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The papilla number is one of the most economically important traits of sea cucumber in the China marketing trade. However, the genetic basis for papilla number diversity in holothurians is still scarce. In the present study, we conducted genome-wide association studies (GWAS) for the trait papilla number of sea cucumbers utilizing a set of 400,186 high-quality SNPs derived from 200 sea cucumbers. Two significant trait-associated SNPs that passed Bonferroni correction (P PATS1 and the genic region of EIF4G, which were reported to play a pivotal role in cell growth and proliferation. The fine-mapping regions around the top two lead SNPs provided precise causative loci/genes related to papilla formation and cellular activity, including PPP2R3C, GBP1, and BCAS3. Potential SNPs with P PATS1, PPP2R3C, and EIF4G that near or cover the two lead SNPs was conducted in papilla tissue of TG (Top papilla number group) and BG (Bottom papilla number group) by qRT-PCR. We found the significantly higher expression profile of PATS1 (3.34-fold), PPP2R3C (4.90-fold), and EIF4G (4.23-fold) in TG, implying their potential function in papilla polymorphism. The present results provide valuable information to decipher the phenotype differences of the papilla trait and will provide a scientific basis for selective breeding in sea cucumbers.

Mathematical oncology provides unique and invaluable insights into tumour growth on both the microscopic and macroscopic levels. This review presents state-of-the-art modelling techniques and focuses on their role in understanding glioblastoma, a malignant form of brain cancer. For each approach, we summarise the scope, drawbacks, and assets. We highlight the potential clinical applications of each modelling technique and discuss the connections between the mathematical models and the molecular and imaging data used to inform them. By doing so, we aim to prime cancer researchers with current and emerging computational tools for understanding tumour progression. Finally, by providing an in-depth picture of the different modelling techniques, we also aim to assist researchers who seek to build and develop their own models and the associated inference frameworks., 30 pages, 3 figures, 3 tables

Genotypic and phenotypic adaptation is the consequence of ongoing natural selection in populations and is key to predicting and preventing drug resistance. Whereas classic antibiotic persistence is all-or-nothing, here we demonstrate that an antibiotic resistance gene displays linear dose-responsive selection for increased expression in proportion to rising antibiotic concentration in growing Escherichia coli populations. Furthermore, we report the potentially wide-spread nature of this form of emergent gene expression (EGE) by instantaneous phenotypic selection process under bactericidal and bacteriostatic antibiotic treatment, as well as an amino acid synthesis pathway enzyme under a range of auxotrophic conditions. We propose an analogy to Ohm’s law in electricity (V = IR), where selection pressure acts similarly to voltage (V), gene expression to current (I), and resistance (R) to cellular machinery constraints and costs. Lastly, mathematical modeling using agent-based models of stochastic gene expression in growing populations and Bayesian model selection reveal that the EGE mechanism requires variability in gene expression within an isogenic population, and a cellular “memory” from positive feedbacks between growth and expression of any fitness-conferring gene. Finally, we discuss the connection of the observed phenomenon to a previously described general fluctuation–response relationship in biology.

