From Binary to Multivalued to Continuous Models: The lac Operon as a Case Study

Abstract

CitationFrom binary to multivalued to continuous models: the lac operon as a case study. SummaryUsing the lac operon as a paradigmatic example for a gene regulatory system in prokaryotes, we demonstrate how qualitative knowledge can be initially captured using simple discrete (Boolean) models and then stepwise refined to multivalued logical models and finally to continuous (ODE) models. At all stages, signal transduction and transcriptional regulation is integrated in the model description. We first show the potential benefit of a discrete binary approach and discuss then problems and limitations due to indeterminacy arising in cyclic networks. These limitations can be partially circumvented by using multilevel logic as generalization of the Boolean framework enabling one to formulate a more realistic model of the lac operon. Ultimately a dynamic description is needed to fully appreciate the potential dynamic behavior that can be induced by regulatory feedback loops. As a very promising method we show how the use of multivariate polynomial interpolation allows transformation of the logical network into a system of ordinary differential equations (ODEs), which then enables the analysis of key features of the dynamic behavior.

Biochemical connectionism

“…A potential solution to both of these problems is to use continuousvalued generalisations of Boolean functions. Such models have the added advantage that they can be used to model quantitative (in addition to qualitative) aspects of real biochemical circuits, an idea that has been explored in a number of recent papers [113,32]. In [113], for example, the authors describe a method of transforming Boolean functions into equivalent ODEs, such that their behaviour is equivalent for concentrations of 0 and 1, but differ for intermediate values.…”

“…A more efficient alternative could be to use multi-valued logics rather than fully-continuous models. In [32], for example, the authors present a method of constructing equivalent multivalued logics from Boolean functions, and observe that a 3-valued logic offers important benefits over a Boolean model in terms of correctly capturing the dynamics of a biological regulatory circuit.…”

Piecewise Linear and Boolean Models of Chemical Reaction Networks

“…In certain limits, interactions between network elements become switch–like [Kauffman (1969); Glass (1975,b); Snoussi (1989); Mochizuki (2005); Alon (2006); Mendoza and Xenarios (2006); Davidich and Bornholdt (2008); Wittmann et al (2009); Franke (2010); Veliz-Cuba et al (2012); Casey et al (2006); Sun et al (2013)]. For example, the Hill function, f ( x ) = x n /( x n + J n ), approaches the Heaviside function, H ( x – J ), in the limit of large n , and the domain on which the network is modeled is naturally split into subdomains.…”

“…To simplify the system further, we can map values of x below the threshold to 0, and the values above the threshold to 1 to obtain a Boolean network (BN), a map $$h=false(h_1,\dots ,h_{N}false):{\{0,1\}}^N\to {\{0,1\}}^N,$$where each function h i describes how variable i qualitatively depends on the other variables [Glass and Kauffman (1973); Snoussi (1989); Thomas and D’Ari (1990); Albert and Othmer (2003); Mendoza and Xenarios (2006); Davidich and Bornholdt (2008); Abou-Jaoudé et al (2009, 2010); Wittmann et al (2009); Franke (2010); Veliz-Cuba et al (2012); Cheng et al (2013); Sun et al (2013)]. Such reduced systems are simpler to analyze, and share the dynamical properties of the original system, if the reduction is done properly.…”

“…In particular, singular perturbation theory can be used to obtain reduced equations within each subdomain and the boundary layers, and global approximations within the entire domain [Ironi et al (2011)]. On the other hand, although BNs have been used to model the dynamics of different biological systems, their relation to more complete models was demonstrated mainly in case studies, heuristically or only for steady states [Glass and Kauffman (1973); Glass (1975,b); Snoussi (1989); Thomas and D’Ari (1990); Albert and Othmer (2003); Mendoza and Xenarios (2006); Davidich and Bornholdt (2008); Abou-Jaoudé et al (2009, 2010); Wittmann et al (2009); Franke (2010); Veliz-Cuba et al (2012); Chaves et al (2010); Edwards et al (2001); Sun et al (2013)]. …”