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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

Our deterministic models are based on fractional occupancy, a quantity which expresses the degree of saturation at the transcription factor binding site (Sauro, 2012). Fractional occupancy can be expressed as a ratio of active binding site states – i.e. states leading to gene expression – to all possible binding site states:

As such, it can be interpreted as a probability of a promoter being active, or a probability that transcription and/or translation will occur. Gene expression in our model, in turn, is proportional to this probability.

Two main possibilities are distinguished in our models depending on the type of the promoter. In case of a minimal promoter, binding of a transcriptional activator (leading to an active state) is required for transcription initiation. Other states – binding site being free or bound by a repressor - are considered inactive. In case of a constitutive promoter, binding of a transcriptional repressor leads to an inactive state, while unbound (or activator-bound) states are active, hence leading to gene expression.

We assumed large species concentrations compared to the number of available binding sites. We also assumed fast transitions between promoter states, i.e. transcription factor binding and unbinding occurs much faster than transcription/translation, which in this type of deterministic models are represented as a single step of protein production. Another simplification made was a representation of multiple repetitions of a binding site as a single binding site (e.g. a specific TAL-A:KRAB binding site may in reality have 10 repetitions, while in our models it is represented as a single site). As the purpose of our deterministic models is to provide an approximate, basic characterization of the proposed logic, we consider these simplifications acceptable. Other models, such as C#Sim models (described later), try to formalize some of these aspects in more detail.

The mathematical framework which describes protein production and into which we incorporate fractional occupancy is represented as a set of ordinary differential equations (ODEs) of the form:

and is roughly based on a framework proposed in (Kaern et al., 2005). Here:

• [Protein] is protein concentration at a given time;
• k is a constant specifying protein production rate in case of an active promoter.
• kb is a constant specifying the amount of leaky gene expression, meaning protein production that takes place even in the case of inactive (i.e. repressed constitutive or non-activated minimal) promoter.
• dg is a constant specifying protein degradation rate.
• P(active) is the probability of a promoter being in an active state – this probability is equal to fractional occupancy f.
• P(inactive) is the probability of a promoter being inactive and is equal to (1-f).

No concrete quantitative data describing e.g. transcription, translation or degradation rates of TAL effectors was available to us. For this reason, the parameter values (specifying e.g production or degradation rates) in our models were assumed based on commonly accepted propositions, such as that production (i.e. transcription and translation) rates of proteins are (usually) higher than their degradation rates. Parameter values that were used in our simulations should hence only be considered in relative terms (e.g. as protein production to degradation ratios). We also assumed parameter symmetry of different TAL effectors used (TAL-A and TAL-B), because we did not expect that one would have e.g. a production rate significantly higher than the other.

All deterministic models were implemented in MATLAB.

Stochastic modeling

Gene expression is an inherently stochastic process and deterministic simulation is often not satisfactory. The required conditions for the two approaches to be similar are large system size (high mRNA and protein numbers) and fast promoter kinetics (Kaern et al., 2005), but these conditions are not always met. Furthermore, stochastic models can capture the individuality of single cells, as opposed to deterministic models that can only predict the average behavior of a cell population (Kaern et al., 2003). For this reason, we performed stochastic simulations of the proposed genetic switches. Our results were obtained using SGN Sim, a stochastic genetic networks simulator (Ribeiro et al., 2007) based on stochastic simulation algorithm (SSA).

Competitive binding between an activator and a repressor exists in our positive feedback loop switch. This can lead to three major minimal promoter states: the promoter being free (i. e. unbound - neither an activator nor a repressor is bound, zero activation is present), bound by a repressor or bound by an activator. When minimal promoter is unbound, only leaky expression occurs. If bound by an activator, this causes high, activated expression levels. Bound repressor causes the expression levels to drop even below the ones achieved for the unbound (non-activated) state.

Our deterministic models made no clear distinction between free (unbound) minimal promoter state and a repressor-bound state – in both cases, inactive promoter state was assumed, leading to only leaky expression. In our stochastic models, we improved this by introducing different expression levels for all three states, achieving a more realistic formalization.

Using stochastic simulation, we also tested how the delay between transcription and translation affects bistability.

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.

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