A general computational method for robustness analysis with applications to synthetic gene networks

Efficient parameter generation for constrained models using MCMC

Mathematical models of complex systems rely on parameter values to produce a desired behavior. As mathematical and computational models increase in complexity, it becomes correspondingly difficult to find parameter values that satisfy system constraints. We propose a Markov Chain Monte Carlo (MCMC) approach for the problem of constrained model parameter generation by designing a Markov chain that efficiently explores a model's parameter space. We demonstrate the use of our proposed methodology to analyze responses of a newly constructed bistability-constrained model of protein phosphorylation to perturbations in the underlying protein network. Our results suggest that parameter generation for constrained models using MCMC provides powerful tools for modeling-aided analysis of complex natural processes.

Assessing Robustness of Morphological Characteristics of Arbitrary Grayscale Images

In our previous work, we introduced an empirical model (EM) of arbitrary binary images and three morphological characteristics: disorder of layer structure (DStr), disorder of layer size (DSize), and pattern complexity (PCom). The basic concept of the EM is that forms of lines play no role as a morphological factor in any narrow area of an arbitrary binary image; instead, the basic factor is the type of line connectivity, i.e., isotropic/anisotropic connections. The goal of the present work is to justify the possibility of making the EM applicable for the processing of grayscale arbitrary images. One of the possible ways to reach this goal is to assess the influence of image binarization on the robustness of DStr and DSize. Images that exhibit high and low edge gradient are used for this experimental study. The robustness of DStr and DSize against the binarization procedure is described in absolute (deviation from average) and relative (Pearson’s coefficient correlation) terms. Images with low edge gradient are converted into binary contour maps by applying the watershed algorithm, and DStr and DSize are then calculated for these maps. The robustness of DStr and DSize were assessed against the image threshold for images with high edge gradient and against the grid size of contour maps and Gaussian blur smoothing for images with low edge gradient. Experiments with grayscale arbitrary patterns, such as the surface of Earth and Mars, tidal sand ripples, turbulent flow, a melanoma, and cloud images, are presented to illustrate the spectrum of problems that may be possible to solve by applying the EM. The majority of our experiments show a high level of robustness for DStr and DSize.

Uncovering the effects of heterogeneity and parameter sensitivity on within-host dynamics of disease: malaria as a case study

Background

The fidelity and reliability of disease model predictions depend on accurate and precise descriptions of processes and determination of parameters. Various models exist to describe within-host dynamics during malaria infection but there is a shortage of clinical data that can be used to quantitatively validate them and establish confidence in their predictions. In addition, model parameters often contain a degree of uncertainty and show variations between individuals, potentially undermining the reliability of model predictions. In this study models were reproduced and analysed by means of robustness, uncertainty, local sensitivity and local sensitivity robustness analysis to establish confidence in their predictions.

Results

Components of the immune system are responsible for the most uncertainty in model outputs, while disease associated variables showed the greatest sensitivity for these components. All models showed a comparable degree of robustness but displayed different ranges in their predictions. In these different ranges, sensitivities were well-preserved in three of the four models.

Conclusion

Analyses of the effects of parameter variations in models can provide a comparative tool for the evaluation of model predictions. In addition, it can assist in uncovering model weak points and, in the case of disease models, be used to identify possible points for therapeutic intervention.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12859-021-04289-z.

Computational Modelling of Metabolic Burden and Substrate Toxicity in Escherichia coli Carrying a Synthetic Metabolic Pathway

In our previous work, we designed and implemented a synthetic metabolic pathway for 1,2,3-trichloropropane (TCP) biodegradation in Escherichia coli. Significant effects of metabolic burden and toxicity exacerbation were observed on single cell and population levels. Deeper understanding of mechanisms underlying these effects is extremely important for metabolic engineering of efficient microbial cell factories for biotechnological processes. In this paper, we present a novel mathematical model of the pathway. The model addresses for the first time the combined effects of toxicity exacerbation and metabolic burden in the context of bacterial population growth. The model is calibrated with respect to the real data obtained with our original synthetically modified E. coli strain. Using the model, we explore the dynamics of the population growth along with the outcome of the TCP biodegradation pathway considering the toxicity exacerbation and metabolic burden. On the methodological side, we introduce a unique computational workflow utilising algorithmic methods of computer science for the particular modelling problem.

