Formulation of a Model for Assembly and Disassembly of the Actin Filament Network
Sirotkin et al. (2010)
collected enough quantitative information about the time courses of protein accumulation and loss in actin patches to formulate and test a mathematical model of the assembly and disassembly of actin and associated proteins (). We based our model on the dendritic nucleation hypothesis, the leading hypothesis for actin endocytic patch formation (Kaksonen et al., 2006
; Galletta and Cooper, 2009
). Because the detailed arrangement of molecules in actin patches is not known and diffusion is fast enough to consider the cytoplasm in the tiny patch to be homogeneous (see below), we made a temporal model with ordinary differential equations and did not attempt a spatial or mechanical model with partial differential equations.
In our simplified macroscopic model the reactions take place in a dimensionless compartment representing the endocytic patch. Only for the purpose of converting the number of molecules to local concentrations, we assumed that patches are spheres with diameters of 300 nm (B), the size of “filasomes” observed by electron microscopy (Kanbe et al., 1989
; Takagi et al., 2003
), although we understand that some proteins are not distributed homogenously in this space (Idrissi et al., 2008
). We assume that this patch is suspended in another dimensionless compartment representing the cytoplasm, which supplies reactants whose concentrations remain constant over time.
Comparison of Simulations with Observations of Actin Filament Assembly in Live Cells
Using the starting set of assumptions, cytoplasmic protein concentrations measured in live cells ( and Tables S2 and S3 in Sirotkin et al. (2010)
) and rate constants measured in diluted solution with purified proteins (Blanchoin and Pollard, 1999
; Fujiwara et al., 2007
; Kuhn and Pollard, 2007
; Beltzner and Pollard, 2008
), numerical simulations of the model assembled, and disassembled actin filaments. However, the amplitudes and timings of the protein peaks were not the same as observed in cells (A). The simulated time courses were even worse if severing was not included and actin filament disassembly depended entirely on dissociation of subunits from pointed ends at 0.25 s−1
To obtain good agreement between the simulated and measured time courses of protein accumulation and loss, we varied model parameters and tested alternative hypotheses about mechanism of patch disassembly. After adjusting two parameters and one mechanistic assumption, the simulated time courses for assembly and disassembly of multiple patch proteins were close to observations in live cells; B shows the main model with optimized parameters from and . Two association reactions must be faster to account for events in the cell. These enhanced association reactions are binding of the ternary complex of actin monomer-WASp-Arp2/3 complex to actin filaments (reaction 5) and barbed end capping (reaction 8). The most dramatic departure of the model from current beliefs is that dispersal of short filament fragments augments the disassembly of the actin filament network (reaction 12). The first-order rate constant for ATP hydrolysis measured in vitro did not need to be adjusted. The following paragraphs consider what the simulations revealed about each of the reactions.
Reaction 2: Recruitment and Activation of Nucleation-promoting Factors.
The available evidence indicates that recruitment and/or activation of WASp is the first step in the actin assembly process, although little is known about the mechanisms. Driving the following reactions with a Gaussian time course of active WASp accumulation and disappearance gave simulated time courses for the other components similar to the experimental observations. These simulations showed that only a small fraction of the total WASp is “active” free WASp, because most of the protein is bound to actin monomers, Arp2/3 complex, or filaments (see below; A).
For comparison with this assumed Gaussian wave, we tested two extreme variations of the time course for WASp accumulation and loss: sudden activation of all WASp in the patch at a specific time point (A) and a square wave (C). Subsequently active WASp is consumed, inactivated, and/or released. Simulations of time courses for the other proteins using these extreme initial conditions did not fit the observations as well as those obtained with a Gaussian initial condition, but shared the fundamental features of the process (, B and D). This leeway was important for our simulations, because microscopic measurements (Sirotkin et al., 2010
) document only the total amount of Wsp1p associated within the patches, not the numbers of active WASp.
When we started with a square wave of Wsp1p and optimized the parameter-driving process, the shape of the Wsp1p wave converged on an almost perfect Gaussian (C). Thus our choice of a Gaussian curve to drive the reactions was optimal but not crucial.
Reactions 3–6: Recruitment and Activation of Arp2/3 Complex.
We assumed that WASp sequentially binds an actin monomer (reaction 3) and an Arp2/3 complex (reaction 4) from the bulk cytoplasm to form an inactive ternary complex that is activated (reaction 6) to form a branch after it binds (reaction 5) to the side of a filament (Marchand et al., 2001
; Beltzner and Pollard, 2008
). The source of filaments to initiate the first branches at the onset of patch assembly is not known, but the simulations indicated that only a few short filaments are required to start the autocatalytic branching process that rapidly builds the filament network, as shown by direct observations in vitro (Achard et al., 2010
For the simulations to assemble actin filaments as fast as observed in patches, the ternary complex of actin monomer-WASp-Arp2/3 complex must bind to actin filaments ~400 times faster than calculated from the local protein concentrations and the very slow association rate constant of the purified proteins measured in bulk in vitro (Beltzner and Pollard, 2008
). Although this difference from initial expectations may seem alarming, this difference is quite informative and entirely reasonable in light of the factors considered in the Discussion
Reaction 7: Actin Filament Elongation.
