Actin Polymerization and Depolymerization Coupled to Cooperative Hydrolysis – EPAPS Appendices Xin Li 1 , Jan Kierfeld 2 , and Reinhard Lipowsky 1 1 Theory & Bio-Systems, Max Planck Institute of Colloids and Interfaces, 14424 Potsdam, Germany and 2 Physics Department, TU Dortmund University, 44221 Dortmund, Germany * (Dated: May 19, 2009) As in the main text, we denote the three protomer (or subunit) species ATP-actin, ADP/P i -actin, and ADP- actin by T, Θ, and D, respectively. Rates for cooperative ATP cleavage For cooperative ATP cleavage of a T protomer, we distinguish the three local neighborhoods TT, TΘ, and TD and the corresponding ATP cleavage rates ω cT , ω cΘ and ω cD . In order to eliminate one parameter, we take ω cΘ = ω cD = ω c and put ω cT = ρ c ω c , see equation (2) of the main text. In principle, an alternative choice seems to be ω cD = ω c and ω cT = ω cΘ = ρ c ω c . However, if we then considered the limiting case of vectorial cleavage with ρ c = 0, this limiting situation would be inconsistent with the hydrol- ysis process (VR) as observed experimentally in [1]. This can be understood as follows. In the vectorial limit with ω cT = ω cΘ = 0, the filament would exhibit only two protomer patterns: (i) a single Θ protomer between the T cap and the D core, and (ii) no Θ protomer between the T cap and the D core. The total cleavage flux J c is then given by J c = ω c ω r /(ω c +ω r ) with the P i release rate ω r for the Θ protomer at the ΘD do- main boundary. This expression implies the asymptotic behavior J c ≈ ω r for ω r ≪ ω c which is inconsistent with the process (VR). Indeed, process (VR) is characterized (i) by ω r ≪ ω c , see Table 1 in the main text, and (ii) by J c = ω c for actin concentrations C T >C T,c , see Fig. 2 in the main text. First protomer at barbed end We now discuss the steady state probabilities P T (1), P Θ (1), and P D (1) that the first protomer at the barbed end is a T, Θ and D protomer, respectively. The func- tional dependence of these probabilities on the actin con- centration C T is shown in Fig. 5 for process (RR) con- sisting of random ATP cleavage followed by random P i release with ρ c = ρ r = 1 as defined in Table 1 of the main text. In the absence of any actin monomers in the surround- ing solution, i.e., for C T = 0, one has P D (1) = 1 and P T (1) = P Θ (1) = 0 corresponding to a continuously de- polymerizing filament that consists only of D protomers. As the concentration C T is increased, the probability P D (1) for a D end decreases and the probability P T (1) FIG. 5: Steady state probabilities PT(1), PΘ(1), and PD(1) that the first protomer at the barbed end is a T, Θ, and D protomer, respectively, as functions of actin concentration CT for the process (RR) as defined in the main text, see Table 1. The three straight lines correspond to the analytical expres- sions that describe the asymptotic behavior for small CT. for a T end increases monotonically whereas the prob- ability P Θ (1) for a Θ end increases for small C T and decreases for large C T , see Fig. 5. The latter figure also shows that the large concentration regime is character- ized by P T (1) ≈ 1 and P D (1) ≈ P Θ (1) ≈ 0 as ex- pected. For small C T , one has P T (1) ∼ P Θ (1) ∼ C T and 1 − P D (1) ∼ C T . It is not difficult to calculate the corre- sponding expansion coefficients. As shown in Fig. 5, the resulting expressions are in very good agreement with the simulation data for small concentration C T . Flux balance relations for steady states Next, we express the global fluxes as provided by the filament growth rate J g , the total ATP cleavage flux J c , and the total P i release flux J r in terms of the three probabilities P T (1), P Θ (1), and P D (1). In the steady state, the filament growth (or elongation) rate J g , which is measured in units of protomers per unit time, has the general form J g = ω on − P T (1) ω off,T − P Θ (1) ω off,Θ − P D (1) ω off,D (1) with ω on = κ on C T as defined in the main text. Likewise, the total ATP cleavage flux J c is given by J c = ω on − P T (1) ω off,T and the total P i release flux J r by J r = ω on −