Mathematical modeling of biological networks can help to integrate a large body of information into a consistent framework, which can then be used to arrive at novel mechanistic insight and predictions. We have previously developed a detailed, mechanistic model for the regulation of σ F during sporulation in Bacillus subtilis. The model was based on a wide range of quantitative data, and once fitted to the data, the model made predictions that could be confirmed in experiments. However, the analysis was based on a single optimal parameter set. We wondered whether the predictions of the model would be stable for all optimal parameter sets. To that end we conducted a global parameter screen within the physiological parameter ranges. The screening approach allowed us to identify sensitive and sloppy parameters, and highlighted further required datasets during the optimization. Eventually, all parameter sets that reproduced all available data predicted the physiological situation correctly.
The Min system plays an important role in ensuring that cell division occurs at mid-cell in rod-shaped bacteria. In Escherichia coli, pole-to-pole oscillation of the Min proteins specifically inhibits polar septation. This system also prevents polar division in Bacillus subtilis during vegetative growth; however, the Min proteins do not oscillate in this organism. The Min system of B. subtilis plays a distinct role during sporulation, a process of differentiation which begins with an asymmetrical cell division. Here, we show that oscillation of the E. coli Min proteins can be reproduced following their introduction into B. subtilis cells. Further, we present evidence that the oscillatory behaviour of the Min system inhibits sporulation. We propose that an alternative Min system mechanism avoiding oscillation is evolutionarily important because oscillation of the Min system is incompatible with efficient asymmetrical septum formation and sporulation.
Sporulation in low-G+C gram-positive bacteria (Firmicutes) is an important survival mechanism that involves up to 150 genes, acting in a highly regulated manner. Many sporulation genes have close homologs in non-sporulating bacteria, including cyanobacteria, proteobacteria and spirochaetes, indicating that their products play a wider biological role. Most of them have been characterized as regulatory proteins or enzymes of peptidoglycan turnover; functions of others remain unknown but they are likely to have a general role in cell division and/or development. We have compiled a list of such widely conserved sporulation and germination proteins with poorly characterized functions, ranked them by the width of their phylogenetic distribution, and performed detailed sequence analysis and, where possible, structural modeling aimed at estimating their potential functions. Here we report the results of sequence analysis of Bacillus subtilis spore germination protein GerM, suggesting that it is a widespread cell development protein, whose function might involve binding to peptidoglycan. GerM consists of two tandem copies of a new domain (designated the GERMN domain) that forms phylum-specific fusions with two other newly described domains, GERMN-associated domains 1 and 2 (GMAD1 and GMAD2). Fold recognition reveals a β-propeller fold for GMAD1, while ab initio modeling suggests that GMAD2 adopts a fibronectin type III fold. SpoVS is predicted to adopt the AlbA archaeal chromatin protein fold, which suggests that it is a DNA-binding protein, most likely a novel transcriptional regulator.
Supplementary information: Supplementary data are available at ftp://ftp.ncbi.nih.gov/pub/galperin/Sporulation.html
Rifampin-resistant mutants of Bacillus subtilis that are conditionally temperature sensitive during sporulation have been isolated and characterized. The mutants can grow at the same rate as the wild type at the nonpermissive temperature but cannot sporulate. Depending on the mutation, they are blocked at either stage 0 to I, II, II to III, or IV of sporulation. The mutants showed an altered pattern of RNA synthesis after the stage at which they were blocked. The effect of rifampin on the activity of enzymes from mutant vegetative cells and sporulating cells was significantly different, suggesting that the RNA polymerase from sporulating cells was different from the RNA polymerase of vegetative cells. These results suggest that the conformation of the RNA polymerase core plays an important role in determining correct transcription during sporulation.
The ability of bacteria to responsively regulate the expression of translation components is crucial for rapid adaptation to fluctuating environments. Utilizing Bacillus subtilis (B. subtilis) as a model organism, we followed the dynamics of the translational machinery at a single cell resolution during growth and differentiation. By comprehensive monitoring the activity of the major rrn promoters and ribosomal protein production, we revealed diverse dynamics between cells grown in rich and poor medium, with the most prominent dissimilarities exhibited during deep stationary phase. Further, the variability pattern of translational activity varied among the cells, being affected by nutrient availability. We have monitored for the first time translational dynamics during the developmental process of sporulation within the two distinct cellular compartments of forespore and mother-cell. Our study uncovers a transient forespore specific increase in expression of translational components. Finally, the contribution of each rrn promoter throughout the bacterium life cycle was found to be relatively constant, implying that differential expression is not the main purpose for the existence of multiple rrn genes. Instead, we propose that coordination of the rrn operons serves as a strategy to rapidly fine tune translational activities in a synchronized fashion to achieve an optimal translation level for a given condition.
