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We analyze a class of chemical reaction networks for which all moments can be computed by finite-dimensional linear differential equations. This class allows second and higher order reactions, but only under special assumptions on structure and/or conservation laws.
Cellular signalling processes can exhibit pronounced cell-to-cell variability in genetically identical cells. This affects how individual cells respond differentially to the same environmental stimulus. However, the origins of cell-to-cell variability in cellular signalling systems remain poorly understood. Here we measure the temporal evolution of phosphorylated MEK and ERK dynamics across populations of cells and quantify the levels of population heterogeneity over time using high-throughput image cytometry. We use a statistical modelling framework to show that upstream noise is the dominant factor causing cell-to-cell variability in ERK phosphorylation, rather than stochasticity in the phosphorylation/dephosphorylation of ERK. In particular, the cell-to-cell variability during sustained phosphorylation stems from random fluctuations in the background upstream signalling processes, while during transient phosphorylation, the heterogeneity is primarily due to noise in the intensity of the upstream signal(s). We show that the core MEK/ERK system uses kinetic proof-reading to faithfully and robustly transmits these variable inputs. The MAPK cascade thus propagates cell-to-cell variability at the population level, rather than attenuating or increasing it.
Within populations of cells, fate decisions are controlled by an indeterminate combination of cell-intrinsic and cell-extrinsic factors. In the case of stem cells, the stem cell niche is believed to maintain \"stemness\" through communication and interactions between the stem cells and one or more other cell-types that contribute to the niche conditions. To investigate the robustness of cell fate decisions in the stem cell hierarchy and the role that the niche plays, we introduce simple mathematical models of stem and progenitor cells, their progeny and their interplay in the niche. These models capture the fundamental processes of proliferation and differentiation and allow us to consider alternative possibilities regarding how niche-mediated signalling feedback regulates the niche dynamics. Generalised stability analysis of these stem cell niche systems enables us to describe the stability properties of each model. We find that although the number of feasible states depends on the model, their probabilities of stability in general do not: stem cell-niche models are stable across a wide range of parameters. We demonstrate that niche-mediated feedback increases the number of stable steady states, and show how distinct cell states have distinct branching characteristics. The ecological feedback and interactions mediated by the stem cell niche thus lend (surprisingly) high levels of robustness to the stem and progenitor cell population dynamics. Furthermore, cell-cell interactions are sufficient for populations of stem cells and their progeny to achieve stability and maintain homeostasis. We show that the robustness of the niche -- and hence of the stem cell pool in the niche -- depends only weakly, if at all, on the complexity of the niche make-up: simple as well as complicated niche systems are capable of supporting robust and stable stem cell dynamics.
Osteosarcoma is the most common primary bone tumor in pediatric and young adult patients. Successful treatment of osteosarcomas requires a combination of surgical resection and systemic chemotherapy, both neoadjuvant (prior to surgery) and adjuvant (after surgery). The degree of necrosis following neoadjuvant chemotherapy correlates with the subsequent probability of disease-free survival. Tumors with less than 10% of viable cells after treatment represent patients with a more favorable prognosis. However, being able to predict early, such as at the time of the pre-treatment tumor biopsy, how the patient will respond to the standard chemotherapy would provide an opportunity for more personalized patient care. Patients with unfavorable predictions could be studied in a protocol, rather than a standard setting, towards improving therapeutic success. The onset of necrotic cells in osteosarcomas treated with chemotherapeutic agents is a measure of tumor sensitivity to the drugs. We hypothesize that the remaining viable cells, i.e., cells that have not responded to the treatment, are chemoresistant, and that the pathological characteristics of these chemoresistant tumor cells within the osteosarcoma pre-treatment biopsy can predict tumor response to the standard-of-care chemotherapeutic treatment. This hypothesis can be tested by comparing patient histopathology samples before, as well as after treatment to identify both morphological and immunochemical cellular features that are characteristic of chemoresistant cells, i.e., cells that survived treatment. Consequently, using computational simulations of dynamic changes in tumor pathology under the simulated standard of care chemotherapeutic treatment, one can couple the pre- and post-treatment morphological and spatial patterns of chemoresistant cells, and correlate them with patient clinical diagnoses. This procedure, that we named Virtual Clinical Trials, can serve as a potential predictive biomarker providing a novel value added decision support tool for oncologists.