CHARACTERIZATION OF INTRINSIC AND EXTRINSIC NOISE EFFECTS IN POSITIVELY REGULATED GENES

Gene regulation is fundamental for cell survival. This regulation must be both robust to noise and sensitive enough to external stimuli to elicit the proper responses. In this work, we study, through stochastic numerical simulations, how a gene regulatory network with a positive feedback loop responds to environmental changes in the presence of intrinsic and extrinsic noises. Noise effects were characterized by measuring the statistical differences between two protein time series resulting from identical systems subject to the same source of extrinsic noise. A robust analysis was implemented by modifying the kinetic system parameters. We found that the common source of time-varying extrinsic fluctuations leads to a correlation in the systems it affects. The correlation and the extrinsic and intrinsic noise components are modulated by the update period and noise intensity parameters. Our results suggest that noise perception is controlled through the parameters associated with the response time: degradation rates and promoter dissociation constant.

CRA toolbox: software package for conditional robustness analysis of cancer systems biology models in MATLAB

Background

In cancer research, robustness of a complex biochemical network is one of the most relevant properties to investigate for the development of novel targeted therapies. In cancer systems biology, biological networks are typically modeled through Ordinary Differential Equation (ODE) models. Hence, robustness analysis consists in quantifying how much the temporal behavior of a specific node is influenced by the perturbation of model parameters. The Conditional Robustness Algorithm (CRA) is a valuable methodology to perform robustness analysis on a selected output variable, representative of the proliferation activity of cancer disease.

Results

Here we introduce our new freely downloadable software, the CRA Toolbox. The CRA Toolbox is an Object-Oriented MATLAB package which implements the features of CRA for ODE models. It offers the users the ability to import a mathematical model in Systems Biology Markup Language (SBML), to perturb the model parameter space and to choose the reference node for the robustness analysis. The CRA Toolbox allows the users to visualize and save all the generated results through a user-friendly Graphical User Interface (GUI). The CRA Toolbox has a modular and flexible architecture since it is designed according to some engineering design patterns. This tool has been successfully applied in three nonlinear ODE models: the Prostate-specific Pten^−/− mouse model, the Pulse Generator Network and the EGFR-IGF1R pathway.

Conclusions

The CRA Toolbox for MATLAB is an open-source tool implementing the CRA to perform conditional robustness analysis. With its unique set of functions, the CRA Toolbox is a remarkable software for the topological study of biological networks. The source and example code and the corresponding documentation are freely available at the web site: http://gitlab.ict4life.com/SysBiOThe/CRA-Matlab.

Electronic supplementary material

The online version of this article (10.1186/s12859-019-2933-z) contains supplementary material, which is available to authorized users.

Computer-aided biochemical programming of synthetic microreactors as diagnostic devices

Biological systems have evolved efficient sensing and decision-making mechanisms to maximize fitness in changing molecular environments. Synthetic biologists have exploited these capabilities to engineer control on information and energy processing in living cells. While engineered organisms pose important technological and ethical challenges, de novo assembly of non-living biomolecular devices could offer promising avenues toward various real-world applications. However, assembling biochemical parts into functional information processing systems has remained challenging due to extensive multidimensional parameter spaces that must be sampled comprehensively in order to identify robust, specification compliant molecular implementations. We introduce a systematic methodology based on automated computational design and microfluidics enabling the programming of synthetic cell-like microreactors embedding biochemical logic circuits, or protosensors, to perform accurate biosensing and biocomputing operations in vitro according to temporal logic specifications. We show that proof-of-concept protosensors integrating diagnostic algorithms detect specific patterns of biomarkers in human clinical samples. Protosensors may enable novel approaches to medicine and represent a step toward autonomous micromachines capable of precise interfacing of human physiology or other complex biological environments, ecosystems, or industrial bioprocesses.

