Systematic calibration of a cell signaling network model

Numerical and computational analysis on a dissipative dynamical system: Slow invariant manifold for complex chemical mechanism

This article presents the notion about Slow Invariant Manifold (SIM) and their fundamental role in model reduction techniques (MRTs) for challenges encountered in mechanical engineering within dissipative systems of chemical kinetics. Focusing on the reaction routes of complex mechanisms, we construct and compare primary approximations of the SIM through MRTs, including the Spectral Quasi Equilibrium Manifold (SQEM) and Intrinsic Low Dimensional Manifold (ILDM). These methods effectively transform high-dimensional complex problems into lower dimensions, solving them without compromising crucial information about the complex systems modified for homogeneous reactive systems. Employing the sensitivity analysis by using the MATLAB's toolbox, we present the numerical findings in a tabular format obtained through MRTs. This study provides the understanding about the accessible exploration of numerical solutions, improving insights of the complex variation within the system.

Network-driven cancer cell avatars for combination discovery and biomarker identification for DNA damage response inhibitors

Combination therapy is well established as a key intervention strategy for cancer treatment, with the potential to overcome monotherapy resistance and deliver a more durable efficacy. However, given the scale of unexplored potential target space and the resulting combinatorial explosion, identifying efficacious drug combinations is a critical unmet need that is still evolving. In this paper, we demonstrate a network biology-driven, simulation-based solution, the Simulated Cell™. Integration of omics data with a curated signaling network enables the accurate and interpretable prediction of 66,348 combination-cell line pairs obtained from a large-scale combinatorial drug sensitivity screen of 684 combinations across 97 cancer cell lines (BAC = 0.62, AUC = 0.7). We highlight drug combination pairs that interact with DNA Damage Response pathways and are predicted to be synergistic, and deep network insight to identify biomarkers driving combination synergy. We demonstrate that the cancer cell 'avatars' capture the biological complexity of their in vitro counterparts, enabling the identification of pathway-level mechanisms of combination benefit to guide clinical translatability.

Hip osteoarthritis: A novel network analysis of subchondral trabecular bone structures

Hip osteoarthritis (HOA) is a degenerative joint disease that leads to the progressive destruction of subchondral bone and cartilage at the hip joint. Development of effective treatments for HOA remains an open problem, primarily due to the lack of knowledge of its pathogenesis and a typically late-stage diagnosis. We describe a novel network analysis methodology for microcomputed tomography (micro-CT) images of human trabecular bone. We explored differences between the trabecular bone microstructure of femoral heads with and without HOA. Large-scale automated extraction of the network formed by trabecular bone revealed significant network properties not previously reported for bone. Profound differences were discovered, particularly in the proximal third of the femoral head, where HOA networks demonstrated elevated numbers of edges, vertices, and graph components. When further differentiating healthy joint and HOA networks, the latter showed fewer small-world network properties, due to decreased clustering coefficient and increased characteristic path length. Furthermore, we found that HOA networks had reduced length of edges, indicating the formation of compressed trabecular structures. In order to assess our network approach, we developed a deep learning model for classifying HOA and control cases, and we fed it with two separate inputs: (i) micro-CT images of the trabecular bone, and (ii) the network extracted from them. The model with plain micro-CT images achieves 74.6% overall accuracy while the trained model with extracted networks attains 96.5% accuracy. We anticipate our findings to be a starting point for a novel description of bone microstructure in HOA, by considering the phenomenon from a graph theory viewpoint.

Petri nets and ODEs as complementary methods for comprehensive analysis on an example of the ATM-p53-NF-[Formula: see text]B signaling pathways

Intracellular processes are cascades of biochemical reactions, triggered in response to various types of stimuli. Mathematical models describing their dynamics have become increasingly popular in recent years, as tools supporting experimental work in analysis of pathways and regulatory networks. Not only do they provide insights into general properties of these systems, but also help in specific tasks, such as search for drug molecular targets or treatment protocols. Different tools and methods are used to model complex biological systems. In this work, we focus on ordinary differential equations (ODEs) and Petri nets. We consider specific methods of analysis of such models, i.e., sensitivity analysis (SA) and significance analysis. So far, they have been applied separately, with different goals. In this paper, we show that they can complement each other, combining the sensitivity of ODE models and the significance analysis of Petri nets. The former is used to find parameters, whose change results in the greatest quantitative and qualitative changes in the model response, while the latter is a structural analysis and allows indicating the most important subprocesses in terms of information flow in Petri net. Ultimately, both methods facilitate finding the essential processes in a given signaling pathway or regulatory network and may be used to support medical therapy development. In the paper, the use of dual modeling is illustrated with an example of ATM/p53/NF-[Formula: see text]B pathway. Each method was applied to analyze this system, resulting in finding different subsets of important processes that might be prospective targets for changing this system behavior. While some of the processes were indicated in each of the approaches, others were found by one method only and would be missed if only that method was applied. This leads to the conclusion about the complementarity of the methods under investigation. The dual modeling approach of comprehensive structural and parametric analysis yields results that would not be possible if these two modeling approaches were applied separately. The combined approach, proposed in this paper, facilitates finding not only key processes, with which significant parameters are associated, but also significant modules, corresponding to subsystems of regulatory networks. The results provide broader insight into therapy targets in diseases in which the natural control of intracellular processes is disturbed, leading to the development of more effective therapies in medicine.

