Stability, controllability, and observability of the "four state" model for the sarcomeric control of contraction

A model of the sarcomeric control of contraction at various loading conditions has to maintain three cardinal features: stability, controllability (where the output can be controlled by the input), and observability (where the output reflects the effects of all the state variables). The suggested model of the sarcomere couples calcium kinetics with cross-bridge (XB) cycling and comprises two feedback mechanisms: (i) the cooperativity, whereby the number of force-generating (strong) XBs determines calcium affinity, regulates XB recruitment, and (ii) the mechanical feedback, whereby shortening velocity determines XBs cycling rate, controls the XBs contractile efficiency. The sarcomere is described by a set of four first-order nonlinear differential equations, utilizing the Matlab's Simulink software. Small oscillatory input was imposed when the state variables trajectories reached a steady state. The linearized state-space representations of the model were calculated for various initial sarcomere lengths. The analysis of the state-space representation validates the controllability and observability of the model. The model has four poles: three at the left side of the complex plane and one integrating pole at the origin. Therefore, the system is marginally stable. The Laplace transform confirms that the state representation is minimal and is therefore observable and controllable. The extension of the model to a multi-sarcomere lattice was explored, and the effects of inhomogeneity and nonuniform activation were described.