Mesenchymal stromal cells (MSC) are widely used, isolated from a variety of tissues and increasingly adopted for cell therapy, but the identity of these cells is poorly defined and commonalities between MSC from different tissues sources is controversial. Here we undertook a comprehensive review of all public MSC expression studies to assess whether cells derived from different sources shared any common molecular attributes. In doing so, we discovered an over-arching transcriptional phenotype shared by a wide variety of MSC, freshly isolated or cultured cells, and under a variety of growth conditions. We developed a modified variable selection protocol that included cross platform normalisation, and assessment of the selected gene stability and informativeness. A 16-gene signature classified MSC with >97% accuracy, discriminating these from fibroblasts, other adult stem/progenitor cell types and differentiated cells. The genes form part of a protein-interaction network, and mutations in more than 65% of this network were associated with Mendelian disorders of skeletal growth or metabolism. The signature and accompanying datasets are provided as a community resource at www.stemformatics.org resource, and the method is available from the CRAN repository.
The diversity of a persons B- and T-cell repertoires is both clinically important and a key measure of immunological complexity. However, diversity is hard to estimate by current methods, due to inherent uncertainty in the number of B- and T-cell clones that will be missing from a blood or tissue sample by chance (the missing-species problem), inevitable sampling bias, and experimental noise. To solve this problem we developed Recon, a modified maximum-likelihood method that outputs the overall diversity of a repertoire from measurements on a sample. Recon outputs accurate, robust estimates by any of a vast set of complementary diversity measures, including species richness and entropy, at fractional repertoire coverage. It also outputs error bars and power tables, allowing robust comparisons of diversity between individuals and over time. We apply Recon to in silico and experimental immune-repertoire sequencing datasets as proof of principle for measuring diversity in large, complex systems.
Ubiquitination is an essential post-translational regulatory process that can control protein stability, localization, and activity. Ubiquitin is essential for eukaryotic life and is highly conserved, varying in only 3 amino acid positions between yeast and humans. However, recent deep sequencing studies in S. cerevisiae indicate that ubiquitin is highly tolerant to single amino acid mutations. To resolve this paradox, we hypothesized that the set of tolerated substitutions would be reduced when the cultures are not grown in rich media conditions and that chemically induced physiologic perturbations might unmask constraints on the ubiquitin sequence. To test this hypothesis, a class of first year UCSF graduate students employed a deep mutational scanning procedure to determine the fitness landscape of a library of all possible single amino acid mutations of ubiquitin in the presence of one of five small molecule perturbations: MG132, Dithiothreitol (DTT), Hydroxyurea (HU), Caffeine, and DMSO. Our data reveal that the number of tolerated substitutions is greatly reduced by DTT, HU, or Caffeine, and that these perturbations uncover \"shared sensitized positions\" localized to areas around the hydrophobic patch and to the C-terminus. We also show perturbation specific effects including the sensitization of His68 in HU and tolerance to mutation at Lys63 in DTT. Taken together, our data suggest that chemical stress reduces buffering effects in the ubiquitin proteasome system, revealing previously hidden fitness defects. By expanding the set of chemical perturbations assayed, potentially by other classroom-based experiences, we will be able to further address the apparent dichotomy between the extreme sequence conservation and the experimentally observed mutational tolerance of ubiquitin. Finally, this study demonstrates the realized potential of a project lab-based interdisciplinary graduate curriculum.
There is a gradual change towards explicitly considering landscapes in regulatory risk assessment. To realise the objective of developing representative scenarios for risk assessment it is necessary to know how detailed a landscape representation is needed to generate a realistic risk assessment, and indeed how to generate such landscapes. This paper evaluates the contribution of landscape and farming components to a model based risk assessment of a fictitious endocrine disruptor on hares. In addition, we present methods and code examples for generation of landscape structures and farming simulation from data collected primarily for EU agricultural subsidy support and GIS map data.\n\nTen different Danish landscapes were generated and the ERA carried out for each landscape using two different assumed toxicities. The results showed negative impacts in all cases, but the extent and form in terms of impacts on abundance or occupancy differed greatly between landscapes. A meta-model was created, predicting impact from landscape and farming characteristics. Scenarios based on all combinations of farming and landscape for five landscapes representing extreme and middle impacts were created. The meta-models developed from the 10 real landscapes failed to predict impacts for these 25 scenarios. Landscape, farming, and the emergent density of hares all influenced the results of the risk assessment considerably.\n\nThe study indicates that prediction of a reasonable worst case scenario is difficult from structural, farming or population metrics; rather the emergent properties generated from interactions between landscape, management and ecology are needed. Meta-modelling may also fail to predict impacts, even when restricting inputs to combinations of those used to create the model. Future ERA may therefore need to make use of multiple scenarios representing a wide range of conditions to avoid locally unacceptable risks. This approach could now be feasible Europe wide given the landscape generation methods presented.
