Robustness and state-space structure of Boolean gene regulatory models

Robustness to perturbation is an important characteristic of genetic regulatory systems, but the relationship between robustness and model dynamics has not been clearly quantified. We propose a method for quantifying both robustness and dynamics in terms of state-space structures, for Boolean models of genetic regulatory systems. By investigating existing models of the Drosophila melanogaster segment polarity network and the Saccharomyces cerevisiae cell-cycle network, we show that the structure of attractor basins can yield insight into the underlying decision making required of the system, and also the way in which the system maximises its robustness. In particular, gene networks implementing decisions based on a few genes have simple state-space structures, and their attractors are robust by virtue of their simplicity. Gene networks with decisions that involve many interacting genes have correspondingly more complicated state-space structures, and robustness cannot be achieved through the structure of the attractor basins, but is achieved by larger attractor basins that dominate the state space. These different types of robustness are demonstrated by the two models: the D. melanogaster segment polarity network is robust due to simple attractor basins that implement decisions based on spatial signals; the S. cerevisiae cell-cycle network has a complicated state-space structure, and is robust only due to a giant attractor basin that dominates the state space.     

The dynamics of T-cell fratricide: application of a robust approach to mathematical modelling in immunology

Fratricide between CD8(+) T lymphocytes is known to occur in HTLV-I and possibly HSV-1 and HIV-1 infection. However it is not known what effect, if any, T-cell fratricide has on the course of infection. Here we present simple mathematical techniques to investigate T-cell fratricide with particular reference to HTLV-I infection. Using a general model we predict the qualitative and quantitative effect of fratricide on HTLV-I equilibrium proviral load. We also investigate the effect of fratricide on the probability of viral clearance. We show that, surprisingly, fratricide can lead either to an increase or a decrease in equilibrium proviral load. We derive the conditions necessary for fratricide to cause a decrease in load and deduce that, for the five HTLV-I-positive patients considered here, fratricide has probably caused an increase in equilibrium load. We also estimate the percentage increase in load that is attributable to fratricide and determine the parameters that should be measured in order to improve this estimate. Finally, we show that fratricide reduces the probability of viral clearance. Mathematical modelling of HTLV-I infection, as is often the case in biology, is severely hampered by a lack of experimental data. Consequently it is difficult to know what functional form a model should take. The behaviour of complex nonlinear systems is highly model-dependent. Predictions based on theoretical models are therefore sensitive to the choice of model; this is a very severe problem that undermines and limits the success of the application of mathematics to immunology. In this paper we reduce the model dependency of the results in two ways-by considering (analytically) a general model with a minimal number of assumptions and, where this is not possible, by checking (numerically) that a wide range of models yield the same results. We therefore begin to develop two practical methods for dealing with the problem of robustness in mathematical models of the immune system.     

A mathematical framework for inferring connectivity in probabilistic neuronal networks

We describe an approach for determining causal connections among nodes of a probabilistic network even when many nodes remain unobservable. The unobservable nodes introduce ambiguity into the estimate of the causal structure. However, in some experimental contexts, such as those commonly used in neuroscience, this ambiguity is present even without unobservable nodes. The analysis is presented in terms of a point process model of a neuronal network, though the approach can be generalized to other contexts. The analysis depends on the existence of a model that captures the relationship between nodal activity and a set of measurable external variables. The mathematical framework is sufficiently general to allow a large class of such models. The results are modestly robust to deviations from model assumptions, though additional validation methods are needed to assess the success of the results.     

