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Comparison of deterministic and stochastic models of the lac operon genetic network.

ABSTRACT: The lac operon has been a paradigm for genetic regulation with positive feedback, and several modeling studies have described its dynamics at various levels of detail. However, it has not yet been analyzed how stochasticity can enrich the system's behavior, creating effects that are not observed in the deterministic case. To address this problem we use a comparative approach. We develop a reaction network for the dynamics of the lac operon genetic switch and derive corresponding deterministic and stochastic models that incorporate biological details. We then analyze the effects of key biomolecular mechanisms, such as promoter strength and binding affinities, on the behavior of the models. No assumptions or approximations are made when building the models other than those utilized in the reaction network. Thus, we are able to carry out a meaningful comparison between the predictions of the two models to demonstrate genuine effects of stochasticity. Such a comparison reveals that in the presence of stochasticity, certain biomolecular mechanisms can profoundly influence the region where the system exhibits bistability, a key characteristic of the lac operon dynamics. For these cases, the temporal asymptotic behavior of the deterministic model remains unchanged, indicating a role of stochasticity in modulating the behavior of the system.

SUBMITTER: Stamatakis M

PROVIDER: S-EPMC2996131 | biostudies-other | 2009 Feb

REPOSITORIES: biostudies-other

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Comparison of deterministic and stochastic models of the lac operon genetic network.

Stamatakis Michail M Mantzaris Nikos V NV

Biophysical journal 20090201 3

The lac operon has been a paradigm for genetic regulation with positive feedback, and several modeling studies have described its dynamics at various levels of detail. However, it has not yet been analyzed how stochasticity can enrich the system's behavior, creating effects that are not observed in the deterministic case. To address this problem we use a comparative approach. We develop a reaction network for the dynamics of the lac operon genetic switch and derive corresponding deterministic an ...[more]

PMID: 19186128

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Project description:Experimental studies of the flowering of Arabidopsis thaliana have shown that a large complex gene regulatory network (GRN) is responsible for its regulation. This process has been mathematically modelled with deterministic differential equations by considering the interactions between gene activators and inhibitors (Valentim et al. in PLoS ONE 10(2):e0116973, 2015; van Mourik et al. in BMC Syst Biol 4(1):1, 2010). However, due to complexity of the model, the properties of the network and the roles of the individual genes cannot be deducted from the numerical solution the published work offers. Here, we propose simplifications of the model, based on decoupling of the original GRN to motifs, described with three and two differential equations. A stable solution of the original model is sought by linearisation of the original model which contributes to further investigation of the role of the individual genes to the flowering. Furthermore, we study the role of noise by introducing and investigating two types of stochastic elements into the model. The deterministic and stochastic nonlinear dynamic models of Arabidopsis flowering time are considered by following the deterministic delayed model introduced in Valentim et al. (2015). Steady-state regimes and stability of the deterministic original model are investigated analytically and numerically. By decoupling some concentrations, the system was reduced to emphasise the role played by the transcription factor Suppressor of Overexpression of Constants1 ([Formula: see text]) and the important floral meristem identity genes, Leafy ([Formula: see text]) and Apetala1 ([Formula: see text]). Two-dimensional motifs, based on the dynamics of [Formula: see text] and [Formula: see text], are obtained from the reduced network and parameter ranges ensuring flowering are determined. Their stability analysis shows that [Formula: see text] and [Formula: see text] are regulating each other for flowering, matching experimental findings. New sufficient conditions of mean square stability in the stochastic model are obtained using a stochastic Lyapunov approach. Our numerical simulations demonstrate that the reduced models of Arabidopsis flowering time, describing specific motifs of the GRN, can capture the essential behaviour of the full system and also introduce the conditions of flowering initiation. Additionally, they show that stochastic effects can change the behaviour of the stability region through a stability switch. This study thus contributes to a better understanding of the role of [Formula: see text] and [Formula: see text] in Arabidopsis flowering.

Project description:BackgroundGillespie's stochastic simulation algorithm (SSA) for chemical reactions admits three kinds of elementary processes, namely, mass action reactions of 0th, 1st or 2nd order. All other types of reaction processes, for instance those containing non-integer kinetic orders or following other types of kinetic laws, are assumed to be convertible to one of the three elementary kinds, so that SSA can validly be applied. However, the conversion to elementary reactions is often difficult, if not impossible. Within deterministic contexts, a strategy of model reduction is often used. Such a reduction simplifies the actual system of reactions by merging or approximating intermediate steps and omitting reactants such as transient complexes. It would be valuable to adopt a similar reduction strategy to stochastic modelling. Indeed, efforts have been devoted to manipulating the chemical master equation (CME) in order to achieve a proper propensity function for a reduced stochastic system. However, manipulations of CME are almost always complicated, and successes have been limited to relative simple cases.ResultsWe propose a rather general strategy for converting a deterministic process model into a corresponding stochastic model and characterize the mathematical connections between the two. The deterministic framework is assumed to be a generalized mass action system and the stochastic analogue is in the format of the chemical master equation. The analysis identifies situations: where a direct conversion is valid; where internal noise affecting the system needs to be taken into account; and where the propensity function must be mathematically adjusted. The conversion from deterministic to stochastic models is illustrated with several representative examples, including reversible reactions with feedback controls, Michaelis-Menten enzyme kinetics, a genetic regulatory motif, and stochastic focusing.ConclusionsThe construction of a stochastic model for a biochemical network requires the utilization of information associated with an equation-based model. The conversion strategy proposed here guides a model design process that ensures a valid transition between deterministic and stochastic models.

Project description:Mechanical stimulation has been shown to reduce apnea of prematurity (AOP), a major concern in preterm infants. Previous work suggested that the underlying mechanism is stochastic resonance, amplification of a subthreshold signal by stochastic stimulation. We hypothesized that the mechanism behind the reduction of apnea length may not be a solely stochastic phenomenon, and suggest that a purely deterministic, non-random mechanical stimulation could be equally as effective. Mice and rats were anesthetized, tracheostomized, and mechanically ventilated to halt spontaneous breathing. Two miniature motors controlled by a microcontroller were attached around the abdomen. Ventilation was paused, stimulations were applied, and the time to the rodent's first spontaneous breath (T) was measured. Six spectrally different signals were compared to one another and the no-stimulation control in mice. The most successful deterministic stimulation (D) at reducing apnea was then compared to a pseudo-random noise (PRN) signal of comparable amplitude and frequency. CO2%, CO2 stabilization time (Ts), O2 saturation (SpO2%), and T were also measured. D significantly reduced T compared to no stimulation for medium and high amplitudes. PRN also reduced T, without a difference between D and PRN. Furthermore, both stimulations significantly reduced Ts with no significant differences between the respective stimulations. However, there was no effect of D or PRN on SpO2%. The lack of differences between D and PRN led to an additional series of experiment comparing the same D to a band-limited white noise (WN) signal in young rats. Both D and WN were shown to significantly reduce T, with D showing statistical superiority in reduction of apnea. We further speculate that both deterministic and stochastic mechanical stimulations induce some form of mechanotransduction which is responsible for their efficacy, and our findings suggest that mechanical stimulation may be effective in treating AOP.Supplementary informationThe online version contains supplementary material available at 10.1007/s13534-021-00203-x.

Dataset Information

Comparison of deterministic and stochastic models of the lac operon genetic network.

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Comparison of deterministic and stochastic models of the lac operon genetic network.

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