An ensemble method for identifying regulatory circuits with special reference to the qa gene cluster of Neurospora crassa

SAND: Automated Time-Domain Modeling of NMR Spectra Applied to Metabolite Quantification

Developments in untargeted nuclear magnetic resonance (NMR) metabolomics enable the profiling of thousands of biological samples. The exploitation of this rich source of information requires a detailed quantification of spectral features. However, the development of a consistent and automatic workflow has been challenging because of extensive signal overlap. To address this challenge, we introduce the software Spectral Automated NMR Decomposition (SAND). SAND follows on from the previous success of time-domain modeling and automatically quantifies entire spectra without manual interaction. The SAND approach uses hybrid optimization with Markov chain Monte Carlo methods, employing subsampling in both time and frequency domains. In particular, SAND randomly divides the time-domain data into training and validation sets to help avoid overfitting. We demonstrate the accuracy of SAND, which provides a correlation of ∼0.9 with ground truth on cases including highly overlapped simulated data sets, a two-compound mixture, and a urine sample spiked with different amounts of a four-compound mixture. We further demonstrate an automated annotation using correlation networks derived from SAND decomposed peaks, and on average, 74% of peaks for each compound can be recovered in single clusters. SAND is available in NMRbox, the cloud computing environment for NMR software hosted by the Network for Advanced NMR (NAN). Since the SAND method uses time-domain subsampling (i.e., random subset of time-domain points), it has the potential to be extended to a higher dimensionality and nonuniformly sampled data.

Uncovering in vivo biochemical patterns from time-series metabolic dynamics

System biology relies on holistic biomolecule measurements, and untangling biochemical networks requires time-series metabolomics profiling. With current metabolomic approaches, time-series measurements can be taken for hundreds of metabolic features, which decode underlying metabolic regulation. Such a metabolomic dataset is untargeted with most features unannotated and inaccessible to statistical analysis and computational modeling. The high dimensionality of the metabolic space also causes mechanistic modeling to be rather cumbersome computationally. We implemented a faster exploratory workflow to visualize and extract chemical and biochemical dependencies. Time-series metabolic features (about 300 for each dataset) were extracted by Ridge Tracking-based Extract (RTExtract) on measurements from continuous in vivo monitoring of metabolism by NMR (CIVM-NMR) in Neurospora crassa under different conditions. The metabolic profiles were then smoothed and projected into lower dimensions, enabling a comparison of metabolic trends in the cultures. Next, we expanded incomplete metabolite annotation using a correlation network. Lastly, we uncovered meaningful metabolic clusters by estimating dependencies between smoothed metabolic profiles. We thus sidestepped the processes of time-consuming mechanistic modeling, difficult global optimization, and labor-intensive annotation. Multiple clusters guided insights into central energy metabolism and membrane synthesis. Dense connections with glucose 1-phosphate indicated its central position in metabolism in N. crassa. Our approach was benchmarked on simulated random network dynamics and provides a novel exploratory approach to analyzing high-dimensional metabolic dynamics.

The macroscopic limit to synchronization of cellular clocks in single cells of Neurospora crassa

We determined the macroscopic limit for phase synchronization of cellular clocks in an artificial tissue created by a “big chamber” microfluidic device to be about 150,000 cells or less. The dimensions of the microfluidic chamber allowed us to calculate an upper limit on the radius of a hypothesized quorum sensing signal molecule of 13.05 nm using a diffusion approximation for signal travel within the device. The use of a second microwell microfluidic device allowed the refinement of the macroscopic limit to a cell density of 2166 cells per fixed area of the device for phase synchronization. The measurement of averages over single cell trajectories in the microwell device supported a deterministic quorum sensing model identified by ensemble methods for clock phase synchronization. A strong inference framework was used to test the communication mechanism in phase synchronization of quorum sensing versus cell-to-cell contact, suggesting support for quorum sensing. Further evidence came from showing phase synchronization was density-dependent.

Current Approaches of Building Mechanistic Pharmacodynamic Drug-Target Binding Models

Mechanistic pharmacodynamic models that incorporate the binding kinetics of drug-target interactions have several advantages in understanding target engagement and the efficacy of a drug dose. However, guidelines on how to build and interpret mechanistic pharmacodynamic drug-target binding models considering both biological and computational factors are still missing in the literature. In this chapter, current approaches of building mechanistic PD models and their advantages are discussed. We also present a methodology on how to select a suitable model considering both biological and computational perspectives, as well as summarize the challenges of current mechanistic PD models.

Identifying a stochastic clock network with light entrainment for single cells of Neurospora crassa

Stochastic networks for the clock were identified by ensemble methods using genetic algorithms that captured the amplitude and period variation in single cell oscillators of Neurospora crassa. The genetic algorithms were at least an order of magnitude faster than ensemble methods using parallel tempering and appeared to provide a globally optimum solution from a random start in the initial guess of model parameters (i.e., rate constants and initial counts of molecules in a cell). The resulting goodness of fit [Formula: see text] was roughly halved versus solutions produced by ensemble methods using parallel tempering, and the resulting [Formula: see text] per data point was only [Formula: see text] = 2,708.05/953 = 2.84. The fitted model ensemble was robust to variation in proxies for "cell size". The fitted neutral models without cellular communication between single cells isolated by microfluidics provided evidence for only one Stochastic Resonance at one common level of stochastic intracellular noise across days from 6 to 36 h of light/dark (L/D) or in a D/D experiment. When the light-driven phase synchronization was strong as measured by the Kuramoto (K), there was degradation in the single cell oscillations away from the stochastic resonance. The rate constants for the stochastic clock network are consistent with those determined on a macroscopic scale of 10⁷ cells.

Unravelling the γ-butyrolactone network in Streptomyces coelicolor by computational ensemble modelling

Antibiotic production is coordinated in the Streptomyces coelicolor population through the use of diffusible signaling molecules of the γ-butyrolactone (GBL) family. The GBL regulatory system involves a small, and not completely defined two-gene network which governs a potentially bi-stable switch between the “on” and “off” states of antibiotic production. The use of this circuit as a tool for synthetic biology has been hampered by a lack of mechanistic understanding of its functionality. We here present the creation and analysis of a versatile and adaptable ensemble model of the Streptomyces GBL system (detailed information on all model mechanisms and parameters is documented in http://www.systemsbiology.ls.manchester.ac.uk/wiki/index.php/Main_Page). We use the model to explore a range of previously proposed mechanistic hypotheses, including transcriptional interference, antisense RNA interactions between the mRNAs of the two genes, and various alternative regulatory activities. Our results suggest that transcriptional interference alone is not sufficient to explain the system’s behavior. Instead, antisense RNA interactions seem to be the system's driving force, combined with an aggressive scbR promoter. The computational model can be used to further challenge and refine our understanding of the system’s activity and guide future experimentation.

Streptomyces species are Gram-positive soil-dwelling bacteria, which are known as a prolific source of secondary metabolites, such as antibiotics. Antibiotic production is coordinated in the bacterial population through the use of diffusible signalling molecules of the γ-butyrolactone (GBL) family. The GBL regulatory system involves a small, yet complex two-gene network, the mechanism of which has not yet been completely defined. The complete elucidation of this system could potentially lead to the ability to design reliable and sensitive engineered cellular switches. We therefore designed a versatile model of the GBL system in order to investigate the feasibility of various hypothesized mechanisms. The ensemble modelling analysis that we performed revealed that antisense RNA interactions seem to be the system’s driving force, together with an aggressive scbR promoter. Transcriptional interference is also significant; however, it is not sufficient to explain the system’s behavior by itself. Finally, the model indicates key experiments, which could completely elucidate the role of the system and the interactions of its components and potentially lead to the design of reliable and sensitive systems with significant applications as orthologous regulatory circuits in synthetic biology and biotechnology.

