Investigating the molecular mechanism of the Frank-Starling law and pressure-volume loops in tubular organs

The sarcomere is the repeating unit of striated muscle fibers that is responsible for contraction and force production of muscles in organs, including the heart and the esophagus. The stress produced by a single sarcomere results from an active stress component, produced by actin-myosin crossbridging during muscle activation, and a passive stress component, produced from stretching the elastic titin filament that spans the half sarcomere. Additionally, when a sarcomere is activated before stretch, a portion of titin binds to actin, effectively shortening the rest length of the titin, enhancing the passive stress. The pressure in tubular organs can then be solved by considering the hoop stress of the sarcomeres in a “loop” geometry. The coordinated action of the sarcomeres in response to muscle activation leads to gradients in internal pressure, driving fluid flow. This leads to characteristic patterns in each of these organs: in the heart, it’s the pressure-volume loops; in the esophagus, a traveling contraction wave to transport a bolus gives rise to pressure-area loops at cross sections. Prior studies have shown that the length dependent activation of sarcomeres is a necessary, but not a sufficient molecular mechanism behind the Frank-Starling law and the pressure-volume loops. Here, in addition to length dependent activation, we explore the effect of the shortening of the rest length of titin. A sarcomere level “tube” law for pressure was derived and solved in conjunction with the hyperbolic conservation laws for fluid flow in a 1D, tubular geometry. The tube law is a sum of the active stress of actin-myosin interactions, the passive stress of titin stretching, and the passive stress enhancement due to titin-actin binding. The fluid equations were simulated by solving for the characteristic variables of the hyperbolic system, then converted to fluid velocity and cross sectional area as functions of space and time. Pressure-volume relationships in the heart and esophagus were reproduced by resolving the sarcomeric mechanisms. Consequently, we provide a unified understanding of the pressure-volume loops in the heart and the pressure-area loops in the esophagus. This can also be used to explore similar phenomena in other tubular organs.