In this case , where the components of are built from rate constants for the transition from state to state . The diagonal elements of matrix are defined as where if and 0 otherwise; and off diagonal terms as where if and 0 otherwise. The rate constants have finite positive values (>0) only if transition between states and are possible otherwise they are equal to 0. Each component of the vector is the probability of finding the myosin head in one of its states at time and at strain (Huxley, 1957). The operator is the material derivative , where is the shortening velocity of the actin filament relative to myosin filament. Note that for simplicity, here we assume uniform shortening velocity along the filaments, i.e. . This is achieved, without loss of generality, by including a series elastic component equivalent to the elasticity of the filaments and apparently rigid filaments in the overlap region where myosin interacts with actin. The normalized shortening velocity, , denotes in half sarcomere lengths per second.
The state transition matrix , formed exclusively from rate constants , is singular because the sum of the elements in each column is equal to zero, i.e. determinant . Thus, to find a unique solution of Eq. 5, it is necessary to replace one row in by a constraint equation. In the original Huxley 57 model (Huxley, 1957), the constraint is that at any instant of time , the sum of all state probability densities at particular is equal to 1, i.e. . This constraint was later used in almost all sliding filament models (Hill, 1974; Pate and Cooke, 1989; Smith and Geeves,1995a, Smith and Geeves,1995b; Mijailovich et al., 1996). Examples of state transition matrices are shown here.
