Ising Models of Cooperativity in Muscle Contraction

Abstract

Regulation of contraction in striated muscle is controlled by a dual mechanism involving both thin filaments containing actin and thick filaments containing myosin. The thin filament is activated by calcium ions binding to troponin, leading to tropomyosin azimuthal displacement which allows the activation of a regulatory unit (composed of one troponin, one tropomyosin and seven actin monomers) that exposes the actin sites for interaction with the myosin motors. Motor attachment to actin contributes to spreading activation within and beyond a regulatory unit along the thin filament through a cooperative mechanism. We introduce a one-dimensional Ising model to elucidate the mechanism of cooperativity in thin filament activation in relation to the force generated by the attached myosin motor. The model characterizes thin filament activation and cooperativity using only two parameters: one related to calcium concentration and the other to the force exerted by the attached myosin motor, which is modulated by temperature. At any force, the model is able to determine the extent of actin-myosin interactions on a correlation length ranging from two to seven actin monomers in addition to the seven actin monomers of the regulatory unit. Our theoretical predictions are successfully tested on experimental data, and our tests also include the condition of hindered filament activation by the use of the specific drug Omecamtiv Mecarbil (OM). According to our model, the effect of OM results in an anti-cooperativity mechanism accounting for the experimental data.

Figures (6)

• Figure 1: (a) Schematic of a muscle sarcomere. The red rectangle surrounds interacting myosin and actin filaments. (b) Activation-contraction mechanism: Ca^2+ ions bind to regulatory units (RUs) to allow molecular motor attachment and force generation; we stress that the RUs are only pictorially represented. (c) Typical force-pCa relation (red continuous line). The maximum slope of the relation (black dashed line), associated to the Hill coefficient $n_{\text{H}}$, quantifies the degree of cooperativity. (d) Two-state Ising model used in this paper.
• Figure 2: pCa-dependence of the normalized force $f$ in panel (a) and $\log_{10}\left(\frac{f}{1-f}\right)$ in panel (b) (black continuous line) obtained after calibration of the model to experimental data (symbols) measured at 25℃. The experimental data are normalized by the average tension measured at $\text{pCa}=4.5$. The insert in (b) shows the data chosen to fit the parameter $n_{\text{H}}$ using linearization at $c=1$, so that the results are not affected by the deviation of the data from the prediction at the extremes of the interval. The best fit [red dashed line in (b)] is achieved setting $\ce{[Ca^{2+}]_{50}} = \qty[parse-numbers = false]{10^{-6.5}}{\molar}$ and $n_{\text{H}}=3.14$. For one fiber, the value of the force at $\text{pCa}=9.0$ was measured twice. For that fiber, the mean of the two values was subtracted from the force to all data points generating a negative and a positive value of the same amplitude at pCa 9.
• Figure 3: Dependence of the coupling constant $J$ (a) and of the Hill coefficient $n_{\text{H}}$ (b) on the motor force modulated by temperature without OM (circles) and in the presence of 1µ OM (diamonds). Each symbol represents measurement from a single muscle fiber force_pca_data. The results are shown as function of the force per motor $F_0$ measured for different fibers. Experimental data are taken from caremani-2022. Error bars indicate the error in the calibration. In total, fifteen and sixteen measurements were performed with and without OM, respectively. The number of fibers used for each temperature are indicated in parentheses in the legend.
• Figure 4: (a) Correlation function at different calcium concentrations obtained from the model fitted to the experimental data at 12℃. (b) Correlation function at $c=1$ for the temperatures 12;17;25;35℃ presented in Fig. \ref{['fig-hillcoefficient']} in control conditions.
• Figure 5: Correlation length $\xi$ as a function of the force per motor $F_0$ modulated by temperature.