Continuous in vivo Metabolism by NMR

Dense time-series metabolomics data are essential for unraveling the underlying dynamic properties of metabolism. Here we extend high-resolution-magic angle spinning (HR-MAS) to enable continuous in vivo monitoring of metabolism by NMR (CIVM-NMR) and provide analysis tools for these data. First, we reproduced a result in human chronic lymphoid leukemia cells by using isotope-edited CIVM-NMR to rapidly and unambiguously demonstrate unidirectional flux in branched-chain amino acid metabolism. We then collected untargeted CIVM-NMR datasets for Neurospora crassa, a classic multicellular model organism, and uncovered dynamics between central carbon metabolism, amino acid metabolism, energy storage molecules, and lipid and cell wall precursors. Virtually no sample preparation was required to yield a dynamic metabolic fingerprint over hours to days at ~4-min temporal resolution with little noise. CIVM-NMR is simple and readily adapted to different types of cells and microorganisms, offering an experimental complement to kinetic models of metabolism for diverse biological systems.

Modeling differentiation-state transitions linked to therapeutic escape in triple-negative breast cancer

Drug resistance in breast cancer cell populations has been shown to arise through phenotypic transition of cancer cells to a drug-tolerant state, for example through epithelial-to-mesenchymal transition or transition to a cancer stem cell state. However, many breast tumors are a heterogeneous mixture of cell types with numerous epigenetic states in addition to stem-like and mesenchymal phenotypes, and the dynamic behavior of this heterogeneous mixture in response to drug treatment is not well-understood. Recently, we showed that plasticity between differentiation states, as identified with intracellular markers such as cytokeratins, is linked to resistance to specific targeted therapeutics. Understanding the dynamics of differentiation-state transitions in this context could facilitate the development of more effective treatments for cancers that exhibit phenotypic heterogeneity and plasticity. In this work, we develop computational models of a drug-treated, phenotypically heterogeneous triple-negative breast cancer (TNBC) cell line to elucidate the feasibility of differentiation-state transition as a mechanism for therapeutic escape in this tumor subtype. Specifically, we use modeling to predict the changes in differentiation-state transitions that underlie specific therapy-induced changes in differentiation-state marker expression that we recently observed in the HCC1143 cell line. We report several statistically significant therapy-induced changes in transition rates between basal, luminal, mesenchymal, and non-basal/non-luminal/non-mesenchymal differentiation states in HCC1143 cell populations. Moreover, we validate model predictions on cell division and cell death empirically, and we test our models on an independent data set. Overall, we demonstrate that changes in differentiation-state transition rates induced by targeted therapy can provoke distinct differentiation-state aggregations of drug-resistant cells, which may be fundamental to the design of improved therapeutic regimens for cancers with phenotypic heterogeneity.

A novel approach to assessing the ecosystem-wide impacts of reintroductions

Reintroducing a species to an ecosystem can have significant impacts on the recipient ecological community. Although reintroductions can have striking and positive outcomes, they also carry risks; many well-intentioned conservation actions have had surprising and unsatisfactory outcomes. A range of network-based mathematical methods has been developed to make quantitative predictions of how communities will respond to management interventions. These methods are based on the limited knowledge of which species interact with each other and in what way. However, expert knowledge isn't perfect and can only take models so far. Fortunately, other types of data, such as abundance time series, is often available, but, to date, no quantitative method exists to integrate these various data types into these models, allowing more precise ecosystem-wide predictions. In this paper, we develop mathematical methods that combine time-series data of multiple species with knowledge of species interactions and we apply it to proposed reintroductions at Booderee National Park in Australia. There have been large fluctuations in species abundances at Booderee National Park in recent history, following intense feral fox (Vulpes vulpes) control, including the local extinction of the greater glider (Petauroides volans). These fluctuations can provide information about the system isn't readily obtained from a stable system, and we use them to inform models that we then use to predict potential outcomes of eastern quoll (Dasyurus viverrinus) and long-nosed potoroo (Potorous tridactylus) reintroductions. One of the key species of conservation concern in the park is the Eastern Bristlebird (Dasyornis brachypterus), and we find that long-nosed potoroo introduction would have very little impact on the Eastern Bristlebird population, while the eastern quoll introduction increased the likelihood of Eastern Bristlebird decline, although that depends on the strength and form of any possible interaction.

Bayesian Parameter Identification for Turing Systems on Stationary and Evolving Domains

In this study, we apply the Bayesian paradigm for parameter identification to a well-studied semi-linear reaction-diffusion system with activator-depleted reaction kinetics, posed on stationary as well as evolving domains. We provide a mathematically rigorous framework to study the inverse problem of finding the parameters of a reaction-diffusion system given a final spatial pattern. On the stationary domain the parameters are finite-dimensional, but on the evolving domain we consider the problem of identifying the evolution of the domain, i.e. a time-dependent function. Whilst others have considered these inverse problems using optimisation techniques, the Bayesian approach provides a rigorous mathematical framework for incorporating the prior knowledge on uncertainty in the observation and in the parameters themselves, resulting in an approximation of the full probability distribution for the parameters, given the data. Furthermore, using previously established results, we can prove well-posedness results for the inverse problem, using the well-posedness of the forward problem. Although the numerical approximation of the full probability is computationally expensive, parallelised algorithms make the problem solvable using high-performance computing.

Global optimization using Gaussian processes to estimate biological parameters from image data

Parameter estimation is a major challenge in computational modeling of biological processes. This is especially the case in image-based modeling where the inherently quantitative output of the model is measured against image data, which is typically noisy and non-quantitative. In addition, these models can have a high computational cost, limiting the number of feasible simulations, and therefore rendering most traditional parameter estimation methods unsuitable. In this paper, we present a pipeline that uses Gaussian process learning to estimate biological parameters from noisy, non-quantitative image data when the model has a high computational cost. This approach is first successfully tested on a parametric function with the goal of retrieving the original parameters. We then apply it to estimating parameters in a biological setting by fitting artificial in-situ hybridization (ISH) data of the developing murine limb bud. We expect that this method will be of use in a variety of modeling scenarios where quantitative data is missing and the use of standard parameter estimation approaches in biological modeling is prohibited by the computational cost of the model.

Ensemble methods for stochastic networks with special reference to the biological clock of Neurospora crassa

A major challenge in systems biology is to infer the parameters of regulatory networks that operate in a noisy environment, such as in a single cell. In a stochastic regime it is hard to distinguish noise from the real signal and to infer the noise contribution to the dynamical behavior. When the genetic network displays oscillatory dynamics, it is even harder to infer the parameters that produce the oscillations. To address this issue we introduce a new estimation method built on a combination of stochastic simulations, mass action kinetics and ensemble network simulations in which we match the average periodogram and phase of the model to that of the data. The method is relatively fast (compared to Metropolis-Hastings Monte Carlo Methods), easy to parallelize, applicable to large oscillatory networks and large (~2000 cells) single cell expression data sets, and it quantifies the noise impact on the observed dynamics. Standard errors of estimated rate coefficients are typically two orders of magnitude smaller than the mean from single cell experiments with on the order of ~1000 cells. We also provide a method to assess the goodness of fit of the stochastic network using the Hilbert phase of single cells. An analysis of phase departures from the null model with no communication between cells is consistent with a hypothesis of Stochastic Resonance describing single cell oscillators. Stochastic Resonance provides a physical mechanism whereby intracellular noise plays a positive role in establishing oscillatory behavior, but may require model parameters, such as rate coefficients, that differ substantially from those extracted at the macroscopic level from measurements on populations of millions of communicating, synchronized cells.