The modelling of many real-world problems relies on computationally heavy simulations. Since statistical inference rests on repeated simulations to sample the parameter space, the high computational expense of these simulations can become a stumbling block. In this paper, we compare two ways to mitigate this issue based on machine learning methods. One approach is to construct lightweight surrogate models to substitute the simulations used in inference. Alternatively, one might altogether circumnavigate the need for Bayesian sampling schemes and directly estimate the posterior distribution. We focus on stochastic simulations that track autonomous agents and present two case studies of real-world applications: tumour growths and the spread of infectious diseases. We demonstrate that good accuracy in inference can be achieved with a relatively small number of simulations, making our machine learning approaches orders of magnitude faster than classical simulation-based methods that rely on sampling the parameter space. However, we find that while some methods generally produce more robust results than others, no algorithm offers a one-size-fits-all solution when attempting to infer model parameters from observations. Instead, one must choose the inference technique with the specific real-world application in mind. The stochastic nature of the considered real-world phenomena poses an additional challenge that can become insurmountable for some approaches. Overall, we find machine learning approaches that create direct inference machines to be promising for real-world applications. We present our findings as general guidelines for modelling practitioners. Author summaryComputer simulations play a vital role in modern science as they are commonly used to compare theory with observations. One can thus infer the properties of a observed system by comparing the data to the predicted behaviour in different scenarios. Each of these scenarios corresponds to a simulation with slightly different settings. However, since real-world problems are highly complex, the simulations often require extensive computational resources, making direct comparisons with data challenging, if not insurmountable. It is, therefore, necessary to resort to inference methods that mitigate this issue, but it is not clear-cut what path to choose for any specific research problem. In this paper, we provide general guidelines for how to make this choice. We do so by studying examples from oncology and epidemiology and by taking advantage of developments in machine learning. More specifically, we focus on simulations that track the behaviour of autonomous agents, such as single cells or individuals. We show that the best way forward is problem-dependent and highlight the methods that yield the most robust results across the different case studies. We demonstrate that these methods are highly promising and produce reliable results in a small fraction of the time required by classic approaches that rely on comparisons between data and individual simulations. Rather than relying on a single inference technique, we recommend employing several methods and selecting the most reliable based on predetermined criteria.

The modelling of many real-world problems relies on computationally heavy simulations of randomly interacting individuals or agents. However, the values of the parameters that underlie the interactions between agents are typically poorly known, and hence they need to be inferred from macroscopic observations of the system. Since statistical inference rests on repeated simulations to sample the parameter space, the high computational expense of these simulations can become a stumbling block. In this paper, we compare two ways to mitigate this issue in a Bayesian setting through the use of machine learning methods: One approach is to construct lightweight surrogate models to substitute the simulations used in inference. Alternatively, one might altogether circumvent the need for Bayesian sampling schemes and directly estimate the posterior distribution. We focus on stochastic simulations that track autonomous agents and present two case studies: tumour growths and the spread of infectious diseases. We demonstrate that good accuracy in inference can be achieved with a relatively small number of simulations, making our machine learning approaches orders of magnitude faster than classical simulation-based methods that rely on sampling the parameter space. However, we find that while some methods generally produce more robust results than others, no algorithm offers a one-size-fits-all solution when attempting to infer model parameters from observations. Instead, one must choose the inference technique with the specific real-world application in mind. The stochastic nature of the considered real-world phenomena poses an additional challenge that can become insurmountable for some approaches. Overall, we find machine learning approaches that create direct inference machines to be promising for real-world applications. We present our findings as general guidelines for modelling practitioners.

BackgroundSingle cell RNA-sequencing (scRNA-seq) has very rapidly become the new workhorse of modern biology providing an unprecedented global view on cellular diversity and heterogeneity. In particular, the structure of gene-gene expression correlation contains information on the underlying gene regulatory networks. However, interpretation of scRNA-seq data is challenging due to specific experimental error and biases that are unique to this kind of data including drop-out (or technical zeros).MethodsTo deal with this problem several methods for imputation of zeros for scRNA-seq have been developed. However, it is not clear how these processing steps affect inference of genetic networks from single cell data. Here, we introduce Biomodelling.jl, a tool for generation of synthetic scRNA-seq data using multiscale modelling of stochastic gene regulatory networks in growing and dividing cells.ResultsOur tool produces realistic transcription data with a known ground truth network topology that can be used to benchmark different approaches for gene regulatory network inference. Using this tool we investigate the impact of different imputation methods on the performance of several network inference algorithms.ConclusionsBiomodelling.jl provides a versatile and useful tool for future development and benchmarking of network inference approaches using scRNA-seq data.

