or without bound nucleotide (1Zheng X. Diraviyam K. Sept D. Biophys. J. 2007; 93: 1277-1283Abstract Full Text Full Text PDF PubMed Scopus (66) Google Scholar, 2Dalhaimer P. Pollard T.D. Nolen B. J. Mol. Biol. 2008; 376: 166-183Crossref PubMed Scopus (46) Google Scholar, 3Chu J.W. Voth G.A. Proc. Natl. Acad. Sci. U. S. A. 2005; 102: 13111-13116Crossref PubMed Scopus (165) Google Scholar), whereas the cleft of the actin-related protein Arp3 tends to open. This difference depends on a C-terminal extension of Arp3, which fits into the groove between subdomains 1 and 3 and stabilizes the open conformation (2Dalhaimer P. Pollard T.D. Nolen B. J. Mol. Biol. 2008; 376: 166-183Crossref PubMed Scopus (46) Google Scholar). Profilin binding in this groove also promotes cleft opening and nucleotide exchange (4Minehardt T.J. Kollman P.A. Cooke R. Pate E. Biophys. J. 2006; 90: 2445-2449Abstract Full Text Full Text PDF PubMed Scopus (22) Google Scholar). Actin Filament Nucleation and Elongation—Pure actin monomers (Fig. 1) spontaneously polymerize into helical filaments under physiological conditions. Kinetic simulations of the complete time course of polymerization of actin monomers showed that formation of dimers and trimers is extremely unfavorable (5Frieden C. Proc. Natl. Acad. Sci. U. S. A. 1983; 80: 6513-6517Crossref PubMed Scopus (105) Google Scholar). Brownian dynamics simulations showed that the long pitch (end-to-end) dimer is favored over the short pitch dimer and that the third subunit binds laterally to form a trimer nucleus (6Sept D. McCammon J.A. Biophys. J. 2001; 81: 667-674Abstract Full Text Full Text PDF PubMed Scopus (187) Google Scholar). Brownian dynamics simulations showed that electrostatic forces favor elongation at the barbed end over the pointed end as observed (7Sept D. Elcock A.H. McCammon J.A. J. Mol. Biol. 1999; 294: 1181-1189Crossref PubMed Scopus (60) Google Scholar). Effect of Bound Nucleotide on Polymerization—The nucleotide bound to actin influences every aspect of polymerization. In cells, actin monomers are saturated with ATP. When incorporated into a filament, actin hydrolyzes bound ATP 40,000-fold faster than monomeric actin (8Rould M.A. Wan Q. Joel P.B. Lowey S. Trybus K.M. J. Biol. Chem. 2006; 281: 31909-31919Abstract Full Text Full Text PDF PubMed Scopus (92) Google Scholar). Despite enough crystal structures and MD simulations to formulate a hypothesis (2Dalhaimer P. Pollard T.D. Nolen B. J. Mol. Biol. 2008; 376: 166-183Crossref PubMed Scopus (46) Google Scholar, 9Wriggers W. Schulten K. Proteins. 1999; 35: 262-273Crossref PubMed Scopus (99) Google Scholar, 10Nolen B.J. Pollard T.D. Mol. Cell. 2007; 26: 449-457Abstract Full Text Full Text PDF PubMed Scopus (61) Google Scholar) for conformational changes associated with the ATPase cycle, we do not understand how polymerization stimulates hydrolysis or how the presence of γ-phosphate influences the affinity of actin for profilin, thymosin-ß4, and cofilin. At the fast-growing barbed end of filaments, ADP-actin binds slower and dissociates faster than ATP-actin. ADP-Pi-actin associates only slightly faster than ADP-actin but dissociates much slower (11Fujiwara I. Vavylonis D. Pollard T.D. Proc. Natl. Acad. Sci. U. S. A. 2007; 104: 8827-8832Crossref PubMed Scopus (148) Google Scholar). All of these reactions are slower at the pointed ends of filaments. An enduring mystery has been how ATP hydrolysis by polymerized actin makes the critical concentration for elongation ∼10 times more favorable at barbed ends than at pointed ends. Analytical models and Monte Carlo simulations consistent with experimental data show that the difference arises from faster dissociation of phosphate from ADP-Pi subunits near both ends than from the interior subunits and lower affinity of phosphate for terminal subunits at pointed ends than at barbed ends (11Fujiwara I. Vavylonis D. Pollard T.D. Proc. Natl. Acad. Sci. U. S. A. 2007; 104: 8827-8832Crossref PubMed Scopus (148) Google Scholar). At steady state with ATP in the medium, these reactions coupled with random ATP hydrolysis and Pi release in filaments produce gradients of subunits containing bound ATP, ADP-Pi, and ADP from the ends toward the interior of filaments (12Bindschadler M. Osborn E.A. Dewey Jr., C.F. McGrath J.L. Biophys. J. 2004; 86: 