Some Statistical Mechanical Models of Elastic Polyelectrolytes and Proteins

Abstract

Three statistical mechanical models are developed for use in connection with the problem of muscle action. In addition, the models should apply to certain other systems, for example, polyelectrolyte fibers or gels, fibrous proteins in general, and ion exchange resins. The first model is a statistical mechanical version of the ``autone'' model of Varga, Szent-Gyorgyi, Gergely, and Laki, but we extend the theory to include the effect of an adsorbed substance (e.g., ATP) on the length of the elastic element. The second model is a charged rubber-like cylindrical fiber. The charges on the molecular chains of the network may be permanent or may result from the adsorption of ions from solution (e.g., hydrogen ions, or ATP). The approximate free energy of the system is computed by combining the Flory-Rehner swelling theory, the Fowler-Langmuir adsorption model, and an electrostatic free energy based on a ``smeared'' uniform charge density inside the fiber (electrolyte is also present), as suggested by the work of Hermans and Overbeeck. If the radius of the cylinder is large compared to the radius of the Debye-Hückel ionic atmosphere, it is not necessary to linearize the Boltzmann factor in the Poisson-Boltzmann equation. From the free energy of the system one obtains the interconnected pressure-volume, length-tension and adsorption isotherm relations. Important special cases are isotropic swelling (zero tension) with the gel under pressure or not, and the elasticity of the fiber (with adsorption) under zero pressure. When the fiber is constrained to a constant radius, it is possible to get a phase change in the length-tension curve (see below). The third model is an equilibrium statistical theory of the well-known α—β transformation in fibrous proteins (such as keratin and myosin). Adsorption of ions from solution and electrostatic effects can be included here as well as in the preceding model, but the discussion is concerned primarily with an analysis of the elasticity of a sheet of polypeptide chains with α- and β-units statistically scrambled, taking into account as a nearest-neighbor problem in statistical mechanics (using the quasi-chemical method) lateral β—β hydrogen bonds and vertical α—α hydrogen bonds. From the theory it would appear that critical phenomena, phase changes (for example, a considerable increase in length at constant tension, representing a phase change from primarily vertical α—α crystallization to primarily lateral β—β crystallization) and hysteresis are to be expected. These phenomena are observed experimentally, though modified in most cases by superimposed non equilibrium effects. It is pointed out that a fiber operating (on stretching and retraction) between the two ``ends'' of a phase change would be particularly suitable in the muscle problem, because of the sensitivity of the length of the fiber at constant tension to small changes in environment (e.g., pH). Although a few illustrative examples are given, further applications are reserved for a later paper.