Repeated Decision Stumping Distils Simple Rules from Single-Cell Data

Single-cell data afford unprecedented insights into molecular processes. But the complexity and size of these data sets have proved challenging and given rise to a large armory of statistical and machine learning approaches. The majority of approaches focuses on either describing features of these data, or making predictions and classifying unlabeled samples. In this study, we introduce repeated decision stumping (ReDX) as a method to distill simple models from single-cell data. We develop decision trees of depth one-hence "stumps"-to identify in an inductive manner, gene products involved in driving cell fate transitions, and in applications to published data we are able to discover the key players involved in these processes in an unbiased manner without prior knowledge. Our algorithm is deliberately targeting the simplest possible candidate hypotheses that can be extracted from complex high-dimensional data. There are three reasons for this: (1) the predictions become straightforwardly testable hypotheses; (2) the identified candidates form the basis for further mechanistic model development, for example, for engineering and synthetic biology interventions; and (3) this approach complements existing descriptive modeling approaches and frameworks. The approach is computationally efficient, has remarkable predictive power, including in simulation studies where the ground truth is known, and yields robust and statistically stable predictors; the same set of candidates is generated by applying the algorithm to different subsamples of experimental data.

BondGraphs.jl: composable energy-based modelling in systems biology

SUMMARY

BondGraphs.jl is a Julia implementation of bond graphs. Bond graphs provide a modelling framework that describes energy flow through a physical system and by construction enforce thermodynamic constraints. The framework is widely used in engineering and has recently been shown to be a powerful approach for modelling biology. Models are mutable, hierarchical, multiscale, and multiphysics, and BondGraphs.jl is compatible with the Julia modelling ecosystem.

AVAILABILITY AND IMPLEMENTATION

BondGraphs.jl is freely available under the MIT license. Source code and documentation can be found at https://github.com/jedforrest/BondGraphs.jl.

Julia for biologists

Major computational challenges exist in relation to the collection, curation, processing and analysis of large genomic and imaging datasets, as well as the simulation of larger and more realistic models in systems biology. Here we discuss how a relative newcomer among programming languages-Julia-is poised to meet the current and emerging demands in the computational biosciences and beyond. Speed, flexibility, a thriving package ecosystem and readability are major factors that make high-performance computing and data analysis available to an unprecedented degree. We highlight how Julia's design is already enabling new ways of analyzing biological data and systems, and we provide a list of resources that can facilitate the transition into Julian computing.

Open problems in mathematical biology

Biology is data-rich, and it is equally rich in concepts and hypotheses. Part of trying to understand biological processes and systems is therefore to confront our ideas and hypotheses with data using statistical methods to determine the extent to which our hypotheses agree with reality. But doing so in a systematic way is becoming increasingly challenging as our hypotheses become more detailed, and our data becomes more complex. Mathematical methods are therefore gaining in importance across the life- and biomedical sciences. Mathematical models allow us to test our understanding, make testable predictions about future behaviour, and gain insights into how we can control the behaviour of biological systems. It has been argued that mathematical methods can be of great benefit to biologists to make sense of data. But mathematics and mathematicians are set to benefit equally from considering the often bewildering complexity inherent to living systems. Here we present a small selection of open problems and challenges in mathematical biology. We have chosen these open problems because they are of both biological and mathematical interest.

More is different with a vengeance

In his 1972 landmark paper "More is Different," Philip W. Anderson established "complexity" as a fundamentally important subject of inquiry. He highlighted the profound limitations of reductionist approaches in understanding nature's complexity, and he set in motion new lines of investigation that have, among other things, led to systems biology.

HyperGraphs.jl - representing high-order relationships in Julia

Summary

HyperGraphs.jl is a Julia package that implements hypergraphs. These are a generalization of graphs that allow us to represent n-ary relationships and not just binary, pairwise relationships. High-order interactions are commonplace in biological systems and are of critical importance to their dynamics; hypergraphs thus offer a natural way to accurately describe and model these systems.

Availability and implementation

HyperGraphs.jl is freely available under the MIT license. Source code and documentation can be found at https://github.com/lpmdiaz/HyperGraphs.jl.

Supplementary information

Supplementary data are available at Bioinformatics online.

Turing pattern design principles and their robustness

Turing patterns have morphed from mathematical curiosities into highly desirable targets for synthetic biology. For a long time, their biological significance was sometimes disputed but there is now ample evidence for their involvement in processes ranging from skin pigmentation to digit and limb formation. While their role in developmental biology is now firmly established, their synthetic design has so far proved challenging. Here, we review recent large-scale mathematical analyses that have attempted to narrow down potential design principles. We consider different aspects of robustness of these models and outline why this perspective will be helpful in the search for synthetic Turing-patterning systems. We conclude by considering robustness in the context of developmental modelling more generally. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

The design principles of discrete turing patterning systems

The formation of spatial structures lies at the heart of developmental processes. However, many of the underlying gene regulatory and biochemical processes remain poorly understood. Turing patterns constitute a main candidate to explain such processes, but they appear sensitive to fluctuations and variations in kinetic parameters, raising the question of how they may be adopted and realised in naturally evolved systems. The vast majority of mathematical studies of Turing patterns have used continuous models specified in terms of partial differential equations. Here, we complement this work by studying Turing patterns using discrete cellular automata models. We perform a large-scale study on all possible two-species networks and find the same Turing pattern producing networks as in the continuous framework. In contrast to continuous models, however, we find these Turing pattern topologies to be substantially more robust to changes in the parameters of the model. We also find that diffusion-driven instabilities are substantially weaker predictors for Turing patterns in our discrete modelling framework in comparison to the continuous case, in the sense that the presence of an instability does not guarantee a pattern emerging in simulations. We show that a more refined criterion constitutes a stronger predictor. The similarity of the results for the two modelling frameworks suggests a deeper underlying principle of Turing mechanisms in nature. Together with the larger robustness in the discrete case this suggests that Turing patterns may be more robust than previously thought.

