A Geometric Clustering Tool (AGCT) to robustly unravel the inner cluster structures of time-series gene expressions.

Systems biology aims at holistically understanding the complexity of biological systems. In particular, nowadays with the broad availability of gene expression measurements, systems biology challenges the deciphering of the genetic cell machinery from them. In order to help researchers, reverse engineer the genetic cell machinery from these noisy datasets, interactive exploratory clustering methods, pipelines and gene clustering tools have to be specifically developed. Prior methods/tools for time series data, however, do not have the following four major ingredients in analytic and methodological view point: (i) principled time-series feature extraction methods, (ii) variety of manifold learning methods for capturing high-level view of the dataset, (iii) high-end automatic structure extraction, and (iv) friendliness to the biological user community. With a view to meet the requirements, we present AGCT (A Geometric Clustering Tool), a software package used to unravel the complex architecture of large-scale, non-necessarily synchronized time-series gene expression data. AGCT capture signals on exhaustive wavelet expansions of the data, which are then embedded on a low-dimensional non-linear map using manifold learning algorithms, where geometric proximity captures potential interactions. Post-processing techniques, including hard and soft information geometric clustering algorithms, facilitate the summarizing of the complete map as a smaller number of principal factors which can then be formally identified using embedded statistical inference techniques. Three-dimension interactive visualization and scenario recording over the processing helps to reproduce data analysis results without additional time. Analysis of the whole-cell Yeast Metabolic Cycle (YMC) moreover, Yeast Cell Cycle (YCC) datasets demonstrate AGCT's ability to accurately dissect all stages of metabolism and the cell cycle progression, independently of the time course and the number of patterns related to the signal. Analysis of Pentachlorophenol iduced dataset demonstrat how AGCT dissects data to identify two networks: Interferon signaling and NRF2-signaling networks.

Time-dependent product-form Poisson distributions for reaction networks with higher order complexes.

It is well known that stochastically modeled reaction networks that are complex balanced admit a stationary distribution that is a product of Poisson distributions. In this paper, we consider the following related question: supposing that the initial distribution of a stochastically modeled reaction network is a product of Poissons, under what conditions will the distribution remain a product of Poissons for all time? By drawing inspiration from Crispin Gardiner's "Poisson representation" for the solution to the chemical master equation, we provide a necessary and sufficient condition for such a product-form distribution to hold for all time. Interestingly, the condition is a dynamical "complex-balancing" for only those complexes that have multiplicity greater than or equal to two (i.e. the higher order complexes that yield non-linear terms to the dynamics). We term this new condition the "dynamical and restricted complex balance" condition (DR for short).

Measuring the complexity of directed graphs: A polynomial-based approach.

In this paper, we define novel graph measures for directed networks. The measures are based on graph polynomials utilizing the out- and in-degrees of directed graphs. Based on these polynomial, we define another polynomial and use their positive zeros as graph measures. The measures have meaningful properties that we investigate based on analytical and numerical results. As the computational complexity to compute the measures is polynomial, our approach is efficient and can be applied to large networks. We emphasize that our approach clearly complements the literature in this field as, to the best of our knowledge, existing complexity measures for directed graphs have never been applied on a large scale.

Fast uncertainty quantification for dynamic flux balance analysis using non-smooth polynomial chaos expansions.

We present a novel surrogate modeling method that can be used to accelerate the solution of uncertainty quantification (UQ) problems arising in nonlinear and non-smooth models of biological systems. In particular, we focus on dynamic flux balance analysis (DFBA) models that couple intracellular fluxes, found from the solution of a constrained metabolic network model of the cellular metabolism, to the time-varying nature of the extracellular substrate and product concentrations. DFBA models are generally computationally expensive and present unique challenges to UQ, as they entail dynamic simulations with discrete events that correspond to switches in the active set of the solution of the constrained intracellular model. The proposed non-smooth polynomial chaos expansion (nsPCE) method is an extension of traditional PCE that can effectively capture singularities in the DFBA model response due to the occurrence of these discrete events. The key idea in nsPCE is to use a model of the singularity time to partition the parameter space into two elements on which the model response behaves smoothly. Separate PCE models are then fit in both elements using a basis-adaptive sparse regression approach that is known to scale well with respect to the number of uncertain parameters. We demonstrate the effectiveness of nsPCE on a DFBA model of an E. coli monoculture that consists of 1075 reactions and 761 metabolites. We first illustrate how traditional PCE is unable to handle problems of this level of complexity. We demonstrate that over 800-fold savings in computational cost of uncertainty propagation and Bayesian estimation of parameters in the substrate uptake kinetics can be achieved by using the nsPCE surrogates in place of the full DFBA model simulations. We then investigate the scalability of the nsPCE method by utilizing it for global sensitivity analysis and maximum a posteriori estimation in a synthetic metabolic network problem with a larger number of parameters related to both intracellular and extracellular quantities.

