Stochastic models for regulatory networks of the genetic toggle switch

Bistability arises within a wide range of biological systems from the λ phage switch in bacteria to cellular signal transduction pathways in mammalian cells. Changes in regulatory mechanisms may result in genetic switching in a bistable system. Recently, more and more experimental evidence in the form of bimodal population distributions indicates that noise plays a very important role in the switching of bistable systems. Although deterministic models have been used for studying the existence of bistability properties under various system conditions, these models cannot realize cell-to-cell fluctuations in genetic switching. However, there is a lag in the development of stochastic models for studying the impact of noise in bistable systems because of the lack of detailed knowledge of biochemical reactions, kinetic rates, and molecular numbers. In this work, we develop a previously undescribed general technique for developing quantitative stochastic models for large-scale genetic regulatory networks by introducing Poisson random variables into deterministic models described by ordinary differential equations. Two stochastic models have been proposed for the genetic toggle switch interfaced with either the SOS signaling pathway or a quorum-sensing signaling pathway, and we have successfully realized experimental results showing bimodal population distributions. Because the introduced stochastic models are based on widely used ordinary differential equation models, the success of this work suggests that this approach is a very promising one for studying noise in large-scale genetic regulatory networks.

Numerical methods for second-order stochastic differential equations

We seek numerical methods for second‐order stochastic differential equations that reproduce the stationary density accurately for all values of damping. A complete analysis is possible for scalar linear second‐order equations (damped harmonic oscillators with additive noise), where the statistics are Gaussian and can be calculated exactly in the continuous‐time and discrete‐time cases. A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep. The only Runge–Kutta method with a nonsingular tableau matrix that gives the exact steady state density for all values of damping is the implicit midpoint rule. Numerical experiments, comparing the implicit midpoint rule with Heun and leapfrog methods on nonlinear equations with additive or multiplicative noise, produce behavior similar to the linear case.

Stochastic simulation of chemical reactions in spatially complex media

Discrete stochastic simulations, via techniques such as the Stochastic Simulation Algorithm (SSA) are a powerful tool for understanding the dynamics of chemical kinetics when there are low numbers of certain molecular species. However, an important constraint is the assumption of well-mixedness and homogeneity. In this paper, we show how to use Monte Carlo simulations to estimate an anomalous diffusion parameter that encapsulates the crowdedness of the spatial environment. We then use this parameter to replace the rate constants of bimolecular reactions by a time-dependent power law to produce an SSA valid in cases where anomalous diffusion occurs or the system is not well-mixed (ASSA). Simulations then show that ASSA can successfully predict the temporal dynamics of chemical kinetics in a spatially constrained environment.

Oscillatory regulation of Hes1: Discrete stochastic delay modelling and simulation

Discrete stochastic simulations are a powerful tool for understanding the dynamics of chemical kinetics when there are small-to-moderate numbers of certain molecular species. In this paper we introduce delays into the stochastic simulation algorithm, thus mimicking delays associated with transcription and translation. We then show that this process may well explain more faithfully than continuous deterministic models the observed sustained oscillations in expression levels of hes1 mRNA and Hes1 protein.

An examination of monotony and hypovigilance, independent of fatigue : relevance to road safety

Driving is a vigilance task, requiring sustained attention to maintain performance and avoid crashes. Hypovigilance (i.e., marked reduction in vigilance) while driving manifests as poor driving performance and is commonly attributed to fatigue (Dinges, 1995). However, poor driving performance has been found to be more frequent when driving in monotonous road environments, suggesting that monotony plays a role in generating hypovigilance (Thiffault & Bergeron, 2003b). Research to date has tended to conceptualise monotony as a uni-dimensional task characteristic, typically used over a prolonged period of time to facilitate other factors under investigation, most notably fatigue. However, more often than not, more than one exogenous factor relating to the task or operating environment is manipulated to vary or generate monotony (Mascord & Heath, 1992). Here we aimed to explore whether monotony is a multi-dimensional construct that is determined by characteristics of both the task proper and the task environment. The general assumption that monotony is a task characteristic used solely to elicit hypovigilance or poor performance related to fatigue appears to have led to there being little rigorous investigation into the exact nature of the relationship. While the two concepts are undoubtedly linked, the independent effect of monotony on hypovigilance remains largely ignored. Notwithstanding, there is evidence that monotony effects can emerge very early in vigilance tasks and are not necessarily accompanied by fatigue (see Meuter, Rakotonirainy, Johns, & Wagner, 2005). This phenomenon raises a largely untested, empirical question explored in two studies: Can hypovigilance emerge as a consequence of task and/or environmental monotony, independent of time on task and fatigue? In Study 1, using a short computerised vigilance task requiring responses to be withheld to infrequent targets, we explored the differential impacts of stimuli and task demand manipulations on the development of a monotonous context and the associated effects on vigilance performance (as indexed by respone errors and response times), independent of fatigue and time on task. The role of individual differences (sensation seeking, extroversion and cognitive failures) in moderating monotony effects was also considered. The results indicate that monotony affects sustained attention, with hypovigilance and associated performance worse in monotonous than in non-monotonous contexts. Critically, performance decrements emerged early in the task (within 4.3 minutes) and remained consistent over the course of the experiment (21.5 minutes), suggesting that monotony effects can operate independent of time on task and fatigue. A combination of low task demands and low stimulus variability form a monotonous context characterised by hypovigilance and poor task performance. Variations to task demand and stimulus variability were also found to independently affect performance, suggesting that monotony is a multi-dimensional construct relating to both task monotony (associated with the task itself) and environmental monotony (related to characteristics of the stimulus). Consequently, it can be concluded that monotony is multi-dimensional and is characterised by low variability in stimuli and/or task demands. The proposition that individual differences emerge under conditions of varying monotony with high sensation seekers and/or extroverts performing worse in monotonous contexts was only partially supported. Using a driving simulator, the findings of Study 1 were extended to a driving context to identify the behavioural and psychophysiological indices of monotony-related hypovigilance associated with variations to road design and road side scenery (Study 2). Supporting the proposition that monotony is a multi-dimensional construct, road design variability emerged as a key moderating characteristic of environmental monotony, resulting in poor driving performance indexed by decrements in steering wheel measures (mean lateral position). Sensation seeking also emerged as a moderating factor, where participants high in sensation seeking tendencies displayed worse driving behaviour in monotonous conditions. Importantly, impaired driving performance was observed within 8 minutes of commencing the driving task characterised by environmental monotony (low variability in road design) and was not accompanied by a decline in psychophysiological arousal. In addition, no subjective declines in alertness were reported. With fatigue effects associated with prolonged driving (van der Hulst, Meijman, & Rothengatter, 2001) and indexed by drowsiness, this pattern of results indicates that monotony can affect driver vigilance, independent of time on task and fatigue. Perceptual load theory (Lavie, 1995, 2005) and mindlessness theory (Robertson, Manly, Andrade, Baddley, & Yiend, 1997) provide useful theoretical frameworks for explaining and predicting monotony effects by positing that the low load (of task and/or stimuli) associated with a monotonous task results in spare attentional capacity which spills over involuntarily, resulting in the processing of task-irrelevant stimuli or task unrelated thoughts. That is, individuals – even when not fatigued - become easily distracted when performing a highly monotonous task, resulting in hypovigilance and impaired performance. The implications for road safety, including the likely effectiveness of fatigue countermeasures to mitigate monotony-related driver hypovigilance are discussed.