Modeling the interactions of sense and antisense Period transcripts in the mammalian circadian clock network

In recent years, it has become increasingly apparent that antisense transcription plays an important role in the regulation of gene expression. The circadian clock is no exception: an antisense transcript of the mammalian core-clock gene PERIOD2 (PER2), which we shall refer to as Per2AS RNA, oscillates with a circadian period and a nearly 12 h phase shift from the peak expression of Per2 mRNA. In this paper, we ask whether Per2AS plays a regulatory role in the mammalian circadian clock by studying in silico the potential effects of interactions between Per2 and Per2AS RNAs on circadian rhythms. Based on the antiphasic expression pattern, we consider two hypotheses about how Per2 and Per2AS mutually interfere with each other's expression. In our pre-transcriptional model, the transcription of Per2AS RNA from the non-coding strand represses the transcription of Per2 mRNA from the coding strand and vice versa. In our post-transcriptional model, Per2 and Per2AS transcripts form a double-stranded RNA duplex, which is rapidly degraded. To study these two possible mechanisms, we have added terms describing our alternative hypotheses to a published mathematical model of the molecular regulatory network of the mammalian circadian clock. Our pre-transcriptional model predicts that transcriptional interference between Per2 and Per2AS can generate alternative modes of circadian oscillations, which we characterize in terms of the amplitude and phase of oscillation of core clock genes. In our post-transcriptional model, Per2/Per2AS duplex formation dampens the circadian rhythm. In a model that combines pre- and post-transcriptional controls, the period, amplitude and phase of circadian proteins exhibit non-monotonic dependencies on the rate of expression of Per2AS. All three models provide potential explanations of the observed antiphasic, circadian oscillations of Per2 and Per2AS RNAs. They make discordant predictions that can be tested experimentally in order to distinguish among these alternative hypotheses.

The best models of metabolism

Biochemical systems are among of the oldest application areas of mathematical modeling. Spanning a time period of over one hundred years, the repertoire of options for structuring a model and for formulating reactions has been constantly growing, and yet, it is still unclear whether or to what degree some models are better than others and how the modeler is to choose among them. In fact, the variety of options has become overwhelming and difficult to maneuver for novices and experts alike. This review outlines the metabolic model design process and discusses the numerous choices for modeling frameworks and mathematical representations. It tries to be inclusive, even though it cannot be complete, and introduces the various modeling options in a manner that is as unbiased as that is feasible. However, the review does end with personal recommendations for the choices of default models. WIREs Syst Biol Med 2017, 9:e1391. doi: 10.1002/wsbm.1391 For further resources related to this article, please visit the WIREs website.

Systems Level Modeling of the Cell Cycle Using Budding Yeast

Proteins involved in the regulation of the cell cycle are highly conserved across all eukaryotes, and so a relatively simple eukaryote such as yeast can provide insight into a variety of cell cycle perturbations including those that occur in human cancer. To date, the budding yeast Saccharomyces cerevisiae has provided the largest amount of experimental and modeling data on the progression of the cell cycle, making it a logical choice for in-depth studies of this process. Moreover, the advent of methods for collection of high-throughput genome, transcriptome, and proteome data has provided a means to collect and precisely quantify simultaneous cell cycle gene transcript and protein levels, permitting modeling of the cell cycle on the systems level. With the appropriate mathematical framework and sufficient and accurate data on cell cycle components, it should be possible to create a model of the cell cycle that not only effectively describes its operation, but can also predict responses to perturbations such as variation in protein levels and responses to external stimuli including targeted inhibition by drugs. In this review, we summarize existing data on the yeast cell cycle, proteomics technologies for quantifying cell cycle proteins, and the mathematical frameworks that can integrate this data into representative and effective models. Systems level modeling of the cell cycle will require the integration of high-quality data with the appropriate mathematical framework, which can currently be attained through the combination of dynamic modeling based on proteomics data and using yeast as a model organism.

Formulation, construction and analysis of kinetic models of metabolism: A review of modelling frameworks

Kinetic models are critical to predict the dynamic behaviour of metabolic networks. Mechanistic kinetic models for large networks remain uncommon due to the difficulty of fitting their parameters. Recent modelling frameworks promise new ways to overcome this obstacle while retaining predictive capabilities. In this review, we present an overview of the relevant mathematical frameworks for kinetic formulation, construction and analysis. Starting with kinetic formalisms, we next review statistical methods for parameter inference, as well as recent computational frameworks applied to the construction and analysis of kinetic models. Finally, we discuss opportunities and limitations hindering the development of larger kinetic reconstructions.

Mathematical model reveals role of nucleotide signaling in airway surface liquid homeostasis and its dysregulation in cystic fibrosis

Mucociliary clearance is composed of three components (i.e., mucin secretion, airway surface hydration, and ciliary-activity) which function coordinately to clear inhaled microbes and other foreign particles from airway surfaces. Airway surface hydration is maintained by water fluxes driven predominantly by active chloride and sodium ion transport. The ion channels that mediate electrogenic ion transport are regulated by extracellular purinergic signals that signal through G protein-coupled receptors. These purinoreceptors and the signaling pathways they activate have been identified as possible therapeutic targets for treating lung disease. A systems-level description of airway surface liquid (ASL) homeostasis could accelerate development of such therapies. Accordingly, we developed a mathematical model to describe the dynamic coupling of ion and water transport to extracellular purinergic signaling. We trained our model from steady-state and time-dependent experimental measurements made using normal and cystic fibrosis (CF) cultured human airway epithelium. To reproduce CF conditions, reduced chloride secretion, increased potassium secretion, and increased sodium absorption were required. The model accurately predicted ASL height under basal normal and CF conditions and the collapse of surface hydration due to the accelerated nucleotide metabolism associated with CF exacerbations. Finally, the model predicted a therapeutic strategy to deliver nucleotide receptor agonists to effectively rehydrate the ASL of CF airways.

Ensemble ecosystem modeling for predicting ecosystem response to predator reintroduction

Introducing a new or extirpated species to an ecosystem is risky, and managers need quantitative methods that can predict the consequences for the recipient ecosystem. Proponents of keystone predator reintroductions commonly argue that the presence of the predator will restore ecosystem function, but this has not always been the case, and mathematical modeling has an important role to play in predicting how reintroductions will likely play out. We devised an ensemble modeling method that integrates species interaction networks and dynamic community simulations and used it to describe the range of plausible consequences of 2 keystone-predator reintroductions: wolves (Canis lupus) to Yellowstone National Park and dingoes (Canis dingo) to a national park in Australia. Although previous methods for predicting ecosystem responses to such interventions focused on predicting changes around a given equilibrium, we used Lotka-Volterra equations to predict changing abundances through time. We applied our method to interaction networks for wolves in Yellowstone National Park and for dingoes in Australia. Our model replicated the observed dynamics in Yellowstone National Park and produced a larger range of potential outcomes for the dingo network. However, we also found that changes in small vertebrates or invertebrates gave a good indication about the potential future state of the system. Our method allowed us to predict when the systems were far from equilibrium. Our results showed that the method can also be used to predict which species may increase or decrease following a reintroduction and can identify species that are important to monitor (i.e., species whose changes in abundance give extra insight into broad changes in the system). Ensemble ecosystem modeling can also be applied to assess the ecosystem-wide implications of other types of interventions including assisted migration, biocontrol, and invasive species eradication.

