Mathematical models of coagulation-are we there yet?

BACKGROUND: Mathematical models of coagulation have been developed to mirror thrombin generation in plasma, with the aim of investigating how variation in coagulation factor levels regulates hemostasis. However, current models vary in the reactions they capture and the reaction rates used, and their validation is restricted by a lack of large coherent datasets, resulting in questioning of their utility. OBJECTIVES: To address this debate, we systematically assessed current models against a large dataset, using plasma coagulation factor levels from 348 individuals with normal hemostasis to identify the causes of these variations. METHODS: We compared model predictions with measured thrombin generation, quantifying and comparing the ability of each model to predict thrombin generation, the contributions of the individual reactions, and their dependence on reaction rates. RESULTS: We found that no current model predicted the hemostatic response across the whole cohort and all produced thrombin generation curves that did not resemble those obtained experimentally. Our analysis has identified the key reactions that lead to differential model predictions, where experimental uncertainty leads to variability in predictions, and we determined reactions that have a high influence on measured thrombin generation, such as the contribution of factor XI. CONCLUSION: This systematic assessment of models of coagulation, using large dataset inputs, points to ways in which these models can be improved. A model that accurately reflects the effects of the multiple subtle variations in an individual's hemostatic profile could be used for assessing antithrombotics or as a tool for precision medicine.

Fluctuations in auxin levels depend upon synchronicity of cell divisions in a one-dimensional model of auxin transport.

Auxin is a well-studied plant hormone, the spatial distribution of which remains incompletely understood. Here, we investigate the effects of cell growth and divisions on the dynamics of auxin patterning, using a combination of mathematical modelling and experimental observations. In contrast to most prior work, models are not designed or tuned with the aim to produce a specific auxin pattern. Instead, we use well-established techniques from dynamical systems theory to uncover and classify ranges of auxin patterns as exhaustively as possible as parameters are varied. Previous work using these techniques has shown how a multitude of stable auxin patterns may coexist, each attainable from a specific ensemble of initial conditions. When a key parameter spans a range of values, these steady patterns form a geometric curve with successive folds, often nicknamed a snaking diagram. As we introduce growth and cell division into a one-dimensional model of auxin distribution, we observe new behaviour which can be explained in terms of this diagram. Cell growth changes the shape of the snaking diagram, and this corresponds in turn to deformations in the patterns of auxin distribution. As divisions occur this can lead to abrupt creation or annihilation of auxin peaks. We term this phenomenon 'snake-jumping'. Under rhythmic cell divisions, we show how this can lead to stable oscillations of auxin. We also show that this requires a high level of synchronisation between cell divisions. Using 18 hour time-lapse imaging of the auxin reporter DII:Venus in roots of Arabidopsis thaliana, we show auxin fluctuates greatly, both in terms of amplitude and periodicity, consistent with the snake-jumping events observed with non-synchronised cell divisions. Periodic signals downstream of the auxin signalling pathway have previously been recorded in plant roots. The present work shows that auxin alone is unlikely to play the role of a pacemaker in this context.

A genome-scale metabolic model of Cupriavidus necator H16 integrated with TraDIS and transcriptomic data reveals metabolic insights for biotechnological applications.

Exploiting biological processes to recycle renewable carbon into high value platform chemicals provides a sustainable and greener alternative to current reliance on petrochemicals. In this regard Cupriavidus necator H16 represents a particularly promising microbial chassis due to its ability to grow on a wide range of low-cost feedstocks, including the waste gas carbon dioxide, whilst also naturally producing large quantities of polyhydroxybutyrate (PHB) during nutrient-limited conditions. Understanding the complex metabolic behaviour of this bacterium is a prerequisite for the design of successful engineering strategies for optimising product yields. We present a genome-scale metabolic model (GSM) of C. necator H16 (denoted iCN1361), which is directly constructed from the BioCyc database to improve the readability and reusability of the model. After the initial automated construction, we have performed extensive curation and both theoretical and experimental validation. By carrying out a genome-wide essentiality screening using a Transposon-directed Insertion site Sequencing (TraDIS) approach, we showed that the model could predict gene knockout phenotypes with a high level of accuracy. Importantly, we indicate how experimental and computational predictions can be used to improve model structure and, thus, model accuracy as well as to evaluate potential false positives identified in the experiments. Finally, by integrating transcriptomics data with iCN1361 we create a condition-specific model, which, importantly, better reflects PHB production in C. necator H16. Observed changes in the omics data and in-silico-estimated alterations in fluxes were then used to predict the regulatory control of key cellular processes. The results presented demonstrate that iCN1361 is a valuable tool for unravelling the system-level metabolic behaviour of C. necator H16 and can provide useful insights for designing metabolic engineering strategies.