The cell cycle is one of the biological processes most frequently investigated in systems biology studies and it involves the knowledge of a large number of genes and networks of protein interactions. A deep knowledge of the molecular aspect of this biological process can contribute to making cancer research more accurate and innovative. In this context the mathematical modelling of the cell cycle has a relevant role to quantify the behaviour of each component of the systems. The mathematical modelling of a biological process such as the cell cycle allows a systemic description that helps to highlight some features such as emergent properties which could be hidden when the analysis is performed only from a reductionism point of view. Moreover, in modelling complex systems, a complete annotation of all the components is equally important to understand the interaction mechanism inside the network: for this reason data integration of the model components has high relevance in systems biology studies.
In this work, we present a resource, the Cell Cycle Database, intended to support systems biology analysis on the Cell Cycle process, based on two organisms, yeast and mammalian. The database integrates information about genes and proteins involved in the cell cycle process, stores complete models of the interaction networks and allows the mathematical simulation over time of the quantitative behaviour of each component. To accomplish this task, we developed, a web interface for browsing information related to cell cycle genes, proteins and mathematical models. In this framework, we have implemented a pipeline which allows users to deal with the mathematical part of the models, in order to solve, using different variables, the ordinary differential equation systems that describe the biological process.
This integrated system is freely available in order to support systems biology research on the cell cycle and it aims to become a useful resource for collecting all the information related to actual and future models of this network. The flexibility of the database allows the addition of mathematical data which are used for simulating the behavior of the cell cycle components in the different models. The resource deals with two relevant problems in systems biology: data integration and mathematical simulation of a crucial biological process related to cancer, such as the cell cycle. In this way the resource is useful both to retrieve information about cell cycle model components and to analyze their dynamical properties. The Cell Cycle Database can be used to find system-level properties, such as stable steady states and oscillations, by coupling structure and dynamical information about models.
One of the main challenges in the development of mathematical and computational models of biological systems is the precise estimation of parameter values. Understanding the effects of uncertainties in parameter values on model behaviour is crucial to the successful use of these models. Global sensitivity analysis (SA) can be used to quantify the variability in model predictions resulting from the uncertainty in multiple parameters and to shed light on the biological mechanisms driving system behaviour. We present a new methodology for global SA in systems biology which is computationally efficient and can be used to identify the key parameters and their interactions which drive the dynamic behaviour of a complex biological model. The approach combines functional principal component analysis with established global SA techniques. The methodology is applied to a model of the insulin signalling pathway, defects of which are a major cause of type 2 diabetes and a number of key features of the system are identified.
sensitivity analysis; computational modelling; insulin signalling
We have discovered that 6S-1 RNA (encoded by bsrA) is important for appropriate timing of sporulation in Bacillus subtilis in that cells lacking 6S-1 RNA sporulate earlier than wild-type cells. The time to generate a mature spore once the decision to sporulate has been made is unaffected by 6S-1 RNA, and, therefore, we propose that it is the timing of onset of sporulation that is altered. Interestingly, the presence of cells lacking 6S-1 RNA in coculture leads to all cell types exhibiting an early-sporulation phenotype. We propose that cells lacking 6S-1 RNA modify their environment in a manner that promotes early sporulation. In support of this model, resuspension of wild-type cells in conditioned medium from ΔbsrA cultures also resulted in early sporulation. Use of Escherichia coli growth as a reporter of the nutritional status of conditioned media suggested that B. subtilis cells lacking 6S-1 RNA reduce the nutrient content of their environment earlier than wild-type cells. Several pathways known to impact the timing of sporulation, such as the skf- and sdp-dependent cannibalism pathways, were eliminated as potential targets of 6S-1 RNA-mediated changes, suggesting that 6S-1 RNA activity defines a novel mechanism for altering the timing of onset of sporulation. In addition, 6S-2 RNA does not influence the timing of sporulation, providing further evidence of the independent influences of these two related RNAs on cell physiology.