JuPOETs: a constrained multiobjective optimization approach to estimate biochemical model ensembles in the Julia programming language

Background

Ensemble modeling is a promising approach for obtaining robust predictions and coarse grained population behavior in deterministic mathematical models. Ensemble approaches address model uncertainty by using parameter or model families instead of single best-fit parameters or fixed model structures. Parameter ensembles can be selected based upon simulation error, along with other criteria such as diversity or steady-state performance. Simulations using parameter ensembles can estimate confidence intervals on model variables, and robustly constrain model predictions, despite having many poorly constrained parameters.

Results

In this software note, we present a multiobjective based technique to estimate parameter or models ensembles, the Pareto Optimal Ensemble Technique in the Julia programming language (JuPOETs). JuPOETs integrates simulated annealing with Pareto optimality to estimate ensembles on or near the optimal tradeoff surface between competing training objectives. We demonstrate JuPOETs on a suite of multiobjective problems, including test functions with parameter bounds and system constraints as well as for the identification of a proof-of-concept biochemical model with four conflicting training objectives. JuPOETs identified optimal or near optimal solutions approximately six-fold faster than a corresponding implementation in Octave for the suite of test functions. For the proof-of-concept biochemical model, JuPOETs produced an ensemble of parameters that gave both the mean of the training data for conflicting data sets, while simultaneously estimating parameter sets that performed well on each of the individual objective functions.

Conclusions

JuPOETs is a promising approach for the estimation of parameter and model ensembles using multiobjective optimization. JuPOETs can be adapted to solve many problem types, including mixed binary and continuous variable types, bilevel optimization problems and constrained problems without altering the base algorithm. JuPOETs is open source, available under an MIT license, and can be installed using the Julia package manager from the JuPOETs GitHub repository

Synchronizing stochastic circadian oscillators in single cells of Neurospora crassa

The synchronization of stochastic coupled oscillators is a central problem in physics and an emerging problem in biology, particularly in the context of circadian rhythms. Most measurements on the biological clock are made at the macroscopic level of millions of cells. Here measurements are made on the oscillators in single cells of the model fungal system, Neurospora crassa, with droplet microfluidics and the use of a fluorescent recorder hooked up to a promoter on a clock controlled gene-2 (ccg-2). The oscillators of individual cells are stochastic with a period near 21 hours (h), and using a stochastic clock network ensemble fitted by Markov Chain Monte Carlo implemented on general-purpose graphical processing units (or GPGPUs) we estimated that >94% of the variation in ccg-2 expression was stochastic (as opposed to experimental error). To overcome this stochasticity at the macroscopic level, cells must synchronize their oscillators. Using a classic measure of similarity in cell trajectories within droplets, the intraclass correlation (ICC), the synchronization surface ICC is measured on >25,000 cells as a function of the number of neighboring cells within a droplet and of time. The synchronization surface provides evidence that cells communicate, and synchronization varies with genotype.

Modelling and analysis of an ensemble of eukaryotic translation initiation models

Programmed protein synthesis plays an important role in the cell cycle. Deregulated translation has been observed in several cancers. In this study, the authors constructed an ensemble of mathematical models describing the integration of growth factor signals with translation initiation. Using these models, the authors estimated critical structural features of the translation architecture. Sensitivity and robustness analysis with and without growth factors suggested that a balance between competing regulatory programmes governed translation initiation. Proteins such as Akt and mTor promoted initiation by integrating growth factor signals with the assembly of the 80S initiation complex. However, negative regulators such as PTEN and 4EBP1 restrained initiation in the absence of stimulation. Other proteins such as eIF4E were also found to be structurally critical as deletion of amplification of these components resulted in a network incapable of nominal operation. These findings could help understand the molecular basis of translation deregulation observed in cancer and perhaps lead to new anti-cancer therapeutic strategies. [Includes supplementary material].

The inflammatory response to influenza A virus (H1N1): An experimental and mathematical study

Mortality from influenza infections continues as a global public health issue, with the host inflammatory response contributing to fatalities related to the primary infection. Based on Ordinary Differential Equation (ODE) formalism, a computational model was developed for the in-host response to influenza A virus, merging inflammatory, innate, adaptive and humoral responses to virus and linking severity of infection, the inflammatory response, and mortality. The model was calibrated using dense cytokine and cell data from adult BALB/c mice infected with the H1N1 influenza strain A/PR/8/34 in sublethal and lethal doses. Uncertainty in model parameters and disease mechanisms was quantified using Bayesian inference and ensemble model methodology that generates probabilistic predictions of survival, defined as viral clearance and recovery of the respiratory epithelium. The ensemble recovers the expected relationship between magnitude of viral exposure and the duration of survival, and suggests mechanisms primarily responsible for survival, which could guide the development of immuno-modulatory interventions as adjuncts to current anti-viral treatments. The model is employed to extrapolate from available data survival curves for the population and their dependence on initial viral aliquot. In addition, the model allows us to illustrate the positive effect of controlled inflammation on influenza survival.

Approximate Bayesian computation schemes for parameter inference of discrete stochastic models using simulated likelihood density

BACKGROUND

Mathematical modeling is an important tool in systems biology to study the dynamic property of complex biological systems. However, one of the major challenges in systems biology is how to infer unknown parameters in mathematical models based on the experimental data sets, in particular, when the data are sparse and the regulatory network is stochastic.

RESULTS

To address this issue, this work proposed a new algorithm to estimate parameters in stochastic models using simulated likelihood density in the framework of approximate Bayesian computation. Two stochastic models were used to demonstrate the efficiency and effectiveness of the proposed method. In addition, we designed another algorithm based on a novel objective function to measure the accuracy of stochastic simulations.

CONCLUSIONS

Simulation results suggest that the usage of simulated likelihood density improves the accuracy of estimates substantially. When the error is measured at each observation time point individually, the estimated parameters have better accuracy than those obtained by a published method in which the error is measured using simulations over the entire observation time period.

A MINE alternative to D-optimal designs for the linear model

Doing large-scale genomics experiments can be expensive, and so experimenters want to get the most information out of each experiment. To this end the Maximally Informative Next Experiment (MINE) criterion for experimental design was developed. Here we explore this idea in a simplified context, the linear model. Four variations of the MINE method for the linear model were created: MINE-like, MINE, MINE with random orthonormal basis, and MINE with random rotation. Each method varies in how it maximizes the MINE criterion. Theorem 1 establishes sufficient conditions for the maximization of the MINE criterion under the linear model. Theorem 2 establishes when the MINE criterion is equivalent to the classic design criterion of D-optimality. By simulation under the linear model, we establish that the MINE with random orthonormal basis and MINE with random rotation are faster to discover the true linear relation with regression coefficients and observations when . We also establish in simulations with , , and 1000 replicates that these two variations of MINE also display a lower false positive rate than the MINE-like method and additionally, for a majority of the experiments, for the MINE method.

A Bayesian active learning strategy for sequential experimental design in systems biology

BackgroundDynamical models used in systems biology involve unknown kinetic parameters. Setting these parameters is a bottleneck in many modeling projects. This motivates the estimation of these parameters from empirical data. However, this estimation problem has its own difficulties, the most important one being strong ill-conditionedness. In this context, optimizing experiments to be conducted in order to better estimate a system¿s parameters provides a promising direction to alleviate the difficulty of the task.ResultsBorrowing ideas from Bayesian experimental design and active learning, we propose a new strategy for optimal experimental design in the context of kinetic parameter estimation in systems biology. We describe algorithmic choices that allow to implement this method in a computationally tractable way and make it fully automatic. Based on simulation, we show that it outperforms alternative baseline strategies, and demonstrate the benefit to consider multiple posterior modes of the likelihood landscape, as opposed to traditional schemes based on local and Gaussian approximations.ConclusionThis analysis demonstrates that our new, fully automatic Bayesian optimal experimental design strategy has the potential to support the design of experiments for kinetic parameter estimation in systems biology.