A computational model of hemodynamic parameters in cortical capillary networks

The analysis of hemodynamic parameters and functional reactivity of cerebral capillaries is still controversial. To assess the hemodynamic parameters in the cortical capillary network, a generic model was created using 2D voronoi tessellation in which each edge represents a capillary segment. This method is capable of creating an appropriate generic model of cerebral capillary network relating to each part of the brain cortex because the geometric model is able to vary the capillary density. The modeling presented here is based on morphometric parameters extracted from physiological data of the human cortex. The pertinent hemodynamic parameters were obtained by numerical simulation based on effective blood viscosity as a function of hematocrit and microvessel diameter, phase separation and plasma skimming effects. The hemodynamic parameters of capillary networks with two different densities (consistent with the variation of the morphometric data in the human cortical capillary network) were analyzed. The results show pertinent hemodynamic parameters for each model. The heterogeneity (coefficient variation) and the mean value of hematocrits, flow rates and velocities of the both network models were specified. The distributions of blood flow throughout the both models seem to confirm the hypothesis in which all capillaries in a cortical network are recruited at rest (normal condition). The results also demonstrate a discrepancy of the network resistance between two models, which are derived from the difference in the number density of capillary segments between the models.     

Elementary network reconstruction: A framework for the analysis of regulatory networks in biological systems

Complexity of regulatory networks arises from the high degree of interaction between network components such as DNA, RNA, proteins, and metabolites. We have developed a modeling tool, elementary network reconstruction (ENR), to characterize these networks. ENR is a knowledge-driven, steady state, deterministic, quantitative modeling approach based on linear perturbation theory. In ENR we demonstrate a novel means of expressing control mechanisms by way of dimensionless steady state gains relating input and output variables, which are purely in terms of species abundances (extensive variables). As a result of systematic enumeration of network species in n×n matrix, the two properties of linear perturbation are manifested in graphical representations: transitive property is evident in a special L-shape structure, and additive property is evident in multiple L-shape structures arriving at the same matrix cell. Upon imposing mechanistic (lowest-level) gains, network self-assembly through transitive and additive properties results in elucidation of inherent topology and explicit cataloging of higher level gains, which in turn can be used to predict perturbation results. Application of ENR to the regulatory network behind carbon catabolite repression in Escherichia coli is presented. Through incorporation of known molecular mechanisms governing transient and permanent repressions, the ENR model correctly predicts several key features of this regulatory network, including a 50% downshift in intracellular cAMP level upon exposure to glucose. Since functional genomics studies are mainly concerned with redistribution of species abundances in perturbed systems, ENR could be exploited in the system-level analysis of biological systems.     

Dynamical analysis of cellular networks based on the Green's function matrix

The complexity of cellular networks often limits human intuition in understanding functional regulations in a cell from static network diagrams. To this end, mathematical models of ordinary differential equations (ODEs) have commonly been used to simulate dynamical behavior of cellular networks, to which a quantitative model analysis can be applied in order to gain biological insights. In this paper, we introduce a dynamical analysis based on the use of Green's function matrix (GFM) as sensitivity coefficients with respect to initial concentrations. In contrast to the classical (parametric) sensitivity analysis, the GFM analysis gives a dynamical, molecule-by-molecule insight on how system behavior is accomplished and complementarily how (impulse) signal propagates through the network. The knowledge gained will have application from model reduction and validation to drug discovery research in identifying potential drug targets, studying drug efficacy and specificity, and optimizing drug dosing and timing. The efficacy of the method is demonstrated through applications to common network motifs and a Fas-induced programmed cell death model in Jurkat T cell line.     

Analysis of Brassica napus dehydrins and their Co-Expression regulatory networks in relation to cold stress

Dehydrins (DHNs) are plant specific cold and drought stress-responsive proteins that belong to late embryogenesis abundant (LEA) protein families. B. napus DHNs (BnDHNs) were computationally analyzed to establish gene regulatory- and protein-protein interaction networks. Promoter analyses suggested functionality of phytohormones in BnDHNs gene network. The relative expressions of some BnDHNs were analyzed using qRT-PCR in seedling leaves of both cold-tolerant (Zarfam) and -sensitive (Sari Gul) canola treated/untreated by cold. Our expression data were indicative of the importance of BnDHNs in cold tolerance in Zarfam. BnDHNs were classified into three classes according to the expression pattern. Moreover, expression of three BnDHN types, SK_n (BnLEA10 and BnLEA18), Y_nK_n (BnLEA90) and Y_nSK_n (BnLEA104) were significantly high in the tolerant cultivar at 12 h of cold treatment. Our findings put forward the possibility of considering these genes as screening biomarker to determine cold-tolerant breeding lines; something that needs to be further corroborated. Furthermore, these genes may have some implications in developing such tolerant lines via transgenesis.     

