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

Wolfram Weckwerth

System theory has its roots in mathematical formalisms developed by mathematicians and physicists, such as Leibniz, Euler, and Newton, and applied by congenial chemists 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 (ODEs) 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 vs. a Signal–Input–Output (SIO) model of an electrical engineer. The drawback of this “Gedankenexperiment” is that two rather different approaches are compared—a forward model predictive control approach in the case of the SIO model by an engineer and an inverse or reverse approach by the biologist or ecologist. Biological and ecological systems are much too complex to estimate all the underlying ODE's, parameter and input signals that generate a probability distribution. Thus, the combination of inverse data-driven modeling and stochastic simulation is a key process for understanding 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 have developed a hybrid modeling approach based on the stochastic Lyapunov matrix equation for the analysis of genome-scale molecular data. This workflow connects forward and inverse 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 multiomics data. Ultimately, this workflow is linked to physiology and phenotype (the output) to unambiguously define the genotype–environment–phenotype relationship. This system-theoretical formalism establishes the generic analysis of the genotype–environment–phenotype relationship to finally result in predictability of organismal function in the environmental context. The approach is based on fundamental mathematical control theory for the analysis of dynamical systems using eigenvalues and matrix algebra, stochastic differential equations (SDEs), and Langevin- and Fokker–Planck-type equations eventually leading to the continuous stochastic Lyapunov matrix equation. The stochastic Lyapunov matrix equation is also a fundamental approach for the analysis and control of artificial intelligence systems in model predictive control and thus opens up completely new perspectives for the integration of systems engineering and systems biology. Furthermore, similar mathematical formalisms—using a community matrix instead of a stoichiometric matrix of a metabolic network—were also conceptually developed and applied by ecologists such as Levins and May in the analysis of stability and complexity of model ecosystems. Thus, the generalization of this hybrid forward–inverse approach spans from biology to ecology and promises to be a systematic iterative process that finally leads to functional units able to explain living systems up to their interaction in complex ecosystems.

Functional and Evolutionary Ecology, Large-Instrument Facility for Mass Spectrometry in Life Sciences, Department of Ecogenomics and Systems Biology
Frontiers in Applied Mathematics and Statistics
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Peer reviewed
Austrian Fields of Science 2012
Biomathematics, Systems biology, Mathematical modelling
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