A continuous optimization approach for inferring parameters in mathematical models of regulatory networks

BACKGROUND

The advances of systems biology have raised a large number of sophisticated mathematical models for describing the dynamic property of complex biological systems. One of the major steps in developing mathematical models is to estimate unknown parameters of the model based on experimentally measured quantities. However, experimental conditions limit the amount of data that is available for mathematical modelling. The number of unknown parameters in mathematical models may be larger than the number of observation data. The imbalance between the number of experimental data and number of unknown parameters makes reverse-engineering problems particularly challenging.

RESULTS

To address the issue of inadequate experimental data, we propose a continuous optimization approach for making reliable inference of model parameters. This approach first uses a spline interpolation to generate continuous functions of system dynamics as well as the first and second order derivatives of continuous functions. The expanded dataset is the basis to infer unknown model parameters using various continuous optimization criteria, including the error of simulation only, error of both simulation and the first derivative, or error of simulation as well as the first and second derivatives. We use three case studies to demonstrate the accuracy and reliability of the proposed new approach. Compared with the corresponding discrete criteria using experimental data at the measurement time points only, numerical results of the ERK kinase activation module show that the continuous absolute-error criteria using both function and high order derivatives generate estimates with better accuracy. This result is also supported by the second and third case studies for the G1/S transition network and the MAP kinase pathway, respectively. This suggests that the continuous absolute-error criteria lead to more accurate estimates than the corresponding discrete criteria. We also study the robustness property of these three models to examine the reliability of estimates. Simulation results show that the models with estimated parameters using continuous fitness functions have better robustness properties than those using the corresponding discrete fitness functions.

CONCLUSIONS

The inference studies and robustness analysis suggest that the proposed continuous optimization criteria are effective and robust for estimating unknown parameters in mathematical models.

A mathematical model of intrahost pneumococcal pneumonia infection dynamics in murine strains

The seriousness of pneumococcal pneumonia in mouse models has been shown to depend both on bacterial serotype and murine strain. We here present a simple ordinary differential equation model of the intrahost immune response to bacterial pneumonia that is capable of capturing diverse experimentally determined responses of various murine strains. We discuss the main causes of such differences while accounting for the uncertainty in the estimation of model parameters. We model the bacterial population in both the lungs and blood, the cellular death caused by the infection, and the activation and immigration of phagocytes to the infected tissue. The ensemble model suggests that inter-strain differences in response to streptococcus pneumonia inoculation reside in the strength of nonspecific immune response and the rate of extrapulmonary phagocytosis.

Computational model for the regulation of extracellular ATP and adenosine in airway epithelia

Extracellular nucleotides are key components of the signaling network regulating airway clearance. They are released by the epithelium into the airway surface liquid (ASL) to stimulate cilia beating activity, mucus secretion and airway hydration. Understanding the factors affecting their availability for purinoceptor activation is an important step toward the development of new therapies for obstructive lung diseases. This chapter presents a mathematical model developed to gain predictive insights into the regulation of ASL nucleotide concentrations on human airway epithelia. The parameters were estimated from experimental data collected on polarized primary cultures of human nasal and bronchial epithelial cells. This model reproduces major experimental observations: (1) the independence of steady-state nucleotide concentrations on ASL height, (2) the impact of selective ectonucleotidase inhibitors on their steady-state ASL concentrations, (3) the changes in ASL composition caused by mechanical stress mimicking normal breathing, (4) and the differences in steady-state concentrations existing between nasal and bronchial epithelia. In addition, this model launched the study of nucleotide release into uncharted territories, which led to the discovery that airway epithelia release, not only ATP, but also ADP and AMP. This study shows that computational modeling, coupled to experimental validation, provides a powerful approach for the identification of key therapeutic targets for the improvement of airway clearance in obstructive respiratory diseases.

Determination of parameter identifiability in nonlinear biophysical models: A Bayesian approach

A major goal of biophysics is to understand the physical mechanisms of biological molecules and systems. Mechanistic models are evaluated based on their ability to explain carefully controlled experiments. By fitting models to data, biophysical parameters that cannot be measured directly can be estimated from experimentation. However, it might be the case that many different combinations of model parameters can explain the observations equally well. In these cases, the model parameters are not identifiable: the experimentation has not provided sufficient constraining power to enable unique estimation of their true values. We demonstrate that this pitfall is present even in simple biophysical models. We investigate the underlying causes of parameter non-identifiability and discuss straightforward methods for determining when parameters of simple models can be inferred accurately. However, for models of even modest complexity, more general tools are required to diagnose parameter non-identifiability. We present a method based in Bayesian inference that can be used to establish the reliability of parameter estimates, as well as yield accurate quantification of parameter confidence.

Reverse engineering and identification in systems biology: strategies, perspectives and challenges

The interplay of mathematical modelling with experiments is one of the central elements in systems biology. The aim of reverse engineering is to infer, analyse and understand, through this interplay, the functional and regulatory mechanisms of biological systems. Reverse engineering is not exclusive of systems biology and has been studied in different areas, such as inverse problem theory, machine learning, nonlinear physics, (bio)chemical kinetics, control theory and optimization, among others. However, it seems that many of these areas have been relatively closed to outsiders. In this contribution, we aim to compare and highlight the different perspectives and contributions from these fields, with emphasis on two key questions: (i) why are reverse engineering problems so hard to solve, and (ii) what methods are available for the particular problems arising from systems biology?

Prediction stability in a data-based, mechanistic model of σF regulation during sporulation in Bacillus subtilis

Mathematical modeling of biological networks can help to integrate a large body of information into a consistent framework, which can then be used to arrive at novel mechanistic insight and predictions. We have previously developed a detailed, mechanistic model for the regulation of σ ^F during sporulation in Bacillus subtilis. The model was based on a wide range of quantitative data, and once fitted to the data, the model made predictions that could be confirmed in experiments. However, the analysis was based on a single optimal parameter set. We wondered whether the predictions of the model would be stable for all optimal parameter sets. To that end we conducted a global parameter screen within the physiological parameter ranges. The screening approach allowed us to identify sensitive and sloppy parameters, and highlighted further required datasets during the optimization. Eventually, all parameter sets that reproduced all available data predicted the physiological situation correctly.

Biophysical model of ion transport across human respiratory epithelia allows quantification of ion permeabilities

Lung health and normal mucus clearance depend on adequate hydration of airway surfaces. Because transepithelial osmotic gradients drive water flows, sufficient hydration of the airway surface liquid depends on a balance between ion secretion and absorption by respiratory epithelia. In vitro experiments using cultures of primary human nasal epithelia and human bronchial epithelia have established many of the biophysical processes involved in airway surface liquid homeostasis. Most experimental studies, however, have focused on the apical membrane, despite the fact that ion transport across respiratory epithelia involves both cellular and paracellular pathways. In fact, the ion permeabilities of the basolateral membrane and paracellular pathway remain largely unknown. Here we use a biophysical model for water and ion transport to quantify ion permeabilities of all pathways (apical, basolateral, paracellular) in human nasal epithelia cultures using experimental (Ussing Chamber and microelectrode) data reported in the literature. We derive analytical formulas for the steady-state short-circuit current and membrane potential, which are for polarized epithelia the equivalent of the Goldman-Hodgkin-Katz equation for single isolated cells. These relations allow parameter estimation to be performed efficiently. By providing a method to quantify all the ion permeabilities of respiratory epithelia, the model may aid us in understanding the physiology that regulates normal airway surface hydration.