Mathematical Modelling Highlights the Potential for Genetic Manipulation as an Adjuvant to Counter Efflux-Mediated MDR in Salmonella.

Bacteria have developed resistance to antibiotics by various mechanisms, notable amongst these is the use of permeation barriers and the expulsion of antibiotics via efflux pumps. The resistance-nodulation-division (RND) family of efflux pumps is found in Gram-negative bacteria and a major contributor to multidrug resistance (MDR). In particular, Salmonella encodes five RND efflux pump systems: AcrAB, AcrAD, AcrEF, MdsAB and MdtAB which have different substrate ranges including many antibiotics. We produce a spatial partial differential equation (PDE) model governing the diffusion and efflux of antibiotic in Salmonella, via these RND efflux pumps. Using parameter fitting techniques on experimental data, we are able to establish the behaviour of multiple wild-type and efflux mutant Salmonella strains, which enables us to produce efflux profiles for each individual efflux pump system. By combining the model with a gene regulatory network (GRN) model of efflux regulation, we simulate how the bacteria respond to their environment. Finally, performing a parameter sensitivity analysis, we look into various different targets to inhibit the efflux pumps. The model provides an in silico framework with which to test these potential adjuvants to counter MDR.

Endothelial Cell RNA-Seq Data: Differential Expression and Functional Enrichment Analyses to Study Phenotypic Switching.

RNA-seq is a common approach used to explore gene expression data between experimental conditions or cell types and ultimately leads to information that can shed light on the biological processes involved and inform further hypotheses. While the protocols required to generate samples for sequencing can be performed in most research facilities, the resulting computational analysis is often an area in which researchers have little experience. Here we present a user-friendly bioinformatics workflow which describes the methods required to take raw data produced by RNA sequencing to interpretable results. Widely used and well documented tools are applied. Data quality assessment and read trimming were performed by FastQC and Cutadapt, respectively. Following this, STAR was utilized to map the trimmed reads to a reference genome and the alignment was analyzed by Qualimap. The subsequent mapped reads were quantified by featureCounts. DESeq2 was used to normalize and perform differential expression analysis on the quantified reads, identifying differentially expressed genes and preparing the data for functional enrichment analysis. Gene set enrichment analysis identified enriched gene sets from the normalized count data and clusterProfiler was used to perform functional enrichment against the GO, KEGG, and Reactome databases. Example figures of the functional enrichment analysis results were also generated. The example data used in the workflow are derived from HUVECs, an in vitro model used in the study of endothelial cells, published and publicly available for download from the European Nucleotide Archive.

Parameter estimation in fluorescence recovery after photobleaching: quantitative analysis of protein binding reactions and diffusion.

Fluorescence recovery after photobleaching (FRAP) is a common experimental method for investigating rates of molecular redistribution in biological systems. Many mathematical models of FRAP have been developed, the purpose of which is usually the estimation of certain biological parameters such as the diffusivity and chemical reaction rates of a protein, this being accomplished by fitting the model to experimental data. In this article, we consider a two species reaction-diffusion FRAP model. Using asymptotic analysis, we derive new FRAP recovery curve approximation formulae, and formally re-derive existing ones. On the basis of these formulae, invoking the concept of Fisher information, we predict, in terms of biological and experimental parameters, sufficient conditions to ensure that the values all model parameters can be estimated from data. We verify our predictions with extensive computational simulations. We also use computational methods to investigate cases in which some or all biological parameters are theoretically inestimable. In these cases, we propose methods which can be used to extract the maximum possible amount of information from the FRAP data.

An optogenetic method for interrogating YAP1 and TAZ nuclear-cytoplasmic shuttling.

The shuttling of transcription factors and transcriptional regulators into and out of the nucleus is central to the regulation of many biological processes. Here we describe a new method for studying the rates of nuclear entry and exit of transcriptional regulators. A photo-responsive LOV (light-oxygen-voltage) domain from Avena sativa is used to sequester fluorescently labelled transcriptional regulators YAP1 and TAZ (also known as WWTR1) on the surface of mitochondria and to reversibly release them upon blue light illumination. After dissociation, fluorescent signals from the mitochondria, cytoplasm and nucleus are extracted by a bespoke app and used to generate rates of nuclear entry and exit. Using this method, we demonstrate that phosphorylation of YAP1 on canonical sites enhances its rate of nuclear export. Moreover, we provide evidence that, despite high intercellular variability, YAP1 import and export rates correlate within the same cell. By simultaneously releasing YAP1 and TAZ from sequestration, we show that their rates of entry and exit are correlated. Furthermore, combining the optogenetic release of YAP1 with lattice light-sheet microscopy reveals high heterogeneity of YAP1 dynamics within different cytoplasmic regions, demonstrating the utility and versatility of our tool to study protein dynamics. This article has an associated First Person interview with Anna M. Dowbaj, joint first author of the paper.