FAMoS: A Flexible and dynamic Algorithm for Model Selection to analyse complex systems dynamics.

Most biological systems are difficult to analyse due to a multitude of interacting components and the concomitant lack of information about the essential dynamics. Finding appropriate models that provide a systematic description of such biological systems and that help to identify their relevant factors and processes can be challenging given the sheer number of possibilities. Model selection algorithms that evaluate the performance of a multitude of different models against experimental data provide a useful tool to identify appropriate model structures. However, many algorithms addressing the analysis of complex dynamical systems, as they are often used in biology, compare a preselected number of models or rely on exhaustive searches of the total model space which might be unfeasible dependent on the number of possibilities. Therefore, we developed an algorithm that is able to perform model selection on complex systems and searches large model spaces in a dynamical way. Our algorithm includes local and newly developed non-local search methods that can prevent the algorithm from ending up in local minima of the model space by accounting for structurally similar processes. We tested and validated the algorithm based on simulated data and showed its flexibility for handling different model structures. We also used the algorithm to analyse experimental data on the cell proliferation dynamics of CD4+ and CD8+ T cells that were cultured under different conditions. Our analyses indicated dynamical changes within the proliferation potential of cells that was reduced within tissue-like 3D ex vivo cultures compared to suspension. Due to the flexibility in handling various model structures, the algorithm is applicable to a large variety of different biological problems and represents a useful tool for the data-oriented evaluation of complex model spaces.

Asymptotic State of a Two-Patch System with Infinite Diffusion.

Mathematical theory has predicted that populations diffusing in heterogeneous environments can reach larger total size than when not diffusing. This prediction was tested in a recent experiment, which leads to extension of the previous theory to consumer-resource systems with external resource input. This paper studies a two-patch model with diffusion that characterizes the experiment. Solutions of the model are shown to be nonnegative and bounded, and global dynamics of the subsystems are completely exhibited. It is shown that there exist stable positive equilibria as the diffusion rate is large, and the equilibria converge to a unique positive point as the diffusion tends to infinity. Rigorous analysis on the model demonstrates that homogeneously distributed resources support larger carrying capacity than heterogeneously distributed resources with or without diffusion, which coincides with experimental observations but refutes previous theory. It is shown that spatial diffusion increases total equilibrium population abundance in heterogeneous environments, which coincides with real data and previous theory while a new insight is exhibited. A novel prediction of this work is that these results hold even with source-sink populations and increasing diffusion rate of consumer could change its persistence to extinction in the same-resource environments.

Conceptual and computational framework for logical modelling of biological networks deregulated in diseases.

Mathematical models can serve as a tool to formalize biological knowledge from diverse sources, to investigate biological questions in a formal way, to test experimental hypotheses, to predict the effect of perturbations and to identify underlying mechanisms. We present a pipeline of computational tools that performs a series of analyses to explore a logical model's properties. A logical model of initiation of the metastatic process in cancer is used as a transversal example. We start by analysing the structure of the interaction network constructed from the literature or existing databases. Next, we show how to translate this network into a mathematical object, specifically a logical model, and how robustness analyses can be applied to it. We explore the visualization of the stable states, defined as specific attractors of the model, and match them to cellular fates or biological read-outs. With the different tools we present here, we explain how to assign to each solution of the model a probability and how to identify genetic interactions using mutant phenotype probabilities. Finally, we connect the model to relevant experimental data: we present how some data analyses can direct the construction of the network, and how the solutions of a mathematical model can also be compared with experimental data, with a particular focus on high-throughput data in cancer biology. A step-by-step tutorial is provided as a Supplementary Material and all models, tools and scripts are provided on an accompanying website: https://github.com/sysbio-curie/Logical_modelling_pipeline.

Bayesian inference of hub nodes across multiple networks.

