# Force transients and minimum cross-bridge models in muscular contraction

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DOI: 10.1007/s10974-008-9131-3

- Cite this article as:
- Kawai, M. & Halvorson, H.R. J Muscle Res Cell Motil (2007) 28: 371. doi:10.1007/s10974-008-9131-3

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## Abstract

Two- and three-state cross-bridge models are considered and examined with respect to their ability to predict three distinct phases of the force transients that occur in response to step change in muscle fiber length. Particular attention is paid to satisfying the Le Châtelier–Brown Principle. This analysis shows that the two-state model can account for phases 1 and 2 of a force transient, but is barely adequate to account for phase 3 (delayed force) unless a stretch results in a sudden increase in the number of cross-bridges in the detached state. The three-state model \(({\mathbf{A \rightarrow B\rightarrow C\rightarrow A}})\) makes it possible to account for all three phases if we assume that the \({\mathbf{A\rightarrow B}}\) transition is fast (corresponding to phase 2), the \({\mathbf{B \rightarrow C}}\) transition is of intermediate speed (corresponding to phase 3), and the \({\mathbf{C \rightarrow A}}\) transition is slow; in such a scenario, states **A** and **C** can support or generate force (high force states) but state **B** cannot (detached, or low-force state). This model involves at least one ratchet mechanism. In this model, force can be generated by either of two transitions: \({\mathbf{B\rightarrow A}}\) or \({\mathbf{B\rightarrow C}}\). To determine which of these is the major force-generating step that consumes ATP and transduces energy, we examine the effects of ATP, ADP, and phosphate (Pi) on force transients. In doing so, we demonstrate that the fast transition (phase 2) is associated with the nucleotide-binding step, and that the intermediate-speed transition (phase 3) is associated with the Pi-release step. To account for all the effects of ligands, it is necessary to expand the three-state model into a six-state model that includes three ligand-bound states. The slowest phase of a force transient (phase 4) cannot be explained by any of the models described unless an additional mechanism is introduced. Here we suggest a role of series compliance to account for this phase, and propose a model that correlates the slowest step of the cross-bridge cycle (transition \({\mathbf{C\rightarrow A}}\)) to: phase 4 of step analysis, the rate constant *k*_{tr} of the quick-release and restretch experiment, and the rate constant *k*_{act} for force development time course following Ca^{2+} activation.