2720-2739Abstract Full Text Full Text PDF PubMed Scopus (74) Google Scholar), small length fluctuations at barbed ends (13Vavylonis D. Yang Q. O'Shaughnessy B. Proc. Natl. Acad. Sci. U. S. A. 2005; 102: 8543-8548Crossref PubMed Scopus (103) Google Scholar, 14Stukalin E.B. Kolomeisky A.B. Biophys. J. 2006; 87: 3679-3689Google Scholar), and net addition of subunits at barbed ends balanced by net loss of subunits at pointed ends (∼0.1 s–1). Force Production by Polymerizing Actin—Experiments and theory agree that polymerizing actin filaments produce a few piconewtons (pN) of force. Single filaments as short as 700 nm long buckle when they elongate between two attachment sites (15Kovar D.R. Pollard T.D. Proc. Natl. Acad. Sci. U. S. A. 2004; 101: 14725-14730Crossref PubMed Scopus (357) Google Scholar). Given the stiffness of actin filaments, the force is >1 pN (16Berro J. Michelot A. Blanchoin L. Kovar D.R. Martiel J.L. Biophys. J. 2007; 92: 2546-2558Abstract Full Text Full Text PDF PubMed Scopus (36) Google Scholar). Pioneering studies (17Peskin C.S. Odell G.M. Oster G.F. Biophys. J. 1993; 65: 316-324Abstract Full Text PDF PubMed Scopus (733) Google Scholar, 18Mogilner A. Oster G. Biophys. J. 1996; 71: 3030-3045Abstract Full Text PDF PubMed Scopus (714) Google Scholar) proposed that elongating filaments move objects by a Brownian ratchet mechanism. The original model considered how elongation of rigid filaments could rectify the thermal motion of a diffusing object. A later elastic Brownian ratchet model considered not only the motion of the object but also the thermal motion of flexible elastic filaments (Fig. 2C). When diffusion opens a gap between the end of the polymer and an object, insertion of a subunit prevents the object from reentering this space. The elongation rate of a single filament against such a load is the elongation rate of a free filament times the probability that a gap exists between the tip and the load, which is given by the Boltzmann term e–ΔE/kT, where ΔE is the energy required to create the gap, k is the Boltzmann's constant, and T is absolute temperature. Calculation of the distribution of positions of the object over time gave a logarithmic dependence of force (in the pN range) on the rate of elongation (in the range of 0–110 subunits/s). The velocity depends on actin monomer concentration, elongation rate constant, length of the filaments, and angle of incidence between the filament and the barrier, with an optimum angle near 45°. Remarkably filament growth and branching at the leading edge of motile cells select filaments oriented at angles near 45° relative to the membrane (19Schaus T.E. Taylor E.W. Borisy G.G. Proc. Natl. Acad. Sci. U. S. A. 2007; 104: 7086-7091Crossref PubMed Scopus (83) Google Scholar). Note that long filaments buckle, filaments parallel to the barrier exert no force, and bending of filaments normal to the barrier opens only a small gap for elongation. Langevin dynamics simulations (20Carlsson A.E. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics. 2000; 62: 7082-7091Crossref PubMed Scopus (23) Google Scholar) and Monte Carlo simulations (21Burroughs N.J. Marenduzzo D. J. Chem. Phys. 2005; 123: 174908Crossref PubMed Scopus (19) Google Scholar) confirmed the general features of the elastic Brownian ratchet mechanism. Physical Properties of Actin Filaments—Massive all-atom MD simulations and normal mode analysis of coarse-grained models (3Chu J.W. Voth G.A. Proc. Natl. Acad. Sci. U. S. A. 2005; 102: 13111-13116Crossref PubMed Scopus (165) Google Scholar) reproduced the observed stiffness of ATP- and ADP-actin filaments. The α-helical DNase loop assumed for ADP-actin has weaker short pitch interactions and no long pitch interactions, accounting for the greater flexibility of ADP-actin filaments. However, we do not understand how γ-phosphate dissociation from filaments alters their structure and influences subunit reactions at the ends. Cells use dozens of proteins to regulate the time and place of actin polymerization. Other proteins shape and reinforce structures composed of actin filaments. Here, we use two proteins that direct actin assembly and one that promotes disassembly to illustrate how modeling contributes to research on actin-binding proteins. Arp2/3 Complex—The Arp2/3 complex nucleates