Functional redundancy in the NF-κB signalling pathway Micha l W lodarczyk Faculty of Mathematics Informatics and Mechanics, University of Warsaw, ul. Banacha 2 02-097 Warsaw, Poland Tomasz Lipniacki * and Michal Komorowski † Institute of Fundamental Technological Research, Polish Academy of Sciences, ul. Pawi´ nskiego 5B 02-106 Warsaw, Poland (Dated: March 14, 2013) The ability to represent intracellular biochemical dynamics via deterministic and stochastic modelling is one of the crucial components to move biological sciences in the observe-predict-control- design knowledge ladder. Compared to the engineering or physics problems, dynamical models in quantitative biology typically dependent on a relatively large number of parameters. Therefore, the relationship between model parameters and dynamics is often prohibitively difficult to determine. We developed a method to depict the input-output relationship for multi-parametric stochastic and deterministic models via information-theoretic quantification of similarity between model parameters and modules. Identification of most information-theoretically orthogonal biological components provided mathematical language to precisely communicate and visualise compensation like phenomena such as biological robustness, sloppiness and statistical non-identifiability. A comprehensive analysis of the multi-parameter NF-κB signalling pathway demonstrates that the information-theoretic similarity reflects a topological structure of the network. Examination of the currently available experimental data on this system reveals the number of identifiable parameters and suggests informative experimental protocols. Supplementary Information available at: http://www.ippt.gov.pl/ ~ mkomor/redundancySI.pdf Last decades accumulated sufficient evidence that a num- ber of biological phenomena, in particular those related to intra-cellular dynamics, noise management, biochem- ical signalling cannot be understood by intuition alone and require mathematical formalism to explain and sum- marise available data. Expectably, mathematical mod- elling will help in prediction, control and design of bio- chemical networks. Therefore adaption of conventional modelling techniques is required to suit the specificity of these problems. Models of biochemical dynamics are dif- ferent from classical models of engineering and physics in a number of ways. Primarily they involve substantially larger relative number of parameters compared to avail- able data size. This challenge has given rise to a number of approaches aimed at improving our ability to develop, verify and apply multi-parameter mechanistic models of such systems. We can loosely group these methods into those aimed at determining model sensitivities to param- eter values [1–3], tools to estimate rate parameters [4–8], and techniques focused on maximisation of the informa- tion content of the experimental data [9–12]. The in- put (parameters) - output (dynamics) dependencies is the main considered object of the above methods. The concept of information, which in the Fisher sense is a sensitivity of an output to parameters, establishes a nat- ural language to communicate a number input-output * tlipnia@ippt.gov.pl † mkomor@ippt.gov.pl phenomena. Sensitive parameters exert strong impact on output and therefore are relatively easy to infer. In consequence, when aiming at parameters estimation, ex- perimental settings, which render model parameters sen- sitive, should be searched. A number of studies have re- ported the intrinsic feature of dynamic multi-parameter models of biochemical dynamics to be sensitive only to a small number of linear combinations of parameters [2, 13– 15]. The developed methodology substantial enriched our repertoire of techniques to investigate input-output rela- tionship in multi-parameter models [1–3, 11, 16–21]. In this paper we built upon these findings to take a com- prehensive view at the problem of sensitivities in multi- parameter models. A notion of functional redundancy between individual parameters and their groups (mod- ules) is introduced with Shannon Information being a measure of its strength. As a result we propose a natural and general mathematical language to precisely commu- nicate and visualise all types of compensation like phe- nomena i.e. multi-parameter sensitivity, biological ro- bustness, sloppiness and statistical non-identifiability. It allows for a more insightful interpretation of sensitivity coefficients, detection and elimination of non-identifiable parameters and guided design of experiments aiming at maximising the number of identifiable parameters. We also find two efficient ways to evaluate functional redun- dancy. One is based on the Fisher Information (FI), therefore is local in the parameters space and requires parameter values as input; second is local in the space of experimental results and is based on posterior distribu- arXiv:1303.3109v1 [q-bio.QM] 13 Mar 2013