Mathematical models on muscle contraction

The force a whole muscle generates is dependent on its activation, the shortening velocity, its length and its cross- sectional area. On the molecular level the active force depends on the interaction between the two contractile proteins actin and myosin. The force generation of the muscle can be described and calculated by mathematical models. The currently accepted model for muscle contraction on the molecular level is the so called cross-bridge model introduced by Andrew Huxley (1957). Although the model gives good descriptions of the muscle in many conditions there are experimental setups where the force cannot be predicted correctly by the model, neither quantitatively and nor qualitatively. Based on the cross-bridge model the aims of this study are twofold: First, we want to explain recent experimental results with an improved muscle model. New experiments on the level of myofibrils by Prof. Herzog`s group from the Human Performance Laboratory, Calgary, show the need of a new model involving the giant protein titin and its interaction with actin. This interaction is not trivial and during this project we will focus on the development of a structural model. One possible application of the model is in the field of cerebral palsy, since pathological altered passive forces might be explainable by the model. Another possible field of medical application is dilated cardiomyopathy. Our second research focus is given by the exact mathematical formulation of the basic idea about the interaction of the contractile proteins. This interaction between a single myosin head and actin is based on the random behavior of the myosin head and can be modeled by a so-called hybrid regime-switching model, in which the continuous dynamics are intertwined with discrete events and subject to change by these events. Our aim is to investigate the relation between the cross-bridge model and regime switching models.

The current paradigm of muscle contraction of the striated muscle is the so-called cross-bridge theory introduced by Andrew Huxley in 1957. Although the basic two-state model introduced in 1957 has been elaborated in a significant number of reformulations, the basic mechanisms remains untouched: Briefly, muscle contraction is exclusively governed by the contractile proteins, actin and myosin. Contraction and force production is achieved by extensions from the myosin filament, so-called cross-bridges that interact cyclically with the actin filament thereby exerting force between actin and myosin. Attachment and detachment cycles are associated with ATP hydrolysis. Structural proteins like titin provide only passive forces at longer muscle lengths. The mathematical formulation of a large ensemble of myosin heads and actin filaments allows force and energy predictions based on the cross-bridge theory. While these predictions are nearly flawless when a muscle is kept at a fixed length (isometric contraction) or is allowed to shorten (concentric contraction), experiments and model predictions differ quantitatively and more important qualitatively when a muscle is stretched actively (eccentric contraction). We overcame the shortcomings by introducing and analyzing a so-called three-filament model where the giant protein titin is allowed to bind to actin in a force-dependent manner, thereby shortening its free length and changing force and stiffness of a muscle. We built a complex titin model based on the microscopic structure of the protein and coupled the model to an extended cross-bridge model. Finally, we compared model predictions to experiments performed on myofibrils, small intact muscle structures with a diameter of around 1m and forces in the magnitude of nanonewtons. The comparison showed that a model comprising activation dependent attachment of titin to the actin filament is able to explain some open mysteries in muscle research in an intriguingly simple way while leaving the flawless predictions of the cross-bridge theory for concentric and isometric contractions untouched.