A simple network thermodynamic method for modeling series-parallel coupled flows I. The linear case

Abstract

A linear network thermodynamic approach to coupled flows through biological structures is developed based on the techniques introduced by Peusner (1970) who developed a means by which electrical network diagrams could be used to represent the flow-force equations of non-equilibrium thermodynamics. In this work, Peusner's methods are extended and elaborated, and a complete linear theory is presented. In further work in this series a non-linear theory is built upon the linear theory and applied to coupled flows through such tissues as epithelia. The network technique used here resembles the equivalent circuit approach used extensively to treat ion flows in cell and epithelial membranes (Finkelstein & Mauro, 1963). The major distinction is that the equivalent circuit approach was developed for independent ionic currents while the network thermodynamic approach deals explicity with the coupling in a multicompartment system. Using an obvious analogy between Kirchhoff's laws in electrical circuits and the conservation laws of continuum physics as applied to mass transport and chemical reactions, a purely topological argument leads to a proof of Tellegen's quasi-power theorem. From this Onsager reciprocity follows almost trivially for this kind of linear system and can be extended into the non-linear domain for certain non-linear systems. Tellegen's theorem also leads to a version of the minimum dissipation theorem in a rather elegant manner. Apart from the ease in obtaining these general theorems, the network approach has the advantage of giving a method for examining the effects of organizational or topological influences in any experimental design and allows for sorting out effects which arise from specific molecular properties (the properties of the hardware in the circuits) and those properties which arise from the way the systems are connected (topology). Methods are given for combining linear n-ports in series and parallel, reducing them symbolically to 1-ports connected in the same topology, Making sure that the computations are compatible with matrix algebra, the result of ordinary circuit analysis is used to deduce the matrix algebraic analysis appropriate for the n-port system from the solution to the 1-port circuit. The methods are applied to the Curran-MacIntosh model for isotonic transport in a series membrane system. The result shows how the membrane structure determines the composition of transported fluid independent of pump rate. In Appendix A, the relation between the network approach and Lagrangian mechanics is developed. In Appendix B, the method for dealing with time dependencies by using capacitors is illustrated using the osmotic transient as an example. In the thirdand final Appendix, the relationship between network thermodynamics and the reductionist philosophy that underlies so much of modern molecular biology is briefly examined.