Response experiments for nonlinear systems with application to reaction kinetics and genetics.

Abstract

A unified type of response experiment is suggested for complex systems made up of individual species (atoms, molecules, quasi-particles, biological organisms, etc.). We make the following assumptions: (i) some of the species may exist in two forms, labeled and unlabeled, respectively; (ii) the kinetic and transport properties of the labeled and unlabeled species are the same, respectively (neutrality assumption); (iii) the experiment preserves the total input and output fluxes; only the fractions of the labeled compounds in the input and output fluxes are varied. Under these circumstances a linear integral superposition law connects the fractions of labeled species in the input and output fluxes. This linear superposition law is valid for homogeneous and inhomogeneous systems and for systems with intrinsic (hidden) state variables; it arises from the neutrality condition and holds even though the underlying dynamics of the process may be highly nonlinear. Because this response law does not involve the linearization of the evolution equations it has great potential for the analysis of complex physical, chemical, and biological systems. We compare our approach with the linearization techniques used in biochemistry and genetics. We consider a simple reaction network involving replication, transformation, and disappearance steps and study the influence of experimental (measurement) and linearization errors on the evaluated values of rate coefficients. We show that the method involving the linearization of the kinetic equations leads to unpredictable results; because of the interference between measurement and linearization errors, either error compensation or error amplification occurs. Although our approach does not eliminate the effects of measurement errors, it leads to more consistent results. For a broad range of input fractions no error amplification or compensation occurs, and the error range for the rate coefficients is about the same as the error range of the measurements.