NPFs concentrate actin monomers at the membrane. From there they transfer to growing filaments or nucleate new ones. Capping protein prevents further polymerization. The Arp2/3 branching factor interacts with a region of the NPF to transfer an actin monomer to activate the branching factor. Free ends can also polymerize with actin monomers from the solution, but grow four times faster with monomers transferred from the NPF [24], ensuring that network growth remains directed toward the membrane. Free ends interact with the membrane at angle θ. The on and off rates describe the association and dissociation rates for each of the binding events.

Depiction of full model. (a) Representation of a state in the full model. The full order of events that have been incorporated into the chain is recorded, emulating a path traced through a full branched network growing. Desaturated components are what is approximated to be the structure of the complete network based on the composition of the filament of interest, resulting in some number of active free ends at the membrane. (b) Transition rates in the full model. X ∈ {N, P, C} Grey squares can represent any type of event, as addition and removal rates depend only on the current number of free ends and most recent event.

The force response of the network impacts both the steady state average and the level of fluctuations around that steady state. (a) The distribution of the number of free ends is symmetrically distributed around the steady state average. As the total network density increases and the polymerization rate slows, the increasing width of the free ends distributions indicates greater fluctuations and a fractionally lower effect of the change of the single discrete nucleation or capping event given the higher overall number of free ends. (b) There is good agreement between the simulated E values, and the self consistent steady state solution. The trends of free ends E and load velocity , with increasing load are similar to experimental ones. The divergence of the simulated and analytic solution at high loads reflects the increasingly wide distribution of free ends with increasing load as seen in (a). However this high load state is very close to the stall load as demonstrated by the vanishingly small load velocity in these cases, and might be considered under a different regime.

(a) The instantaneous force response of a network grown against a load with constant stiffness. In this case the load exerts a force f (t) = x(t)kspring, where x(t) is the current displacement of the membrane. The energy required to move the membrane on monomer gap size remains Cf (t) = f (t)Δ. This system cannot be solved for a non equilibrium steady state due to the constant change in Cf until the network reaches stall. However the response with an initial plateau at low Cf followed by decay of load velocity shows trends similar to experimental findings in cases with spring-like loads. (b) After the initial drop in monomer addition rate, it recovers as the system begins to approach stall and shift the system composition. [12, 13]

Steady state behavior in various parameter regimes of c = ∑kx and kn/kc. (a) Load velocity profiles normalized by c (top) and steady state polymerization probability (bottom) for various values of c (b) Steady state free ends (top) and polymerization probability (bottom) profiles for various values of kn/kc

The entropy production imposes thermodynamic bounds on the nonequilibrium structure of the system, characterized by E. The grey shaded area indicate inaccessible configurations. (a) shows bound on ϵload from Eq. 8, while uses the bound in Eq. 9, both parametrically derived from the entropy production in Eq. 6. For a given average system density, there is some maximum load or monomer addition cost that the system has been assembling under. The distance between The model prediction and the mimic process bound reflects the amount of energy dissipated by the growing system. Yellow points in both plots represent values calculated from the experimental data of Li et al. [10] (Fig 1D bottom), scaled as described in Eq 13. Scaling parameters were s = 19.1, β = −9.5, and κ = .093

The system adapts dynamically to changes in applied load. (a) The fast response system, vload drops immediately when the load is increased, while the free end density changes more slowly, acting as the mechanism of adaptation. The mimic process fully reconstructs the load velocity, even away from steady states. The effect of the shift in free ends and system composition can be seen in the yellow line, which calculates the mimic process using the unadapted E and Pp values. (b) The adaptation error (Δvload/vload0) and rate follow the trends expected in a tradeoff with energy dissipation. Adaptation rate was determined by fitting a logistic function to the free end response curve. Energy dissipation is calculated by the entropy production per monomer addition before the perturbation.