Horizontal gene transfer (HGT) plays a central role in bacterial evolution, yet the molecular and cellular constraints on functional integration of the foreign genes are poorly understood. Here we performed inter-species replacement of the chromosomal folA gene, encoding an essential metabolic enzyme dihydrofolate reductase (DHFR), with orthologs from 35 other mesophilic bacteria. The orthologous inter-species replacements caused a marked drop (in the range 10-90%) in bacterial growth rate despite the fact that most orthologous DHFRs are as stable as E.coli DHFR at 37{degrees} C and are more catalytically active than E. coli DHFR. Although phylogenetic distance between E. coli and orthologous DHFRs as well as their individual molecular properties correlate poorly with growth rates, the product of the intracellular DHFR abundance and catalytic activity (kcat/KM), correlates strongly with growth rates, indicating that the drop in DHFR abundance constitutes the major fitness barrier to HGT. Serial propagation of the orthologous strains for [~]600 generations dramatically improved growth rates by largely alleviating the fitness barriers. Whole genome sequencing and global proteome quantification revealed that the evolved strains with the largest fitness improvements have accumulated mutations that inactivated the ATP-dependent Lon protease, causing an increase in the intracellular DHFR abundance. In one case DHFR abundance increased further due to mutations accumulated in folA promoter, but only after the lon inactivating mutations were fixed in the population. Thus, by apparently distinguishing between self and non-self proteins, protein homeostasis imposes an immediate and global barrier to the functional integration of foreign genes by decreasing the intracellular abundance of their products. Once this barrier is alleviated, more fine-tuned evolution occurs to adjust the function/expression of the transferred proteins to the constraints imposed by the intracellular environment of the host organism.
Inside individual cells, expression of genes is inherently stochastic and manifests as cell-to-cell variability or noise in protein copy numbers. Since proteins half-lives can be comparable to the cell-cycle length, randomness in cell-division times generates additional intercellular variability in protein levels. Moreover, as many mRNA/protein species are expressed at low-copy numbers, errors incurred in partitioning of molecules between the mother and daughter cells are significant. We derive analytical formulas for the total noise in protein levels for a general class of cell-division time and partitioning error distributions. Using a novel hybrid approach the total noise is decomposed into components arising from i) stochastic expression; ii) partitioning errors at the time of cell-division and iii) random cell-division events. These formulas reveal that random cell-division times not only generate additional extrinsic noise but also critically affect the mean protein copy numbers and intrinsic noise components. Counter intuitively, in some parameter regimes noise in protein levels can decrease as cell-division times become more stochastic. Computations are extended to consider genome duplication, where the gene dosage is increased by two-fold at a random point in the cell-cycle. We systematically investigate how the timing of genome duplication influences different protein noise components. Intriguingly, results show that noise contribution from stochastic expression is minimized at an optimal genome duplication time. Our theoretical results motivate new experimental methods for decomposing protein noise levels from single-cell expression data. Characterizing the contributions of individual noise mechanisms will lead to precise estimates of gene expression parameters and techniques for altering stochasticity to change phenotype of individual cells.
In the past decade, researchers have been able to obtain pluripotent stem cells directly from an organisms differentiated cells through a process called cell reprogramming. This opens the way to potentially groundbreaking applications in regenerative and personalized medicine, in which ill patients could use self-derived induced pluripotent stem (iPS) cells where needed. While the process of reprogramming has been shown to be possible, its efficiency remains so low after almost ten years since its conception as to render its applicability limited to laboratory research. In this paper, we study a mathematical model of the core transcriptional circuitry among a set of key transcription factors, which is thought to determine the switch among pluripotent and differentiated cell types. By employing standard tools from dynamical systems theory, we analyze the effects on the systems dynamics of overexpressing the core factors, which is what is performed during the reprogramming process. We demonstrate that the structure of the system is such that it can render the switch from an initial stable steady state (differentiated cell type) to the desired stable steady state (pluripotent cell type) highly unlikely. This finding provides insights into a possible reason for the low efficiency of current reprogramming approaches.
