Mathematical Modeling of the λ Switch: A Fuzzy Logic Approach

1. Introduction

Gene regulation plays a fundamental role in the development and evolution of organisms. Understanding gene regulation within living cells is a major scientific challenge in the post-genome era. Indeed, the analysis of gene regulating networks may have important implications in many fields of science, including biology and gene therapy. It may also lead to methods of synthesizing artificial networks with applications in biotechnology and biocomputing (Gardner, Cantor, & Collins, 2000).

The λswitch (Ptashne, 2004) is a relatively simple gene regulating network that controls two alternative patterns of gene expression in the bacterial virus λ. This epigenetic switch ensures an efficient change from one pattern to the other in response to suitable environmental cues. Bistable switches are common motifs in gene regulation networks, and the λ switch provides a convenient test case, as the virus is one of nature’s simplest organisms. In a recent survey paper, (Zhu et al., 2007) point out that the λ switch “has indeed established itself as one of the fundamental elements in biological processes and as a paradigm for both experimental and theoretical studies in biology.”

Developing suitable mathematical models for gene regulating networks is a non-trivial task. Several researchers have tried to gain a deeper understanding of the λ switch by deriving mathematical models for its dynamic behavior (see the review in Section 2.4 below). Most of the models are quite complex and, consequently, can be studied primarily using simulations and numerical analysis.

In this chapter, we apply fuzzy modeling (FM) to derive a new mathematical model for the λ switch. FM plays an important role in the fields of artificial intelligence and computational intelligence (Zadeh, 1994; Klir & Yuan, 1995). It is routinely used to transform the knowledge of a human expert, stated in natural language, into an artificial expert system (AES) that imitates the human expert’s functioning (Siler & Buckley, 2004; Kandel, 1992). Indeed, the real power of fuzzy logic lies in its ability to handle and manipulate linguistic information based on perceptions (Dubois, Nguyen, Prade, & Sugeno, 1998; Margaliot & Langholz, 1999, 2000; Zadeh, 1996; Novak, 2005). FM provides a simple yet highly efficient approach for transforming verbal descriptions into well-defined mathematical models or algorithms.

Recently, FM has been used to derive mathematical models for biological phenomena. Biologists often provide verbal descriptions and explanations of the phenomena they study. FM provides a convenient tool for transforming these verbal descriptions into well-defined mathematical models. Note that this application of FM is somewhat different than the typical approach applied in the construction of AESs. The motivation is not to replace the human expert with an automatic algorithm, but rather to assist a human expert in transforming his/her knowledge concerning a biological phenomenon, stated in words, into a well-defined mathematical model. The usefulness of this approach was demonstrated by developing mathematical models for animal behavior (Tron & Margaliot, 2004, 2005; Bajec, Zimic, & Mraz, 2005; Rashkovsky & Margaliot, 2007; Rozin & Margaliot, 2007; Margaliot, 2007).

Fuzzy modeling of biological systems offers several advantages (Margaliot, 2008). The resulting model represents the real system in a form that corresponds closely to the way humans perceive it. Thus, the model is understandable, even by non-professionals, and each parameter has a readily perceivable meaning. The model can be easily altered to incorporate new phenomena, and if its behavior is different than expected, it is usually possible to determine which rule/term should be modified and how.