Model comparison via simplicial complexes and persistent homology

In many scientific and technological contexts, we have only a poor understanding of the structure and details of appropriate mathematical models. We often, therefore, need to compare different models. With available data we can use formal statistical model selection to compare and contrast the ability of different mathematical models to describe such data. There is, however, a lack of rigorous methods to compare different models a priori. Here, we develop and illustrate two such approaches that allow us to compare model structures in a systematic way by representing models as simplicial complexes. Using well-developed concepts from simplicial algebraic topology, we define a distance between models based on their simplicial representations. Employing persistent homology with a flat filtration provides for alternative representations of the models as persistence intervals, which represent model structure, from which the model distances are also obtained. We then expand on this measure of model distance to study the concept of model equivalence to determine the conceptual similarity of models. We apply our methodology for model comparison to demonstrate an equivalence between a positional-information model and a Turing-pattern model from developmental biology, constituting a novel observation for two classes of models that were previously regarded as unrelated.

Collocation based training of neural ordinary differential equations

The predictive power of machine learning models often exceeds that of mechanistic modeling approaches. However, the interpretability of purely data-driven models, without any mechanistic basis is often complicated, and predictive power by itself can be a poor metric by which we might want to judge different methods. In this work, we focus on the relatively new modeling techniques of neural ordinary differential equations. We discuss how they relate to machine learning and mechanistic models, with the potential to narrow the gulf between these two frameworks: they constitute a class of hybrid model that integrates ideas from data-driven and dynamical systems approaches. Training neural ODEs as representations of dynamical systems data has its own specific demands, and we here propose a collocation scheme as a fast and efficient training strategy. This alleviates the need for costly ODE solvers. We illustrate the advantages that collocation approaches offer, as well as their robustness to qualitative features of a dynamical system, and the quantity and quality of observational data. We focus on systems that exemplify some of the hallmarks of complex dynamical systems encountered in systems biology, and we map out how these methods can be used in the analysis of mathematical models of cellular and physiological processes.

Phylodynamics for cell biologists

Multicellular organisms are composed of cells connected by ancestry and descent from progenitor cells. The dynamics of cell birth, death, and inheritance within an organism give rise to the fundamental processes of development, differentiation, and cancer. Technical advances in molecular biology now allow us to study cellular composition, ancestry, and evolution at the resolution of individual cells within an organism or tissue. Here, we take a phylogenetic and phylodynamic approach to single-cell biology. We explain how "tree thinking" is important to the interpretation of the growing body of cell-level data and how ecological null models can benefit statistical hypothesis testing. Experimental progress in cell biology should be accompanied by theoretical developments if we are to exploit fully the dynamical information in single-cell data.

Noise and the molecular processes underlying cell fate decision-making

Cell fate decision-making events involve the interplay of many molecular processes, ranging from signal transduction to genetic regulation, as well as a set of molecular and physiological feedback loops. Each aspect offers a rich field of investigation in its own right, but to understand the whole process, even in simple terms, we need to consider them together. Here we attempt to characterise this process by focussing on the roles of noise during cell fate decisions. We use a range of recent results to develop a view of the sequence of events by which a cell progresses from a pluripotent or multipotent to a differentiated state: chromatin organisation, transcription factor stoichiometry, and cellular signalling all change during this progression, and all shape cellular variability, which becomes maximal at the transition state.

Non-equilibrium statistical physics, transitory epigenetic landscapes, and cell fate decision dynamics

Statistical physics provides a useful perspective for the analysis of many complex systems; it allows us to relate microscopic fluctuations to macroscopic observations. Developmental biology, but also cell biology more generally, are examples where apparently robust behaviour emerges from highly complex and stochastic sub-cellular processes. Here we attempt to make connections between different theoretical perspectives to gain qualitative insights into the types of cell-fate decision making processes that are at the heart of stem cell and developmental biology. We discuss both dynamical systems as well as statistical mechanics perspectives on the classical Waddington or epigenetic landscape. We find that non-equilibrium approaches are required to overcome some of the shortcomings of classical equilibrium statistical thermodynamics or statistical mechanics in order to shed light on biological processes, which, almost by definition, are typically far from equilibrium.

Exactly solvable models of stochastic gene expression

Stochastic models are key to understanding the intricate dynamics of gene expression. However, the simplest models that only account for active and inactive states of a gene fail to capture common observations in both prokaryotic and eukaryotic organisms. Here, we consider multistate models of gene expression that generalize the canonical Telegraph process and are capable of capturing the joint effects of transcription factors, heterochromatin state, and DNA accessibility (or, in prokaryotes, sigma-factor activity) on transcript abundance. We propose two approaches for solving classes of these generalized systems. The first approach offers a fresh perspective on a general class of multistate models and allows us to "decompose" more complicated systems into simpler processes, each of which can be solved analytically. This enables us to obtain a solution of any model from this class. Next, we develop an approximation method based on a power series expansion of the stationary distribution for an even broader class of multistate models of gene transcription. We further show that models from both classes cannot have a heavy-tailed distribution in the absence of extrinsic noise. The combination of analytical and computational solutions for these realistic gene expression models also holds the potential to design synthetic systems and control the behavior of naturally evolved gene expression systems in guiding cell-fate decisions.