A Novel Method to Verify Multilevel Computational Models of Biological Systems Using Multiscale Spatio-Temporal Meta Model Checking

Insights gained from multilevel computational models of biological systems can be translated into real-life applications only if the model correctness has been verified first. One of the most frequently employed in silico techniques for computational model verification is model checking. Traditional model checking approaches only consider the evolution of numeric values, such as concentrations, over time and are appropriate for computational models of small scale systems (e.g. intracellular networks). However for gaining a systems level understanding of how biological organisms function it is essential to consider more complex large scale biological systems (e.g. organs). Verifying computational models of such systems requires capturing both how numeric values and properties of (emergent) spatial structures (e.g. area of multicellular population) change over time and across multiple levels of organization, which are not considered by existing model checking approaches. To address this limitation we have developed a novel approximate probabilistic multiscale spatio-temporal meta model checking methodology for verifying multilevel computational models relative to specifications describing the desired/expected system behaviour. The methodology is generic and supports computational models encoded using various high-level modelling formalisms because it is defined relative to time series data and not the models used to generate it. In addition, the methodology can be automatically adapted to case study specific types of spatial structures and properties using the spatio-temporal meta model checking concept. To automate the computational model verification process we have implemented the model checking approach in the software tool Mule (http://mule.modelchecking.org). Its applicability is illustrated against four systems biology computational models previously published in the literature encoding the rat cardiovascular system dynamics, the uterine contractions of labour, the Xenopus laevis cell cycle and the acute inflammation of the gut and lung. Our methodology and software will enable computational biologists to efficiently develop reliable multilevel computational models of biological systems.

Conditional robustness analysis for fragility discovery and target identification in biochemical networks and in cancer systems biology

Background

The study of cancer therapy is a key issue in the field of oncology research and the development of target therapies is one of the main problems currently under investigation. This is particularly relevant in different types of tumor where traditional chemotherapy approaches often fail, such as lung cancer.

Results

We started from the general definition of robustness introduced by Kitano and applied it to the analysis of dynamical biochemical networks, proposing a new algorithm based on moment independent analysis of input/output uncertainty. The framework utilizes novel computational methods which enable evaluating the model fragility with respect to quantitative performance measures and parameters such as reaction rate constants and initial conditions. The algorithm generates a small subset of parameters that can be used to act on complex networks and to obtain the desired behaviors. We have applied the proposed framework to the EGFR-IGF1R signal transduction network, a crucial pathway in lung cancer, as an example of Cancer Systems Biology application in drug discovery. Furthermore, we have tested our framework on a pulse generator network as an example of Synthetic Biology application, thus proving the suitability of our methodology to the characterization of the input/output synthetic circuits.

Conclusions

The achieved results are of immediate practical application in computational biology, and while we demonstrate their use in two specific examples, they can in fact be used to study a wider class of biological systems.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-015-0216-5) contains supplementary material, which is available to authorized users.

Invariants reveal multiple forms of robustness in bifunctional enzyme systems

Experimental and theoretical studies have suggested that bifunctional enzymes catalyzing opposing modification and demodification reactions can confer steady-state concentration robustness to their substrates. However, the types of robustness and the biochemical basis for them have remained elusive. Here we report a systematic study of the most general biochemical reaction network for a bifunctional enzyme acting on a substrate with one modification site, along with eleven sub-networks with more specialized biochemical assumptions. We exploit ideas from computational algebraic geometry, introduced in previous work, to find a polynomial expression (an invariant) between the steady state concentrations of the modified and unmodified substrate for each network. We use these invariants to identify five classes of robust behavior: robust upper bounds on concentration, robust two-sided bounds on concentration ratio, hybrid robustness, absolute concentration robustness (ACR), and robust concentration ratio. This analysis demonstrates that robustness can take a variety of forms and that the type of robustness is sensitive to many biochemical details, with small changes in biochemistry leading to very different steady-state behaviors. In particular, we find that the widely-studied ACR requires highly specialized assumptions in addition to bifunctionality. An unexpected result is that the robust bounds derived from invariants are strictly tighter than those derived by ad hoc manipulation of the underlying differential equations, confirming the value of invariants as a tool to gain insight into biochemical reaction networks. Furthermore, invariants yield multiple experimentally testable predictions and illuminate new strategies for inferring enzymatic mechanisms from steady-state measurements.