Modeling and analysis of the macronutrient signaling network in budding yeast

Adaptive modulation of the global cellular growth state of unicellular organisms is crucial for their survival in fluctuating nutrient environments. Because these organisms must be able to respond reliably to ever varying and unpredictable nutritional conditions, their nutrient signaling networks must have a certain inbuilt robustness. In eukaryotes, such as the budding yeast Saccharomyces cerevisiae, distinct nutrient signals are relayed by specific plasma membrane receptors to signal transduction pathways that are interconnected in complex information-processing networks, which have been well characterized. However, the complexity of the signaling network confounds the interpretation of the overall regulatory "logic" of the control system. Here, we propose a literature-curated molecular mechanism of the integrated nutrient signaling network in budding yeast, focusing on early temporal responses to carbon and nitrogen signaling. We build a computational model of this network to reconcile literature-curated quantitative experimental data with our proposed molecular mechanism. We evaluate the robustness of our estimates of the model's kinetic parameter values. We test the model by comparing predictions made in mutant strains with qualitative experimental observations made in the same strains. Finally, we use the model to predict nutrient-responsive transcription factor activities in a number of mutant strains undergoing complex nutrient shifts.

The development of tumour vascular networks

The growth of solid tumours relies on an ever-increasing supply of oxygen and nutrients that are delivered via vascular networks. Tumour vasculature includes endothelial cell lined angiogenesis and the less common cancer cell lined vasculogenic mimicry (VM). To study and compare the development of vascular networks formed during angiogenesis and VM (represented here by breast cancer and pancreatic cancer cell lines) a number of in vitro assays were utilised. From live cell imaging, we performed a large-scale automated extraction of network parameters and identified properties not previously reported. We show that for both angiogenesis and VM, the characteristic network path length reduces over time; however, only endothelial cells increase network clustering coefficients thus maintaining small-world network properties as they develop. When compared to angiogenesis, the VM network efficiency is improved by decreasing the number of edges and vertices, and also by increasing edge length. Furthermore, our results demonstrate that angiogenic and VM networks appear to display similar properties to road traffic networks and are also subject to the well-known Braess paradox. This quantitative measurement framework opens up new avenues to potentially evaluate the impact of anti-cancer drugs and anti-vascular therapies.

Fouladzadeh, Dorraki and colleagues investigate the development of angiogenic networks for in vitro cancer cell lines. They demonstrate that during the growth stages of vasculogenic mimicry, the number of edges and vertices decreases but the edge length increases resulting in improved network efficiency.

Regulation of CSF and Brain Tissue Sodium Levels by the Blood-CSF and Blood-Brain Barriers During Migraine

Cerebrospinal fluid (CSF) and brain tissue sodium levels increase during migraine. However, little is known regarding the underlying mechanisms of sodium homeostasis disturbance in the brain during the onset and propagation of migraine. Exploring the cause of sodium dysregulation in the brain is important, since correction of the altered sodium homeostasis could potentially treat migraine. Under the hypothesis that disturbances in sodium transport mechanisms at the blood-CSF barrier (BCSFB) and/or the blood-brain barrier (BBB) are the underlying cause of the elevated CSF and brain tissue sodium levels during migraines, we developed a mechanistic, differential equation model of a rat's brain to compare the significance of the BCSFB and the BBB in controlling CSF and brain tissue sodium levels. The model includes the ventricular system, subarachnoid space, brain tissue and blood. Sodium transport from blood to CSF across the BCSFB, and from blood to brain tissue across the BBB were modeled by influx permeability coefficients P_BCSFB and P_BBB, respectively, while sodium movement from CSF into blood across the BCSFB, and from brain tissue to blood across the BBB were modeled by efflux permeability coefficients PBCSFB′ and PBBB′, respectively. We then performed a global sensitivity analysis to investigate the sensitivity of the ventricular CSF, subarachnoid CSF and brain tissue sodium concentrations to pathophysiological variations in P_BCSFB, P_BBB, PBCSFB′ and PBBB′. Our results show that the ventricular CSF sodium concentration is highly influenced by perturbations of P_BCSFB, and to a much lesser extent by perturbations of PBCSFB′. Brain tissue and subarachnoid CSF sodium concentrations are more sensitive to pathophysiological variations of P_BBB and PBBB′ than variations of P_BCSFB and PBCSFB′ within 30 min of the onset of the perturbations. However, P_BCSFB is the most sensitive model parameter, followed by P_BBB and PBBB′, in controlling brain tissue and subarachnoid CSF sodium levels within 3 h of the perturbation onset.

