Joint Modeling of Multiple Social Networks to Elucidate Primate Social Dynamics: I. Maximum Entropy Principle and Network-Based Interactions

In a complex behavioral system, such as an animal society, the dynamics of the system as a whole represent the synergistic interaction among multiple aspects of the society. We constructed multiple single-behavior social networks for the purpose of approximating from multiple aspects a single complex behavioral system of interest: rhesus macaque society. Instead of analyzing these networks individually, we describe a new method for jointly analyzing them in order to gain comprehensive understanding about the system dynamics as a whole. This method of jointly modeling multiple networks becomes valuable analytical tool for studying the complex nature of the interaction among multiple aspects of any system. Here we develop a bottom-up, iterative modeling approach based upon the maximum entropy principle. This principle is applied to a multi-dimensional link-based distributional framework, which is derived by jointly transforming the multiple directed behavioral social network data, for extracting patterns of synergistic inter-behavioral relationships. Using a rhesus macaque group as a model system, we jointly modeled and analyzed four different social behavioral networks at two different time points (one stable and one unstable) from a rhesus macaque group housed at the California National Primate Research Center (CNPRC). We report and discuss the inter-behavioral dynamics uncovered by our joint modeling approach with respect to social stability.     

A Markov chain model for N-linked protein glycosylation – towards a low-parameter tool for model-driven glycoengineering

Glycosylation is a critical quality attribute of most recombinant biotherapeutics. Consequently, drug development requires careful control of glycoforms to meet bioactivity and biosafety requirements. However, glycoengineering can be extraordinarily difficult given the complex reaction networks underlying glycosylation and the vast number of different glycans that can be synthesized in a host cell. Computational modeling offers an intriguing option to rationally guide glycoengineering, but the high parametric demands of current modeling approaches pose challenges to their application. Here we present a novel low-parameter approach to describe glycosylation using flux-balance and Markov chain modeling. The model recapitulates the biological complexity of glycosylation, but does not require user-provided kinetic information. We use this method to predict and experimentally validate glycoprofiles on EPO, IgG as well as the endogenous secretome following glycosyltransferase knock-out in different Chinese hamster ovary (CHO) cell lines. Our approach offers a flexible and user-friendly platform that can serve as a basis for powerful computational engineering efforts in mammalian cell factories for biopharmaceutical production.     

The systems biology of signaling pathways

Signaling networks play the central role in the regulation of processes in a single cell and in the entire body. A recent breakthrough in technologies for systems biology, which combine experimental and mathematical methods, permits scientists to model signaling pathways in an individual cell and in cell populations. This approach provides new information on mechanisms that regulate a variety of biological processes. Here we discuss the mathematical formalisms that are applied to signaling pathway modeling and relevant experimental methods.     

Modeling of Hysteresis in Gene Regulatory Networks

Hysteresis, observed in many gene regulatory networks, has a pivotal impact on biological systems, which enhances the robustness of cell functions. In this paper, a general model is proposed to describe the hysteretic gene regulatory network by combining the hysteresis component and the transient dynamics. The Bouc-Wen hysteresis model is modified to describe the hysteresis component in the mammalian gene regulatory networks. Rigorous mathematical analysis on the dynamical properties of the model is presented to ensure the bounded-input-bounded-output (BIBO) stability and demonstrates that the original Bouc-Wen model can only generate a clockwise hysteresis loop while the modified model can describe both clockwise and counter clockwise hysteresis loops. Simulation studies have shown that the hysteresis loops from our model are consistent with the experimental observations in three mammalian gene regulatory networks and two E.coli gene regulatory networks, which demonstrate the ability and accuracy of the mathematical model to emulate natural gene expression behavior with hysteresis. A comparison study has also been conducted to show that this model fits the experiment data significantly better than previous ones in the literature. The successful modeling of the hysteresis in all the five hysteretic gene regulatory networks suggests that the new model has the potential to be a unified framework for modeling hysteresis in gene regulatory networks and provide better understanding of the general mechanism that drives the hysteretic function.     

