Steady-state values for filament concentration, filament length, end turnover, and total monomeric actin as a function of Cap, cofilin, and profilin. The rate for activation of Arp2/3 by N-WASP was set to 0 for these simulations. Total actin was set to 200 μM and thymosin-β4 were set at 200 μM and 100 μM, respectively, at the beginning of each simulation and allowed to equilibrate for 10,000 s at which time all the species had attained steady-state levels. Free ATP, ADP, and Pi were clamped at 10 mM, 2 mM, and 2 mM, respectively. (A) Filament concentration is calculated by taking the sum of the concentrations of the three nucleotide forms of pointed ends (equal to the sum of nucleotide forms of capped and uncapped barbed ends). (B) The filament length is calculated from the state variables by taking the sum of all forms of F-actin and dividing it by the filament concentration. (C) The barbed end turnover was calculated from the sum of 36 individual rates (18 reactions) for the addition (given a positive value) and dissociation (assigned a negative value) of the three forms of G-actin and the three forms of profilin-G-actin to the three forms of barbed ends; these positive rates were exactly balanced by the negative rates associated with the corresponding addition and dissociation of G-actin at pointed ends. (D) The total G-actin is the sum of all nucleotide forms including free and bound to profilin and thymosin-β4.

The cell-like spatial geometry for the reaction-diffusion-advection model. The geometry is adapted from a public Virtual Cell 3D geometry originally formulated by Jason Haugh (see Biomodel “PDGF Gradient Sensing” under username “jmhaugh”). The expression defining the geometry is shown at the top (units of x,y,z is μm) and the image depicts a surface rendering; the cytosolic compartment is defined at all points where the expression evaluates to TRUE (assigned a numerical value of 1), and the plasma membrane is the surface where the expression transitions from true to false (i.e., a value of 0). Polymerization is triggered at the leading edge by ActiveNWASP (molecules/μm²) as depicted by the color scale and as defined by the expression at the bottom of the figure. This corresponds to 7246 molecules of ActiveNWASP molecules in a graded distribution within a 46 μm² band of membrane.

The rate of filament growth displays a sharp boundary between polymerization and depolymerization 1 μm from the cell edge. (a) A slice from the bottom xy plane of the 3D geometry shows the steady-state rate of assembly sharply peaking at the activated edge of the cell. (b) The same values as in a are mapped to a new color scale in which all the positive values are white and negative values are mapped from blue (−1.2 μM/s) to red (balanced assembly and disassembly). (c) The steady-state distribution of free barbed ends and free pointed ends mirror the regions of highest actin assembly and disassembly, respectively (top, lowest XY plane; bottom, central YZ plane. The plot to the right shows the concentration of ends as a function of distance along a line emanating from the edge toward the middle of the cell (superposed on the image in the upper left panel). (d) Steady-state distribution of assembly rates when the rate constants for dissociation of branches from ATP-bound and ADPPi-bound subunits on mother filaments are increased by a factor of 5 to match the normally faster rate constant for dissociation from ADP-bound subunits (24) (left) or when all the debranching rate constants are decreased by a factor of 4 (right). (e) Steady-state distribution of assembly rates when F-actin flow is turned off (left) or when the velocity field is set equal to 3200 nm/min ^∗BranchFraction (right); the latter is four times the nominal flow rate that was used for all the other simulations. (f) Steady-state distribution of assembly rates when the proportionality constant, D_GActin in the equation for F-actin diffusion coefficient is changed from the nominal value of 5 μm²/s as indicated; note that the actual diffusion coefficient of monomeric species remained at 5 μm²/s in these simulations. The initial concentrations and simulation details were identical to those of Fig. 5.

Reaction diagram for the dendritic actin nucleation pathway taken from the Virtual Cell model. Each green circle represents a distinct species or state variable. Each yellow oval represents a transformation or reaction. The solid lines connecting the species to the reactions represent the stoichiometry of each reaction. The dashed lines are directed from species that modulate a reaction without actually being consumed or produced. The reactions labeled “At the Membrane” depict species that are confined to the membrane surface but can interact with several species in the cytosol, as shown. The blue-framed boxes are labels corresponding to the particular mechanisms of the dendritic nucleation model in that general region of the diagram. Each of these is associated with the corresponding literature citations in Table S2. The expressions and rate constants for each reaction are also provided in Table S2. The rate expressions underlying each reaction are also accessible by double clicking their respective yellow icons in the public Biomodel called “Actin Dendritic Nucleation_Detailed Branching” under username “les” in the Virtual Cell database (http://vcell.org).

The effects of Cap and cofilin on actin polymerization after Arp2/3 activation. All these simulations were carried out with concentrations of total Arp2/3 (inactive + active + incorporated in branches) of 1 μM; ActiveNWASP of 1000 molecules/μm²; total actin, 200 μM; thymosin-β4, 100 μM; profilin, 20 μM. (A) Freebarbed ends. (B) G-actin. (C) Change in total F-actin relative to the steady state before activation. (D) Concentration of branches.

Steady-state spatial distributions of key model variables upon activation of dendritic nucleation at the cell edge. The color scale for the minimum (blue) and maximum (red) for each variable is provided under the name of each variable. Total F-actin corresponds to the sum of all species, including ends, that are part of polymerized actin and is given in units of concentration of subunits. The initial uniform Total FActin before activation of dendritic nucleation was 164 μM, so the rear of the cell loses FActin as the front gains. In the model, Velocity is proportional to Branch Fraction, so both show identical distributions but with differing scales; the Branch Fraction is derived from the state variables by dividing the total number of Arp2/3-capped pointed ends (which are part of a branch) by the total concentration of pointed ends. The Average Filament Length is derived by dividing Total FActin by the total concentration of pointed ends; the initial uniform value for this variable before activation is 322 subunits/filament. We also display the distribution of GATProf, that is G-actin associated with ATP and profilin; this is the most active monomeric G-actin species for assembly to barbed ends and is initially at 7.2 μM before activation of dendritic nucleation. Movie S1 shows the approach of all four variables to these steady-state distribution over the first 10 min after activation. The average concentration of the key molecules (sum of all their forms) were: actin, 200 μM; thymosin-β4, 100 μM; profilin, 10 μM; Cap, 1 μM; ADF/cofilin, 0 μM; Arp2/3, 1 μM and ActiveNWASP as specified by the distribution in Fig. 4. The initial concentrations for all the 46 individual species are derived form the corresponding steady state in the absence of ActiveNWASP as determined with a nonspatial (ODE) simulation; these concentrations are provided in Table S3 in the Supporting Material. The geometry was contained in a rectangular domain with x,y,z-dimensions of 30.4 μm, 15.2 μm, and 7.4 μm, respectively; the grid for the numerical simulations contained 90,090 cubic elements with dimensions of 334 nm.