Fig 1.
Simplified cross-bridge cycle of myosin Ic.
We combine the unbound ADP⋅Pi state with the unbound ATP state as a single detached state (1). The four remaining actin-bound states are characterized by the nucleotides bound to the myosin head. The unbound or weakly bound states are shown with a blue background and the strongly bound states with a yellow background. Transitions between the states occur upon the binding and release of nucleotides or upon myosin’s attaching to or detaching from the actin filament. A transition from state (i) to state (j) is described by the transition rate ωij.
Table 1.
The transition rates and parameter values used in our description.
Although the values of most parameters have been measured or can be estimated in the force-free case, the values describing the force dependence have been determined by fitting our model to the experimental data. Note that the transition rate for ATP release is determined by the balance condition of Eq 43 and is not a free parameter of the description. Because the reported values have been measured at room temperature, we use kBT = 4 zJ. Rate constants with a superscribed zero indicate the value in the absence of an external force. The carets denote second-order rate constants with units of s−1M−1.
Fig 2.
The nucleotide-dependent unbinding rate of a single myosin Ic head from the strongly bound states as a function of force.
In (a), we fit the theoretical unbinding rate (Eq 53) (solid lines) to the experimental data (dots) for two different nucleotide concentrations. The orange and blue areas around the experimental data represent the 95% confidence intervals. The parameter values from this fit are summarized in Table 1 and the experimental data have been published [20]. (b) The modeled unbinding rate
as a function of force F for different nucleotide concentrations. Although an increased Pi concentration has no effect compared to (a), raising the ADP concentration reduces the unbinding rate. This reduction can be reversed by increasing the ATP concentration.
Fig 3.
Modeled probability distributions of the states of myosin Ic’s cross-bridge cycle.
(a) For physiological nucleotide concentrations and 100 μM actin, myosin Ic is largely unbound from actin under small forces. Considering only the strongly bound states, the ADP state (3) dominates for small forces and the ATP state (5) for forces larger than 1.5 pN. (b) A highly increased actin concentration of 10 mM populates the weakly bound ADP⋅Pi state (2) for small forces and myosin is mostly bound to the filament. (c) The occupancy of the ADP state (3) is increased by a raised ADP concentration of 250 mM and dominates for forces smaller than 2 pN and larger than 4 pN.
Fig 4.
Effective force-velocity relation.
The effective velocity v depends on the force F and on the actin concentrations. A large actin availability enhances the binding of the myosin head and therefore decreases the cycling time, which in turn increases the velocity. The inset shows the effective velocity in the absence of force as a function of the actin concentration.
Fig 5.
The modeled probability Psb of occupying the strongly bound states and the probability Pon of being bound to actin.
(a,c) Changing the nucleotide concentrations or (b,d) the actin concentration is qualitatively similar for probabilities Psb and Pon; an increased ADP concentration increases both probabilities. More actin in the vicinity of myosin also enhances both probabilities. However, the probability Psb of occupying strongly bound states saturates and is then limited by Pi binding, whereas the probability Pon of being bound to actin reaches unity for high actin concentrations.
Fig 6.
Ensemble of coupled myosin heads.
(a) A simplified ensemble model for determining the average number of myosin heads bound to the actin filament. Each of the N myosin heads can either bind to the filament with binding rate kon or unbind with unbinding rate koff. The system can be described as a finite Markov chain in which each state is associated with the number of bound myosin heads. The transition rates and
between these states are effective rates that depend on the number n of bound myosin heads. (b) The average number of bound myosin heads for an ensemble of N = 30 myosin molecules for a cooperative model (blue, solid line) in which the force per molecule f = F/n is the total external force F divided by the number n of bound myosin heads. In the two non-cooperative models (red and yellow solid lines) the force per molecule is either constant, f = F/N, or varies according to
, in which
is an estimated average number of bound motors. The results of a full Monte Carlo simulation of 30 elastically coupled myosin heads are shown as the broken lines for two different coupling stiffnesses: κ = 200 μN/m (orange, broken line) and κ = 500 μN/m (green, broken line).
Fig 7.
Number of engaged myosin heads.
(a) The fraction 〈n〉/N of bound myosin heads depends on the total number N of myosin molecules. A larger fraction of the heads is bound at low forces for a small ensemble than for a large ensemble. (b) In an ensemble of N = 30 myosin heads, the average number 〈n〉 of bound myosin molecules increases with an increasing actin concentration. In both figures, the analytic results (solid lines) are compared to Monte Carlo simulations (broken lines) with an elastic coupling stiffness of κ = 500 μN/m.
Fig 8.
Elastic properties of an ensemble of myosin Ic molecules.
(a) The extension x of an ensemble of N myosin molecules as a function of the applied force. The slope of this effective force-extension relation of an ensemble of myosin Ic molecules increases as the total number of molecules declines. We assume a spring constant for myosin Ic of κ = 500 μN/m. Because each head bound to the actin filament contributes to the stiffness, larger ensembles are stiffer. (b) The release Δx after reducing the binding probability by β-fold as a function of force for different numbers of myosin heads. For a constant force of 20 pN an ensemble of N = 10 heads relaxes about 20 nm after decreasing the binding probability by a 100-fold (red line). (c) The release Δx after reducing the binding probability by 100-fold and the elasticity of individual myosin molecules by 10-fold as a function of force. Increasing the total number N of myosin heads weakens the dependence on the force.
Fig 9.
Diagram modified from Fig 1 to calculate the average time in different states.
(a) To determine the time that myosin spends in the strongly bound states, we promote state (1) and (2) to absorbing states by eliminating the transitions from these states. (b) A closed diagram is obtained from (a) by redirecting the arrows into the absorbing states (1) and (2) back to the starting states (3) and (5) weighted with the starting probabilities π3 and (1 − π3), respectively. In the same manner, we construct the diagrams shown in (c) and (d) to determine the time that myosin is attached to actin.