Towards a Unified Theory of Muscle Contraction. I: Foundations

Abstract

Molecular models of contractility in striated muscle require an integrated description of the action of myosin motors, firstly in the filament lattice of the half-sarcomere. Existing models do not adequately reflect the biochemistry of the myosin motor and its sarcomeric environment. The biochemical actin–myosin–ATP cycle is reviewed, and we propose a model cycle with two 4- to 5-nm working strokes, where phosphate is released slowly after the first stroke. A smaller third stroke is associated with ATP-induced detachment from actin. A comprehensive model is defined by applying such a cycle to all myosin-S1 heads in the half-sarcomere, subject to generic constraints as follows: (a) all strain-dependent kinetics required for actin–myosin interactions are derived from reaction-energy landscapes and applied to dimeric myosin, (b) actin–myosin interactions in the half-sarcomere are controlled by matching rules derived from the structure of the filaments, so that each dimer may be associated with a target zone of three actin sites, and (c) the myosin and actin filaments are treated as elastically extensible. Numerical predictions for such a model are presented in the following paper.

Appendix

The Simplest Matching Conditions

For a given pair of nearest-neighbor myosin and actin filaments, the following prescription for matching heads to actin sites uses selection rules, defined by discrete cut-offs for the extent of longitudinal and azimuthal mismatches. The chosen site is assumed to be the center of a 3-site target zone. These rules are formulated for single myosin heads; they also apply to dimers with the obvious restriction that the two heads compete for a given site.

Longitudinal matching is achieved if ${\vert}x\vert < d_{\rm co}$, a cut-off distance, or

$$ \vert X_{n}^{\rm A}-X_{m}^{\rm M}\vert < d_{\rm co}\quad (\hbox{axial head-site match})$$

(A1)

in the relaxed lattice. Heads in adjacent layers cannot compete for the same site if d _co < a/6 (7.15 nm), a condition which seems eminently reasonable for myosin II.

Angular mismatches in the azimuthal plane involve the orientations of head and actin site relative to the rungs of the ladder (Fig. 7). The maximum head-ladder mismatch Δϕ_co should be above 20° to avoid a high fraction of inactive heads, and less than 30° to stop the head from exploring different F-actins, in which case the mapping would not be unique. The chosen actin site can be matched to the ladder with an angular tolerance of 13°, since adjacent monomeric sites on the same strand differ in angle by 2π/13 (27.7°). Ladders and F-actins in the unit cell are labeled by k and α as described in the main text. Let θ_kα be the angle of the kth ladder radiating from the F-actin labeled by α, so θ_k1 = (k−1)2π/3 and θ_k2 = θ_k1 + π (measured clockwise from the vertical in Fig. 7). In terms of the polar coordinates of Eqs. (9) and (10), these matching conditions are

$$ |\phi_{l\mu}^{\rm M}-\theta_{{k}\alpha}-\pi| < \Updelta \phi_{\rm co}\quad (\hbox{angular head-ladder match}), $$

(A2a)

$$ \hbox{min}|\phi_{n}^{\rm A}-\theta_{{k}\alpha}| \quad (\hbox{angular ladder-site match}) $$

(A2b)

provided angular differences are adjusted to [−π,π). Equation (A2a) gives the head-ladder map μ = μ _k α(l) of the main text, and is tabulated in Table 1. This mapping is hard-wired by the lattice and the starting orientation of the crowns. Taken together, Eqs. (A1) and (A2) give the mapping n = n _map(m,k,α) from head m to site n via the ladder (k,α); this mapping changes with the length of the sarcomere.

Table 1 The angular ladder-head map for the 2:1 filament lattice with ‘3-start’ F-myosin crowns and ϕ ^M₁₁ = 0

To summarize: as long as d _co < 7 nm and Δϕ_co < 30°, the mapping is 1–0 or 1–1; not all heads can be mapped to sites, but two heads cannot be mapped to the same site. In detail: (i) Each head can be matched to only one ladder, so it cannot map to sites on different F-actins. (ii) Each ‘rung’ of the ladder can map to only actin site at most. (iii) Different heads on the same crown cannot select the same actin site because they select different ladders. (iv) Heads in adjacent layers on the same F-myosin cannot select the same site, although their 40° difference in azimuthal orientation may allow them to select the same ladder. (v) Heads on the three F-myosins surrounding a given F-actin generally address different actin sites, because angular matching to ladders whose orientations differ by 120° implies a mean spacing of 120/27.7 = 4.33 sites. This is sufficient to accommodate nonoverlapping zones of three sites. (vi) Because the 43-nm F-myosin repeat exceeds the 36-nm F-actin repeat, some heads cannot be longitudinally matched to any site (Fig. 1c).

The above matching rules are not unique. They were constructed to give tight angular matching of heads to sites via the azimuthal orientation of the ladder. With this condition, the range of longitudinal matching within the 7.0-nm limit can be determined kinetically by the strain dependence of the binding rate. Loose longitudinal binding occurs with the ‘swing-roll-lock’ mechanism when it $\eta \gg 1$.

In the presence of filament motion, Eq. (A1) is replaced by

$$ |x_{{m,n}\alpha}(t)| < d_{\rm co}, $$

(A3)

with x _m,nα(t) defined by Eq. (15). In modeling, this condition should be applied to all detached heads after each time-integration step.