Cancers arise from genetic abberations but also consistently display high levels of intra-tumor heterogeneity and evolve according to Darwinian dynamics. This makes evolutionary game theory an ideal tool in which to mathematically capture these cell-cell interactions and in which to investigate how they impact evolutionary dynamics. In this chapter we present some examples of how evolutionary game theory can elucidate cancer evolution.
Epithelial to mesenchymal transition (EMT) is an essential differentiation program during tissue morphogenesis and remodeling. EMT is induced by soluble transforming growth factor {beta} (TGF-{beta}) family members, and restricted by vascular endothelial growth factor family members. While many downstream molecular regulators of EMT have been identified, these have been largely evaluated individually without considering potential crosstalk. In this study, we created an ensemble of dynamic mathematical models describing TGF-{beta} induced EMT to better understand the operational hierarchy of this complex molecular program. We used ordinary differential equations (ODEs) to describe the transcriptional and post-translational regulatory events driving EMT. Model parameters were estimated from multiple data sets using multiobjective optimization, in combination with cross-validation. TGF-{beta} exposure drove the model population toward a mesenchymal phenotype, while an epithelial phenotype was enhanced following vascular endothelial growth factor A (VEGF-A) exposure. Simulations predicted that the transcription factors phosphorylated SP1 and NFAT were master regulators promoting or inhibiting EMT, respectively. Surprisingly, simulations also predicted that a cellular population could exhibit phenotypic heterogeneity (characterized by a significant fraction of the population with both high epithelial and mesenchymal marker expression) if treated simultaneously with TGF-{beta} and VEGF-A. We tested this prediction experimentally in both MCF10A and DLD1 cells and found that upwards of 45% of the cellular population acquired this hybrid state in the presence of both TGF-{beta} and VEGF-A. We experimentally validated the predicted NFAT/Sp1 signaling axis for each phenotype response. Lastly, we found that cells in the hybrid state had significantly different functional behavior when compared to VEGF-A or TGF-{beta} treatment alone. Together, these results establish a predictive mechanistic model of EMT susceptibility, and potentially reveal a novel signaling axis which regulates carcinoma progression through an EMT versus tubulogenesis response.\n\nAuthor SummaryTissue formation and remodeling requires a complex and dynamic balance of interactions between epithelial cells, which reside on the surface, and mesenchymal cells that reside in the tissue interior. During embryonic development, wound healing, and cancer, epithelial cells transform into a mesenchymal cell to form new types of tissues. It is important to understand this process so that it can be controlled to generate beneficial effects and limit pathological differentiation. Much research over the past 20 years has identified many different molecular species that are relevant, but these have mainly been studied one at a time. In this study, we developed and implemented a novel computational strategy to interrogate the key players in this transformation process to identify which are the major bottlenecks. We determined that NFATc1 and pSP1 are essential for promoting epithelial or mesenchymal differentiation, respectively. We then predicted the existence of a partially transformed cell that exhibits both epithelial and mesenchymal characteristics. We found this partial cell type develops a network of invasive but stunted vascular structures that may be a unique cell target for understanding cancer progression and angiogenesis.
