An Integrative Approach to Computational Modelling of the Gene Regulatory Network Controlling Clostridium botulinum Type A1 Toxin Production.

Clostridium botulinum produces botulinum neurotoxins (BoNTs), highly potent substances responsible for botulism. Currently, mathematical models of C. botulinum growth and toxigenesis are largely aimed at risk assessment and do not include explicit genetic information beyond group level but integrate many component processes, such as signalling, membrane permeability and metabolic activity. In this paper we present a scheme for modelling neurotoxin production in C. botulinum Group I type A1, based on the integration of diverse information coming from experimental results available in the literature. Experiments show that production of BoNTs depends on the growth-phase and is under the control of positive and negative regulatory elements at the intracellular level. Toxins are released as large protein complexes and are associated with non-toxic components. Here, we systematically review and integrate those regulatory elements previously described in the literature for C. botulinum Group I type A1 into a population dynamics model, to build the very first computational model of toxin production at the molecular level. We conduct a validation of our model against several items of published experimental data for different wild type and mutant strains of C. botulinum Group I type A1. The result of this process underscores the potential of mathematical modelling at the cellular level, as a means of creating opportunities in developing new strategies that could be used to prevent botulism; and potentially contribute to improved methods for the production of toxin that is used for therapeutics.

Time Series Analysis of the Bacillus subtilis Sporulation Network Reveals Low Dimensional Chaotic Dynamics.

Chaotic behavior refers to a behavior which, albeit irregular, is generated by an underlying deterministic process. Therefore, a chaotic behavior is potentially controllable. This possibility becomes practically amenable especially when chaos is shown to be low-dimensional, i.e., to be attributable to a small fraction of the total systems components. In this case, indeed, including the major drivers of chaos in a system into the modeling approach allows us to improve predictability of the systems dynamics. Here, we analyzed the numerical simulations of an accurate ordinary differential equation model of the gene network regulating sporulation initiation in Bacillus subtilis to explore whether the non-linearity underlying time series data is due to low-dimensional chaos. Low-dimensional chaos is expectedly common in systems with few degrees of freedom, but rare in systems with many degrees of freedom such as the B. subtilis sporulation network. The estimation of a number of indices, which reflect the chaotic nature of a system, indicates that the dynamics of this network is affected by deterministic chaos. The neat separation between the indices obtained from the time series simulated from the model and those obtained from time series generated by Gaussian white and colored noise confirmed that the B. subtilis sporulation network dynamics is affected by low dimensional chaos rather than by noise. Furthermore, our analysis identifies the principal driver of the networks chaotic dynamics to be sporulation initiation phosphotransferase B (Spo0B). We then analyzed the parameters and the phase space of the system to characterize the instability points of the network dynamics, and, in turn, to identify the ranges of values of Spo0B and of the other drivers of the chaotic dynamics, for which the whole system is highly sensitive to minimal perturbation. In summary, we described an unappreciated source of complexity in the B. subtilis sporulation network by gathering evidence for the chaotic behavior of the system, and by suggesting candidate molecules driving chaos in the system. The results of our chaos analysis can increase our understanding of the intricacies of the regulatory network under analysis, and suggest experimental work to refine our behavior of the mechanisms underlying B. subtilis sporulation initiation control.

New Elements To Consider When Modeling the Hazards Associated with Botulinum Neurotoxin in Food.

Botulinum neurotoxins (BoNTs) produced by the anaerobic bacterium Clostridium botulinum are the most potent biological substances known to mankind. BoNTs are the agents responsible for botulism, a rare condition affecting the neuromuscular junction and causing a spectrum of diseases ranging from mild cranial nerve palsies to acute respiratory failure and death. BoNTs are a potential biowarfare threat and a public health hazard, since outbreaks of foodborne botulism are caused by the ingestion of preformed BoNTs in food. Currently, mathematical models relating to the hazards associated with C. botulinum, which are largely empirical, make major contributions to botulinum risk assessment. Evaluated using statistical techniques, these models simulate the response of the bacterium to environmental conditions. Though empirical models have been successfully incorporated into risk assessments to support food safety decision making, this process includes significant uncertainties so that relevant decision making is frequently conservative and inflexible. Progression involves encoding into the models cellular processes at a molecular level, especially the details of the genetic and molecular machinery. This addition drives the connection between biological mechanisms and botulism risk assessment and hazard management strategies. This review brings together elements currently described in the literature that will be useful in building quantitative models of C. botulinum neurotoxin production. Subsequently, it outlines how the established form of modeling could be extended to include these new elements. Ultimately, this can offer further contributions to risk assessments to support food safety decision making.

The pattern of growth observed for Clostridium botulinum type A1 strain ATCC 19397 is influenced by nutritional status and quorum sensing: a modelling perspective.

