Linear compartmental systems. I. kinetic analysis and derivation of their optimized symbolic equations

Abstract

The study of many biological systems requires the application of a compartmental analysis, together with the use of isotopic tracers, parameter identification and methods to evaluate the mean parameters. For all this, the kinetic equations of the compartmental system as a function of its parameters are needed. In this paper, we present some considerations on the diagrams of connectivity of linear compartmental systems and obtain new properties from the matrix corresponding to the ordinary first-order linear differential equation systems which describe their kinetic behaviour. Using these properties, symbolic equations are obtained in a simplified form. These equations provide the instantaneous amount of substance in any compartment of the system when zero input is injected into one or more of the system compartments, 1599 solely as a function of those parameters of compartmental systems which really have an influence on the sought expression. This is unlike what happens in the other symbolic equations obtained in a previous contribution that included all the fractional transfer coefficients involved in the compartmental system, regardless of whether or not they had an influence on the instantaneous amount of substance.

Kinetics of irreversible consecutive processes with first order second steps: analytical solutions

“…In fact, this model is a simple case of a linear compartmental system, which can always be analytically solved irrespectively of the size of the problem [7][8][9].…”

“…While the solutions of the rate equations for single step reactions are often well known and available in widely used textbooks (the relevant chapters of the textbook written by Espenson [1] are still the most authoritative source for kineticists), this is seldom the case for reaction sequences. Basically, the only case when analytical solutions are known is the one of first order reaction networks [2][3][4][5][6][7][8][9]. In such networks, each reaction rate is directly proportional to a single concentration, so the overall system of differential equations remains linear, and, therefore, reasonably easy to solve.…”

Persistence and stability of a class of kinetic compartmental models

“…The observability, controllability, realizability and identifiability of compartmental systems are summarized in [10] focusing mainly on linear models. The analytic solution of linear compartmental ODEs is studied in [15] in a kinetic context. A fundamental reference on the qualitative analysis of a wide class of general nonlinear compartmental models is [20], where important results can be found on the structure of equilibria and stability.…”

Linear compartmental systems: II—A software to obtain the symbolic kinetic equations

“…Yet these contributions are limited in that the symbolic equations obtained refer to the entire linear compartmental system, including all the fractional transfer coefficients and zero inputs, regardless of whether or not they influence the instant amount of substance in the desired system compartment. In part I of this series [5], we named these equations general non optimized kinetic equations, as indicated in Table 1 of the aforementioned work. One limitation of this software in connection with the equations implemented in the references indicated, in addition to those inherent in the BASIC language, is that it does not offer the equation form, but the expressions of the coefficients involved in it.…”