Global dynamics of two-compartment models for cell production systems with regulatory mechanisms

We present a global stability analysis of two-compartment models of a hierarchical cell production system with a nonlinear regulatory feedback loop. The models describe cell differentiation processes with the stem cell division rate or the self-renewal fraction regulated by the number of mature cells. The two-compartment systems constitute a basic version of the multicompartment models proposed recently by Marciniak-Czochra and collaborators [25] to investigate the dynamics of the hematopoietic system. Using global stability analysis, we compare different regulatory mechanisms. For both models, we show that there exists a unique positive equilibrium that is globally asymptotically stable if and only if the respective reproduction numbers exceed one. The proof is based on constructing Lyapunov functions, which are appropriate to handle the specific nonlinearities of the model. Additionally, we propose a new model to test biological hypothesis on the regulation of the fraction of differentiating cells. We show that such regulatory mechanism is incapable of maintaining homeostasis and leads to unbounded cell growth. Potential biological implications are discussed.     

A network-based approach for resistance transmission in bacterial populations

Horizontal transfer of mobile genetic elements (conjugation) is an important mechanism whereby resistance is spread through bacterial populations. The aim of our work is to develop a mathematical model that quantitatively describes this process, and to use this model to optimize antimicrobial dosage regimens to minimize resistance development. The bacterial population is conceptualized as a compartmental mathematical model to describe changes in susceptible, resistant, and transconjugant bacteria over time. This model is combined with a compartmental pharmacokinetic model to explore the effect of different plasma drug concentration profiles. An agent-based simulation tool is used to account for resistance transfer occurring when two bacteria are adjacent or in close proximity. In addition, a non-linear programming optimal control problem is introduced to minimize bacterial populations as well as the drug dose. Simulation and optimization results suggest that the rapid death of susceptible individuals in the population is pivotal in minimizing the number of transconjugants in a population. This supports the use of potent antimicrobials that rapidly kill susceptible individuals and development of dosage regimens that maintain effective antimicrobial drug concentrations for as long as needed to kill off the susceptible population. Suggestions are made for experiments to test the hypotheses generated by these simulations.     

Theoretical study for regulatory property of scaffold protein on MAPK cascade: A qualitative modeling

An integrated mathematical model, which incorporates scaffold proteins into a mitogen-activated protein kinases cascade, is constructed. By employing Monte Carlo simulation, regulatory property of scaffold protein on signaling ability for the mitogen-activated protein kinases cascade is investigated theoretically. It is found that (i) scaffold binding increases signal amplification if dephosphorylation is slow and decreases amplification if dephosphorylation is rapid. Also, increasing the number of scaffold decreases amplification if dephosphorylation is slow. (ii) The scaffold number can control the timing of kinase activation so that the time flexibility of signaling is enhanced. (iii) It is observed that for slow dephosphorylation case, scaffolds decrease the sharpness of the dose–response curves. While for fast dephosphorylation case, increasing scaffold number decreases the height of response, but the shape of graded response is sustained. Furthermore, the underlying mechanism and the correlation of our results with real biological systems are clarified.     