Detailed analysis of the dynamic interactions among biological, environmental, social, and economic factors that favour the spread of certain diseases is extremely useful for designing effective control strategies. Diseases like tuberculosis that kills somebody every 15 seconds in the world, require methods that take into account the disease dynamics to design truly efficient control and surveillance strategies. The usual and well established statistical approaches provide insights into the cause-effect relationships that favour disease transmission but they only estimate risk areas, spatial or temporal trends. Here we introduce a novel approach that allows figuring out the dynamical behaviour of the disease spreading. This information can subsequently be used to validate mathematical models of the dissemination process from which the underlying mechanisms that are responsible for this spreading could be inferred.
The method presented here is based on the analysis of the spread of tuberculosis in a Brazilian endemic city during five consecutive years. The detailed analysis of the spatio-temporal correlation of the yearly geo-referenced data, using different characteristic times of the disease evolution, allowed us to trace the temporal path of the aetiological agent, to locate the sources of infection, and to characterize the dynamics of disease spreading. Consequently, the method also allowed for the identification of socio-economic factors that influence the process.
The information obtained can contribute to more effective budget allocation, drug distribution and recruitment of human skilled resources, as well as guiding the design of vaccination programs. We propose that this novel strategy can also be applied to the evaluation of other diseases as well as other social processes.
The initiation of sporulation in Bacillus species is regulated by the phosphorelay signal transduction pathway, which is activated by several histidine sensor kinases in response to cellular and metabolic signals. Comparison of the protein components of the phosphorelay between Bacillus subtilis and Bacillus anthracis revealed high homology in the phosphorelay orthologs of Spo0F, Spo0B, and Spo0A. The sensor domains of sensor histidine kinases are poorly conserved between species, making ortholog recognition tenuous. Putative sporulation sensor histidine kinases of B. anthracis were identified by homology to the HisKA domain of B. subtilis sporulation sensor histidine kinases, which interacts with Spo0F. Nine possible kinases were uncovered, and their genes were assayed for complementation of kinase mutants of B. subtilis, for ability to drive lacZ expression in B. subtilis and B. anthracis, and for the effect of deletion of each on the sporulation of B. anthracis. Five of the nine sensor histidine kinases were inferred to be capable of inducing sporulation in B. anthracis. Four of the sensor kinases could not be shown to induce sporulation; however, the genes for two of these were frameshifted in all B. anthracis strains and one of these was also frameshifted in the pathogenic pXO1-bearing Bacillus cereus strain G9241. It is proposed that acquisition of plasmid pXO1 and pathogenicity may require a dampening of sporulation regulation by mutational selection of sporulation sensor histidine kinase defects. The sporulation of B. anthracis ex vivo appears to result from any one or a combination of the sporulation sensor histidine kinases remaining.
A constant dilemma in theoretical ecology is knowing whether model predictions corrspond to real phenomena or whether they are artifacts of the modelling framework. The frequent absence of detailed ecological data against which models can be tested gives this issue particular importance. We address this question in the specific case of invasion in a predator-prey system with oscillatory population kinetics, in which both species exhibit local random movement. Given only these two basic qualitative features, we consider whether we can deduce any properties of the behaviour following invasion. To do this we study four different types of mathematical model, which have no formal relationship, but which all reflect our two qualitative ingredients. The models are: reaction-diffusion equations, coupled map lattices, deterministic cellular automata, and integrodifference equations. We present results of numerical simulations of the invasion of prey by predators for each model, and show that although there are certain differences, the main qualitative features of the behaviour behind invasion are the same for all the models. Specifically, there are either irregular spatiotemporal oscillations behind the invasion, or regular spatiotemporal oscillations with the form of a periodic travelling 'wake', depending on parameter values. The observation of this behaviour in all types of model strongly suggests that it is a direct consequence of our basic qualitative assumptions, and as such is an ecological reality which will always occur behind invasion in actual oscillatory predator-prey systems.