Parameter estimation in models of biological oscillators: an automated regularised estimation approach

BACKGROUND

Dynamic modelling is a core element in the systems biology approach to understanding complex biosystems. Here, we consider the problem of parameter estimation in models of biological oscillators described by deterministic nonlinear differential equations. These problems can be extremely challenging due to several common pitfalls: (i) a lack of prior knowledge about parameters (i.e. massive search spaces), (ii) convergence to local optima (due to multimodality of the cost function), (iii) overfitting (fitting the noise instead of the signal) and (iv) a lack of identifiability. As a consequence, the use of standard estimation methods (such as gradient-based local ones) will often result in wrong solutions. Overfitting can be particularly problematic, since it produces very good calibrations, giving the impression of an excellent result. However, overfitted models exhibit poor predictive power. Here, we present a novel automated approach to overcome these pitfalls. Its workflow makes use of two sequential optimisation steps incorporating three key algorithms: (1) sampling strategies to systematically tighten the parameter bounds reducing the search space, (2) efficient global optimisation to avoid convergence to local solutions, (3) an advanced regularisation technique to fight overfitting. In addition, this workflow incorporates tests for structural and practical identifiability.

RESULTS

We successfully evaluate this novel approach considering four difficult case studies regarding the calibration of well-known biological oscillators (Goodwin, FitzHugh-Nagumo, Repressilator and a metabolic oscillator). In contrast, we show how local gradient-based approaches, even if used in multi-start fashion, are unable to avoid the above-mentioned pitfalls.

CONCLUSIONS

Our approach results in more efficient estimations (thanks to the bounding strategy) which are able to escape convergence to local optima (thanks to the global optimisation approach). Further, the use of regularisation allows us to avoid overfitting, resulting in more generalisable calibrated models (i.e. models with greater predictive power).

Strategies for calibrating models of biology

Computational and mathematical modelling has become a valuable tool for investigating biological systems. Modelling enables prediction of how biological components interact to deliver system-level properties and extrapolation of biological system performance to contexts and experimental conditions where this is unknown. A model's value hinges on knowing that it faithfully represents the biology under the contexts of use, or clearly ascertaining otherwise and thus motivating further model refinement. These qualities are evaluated through calibration, typically formulated as identifying model parameter values that align model and biological behaviours as measured through a metric applied to both. Calibration is critical to modelling but is often underappreciated. A failure to appropriately calibrate risks unrepresentative models that generate erroneous insights. Here, we review a suite of strategies to more rigorously challenge a model's representation of a biological system. All are motivated by features of biological systems, and illustrative examples are drawn from the modelling literature. We examine the calibration of a model against distributions of biological behaviours or outcomes, not only average values. We argue for calibration even where model parameter values are experimentally ascertained. We explore how single metrics can be non-distinguishing for complex systems, with multiple-component dynamic and interaction configurations giving rise to the same metric output. Under these conditions, calibration is insufficiently constraining and the model non-identifiable: multiple solutions to the calibration problem exist. We draw an analogy to curve fitting and argue that calibrating a biological model against a single experiment or context is akin to curve fitting against a single data point. Though useful for communicating model results, we explore how metrics that quantify heavily emergent properties may not be suitable for use in calibration. Lastly, we consider the role of sensitivity and uncertainty analysis in calibration and the interpretation of model results. Our goal in this manuscript is to encourage a deeper consideration of calibration, and how to increase its capacity to either deliver faithful models or demonstrate them otherwise.

