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The Effect of Negative Feedback Loops on the Dynamics of Boolean Networks

Eduardo Sontag ^*, Alan Veliz-Cuba , Reinhard Laubenbacher  and Abdul Salam Jarrah 

^* Mathematics Department, Rutgers University, Piscataway, New Jersey; and Virginia Bioinformatics Institute/Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia

Correspondence: Address reprint requests to Eduardo Sontag, Tel.: 732-445-3072; E-mail: sontag{at}math.rutgers.edu.

Feedback loops play an important role in determining the dynamics of biological networks. To study the role of negative feedback loops, this article introduces the notion of distance-to-positive-feedback which, in essence, captures the number of independent negative feedback loops in the network, a property inherent in the network topology. Through a computational study using Boolean networks, it is shown that distance-to-positive-feedback has a strong influence on network dynamics and correlates very well with the number and length of limit cycles in the phase space of the network. To be precise, it is shown that, as the number of independent negative feedback loops increases, the number (length) of limit cycles tends to decrease (increase). These conclusions are consistent with the fact that certain natural biological networks exhibit generally regular behavior and have fewer negative feedback loops than randomized networks with the same number of nodes and same connectivity.