A Model to Investigate the Impact of Farm Practice on Antimicrobial Resistance in UK Dairy Farms.

The ecological and human health impact of antibiotic use and the related antimicrobial resistance (AMR) in animal husbandry is poorly understood. In many countries, there has been considerable pressure to reduce overall antibiotic use in agriculture or to cease or minimise use of human critical antibiotics. However, a more nuanced approach would consider the differential impact of use of different antibiotic classes; for example, it is not known whether reduced use of bacteriostatic or bacteriolytic classes of antibiotics would be of greater value. We have developed an ordinary differential equation model to investigate the effects of farm practice on the spread and persistence of AMR in the dairy slurry tank environment. We model the chemical fate of bacteriolytic and bacteriostatic antibiotics within the slurry and their effect on a population of bacteria, which are capable of resistance to both types of antibiotic. Through our analysis, we find that changing the rate at which a slurry tank is emptied may delay the proliferation of multidrug-resistant bacteria by up to five years depending on conditions. This finding has implications for farming practice and the policies that influence waste management practices. We also find that, within our model, the development of multidrug resistance is particularly sensitive to the use of bacteriolytic antibiotics, rather than bacteriostatic antibiotics, and this may be cause for controlling the usage of bacteriolytic antibiotics in agriculture.

Time dependent asymptotic analysis of the gene regulatory network of the AcrAB-TolC efflux pump system in gram-negative bacteria.

Efflux pumps are a mechanism of intrinsic and evolved resistance in bacteria. If an efflux pump can expel an antibiotic so that its concentration within the cell is below a killing threshold the bacteria are resistant to the antibiotic. Efflux pumps may be specific or they may pump various different substances. This is why many efflux pumps confer multi drug resistance (MDR). In particular over expression of the AcrAB-TolC efflux pump system confers MDR in both Salmonella and Escherichia coli. We consider the complex gene regulation network that controls expression of genes central to controlling the efflux associated genes acrAB and acrEF in Salmonella. We present the first mathematical model of this gene regulatory network in the form of a system of ordinary differential equations. Using a time dependent asymptotic analysis, we examine in detail the behaviour of the efflux system on various different timescales. Asymptotic approximations of the steady states provide an analytical comparison of targets for efflux inhibition.

Mathematical modelling of a microRNA-regulated gene network in Caenorhabditis elegans.

MicroRNAs are known to regulate gene expression either by repressing translation or by directing sequence-specific degradation of target mRNAs, and are therefore considered to be key regulators of gene expression. A gene-regulatory pathway involving heterochronic genes controls the temporal pattern of Caenorhabditis elegans postembryonic cell lineages. Based on experimental data, we propose and analyze a mathematical model of a gene-regulatory module in this nematode involving two heterochronic genes, lin-14 and lin-28, which are both regulated by lin-4, encoding a microRNA. The conditions under which the model experiences bifurcations are investigated. We determine the parameter regimes for which the system exhibits monostability and bistability, the latter associated with a biological switch. We observe in particular that bistability occurs without co-operativity, in keeping with knowledge about the regulatory behaviour of lin-14 and lin-28. The analytical results are confirmed by numerical simulations that illustrate how the microRNA lin-4 plays a crucial role in determining of the qualitative dynamics of the model.

Using singular perturbation theory to determine kinetic parameters in a non-standard coupled enzyme assay.

We investigate how to characterize the kinetic parameters of an aminotransaminase using a non-standard coupled (or auxiliary) enzyme assay, where the peculiarity arises for two reasons. First, one of the products of the auxiliary enzyme is a substrate for the primary enzyme and, second, we explicitly account for the reversibility of the auxiliary enzyme reaction. Using singular perturbation theory, we characterize the two distinguished asymptotic limits in terms of the strength of the reverse reaction, which allows us to determine how to deduce the kinetic parameters of the primary enzyme for a characterized auxiliary enzyme. This establishes a parameter-estimation algorithm that is applicable more generally to similar reaction networks. We demonstrate the applicability of our theory by performing enzyme assays to characterize a novel putative aminotransaminase enzyme, CnAptA (UniProtKB Q0KEZ8) from Cupriavidus necator H16, for two different omega-amino acid substrates.

An Asymptotic Analysis of the Malonyl-CoA Route to 3-Hydroxypropionic Acid in Genetically Engineered Microbes.