Perturbation analysis analyzed—mathematical modeling of intact and perturbed gene regulatory circuits for animal development

Gene regulatory networks for animal development are the underlying mechanisms controlling cell fate specification and differentiation. The architecture of gene regulatory circuits determines their information processing properties and their developmental function. It is a major task to derive realistic network models from exceedingly advanced high throughput experimental data. Here we use mathematical modeling to study the dynamics of gene regulatory circuits to advance the ability to infer regulatory connections and logic function from experimental data. This study is guided by experimental methodologies that are commonly used to study gene regulatory networks that control cell fate specification. We study the effect of a perturbation of an input on the level of its downstream genes and compare between the cis-regulatory execution of OR and AND logics. Circuits that initiate gene activation and circuits that lock on the expression of genes are analyzed. The model improves our ability to analyze experimental data and construct from it the network topology. The model also illuminates information processing properties of gene regulatory circuits for animal development.     

Signaling networks in Leishmania macrophages deciphered through integrated systems biology: a mathematical modeling approach

Network of signaling proteins and functional interaction between the infected cell and the leishmanial parasite, though are not well understood, may be deciphered computationally by reconstructing the immune signaling network. As we all know signaling pathways are well-known abstractions that explain the mechanisms whereby cells respond to signals, collections of pathways form networks, and interactions between pathways in a network, known as cross-talk, enables further complex signaling behaviours. In silico perturbations can help identify sensitive crosstalk points in the network which can be pharmacologically tested. In this study, we have developed a model for immune signaling cascade in leishmaniasis and based upon the interaction analysis obtained through simulation, we have developed a model network, between four signaling pathways i.e., CD14, epidermal growth factor (EGF), tumor necrotic factor (TNF) and PI3 K mediated signaling. Principal component analysis of the signaling network showed that EGF and TNF pathways can be potent pharmacological targets to curb leishmaniasis. The approach is illustrated with a proposed workable model of epidermal growth factor receptor (EGFR) that modulates the immune response. EGFR signaling represents a critical junction between inflammation related signal and potent cell regulation machinery that modulates the expression of cytokines.     

A novel approach for large-scale polypeptide folding based on elastic networks using continuous optimization

We present a new computationally efficient method for large-scale polypeptide folding using coarse-grained elastic networks and gradient-based continuous optimization techniques. The folding is governed by minimization of energy based on Miyazawa-Jernigan contact potentials. Using this method we are able to substantially reduce the computation time on ordinary desktop computers for simulation of polypeptide folding starting from a fully unfolded state. We compare our results with available native state structures from Protein Data Bank (PDB) for a few de-novo proteins and two natural proteins, Ubiquitin and Lysozyme. Based on our simulations we are able to draw the energy landscape for a small de-novo protein, Chignolin. We also use two well known protein structure prediction software, MODELLER and GROMACS to compare our results. In the end, we show how a modification of normal elastic network model can lead to higher accuracy and lower time required for simulation.     

Attractor analysis of asynchronous Boolean models of signal transduction networks

Prior work on the dynamics of Boolean networks, including analysis of the state space attractors and the basin of attraction of each attractor, has mainly focused on synchronous update of the nodes’ states. Although the simplicity of synchronous updating makes it very attractive, it fails to take into account the variety of time scales associated with different types of biological processes. Several different asynchronous update methods have been proposed to overcome this limitation, but there have not been any systematic comparisons of the dynamic behaviors displayed by the same system under different update methods. Here we fill this gap by combining theoretical analysis such as solution of scalar equations and Markov chain techniques, as well as numerical simulations to carry out a thorough comparative study on the dynamic behavior of a previously proposed Boolean model of a signal transduction network in plants. Prior evidence suggests that this network admits oscillations, but it is not known whether these oscillations are sustained. We perform an attractor analysis of this system using synchronous and three different asynchronous updating schemes both in the case of the unperturbed (wild-type) and perturbed (node-disrupted) systems. This analysis reveals that while the wild-type system possesses an update-independent fixed point, any oscillations eventually disappear unless strict constraints regarding the timing of certain processes and the initial state of the system are satisfied. Interestingly, in the case of disruption of a particular node all models lead to an extended attractor. Overall, our work provides a roadmap on how Boolean network modeling can be used as a predictive tool to uncover the dynamic patterns of a biological system under various internal and environmental perturbations.