Predator-Prey Spatiotemporal Chaos Cellular Automata Coupled Map Lattice Reaction-Diffusion Integrodifference Equations
Clostridium perfringens type A food poisoning is the second most commonly identified bacterial food-borne illness. Sporulation contributes to this disease in two ways: (i) most food-poisoning strains form exceptionally resistant spores to facilitate their survival of food-associated stresses, and (ii) the enterotoxin (CPE) responsible for the symptoms of this food poisoning is synthesized only during sporulation. In Bacillus subtilis, four alternative sigma factors mediate sporulation. The same four sigma factors are encoded by C. perfringens genomes, and two (SigE and SigK) have previously been shown to be necessary for sporulation and CPE production by SM101, a transformable derivative of a C. perfringens food-poisoning strain (K. H. Harry, R. Zhou, L. Kroos, and S. B. Melville, J. Bacteriol. 2009, 191:2728-2742). However, the importance of SigF and SigG for C. perfringens sporulation or CPE production had not yet been assessed. In the current study, after confirming that sporulating wild-type SM101 cultures produce SigF (from a tricistronic operon) and SigG, we prepared isogenic sigF- or sigG-null mutants. Whereas SM101 formed heat-resistant, phase-refractile spores, spore formation was blocked in the sigF- and sigG-null mutants. Complementation fully restored sporulation by both mutants. By use of these mutants and complementing strains, CPE production was shown to be SigF dependent but SigG independent. This finding apparently involved regulation of the production of SigE and SigK, which Harry et al. showed to be necessary for CPE synthesis, by SigF. By combining these findings with those previous results, it is now apparent that all four alternative sigma factors are necessary for C. perfringens sporulation, but only SigE, SigF, and SigK are needed for CPE synthesis.
Cells must make reliable decisions under fluctuating extracellular conditions, but also be flexible enough to adapt to such changes. How cells reconcile these seemingly contradictory requirements through the dynamics of cellular decision-making is poorly understood. To study this issue we quantitatively measured gene expression and protein localization in single cells of the model organism Bacillus subtilis during the progression to spore formation. We found that sporulation proceeded through noisy and reversible steps towards an irreversible, all-or-none commitment point. Specifically, we observed cell-autonomous and spontaneous bursts of gene expression and transient protein localization events during sporulation. Based on these measurements we developed mathematical population models to investigate how the degree of reversibility affects cellular decision-making. In particular, we evaluated the effect of reversibility on the 1) reliability in the progression to sporulation, and 2) adaptability under changing extracellular stress conditions. Results show that reversible progression allows cells to remain responsive to long-term environmental fluctuations. In contrast, the irreversible commitment point supports reliable execution of cell fate choice that is robust against short-term reductions in stress. This combination of opposite dynamic behaviors (reversible and irreversible) thus maximizes both adaptable and reliable decision-making over a broad range of changes in environmental conditions. These results suggest that decision-making systems might employ a general hybrid strategy to cope with unpredictably fluctuating environmental conditions.
Cells must continuously make decisions in response to changes in their environment. These decisions must be irreversible, to prevent cells from reverting back to unfit cellular states, but also be flexible, to allow cells to go back to their previous state upon environmental changes. Using single-cell time-lapse fluorescence microscopy, we show that these seemingly contradictory properties coexist in Bacillus subtilis cells during their progression to spore formation. We suggest, on the basis of a mathematical population model, that reversible progression towards the irreversible decision to sporulate optimizes respectively adaptability and reliability of decision-making over a broad range of changes in environmental conditions.
Genome sequencing of multiple species makes it possible to understand the main principles behind the evolution of developmental regulatory networks. It is especially interesting to analyze the evolution of well-defined model systems in which conservation patterns can be directly correlated with the functional roles of various network components. Endospore formation (sporulation), extensively studied in Bacillus subtilis, is driven by such a model bacterial network of cellular development and differentiation. In this review, we analyze the evolution of the sporulation network in multiple endospore-forming bacteria. Importantly, the network evolution is not random but primarily follows the hierarchical organization and functional logic of the sporulation process. Specifically, the sporulation sigma factors and the master regulator of sporulation, Spo0A, are conserved in all considered spore-formers. The sequential activation of these global regulators is also strongly conserved. The feed-forward loops, which are likely used to fine-tune waves of gene expression within regulatory modules, show an intermediate level of conservation. These loops are less conserved than the sigma factors but significantly more than the structural sporulation genes, which form the lowest level in the functional and evolutionary hierarchy of the sporulation network. Interestingly, in spore-forming bacteria, gene regulation is more conserved than gene presence for sporulation genes, while the opposite is true for non-sporulation genes. The observed patterns suggest that by understanding the functional organization of a developmental network in a model organism, it is possible to understand the logic behind the evolution of this network in multiple related species.