GpABC: a Julia package for approximate Bayesian computation with Gaussian process emulation

MOTIVATION

Approximate Bayesian computation (ABC) is an important framework within which to infer the structure and parameters of a systems biology model. It is especially suitable for biological systems with stochastic and nonlinear dynamics, for which the likelihood functions are intractable. However, the associated computational cost often limits ABC to models that are relatively quick to simulate in practice.

RESULTS

We here present a Julia package, GpABC, that implements parameter inference and model selection for deterministic or stochastic models using (i) standard rejection ABC or sequential Monte Carlo ABC or (ii) ABC with Gaussian process emulation. The latter significantly reduces the computational cost.

AVAILABILITY AND IMPLEMENTATION

https://github.com/tanhevg/GpABC.jl.

Multi-model and network inference based on ensemble estimates: avoiding the madness of crowds

Recent progress in theoretical systems biology, applied mathematics and computational statistics allows us to compare the performance of different candidate models at describing a particular biological system quantitatively. Model selection has been applied with great success to problems where a small number-typically less than 10-of models are compared, but recent studies have started to consider thousands and even millions of candidate models. Often, however, we are left with sets of models that are compatible with the data, and then we can use ensembles of models to make predictions. These ensembles can have very desirable characteristics, but as I show here are not guaranteed to improve on individual estimators or predictors. I will show in the cases of model selection and network inference when we can trust ensembles, and when we should be cautious. The analyses suggest that the careful construction of an ensemble-choosing good predictors-is of paramount importance, more than had perhaps been realized before: merely adding different methods does not suffice. The success of ensemble network inference methods is also shown to rest on their ability to suppress false-positive results. A Jupyter notebook which allows carrying out an assessment of ensemble estimators is provided.

Extrinsic Noise and Heavy-Tailed Laws in Gene Expression

Noise in gene expression is one of the hallmarks of life at the molecular scale. Here we derive analytical solutions to a set of models describing the molecular mechanisms underlying transcription of DNA into RNA. Our ansatz allows us to incorporate the effects of extrinsic noise-encompassing factors external to the transcription of the individual gene-and discuss the ramifications for heterogeneity in gene product abundance that has been widely observed in single cell data. Crucially, we are able to show that heavy-tailed distributions of RNA copy numbers cannot result from the intrinsic stochasticity in gene expression alone, but must instead reflect extrinsic sources of variability.

Parameter inference in dynamical systems with co-dimension 1 bifurcations

Dynamical systems with intricate behaviour are all-pervasive in biology. Many of the most interesting biological processes indicate the presence of bifurcations, i.e. phenomena where a small change in a system parameter causes qualitatively different behaviour. Bifurcation theory has become a rich field of research in its own right and evaluating the bifurcation behaviour of a given dynamical system can be challenging. An even greater challenge, however, is to learn the bifurcation structure of dynamical systems from data, where the precise model structure is not known. Here, we study one aspects of this problem: the practical implications that the presence of bifurcations has on our ability to infer model parameters and initial conditions from empirical data; we focus on the canonical co-dimension 1 bifurcations and provide a comprehensive analysis of how dynamics, and our ability to infer kinetic parameters are linked. The picture thus emerging is surprisingly nuanced and suggests that identification of the qualitative dynamics-the bifurcation diagram-should precede any attempt at inferring kinetic parameters.

Gene Regulatory Network Inference from Single-Cell Data Using Multivariate Information Measures

While single-cell gene expression experiments present new challenges for data processing, the cell-to-cell variability observed also reveals statistical relationships that can be used by information theory. Here, we use multivariate information theory to explore the statistical dependencies between triplets of genes in single-cell gene expression datasets. We develop PIDC, a fast, efficient algorithm that uses partial information decomposition (PID) to identify regulatory relationships between genes. We thoroughly evaluate the performance of our algorithm and demonstrate that the higher-order information captured by PIDC allows it to outperform pairwise mutual information-based algorithms when recovering true relationships present in simulated data. We also infer gene regulatory networks from three experimental single-cell datasets and illustrate how network context, choices made during analysis, and sources of variability affect network inference. PIDC tutorials and open-source software for estimating PID are available. PIDC should facilitate the identification of putative functional relationships and mechanistic hypotheses from single-cell transcriptomic data.

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PIDC infers gene regulatory networks from single-cell transcriptomic data

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Multivariate information measures and context in PIDC improve network inference

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Heterogeneity in single-cell data carries information about gene-gene interactions

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Fast, efficient, open-source software is made freely available

Quantifying Dynamic Regulation in Metabolic Pathways with Nonparametric Flux Inference

One of the central tasks in systems biology is to understand how cells regulate their metabolism. Hierarchical regulation analysis is a powerful tool to study this regulation at the metabolic, gene-expression, and signaling levels. It has been widely applied to study steady-state regulation, but analysis of the metabolic dynamics remains challenging because it is difficult to measure time-dependent metabolic flux. Here, we develop a nonparametric method that uses Gaussian processes to accurately infer the dynamics of a metabolic pathway based only on metabolite measurements; from this, we then go on to obtain a dynamical view of the hierarchical regulation processes invoked over time to control the activity in a pathway. Our approach allows us to use hierarchical regulation analysis in a dynamic setting but without the need for explicitly time-dependent flux measurements.