actin filaments as 78° branches on the sides of pre-existing actin filaments. Five protein subunits hold the two actin-related proteins, Arp2 and Arp3, close together but separated enough to prevent them from initiating an actin filament. Kinetic simulations (22Beltzner C.C. Pollard T.D. J. Biol. Chem. 2008; 283: 7135-7144Abstract Full Text Full Text PDF PubMed Scopus (77) Google Scholar) based on a partial set of rate constants showed that the favored pathway begins with a nucleation-promoting factor such as WASp binding actin and then the Arp2/3 complex. This ternary complex has no nucleation activity until it binds very slowly to the side of a filament. Then, a daughter filament grows at its free barbed end from the side of the “mother” filament (Fig. 2). Formins—Formins are homodimers with multiple domains, including formin homology (FH)-2 domains that associate with barbed ends of actin filaments (reviewed in Ref. 23Goode B.L. Eck M.J. Annu. Rev. Biochem. 2007; 76: 593-627Crossref PubMed Scopus (614) Google Scholar). Formins stimulate formation of unbranched actin filaments in cables, filopodia, and cytokinetic contractile rings. Simulations of experimental data suggest that FH2 domains nucleate filaments by stabilizing actin dimers (24Pring M. Evangelista M. Boone C. Yang C. Zigmond S.H. Biochemistry. 2003; 42: 486-496Crossref PubMed Scopus (190) Google Scholar) and show that free actin monomers account for all nucleation in the presence of profilin (25Paul A.S. Pollard T.D. Curr. Biol. 2008; 18: 9-19Abstract Full Text Full Text PDF PubMed Scopus (162) Google Scholar). 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Nature. 2005; 433: 488-494Crossref PubMed Scopus (290) Google Scholar) with solid theoretical support (29Shemesh T. Kozlov M.M. Biophys. J. 2007; 92: 1512-1521Abstract Full Text Full Text PDF PubMed Scopus (21) Google Scholar) is that the leading FH2 domain steps off the end the actin subunit binds (Fig. the may addition of actin subunit (Fig. (25Paul A.S. Pollard T.D. Curr. Biol. 2008; 18: 9-19Abstract Full Text Full Text PDF PubMed Scopus (162) Google Scholar). One an FH2 dimer to the of the growing actin but this is not observed the formin and of the filament are both to a (15Kovar D.R. Pollard T.D. Proc. Natl. Acad. Sci. U. S. A. 2004; 101: 14725-14730Crossref PubMed Scopus (357) Google Scholar). One idea is that an FH2 domain with the growing helix for about subunits and then steps the filament in the to T. Otomo T. Rosen M.K. Kozlov M.M. J. Biol. 2005; PubMed Scopus Google Scholar). domains to FH2 domains multiple that of and actin the barbed end of the filament (Fig. at rates of s–1 D. Kovar D.R. O'Shaughnessy B. Pollard T.D. Mol. Cell. 2006; Full Text Full Text PDF PubMed Scopus Google Scholar). is more favorable from than (25Paul A.S. Pollard T.D. Curr. Biol. 2008; 18: 9-19Abstract Full Text Full Text PDF PubMed Scopus (162) Google Scholar). work Biophys. J. 2006; Full Text Full Text PDF PubMed Scopus Google Scholar) showed that elongation in with a protein an FH2 domain produce more force subunit addition is coupled to hydrolysis of ATP bound to However, use ADP-actin monomers to elongate filaments D.R. Harris E.S. Mahaffy R. Higgs H.N. Pollard T.D. 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When the concentration of growing filaments is polymerization actin monomers and a of monomers at the leading such that diffusion of actin monomers bound to and profilin to the leading edge is for The rate of depends on the of growing filaments, an optimum of with on the polymerization at end and monomer polymerization at end models filament in a This of how the work by filament. A.E. Biophys. J. 2001; 81: Full Text Full Text PDF PubMed Scopus Google Scholar) Monte Carlo simulations of the growth of of actin filaments against a rigid the reactions in the nucleation model and into account the positions of every subunit in filament. assumed a concentration of formation only near the rates to to that in cells), and to polymerization to a Brownian Simulations on such as branching from the sides of filaments or only at barbed ends. of observed in with because the leading filaments, more filaments to the (19Schaus T.E. Taylor E.W. Borisy G.G. Proc. Natl. Acad. Sci. U. S. 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