Quantitative stochastic models of gene regulatory networks are important tools for studying cellular regulation. Such models can be formulated at many different levels of fidelity. A practical challenge is to determine what model fidelity to use in order to get accurate and representative results. The choice is important, because models of successively higher fidelity come at a rapidly increasing computational cost. In some situations, the level of detail is clearly motivated by the question under study. In many situations however, many model options could qualitatively agree with available data, depending on the amount of data and the nature of the observations. Here, an important distinction is whether we are interested in inferring the true (but unknown) physical parameters of the model or if it is sufficient to be able to capture and explain available data. The situation becomes complicated from a computational perspective because inference and model selection need to be approximate. Most often it is based on likelihood-free Approximate Bayesian Computation (ABC) and here determining which summary statistics to use, as well as how much data is needed to reach the desired level of accuracy, are difficult tasks. Ultimately, all of these aspects - the model fidelity, the available data, and the numerical choices for inference and model selection - interplay in a complex manner. In this paper we develop a computational pipeline designed to systematically evaluate inference accuracy for a wide range of true known parameters. We then use it to explore inference settings for negative feedback gene regulation. In particular, we compare a spatial stochastic model, a coarse-grained multiscale model, and a simple well-mixed model for several data-scenarios and for multiple numerical options for parameter inference. Practically speaking, this pipeline can be used as a preliminary step to guide modelers prior to gathering experimental data. By training Gaussian processes to approximate the distance metric, we are able to significantly reduce the computational cost of running the pipeline.

SummaryA common cellular task is to match gene expression dynamically to a range of concentrations of a regulatory molecule. Studying glucose transport in budding yeast, we determine mechanistically how such matching occurs for seven hexose transporters. By combining time-lapse microscopy with mathematical modelling, we find that levels of transporters are history-dependent and are regulated by a push-pull system comprising two types of repressors. Repression by these two types varies with glucose in opposite ways, and not only matches the expression of transporters by their affinity to a range of glucose concentrations, but also the expression of some to how glucose is changing. We argue that matching is favoured by a rate-affinity trade-off and that the regulatory system allows yeast to import glucose rapidly enough to starve competitors. Matching expression to a pattern of input is fundamental, and we believe that push-pull repression is widespread.

Glioblastoma originates in the brain and is one of the most aggressive cancer types. Glioblastoma represents 15% of all brain tumours, with a median survival of 15 months. Although the current standard of care for such a tumour (the Stupp protocol) has shown positive results for the prognosis of patients, O-6-methylguanine-DNA methyltransferase (MGMT) driven drug resistance has been an issue of increasing concern and hence requires innovative approaches. In addition to the well established drug resistance factors such as tumour location and blood brain barriers, it is also important to understand how the genetic and epigenetic dynamics of the glioblastoma cells can play a role. One important aspect of this is the study of methylation status of MGMT following administration of temozolomide. In this paper, we extend our previously published model that simulated MGMT expression in glioblastoma cells to incorporate the promoter methylation status of MGMT. This methylation status has clinical significance and is used as a marker for patient outcomes. Using this model, we investigate the causative relationship between temozolomide treatment and the methylation status of the MGMT promoter in a population of cells. In addition by constraining the model to relevant biological data using Approximate Bayesian Computation, we were able to identify parameter regimes that yield different possible modes of resistances, namely, phenotypic selection of MGMT, a downshift in the methylation status of the MGMT promoter or both simultaneously. We analysed each of the parameter sets associated with the different modes of resistance, presenting representative solutions as well as discovering some similarities between them as well as unique requirements for each of them. Finally, we used them to devise optimal strategies for inhibiting MGMT expression with the aim of minimising live glioblastoma cell numbers.

Glioblastoma (GBM) is the most aggressive malignant primary brain tumour with a median overall survival of 15 months. To treat GBM, patients currently undergo a surgical resection followed by exposure to radiotherapy and concurrent and adjuvant temozolomide (TMZ) chemotherapy. However, this protocol often leads to treatment failure, with drug resistance being the main reason behind this. To date, many studies highlight the role of O-6-methylguanine-DNA methyltransferase (MGMT) in conferring drug resistance. The mechanism through which MGMT confers resistance is not well studied—particularly in terms of computational models. With only a few reasonable biological assumptions, we were able to show that even a minimal model of MGMT expression could robustly explain TMZ-mediated drug resistance. In particular, we showed that for a wide range of parameter values constrained by novel cell growth and viability assays, a model accounting for only stochastic gene expression of MGMT coupled with cell growth, division, partitioning and death was able to exhibit phenotypic selection of GBM cells expressing MGMT in response to TMZ. Furthermore, we found this selection allowed the cells to pass their acquired phenotypic resistance onto daughter cells in a stable manner (as long as TMZ is provided). This suggests that stochastic gene expression alone is enough to explain the development of chemotherapeutic resistance.