Proteolytic dynamics of human 20S thymoproteasome

An efficient immunosurveillance of CD8⁺ T cells in the periphery depends on positive/negative selection of thymocytes and thus on the dynamics of antigen degradation and epitope production by thymoproteasome and immunoproteasome in the thymus. Although studies in mouse systems have shown how thymoproteasome activity differs from that of immunoproteasome and strongly impacts the T cell repertoire, the proteolytic dynamics and the regulation of human thymoproteasome are unknown. By combining biochemical and computational modeling approaches, we show here that human 20S thymoproteasome and immunoproteasome differ not only in the proteolytic activity of the catalytic sites but also in the peptide transport. These differences impinge upon the quantity of peptide products rather than where the substrates are cleaved. The comparison of the two human 20S proteasome isoforms depicts different processing of antigens that are associated to tumors and autoimmune diseases.

Parametric and non-parametric gradient matching for network inference: a comparison

Background

Reverse engineering of gene regulatory networks from time series gene-expression data is a challenging problem, not only because of the vast sets of candidate interactions but also due to the stochastic nature of gene expression. We limit our analysis to nonlinear differential equation based inference methods. In order to avoid the computational cost of large-scale simulations, a two-step Gaussian process interpolation based gradient matching approach has been proposed to solve differential equations approximately.

Results

We apply a gradient matching inference approach to a large number of candidate models, including parametric differential equations or their corresponding non-parametric representations, we evaluate the network inference performance under various settings for different inference objectives. We use model averaging, based on the Bayesian Information Criterion (BIC), to combine the different inferences. The performance of different inference approaches is evaluated using area under the precision-recall curves.

Conclusions

We found that parametric methods can provide comparable, and often improved inference compared to non-parametric methods; the latter, however, require no kinetic information and are computationally more efficient.

Electronic supplementary material

The online version of this article (10.1186/s12859-018-2590-7) contains supplementary material, which is available to authorized users.

Gene Regulatory Networks from Single Cell Data for Exploring Cell Fate Decisions

Single cell experimental techniques now allow us to quantify gene expression in up to thousands of individual cells. These data reveal the changes in transcriptional state that occur as cells progress through development and adopt specialized cell fates. In this chapter we describe in detail how to use our network inference algorithm (PIDC)-and the associated software package NetworkInference.jl-to infer functional interactions between genes from the observed gene expression patterns. We exploit the large sample sizes and inherent variability of single cell data to detect statistical dependencies between genes that indicate putative (co-)regulatory relationships, using multivariate information measures that can capture complex statistical relationships. We provide guidelines on how best to combine this analysis with other complementary methods designed to explore single cell data, and how to interpret the resulting gene regulatory network models to gain insight into the processes regulating cell differentiation.

PEITH(Θ): perfecting experiments with information theory in Python with GPU support

Motivation

Different experiments provide differing levels of information about a biological system. This makes it difficult, a priori, to select one of them beyond mere speculation and/or belief, especially when resources are limited. With the increasing diversity of experimental approaches and general advances in quantitative systems biology, methods that inform us about the information content that a given experiment carries about the question we want to answer, become crucial.

Results

PEITH(Θ) is a general purpose, Python framework for experimental design in systems biology. PEITH(Θ) uses Bayesian inference and information theory in order to derive which experiments are most informative in order to estimate all model parameters and/or perform model predictions.

Availability and implementation

https://github.com/MichaelPHStumpf/Peitho

Transition state characteristics during cell differentiation

Models describing the process of stem-cell differentiation are plentiful, and may offer insights into the underlying mechanisms and experimentally observed behaviour. Waddington's epigenetic landscape has been providing a conceptual framework for differentiation processes since its inception. It also allows, however, for detailed mathematical and quantitative analyses, as the landscape can, at least in principle, be related to mathematical models of dynamical systems. Here we focus on a set of dynamical systems features that are intimately linked to cell differentiation, by considering exemplar dynamical models that capture important aspects of stem cell differentiation dynamics. These models allow us to map the paths that cells take through gene expression space as they move from one fate to another, e.g. from a stem-cell to a more specialized cell type. Our analysis highlights the role of the transition state (TS) that separates distinct cell fates, and how the nature of the TS changes as the underlying landscape changes-change that can be induced by e.g. cellular signaling. We demonstrate that models for stem cell differentiation may be interpreted in terms of either a static or transitory landscape. For the static case the TS represents a particular transcriptional profile that all cells approach during differentiation. Alternatively, the TS may refer to the commonly observed period of heterogeneity as cells undergo stochastic transitions.

Construction of quasipotentials for stochastic dynamical systems: An optimization approach

The construction of effective and informative landscapes for stochastic dynamical systems has proven a long-standing and complex problem. In many situations, the dynamics may be described by a Langevin equation while constructing a landscape comes down to obtaining the quasipotential, a scalar function that quantifies the likelihood of reaching each point in the state space. In this work we provide a novel method for constructing such landscapes by extending a tool from control theory: the sum-of-squares method for generating Lyapunov functions. Applicable to any system described by polynomials, this method provides an analytical polynomial expression for the potential landscape, in which the coefficients of the polynomial are obtained via a convex optimization problem. The resulting landscapes are based on a decomposition of the deterministic dynamics of the original system, formed in terms of the gradient of the potential and a remaining "curl" component. By satisfying the condition that the inner product of the gradient of the potential and the remaining dynamics is everywhere negative, our derived landscapes provide both upper and lower bounds on the true quasipotential; these bounds becoming tight if the decomposition is orthogonal. The method is demonstrated to correctly compute the quasipotential for high-dimensional linear systems and also for a number of nonlinear examples.

