COMPUTATIONAL ABSTRACTION OF MUSCLE THIN FILAMENTS

The contraction of muscle results from the interaction of two sets of filaments: the thick filaments, composed mainly of myosin; and the thin filaments, composed mainly of of actin, tropomyosin, and troponin.

On the thin filament, actin monomers are arranged in a long helical polymer known as F-actin. Tropomyosin molecules form head-to-tail connections and bind to actin. Troponin molecules are bound to each tropomyosin molecule along F-actin. There exists a stoichiometry of seven actin molecules to one tropomyosin and one troponin.

It is generally accepted that the forces which cause filament motion are generated by cross-bridges (myosin S1) that extend radially from the thick filaments and interact cyclically with the thin filaments while hydrolyzing ATP. The regulation of muscle contraction, which is largely due to the association and dissociation of to and from TnC, has resulted in several models [6-13] all with the common feature that there is strong cooperative behavior between the various components [14].

The computational abstraction of the thin filament is achieved by considering each actin or troponin molecule as a finite automaton, namely actin automaton or troponin automaton, and the tropomyosin molecule as a local connection that passes the state information of a finite automaton to its nearby neighbors.

With respect to the binding of myosin S1, an actin monomer on the thin filament has at least two stable configurations. The inactive configuration denoted by corresponds to weakly bound S1 or no S1 bound and the active configuration denoted by corresponds to the strongly bound S1. The configurational change of an actin monomer corresponds to the association or dissociation of S1 and can be described as:

where and are association and dissociation rate constants, respectively. Eq. (1) can also be considered as the description of state transitions of an actin automaton. Within a unit time, the transition probability of an actin automaton from state to is and from state to is , where [S1] stands for the concentration of S1. The state transition of an actin automaton will be influenced by the states of its neighbors (the neighborhood-configuration). The rate constant of a given actin automaton depends on the neighborhood-configuration of the actin automaton and can be described by a set of local transition rules [1].

With respect to the binding of , each troponin molecule on the thin filament also has at least two stable configurations. The inhibiting configuration denoted by corresponds to non--bound TnC and the facilitating configuration denoted by corresponds to -bound TnC. The configurational change of troponin molecule can be described as:

where and are the association and dissociation rate constants of to and from TnC, respectively. As before, Eq. (2) describes the the state transition of an troponin automaton. The transition rate constant of a given troponin automaton depends on the neighborhood-configuration of the troponin automaton and is also described by a set of local transition rules.

The exact form of the local transition rules, once identified, will provide a quantitative understanding of the cooperativity between the constituent protein molecules on the thin filament, and at the same time, allow the automata system mimics muscle filaments under a wide range of experimental conditions [1]. The transition rate constants in the rules are fundamentally different from simple data-fitting parameters in the following ways: (1) the transition rate constants have clear physical meaning and correspond to well defined chemical processes; (2) the transition rate constants can be independently measured, in principle, without referring to any specific model; and (3) once the values of the rate constants are determined, either by direct experiment or by comparisons of the model with certain data sets, they are not allowed to have multiple values for comparison with different experimental data.

In the real system, molecules communicate with each other through physical interactions that are largely expressed by tropomyosin molecules. In the corresponding computational system, these physical interactions are represented by local connections (Fig. 1).

The computational simulation of the thin filament begins with arrays of finite automata distributed in states: , , , or , that represents the initial condition of the thin filament. There is a set of parameters that represents , [S1], [ATP], etc and can be read by every finite automata. At time step , all automata start to check the states of their neighbors, then make decisions about the their state transitions according to the corresponding local transition rules. After all state transitions are decided, the automata in the system update their states at the same time, which results a new generation of automata arrays at time . By repeating the above procedure, we can generate a sequence of automata arrays at discrete time intervals. This sequence, if it contains enough automata, can simulate both the equilibrium and dynamic experimental measurements of the thin filament. Simulations corresponding to various types of experiments were performed which showed excellent agreement with the real data [1].