Gene expression is an inherently noisy process. This noise is generally thought to be deleterious as precise internal regulation of biochemical reactions is essential for cell growth and survival. Self-repression of gene expression, which is the simplest form of a negative feedback loop, is commonly believed to be employed by cellular systems to decrease the stochastic fluctuations in gene expression. When there is some delay in autoregulation, it is also believed that this system can generate oscillations. In eukaryotic cells, mRNAs that are synthesised in the nucleus must be exported to the cytoplasm to function in protein synthesis, whereas proteins must be transported into the nucleus from the cytoplasm to regulate the expression levels of genes. Nuclear transport thus plays a critical role in eukaryotic gene expression and regulation. Some recent studies have suggested that nuclear retention of mRNAs can control noise in mRNA expression. However, the effect of nuclear transport on protein noise and its interplay with negative feedback regulation is not completely understood. In this paper, we systematically compare four different simple models of gene expression. By using simulations and applying the linear noise approximation to the corresponding chemical master equations, we investigate the influence of nuclear import and export on noise in gene expression in a negative autoregulatory feedback loop. We first present results consistent with the literature, i.e., that negative feedback can effectively buffer the variability in protein levels, and nuclear retention can decrease mRNA noise levels. Interestingly we find that when negative feedback is combined with nuclear retention, an amplification in gene expression noise can be observed and is dependant on nuclear translocation rates. Finally, we investigate the effect of nuclear compartmentalisation on the ability of self-repressing genes to exhibit stochastic oscillatory dynamics.

Ligand binding to death receptors activates apoptosis in cancer cells. Stimulation of death receptors results in the formation of intracellular multiprotein platforms that either activate the apoptotic initiator Caspase-8 to trigger cell death, or signal through kinases to initiate inflammatory and cell survival signalling. Two of these platforms, the Death-Inducing Signalling Complex (DISC) and the RIPoptosome, also initiate necroptosis by building filamentous scaffolds that lead to the activation of mixed lineage kinase domain-like pseudokinase. To explain cell decision making downstream of death receptor activation, we developed a semi-stochastic model of DISC/RIPoptosome formation. The model is a hybrid of a direct Gillespie stochastic simulation algorithm for slow assembly of the RIPoptosome and a deterministic model of downstream caspase activation. The model explains how alterations in the level of death receptor-ligand complexes, their clustering properties and intrinsic molecular fluctuations in RIPoptosome assembly drive heterogeneous dynamics of Caspase-8 activation. The model highlights how kinetic proofreading leads to heterogeneous cell responses and results in fractional cell killing at low levels of receptor stimulation. It reveals that the noise in Caspase-8 activation—exclusively caused by the stochastic molecular assembly of the DISC/RIPoptosome platform—has a key function in extrinsic apoptotic stimuli recognition., Author summary Death receptors are targets of novel cancer therapeutics. Most of them signal through flexible multiprotein platforms to either activate apoptotic or necroptotic cell death, or propagate cell survival and pro-inflammatory signals. We focused our study on the role of dynamic assembly and composition of these platforms in the initiation of cell death at the single cell level. Since the assembly is slow through the competitive nature of protein binding within the platforms core we developed a stochastic mathematical model of the death inducing signalling platform. Our model provided an explanation for delayed cell death and fractional killing upon the death receptor stimulation. Additionally, we found that the variability in the cell death response arises through the random assembly initiates a slow noise-prone ramp activation of initiator Caspase-8 spontaneously triggering the apoptotic cascade. Our computational simulations predicted high variation in the time required for cell death induction at the single cell level and highlighted a significant role of death receptor clustering in effective Caspase-8 activation. Our knowledge and data driven model captures detailed processes governing the early events of cell death initiation and can be used to guide the development of more rational combinational treatments against cancer.