Control mechanisms for stochastic biochemical systems via computation of reachable sets

Controlling the behaviour of cells by rationally guiding molecular processes is an overarching aim of much of synthetic biology. Molecular processes, however, are notoriously noisy and frequently nonlinear. We present an approach to studying the impact of control measures on motifs of molecular interactions that addresses the problems faced in many biological systems: stochasticity, parameter uncertainty and nonlinearity. We show that our reachability analysis formalism can describe the potential behaviour of biological (naturally evolved as well as engineered) systems, and provides a set of bounds on their dynamics at the level of population statistics: for example, we can obtain the possible ranges of means and variances of mRNA and protein expression levels, even in the presence of uncertainty about model parameters.

How to deal with parameters for whole-cell modelling

Dynamical systems describing whole cells are on the verge of becoming a reality. But as models of reality, they are only useful if we have realistic parameters for the molecular reaction rates and cell physiological processes. There is currently no suitable framework to reliably estimate hundreds, let alone thousands, of reaction rate parameters. Here, we map out the relative weaknesses and promises of different approaches aimed at redressing this issue. While suitable procedures for estimation or inference of the whole (vast) set of parameters will, in all likelihood, remain elusive, some hope can be drawn from the fact that much of the cellular behaviour may be explained in terms of smaller sets of parameters. Identifying such parameter sets and assessing their behaviour is now becoming possible even for very large systems of equations, and we expect such methods to become central tools in the development and analysis of whole-cell models.

A Comprehensive Evaluation of Nasal and Bronchial Cytokines and Chemokines Following Experimental Rhinovirus Infection in Allergic Asthma: Increased Interferons (IFN-γ and IFN-λ) and Type 2 Inflammation (IL-5 and IL-13)

Background

Rhinovirus infection is a major cause of asthma exacerbations.

Objectives

We studied nasal and bronchial mucosal inflammatory responses during experimental rhinovirus-induced asthma exacerbations.

Methods

We used nasosorption on days 0, 2–5 and 7 and bronchosorption at baseline and day 4 to sample mucosal lining fluid to investigate airway mucosal responses to rhinovirus infection in patients with allergic asthma (n = 28) and healthy non-atopic controls (n = 11), by using a synthetic absorptive matrix and measuring levels of 34 cytokines and chemokines using a sensitive multiplex assay.

Results

Following rhinovirus infection asthmatics developed more upper and lower respiratory symptoms and lower peak expiratory flows compared to controls (all P < 0.05). Asthmatics also developed higher nasal lining fluid levels of an anti-viral pathway (including IFN-γ, IFN-λ/IL-29, CXCL11/ITAC, CXCL10/IP10 and IL-15) and a type 2 inflammatory pathway (IL-4, IL-5, IL-13, CCL17/TARC, CCL11/eotaxin, CCL26/eotaxin-3) (area under curve day 0–7, all P < 0.05). Nasal IL-5 and IL-13 were higher in asthmatics at day 0 (P < 0.01) and levels increased by days 3 and 4 (P < 0.01). A hierarchical correlation matrix of 24 nasal lining fluid cytokine and chemokine levels over 7 days demonstrated expression of distinct interferon-related and type 2 pathways in asthmatics. In asthmatics IFN-γ, CXCL10/IP10, CXCL11/ITAC, IL-15 and IL-5 increased in bronchial lining fluid following viral infection (all P < 0.05).

Conclusions

Precision sampling of mucosal lining fluid identifies robust interferon and type 2 responses in the upper and lower airways of asthmatics during an asthma exacerbation. Nasosorption and bronchosorption have potential to define asthma endotypes in stable disease and at exacerbation.

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Following rhinovirus infection asthmatics have increased interferons and type 2 inflammation in airway mucosal lining fluid.

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Nasosorption cytokines and chemokines showed distinct pathways of interferon and type 2 inflammation in asthma.

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Precision mucosal sampling has potential for stratifying molecular endotypes of asthma.

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Validation of nasosorption and bronchosorption will be required for selection of asthmatics for therapy with biologics.

Experimental human rhinovirus (HRV) infection causes more severe upper and lower respiratory tract symptoms in allergic asthmatics than in healthy controls. There is greater induction of cytokines and chemokines in nasal and bronchial mucosal lining fluid (MLF) of asthmatics: with distinct pathways of type 2 and anti-viral/regulatory inflammation.

Subject to further validation, analysis of MLF may prove useful in stratification of patients with asthma, and the definition of molecular endotypes.

Interpretation

Nasosorption and bronchosorption are precision sampling methods with potential for widespread application in respiratory and other mucosal diseases (e.g. gastrointestinal diseases).

Biomarkers identified in nasosorption and bronchosorption samples will need to be validated compared to established airway sampling methods, in a range of asthma phenotypes, and with current and novel therapies.

Coalescent models for developmental biology and the spatio-temporal dynamics of growing tissues

Development is a process that needs to be tightly coordinated in both space and time. Cell tracking and lineage tracing have become important experimental techniques in developmental biology and allow us to map the fate of cells and their progeny. A generic feature of developing and homeostatic tissues that these analyses have revealed is that relatively few cells give rise to the bulk of the cells in a tissue; the lineages of most cells come to an end quickly. Computational and theoretical biologists/physicists have, in response, developed a range of modelling approaches, most notably agent-based modelling. These models seem to capture features observed in experiments, but can also become computationally expensive. Here, we develop complementary genealogical models of tissue development that trace the ancestry of cells in a tissue back to their most recent common ancestors. We show that with both bounded and unbounded growth simple, but universal scaling relationships allow us to connect coalescent theory with the fractal growth models extensively used in developmental biology. Using our genealogical perspective, it is possible to study bulk statistical properties of the processes that give rise to tissues of cells, without the need for large-scale simulations.