Botulinum neurotoxins (BoNTs) produced by the anaerobic bacterium Clostridium botulinum are the most poisonous substances known to mankind. However, toxin regulation and signals triggering synthesis as well as the regulatory network and actors controlling toxin production are unknown. Experiments show that the neurotoxin gene is growth phase dependent for C. botulinum type A1 strain ATCC 19397, and toxin production is influenced both by culture conditions and nutritional status of the medium. Building mathematical models to describe the genetic and molecular machinery that drives the synthesis and release of BoNT requires a simultaneous description of the growth of the bacterium in culture. Here, we show four plausible modelling options which could be considered when constructing models describing the pattern of growth observed in a botulinum growth medium. Commonly used bacterial growth models are unsuitable to fit the pattern of growth observed, since they only include monotonic growth behaviour. We find that a model that includes both the nutritional status and the ability of the cells to sense their surroundings in a quorum-sensing manner is most successful at explaining the pattern of growth obtained for C. botulinum type A1 strain ATCC 19397.

Computational modelling and analysis of the molecular network regulating sporulation initiation in Bacillus subtilis.

BACKGROUND: Bacterial spores are important contaminants in food, and the spore forming bacteria are often implicated in food safety and food quality considerations. Spore formation is a complex developmental process involving the expression of more than 500 genes over the course of 6 to 8 hrs. The process culminates in the formation of resting cells capable of resisting environmental extremes and remaining dormant for long periods of time, germinating when conditions promote further vegetative growth. Experimental observations of sporulation and germination are problematic and time consuming so that reliable models are an invaluable asset in terms of prediction and risk assessment. In this report we develop a model which assists in the interpretation of sporulation dynamics. RESULTS: This paper defines and analyses a mathematical model for the network regulating Bacillus subtilis sporulation initiation, from sensing of sporulation signals down to the activation of the early genes under control of the master regulator Spo0A. Our model summarises and extends other published modelling studies, by allowing the user to execute sporulation initiation in a scenario where Isopropyl ß-D-1-thiogalactopyranoside (IPTG) is used as an artificial sporulation initiator as well as in modelling the induction of sporulation in wild-type cells. The analysis of the model results and the comparison with experimental data indicate that the model is good at predicting inducible responses to sporulation signals. However, the model is unable to reproduce experimentally observed accumulation of phosphorelay sporulation proteins in wild type B. subtilis. This model also highlights that the phosphorelay sub-component, which relays the signals detected by the sensor kinases to the master regulator Spo0A, is crucial in determining the response dynamics of the system. CONCLUSION: We show that there is a complex connectivity between the phosphorelay features and the master regulatory Spo0A. Additional we discovered that the experimentally observed regulation of the phosphotransferase Spo0B for wild-type B. subtilis may be playing an important role in the network which suggests that modelling of sporulation initiation may require additional experimental support.

Communicating oscillatory networks: frequency domain analysis.

BACKGROUND: Constructing predictive dynamic models of interacting signalling networks remains one of the great challenges facing systems biology. While detailed dynamical data exists about individual pathways, the task of combining such data without further lengthy experimentation is highly nontrivial. The communicating links between pathways, implicitly assumed to be unimportant and thus excluded, are precisely what become important in the larger system and must be reinstated. To maintain the delicate phase relationships between signals, signalling networks demand accurate dynamical parameters, but parameters optimised in isolation and under varying conditions are unlikely to remain optimal when combined. The computational burden of estimating parameters increases exponentially with increasing system size, so it is crucial to find precise and efficient ways of measuring the behaviour of systems, in order to re-use existing work. RESULTS: Motivated by the above, we present a new frequency domain-based systematic analysis technique that attempts to address the challenge of network assembly by defining a rigorous means to quantify the behaviour of stochastic systems. As our focus we construct a novel coupled oscillatory model of p53, NF-kB and the mammalian cell cycle, based on recent experimentally verified mathematical models. Informed by online databases of protein networks and interactions, we distilled their key elements into simplified models containing the most significant parts. Having coupled these systems, we constructed stochastic models for use in our frequency domain analysis. We used our new technique to investigate the crosstalk between the components of our model and measure the efficacy of certain network-based heuristic measures. CONCLUSIONS: We find that the interactions between the networks we study are highly complex and not intuitive: (i) points of maximum perturbation do not necessarily correspond to points of maximum proximity to influence; (ii) increased coupling strength does not necessarily increase perturbation; (iii) different perturbations do not necessarily sum and (iv) overall, susceptibility to perturbation is amplitude and frequency dependent and cannot easily be predicted by heuristic measures.Our methodology is particularly relevant for oscillatory systems, though not limited to these, and is most revealing when applied to the results of stochastic simulation. The technique is able to characterise precisely the distance in behaviour between different models, different systems and different parts within the same system. It can also measure the difference between different simulation algorithms used on the same system and can be used to inform the choice of dynamic parameters. By measuring crosstalk between subsystems it can also indicate mechanisms by which such systems may be controlled in experiments and therapeutics. We have thus found our technique of frequency domain analysis to be a valuable benchmark systems-biological tool.