Tellurium notebooks-An environment for reproducible dynamical modeling in systems biology

The considerable difficulty encountered in reproducing the results of published dynamical models limits validation, exploration and reuse of this increasingly large biomedical research resource. To address this problem, we have developed Tellurium Notebook, a software system for model authoring, simulation, and teaching that facilitates building reproducible dynamical models and reusing models by 1) providing a notebook environment which allows models, Python code, and narrative to be intermixed, 2) supporting the COMBINE archive format during model development for capturing model information in an exchangeable format and 3) enabling users to easily simulate and edit public COMBINE-compliant models from public repositories to facilitate studying model dynamics, variants and test cases. Tellurium Notebook, a Python–based Jupyter–like environment, is designed to seamlessly inter-operate with these community standards by automating conversion between COMBINE standards formulations and corresponding in–line, human–readable representations. Thus, Tellurium brings to systems biology the strategy used by other literate notebook systems such as Mathematica. These capabilities allow users to edit every aspect of the standards–compliant models and simulations, run the simulations in–line, and re–export to standard formats. We provide several use cases illustrating the advantages of our approach and how it allows development and reuse of models without requiring technical knowledge of standards. Adoption of Tellurium should accelerate model development, reproducibility and reuse.

There is considerable value to systems and synthetic biology in creating reproducible models. An essential element of reproducibility is the use of community standards, an often challenging undertaking for modelers. This article describes Tellurium Notebook, a tool for developing dynamical models that provides an intuitive approach to building and reusing models built with community standards. Tellurium automates embedding human–readable representations of COMBINE archives in literate coding notebooks, bringing to systems biology this strategy central to other literate notebook systems such as Mathematica. We show that the ability to easily edit this human–readable representation enables users to test models under a variety of conditions, thereby providing a way to create, reuse, and modify standard–encoded models and simulations, regardless of the user’s level of technical knowledge of said standards.

Protein abundance of AKT and ERK pathway components governs cell type-specific regulation of proliferation

Signaling through the AKT and ERK pathways controls cell proliferation. However, the integrated regulation of this multistep process, involving signal processing, cell growth and cell cycle progression, is poorly understood. Here, we study different hematopoietic cell types, in which AKT and ERK signaling is triggered by erythropoietin (Epo). Although these cell types share the molecular network topology for pro‐proliferative Epo signaling, they exhibit distinct proliferative responses. Iterating quantitative experiments and mathematical modeling, we identify two molecular sources for cell type‐specific proliferation. First, cell type‐specific protein abundance patterns cause differential signal flow along the AKT and ERK pathways. Second, downstream regulators of both pathways have differential effects on proliferation, suggesting that protein synthesis is rate‐limiting for faster cycling cells while slower cell cycles are controlled at the G1‐S progression. The integrated mathematical model of Epo‐driven proliferation explains cell type‐specific effects of targeted AKT and ERK inhibitors and faithfully predicts, based on the protein abundance, anti‐proliferative effects of inhibitors in primary human erythroid progenitor cells. Our findings suggest that the effectiveness of targeted cancer therapy might become predictable from protein abundance.

Assessing the Impact of Tissue Target Concentration Data on Uncertainty in In Vivo Target Coverage Predictions

Understanding pharmacological target coverage is fundamental in drug discovery and development as it helps establish a sequence of research activities, from laboratory objectives to clinical doses. To this end, we evaluated the impact of tissue target concentration data on the level of confidence in tissue coverage predictions using a site of action (SoA) model for antibodies. By fitting the model to increasing amounts of synthetic tissue data and comparing the uncertainty in SoA coverage predictions, we confirmed that, in general, uncertainty decreases with longitudinal tissue data. Furthermore, a global sensitivity analysis showed that coverage is sensitive to experimentally identifiable parameters, such as baseline target concentration in plasma and target turnover half‐life and fixing them reduces uncertainty in coverage predictions. Overall, our computational analysis indicates that measurement of baseline tissue target concentration reduces the uncertainty in coverage predictions and identifies target‐related parameters that greatly impact the confidence in coverage predictions.

Robust and efficient parameter estimation in dynamic models of biological systems

Background

Dynamic modelling provides a systematic framework to understand function in biological systems. Parameter estimation in nonlinear dynamic models remains a very challenging inverse problem due to its nonconvexity and ill-conditioning. Associated issues like overfitting and local solutions are usually not properly addressed in the systems biology literature despite their importance.

Here we present a method for robust and efficient parameter estimation which uses two main strategies to surmount the aforementioned difficulties: (i) efficient global optimization to deal with nonconvexity, and (ii) proper regularization methods to handle ill-conditioning. In the case of regularization, we present a detailed critical comparison of methods and guidelines for properly tuning them. Further, we show how regularized estimations ensure the best trade-offs between bias and variance, reducing overfitting, and allowing the incorporation of prior knowledge in a systematic way.