Natural selection relies on genotypic and phenotypic adaptation in response to fluctuating environmental conditions and is the key to predicting and preventing drug resistance. Whereas classic persistence is all-or-nothing, here we show for the first time that an antibiotic resistance gene displays linear dose-responsive selection for increased expression in proportion to rising antibiotic concentration in E. coli. Furthermore, we observe the general nature of an instantaneous phenotypic selection process upon bactericidal and bacteriostatic antibiotic treatment, as well as an amino acid synthesis pathway enzyme under a range of auxotrophic conditions. To explain this phenomenon, we propose an analogy to Ohm’s law in electricity (V=IR) where fitness pressure acts similarly to voltage (V), gene expression to current (I), and resistance (R) to cellular machinery constraints. Lastly, mathematical modelling approaches reveal that the emergent gene expression mechanism requires variation in mRNA and protein production within an isogenic population, and cell ‘memory’ from positive feedbacks between growth and expression of any fitness-inducing gene.

Bevacizumab (bvz) is a first choice anti-angiogenic drug in oncology and is primarily administered in combination with chemotherapy. It has been hypothesized that anti-angiogenic drugs enhance efficacy of cytotoxic drugs by “normalizing” abnormal tumor vessels and improving drug penetration. Nevertheless, the clinical relevance of this phenomenon is still unclear with several studies over recent years suggesting an opposing relationship. Herein, we sought to develop a new computational tool to interrogate anti-angiogenic drug scheduling with particular application in the setting of colorectal cancer (CRC). Specifically, we have employed a mathematical model of vascular tumour growth which interrogates the impact of anti-angiogenic treatment and chemotherapeutic treatment on tumour volume. Model predictions were validated using CRC xenografts which underwent treatment with a clinically relevant combinatorial anti-angiogenic regimen. Bayesian model selection revealed the most appropriate term for capturing the effect of treatments on the tumour size, and provided insights into a switch-like dependence of FOLFOX delivery on the tumour vasculature. Our experimental data and mathematical model suggest that delivering chemotherapy prior to bvz may be optimal in the colorectal cancer setting.

Hes1 is a member of the family of basic helix-loop-helix transcription factors and the Hes1 gene regulatory network (GRN) may be described as the canonical example of transcriptional control in eukaryotic cells, since it involves only the Hes1 protein and its own mRNA. Recently, the Hes1 protein has been established as an excellent target for an anti-cancer drug treatment, with the design of a small molecule Hes1 dimerisation inhibitor representing a promising if challenging approach to therapy. In this paper, we extend a previous spatial stochastic model of the Hes1 GRN to include nuclear transport and dimerisation of Hes1 monomers. Initially, we assume that dimerisation occurs only in the cytoplasm, with only dimers being imported into the nucleus. Stochastic simulations of this novel model using the URDME software show that oscillatory dynamics in agreement with experimental studies are retained. Furthermore, we find that our model is robust to changes in the nuclear transport and dimerisation parameters. However, since the precise dynamics of the nuclear import of Hes1 and the localisation of the dimerisation reaction are not known, we consider a second modelling scenario in which we allow for both Hes1 monomers and dimers to be imported into the nucleus, and we allow dimerisation of Hes1 to occur everywhere in the cell. Once again, computational solutions of this second model produce oscillatory dynamics in agreement with experimental studies. We also explore sensitivity of the numerical solutions to nuclear transport and dimerisation parameters. Finally, we compare and contrast the two different modelling scenarios using numerical experiments that simulate dimer disruption, and suggest a biological experiment that could distinguish which model more faithfully captures the Hes1 GRN. Postprint

A correction to this article has been published and is linked from the HTML and PDF versions of this paper. The error has been fixed in the paper.