Biophysical model of ion transport across human respiratory epithelia allows quantification of ion permeabilities

Lung health and normal mucus clearance depend on adequate hydration of airway surfaces. Because transepithelial osmotic gradients drive water flows, sufficient hydration of the airway surface liquid depends on a balance between ion secretion and absorption by respiratory epithelia. In vitro experiments using cultures of primary human nasal epithelia and human bronchial epithelia have established many of the biophysical processes involved in airway surface liquid homeostasis. Most experimental studies, however, have focused on the apical membrane, despite the fact that ion transport across respiratory epithelia involves both cellular and paracellular pathways. In fact, the ion permeabilities of the basolateral membrane and paracellular pathway remain largely unknown. Here we use a biophysical model for water and ion transport to quantify ion permeabilities of all pathways (apical, basolateral, paracellular) in human nasal epithelia cultures using experimental (Ussing Chamber and microelectrode) data reported in the literature. We derive analytical formulas for the steady-state short-circuit current and membrane potential, which are for polarized epithelia the equivalent of the Goldman-Hodgkin-Katz equation for single isolated cells. These relations allow parameter estimation to be performed efficiently. By providing a method to quantify all the ion permeabilities of respiratory epithelia, the model may aid us in understanding the physiology that regulates normal airway surface hydration.

Properties of cell death models calibrated and compared using Bayesian approaches

Using models to simulate and analyze biological networks requires principled approaches to parameter estimation and model discrimination. We use Bayesian and Monte Carlo methods to recover the full probability distributions of free parameters (initial protein concentrations and rate constants) for mass-action models of receptor-mediated cell death. The width of the individual parameter distributions is largely determined by non-identifiability but covariation among parameters, even those that are poorly determined, encodes essential information. Knowledge of joint parameter distributions makes it possible to compute the uncertainty of model-based predictions whereas ignoring it (e.g., by treating parameters as a simple list of values and variances) yields nonsensical predictions. Computing the Bayes factor from joint distributions yields the odds ratio (∼20-fold) for competing 'direct' and 'indirect' apoptosis models having different numbers of parameters. Our results illustrate how Bayesian approaches to model calibration and discrimination combined with single-cell data represent a generally useful and rigorous approach to discriminate between competing hypotheses in the face of parametric and topological uncertainty.

Ensemble kinetic modeling of metabolic networks from dynamic metabolic profiles

Kinetic modeling of metabolic pathways has important applications in metabolic engineering, but significant challenges still remain. The difficulties faced vary from finding best-fit parameters in a highly multidimensional search space to incomplete parameter identifiability. To meet some of these challenges, an ensemble modeling method is developed for characterizing a subset of kinetic parameters that give statistically equivalent goodness-of-fit to time series concentration data. The method is based on the incremental identification approach, where the parameter estimation is done in a step-wise manner. Numerical efficacy is achieved by reducing the dimensionality of parameter space and using efficient random parameter exploration algorithms. The shift toward using model ensembles, instead of the traditional "best-fit" models, is necessary to directly account for model uncertainty during the application of such models. The performance of the ensemble modeling approach has been demonstrated in the modeling of a generic branched pathway and the trehalose pathway in Saccharomyces cerevisiae using generalized mass action (GMA) kinetics.

Multiscale models of breast cancer progression

Breast cancer initiation, invasion and metastasis span multiple length and time scales. Molecular events at short length scales lead to an initial tumorigenic population, which left unchecked by immune action, acts at increasingly longer length scales until eventually the cancer cells escape from the primary tumor site. This series of events is highly complex, involving multiple cell types interacting with (and shaping) the microenvironment. Multiscale mathematical models have emerged as a powerful tool to quantitatively integrate the convective-diffusion-reaction processes occurring on the systemic scale, with the molecular signaling processes occurring on the cellular and subcellular scales. In this study, we reviewed the current state of the art in cancer modeling across multiple length scales, with an emphasis on the integration of intracellular signal transduction models with pro-tumorigenic chemical and mechanical microenvironmental cues. First, we reviewed the underlying biomolecular origin of breast cancer, with a special emphasis on angiogenesis. Then, we summarized the development of tissue engineering platforms which could provide high-fidelity ex vivo experimental models to identify and validate multiscale simulations. Lastly, we reviewed top-down and bottom-up multiscale strategies that integrate subcellular networks with the microenvironment. We present models of a variety of cancers, in addition to breast cancer specific models. Taken together, we expect as the sophistication of the simulations increase, that multiscale modeling and bottom-up agent-based models in particular will become an increasingly important platform technology for basic scientific discovery, as well as the identification and validation of potentially novel therapeutic targets.

Compartmentation of glycogen metabolism revealed from 13C isotopologue distributions

Background

Stable isotope tracers are used to assess metabolic flux profiles in living cells. The existing methods of measurement average out the isotopic isomer distribution in metabolites throughout the cell, whereas the knowledge of compartmental organization of analyzed pathways is crucial for the evaluation of true fluxes. That is why we accepted a challenge to create a software tool that allows deciphering the compartmentation of metabolites based on the analysis of average isotopic isomer distribution.

Results

The software Isodyn, which simulates the dynamics of isotopic isomer distribution in central metabolic pathways, was supplemented by algorithms facilitating the transition between various analyzed metabolic schemes, and by the tools for model discrimination. It simulated ¹³C isotope distributions in glucose, lactate, glutamate and glycogen, measured by mass spectrometry after incubation of hepatocytes in the presence of only labeled glucose or glucose and lactate together (with label either in glucose or lactate). The simulations assumed either a single intracellular hexose phosphate pool, or also channeling of hexose phosphates resulting in a different isotopic composition of glycogen. Model discrimination test was applied to check the consistency of both models with experimental data. Metabolic flux profiles, evaluated with the accepted model that assumes channeling, revealed the range of changes in metabolic fluxes in liver cells.

Conclusions

The analysis of compartmentation of metabolic networks based on the measured ¹³C distribution was included in Isodyn as a routine procedure. The advantage of this implementation is that, being a part of evaluation of metabolic fluxes, it does not require additional experiments to study metabolic compartmentation. The analysis of experimental data revealed that the distribution of measured ¹³C-labeled glucose metabolites is inconsistent with the idea of perfect mixing of hexose phosphates in cytosol. In contrast, the observed distribution indicates the presence of a separate pool of hexose phosphates that is channeled towards glycogen synthesis.

Efficient characterization of high-dimensional parameter spaces for systems biology

Background

A biological system's robustness to mutations and its evolution are influenced by the structure of its viable space, the region of its space of biochemical parameters where it can exert its function. In systems with a large number of biochemical parameters, viable regions with potentially complex geometries fill a tiny fraction of the whole parameter space. This hampers explorations of the viable space based on "brute force" or Gaussian sampling.

Results

We here propose a novel algorithm to characterize viable spaces efficiently. The algorithm combines global and local explorations of a parameter space. The global exploration involves an out-of-equilibrium adaptive Metropolis Monte Carlo method aimed at identifying poorly connected viable regions. The local exploration then samples these regions in detail by a method we call multiple ellipsoid-based sampling. Our algorithm explores efficiently nonconvex and poorly connected viable regions of different test-problems. Most importantly, its computational effort scales linearly with the number of dimensions, in contrast to "brute force" sampling that shows an exponential dependence on the number of dimensions. We also apply this algorithm to a simplified model of a biochemical oscillator with positive and negative feedback loops. A detailed characterization of the model's viable space captures well known structural properties of circadian oscillators. Concretely, we find that model topologies with an essential negative feedback loop and a nonessential positive feedback loop provide the most robust fixed period oscillations. Moreover, the connectedness of the model's viable space suggests that biochemical oscillators with varying topologies can evolve from one another.