Exploring biological network structure using exponential random graph models

MOTIVATION: The functioning of biological networks depends in large part on their complex underlying structure. When studying their systemic nature many modeling approaches focus on identifying simple, but prominent, structural components, as such components are easier to understand, and, once identified, can be used as building blocks to succinctly describe the network. RESULTS: In recent social network studies, exponential random graph models have been used extensively to model global social network structure as a function of their 'local features'. Starting from those studies, we describe the exponential random graph models and demonstrate their utility in modeling the architecture of biological networks as a function of the prominence of local features. We argue that the flexibility, in terms of the number of available local feature choices, and scalability, in terms of the network sizes, make this approach ideal for statistical modeling of biological networks. We illustrate the modeling on both genetic and metabolic networks and provide a novel way of classifying biological networks based on the prevalence of their local features.     

Reverse-engineering of gene networks for regulating early blood development from single-cell measurements

Background

Recent advances in omics technologies have raised great opportunities to study large-scale regulatory networks inside the cell. In addition, single-cell experiments have measured the gene and protein activities in a large number of cells under the same experimental conditions. However, a significant challenge in computational biology and bioinformatics is how to derive quantitative information from the single-cell observations and how to develop sophisticated mathematical models to describe the dynamic properties of regulatory networks using the derived quantitative information.

Methods

This work designs an integrated approach to reverse-engineer gene networks for regulating early blood development based on singel-cell experimental observations. The wanderlust algorithm is initially used to develop the pseudo-trajectory for the activities of a number of genes. Since the gene expression data in the developed pseudo-trajectory show large fluctuations, we then use Gaussian process regression methods to smooth the gene express data in order to obtain pseudo-trajectories with much less fluctuations. The proposed integrated framework consists of both bioinformatics algorithms to reconstruct the regulatory network and mathematical models using differential equations to describe the dynamics of gene expression.

Results

The developed approach is applied to study the network regulating early blood cell development. A graphic model is constructed for a regulatory network with forty genes and a dynamic model using differential equations is developed for a network of nine genes. Numerical results suggests that the proposed model is able to match experimental data very well. We also examine the networks with more regulatory relations and numerical results show that more regulations may exist. We test the possibility of auto-regulation but numerical simulations do not support the positive auto-regulation. In addition, robustness is used as an importantly additional criterion to select candidate networks.

Conclusion

The research results in this work shows that the developed approach is an efficient and effective method to reverse-engineer gene networks using single-cell experimental observations.

     

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.     

A flexible state-space approach for the modeling of metabolic networks II: advanced interrogation of hybridoma metabolism

Having previously introduced the mathematical framework of topological metabolic analysis (TMA) - a novel optimization-based technique for modeling metabolic networks of arbitrary size and complexity - we demonstrate how TMA facilitates unique methods of metabolic interrogation. With the aid of several hybridoma metabolic investigations as case-studies (Bonarius et al., 1995, 1996, 2001), we first establish that the TMA framework identifies biologically important aspects of the metabolic network under investigation. We also show that the use of a structured weighting approach within our objective provides a substantial modeling benefit over an unstructured, uniform, weighting approach. We then illustrate the strength of TAM as an advanced interrogation technique, first by using TMA to prove the existence of (and to quantitatively describe) multiple topologically distinct configurations of a metabolic network that each optimally model a given set of experimental observations. We further show that such alternate topologies are indistinguishable using existing stoichiometric modeling techniques, and we explain the biological significance of the topological variables appearing within our model. By leveraging the manner in which TMA implements metabolite inputs and outputs, we also show that metabolites whose possible metabolic fates are inadequately described by a given network reconstruction can be quickly identified. Lastly, we show how the use of the TMA aggregate objective function (AOF) permits the identification of modeling solutions that can simultaneously consider experimental observations, underlying biological motivations, or even purely engineering- or design-based goals.     