A large fraction of HLA class I ligands are proteasome-generated spliced peptides

The proteasome generates the epitopes presented on human leukocyte antigen (HLA) class I molecules that elicit CD8⁺ T cell responses. Reports of proteasome-generated spliced epitopes exist, but they have been regarded as rare events. Here, however, we show that the proteasome-generated spliced peptide pool accounts for one-third of the entire HLA class I immunopeptidome in terms of diversity and one-fourth in terms of abundance. This pool also represents a unique set of antigens, possessing particular and distinguishing features. We validated this observation using a range of complementary experimental and bioinformatics approaches, as well as multiple cell types. The widespread appearance and abundance of proteasome-catalyzed peptide splicing events has implications for immunobiology and autoimmunity theories and may provide a previously untapped source of epitopes for use in vaccines and cancer immunotherapy.

MEANS: python package for Moment Expansion Approximation, iNference and Simulation

MOTIVATION

Many biochemical systems require stochastic descriptions. Unfortunately these can only be solved for the simplest cases and their direct simulation can become prohibitively expensive, precluding thorough analysis. As an alternative, moment closure approximation methods generate equations for the time-evolution of the system's moments and apply a closure ansatz to obtain a closed set of differential equations; that can become the basis for the deterministic analysis of the moments of the outputs of stochastic systems.

RESULTS

We present a free, user-friendly tool implementing an efficient moment expansion approximation with parametric closures that integrates well with the IPython interactive environment. Our package enables the analysis of complex stochastic systems without any constraints on the number of species and moments studied and the type of rate laws in the system. In addition to the approximation method our package provides numerous tools to help non-expert users in stochastic analysis.

AVAILABILITY AND IMPLEMENTATION

https://github.com/theosysbio/means

CONTACTS

m.stumpf@imperial.ac.uk or e.lakatos13@imperial.ac.uk

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Inferring extrinsic noise from single-cell gene expression data using approximate Bayesian computation

Background

Gene 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.

Results

To 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.

Conclusions

We employ this simulation algorithm with approximate Bayesian computation to jointly infer the model’s rate and noise parameters from published gene expression data. Our analysis indicates that for most genes the extrinsic contributions to noise will be small to moderate but certainly are non-negligible.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-016-0324-x) contains supplementary material, which is available to authorized users.

Accurate Reconstruction of Cell and Particle Tracks from 3D Live Imaging Data

Spatial structures often constrain the 3D movement of cells or particles in vivo, yet this information is obscured when microscopy data are analyzed using standard approaches. Here, we present methods, called unwrapping and Riemannian manifold learning, for mapping particle-tracking data along unseen and irregularly curved surfaces onto appropriate 2D representations. This is conceptually similar to the problem of reconstructing accurate geography from conventional Mercator maps, but our methods do not require prior knowledge of the environments' physical structure. Unwrapping and Riemannian manifold learning accurately recover the underlying 2D geometry from 3D imaging data without the need for fiducial marks. They outperform standard x-y projections, and unlike standard dimensionality reduction techniques, they also successfully detect both bias and persistence in cell migration modes. We demonstrate these features on simulated data and zebrafish and Drosophila in vivo immune cell trajectory datasets. Software packages that implement unwrapping and Riemannian manifold learning are provided.

Systems Analysis of the Dynamic Inflammatory Response to Tissue Damage Reveals Spatiotemporal Properties of the Wound Attractant Gradient

In the acute inflammatory phase following tissue damage, cells of the innate immune system are rapidly recruited to sites of injury by pro-inflammatory mediators released at the wound site. Although advances in live imaging allow us to directly visualize this process in vivo, the precise identity and properties of the primary immune damage attractants remain unclear, as it is currently impossible to directly observe and accurately measure these signals in tissues. Here, we demonstrate that detailed information about the attractant signals can be extracted directly from the in vivo behavior of the responding immune cells. By applying inference-based computational approaches to analyze the in vivo dynamics of the Drosophila inflammatory response, we gain new detailed insight into the spatiotemporal properties of the attractant gradient. In particular, we show that the wound attractant is released by wound margin cells, rather than by the wounded tissue per se, and that it diffuses away from this source at rates far slower than those of previously implicated signals such as H₂O₂ and ATP, ruling out these fast mediators as the primary chemoattractant. We then predict, and experimentally test, how competing attractant signals might interact in space and time to regulate multi-step cell navigation in the complex environment of a healing wound, revealing a period of receptor desensitization after initial exposure to the damage attractant. Extending our analysis to model much larger wounds, we uncover a dynamic behavioral change in the responding immune cells in vivo that is prognostic of whether a wound will subsequently heal or not.

Video Abstract

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Computational modeling of in vivo inflammatory response to tissue damage is applied

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The model infers novel spatiotemporal properties of the wound attractant gradient

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Wound signal is released from the wound edge for 30 min and diffuses at 200 μm²/min

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Modeling two competing wounds reveals a period of immune cell desensitization

Robustness of MEK-ERK Dynamics and Origins of Cell-to-Cell Variability in MAPK Signaling

Cellular signaling 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 signaling systems remain poorly understood. Here, we measure the dynamics of phosphorylated MEK and ERK across cell populations and quantify the levels of population heterogeneity over time using high-throughput image cytometry. We use a statistical modeling framework to show that extrinsic noise, particularly that from upstream MEK, is the dominant factor causing cell-to-cell variability in ERK phosphorylation, rather than stochasticity in the phosphorylation/dephosphorylation of ERK. We furthermore show that without extrinsic noise in the core module, variable (including noisy) signals would be faithfully reproduced downstream, but the within-module extrinsic variability distorts these signals and leads to a drastic reduction in the mutual information between incoming signal and ERK activity.