Cell polarization is a ubiquitous process which results in cellular constituents being organized into discrete intracellular spatial domains. It occurs in a variety of cell types, including epithelial cells, immune system cells and neurons. A key player in this process is the Par protein family whose asymmetric localization to anterior and posterior parts of the cell is crucial for proper division and cell fate specification. In this paper, we explore a stochastic analogue of the temporal model of Par protein interactions first developed in Dawes & Munro (Dawes and Munro 2011 Biophys. J. 101, 1412–1422. ( doi:10.1016/j.bpj.2011.07.030 )). We focus on how protein abundance influences the behaviour of both the deterministic and stochastic versions of the model. In Dawes & Munro (2011), it was found that bistable behaviour in the temporal model of Par protein led to the existence of complementary domains in the corresponding spatio-temporal model. Here, we find that the corresponding temporal stochastic model permits switching behaviour (the model solution ‘jumps’ between steady states) for lower protein abundances, whereas for higher protein abundances the stochastic and deterministic models are in good agreement (the model solution evolves to one of two steady states). This led us to the testable hypothesis that cells with lower abundances of Par protein may be more sensitive to external cues, whereas cells with higher abundances of Par protein may be less sensitive to external cues. In order to gain more control over the precise abundance of Par protein, we proposed and explored a second model (again, examining both deterministic and stochastic versions) in which the total number of Par molecules is conserved. We found that this model required an additional dimerization reaction in the cytoplasm in order for bistable and switching behaviour to be found. Once this additional reaction was included, we found that both the first and second models gave qualitatively similar results but in different regions of the parameter space, suggesting a further regulatory mechanism that cells could potentially use to modulate their response to external signals.

Due to their location, the malignant gliomas of the brain in humans are very difficult to treat in advanced stages. Blood-based biomarkers for glioma are needed for more accurate evaluation of treatment response as well as early diagnosis. However, biomarker research in primary brain tumors is challenging given their relative rarity and genetic diversity. It is further complicated by variations in the permeability of the blood brain barrier that affects the amount of marker released into the bloodstream. Inspired by recent temporal data indicating a possible decrease in serum glucose levels in patients with gliomas yet to be diagnosed, we present an ordinary differential equation model to capture early stage glioma growth. The model contains glioma–glucose-immune interactions and poses a potential mechanism by which this glucose drop can be explained. We present numerical simulations, parameter sensitivity analysis, linear stability analysis and a numerical experiment whereby we show how a dormant glioma can become malignant.

M.A.J.C. and M.S. gratefully acknowledge the support of the ERC Advanced Investigator Grant 227619, “M5CGS — From Mutations to Metastases: Multiscale Mathematical Modelling of Cancer Growth and Spread”. M.S. would also like to thank the support from the Mathematical Biosciences Institute at the Ohio State University and NSF Grant DMS0931642. Gene regulatory networks, i.e. DNA segments in a cell which interact with each other indirectly through their RNA and protein products, lie at the heart of many important intracellular signal transduction processes. In this paper, we analyze a mathematical model of a canonical gene regulatory network consisting of a single negative feedback loop between a protein and its mRNA (e.g. the Hes1 transcription factor system). The model consists of two partial differential equations describing the spatio-temporal interactions between the protein and its mRNA in a one-dimensional domain. Such intracellular negative feedback systems are known to exhibit oscillatory behavior and this is the case for our model, shown initially via computational simulations. In order to investigate this behavior more deeply, we undertake a linearized stability analysis of the steady states of the model. Our results show that the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. This shows that the spatial movement of the mRNA and protein molecules alone is sufficient to cause the oscillations. Our result has implications for transcription factors such as p53, NF-κB and heat shock proteins which are involved in regulating important cellular processes such as inflammation, meiosis, apoptosis and the heat shock response, and are linked to diseases such as arthritis and cancer. Publisher PDF

A gene regulatory network may be defined as a collection of DNA segments which interact with each other indirectly through their RNA and protein products. Such a network is said to contain a negative feedback loop if its products inhibit gene transcription, and a positive feedback loop if a gene product promotes its own production. Negative feedback loops can create oscillations in mRNA and protein levels while positive feedback loops are primarily responsible for signal amplification. It is often the case in real biological systems that both negative and positive feedback loops operate in parameter regimes that result in low copy numbers of gene products. In this paper we investigate the spatio-temporal dynamics of a single feedback loop in a eukaryotic cell. We first develop a simplified spatial stochastic model of a canonical feedback system (either positive or negative). Using a Gillespie's algorithm, we compute sample trajectories and analyse their corresponding statistics. We then derive a system of equations that describe the spatio-temporal evolution of the stochastic means. Subsequently, we examine the spatially homogeneous case and compare the results of numerical simulations with the spatially explicit case. Finally, using a combination of steady-state analysis and data clustering techniques, we explore model behaviour across a subregion of the parameter space that is difficult to access experimentally and compare the parameter landscape of our spatio-temporal and spatially-homogeneous models. Postprint

Individual mouse embryonic stem cells have been found to exhibit highly variable differentiation responses under the same environmental conditions. The noisy cyclic expression of Hes1 and its downstream genes are known to be responsible for this, but the mechanism underlying this variability in expression is not well understood. In this paper, we show that the observed experimental data and diverse differentiation responses can be explained by a spatial stochastic model of the Hes1 gene regulatory network. We also propose experiments to control the precise differentiation response using drug treatment.