Computational approaches for modeling regulatory cellular networks

Cellular components interact with each other to form networks that process information and evoke biological responses. A deep understanding of the behavior of these networks requires the development and analysis of mathematical models. In this article, different types of mathematical representations for modeling signaling networks are described, and the advantages and disadvantages of each type are discussed. Two experimentally well-studied signaling networks are then used as examples to illustrate the insight that could be gained through modeling. Finally, the modeling approach is expanded to describe how signaling networks might regulate cellular machines and evoke phenotypic behaviors.     

A flexible state-space approach for the modeling of metabolic networks I: development of mathematical methods

We introduce a novel, flexible, optimization-based mathematical framework for the modeling of arbitrarily complex metabolic networks: topological metabolic analysis (TMA). The framework is adapted from state-space approaches used by Manousiouthakis and co-workers for the representation of complex heat- and mass-exchanger networks. We offer a thorough discussion of the mathematics and general theory underlying the framework, and discuss certain mathematical advantages of our modeling representation in comparison with other commonly used techniques (MFA and FBA). We employ a novel aggregate objective function for use with our basic constraint model, including a generalized least-squares term (for fitting available experimental measurements) and a linear design term (for representing biological or engineering goals). Using a case-study taken from recent literature (McKinlay et al., 2007), we demonstrate (among other benefits) the ability of this objective to identify alternate distinct-yet-equally optimal solutions for a given modeling problem. We also show that these solutions, obtained using only external metabolite uptake and secretion measurements, provide useful biological insights and compare favorably with solutions obtained on the basis of (13)C isotope-tracing data.     

Simple network motifs can capture key characteristics of the floral transition in Arabidopsis

The floral transition is a key decision during plant development. While different species have evolved diverse pathways to respond to different environmental cues to flower in the correct season, key properties such as irreversibility and robustness to fluctuating signals appear to be conserved. We have used mathematical modeling to demonstrate how minimal regulatory networks of core components are sufficient to capture these behaviors. Simplified models inevitably miss finer details of the biological system, yet they provide a tractable route to understanding the overall system behavior. We combined models with experimental data to qualitatively reproduce characteristics of the floral transition and to quantitatively scale the network to fit with available leaf numbers. Our study highlights the value of pursuing an iterative approach combining modeling with experimental work to capture key features of complex systems.     

Modeling regulatory networks at Virginia Tech

The life of a cell is governed by the physicochemical properties of a complex network of interacting macromolecules (primarily genes and proteins). Hence, a full scientific understanding of and rational engineering approach to cell physiology require accurate mathematical models of the spatial and temporal dynamics of these macromolecular assemblies, especially the networks involved in integrating signals and regulating cellular responses. The Virginia Tech Consortium is involved in three specific goals of DARPA's computational biology program (Bio-COMP): to create effective software tools for modeling gene-protein-metabolite networks, to employ these tools in creating a new generation of realistic models, and to test and refine these models by well-conceived experimental studies. The special emphasis of this group is to understand the mechanisms of cell cycle control in eukaryotes (yeast cells and frog eggs). The software tools developed at Virginia Tech are designed to meet general requirements of modeling regulatory networks and are collected in a problem-solving environment called JigCell.     

Modeling and simulation of biological systems with stochasticity

Mathematical modeling is a powerful approach for understanding the complexity of biological systems. Recently, several successful attempts have been made for simulating complex biological processes like metabolic pathways, gene regulatory networks and cell signaling pathways. The pathway models have not only generated experimentally verifiable hypothesis but have also provided valuable insights into the behavior of complex biological systems. Many recent studies have confirmed the phenotypic variability of organisms to an inherent stochasticity that operates at a basal level of gene expression. Due to this reason, development of novel mathematical representations and simulations algorithms are critical for successful modeling efforts in biological systems. The key is to find a biologically relevant representation for each representation. Although mathematically rigorous and physically consistent, stochastic algorithms are computationally expensive, they have been successfully used to model probabilistic events in the cell. This paper offers an overview of various mathematical and computational approaches for modeling stochastic phenomena in cellular systems.     

