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Analysis of Biological Reaction Networks

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Abstract

Cellular functions are realised by complex networks of chemical reactions. In most cases, however, several reaction schemes can be considered as plausible a priori hypotheses. This project aims at providing a set of methods that can be used to safely discard hypotheses on the basis of qualitative properties or measurement information. Methods include the construction of "safe" approximating automata and, more recently, the application of Feinberg's Chemical Reaction Network Theory.

Cooperation

People involved

Description

In chemical engineering and in biology, reaction networks are of paramount importance. Most "real world" networks are of enormous complexity. In cell biology, e.g., complexity often stems from the large number of strongly interacting components. The dynamics of such systems can rarely be understood by chemical or biological intuition alone. Mathematical modelling is therefore an especially attractive route to understand their functioning. In the process of "building" a mathematical model, however, one often faces uncertainty with respect to both network topology and reaction mechanisms. This usually implies the existence of several plausible hypotheses regarding the investigated reaction scheme; hence, according to Bailey, mathematical modelling basically means the development of families of differently structured models followed by a model discrimination step.

The standard modeling procedure is as follows: each candidate network (hypothesis) is translated into a set of ordinary differential equations with yet unknown parameters. Then an identification scheme is used that fits the unknown parameters in each set of differential equations to experimental data. Clearly, this is an extremely time-consuming procedure. If, for the currently investigated hypothesis, a satisfactory set of parameter values can be established, it is concluded that the hypothesis is a valid explanation of reality, and the corresponding candidate network, together with the resulting set of parameter values, is accepted as a possible dynamical model. However, if the identification procedure does not come up with a satisfactory set of parameters, it is only safe to discard the candidate network under consideration, if the error function is globally convex. As the latter is rarely the case, this procedure is only semi-conclusive and therefore not a sound basis for falsifying network hypotheses. These drawbacks are the motivation for our research on using qualitative network properties to conclusively rule out certain candidate networks. It is pursued in close cooperation with the SBI group at the MPI.

To date we have been investigating two approaches. One method is to construct safe discrete approximations of the continuous dynamic models associated with each reaction network. For this purpose, every network structure is translated into a finite state machine. Under some mild assumptions regarding the properties of the reaction kinetics involved, it can be shown that on a suitable discrete signal space this automaton is a conservative approximation of all possible sets of ordinary differential equations that can be derived for the network. If this automaton fails to explain available experimental data, the corresponding network structure can therefore be safely discarded [1]. This research builds on methods that have been developed within our project on hybrid control systems.

The second method currently under investigation is the incorporation of knowledge on the number of steady states into the process of model discrimination: suppose, e.g., that the existence of multiple steady states has been observed in experiments. It is then natural to ask which of the postulated network structures can, for some conceivable parameter vector, exhibit such a qualitative property. To this end Feinberg's Chemical Reaction Network Theory (CNRT) is applied. CNRT connects qualitative properties of ordinary differential equations corresponding to a reaction network to the network structure. In particular, its assertions are independent of specific parameter values and its only assumption is that all kinetics are of mass-action form. More specifically, we use Chemical Reaction Network Theory to identify those candidate networks that can exhibit multistationarity. Observation of multistationarity in experiments consequently falsifies all other networks.

This procedure has been successfully applied to different reaction networks representing a single layer of the well studied Mitogen-activated protein (MAPK) cascade. For example, recent results by Markevich et al. show that multilayered protein kinase cascades can exhibit multistationarity even on a single cascade level. Using CRNT, it is possible to show that the assumption of a distributive mechanism for double phosphorylation and dephosphorylation is crucial for multistationarity on the single cascade level; reaction networks incorporating different hypotheses for this step can therefore be safeley discarded [2],[3].

Publications

  1. Carsten Conradi, Jörg Stelling, Jörg Raisch. Structure discrimination of continuous models for biochemical reaction networks via finite state machines. In Proc. IEEE Int. Symposium on Intelligent Control, pages 138–143, Mexico City, Mexico, 2001.
  2. Carsten Conradi, Julio Saez-Rodriguez, Ernst D. Gilles, Jörg Raisch. Using chemical reaction network theory to discard a kinetic mechanism hypothesis. IEE Proc. Systems Biology, 152 (4):243–248, 2005.
  3. Carsten Conradi, Julio Saez-Rodriguez, Ernst D. Gilles, Jörg Raisch. Chemical Reaction Network Theory ... a tool for systems biology. In Proc. of the 5th MATHMOD, Vienna, Austria, 2006.

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