Compartment-based (lattice-based) reaction-diffusion algorithms are often used for studying complex stochastic spatio-temporal processes inside cells. In this paper the influence of macromolecular crowding on stochastic reaction-diffusion simulations is investigated. Reaction-diffusion processes are considered on two different kinds of compartmental lattice, a cubic lattice and a hexagonal close packed lattice, and solved using two different algorithms, the stochastic simulation algorithm and the spatiocyte algorithm (Arjunan and Tomita 2010 Syst. Synth. Biol. 4, 35-53). Obstacles (modelling macromolecular crowding) are shown to have substantial effects on the mean squared displacement and average number of molecules in the domain but the nature of these effects is dependent on the choice of lattice, with the cubic lattice being more susceptible to the effects of the obstacles. Finally, improvements for both algorithms are presented.

There are many intracellular signalling pathways where the spatial distribution of the molecular species cannot be neglected. One such class of pathways is those involving transcription factors (e.g. Hes 1, p53-Mdm2, NF-κ B, heat-shock proteins) which often exhibit oscillations in both space and time. In this chapter we present a partial differential equation model of the transcription factor, Hes 1. Our model considers the dynamics of Hes 1 in a 2-dimensional cellular domain including a nucleus, cytoplasm and microtubule-organising centre (MTOC). Spatial movement of the molecules (protein, mRNA) is assumed to be by diffusion, and also convection along microtubules. Through numerical simulations we find ranges of values for the model parameters such that sustained oscillatory dynamics occur, consistent with available experimental measurements. In order to bridge the gap between in vivo and in silico experiments we investigate more realistic cell geometries by using an imported image of a real cell as our computational domain.

There are many intracellular signalling pathways where the spatial distribution of the molecular species cannot be neglected. These pathways often contain negative feedback loops and can exhibit oscillatory dynamics in space and time. Two such pathways are those involving Hes1 and p53–Mdm2, both of which are implicated in cancer. In this paper we further develop the partial differential equation (PDE) models of Sturrock et al. (J. Theor. Biol., 273:15–31, 2011) which were used to study these dynamics. We extend these PDE models by including a nuclear membrane and active transport, assuming that proteins are convected in the cytoplasm towards the nucleus in order to model transport along microtubules. We also account for Mdm2 inhibition of p53 transcriptional activity. Through numerical simulations we find ranges of values for the model parameters such that sustained oscillatory dynamics occur, consistent with available experimental measurements. We also find that our model extensions act to broaden the parameter ranges that yield oscillations. Hence oscillatory behaviour is made more robust by the inclusion of both the nuclear membrane and active transport. In order to bridge the gap between in vivo and in silico experiments, we investigate more realistic cell geometries by using an imported image of a real cell as our computational domain. For the extended p53–Mdm2 model, we consider the effect of microtubule-disrupting drugs and proteasome inhibitor drugs, obtaining results that are in agreement with experimental studies.

