A simplified 1D model for the chemo-mechanical coupling in sarcomere dynamics

Abstract

Muscle tissue is a biological actuator whose performance is controlled by a biochemical activation. This tissue is able to generate force when stimulated either electrically (intact muscle cells and larger sample) or by tuning the concentration of calcium ions in the surrounding uid (skinned cells and smaller samples). The underlying regulatory mechanisms are still poorly understood. The coupling between chemical activation and mechanical contraction is an open problem of great interest to cardiologists and the biomedical profession in general, since muscle diseases like cardiac arrhythmias and muscular dystrophy involve dysfunctions of the regulatory mechanisms. Simpli ed models are of major interest in this endeavour. Such models, not accounting for the molecular details of the contraction mechanism, may be characterized by a relatively small number of parameters, that can be calibrated by comparing the model dynamics with available experimental results. Moreover, the use of simpli ed models makes it possible to compute the long time evolution of large collective systems, thus bridging the gap between the single molecular motor and the sarcomere. In this thesis I have developed a computational method suited to simulate and support advanced experimental studies.To this end, I adapted a simpli ed 1D model to describe the dynamics of myosin II motors in a sarcomere. The model was inspired by the Brownian ratchet model proposed by Julicher et al., who employed it successfully to describe some properties of processive motors active in the cytoskeleton. Each state of the system evolves according to a Langevin stochastic equation. The transition from one state to the other is modelled using Kramers' idea of first-passage time between two metastable states. My focus is mostly on skeletal muscle, since this tissue is hierarchically organized in a very orderly way from the nanoscale to the microscale and beyond. Building on knowledge gained from studies in physiology, chemistry, and physics, I have scaled the model up to a half-sarcomere. The model is also extended to allow for partial activation levels and time-varying stimuli. To this end a switch-like approach is adopted. I implemented my model in C-language to compute its evolution in time. The numerical integration of three di erent setups was realized: i) a single{motor system, ii) a collective system of motors in series representing a pair of actin and myosin laments, and iii) a collective system of laments representing a (decimated) half{sarcomere. System i) has been compared to the pioneering single motor assay. In this assay, the actin lament, directly controlled by an optical tweezer, is brought near to a single myosin motor xed onto a support and the mechanical response to interaction is measured. The numerical results obtained are found to be in good agreement with the experimental data. System ii) consists of a lament pair, where the actin lament is linked harmonically to a xed point. It is found that the mean restraint force on the F-actin depends linearly on the number of myosin heads and on the density of docking sites on the actin lament Moreover, the mean force decreases exponentially for an increasing transition time from the non-interactive to the interactive state. The value of the inverse-transition time, on the contrary, does not a ect signi cantly the mean force. To test whether the collective motor system a ords an appropriate description of the interaction between actin and myosin laments, the model dynamics had to be compared with experimental data. To this end, I opted for an experimental setup where the actin lament is trapped by an optical tweezer and interacts with a bundle of myosin laments. This setup preserves the 3D arrangement realized within a sarcomere, where an F-actin can interact with up to three myosin laments. Also in this case, my numerical results are in good agreement with the experimental data. After ensuring that my model gives reasonable results under tetanized conditions, partial activation was introduced and results analized for the lament setup and the half{sarcomere setup. The partial activation mechanism is studied in the case of complete overlap, when the thick lament is superposed entirely to the thin lament. In this way, the e ects of partial activation are studied independently of the change in the number of myosin heads in the overlap region. The mean force, computed as a funtion of the activation level, exhibits the same qualitative behaviour for the lament pair and the half{sarcomere. Finally, I implemented a time-varying stimulus and computed accordingly the response of a lament pair and of a half{sarcomere. All the numerical results presented in this thesis were obtained with a serial homemade code. The required computer time seriously limits its application: a real-life half-sarcomere contains about ten thousand myosin motors, and a satisfactory model of it would likely require at least a few thousands of myosin heads. To compare, consider that my serial code allows for computation of collective systems comprising at most a few hundreds of myosin heads. This limitation may hopefully be overcome by implementing an e cient parallel version of the code. Once computer time is abated, it will become possible to simulate the experimental setups of sarcomere assays used for studying the intriguing e ects denominated force depression and force enhancement. Another promising line of research that would pro t from this simulation device could be the one intended to shed light on the details of the activation mechanism as a function of calcium concentration. In this case, di erent hypothetical activation mechanisms may be implemented in silico, and a careful comparative study of the numerical results and of the available experimental data may help verifying or falsifying the hypotheses under exam.