Modeling methods

• Introduction
  
• Cooperativity
• Deterministic modeling
• Stochastic modeling
• Quantitative model and stability analysis
• C#Sim - algorithmic modeling

Introduction

In order to examine (i.e. simulate) the proposed genetic switches in silico, different modeling approaches were used. First, a deterministic model based on the probabilistic interpretation of gene regulation was constructed for each type of a genetic switch. Next, a stochastic simulation was performed to take inherent stochastic dynamics of gene expression into account. To further verify the results obtained using these methods, we also developed a quantitative model that builds upon experimental data. Moreover, we developed a modeling algorithm to more explicitly simulate transcription factor binding, considering number of available binding sites and competitive binding. Each modeling approach is discussed in the following sections. We also discuss the notion of cooperativity in the context of bistability.

Cooperativity

It is often assumed that functional cooperativity (e.g. multimeric regulation) is required for bistability. However, it has been shown theoretically that bistability can emerge in systems without multimeric regulation, provided that at least one regulatory autoloop is present. (Widder et al., 2006). Furthermore, in silico analysis has shown the existence of bistable architectures without the transcription factor cooperativity typically associated with switch-like properties. (Siegal-Gaskins et al., 2011). An essential feature of these proposed architectures was the competitive binding of two transcription factors to the promoter.

In terms of modeling, sigmoidal functions (characterizing the rate of change dP/dt) – arising from (Hill) exponents greater than one - are often equated with molecular cooperativity (the way the transcription factor binds to a promoter). However, as non-linearity and multi-stability can arise without assuming molecular cooperativity, it has been suggested this is not an accurate proposition and that mathematical, or functional cooperativity – referring to a sigmoidal function arising from system equations - should not automatically be interpreted as molecular cooperativity (Andrecut et al., 2011). One reason for this is that model equations represent a significant simplification of actual biological dynamics of gene expression, which include a large number of reactions not explicitly considered in modeling, such as reactions describing chromosome opening and transcription initiation.

For this reasons, we believe that sigmoidal behavior alone – arising in some of our models for transcription factors’ exponent values (non-linearity) greater than 1 - should not by default be interpreted as molecular cooperativity. Thus, in the context of modeling, with the term cooperativity we mean functional cooperativity greater than 1. Functional cooperativity equal to 1 is referred to as no cooperativity.

Indeed, in case of our positive feedback loop switch – which contains both competitive binding and regulatory autoloops - even deterministic models predict experimentally-verified bistability at low (i.e. close to 1; deterministic fractional occupancy model) or no (quantitative model) functional cooperativity. Our stochastic model of the positive feedback loop switch also predicts bistability is possible without cooperativity.

Deterministic modeling

Stochastic modeling

Quantitative model and stability analysis

Based on experimental data, we constructed what we refer to as a quantitative model. Please see Quantitative and stability model for details.

C#Sim - algorithmic modeling

References

Andrecut M, Halley JD, Winkler DA, Huang S. (2011) A General Model for Binary Cell Fate Decision Gene Circuits with Degeneracy: Indeterminacy and Switch Behavior in the Absence of Cooperativity. PLoS ONE 6(5): e19358. doi:10.1371/journal.pone.0019358.

Kaern M, Blake WJ, Collins JJ. (2003) The engineering of gene regulatory networks. Annual Review of Biomedical Engineering. 5, 179-206.

Kaern M, Elston TC, Blake WJ, Collins JJ. (2005) Stochasticity in gene expression: from theories to phenotypes. Nature. 6, 451-464.

Ribeiro AS, Lloyd-Price J. (2007) SGN Sim, a Stochastic Genetic Networks Simulator. Bioinformatics. 23 (6): 777-779.

Sauro HM. (2012) Enzyme Kinetics for Systems Biology. Future Skill Software.

Siegal-Gaskins D, Mejia-Guerra MK, Smith GD, Grotewold E. (2011) Emergence of Switch-Like Behavior in a Large Family of Simple Biochemical Networks. PLoS Comput Biol 7(5): e1002039. doi:10.1371/journal.pcbi.1002039.

Widder S, Macía J, Solé R. (2009) Monomeric Bistability and the Role of Autoloops in Gene Regulation. PLoS ONE 4(4): e5399. doi:10.1371/journal.pone.0005399.