Coupling of actin hydrolysis and polymerization: Reduced description with two nucleotide states

The polymerization of actin filaments is coupled to the hydrolysis of adenosine triphosphate (ATP), which involves both the cleavage of ATP and the release of inorganic phosphate. We describe hydrolysis by a reduced two-state model with a cooperative cleavage mechanism, where the cleavage rate depends on the state of the neighboring actin protomer in a filament. We obtain theoretical predictions of experimentally accessible steady state quantities such as the size of the ATP-actin cap, the size distribution of ATP-actin islands, and the cleavage flux for cooperative cleavage mechanisms.

💡 Research Summary

The authors present a minimal two‑state description of actin filament dynamics that captures the essential coupling between polymerization and ATP hydrolysis. In the full three‑state picture a filament monomer can be in the ATP‑bound (T), ADP‑Pi (Θ) or ADP (D) state. The authors merge the Θ and D states into a single “D*” state, thereby reducing the system to a binary sequence of T and D* monomers. Hydrolysis (cleavage of ATP) is modeled as a cooperative process: the cleavage rate of a T monomer depends on the identity of its neighbor on the pointed (minus) side. If that neighbor is a T monomer the cleavage proceeds with rate ω_cT = ρ ω_c, whereas if the neighbor is D* the rate is ω_cD* = ω_c. The parameter ρ (0 ≤ ρ ≤ 1) quantifies cooperativity; ρ = 1 corresponds to random (non‑cooperative) cleavage, ρ = 0 to strictly vectorial cleavage at the T/D* interface, and intermediate values describe partially cooperative mechanisms.

Polymerization occurs at the barbed end with an attachment rate ω_on = κ_on C_T proportional to the concentration C_T of ATP‑actin monomers, and a detachment rate ω_off,T. Because D* monomers bind very weakly, their attachment is neglected. In the regime of high C_T (C_T ≫ C_T,g = ω_off,T/κ_on) the filament tip is almost always a T monomer, allowing the authors to set the probability of a D* at the tip, P_1,D*, to zero. The net growth rate then simplifies to J_T ≈ κ_on C_T − ω_off,T.

The core of the analysis is the master equation for the probability p_k that the ATP‑cap (the “ATP‑tip”) has length k. In steady state the equation reduces to a recursion for the cumulative probability P_k = ∑_{ℓ≥k} p_ℓ. By applying a continuum approximation in k, the recursion is transformed into a second‑order differential equation whose solution involves Airy functions. The exact steady‑state solution is

 P_k = exp(−a k/2) Ai