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Active MEK and ERK levels differ profoundly among genetically identical cells

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A statistical framework is developed to identify the causes of this variability

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Analysis shows that extrinsic noise upstream MEK-ERK module causes cell variability

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Within-module extrinsic variability distorts signals

Systematic tracking of altered haematopoiesis during sporozoite-mediated malaria development reveals multiple response points

Haematopoiesis is the complex developmental process that maintains the turnover of all blood cell lineages. It critically depends on the correct functioning of rare, quiescent haematopoietic stem cells (HSCs) and more numerous, HSC-derived, highly proliferative and differentiating haematopoietic progenitor cells (HPCs). Infection is known to affect HSCs, with severe and chronic inflammatory stimuli leading to stem cell pool depletion, while acute, non-lethal infections exert transient and even potentiating effects. Both whether this paradigm applies to all infections and whether the HSC response is the dominant driver of the changes observed during stressed haematopoiesis remain open questions. We use a mouse model of malaria, based on natural, sporozoite-driven Plasmodium berghei infection, as an experimental platform to gain a global view of haematopoietic perturbations during infection progression. We observe coordinated responses by the most primitive HSCs and multiple HPCs, some starting before blood parasitaemia is detected. We show that, despite highly variable inter-host responses, primitive HSCs become highly proliferative, but mathematical modelling suggests that this alone is not sufficient to significantly impact the whole haematopoietic cascade. We observe that the dramatic expansion of Sca-1(+) progenitors results from combined proliferation of direct HSC progeny and phenotypic changes in downstream populations. We observe that the simultaneous perturbation of HSC/HPC population dynamics is coupled with early signs of anaemia onset. Our data uncover a complex relationship between Plasmodium and its host's haematopoiesis and raise the question whether the variable responses observed may affect the outcome of the infection itself and its long-term consequences on the host.

Information processing by simple molecular motifs and susceptibility to noise

Biological organisms rely on their ability to sense and respond appropriately to their environment. The molecular mechanisms that facilitate these essential processes are however subject to a range of random effects and stochastic processes, which jointly affect the reliability of information transmission between receptors and, for example, the physiological downstream response. Information is mathematically defined in terms of the entropy; and the extent of information flowing across an information channel or signalling system is typically measured by the 'mutual information', or the reduction in the uncertainty about the output once the input signal is known. Here, we quantify how extrinsic and intrinsic noise affects the transmission of simple signals along simple motifs of molecular interaction networks. Even for very simple systems, the effects of the different sources of variability alone and in combination can give rise to bewildering complexity. In particular, extrinsic variability is apt to generate 'apparent' information that can, in extreme cases, mask the actual information that for a single system would flow between the different molecular components making up cellular signalling pathways. We show how this artificial inflation in apparent information arises and how the effects of different types of noise alone and in combination can be understood.

Feedback mechanisms control coexistence in a stem cell model of acute myeloid leukaemia

Haematopoietic stem cell dynamics regulate healthy blood cell production and are disrupted during leukaemia. Competition models of cellular species help to elucidate stem cell dynamics in the bone marrow microenvironment (or niche), and to determine how these dynamics impact leukaemia progression. Here we develop two models that target acute myeloid leukaemia with particular focus on the mechanisms that control proliferation via feedback signalling. It is within regions of parameter space permissive of coexistence that the effects of competition are most subtle and the clinical outcome least certain. Steady state and linear stability analyses identify parameter regions that allow for coexistence to occur, and allow us to characterise behaviour near critical points. Where analytical expressions are no longer informative, we proceed statistically and sample parameter space over a coexistence region. We find that the rates of proliferation and differentiation of healthy progenitors exert key control over coexistence. We also show that inclusion of a regulatory feedback onto progenitor cells promotes healthy haematopoiesis at the expense of leukaemia, and that – somewhat paradoxically – within the coexistence region feedback increases the sensitivity of the system to dominance by one lineage over another.

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Models of competition between cell populations can describe the progression of acute myeloid leukaemia.

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We identify regions of coexistence in which leukaemia and healthy haematopoietic species can coexist in the niche.

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The dynamics of progenitor cells exert key control over species coexistence.

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The introduction of regulatory feedback can promote healthy haematopoiesis and suppress leukaemia.

Great cities look small

Great cities connect people; failed cities isolate people. Despite the fundamental importance of physical, face-to-face social ties in the functioning of cities, these connectivity networks are not explicitly observed in their entirety. Attempts at estimating them often rely on unrealistic over-simplifications such as the assumption of spatial homogeneity. Here we propose a mathematical model of human interactions in terms of a local strategy of maximizing the number of beneficial connections attainable under the constraint of limited individual travelling-time budgets. By incorporating census and openly available online multi-modal transport data, we are able to characterize the connectivity of geometrically and topologically complex cities. Beyond providing a candidate measure of greatness, this model allows one to quantify and assess the impact of transport developments, population growth, and other infrastructure and demographic changes on a city. Supported by validations of gross domestic product and human immunodeficiency virus infection rates across US metropolitan areas, we illustrate the effect of changes in local and city-wide connectivities by considering the economic impact of two contemporary inter- and intra-city transport developments in the UK: High Speed 2 and London Crossrail. This derivation of the model suggests that the scaling of different urban indicators with population size has an explicitly mechanistic origin.

