Lac Operon Boolean Models: Dynamical Robustness and Alternative Improvements

Marco Montalva-Medel,Thomas Ledger,Eric Goles,Gonzalo A Ruz

doi:10.3390/math9060600

Marco Montalva-Medel, Thomas Ledger + Show 2 more

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https://doi.org/10.3390/math9060600

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Abstract

In Veliz-Cuba and Stigler 2011, Boolean models were proposed for the lac operon in Escherichia coli capable of reproducing the operon being OFF, ON and bistable for three (low, medium and high) and two (low and high) parameters, representing the concentration ranges of lactose and glucose, respectively. Of these 6 possible combinations of parameters, 5 produce results that match with the biological experiments of Ozbudak et al., 2004. In the remaining one, the models predict the operon being OFF while biological experiments show a bistable behavior. In this paper, we first explore the robustness of two such models in the sense of how much its attractors change against any deterministic update schedule. We prove mathematically that, in cases where there is no bistability, all the dynamics in both models lack limit cycles while, when bistability appears, one model presents 30% of its dynamics with limit cycles while the other only 23%. Secondly, we propose two alternative improvements consisting of biologically supported modifications; one in which both models match with Ozbudak et al., 2004 in all 6 combinations of parameters and, the other one, where we increase the number of parameters to 9, matching in all these cases with the biological experiments of Ozbudak et al., 2004.

Highlights

The lac operon in Escherichia coli is a paradigmatic example of a genetic regulation system involving the interaction of positive and negative regulatory molecules
The system encodes a pathway for lactose catabolism that is hierarchically controlled by glucose availability, and it has been reported to exhibit a bistable behavior, since the catabolic genes are either uninduced (OFF) or induced (ON) in a single cell, depending on previous activation history and specific extracellular lactose and/or glucose concentrations [1,2]
Observe that it is easy to deduce that the steady states OFF and ON obtained in the dynamics of the original, and reduced models will continue to appear with any other deterministic update schedule because their are known to be invariant under update schemes; limit cycles could appear making the desired modeling less robust

Summary

Introduction

The lac operon in Escherichia coli is a paradigmatic example of a genetic regulation system involving the interaction of positive and negative regulatory molecules. The lac operon has been used as a model system for gene regulation since its initial description in [3], where the concept of an operon was first introduced. The operon includes three genes involved in the uptake and catabolism of lactose and/or structurally similar sugars. The first gene, lacZ, encodes for the enzyme β-galactosidase, which converts lactose to allolactose and to subsequent catabolic intermediates; lacY, the second structural gene of the pathway, encodes for lactose permease, a membrane transporter for lactose uptake. The product of gene lacA is an acetyltransferase that takes part in the degradation and excretion of non-lactose sugars that may be misrouted through the lactose degradation pathway

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