Under conditions of nutrient limitation, Bacillus subtilis cells terminally differentiate into a dormant spore state. Progression to sporulation is controlled by a genetic circuit consisting of a phosphorelay embedded in multiple transcriptional feedback loops, which is used to activate the master regulator Spo0A by phosphorylation. These transcriptional regulatory interactions are “bandpass”-like, in the sense that activation occurs within a limited band of Spo0A∼P concentrations. Additionally, recent results show that the phosphorelay activation occurs in pulses, in a cell-cycle dependent fashion. However, the impact of these pulsed bandpass interactions on the circuit dynamics preceding sporulation remains unclear. In order to address this question, we measured key features of the bandpass interactions at the single-cell level and analyzed them in the context of a simple mathematical model. The model predicted the emergence of a delayed phase shift between the pulsing activity of the different sporulation genes, as well as the existence of a stable state, with elevated Spo0A activity but no sporulation, embedded within the dynamical structure of the system. To test the model, we used time-lapse fluorescence microscopy to measure dynamics of single cells initiating sporulation. We observed the delayed phase shift emerging during the progression to sporulation, while a re-engineering of the sporulation circuit revealed behavior resembling the predicted additional state. These results show that periodically-driven bandpass feedback loops can give rise to complex dynamics in the progression towards sporulation.
Decoyinine, an inhibitor of GMP synthetase, allows sporulation in Bacillus subtilis to initiate and proceed under otherwise catabolite-repressing conditions. The effect of decoyinine on alpha-amylase synthesis in B. subtilis, an event which exhibits regulatory features resembling sporulation initiation, was examined. Decoyinine did not overcome catabolite repression of alpha-amylase synthesis in a wild-type strain of B. subtilis but did cause premature and enhanced synthesis in a mutant strain specifically blocked in catabolite repression of alpha-amylase synthesis. Decoyinine had no effect on alpha-amylase enzymatic activity. Thus, it appears that the catabolite control mechanisms governing alpha-amylase synthesis and sporulation in B. subtilis differ in their responses to decoyinine and hence must consist at least partially of separate components.
Background. Normalisation is a critical step in obtaining meaningful information from the high-dimensional DNA array data. This is particularly important when complex biological hypotheses/questions, such a functional analysis and regulatory interactions within biological systems, are investigated. A nonparametric, intensity-dependent normalisation method based on global identification of self-consistent set (SCS) of genes is proposed here for such systems.
Results. The SCS normalisation is introduced and its behaviour demonstrated for a range of user-defined parameters affecting its performance. It is compared to a standard global normalisation method in terms of noise reduction and signal retention. Conclusions. The SCS normalisation results using 16 macroarray data sets from a Bacillus subtilis experiment confirm that the method is capable of reducing undesirable experimental variation whilst retaining important biological information. The ease and speed of implementation mean that this method can be easily adapted to other multicondition time/strain series single colour array data.
SpoIIGA is a novel type of membrane-associated aspartic protease that responds to a signal from the forespore by cleaving Pro-σE in the mother cell during sporulation of Bacillus subtilis. Very little is known about how SpoIIGA recognizes Pro-σE. By co-expressing proteins in Escherichia coli, it was shown that charge reversal substitutions for acidic residues 24 and 25 of Pro-σE, and for basic residues 245 and 284 of SpoIIGA, impaired cleavage. These results are consistent with a model predicting possible electrostatic interactions between these residues; however, no charge reversal substitution for residue 245 or residue 284 of SpoIIGA restored cleavage of Pro-σE with a charge reversal substitution for residue 24 or residue 25. Bacillus subtilis SpoIIGA cleaved Pro-σE orthologs from Bacillus licheniformis and Bacillus halodurans, but not from Bacillus cereus. A triple substitution in the pro-sequence of B. cereus Pro-σE allowed cleavage by B. subtilis SpoIIGA, indicating that residues distal from the cleavage site contribute to substrate specificity. Co-expression of SpoIIGA and Pro-σE orthologs in different combinations suggested that B. licheniformis SpoIIGA has a relatively narrow substrate specificity as compared with B. subtilis SpoIIGA, whereas B. cereus SpoIIGA and B. halodurans SpoIIGA appear to have broader substrate specificity.