Conclusions

Our algorithm permits an efficient analysis of high-dimensional, nonconvex, and poorly connected viable spaces characteristic of complex biological circuitry. It allows a systematic use of robustness as a tool for model discrimination.

Bayesian design of synthetic biological systems

Here we introduce a new design framework for synthetic biology that exploits the advantages of Bayesian model selection. We will argue that the difference between inference and design is that in the former we try to reconstruct the system that has given rise to the data that we observe, whereas in the latter, we seek to construct the system that produces the data that we would like to observe, i.e., the desired behavior. Our approach allows us to exploit methods from Bayesian statistics, including efficient exploration of models spaces and high-dimensional parameter spaces, and the ability to rank models with respect to their ability to generate certain types of data. Bayesian model selection furthermore automatically strikes a balance between complexity and (predictive or explanatory) performance of mathematical models. To deal with the complexities of molecular systems we employ an approximate Bayesian computation scheme which only requires us to simulate from different competing models to arrive at rational criteria for choosing between them. We illustrate the advantages resulting from combining the design and modeling (or in silico prototyping) stages currently seen as separate in synthetic biology by reference to deterministic and stochastic model systems exhibiting adaptive and switch-like behavior, as well as bacterial two-component signaling systems.

Systems biology of the qa gene cluster in Neurospora crassa

An ensemble of genetic networks that describe how the model fungal system, Neurospora crassa, utilizes quinic acid (QA) as a sole carbon source has been identified previously. A genetic network for QA metabolism involves the genes, qa-1F and qa-1S, that encode a transcriptional activator and repressor, respectively and structural genes, qa-2, qa-3, qa-4, qa-x, and qa-y. By a series of 4 separate and independent, model-guided, microarray experiments a total of 50 genes are identified as QA-responsive and hypothesized to be under QA-1F control and/or the control of a second QA-responsive transcription factor (NCU03643) both in the fungal binuclear Zn(II)2Cys6 cluster family. QA-1F regulation is not sufficient to explain the quantitative variation in expression profiles of the 50 QA-responsive genes. QA-responsive genes include genes with products in 8 mutually connected metabolic pathways with 7 of them one step removed from the tricarboxylic (TCA) Cycle and with 7 of them one step removed from glycolysis: (1) starch and sucrose metabolism; (2) glycolysis/glucanogenesis; (3) TCA Cycle; (4) butanoate metabolism; (5) pyruvate metabolism; (6) aromatic amino acid and QA metabolism; (7) valine, leucine, and isoleucine degradation; and (8) transport of sugars and amino acids. Gene products both in aromatic amino acid and QA metabolism and transport show an immediate response to shift to QA, while genes with products in the remaining 7 metabolic modules generally show a delayed response to shift to QA. The additional QA-responsive cutinase transcription factor-1β (NCU03643) is found to have a delayed response to shift to QA. The series of microarray experiments are used to expand the previously identified genetic network describing the qa gene cluster to include all 50 QA-responsive genes including the second transcription factor (NCU03643). These studies illustrate new methodologies from systems biology to guide model-driven discoveries about a core metabolic network involving carbon and amino acid metabolism in N. crassa.

Modeling and analysis of retinoic acid induced differentiation of uncommitted precursor cells

Manipulation of differentiation programs has therapeutic potential in a spectrum of human cancers and neurodegenerative disorders. In this study, we integrated computational and experimental methods to unravel the response of a lineage uncommitted precursor cell-line, HL-60, to Retinoic Acid (RA). HL-60 is a human myeloblastic leukemia cell-line used extensively to study human differentiation programs. Initially, we focused on the role of the BLR1 receptor in RA-induced differentiation and G1/0-arrest in HL-60. BLR1, a putative G protein-coupled receptor expressed following RA exposure, is required for RA-induced cell-cycle arrest and differentiation and causes persistent MAPK signaling. A mathematical model of RA-induced cell-cycle arrest and differentiation was formulated and tested against BLR1 wild-type (wt) knock-out and knock-in HL-60 cell-lines with and without RA. The current model described the dynamics of 729 proteins and protein complexes interconnected by 1356 interactions. An ensemble strategy was used to compensate for uncertain model parameters. The ensemble of HL-60 models recapitulated the positive feedback between BLR1 and MAPK signaling. The ensemble of models also correctly predicted Rb and p47phox regulation and the correlation between p21-CDK4-cyclin D formation and G1/0-arrest following exposure to RA. Finally, we investigated the robustness of the HL-60 network architecture to structural perturbations and generated experimentally testable hypotheses for future study. Taken together, the model presented here was a first step toward a systematic framework for analysis of programmed differentiation. These studies also demonstrated that mechanistic network modeling can help prioritize experimental directions by generating falsifiable hypotheses despite uncertainty.

Workflow for generating competing hypothesis from models with parameter uncertainty

Mathematical models are increasingly used in life sciences. However, contrary to other disciplines, biological models are typically over-parametrized and loosely constrained by scarce experimental data and prior knowledge. Recent efforts on analysis of complex models have focused on isolated aspects without considering an integrated approach-ranging from model building to derivation of predictive experiments and refutation or validation of robust model behaviours. Here, we develop such an integrative workflow, a sequence of actions expanding upon current efforts with the purpose of setting the stage for a methodology facilitating an extraction of core behaviours and competing mechanistic hypothesis residing within underdetermined models. To this end, we make use of optimization search algorithms, statistical (machine-learning) classification techniques and cluster-based analysis of the state variables' dynamics and their corresponding parameter sets. We apply the workflow to a mathematical model of fat accumulation in the arterial wall (atherogenesis), a complex phenomena with limited quantitative understanding, thus leading to a model plagued with inherent uncertainty. We find that the mathematical atherogenesis model can still be understood in terms of a few key behaviours despite the large number of parameters. This result enabled us to derive distinct mechanistic predictions from the model despite the lack of confidence in the model parameters. We conclude that building integrative workflows enable investigators to embrace modelling of complex biological processes despite uncertainty in parameters.

Time-resolved NMR: extracting the topology of complex enzyme networks

The use of nondestructive NMR spectroscopy for enzymatic studies offers unique opportunities to identify nearly all enzymatic byproducts and detect unstable short-lived products or intermediates at the molecular level; however, numerous challenges must be overcome before it can become a widely used tool. The biosynthesis of acetyl-coenzyme A (acetyl-CoA) by acetyl-CoA synthetase is used here as a case study for the development of an analytical NMR-based time-course assay platform. We describe an algorithm to deconvolve superimposed spectra into spectra for individual molecules, and further develop a model to simulate the acetyl-CoA synthetase enzyme reaction network using the data derived from time-course NMR. Simulation shows indirectly that synthesis of acetyl-CoA is mediated via an enzyme-bound intermediate (possibly acetyl-AMP) and is accompanied by a nonproductive loss from an intermediate. The ability to predict enzyme function based on partial knowledge of the enzymatic pathway topology is also discussed.

