Toward a Unification of System-Theoretical Principles in Biology and Ecology—The Stochastic Lyapunov Matrix Equation and Its Inverse Application

Abstract

System theory has its roots in mathematical formalisms developed by mathematicians and physicians, such as Leibniz, Euler and Newton, and applied by congenial ecologists and biologists such as Lotka and Bertalanffy. In these approaches the dynamical system – may it be either single organisms or populations of organisms in their ecosystems – is defined and formally translated into an interaction matrix and first order ordinary differential equations (ODE) which are then solved. This provides the background for the quantitative analysis of any linear to non-linear system. In his inspiring article “ can a biologist fix a radio?” Lazebnik made the differences very clear between a “guilt-by association” hypothesis of a modern biologist versus a Signal-Input-Output (SIO) model of an electrical engineer. The drawback of this “Gedankenexperiment” is that two different approaches are compared – a forward approach in case of the SIO model by an engineer and an inverse or reverse approach by the biologist or ecologist. The point is that biological and ecological systems are much too complex to estimate the deterministic input signals that generate a probability distribution. Thus, a combination of reverse data-driven modelling with stochastic simulation is a key process to understand the control of a biological or ecological system. The challenge of the next decades of systems biology is to link these approaches more systematically. Over the last years we and others have developed such an hybrid modelling approach based on the stochastic Lyapunov matrix equation for the analysis of genome-scale molecular data which links forward and reverse strategies such as the genome-scale based metabolic reconstruction of an organism and the calculation of dynamics around a quasi steady state using statistical features of large-scale PANOMICS data. PANOMICS combines all molecular and phenotypical levels of an organism from the genome, transcriptome, proteome, metabolome to morphology and physiology to unambiguously define the genotype-phenotype-relationship. This system-theoretical formalism establishes the generic analysis of the genotype-environment-phenotype-relationship to finally result in predictability of organismal function.