The elongation rate constant measured in vitro and the cytoplasmic pool of 22 μM ATP-actin monomers ( in Sirotkin et al., 2010
) accounted for the time course of actin assembly.
A simple calculation, the difference between the experimental measurements of the numbers of Arp2/3 complex (ARPC5) and capping protein (Acp2p) molecules in a patch (Figure 6D in Sirotkin et al., 2010
), established that patches could contain up to 140 free barbed ends, but the simulations showed that actual number is much less, because at every point in time, a large fraction of the Arp2/3 complex has not initiated a branch (C). In fact, a rough estimate showed that only eight barbed ends are required to account for the maximum polymerization rate of about 2000 subunits per second (i.e., eight ends × 22 μM actin monomers × 12 μM−1
). The number of filaments growing at each point (D) in time is small, because the rate of nucleation is finite and capping barbed ends rapidly terminates elongation. In wild-type cells the unpolymerized actin monomer concentration is expected to be ~40 μM, so the filaments should grow twice as fast and the total polymerized actin per patch is expected to be about twofold higher than in mGFP-act1/act1+
diploids (Figure S1). This difference has a modest effect on the other reactions in the model.
Reaction 8: Actin Filament Capping.
We assumed that binding of cytoplasmic capping protein to barbed ends stops elongation. If all of the capping protein assembled in patches is bound to barbed ends, simulations showed that the binding reaction rate in patches is similar to the rate measured for mouse capping protein (Schafer et al., 1996
) and about 10-fold faster than expected from the cytoplasmic concentration of capping protein and the rate constant measured for Schizosacchromyces pombe
capping protein binding chicken actin filaments (Kuhn and Pollard, 2007
Capping at the rate observed in cells terminated elongation in ~0.2 s when the average filament was only a couple hundred of nanometers long. The ratios of polymerized actin to the number of filament ends in our simulations gave a similar estimate of 100–150 nm (D), a length that varied little during patch assembly and disassembly. Filament lengths estimated by these simulations are similar to average lengths observed by electron microscopy of budding yeast patches: 50 nm (corresponding to ~19 subunits) in isolated actin patches (Young et al., 2004
) and 100 nm (38 subunits) in patches on plasma membranes (Rodal et al., 2005
Reaction 9: ATP Hydrolysis by Polymerized Actin.
Fast disassembly of actin patches was consistent with the hydrolysis of ATP bound to polymerized actin at 0.3 s−1
, but Pi
-release had to be much faster than measured in vitro for muscle actin (0.003 s−1
; Blanchoin and Pollard, 1999
). Thus we assumed that Pi
-release from ADP-Pi
-actin is fast, as estimated for budding yeast actin (Bryan and Rubenstein, 2005
), and keeps pace with hydrolysis.
Reactions 10–12: Patch Disassembly.
The rapid disappearance of actin and all associated proteins in less than 10 s provided the greatest challenge to explain mechanistically. Assuming branching nucleation by Arp2/3 complex and termination by capping protein, most of the filaments will be capped on both ends. In vitro these caps dissociate slowly (Mullins et al., 1998
; Kuhn and Pollard, 2007
), limiting subunit dissociation.
Therefore, we postulated that patch disassembly depends on actin filament severing. Cofilin is expected to be a major contributor to severing, because cofilin mutations slow actin patch turnover (Lappalainen and Drubin, 1997
; Nakano and Mabuchi, 2006
; Okreglak and Drubin, 2007
). However, simulations showed that filament severing by cofilin and subsequent actin subunit dissociation from the ends were not sufficient to account for the rapid disappearance of actin from patches, owing to the intrinsically slow rates of ADP-actin dissociation from filament ends. Even if cofilin mediates rapid debranching and Arp2/3 complex dissociates rapidly, to depolymerize the actin in 10 s ADP-actin would have to dissociate from pointed ends at 83 s−1
(D) rather than the rate of 0.25 s−1
observed in vitro (Fujiwara et al., 2007
A simple solution is to assume that oligomers of actin severed from the ends of filaments diffuse out of patches into the cytoplasm (reaction 12), whereas the proximal stump remains anchored in the patch, as observed in vitro (Michelot et al., 2007
; Roland et al., 2008
). We tested several reaction rates to account for the time course of actin subunit loss from patches, and because they all gave the same qualitative results, we chose a simple kinetic mechanism for the loss of subunits. We assumed that the probability of severing is proportional to the concentration of polymerized actin subunits with bound cofilin. We assumed that fragments severed from a filament consist of the same uniform mixture of subunits (actin-ATP, actin-ADP/actin-ADP-Pi
, and actin-ADP-cofilin) as the whole population of filaments.