Stochastic simulation of the spatio-temporal dynamics of reaction-diffusion systems: the case for the bicoid gradient.

Reaction-diffusion systems are mathematical models that describe how the concentrations of substances distributed in space change under the influence of local chemical reactions, and diffusion which causes the substances to spread out in space. The classical representation of a reaction-diffusion system is given by semi-linear parabolic partial differential equations, whose solution predicts how diffusion causes the concentration field to change with time. This change is proportional to the diffusion coefficient. If the solute moves in a homogeneous system in thermal equilibrium, the diffusion coefficients are constants that do not depend on the local concentration of solvent and solute. However, in nonhomogeneous and structured media the assumption of constant intracellular diffusion coefficient is not necessarily valid, and, consequently, the diffusion coefficient is a function of the local concentration of solvent and solutes. In this paper we propose a stochastic model of reaction-diffusion systems, in which the diffusion coefficients are function of the local concentration, viscosity and frictional forces. We then describe the software tool Redi (REaction-DIffusion simulator) which we have developed in order to implement this model into a Gillespie-like stochastic simulation algorithm. Finally, we show the ability of our model implemented in the Redi tool to reproduce the observed gradient of the bicoid protein in the Drosophila Melanogaster embryo. With Redi, we were able to simulate with an accuracy of 1% the experimental spatio-temporal dynamics of the bicoid protein, as recorded in time-lapse experiments obtained by direct measurements of transgenic bicoidenhanced green fluorescent protein.

Elucidation of functional consequences of signalling pathway interactions.

BACKGROUND: A great deal of data has accumulated on signalling pathways. These large datasets are thought to contain much implicit information on their molecular structure, interaction and activity information, which provides a picture of intricate molecular networks believed to underlie biological functions. While tremendous advances have been made in trying to understand these systems, how information is transmitted within them is still poorly understood. This ever growing amount of data demands we adopt powerful computational techniques that will play a pivotal role in the conversion of mined data to knowledge, and in elucidating the topological and functional properties of protein - protein interactions. RESULTS: A computational framework is presented which allows for the description of embedded networks, and identification of common shared components thought to assist in the transmission of information within the systems studied. By employing the graph theories of network biology - such as degree distribution, clustering coefficient, vertex betweenness and shortest path measures - topological features of protein-protein interactions for published datasets of the p53, nuclear factor kappa B (NF-kappaB) and G1/S phase of the cell cycle systems were ascertained. Highly ranked nodes which in some cases were identified as connecting proteins most likely responsible for propagation of transduction signals across the networks were determined. The functional consequences of these nodes in the context of their network environment were also determined. These findings highlight the usefulness of the framework in identifying possible combination or links as targets for therapeutic responses; and put forward the idea of using retrieved knowledge on the shared components in constructing better organised and structured models of signalling networks. CONCLUSION: It is hoped that through the data mined reconstructed signal transduction networks, well developed models of the published data can be built which in the end would guide the prediction of new targets based on the pathway's environment for further analysis. Source code is available upon request.

Bridging the gap between in silico and cell-based analysis of the nuclear factor-kappaB signaling pathway by in vitro studies of IKK2.

Previously, we have shown by sensitivity analysis, that the oscillatory behavior of nuclear factor (NF-kappaB) is coupled to free IkappaB kinase-2 (IKK2) and IkappaBalpha(IkappaBalpha), and that the phosphorylation of IkappaBalpha by IKK influences the amplitude of NF-kappaB oscillations. We have performed further analyses of the behavior of NF-kappaB and its signal transduction network to understand the dynamics of this system. A time lapse study of NF-kappaB translocation in 10,000 cells showed discernible oscillations in levels of nuclear NF-kappaB amongst cells when stimulated with interleukin (IL-1alpha), which suggests a small degree of synchronization amongst the cell population. When the kinetics for the phosphorylation of IkappaBalpha by IKK were measured, we found that the values for the affinity and catalytic efficiency of IKK2 for IkappaBalpha were dependent on assay conditions. The application of these kinetic parameters in our computational model of the NF-kappaB pathway resulted in significant differences in the oscillatory patterns of NF-kappaB depending on the rate constant value used. Hence, interpretation of in silico models should be made in the context of this uncertainty.