Evolving robust gene regulatory networks

Design and implementation of robust network modules is essential for construction of complex biological systems through hierarchical assembly of 'parts' and 'devices'. The robustness of gene regulatory networks (GRNs) is ascribed chiefly to the underlying topology. The automatic designing capability of GRN topology that can exhibit robust behavior can dramatically change the current practice in synthetic biology. A recent study shows that Darwinian evolution can gradually develop higher topological robustness. Subsequently, this work presents an evolutionary algorithm that simulates natural evolution in silico, for identifying network topologies that are robust to perturbations. We present a Monte Carlo based method for quantifying topological robustness and designed a fitness approximation approach for efficient calculation of topological robustness which is computationally very intensive. The proposed framework was verified using two classic GRN behaviors: oscillation and bistability, although the framework is generalized for evolving other types of responses. The algorithm identified robust GRN architectures which were verified using different analysis and comparison. Analysis of the results also shed light on the relationship among robustness, cooperativity and complexity. This study also shows that nature has already evolved very robust architectures for its crucial systems; hence simulation of this natural process can be very valuable for designing robust biological systems.

Automatic validation of computational models using pseudo-3D spatio-temporal model checking

BACKGROUND

Computational models play an increasingly important role in systems biology for generating predictions and in synthetic biology as executable prototypes/designs. For real life (clinical) applications there is a need to scale up and build more complex spatio-temporal multiscale models; these could enable investigating how changes at small scales reflect at large scales and viceversa. Results generated by computational models can be applied to real life applications only if the models have been validated first. Traditional in silico model checking techniques only capture how non-dimensional properties (e.g. concentrations) evolve over time and are suitable for small scale systems (e.g. metabolic pathways). The validation of larger scale systems (e.g. multicellular populations) additionally requires capturing how spatial patterns and their properties change over time, which are not considered by traditional non-spatial approaches.

RESULTS

We developed and implemented a methodology for the automatic validation of computational models with respect to both their spatial and temporal properties. Stochastic biological systems are represented by abstract models which assume a linear structure of time and a pseudo-3D representation of space (2D space plus a density measure). Time series data generated by such models is provided as input to parameterised image processing modules which automatically detect and analyse spatial patterns (e.g. cell) and clusters of such patterns (e.g. cellular population). For capturing how spatial and numeric properties change over time the Probabilistic Bounded Linear Spatial Temporal Logic is introduced. Given a collection of time series data and a formal spatio-temporal specification the model checker Mudi ( http://mudi.modelchecking.org ) determines probabilistically if the formal specification holds for the computational model or not. Mudi is an approximate probabilistic model checking platform which enables users to choose between frequentist and Bayesian, estimate and statistical hypothesis testing based validation approaches. We illustrate the expressivity and efficiency of our approach based on two biological case studies namely phase variation patterning in bacterial colony growth and the chemotactic aggregation of cells.

CONCLUSIONS

The formal methodology implemented in Mudi enables the validation of computational models against spatio-temporal logic properties and is a precursor to the development and validation of more complex multidimensional and multiscale models.

Quantitative Analysis of Robustness of Dynamic Response and Signal Transfer in Insulin mediated PI3K/AKT Pathway

Robustness is a critical feature of signaling pathways ensuring signal propagation with high fidelity in the event of perturbations. Here we present a detailed quantitative analysis of robustness in insulin mediated PI3K/AKT pathway, a critical signaling pathway maintaining self-renewal in human embryonic stem cells. Using global sensitivity analysis, we identified robustness promoting mechanisms that ensure (1) maintenance of a first order or overshoot dynamics of self-renewal molecule, p-AKT and (2) robust transfer of signals from oscillatory insulin stimulus to p-AKT in the presence of noise. Our results indicate that negative feedback controls the robustness to most perturbations. Faithful transfer of signal from the stimulating ligand to p-AKT occurs even in the presence of noise, albeit with signal attenuation and high frequency cut-off. Negative feedback contributes to signal attenuation, while positive regulators upstream of PIP₃ contribute to signal amplification. These results establish precise mechanisms to modulate self-renewal molecules like p-AKT.