There has been recent interest in creating an efficient microbial production route for 3-hydroxypropionic acid, an important platform chemical. We develop and solve a mathematical model for the time-dependent metabolite concentrations in the malonyl-CoA pathway for 3-hydroxypropionic acid production in microbes, using a combination of numerical and asymptotic methods. This allows us to identify the most important targets for enzyme regulation therein under conditions of plentiful and sparse pyruvate, and to quantify their relative importance. In our model, we account for sinks of acetyl-CoA and malonyl-CoA to, for example, the citric acid cycle and fatty acid biosynthesis, respectively. Notably, in the plentiful pyruvate case we determine that there is a bifurcation in the asymptotic structure of the system, the crossing of which corresponds to a significant increase in 3-hydroxypropionic acid production. Moreover, we deduce that the most significant increases to 3-hydroxypropionic acid production can be obtained by up-regulating two specific enzymes in tandem, as the inherent nonlinearity of the system means that a solo up-regulation of either does not result in large increases in production. The types of issue arising here are prevalent in synthetic biology applications, and it is hoped that the system considered provides an instructive exemplar for broader applications.

A core mechanism for specifying root vascular patterning can replicate the anatomical variation seen in diverse plant species.

Pattern formation is typically controlled through the interaction between molecular signals within a given tissue. During early embryonic development, roots of the model plant Arabidopsis thaliana have a radially symmetric pattern, but a heterogeneous input of the hormone auxin from the two cotyledons forces the vascular cylinder to develop a diarch pattern with two xylem poles. Molecular analyses and mathematical approaches have uncovered the regulatory circuit that propagates this initial auxin signal into a stable cellular pattern. The diarch pattern seen in Arabidopsis is relatively uncommon among flowering plants, with most species having between three and eight xylem poles. Here, we have used multiscale mathematical modelling to demonstrate that this regulatory module does not require a heterogeneous auxin input to specify the vascular pattern. Instead, the pattern can emerge dynamically, with its final form dependent upon spatial constraints and growth. The predictions of our simulations compare to experimental observations of xylem pole number across a range of species, as well as in transgenic systems in Arabidopsis in which we manipulate the size of the vascular cylinder. By considering the spatial constraints, our model is able to explain much of the diversity seen in different flowering plant species.

Limit-cycle oscillatory coexpression of cross-inhibitory transcription factors: a model mechanism for lineage promiscuity.

Lineage switches are genetic regulatory motifs that govern and maintain the commitment of a developing cell to a particular cell fate. A canonical example of a lineage switch is the pair of transcription factors PU.1 and GATA-1, of which the former is affiliated with the myeloid and the latter with the erythroid lineage within the haematopoietic system. On a molecular level, PU.1 and GATA-1 positively regulate themselves and antagonize each other via direct protein-protein interactions. Here we use mathematical modelling to identify a novel type of dynamic behaviour that can be supported by such a regulatory architecture. Guided by the specifics of the PU.1-GATA-1 interaction, we formulate, using the law of mass action, a system of differential equations for the key molecular concentrations. After a series of systematic approximations, the system is reduced to a simpler one, which is tractable to phase-plane and linearization methods. The reduced system formally resembles, and generalizes, a well-known model for competitive species from mathematical ecology. However, in addition to the qualitative regimes exhibited by a pair of competitive species (exclusivity, bistable exclusivity, stable-node coexpression) it also allows for oscillatory limit-cycle coexpression. A key outcome of the model is that, in the context of cell-fate choice, such oscillations could be harnessed by a differentiating cell to prime alternately for opposite outcomes; a bifurcation-theory approach is adopted to characterize this possibility.

Multi-timescale analysis of a metabolic network in synthetic biology: a kinetic model for 3-hydroxypropionic acid production via beta-alanine.

A biosustainable production route for 3-hydroxypropionic acid (3HP), an important platform chemical, would allow 3HP to be produced without using fossil fuels. We are interested in investigating a potential biochemical route to 3HP from pyruvate through [Formula: see text]-alanine and, in this paper, we develop and solve a mathematical model for the reaction kinetics of the metabolites involved in this pathway. We consider two limiting cases, one where the levels of pyruvate are never replenished, the other where the levels of pyruvate are continuously replenished and thus kept constant. We exploit the natural separation of both the time scales and the metabolite concentrations to make significant asymptotic progress in understanding the system without resorting to computationally expensive parameter sweeps. Using our asymptotic results, we are able to predict the most important reactions to maximize the production of 3HP in this system while reducing the maximum amount of the toxic intermediate compound malonic semi-aldehyde present at any one time, and thus we are able to recommend which enzymes experimentalists should focus on manipulating.

