Detection of biological switches using the method of Gröebner bases

Bistability and ability to switch between two stable states is the hallmark of cellular responses. Cellular signaling pathways often contain bistable switches that regulate the transmission of the extracellular information to the nucleus where important biological functions are executed.

In this work we show how the method of Gröebner bases can be used to detect bistability and output switchability. The method of Gröebner bases can be seen as a multivariate, non-linear generalization of the Gaussian elimination for linear systems which conveniently seperates the variables and drastically simplifies the simultaneous solution of polynomial equations. A necessary condition for fixed-point state bistability is for the Gröbner basis to have three distinct solutions for the state. A sufficient condition is provided by the eigenvalues of the local Jacobians. We also introduce the concept of output switchability which is defined as the ability of an output of a bistable system to switch between two different stable steady-state values. It is shown that bistability does not necessarily guarantee switchability of every state variable of the system. We further show that, for a bistable system, the necessary conditions for output switchability can be derived using the Gröebner basis. The theoretical results are incorporated into an analysis procedure and applied to several systems including the AKT (Protein kinase B), RAS (Rat Sarcoma) and MAPK (Mitogen-activated protein kinase) signal transduction pathways. Results demonstrate that the Gröebner bases can be conveniently used to analyze biological switches by simultaneously detecting bistability and output switchability.

The Gröebner bases provides a novel methodology to analyze bistability. Results clarify the distinction between bistability and output switchability which is lacking in the literature. We have shown that theoretically, it is possible to have an output subspace of an n-dimensional bistable system where certain variables cannot switch. It is possible to construct such systems as we have done with two reaction networks.