Human microbiota plays an important role in determining changes from health to disease. Increasing research activity is dedicated to understand its diversity and variability. We analyse 16S rRNA and whole genome sequencing (WGS) data from the gut microbiota of 97 individuals monitored in time. Temporal fluctuations in the microbiome reveal significant differences due to factors that affect the microbiota such as dietary changes, antibiotic intake, early gut development or disease. Here we show that a fluctuation scaling law describes the temporal variability of the system and that a noise-induced phase transition is central in the route to disease. The universal law distinguishes healthy from sick microbiota and quantitatively characterizes the path in the phase space, which opens up its potential clinical use and, more generally, other technological applications where microbiota plays an important role.\n\nPACS numbers:
BackgroundGene expression is known to be an intrinsically stochastic process which can involve single-digit numbers of mRNA molecules in a cell at any given time. The modelling of such processes calls for the use of exact stochastic simulation methods, most notably the Gillespie algorithm. However, this stochasticity, also termed \"intrinsic noise\", does not account for all the variability between genetically identical cells growing in a homogeneous environment. Despite substantial experimental efforts, determining appropriate model parameters continues to be a challenge. Methods based on approximate Bayesian computation can be used to obtain posterior parameter distributions given the observed data. However, such inference procedures require large numbers of simulations of the model and exact stochastic simulation is computationally costly. In this work we focus on the specific case of trying to infer model parameters describing reaction rates and extrinsic noise on the basis of measurements of molecule numbers in individual cells at a given time point.\n\nResultsTo make the problem computationally tractable we develop an exact, model-specific, stochastic simulation algorithm for the commonly used two-state model of gene expression. This algorithm relies on certain assumptions and favourable properties of the model to forgo the simulation of the whole temporal trajectory of protein numbers in the system, instead returning only the number of protein and mRNA molecules present in the system at a specified time point. The computational gain is proportional to the number of protein molecules created in the system and becomes significant for systems involving hundreds or thousands of protein molecules. We employ this algorithm, approximate Bayesian computation, and published gene expression data for Escherichia coli to simultaneously infer the models rate parameters and parameters describing extrinsic noise for 86 genes.
SummaryIn this technical report we describe a simple extension to python-libSBML that allows users of Python to more easily construct SBML based models. The most commonly used package for constructing SBML models in Python is python-libSBML based on the C/C++ library libSBML. python-libSBML supports a comprehensive set of model types, but is difficult for new users to learn and requires long scripts to create even the simplest models. We present SimpleSBML, a package that allows users to add species, parameters, reactions, events, and rules to a libSBML model with only one command for each. Models can be exported to SBML format, and SBML files can be imported and converted to SimpleSBML commands that creates each element in a new model. This allows users to create new models and edit existing models for use with other software. Accessibility and ImplementationSimpleSBML is publicly available and licensed under the liberal Apache 2.0 open source license. It supports SBML levels 2 and 3. Its only dependency is libSBML. It is supported on Windows and Mac OS X. All code has been deposited at the GitHub site https://github.com/sys-bio/simplesbml Contactcarolc24@u.washington.edu or hsauro@u.washington.edu
Flux balance analysis (FBA) is an increasingly useful approach for modeling the behavior of metabolic systems. However, standard FBA modeling of genetic knockouts can not predict drug combination synergies observed between serial metabolic targets, even though such synergies give rise to some of the most widely used antibiotic treatments. Here we extend FBA modeling to simulate responses to chemical inhibitors at varying concentrations, by diverting enzymatic flux to a waste reaction. This flux diversion yields very similar qualitative predictions to prior methods for single target activity. However, we find very different predictions for combinations, where flux diversion, which mimics the kinetics of competitive metabolic inhibitors, can explain serial target synergies between metabolic enzyme inhibitors that we confirmed in Escherichia coli cultures. FBA flux diversion opens the possibility for more accurate genome-scale predictions of drug synergies, which can be used to suggest treatments for infections and other diseases.
We present a dynamical model of drug accumulation in bacteria. The model captures key features in experimental time courses on ofloxacin accumulation: initial uptake; two-phase response; and long-term acclimation. In combination with experimental data, the model provides estimates of import and export rates in each phase, the time of entry into the second phase, and the decrease of internal drug during acclimation. Global sensitivity analysis, local sensitivity analysis, and Bayesian sensitivity analysis of the model provide information about the robustness of these estimates, and about the relative importance of different parameters in determining the features of the accumulation time courses in three different bacterial species: Escherichia coli, Staphylococcus aureus, and Pseudomonas aeruginosa. The results lead to experimentally testable predictions of the effects of membrane permeability, drug efflux and trapping (e.g., by DNA binding) on drug accumulation. A key prediction is that a sudden increase in ofloxacin accumulation in both E. coli and S. aureus is accompanied by a decrease in membrane permeability.\n\nAuthor SummaryBacteria live or die depending on how much antibiotic gets inside them. Using a simple mathematical model, detailed information about drug import and export can be teased out of time courses of internal drug levels after a sudden exposure. The results suggest that membrane permeability can suddenly decrease during exposure to drug, accompanied by an increase, rather than a decrease, in the internal drug level.