Aspartic protease; bacillus subtilis; sigma factor; signal transduction; sporulation
Endospores of Bacillus subtilis are enclosed in a proteinaceous coat which can be differentiated into a thick, striated outer layer and a thinner, lamellar inner layer. We found that the N-terminal sequence of a 25-kDa protein present in a preparation of spore coat proteins matched that of the Mn-dependent superoxide dismutase (SOD) encoded by the sodA locus. sodA is transcribed throughout the growth and sporulation of a wild-type strain and is responsible for the SOD activity detected in total cell extracts prepared from B. subtilis. Disruption of the sodA locus produced a mutant that lacked any detectable SOD activity during vegetative growth and sporulation. The sodA mutant was not impaired in the ability to form heat- or lysozyme-resistant spores. However, examination of the coat layers of sodA mutant spores revealed increased extractability of the tyrosine-rich outer coat protein CotG. We showed that this condition was not accompanied by augmented transcription of the cotG gene in sporulating cells of the sodA mutant. We conclude that SodA is required for the assembly of CotG into the insoluble matrix of the spore and suggest that CotG is covalently cross-linked into the insoluble matrix by an oxidative reaction dependent on SodA. Ultrastructural analysis revealed that the inner coat formed by a sodA mutant was incomplete. Moreover, the outer coat lacked the characteristic striated appearance of wild-type spores, a pattern that was accentuated in a cotG mutant. These observations suggest that the SodA-dependent formation of the insoluble matrix containing CotG is largely responsible for the striated appearance of this coat layer.
We measured the synthesis of dipicolinic acid (DPA) during sporulation in spo mutants of Bacillus subtilis by a sensitive biological assay based on cross-feeding of a spoVF mutant strain and also chemically. Many spo mutations, including several that block sporulation at stage III, did not prevent synthesis of DPA but instead prevented its incorporation into the spore. In general, strains with mutations in loci that are expressed in the spore compartment synthesized DPA, whereas strains with mutations in most of the loci that are expressed in the mother-cell compartment did not. Transcription of the gerE gene, as measured by DNA-RNA hybridization, followed a dependence pattern very similar to that of DPA synthesis. However, the dependence patterns of the two operons show that at about stage IV of sporulation there is a branch in the sequence of operon expression in the mother cell. One branch leads through spoVC to synthesis of DPA synthetase, and the other leads through spoVD to expression of gerE.
A number of regulatory circuits in biological systems function through the exchange of phosphoryl groups from one protein to another. Spo0F and Spo0B are components of a phosphorelay that control sporulation in the bacterium Bacillus subtilis through the exchange of a phosphoryl group. Using beryllofluoride as a mimic for phosphorylation, we trapped the interaction of the phosphorylated Spo0F with Spo0B in the crystal lattice. The transition state of phosphoryl transfer continues to be a highly debated issue, as to whether it is associative or dissociative in nature. The geometry of Spo0F binding to Spo0B favors an associative mechanism for phosphoryl transfer. In order to visualize the autophosphorylation of the histidine kinase, KinA, and the subsequent phosphoryl transfer to Spo0F, we generated in silico models representing these reaction steps.
In many plants, starch is synthesized during the day and degraded during the night to avoid carbohydrate starvation in darkness. The circadian clock participates in a dynamic adjustment of starch turnover to changing environmental condition through unknown mechanisms. We used mathematical modelling to explore the possible scenarios for the control of starch turnover by the molecular components of the plant circadian clock. Several classes of plausible models were capable of describing the starch dynamics observed in a range of clock mutant plants and light conditions, including discriminating circadian protocols. Three example models of these classes are studied in detail, differing in several important ways. First, the clock components directly responsible for regulating starch degradation are different in each model. Second, the intermediate species in the pathway may play either an activating or inhibiting role on starch degradation. Third, the system may include a light-dependent interaction between the clock and downstream processes. Finally, the clock may be involved in the regulation of starch synthesis. We discuss the differences among the models’ predictions for diel starch profiles and the properties of the circadian regulators. These suggest additional experiments to elucidate the pathway structure, avoid confounding results and identify the molecular components involved.