General applicability of synthetic gene-overexpression for cell-type ratio control via reprogramming

Control of the cell-type ratio in multistable systems requires wide-range control of the initial states of cells. Here, using a synthetic circuit in E. coli, we describe the use of a simple gene-overexpression system combined with a bistable toggle switch, for the purposes of enabling the wide-range control of cellular states and thus generating arbitrary cell-type ratios. Theoretically, overexpression induction temporarily alters the bistable system to a monostable system, in which the location of the single steady state of cells can be manipulated over a wide range by regulating the overexpression levels. This induced cellular state becomes the initial state of the basal bistable system upon overexpression cessation, which restores the original bistable system. We experimentally demonstrated that the overexpression induced a monomodal cell distribution, and subsequent overexpression withdrawal generated a bimodal distribution. Furthermore, as designed theoretically, regulating the overexpression levels by adjusting the concentrations of small molecules generated arbitrary cell-type ratios.

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.

Robustness analysis of stochastic biochemical systems

We propose a new framework for rigorous robustness analysis of stochastic biochemical systems that is based on probabilistic model checking techniques. We adapt the general definition of robustness introduced by Kitano to the class of stochastic systems modelled as continuous time Markov Chains in order to extensively analyse and compare robustness of biological models with uncertain parameters. The framework utilises novel computational methods that enable to effectively evaluate the robustness of models with respect to quantitative temporal properties and parameters such as reaction rate constants and initial conditions. We have applied the framework to gene regulation as an example of a central biological mechanism where intrinsic and extrinsic stochasticity plays crucial role due to low numbers of DNA and RNA molecules. Using our methods we have obtained a comprehensive and precise analysis of stochastic dynamics under parameter uncertainty. Furthermore, we apply our framework to compare several variants of two-component signalling networks from the perspective of robustness with respect to intrinsic noise caused by low populations of signalling components. We have successfully extended previous studies performed on deterministic models (ODE) and showed that stochasticity may significantly affect obtained predictions. Our case studies demonstrate that the framework can provide deeper insight into the role of key parameters in maintaining the system functionality and thus it significantly contributes to formal methods in computational systems biology.

Analysis of the robustness of network-based disease-gene prioritization methods reveals redundancy in the human interactome and functional diversity of disease-genes

Complex biological systems usually pose a trade-off between robustness and fragility where a small number of perturbations can substantially disrupt the system. Although biological systems are robust against changes in many external and internal conditions, even a single mutation can perturb the system substantially, giving rise to a pathophenotype. Recent advances in identifying and analyzing the sequential variations beneath human disorders help to comprehend a systemic view of the mechanisms underlying various disease phenotypes. Network-based disease-gene prioritization methods rank the relevance of genes in a disease under the hypothesis that genes whose proteins interact with each other tend to exhibit similar phenotypes. In this study, we have tested the robustness of several network-based disease-gene prioritization methods with respect to the perturbations of the system using various disease phenotypes from the Online Mendelian Inheritance in Man database. These perturbations have been introduced either in the protein-protein interaction network or in the set of known disease-gene associations. As the network-based disease-gene prioritization methods are based on the connectivity between known disease-gene associations, we have further used these methods to categorize the pathophenotypes with respect to the recoverability of hidden disease-genes. Our results have suggested that, in general, disease-genes are connected through multiple paths in the human interactome. Moreover, even when these paths are disturbed, network-based prioritization can reveal hidden disease-gene associations in some pathophenotypes such as breast cancer, cardiomyopathy, diabetes, leukemia, parkinson disease and obesity to a greater extend compared to the rest of the pathophenotypes tested in this study. Gene Ontology (GO) analysis highlighted the role of functional diversity for such diseases.

