Four-state models and regulation of contraction of smooth muscle. I. Physical considerations, stability, and solutions

Modelling gastrointestinal bioelectric activity

The development of an anatomically realistic biophysically based model of the human gastrointestinal (GI) tract is presented. A major objective of this work is to develop a modelling framework that can be used to integrate the physiological, anatomical and medical knowledge of the GI system. The anatomical model was developed by fitting derivative continuous meshes to digitised data taken from images of the visible man. Structural information, including fibre distributions of the smooth muscle layers and the arrangement of the networks of interstitial cells of Cajal, were incorporated using published information. A continuum modelling framework was used to simulate electrical activity from the single cell to the whole organ and body. Also computed was the external magnetic field generated from the GI electrical activity. The set of governing equations were solved using a combination of numerical techniques. Activity at the (continuum) cell level was solved using a high-resolution trilinear finite element procedure that had been defined from the previously fitted C1 continuous anatomical mesh. Multiple dipolar sources were created from the excitation waves which were embedded within a coupled C1 continuous torso model to produce both the cutaneous electrical field and the external magnetic field. Initial simulations were performed using a simplified geometry to test the implementation of the numerical solution procedure. The numerical procedures were shown to rapidly converge with mesh refinement. In the process of this testing, errors in a long standing analytic solution were identified and are corrected in Appendix B. Results of single cell activity were compared to published results illustrating that the key features of the slow wave activity were successfully replicated. Simulations using a two-dimensional slice through the gastric wall produced slow wave activity that agreed with the known frequency and propagation characteristics. Three-dimensional simulations were also performed using the full stomach mesh and results illustrated the slow wave propagation throughout the stomach musculature.

Weakly-coupled models for motor enzyme function

In strongly-coupled models for motor enzyme function, such as the original Huxley (1957) model for muscle, ATP binding and subsequent hydrolysis are required for the detachment and reattachment of every force-producing cross-bridge. In weakly-coupled models, cross-bridges can be 'mechanically detached' without ATP binding when they have been pushed far beyond their free energy minimum and have accumulated so much strain that the attached state is less stable than the detached state. Weakly-coupled models assume that these mechanically detached cross-bridges can rejoin the pool of detached molecules that can reattach as force-producing cross-bridges, without going through an ATP hydrolysis cycle. This paper bases this assumption on a thermodynamically rigorous model for interaction between a motor enzyme molecule and binding sites on a cytoskeletal protein filament, equivalent to other examples of ligand binding interactions. It attempts to identify more clearly the features that must be added to the idea of ligand binding equilibrium to simulate a weakly-coupled motor enzyme model. Models that assume a vectorial conformational change and a longitudinal series elastic element appear to be incompatible with the assumptions of weakly-coupled cross-bridge models. A stochastic computational method has been used to examine the properties of these models. The computations have examined the behaviour of a model containing a four-state ATPase cycle, but the model is computationally a nine-state model because a force-generating attached state is allowed to equilibrate with different detached states at negative and at positive distortions, and because three adjacent sites are considered as possible attachment sites for each of the two attached states of the ATPase cycle.