Results

We illustrate the performance of the presented method with seven case studies of different nature and increasing complexity, considering several scenarios of data availability, measurement noise and prior knowledge. We show how our method ensures improved estimations with faster and more stable convergence. We also show how the calibrated models are more generalizable. Finally, we give a set of simple guidelines to apply this strategy to a wide variety of calibration problems.

Conclusions

Here we provide a parameter estimation strategy which combines efficient global optimization with a regularization scheme. This method is able to calibrate dynamic models in an efficient and robust way, effectively fighting overfitting and allowing the incorporation of prior information.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-015-0219-2) contains supplementary material, which is available to authorized users.

Sensitivity analysis approaches applied to systems biology models

With the rising application of systems biology, sensitivity analysis methods have been widely applied to study the biological systems, including metabolic networks, signalling pathways and genetic circuits. Sensitivity analysis can provide valuable insights about how robust the biological responses are with respect to the changes of biological parameters and which model inputs are the key factors that affect the model outputs. In addition, sensitivity analysis is valuable for guiding experimental analysis, model reduction and parameter estimation. Local and global sensitivity analysis approaches are the two types of sensitivity analysis that are commonly applied in systems biology. Local sensitivity analysis is a classic method that studies the impact of small perturbations on the model outputs. On the other hand, global sensitivity analysis approaches have been applied to understand how the model outputs are affected by large variations of the model input parameters. In this review, the author introduces the basic concepts of sensitivity analysis approaches applied to systems biology models. Moreover, the author discusses the advantages and disadvantages of different sensitivity analysis methods, how to choose a proper sensitivity analysis approach, the available sensitivity analysis tools for systems biology models and the caveats in the interpretation of sensitivity analysis results.

Model-based global sensitivity analysis as applied to identification of anti-cancer drug targets and biomarkers of drug resistance in the ErbB2/3 network

High levels of variability in cancer-related cellular signalling networks and a lack of parameter identifiability in large-scale network models hamper translation of the results of modelling studies into the process of anti-cancer drug development. Recently global sensitivity analysis (GSA) has been recognised as a useful technique, capable of addressing the uncertainty of the model parameters and generating valid predictions on parametric sensitivities. Here we propose a novel implementation of model-based GSA specially designed to explore how multi-parametric network perturbations affect signal propagation through cancer-related networks. We use area-under-the-curve for time course of changes in phosphorylation of proteins as a characteristic for sensitivity analysis and rank network parameters with regard to their impact on the level of key cancer-related outputs, separating strong inhibitory from stimulatory effects. This allows interpretation of the results in terms which can incorporate the effects of potential anti-cancer drugs on targets and the associated biological markers of cancer. To illustrate the method we applied it to an ErbB signalling network model and explored the sensitivity profile of its key model readout, phosphorylated Akt, in the absence and presence of the ErbB2 inhibitor pertuzumab. The method successfully identified the parameters associated with elevation or suppression of Akt phosphorylation in the ErbB2/3 network. From analysis and comparison of the sensitivity profiles of pAkt in the absence and presence of targeted drugs we derived predictions of drug targets, cancer-related biomarkers and generated hypotheses for combinatorial therapy. Several key predictions have been confirmed in experiments using human ovarian carcinoma cell lines. We also compared GSA-derived predictions with the results of local sensitivity analysis and discuss the applicability of both methods. We propose that the developed GSA procedure can serve as a refining tool in combinatorial anti-cancer drug discovery.

Measuring and modeling apoptosis in single cells

Cell death plays an essential role in the development of tissues and organisms, the etiology of disease, and the responses of cells to therapeutic drugs. Here we review progress made over the last decade in using mathematical models and quantitative, often single-cell, data to study apoptosis. We discuss the delay that follows exposure of cells to prodeath stimuli, control of mitochondrial outer membrane permeabilization, switch-like activation of effector caspases, and variability in the timing and probability of death from one cell to the next. Finally, we discuss challenges facing the fields of biochemical modeling and systems pharmacology.

What's all the noise about developmental stochasticity?

In October 2010, researchers from diverse backgrounds collided at the historic Cumberland Lodge (Windsor, UK) to discuss the role of randomness in cell and developmental biology. Organized by James Briscoe and Alfonso Marinez-Arias, The Company of Biologists' workshop was the latest in a series of meetings aimed at encouraging interdisciplinary interactions between biologists. This aim was reflected in talks at this workshop that ranged from the tissue to the cellular scale, and that integrated experimental and theoretical approaches to examining stochastic behavior in diverse systems.