## Structural Model of Muscle Contraction

**Abstract:**

Mathematical modeling is a powerful method of representing many types of systems, allowing scientists to develop a better understanding of the mechanisms that govern them. This is especially true in biology, where it is often uneconomical, inefficient , or simply impossible to conduct certain experiments. Simulation is now considered to be on equal footing with theory and experiments in the process of scientific discovery.

The mechanisms of muscle contraction are of great interest to many scientists. Consequently, they have been studied extensively. Researchers in the medical field are especially interested in developing a tool that would allow them to further investigate and predict how muscles would respond under a variety of conditions imposed by disease.

The Huxley model, first formulated by A. F. Huxley in 1957 [5], is the accepted paradigm of muscle force production [2]. It was considered to be the first unifying theory of muscle contraction, as it succeeded in simultaneously considering mechanical, chemical, thermal, and structural changes in the muscle during contraction. Throughout this paper, when we refer to the model of muscle contraction, we mean A.F Huxley’s original theory of muscle contraction, and not a specific equation. Although the Huxley equation is still used today, the original mathematical formulation is limiting, and has since been extended to provide a more versatile mathematical model. G. Zahalak is credited with this extension, known as the Huxley-Zahalak equation [12], which is also founded upon A.F. Huxley’s original theory.

This thesis is organized into two main sections. Chapter 2 reviews the biological background upon which A.F. Huxley’s theory of muscle contraction is based. This includes the biochemical and physical structure of muscle, as well as the mechanism of contraction. We present a concise derivation of the Huxley [5] and the HuxleyZahalak equations [12] and discuss how they are used to calculate the total force generated in the muscle.

In Chapter 3 we begin by presenting the complete solution to the Huxley equation. This is followed by a derivation of the force equation from this solution and comparison of the results with A.V. Hill’s famous force-velocity curve [4]. Although a method for solving the Huxley-Zahalak equation is discussed, we only present an outline of its analytical solution. The chapter concludes with a discussion of numerical schemes.

The aim of this thesis is threefold. First to present a comprehensive review of A.F. Huxley’s theory of muscle contraction. Second to introduce the Huxley and the Huxley-Zahalak equations, which describe muscle contraction in mathematical terms. Third, to solve these equations analytically for constant speed of contraction.

**Author:** Hannah McKenzie

**Advisor:** Raymond J. Spiteri

**Download:** hmckenzie_bsc_thesis