A graph theoretical approach to data fusion

The rapid development of high throughput experimental techniques has resulted in a growing diversity of genomic datasets being produced and requiring analysis. Therefore, it is increasingly being recognized that we can gain deeper understanding about underlying biology by combining the insights obtained from multiple, diverse datasets. Thus we propose a novel scalable computational approach to unsupervised data fusion. Our technique exploits network representations of the data to identify similarities among the datasets. We may work within the Bayesian formalism, using Bayesian nonparametric approaches to model each dataset; or (for fast, approximate, and massive scale data fusion) can naturally switch to more heuristic modeling techniques. An advantage of the proposed approach is that each dataset can initially be modeled independently (in parallel), before applying a fast post-processing step to perform data integration. This allows us to incorporate new experimental data in an online fashion, without having to rerun all of the analysis. We first demonstrate the applicability of our tool on artificial data, and then on examples from the literature, which include yeast cell cycle, breast cancer and sporadic inclusion body myositis datasets.

Phosphorelay of non-orthodox two component systems functions through a bi-molecular mechanism in vivo: the case of ArcB

Two-component systems play a central part in bacterial signal transduction. Phosphorelay mechanisms have been linked to more robust and ultra-sensitive signalling dynamics. The molecular machinery that facilitates such a signalling is, however, only understood in outline. In particular the functional relevance of the dimerization of a non-orthodox or hybrid histidine kinase along which the phosphorelay takes place has been a subject of debate. We use a combination of molecular and genetic approaches, coupled to mathematical and statistical modelling, to demonstrate that the different possible intra- and inter-molecular mechanisms of phosphotransfer are formally non-identifiable in Escherichia coli expressing the ArcB non-orthodox histidine kinase used in anoxic redox control. In order to resolve this issue we further analyse the mathematical model in order to identify discriminatory experiments, which are then performed to address cis- and trans-phosphorelay mechanisms. The results suggest that exclusive cis- and trans-mechanisms will not be operating, instead the functional phosphorelay is likely to build around a sequence of allosteric interactions among the domain pairs in the histidine kinase. This is the first detailed mechanistic analysis of the molecular processes involved in non-orthodox two-component signalling and our results suggest strongly that dimerization facilitates more discriminatory proof-reading of external signals, via these allosteric reactions, prior to them being further processed.

Mathematical and Statistical Techniques for Systems Medicine: The Wnt Signaling Pathway as a Case Study

The last decade has seen an explosion in models that describe phenomena in systems medicine. Such models are especially useful for studying signaling pathways, such as the Wnt pathway. In this chapter we use the Wnt pathway to showcase current mathematical and statistical techniques that enable modelers to gain insight into (models of) gene regulation and generate testable predictions. We introduce a range of modeling frameworks, but focus on ordinary differential equation (ODE) models since they remain the most widely used approach in systems biology and medicine and continue to offer great potential. We present methods for the analysis of a single model, comprising applications of standard dynamical systems approaches such as nondimensionalization, steady state, asymptotic and sensitivity analysis, and more recent statistical and algebraic approaches to compare models with data. We present parameter estimation and model comparison techniques, focusing on Bayesian analysis and coplanarity via algebraic geometry. Our intention is that this (non-exhaustive) review may serve as a useful starting point for the analysis of models in systems medicine.

Cellular population dynamics control the robustness of the stem cell niche

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.

Summary: Stem cell niche dynamics are very robust to external and physiological perturbations because proliferation and differentiation are naturally balanced and controlled by the reliance on a shared niche environment.

Quantitative time-resolved analysis reveals intricate, differential regulation of standard- and immuno-proteasomes

Proteasomal protein degradation is a key determinant of protein half-life and hence of cellular processes ranging from basic metabolism to a host of immunological processes. Despite its importance the mechanisms regulating proteasome activity are only incompletely understood. Here we use an iterative and tightly integrated experimental and modelling approach to develop, explore and validate mechanistic models of proteasomal peptide-hydrolysis dynamics. The 20S proteasome is a dynamic enzyme and its activity varies over time because of interactions between substrates and products and the proteolytic and regulatory sites; the locations of these sites and the interactions between them are predicted by the model, and experimentally supported. The analysis suggests that the rate-limiting step of hydrolysis is the transport of the substrates into the proteasome. The transport efficiency varies between human standard- and immuno-proteasomes thereby impinging upon total degradation rate and substrate cleavage-site usage.

DOI:http://dx.doi.org/10.7554/eLife.07545.001

Cells have to be able to reliably destroy or remove molecules from their interior that they no longer need. Structures called proteasomes play a central part in this complex process by cutting up and digesting proteins. Mammals have several different types of proteasomes, each made up of several protein ‘subunits’. For example, when a cell experiences inflammation some proteasomes change some of their subunits and form an immuno-proteasome. These immuno-proteasomes tend to break down proteins more quickly than ‘standard’ proteasomes, but it was not clear how they are able to do so.

Liepe et al. have now combined experiments and mathematical modelling to construct a detailed model of proteasome activity. The model shows that protein transport into and out of the proteasome chamber are the steps that limit how quickly the proteasomes can break down proteins. Furthermore, these transport processes are also to a large extent responsible for the different rates at which standard and immuno-proteasomes process proteins. Liepe et al. were also able to confirm the existence of regulatory sites within the proteasome, and describe how these are arranged.

Problems that alter the rate at which proteasomes break down proteins have been linked to tumors and neurological and autoimmune diseases. Liepe et al.'s model opens up the ability to study how the proteasome's activity is affected by drugs and therefore makes it easier to investigate ways of interfering with this activity for therapeutic purposes.

DOI:http://dx.doi.org/10.7554/eLife.07545.002