A Methodology for Vertical Translation Between Molecular and Organismal Level in Biological Feedback Loops

Abstract

Feedback loops are among the primary network motifs in living organisms, ensuring survival via homeostatic control of key metabolites and physical properties. However, from a scientific perspective, their characterization is unsatisfactory since the usual modelling methodology is incompatible with the physiological and biochemical basis of metabolic networks. Therefore, any "vertical translation", i.e. the study of the correspondence between molecular and organismal levels of causality, is difficult and in most cases impossible. As a viable solution, we demonstrate an alternative modelling platform for biological feedback loops that is based on key biochemical principles, including mass action law, enzyme kinetics, binding of mediators to transporters and receptors, and basic pharmacological properties. Subsequently, we show how this framework can be used for translating from molecular to systems-level behaviour. Basic elements of the proposed modelling platform include Michaelis-Menten kinetics defining nonlinear dependence of the output y(t) on an input signal x(t) with the Hill-Langmuir equation y(t) = G * x(t)n / (D + x(t)n), non-competitive inhibition for linking stimulatory and inhibitory inputs with y(t) = G + x1(t) / ((D + x1(t) * (1 + x2(t) / KI)) and processing structures for distribution and elimination. Depending on the structure of the feedback loop, its equifinal (steady-state) behaviour can be solved in form of polynomials, with a quadratic equation for the simplest case with one feedback loop and a Hill exponent of 1, and higher-grade polynomials for additional feedback loops and/or integer Hill exponents > 1. As a companion to the analytical solution, a flexible class library (CyberUnits) facilitates computer simulations for studying the transitional behaviour of the feedback loop. Unlike other modelling strategies in biocybernetics and systems biology, this platform allows for straightforward translation from the statistical properties of single molecules on a "microscopic" level to the behaviour of the whole feedback loop on an organismal "macroscopic" level. An example is the Michaelis constant D, which is equivalent to (k-1 + k2) / k1, where k1, k-1 and k2 denote the rate constants for the association and dissociation of the enzyme-substrate or receptor-hormone complex, respectively. From the perspective of a single molecule the rate constants represent the probability (per unit time) that the corresponding reaction will happen in the subsequent time interval. Therefore 1/k represents the mean lifetime of the complex. Very similar considerations apply to the other described constants of the feedback loop. In summary, this modelling technique renders the translation from a molecular level to a systems perspective possible. In addition to providing new insights into the physiology of biological feedback loops, it may be a valuable tool for multiple disciplines of biomedical research, including drug design, molecular genetics and investigations on the effects of endocrine disruptors.