Model reduction using piecewise-linear approximations preserves dynamic properties of the carbon starvation response in Escherichia coli

The adaptation of the bacterium Escherichia coli to carbon starvation is controlled by a large network of biochemical reactions involving genes, mRNAs, proteins, and signalling molecules. The dynamics of these networks is difficult to analyze, notably due to a lack of quantitative information on parameter values. To overcome these limitations, model reduction approaches based on quasi-steady-state (QSS) and piecewise-linear (PL) approximations have been proposed, resulting in models that are easier to handle mathematically and computationally. These approximations are not supposed to affect the capability of the model to account for essential dynamical properties of the system, but the validity of this assumption has not been systematically tested. In this paper, we carry out such a study by evaluating a large and complex PL model of the carbon starvation response in E. coli using an ensemble approach. The results show that, in comparison with conventional nonlinear models, the PL approximations generally preserve the dynamics of the carbon starvation response network, although with some deviations concerning notably the quantitative precision of the model predictions. This encourages the application of PL models to the qualitative analysis of bacterial regulatory networks, in situations where the reference time scale is that of protein synthesis and degradation.

Ensembles of uncertain mathematical models can identify network response to therapeutic interventions

The role of mechanistic modeling and systems biology in molecular medicine remains unclear. In this study, we explored whether uncertain models could be used to understand how a network responds to a therapeutic intervention. As a proof of concept, we modeled and analyzed the response of the human coagulation cascade to recombinant factor VIIa (rFVIIa) and prothrombin (fII) addition in normal and hemophilic plasma. An ensemble of parametrically uncertain human coagulation models was developed (N = 437). Each model described the time evolution of 193 proteins and protein complexes interconnected by 301 interactions under quiescent flow. The 467 unknown model parameters were estimated, using multiobjective optimization, from published in vitro coagulation studies. The model ensemble was validated using published in vitro thrombin measurements and thrombin measurements taken from coronary artery disease patients. Sensitivity analysis was then used to rank-order the importance of model parameters as a function of experimental or physiological conditions. A novel strategy for the systematic comparison of ranks identified a family of fX/FXa and fII/FIIa interactions that became more sensitive with decreasing fVIII/fIX. The fragility of these interactions was preserved following the addition of exogenous rFVIIa and fII. This suggested that exogenous rFVIIa did not alter the qualitative operation of the cascade. Rather, exogenous rFVIIa and fII took advantage of existing fluid and interfacial fX/FXa and fII/FIIa sensitivity to restore normal coagulation in low fVIII/fIX conditions. The proposed rFVIIa mechanism of action was consistent with experimental literature not used in model training. Thus, we demonstrated that an ensemble of uncertain models could unravel key facets of the mechanism of action of a focused intervention. Whereas the current study was limited to coagulation, perhaps the general strategy used could be extended to other molecular networks relevant to human health.

Mathematical modeling of monolignol biosynthesis in Populus xylem

Recalcitrance of lignocellulosic biomass to sugar release is a central issue in the production of biofuel as an economically viable energy source. Among all contributing factors, variations in lignin content and its syringyl-guaiacyl monomer composition have been directly linked with the yield of fermentable sugars. While recent advances in genomics and metabolite profiling have significantly broadened our understanding of lignin biosynthesis, its regulation at the pathway level is yet poorly understood. During the past decade, computational and mathematical methods of systems biology have become effective tools for deciphering the structure and regulation of complex metabolic networks. As increasing amounts of data from various organizational levels are being published, the application of these methods to studying lignin biosynthesis appears to be very beneficial for the future development of genetically engineered crops with reduced recalcitrance. Here, we use techniques from flux balance analysis and nonlinear dynamic modeling to construct a mathematical model of monolignol biosynthesis in Populus xylem. Various types of experimental data from the literature are used to identify the statistically most significant parameters and to estimate their values through an ensemble approach. The thus generated ensemble of models yields results that are quantitatively consistent with several transgenic experiments, including two experiments not used in the model construction. Additional model results not only reveal probable substrate saturation at steps leading to the synthesis of sinapyl alcohol, but also suggest that the ratio of syringyl to guaiacyl monomers might not be affected by genetic modulations prior to the reactions involving coniferaldehyde. This latter model prediction is directly supported by data from transgenic experiments. Finally, we demonstrate the applicability of the model in metabolic engineering, where the pathway is to be optimized toward a higher yield of xylose through modification of the relative amounts of the two major monolignols. The results generated by our preliminary model of in vivo lignin biosynthesis are encouraging and demonstrate that mathematical modeling is poised to become an effective and predictive complement to traditional biotechnological and transgenic approaches, not just in microorganisms but also in plants.

Ensembles of signal transduction models using Pareto Optimal Ensemble Techniques (POETs)

Mathematical modeling of complex gene expression programs is an emerging tool for understanding disease mechanisms. However, identification of large models sometimes requires training using qualitative, conflicting or even contradictory data sets. One strategy to address this challenge is to estimate experimentally constrained model ensembles using multiobjective optimization. In this study, we used Pareto Optimal Ensemble Techniques (POETs) to identify a family of proof-of-concept signal transduction models. POETs integrate Simulated Annealing (SA) with Pareto optimality to identify models near the optimal tradeoff surface between competing training objectives. We modeled a prototypical-signaling network using mass-action kinetics within an ordinary differential equation (ODE) framework (64 ODEs in total). The true model was used to generate synthetic immunoblots from which the POET algorithm identified the 117 unknown model parameters. POET generated an ensemble of signaling models, which collectively exhibited population-like behavior. For example, scaled gene expression levels were approximately normally distributed over the ensemble following the addition of extracellular ligand. Also, the ensemble recovered robust and fragile features of the true model, despite significant parameter uncertainty. Taken together, these results suggest that experimentally constrained model ensembles could capture qualitatively important network features without exact parameter information.

Parameter estimation and model selection in computational biology

A central challenge in computational modeling of biological systems is the determination of the model parameters. Typically, only a fraction of the parameters (such as kinetic rate constants) are experimentally measured, while the rest are often fitted. The fitting process is usually based on experimental time course measurements of observables, which are used to assign parameter values that minimize some measure of the error between these measurements and the corresponding model prediction. The measurements, which can come from immunoblotting assays, fluorescent markers, etc., tend to be very noisy and taken at a limited number of time points. In this work we present a new approach to the problem of parameter selection of biological models. We show how one can use a dynamic recursive estimator, known as extended Kalman filter, to arrive at estimates of the model parameters. The proposed method follows. First, we use a variation of the Kalman filter that is particularly well suited to biological applications to obtain a first guess for the unknown parameters. Secondly, we employ an a posteriori identifiability test to check the reliability of the estimates. Finally, we solve an optimization problem to refine the first guess in case it should not be accurate enough. The final estimates are guaranteed to be statistically consistent with the measurements. Furthermore, we show how the same tools can be used to discriminate among alternate models of the same biological process. We demonstrate these ideas by applying our methods to two examples, namely a model of the heat shock response in E. coli, and a model of a synthetic gene regulation system. The methods presented are quite general and may be applied to a wide class of biological systems where noisy measurements are used for parameter estimation or model selection.

Parameter estimation is a key issue in systems biology, as it represents the crucial step to obtaining predictions from computational models of biological systems. This issue is usually addressed by “fitting” the model simulations to the observed experimental data. Such approach does not take the measurement noise into full consideration. We introduce a new method built on the combination of Kalman filtering, statistical tests, and optimization techniques. The filter is well-known in control and estimation theory and has found application in a wide range of fields, such as inertial guidance systems, weather forecasting, and economics. We show how the statistics of the measurement noise can be optimally exploited and directly incorporated into the design of the estimation algorithm in order to achieve more accurate results, and to validate/invalidate the computed estimates. We also show that a significant advantage of our estimator is that it offers a powerful tool for model selection, allowing rejection or acceptance of competing models based on the available noisy measurements. These results are of immediate practical application in computational biology, and while we demonstrate their use for two specific examples, they can in fact be used to study a wide class of biological systems.