starch metabolism; circadian rhythms; Arabidopsis thaliana; mathematical modelling; systems biology; biological clocks
Systems Biology requires that biological modelling is scaled up from small components to system level. This can produce exceedingly complex models, which obscure understanding rather than facilitate it. The successful use of highly simplified models would resolve many of the current problems faced in Systems Biology. This paper questions whether the conclusions of simple mathematical models of biological systems are trustworthy. The simplification of a specific model of calcium oscillations in hepatocytes is examined in detail, and the conclusions drawn from this scrutiny generalized. We formalize our choice of simplification approach through the use of functional ‘building blocks’. A collection of models is constructed, each a progressively more simplified version of a well-understood model. The limiting model is a piecewise linear model that can be solved analytically.
We find that, as expected, in many cases the simpler models produce incorrect results. However, when we make a sensitivity analysis, examining which aspects of the behaviour of the system are controlled by which parameters, the conclusions of the simple model often agree with those of the richer model. The hypothesis that the simplified model retains no information about the real sensitivities of the unsimplified model can be very strongly ruled out by treating the simplification process as a pseudo-random perturbation on the true sensitivity data. We conclude that sensitivity analysis is, therefore, of great importance to the analysis of simple mathematical models in biology. Our comparisons reveal which results of the sensitivity analysis regarding calcium oscillations in hepatocytes are robust to the simplifications necessarily involved in mathematical modelling. For example, we find that if a treatment is observed to strongly decrease the period of the oscillations while increasing the proportion of the cycle during which cellular calcium concentrations are rising, without affecting the inter-spike or maximum calcium concentrations, then it is likely that the treatment is acting on the plasma membrane calcium pump.
modelling; simplification; hepatocytes
Three classes of low-G+C Gram-positive bacteria (Firmicutes), Bacilli, Clostridia and Negativicutes, include numerous members that are capable of producing heat-resistant endospores. Spore-forming firmicutes include many environmentally important organisms, such as insect pathogens and cellulose-degrading industrial strains, as well as human pathogens responsible for such diseases as anthrax, botulism, gas gangrene and tetanus. In the best-studied model organism Bacillus subtilis, sporulation involves over 500 genes, many of which are conserved among other bacilli and clostridia. This work aimed to define the genomic requirements for sporulation through an analysis of the presence of sporulation genes in various firmicutes, including those with smaller genomes than B. subtilis. Cultivable spore-formers were found to have genomes larger than 2300 kb and encompass over 2150 protein-coding genes of which 60 are orthologues of genes that are apparently essential for sporulation in B. subtilis. Clostridial spore-formers lack, among others, spoIIB, sda, spoVID and safA genes and have non-orthologous displacements of spoIIQ and spoIVFA, suggesting substantial differences between bacilli and clostridia in the engulfment and spore coat formation steps. Many B. subtilis sporulation genes, particularly those encoding small acid-soluble spore proteins and spore coat proteins, were found only in the family Bacillaceae, or even in a subset of Bacillus spp. Phylogenetic profiles of sporulation genes, compiled in this work, confirm the presence of a common sporulation gene core, but also illuminate the diversity of the sporulation processes within various lineages. These profiles should help further experimental studies of uncharacterized widespread sporulation genes, which would ultimately allow delineation of the minimal set(s) of sporulation-specific genes in Bacilli and Clostridia.
The Bacillus subtilis gsiA operon was induced rapidly, but transiently, as cells entered the stationary phase in nutrient broth medium. A mutation at the gsiC locus caused sporulation to be defective and expression of gsiA to be elevated and prolonged. The sporulation defect in this strain was apparently due to persistent expression of gsiA, since a gsiA null mutation restored sporulation to wild-type levels. Detailed mapping experiments revealed that the gsiC82 mutation lies within the kinA gene, which encodes the histidine protein kinase member of a two-component regulatory system. Since mutations in this gene caused a substantial blockage in expression of spoIIA, spoIIG, and spoIID genes, it seems that accumulation of a product of the gsiA operon interferes with sporulation by blocking the completion of stage II. It apparently does so by inhibiting or counteracting the activity of KinA.