Published: April 27, 2011 r2011 American Chemical Society 6272 dx.doi.org/10.1021/jp200578e | J. Phys. Chem. B 2011, 115, 6272–6278 ARTICLE pubs.acs.org/JPCB On the Relationship between Sensitivity Coeffcients and Transfer Functions of Reaction Kinetic Networks Tormod Drengstig, † Thomas Kjosmoen, † and Peter Ruoff * ,‡ † Department of Electrical Engineering and Computer Science and ‡ Centre for Organelle Research, Faculty of Science and Technology, University of Stavanger, Stavanger, Norway b S Supporting Information ’ INTRODUCTION Methods such as general and biological systems theory, 13 biochemical systems theory, 4,5 and metabolic control analysis and its extensions 616 provide important tools in the analysis of complex oscillatory or nonoscillatory reaction systems. 1720 We became interested in how adaptation processes in reaction kinetic networks can be characterized and identified by using both metabolic control analysis and methods from control engineering. 2126 For example, a condition for robust (rate parameter independent) perfect adaptation for a certain network site requires that the respective sensitivity coefficient between an input (for example, a changed rate constant) and an output (for example, a concentration) becomes zero. Using methods from control engineering, a sufficient condition for robust perfect adaptation can be formulated, which requires that the transfer function between input and output has a zero in origo indepen- dently of rate constant values. 26 Moreover, if the transfer func- tion has a zero in origo only for a given combination of rate constants or activation energies, the output still shows perfect adaptation, but since the adaptation behavior depends on certain parameter values, this situation is “nonrobust”. A question which repeatedly occurred to us was how control coefficients in particular, or sensitivity coefficients in general, may be related to transfer functions. In this work, we describe the relationship between sensitivity coefficients and their corresponding transfer functions from a general kinetic (state space) perspective, i.e., considering an open system model with independent rate con- stant parameters or temperature as input and species concentra- tions or fluxes as output. Ingalls 15 analyzed the relationship between response/control coefficients and transfer functions on the basis of stoichiometric network presentation. 9 However, as we show in the Supporting Information, Ingalls’ results may differ from the input/output state space formulation, because in the stoichiometric network formulation perturbations are not applied to individual rate constants but to reaction velocities and their linear combinations. The formulation of control coefficients by using stoichiometric networks 15 is dependent on the choice of model output, e.g., whether the outputs are concentrations or fluxes, and no formulation has been provided for how other inputs than reactions rates, for example temperature, can be included in the analysis. Received: January 19, 2011 Revised: March 25, 2011 ABSTRACT: Metabolic control analysis (MCA) and biochemical systems theory (BST) have become established methods when analyzing the behavior/kinetics of biochemical reaction systems. While the usage of MCA and BST involves the determination of sensitivities, e.g., steady state control coefficients (CCs), typically between reaction rates and concentra- tions/fluxes, transfer functions (TFs) from control engineering allow to analyze the connectivity between arbitrary input signals (e.g., rate constants or temperature) and arbitrary output signals (e.g., concentrations or fluxes) in the complex-valued s- or frequency domain. As CCs generally do not provide information about the connectivity between input and output signals, we became interested in the question of how CCs and TFs, or more generally, how arbitrary sensitivity coefficients (SCs) and TFs are related to each other. In this work, we describe a general relationship between SCs and their corresponding TFs from a general kinetic (state space) approach and show that the state space approach can describe the SC-TF relationship by a single equation. During our work, we became aware of an alternative method which relates CCs and TFs by using a stoichiometric network approach. In this work, we describe a procedure to identify conditions to determine whether a receptor-mediated input to a reaction kinetic network can show robust (perturbation independent) or nonrobust (balanced or perturbation dependent) adaptive or homeostatic behavior in an output. Compared to the stoichiometric network approach, the here described method allows for dealing with arbitrary (including empirically identified) kinetic expressions.