The correct localisation of transcription factors is vitally important for the proper functioning of many intracellular signalling pathways. Experimental data has shown that many pathways exhibit oscillations in concentrations of the substances involved, both temporally and spatially. Negative feedback loops are important components of these oscillations, providing fine regulation for the factors involved. In this paper we consider mathematical models of two such pathways—Hes1 and p53-Mdm2. Building on previous mathematical modelling approaches, we derive systems of partial differential equations to capture the evolution in space and time of the variables in the Hes1 and p53-Mdm2 systems. Through computational simulations we show that our reaction-diffusion models are able to produce sustained oscillations both spatially and temporally, accurately reflecting experimental evidence and advancing previous models. The simulations of our models also allow us to calculate a diffusion coefficient range for the variables in each mRNA and protein system, as well as ranges for other key parameters of the models, where sustained oscillations are observed. Finally, by exploiting the explicitly spatial nature of the partial differential equations, we are also able to manipulate mathematically the spatial location of the ribosomes, thus controlling where the proteins are synthesized within the cytoplasm. The results of these simulations predict an optimal distance outside the nucleus where protein synthesis should take place in order to generate sustained oscillations. Using partial differential equation models, new information can be gained about the precise spatio-temporal dynamics of mRNA and proteins. The ability to determine spatial localisation of proteins within the cell is likely to yield fresh insight into a range of cellular diseases such as diabetes and cancer.

Background: The p53 transcription factor is a well-established regulator of the cell cycle and key to preventing cancer. In response to a variety of cellular stresses (e.g., DNA damage or oncogene activation) p53 is activated and translocates to the nucleus where it activates transcription of target genes which can induce cell cycle arrest, senescence or apoptosis. One of the target genes for p53 is the gene for Mdm2, a negative regulator of p53 function in cells. Mdm2 represses p53 function through two main mechanisms: by promoting p53 ubiquitination and proteasomal degradation and, through direct inhibition of p53 transcriptional activity. Experimental data have revealed that in response to gamma irradiation, p53 and Mdm2 concentrations exhibit spatial and temporal oscillatory dynamics in individual MCF7 breast cancer cells. The precise function of these oscillations is still under investigation. Method: Using in vitro laboratory data, we have developed a computational model to study the spatio-temporal evolution of the p53-Mdm2 pathway. The model comprises a system of coupled nonlinear partial differential equations, including transport terms and reaction kinetics. The transport is assumed to include both random (diffusion) and active mechanisms with proteins convected in the cytoplasm toward the nucleus, modeling transport along microtubules. Internal structures such as ribosomes and the nuclear membrane are also explicitly modeled. Results: Using computational simulations we have found ranges of values for the model parameters such that sustained oscillatory dynamics occur, consistent with available experimental measurements. To bridge the gap between in vitro and in silico experiments realistic cell geometries using imported images of cells have informed our computational domain. The effects of microtubule-disrupting drugs, proteasome inhibitor drugs, and mutations in the DNA binding site of p53, have been modeled yielding data in agreement with published experimental laboratory studies. Conclusions: This mathematical computational model enables us to compare the effects of clinically relevant therapeutic approaches at the single cell level. In particular, the precise temporal and spatial distributions of p53 can be determined and the cellular consequences modeled. Citation Information: Cancer Res 2012;72(24 Suppl):Abstract nr P5-05-06.

In the vertebrate embryo, tissue blocks called somites are laid down in head-to-tail succession, a process known as somitogenesis. Research into somitogenesis has been both experimental and mathematical. For zebrafish, there is experimental evidence for oscillatory gene expression in cells in the presomitic mesoderm (PSM) as well as evidence that Notch signalling synchronises the oscillations in neighbouring PSM cells. A biological mechanism has previously been proposed to explain these phenomena. Here we have converted this mechanism into a mathematical model of partial differential equations in which the nuclear and cytoplasmic diffusion of protein and mRNA molecules is explicitly considered. By performing simulations, we have found ranges of values for the model parameters (such as diffusion and degradation rates) that yield oscillatory dynamics within PSM cells and that enable Notch signalling to synchronise the oscillations in two touching cells. Our model contains a Hill coefficient that measures the co-operativity between two proteins (Her1, Her7) and three genes (her1, her7, deltaC) which they inhibit. This coefficient appears to be bounded below by the requirement for oscillations in individual cells and bounded above by the requirement for synchronisation. Consistent with experimental data and a previous spatially non-explicit mathematical model, we have found that signalling can increase the average level of Her1 protein. Biological pattern formation would be impossible without a certain robustness to variety in cell shape and size; our results possess such robustness. Our spatially-explicit modelling approach, together with new imaging technologies that can measure intracellular protein diffusion rates, is likely to yield significant new insight into somitogenesis and other biological processes.