Symbolic modeling of epistasis

The workhorse of modern genetic analysis is the parametric linear model. The advantages of the linear modeling framework are many and include a mathematical understanding of the model fitting process and ease of interpretation. However, an important limitation is that linear models make assumptions about the nature of the data being modeled. This assumption may not be realistic for complex biological systems such as disease susceptibility where nonlinearities in the genotype to phenotype mapping relationship that result from epistasis, plastic reaction norms, locus heterogeneity, and phenocopy, for example, are the norm rather than the exception. We have previously developed a flexible modeling approach called symbolic discriminant analysis (SDA) that makes no assumptions about the patterns in the data. Rather, SDA lets the data dictate the size, shape, and complexity of a symbolic discriminant function that could include any set of mathematical functions from a list of candidates supplied by the user. Here, we outline a new five step process for symbolic model discovery that uses genetic programming (GP) for coarse-grained stochastic searching, experimental design for parameter optimization, graphical modeling for generating expert knowledge, and estimation of distribution algorithms for fine-grained stochastic searching. Finally, we introduce function mapping as a new method for interpreting symbolic discriminant functions. We show that function mapping when combined with measures of interaction information facilitates statistical interpretation by providing a graphical approach to decomposing complex models to highlight synergistic, redundant, and independent effects of polymorphisms and their composite functions. We illustrate this five step SDA modeling process with a real case-control dataset.     

Visinets: A Web-Based Pathway Modeling and Dynamic Visualization Tool

In this report we describe a novel graphically oriented method for pathway modeling and a software package that allows for both modeling and visualization of biological networks in a user-friendly format. The Visinets mathematical approach is based on causal mapping (CMAP) that has been fully integrated with graphical interface. Such integration allows for fully graphical and interactive process of modeling, from building the network to simulation of the finished model. To test the performance of Visinets software we have applied it to: a) create executable EGFR-MAPK pathway model using an intuitive graphical way of modeling based on biological data, and b) translate existing ordinary differential equation (ODE) based insulin signaling model into CMAP formalism and compare the results. Our testing fully confirmed the potential of the CMAP method for broad application for pathway modeling and visualization and, additionally, showed significant advantage in computational efficiency. Furthermore, we showed that Visinets web-based graphical platform, along with standardized method of pathway analysis, may offer a novel and attractive alternative for dynamic simulation in real time for broader use in biomedical research. Since Visinets uses graphical elements with mathematical formulas hidden from the users, we believe that this tool may be particularly suited for those who are new to pathway modeling and without the in-depth modeling skills often required when using other software packages.     

Experimental and mathematical approaches to modeling plant metabolic networks

To support their sessile and autotrophic lifestyle higher plants have evolved elaborate networks of metabolic pathways. Dynamic changes in these metabolic networks are among the developmental forces underlying the functional differentiation of organs, tissues and specialized cell types. They are also important in the various interactions of a plant with its environment. Further complexity is added by the extensive compartmentation of the various interconnected metabolic pathways in plants. Thus, although being used widely for assessing the control of metabolic flux in microbes, mathematical modeling approaches that require steady-state approximations are of limited utility for understanding complex plant metabolic networks. However, considerable progress has been made when manageable metabolic subsystems were studied. In this article, we will explain in general terms and using simple examples the concepts underlying stoichiometric modeling (metabolic flux analysis and metabolic pathway analysis) and kinetic approaches to modeling (including metabolic control analysis as a special case). Selected studies demonstrating the prospects of these approaches, or combinations of them, for understanding the control of flux through particular plant pathways are discussed. We argue that iterative cycles of (dry) mathematical modeling and (wet) laboratory testing will become increasingly important for simulating the distribution of flux in plant metabolic networks and deriving rational experimental designs for metabolic engineering efforts.