Mathematical modelling of the MAP kinase pathway using proteomic datasets

The advances in proteomics technologies offer an unprecedented opportunity and valuable resources to understand how living organisms execute necessary functions at systems levels. However, little work has been done up to date to utilize the highly accurate spatio-temporal dynamic proteome data generated by phosphoprotemics for mathematical modeling of complex cell signaling pathways. This work proposed a novel computational framework to develop mathematical models based on proteomic datasets. Using the MAP kinase pathway as the test system, we developed a mathematical model including the cytosolic and nuclear subsystems; and applied the genetic algorithm to infer unknown model parameters. Robustness property of the mathematical model was used as a criterion to select the appropriate rate constants from the estimated candidates. Quantitative information regarding the absolute protein concentrations was used to refine the mathematical model. We have demonstrated that the incorporation of more experimental data could significantly enhance both the simulation accuracy and robustness property of the proposed model. In addition, we used the MAP kinase pathway inhibited by phosphatases with different concentrations to predict the signal output influenced by different cellular conditions. Our predictions are in good agreement with the experimental observations when the MAP kinase pathway was inhibited by phosphatase PP2A and MKP3. The successful application of the proposed modeling framework to the MAP kinase pathway suggests that our method is very promising for developing accurate mathematical models and yielding insights into the regulatory mechanisms of complex cell signaling pathways.

An overview of existing modeling tools making use of model checking in the analysis of biochemical networks

Model checking is a well-established technique for automatically verifying complex systems. Recently, model checkers have appeared in computer tools for the analysis of biochemical (and gene regulatory) networks. We survey several such tools to assess the potential of model checking in computational biology. Next, our overview focuses on direct applications of existing model checkers, as well as on algorithms for biochemical network analysis influenced by model checking, such as those using binary decision diagrams (BDDs) or Boolean-satisfiability solvers. We conclude with advantages and drawbacks of model checking for the analysis of biochemical networks.

Competing G protein-coupled receptor kinases balance G protein and β-arrestin signaling

The molecular mechanisms and hidden dynamics governing ERK activation by the angiotensin II type 1A receptor are studied and deciphered, revealing a signal balancing mechanism that is found to be relevant to a range of other seven transmembrane receptors.

An ODE-based dynamical model of ERK activation by the prototypical angiotensin II type-1A seven transmembrane receptor has been built and validated.

In order to deal with a limited number of experimental read-outs, unknown parameters have been inferred by simultaneously fitting control and perturbed conditions.

In addition to its well-established function in G-protein uncoupling, G protein-coupled receptor kinase 2 has been shown to exert a strong negative effect on β-arrestin-dependent signaling and by doing so, to balance G-protein and β-arrestin signaling.

This novel function of G protein-coupled receptor kinase 2 has also been evidenced in primary vascular smooth muscle cells naturally expressing the AT_1AR and in HEK293 cells expressing other 7TMRs.

Seven-transmembrane receptors (7TMRs) are involved in nearly all aspects of chemical communications and represent major drug targets. 7TMRs transmit their signals not only via heterotrimeric G proteins but also through β-arrestins, whose recruitment to the activated receptor is regulated by G protein-coupled receptor kinases (GRKs). In this paper, we combined experimental approaches with computational modeling to decipher the molecular mechanisms as well as the hidden dynamics governing extracellular signal-regulated kinase (ERK) activation by the angiotensin II type 1A receptor (AT_1AR) in human embryonic kidney (HEK)293 cells. We built an abstracted ordinary differential equations (ODE)-based model that captured the available knowledge and experimental data. We inferred the unknown parameters by simultaneously fitting experimental data generated in both control and perturbed conditions. We demonstrate that, in addition to its well-established function in the desensitization of G-protein activation, GRK2 exerts a strong negative effect on β-arrestin-dependent signaling through its competition with GRK5 and 6 for receptor phosphorylation. Importantly, we experimentally confirmed the validity of this novel GRK2-dependent mechanism in both primary vascular smooth muscle cells naturally expressing the AT_1AR, and HEK293 cells expressing other 7TMRs.