Applying asymptotic methods to synthetic biology: Modelling the reaction kinetics of the mevalonate pathway.

The mevalonate pathway is normally found in eukaryotes, and allows for the production of isoprenoids, a useful class of organic compounds. This pathway has been successfully introduced to Escherichia coli, enabling a biosynthetic production route for many isoprenoids. In this paper, we develop and solve a mathematical model for the concentration of metabolites in the mevalonate pathway over time, accounting for the loss of acetyl-CoA to other metabolic pathways. Additionally, we successfully test our theoretical predictions experimentally by introducing part of the pathway into Cupriavidus necator. In our model, we exploit the natural separation of time scales as well as of metabolite concentrations to make significant asymptotic progress in understanding the system. We confirm that our asymptotic results agree well with numerical simulations, the former enabling us to predict the most important reactions to increase isopentenyl diphosphate production whilst minimizing the levels of HMG-CoA, which inhibits cell growth. Thus, our mathematical model allows us to recommend the upregulation of certain combinations of enzymes to improve production through the mevalonate pathway.

Root hydrotropism is controlled via a cortex-specific growth mechanism.

Plants can acclimate by using tropisms to link the direction of growth to environmental conditions. Hydrotropism allows roots to forage for water, a process known to depend on abscisic acid (ABA) but whose molecular and cellular basis remains unclear. Here we show that hydrotropism still occurs in roots after laser ablation removed the meristem and root cap. Additionally, targeted expression studies reveal that hydrotropism depends on the ABA signalling kinase SnRK2.2 and the hydrotropism-specific MIZ1, both acting specifically in elongation zone cortical cells. Conversely, hydrotropism, but not gravitropism, is inhibited by preventing differential cell-length increases in the cortex, but not in other cell types. We conclude that root tropic responses to gravity and water are driven by distinct tissue-based mechanisms. In addition, unlike its role in root gravitropism, the elongation zone performs a dual function during a hydrotropic response, both sensing a water potential gradient and subsequently undergoing differential growth.

Hybrid vertex-midline modelling of elongated plant organs.

We describe a method for the simulation of the growth of elongated plant organs, such as seedling roots. By combining a midline representation of the organ on a tissue scale and a vertex-based representation on the cell scale, we obtain a multiscale method, which is able to both simulate organ growth and incorporate cell-scale processes. Equations for the evolution of the midline are obtained, which depend on the cell-wall properties of individual cells through appropriate averages over the vertex-based representation. The evolution of the organ midline is used to deform the cellular-scale representation. This permits the investigation of the regulation of organ growth through the cell-scale transport of the plant hormone auxin. The utility of this method is demonstrated in simulating the early stages of the response of a root to gravity, using a vertex-based template acquired from confocal imaging. Asymmetries in the concentrations of auxin between the upper and lower sides of the root lead to bending of the root midline, reflecting a gravitropic response.

Dynamic regulation of auxin oxidase and conjugating enzymes AtDAO1 and GH3 modulates auxin homeostasis.

The hormone auxin is a key regulator of plant growth and development, and great progress has been made understanding auxin transport and signaling. Here, we show that auxin metabolism and homeostasis are also regulated in a complex manner. The principal auxin degradation pathways in Arabidopsis include oxidation by Arabidopsis thaliana gene DIOXYGENASE FOR AUXIN OXIDATION 1/2 (AtDAO1/2) and conjugation by Gretchen Hagen3s (GH3s). Metabolic profiling of dao1-1 root tissues revealed a 50% decrease in the oxidation product 2-oxoindole-3-acetic acid (oxIAA) and increases in the conjugated forms indole-3-acetic acid aspartic acid (IAA-Asp) and indole-3-acetic acid glutamic acid (IAA-Glu) of 438- and 240-fold, respectively, whereas auxin remains close to the WT. By fitting parameter values to a mathematical model of these metabolic pathways, we show that, in addition to reduced oxidation, both auxin biosynthesis and conjugation are increased in dao1-1 Transcripts of AtDAO1 and GH3 genes increase in response to auxin over different timescales and concentration ranges. Including this regulation of AtDAO1 and GH3 in an extended model reveals that auxin oxidation is more important for auxin homoeostasis at lower hormone concentrations, whereas auxin conjugation is most significant at high auxin levels. Finally, embedding our homeostasis model in a multicellular simulation to assess the spatial effect of the dao1-1 mutant shows that auxin increases in outer root tissues in agreement with the dao1-1 mutant root hair phenotype. We conclude that auxin homeostasis is dependent on AtDAO1, acting in concert with GH3, to maintain auxin at optimal levels for plant growth and development.