In simulations using dissociation of severed oligomers and an estimate of ~40 μM cytoplasmic cofilin measured on immunoblots (V. Sirotkin, unpublished observation), actin patches disassembled as quickly as in cells, if the rate constant for subunit loss by severing kChop was 0.003 μM−1 s−1 (B). Thus the modeling and simulations show that depolymerization alone cannot account for the turnover of actin and associated proteins, but severing and fragment dissociation is a reasonable alternative hypothesis consistent with the experimental measurements.
Diffusion of Molecules Does Not Limit Patch Assembly
Our model does not take into account the geometry of the actin network around the patch, but the actin filament network might influence diffusion of the reactants and limit the rates of the reactions. However, the following calculations show that diffusion of small proteins does not limit actin patch assembly. We estimate from our simulations that proteins from the cytoplasm are consumed at the maximum following rates: actin monomers 4462 s−1, Arp2/3 complex 70 s−1, capping protein 60 s−1, and cofilin 510 s−1. On the other hand, if we approximate the geometry of the membrane where polymerization occurs as a sphere of radius r + 25 nm, a minimum value for the flux of proteins reaching its surface can be estimated from the diffusion equation as 4πDArCmax, where D is the diffusion coefficient, A is Avogadro's number, and Cmax is the bulk concentration of each protein. If we use a low estimate for D + 2 μm2 s−1 in a crowded environment, the following numbers of proteins collide with the surface of the sphere each second: ~8000 actin monomers, 500 Arp2/3 complexes, 300 capping proteins, and 15,000 cofilins. In the worst case, this is 2–30 times more than the numbers of proteins consumed in the reactions. Thus our assumption regarding diffusion is valid even if crowding reduces diffusion up to one order of magnitude. However, this is unlikely, because we estimate that the actin network occupies <3% of the volume of the patch. This idea agrees well with estimates of diffusion and actin assembly in similar actin networks (Plastino et al., 2004; Rafelski et al., 2009).
Robustness of the Model
To test the robustness of the mechanism based on the dendritic nucleation hypothesis, we varied each parameter value, one at a time from 0.1 to 10 times the best value and simulated the time courses of actin, WASp, Arp2/3 complex, and capping protein. For each protein, we calculated for each peak three characteristic values: amplitude (B and Figure S2, E–H), time (A and Figure S2, A–D), and width, defined as the difference in time between the points when the protein number was 25% of the peak value during the assembly and the disassembly phases (C and Figure S2, I–L).
Figure 5. Sensitivity analysis. The model was simulated with each parameter varied ± 1 order of magnitude to determine the sensitivity of the time course of actin accumulation and dispersal on parameter values. (A) Times of protein peaks. (B) Amplitudes (more ...)
Simulations showed that the mechanism in is robust, because it assembles and disassembles actin filaments over at least this hundred-fold range of individual parameter values. Nevertheless, these parameter scans showed that some parameter values influence the amplitudes and times of the protein peaks, especially for actin and capping proteins ( and Figure S2).
The peak amplitude for polymerized actin increases with the rates of ternary complex formation (kWASPGBinding+, kArpComplexFormation+) and with the rate of elongation (kPolymerization+), whereas the rate of capping (kCap+) has the opposite effect. The aging and severing parameters (kHydrolysis, kCOFBinding+, kChop) have little influence on the size of the actin peak. The position in time of the actin filament peak is most sensitive to the last step of Arp2/3 complex activation (kArpActivation+) and the barbed end parameters (kPolymerization+ and kCap+). The duration of the actin filament peak depends mainly on the rate of the ternary complex binding to the side of a filament (kARPGWBindingF+) and its subsequent activation step (kArpActivation+) and less on the severing parameter (kChop) and the barbed end parameters.
Two-dimensional parameter scans (with two parameters varied at the same time) showed that simulations produce reasonable fits to the data if both the polymerization and capping rates vary in parallel up to 10-fold (Figure S3A) and that variation of the rates of cofilin binding to filaments, filament severing, or ATP hydrolysis ± 10-fold can be compensated for by varying either of the two other rates (Figure S3, B–D). Variation in the rates of WASp binding actin monomers and formation of the ternary complex can compensate for each other over a limited range (Figure S3E).