Quantification of degeneracy in biological systems for characterization of functional interactions between modules

There is an evolutionary advantage in having multiple components with overlapping functionality (i.e degeneracy) in organisms. While theoretical considerations of degeneracy have been well established in neural networks using information theory, the same concepts have not been developed for differential systems, which form the basis of many biochemical reaction network descriptions in systems biology. Here we establish mathematical definitions of degeneracy, complexity and robustness that allow for the quantification of these properties in a system. By exciting a dynamical system with noise, the mutual information associated with a selected observable output and the interacting subspaces of input components can be used to define both complexity and degeneracy. The calculation of degeneracy in a biological network is a useful metric for evaluating features such as the sensitivity of a biological network to environmental evolutionary pressure. Using a two-receptor signal transduction network, we find that redundant components will not yield high degeneracy whereas compensatory mechanisms established by pathway crosstalk will. This form of analysis permits interrogation of large-scale differential systems for non-identical, functionally equivalent features that have evolved to maintain homeostasis during disruption of individual components.

On parameter synthesis by parallel model checking

An important problem in current computational systems biology is to analyze models of biological systems dynamics under parameter uncertainty. This paper presents a novel algorithm for parameter synthesis based on parallel model checking. The algorithm is conceptually universal with respect to the modeling approach employed. We introduce the algorithm, show its scalability, and examine its applicability on several biological models.

Robustness analysis and behavior discrimination in enzymatic reaction networks

Characterizing the behavior and robustness of enzymatic networks with numerous variables and unknown parameter values is a major challenge in biology, especially when some enzymes have counter-intuitive properties or switch-like behavior between activation and inhibition. In this paper, we propose new methodological and tool-supported contributions, based on the intuitive formalism of temporal logic, to express in a rigorous manner arbitrarily complex dynamical properties. Our multi-step analysis allows efficient sampling of the parameter space in order to define feasible regions in which the model exhibits imposed or experimentally observed behaviors. In a first step, an algorithmic methodology involving sensitivity analysis is conducted to determine bifurcation thresholds for a limited number of model parameters or initial conditions. In a second step, this boundary detection is supplemented by a global robustness analysis, based on quasi-Monte Carlo approach that takes into account all model parameters. We apply this method to a well-documented enzymatic reaction network describing collagen proteolysis by matrix metalloproteinase MMP2 and membrane type 1 metalloproteinase (MT1-MMP) in the presence of tissue inhibitor of metalloproteinase TIMP2. For this model, our method provides an extended analysis and quantification of network robustness toward paradoxical TIMP2 switching activity between activation or inhibition of MMP2 production. Further implication of our approach is illustrated by demonstrating and analyzing the possible existence of oscillatory behaviors when considering an extended open configuration of the enzymatic network. Notably, we construct bifurcation diagrams that specify key parameters values controlling the co-existence of stable steady and non-steady oscillatory proteolytic dynamics.

Applications of a formal approach to decipher discrete genetic networks

Background

A growing demand for tools to assist the building and analysis of biological networks exists in systems biology. We argue that the use of a formal approach is relevant and applicable to address questions raised by biologists about such networks. The behaviour of these systems being complex, it is essential to exploit efficiently every bit of experimental information. In our approach, both the evolution rules and the partial knowledge about the structure and the behaviour of the network are formalized using a common constraint-based language.

Results

In this article our formal and declarative approach is applied to three biological applications. The software environment that we developed allows to specifically address each application through a new class of biologically relevant queries. We show that we can describe easily and in a formal manner the partial knowledge about a genetic network. Moreover we show that this environment, based on a constraint algorithmic approach, offers a wide variety of functionalities, going beyond simple simulations, such as proof of consistency, model revision, prediction of properties, search for minimal models relatively to specified criteria.

Conclusions

The formal approach proposed here deeply changes the way to proceed in the exploration of genetic and biochemical networks, first by avoiding the usual trial-and-error procedure, and second by placing the emphasis on sets of solutions, rather than a single solution arbitrarily chosen among many others. Last, the constraint approach promotes an integration of model and experimental data in a single framework.