Hub nodes within biological networks play a pivotal role in determining phenotypes and disease outcomes. In the multiple network setting, we are interested in understanding network similarities and differences across different experimental conditions or subtypes of disease. The majority of proposed approaches for joint modeling of multiple networks focus on the sharing of edges across graphs. Rather than assuming the network similarities are driven by individual edges, we instead focus on the presence of common hub nodes, which are more likely to be preserved across settings. Specifically, we formulate a Bayesian approach to the problem of multiple network inference which allows direct inference on shared and differential hub nodes. The proposed method not only allows a more intuitive interpretation of the resulting networks and clearer guidance on potential targets for treatment, but also improves power for identifying the edges of highly connected nodes. Through simulations, we demonstrate the utility of our method and compare its performance to current popular methods that do not borrow information regarding hub nodes across networks. We illustrate the applicability of our method to inference of co-expression networks from The Cancer Genome Atlas ovarian carcinoma dataset.

Variability of Betweenness Centrality and Its Effect on Identifying Essential Genes.

This paper begins to build a theoretical framework that would enable the pharmaceutical industry to use network complexity measures as a way to identify drug targets. The variability of a betweenness measure for a network node is examined through different methods of network perturbation. Our results indicate a robustness of betweenness centrality in the identification of target genes.

Defining informative priors for ensemble modeling in systems biology.

Ensemble modeling in molecular systems biology requires the reproducible translation of kinetic parameter data into informative probability distributions (priors), as well as approaches that sample parameters from these distributions without violating the thermodynamic consistency of the overall model. Although a number of pioneering frameworks for ensemble modeling have been published, the issue of generating informative priors has not yet been addressed. Here, we present a protocol that aims to fill this gap. This protocol discusses the collection of parameter values from a diverse range of sources (literature, databases and experiments), assessment of their plausibility, and creation of log-normal probability distributions that can be used as informative priors in ensemble modeling. Furthermore, the protocol enables sampling from the generated distributions while maintaining thermodynamic consistency. Once all parameter values have been retrieved from literature and databases, the protocol can be implemented within ~5-10 min per parameter. The aim of this protocol is to facilitate the design and use of informative distributions for ensemble modeling, especially in fields such as synthetic biology and systems medicine.

Using both qualitative and quantitative data in parameter identification for systems biology models.

In systems biology, qualitative data are often generated, but rarely used to parameterize models. We demonstrate an approach in which qualitative and quantitative data can be combined for parameter identification. In this approach, qualitative data are converted into inequality constraints imposed on the outputs of the model. These inequalities are used along with quantitative data points to construct a single scalar objective function that accounts for both datasets. To illustrate the approach, we estimate parameters for a simple model describing Raf activation. We then apply the technique to a more elaborate model characterizing cell cycle regulation in yeast. We incorporate both quantitative time courses (561 data points) and qualitative phenotypes of 119 mutant yeast strains (1647 inequalities) to perform automated identification of 153 model parameters. We quantify parameter uncertainty using a profile likelihood approach. Our results indicate the value of combining qualitative and quantitative data to parameterize systems biology models.

Identification of parameters in systems biology.

We consider the inverse problem for the identification of the finite dimensional set of parameters for systems of nonlinear ordinary differential equations (ODEs) arising in systems biology. A numerical method which combines Bellman's quasilinearization with sensitivity analysis and Tikhonov's regularization is implemented. We apply the method to various biological models such as the classical Lotka-Volterra system, bistable switch model in genetic regulatory networks, gene regulation and repressilator models from synthetic biology. The numerical results and application to real data demonstrate the quadratic convergence.

The Crisis of Reproducibility, the Denominator Problem and the Scientific Role of Multi-scale Modeling.

The "Crisis of Reproducibility" has received considerable attention both within the scientific community and without. While factors associated with scientific culture and practical practice are most often invoked, I propose that the Crisis of Reproducibility is ultimately a failure of generalization with a fundamental scientific basis in the methods used for biomedical research. The Denominator Problem describes how limitations intrinsic to the two primary approaches of biomedical research, clinical studies and preclinical experimental biology, lead to an inability to effectively characterize the full extent of biological heterogeneity, which compromises the task of generalizing acquired knowledge. Drawing on the example of the unifying role of theory in the physical sciences, I propose that multi-scale mathematical and dynamic computational models, when mapped to the modular structure of biological systems, can serve a unifying role as formal representations of what is conserved and similar from one biological context to another. This ability to explicitly describe the generation of heterogeneity from similarity addresses the Denominator Problem and provides a scientific response to the Crisis of Reproducibility.

Enter the Matrix: Factorization Uncovers Knowledge from Omics.

Omics data contain signals from the molecular, physical, and kinetic inter- and intracellular interactions that control biological systems. Matrix factorization (MF) techniques can reveal low-dimensional structure from high-dimensional data that reflect these interactions. These techniques can uncover new biological knowledge from diverse high-throughput omics data in applications ranging from pathway discovery to timecourse analysis. We review exemplary applications of MF for systems-level analyses. We discuss appropriate applications of these methods, their limitations, and focus on the analysis of results to facilitate optimal biological interpretation. The inference of biologically relevant features with MF enables discovery from high-throughput data beyond the limits of current biological knowledge - answering questions from high-dimensional data that we have not yet thought to ask.

Maximally informative next experiments for nonlinear models.

Mathematical modeling is a powerful tool in systems biology; we focus here on improving the reliability of model predictions by reducing the uncertainty in model dynamics through experimental design. Model-based experimental design is a process by which experiments can be systematically chosen to reduce dynamic uncertainty in a given model. We discuss the Maximally Informative Next Experiment (MINE) method for group-wise selection of points in an experimental design and present a convergence result for MINE with nonlinear models. As an application, we illustrate the method on polynomial regression and an ODE model for immune system dynamics. The MINE criterion sequentially determines experiments that can be conducted to best refine model dynamics.

Inferring a nonlinear biochemical network model from a heterogeneous single-cell time course data.

Mathematical modeling and analysis of biochemical reaction networks are key routines in computational systems biology and biophysics; however, it remains difficult to choose the most valid model. Here, we propose a computational framework for data-driven and systematic inference of a nonlinear biochemical network model. The framework is based on the expectation-maximization algorithm combined with particle smoother and sparse regularization techniques. In this method, a "redundant" model consisting of an excessive number of nodes and regulatory paths is iteratively updated by eliminating unnecessary paths, resulting in an inference of the most likely model. Using artificial single-cell time-course data showing heterogeneous oscillatory behaviors, we demonstrated that this algorithm successfully inferred the true network without any prior knowledge of network topology or parameter values. Furthermore, we showed that both the regulatory paths among nodes and the optimal number of nodes in the network could be systematically determined. The method presented in this study provides a general framework for inferring a nonlinear biochemical network model from heterogeneous single-cell time-course data.

Black-boxing and cause-effect power.

Reductionism assumes that causation in the physical world occurs at the micro level, excluding the emergence of macro-level causation. We challenge this reductionist assumption by employing a principled, well-defined measure of intrinsic cause-effect power-integrated information (Φ), and showing that, according to this measure, it is possible for a macro level to "beat" the micro level. Simple systems were evaluated for Φ across different spatial and temporal scales by systematically considering all possible black boxes. These are macro elements that consist of one or more micro elements over one or more micro updates. Cause-effect power was evaluated based on the inputs and outputs of the black boxes, ignoring the internal micro elements that support their input-output function. We show how black-box elements can have more common inputs and outputs than the corresponding micro elements, revealing the emergence of high-order mechanisms and joint constraints that are not apparent at the micro level. As a consequence, a macro, black-box system can have higher Φ than its micro constituents by having more mechanisms (higher composition) that are more interconnected (higher integration). We also show that, for a given micro system, one can identify local maxima of Φ across several spatiotemporal scales. The framework is demonstrated on a simple biological system, the Boolean network model of the fission-yeast cell-cycle, for which we identify stable local maxima during the course of its simulated biological function. These local maxima correspond to macro levels of organization at which emergent cause-effect properties of physical systems come into focus, and provide a natural vantage point for scientific inquiries.

Clarifying of the potential mechanism of Sinisan formula for treatment of chronic hepatitis by systems pharmacology method.

Chronic hepatitis is a general designation class of diseases, which results in different degrees of liver necrosis and inflammatory reaction, followed by liver fibrosis, may eventually develop into cirrhosis. However, the molecular pathogenesis of chronic hepatitis is too complex to elucidate. Herbal medicines, featured with multiple targets and compounds, have long displayed therapeutic effect in treating chronic hepatitis, though their molecular mechanisms of contribution remain indistinct. This research utilized the network pharmacology to confirm the molecular pathogenesis of chronic hepatitis through providing a comprehensive analysis of active chemicals, drug targets and pathways' interaction of Sinisan formula for treating chronic hepatitis. The outcomes showed that 80 active ingredients of Sinisan formula interacting with 91 therapeutic proteins were authenticated. Sinisan formula potentially participates in immune modulation, anti-inflammatory and antiviral activities, even has regulating effects on lipid metabolism. These mechanisms directly or indirectly are involved in curing chronic hepatitis by an interaction way. The network pharmacology based analysis demonstrated that Sinisan has multi-scale curative activity in regulating chronic hepatitis related biological processes, which provides a new potential way for modern medicine in the treatment of chronic diseases.