Gene Regulatory Network Modeling of Macrophage Differentiation Corroborates the Continuum Hypothesis of Polarization States.

Macrophages derived from monocyte precursors undergo specific polarization processes which are influenced by the local tissue environment: classically activated (M1) macrophages, with a pro-inflammatory activity and a role of effector cells in Th1 cellular immune responses, and alternatively activated (M2) macrophages, with anti-inflammatory functions and involved in immunosuppression and tissue repair. At least three different subsets of M2 macrophages, namely, M2a, M2b, and M2c, are characterized in the literature based on their eliciting signals. The activation and polarization of macrophages is achieved through many, often intertwined, signaling pathways. To describe the logical relationships among the genes involved in macrophage polarization, we used a computational modeling methodology, namely, logical (Boolean) modeling of gene regulation. We integrated experimental data and knowledge available in the literature to construct a logical network model for the gene regulation driving macrophage polarization to the M1, M2a, M2b, and M2c phenotypes. Using the software GINsim and BoolNet, we analyzed the network dynamics under different conditions and perturbations to understand how they affect cell polarization. Dynamic simulations of the network model, enacting the most relevant biological conditions, showed coherence with the observed behavior of in vivo macrophages. The model could correctly reproduce the polarization toward the four main phenotypes as well as to several hybrid phenotypes, which are known to be experimentally associated to physiological and pathological conditions. We surmise that shifts among different phenotypes in the model mimic the hypothetical continuum of macrophage polarization, with M1 and M2 being the extremes of an uninterrupted sequence of states. Furthermore, model simulations suggest that anti-inflammatory macrophages are resilient to shift back to the pro-inflammatory phenotype.

Statistical ensemble of gene regulatory networks of macrophage differentiation.

BACKGROUND: Macrophages cover a major role in the immune system, being the most plastic cell yielding several key immune functions. METHODS: Here we derived a minimalistic gene regulatory network model for the differentiation of macrophages into the two phenotypes M1 (pro-) and M2 (anti-inflammatory). RESULTS: To test the model, we simulated a large number of such networks as in a statistical ensemble. In other words, to enable the inter-cellular crosstalk required to obtain an immune activation in which the macrophage plays its role, the simulated networks are not taken in isolation but combined with other cellular agents, thus setting up a discrete minimalistic model of the immune system at the microscopic/intracellular (i.e., genetic regulation) and mesoscopic/intercellular scale. CONCLUSIONS: We show that within the mesoscopic level description of cellular interaction and cooperation, the gene regulatory logic is coherent and contributes to the overall dynamics of the ensembles that shows, statistically, the expected behaviour.

A mathematical model of skeletal muscle disease and immune response in the mdx mouse.

Duchenne muscular dystrophy (DMD) is a genetic disease that results in the death of affected boys by early adulthood. The genetic defect responsible for DMD has been known for over 25 years, yet at present there is neither cure nor effective treatment for DMD. During early disease onset, the mdx mouse has been validated as an animal model for DMD and use of this model has led to valuable but incomplete insights into the disease process. For example, immune cells are thought to be responsible for a significant portion of muscle cell death in the mdx mouse; however, the role and time course of the immune response in the dystrophic process have not been well described. In this paper we constructed a simple mathematical model to investigate the role of the immune response in muscle degeneration and subsequent regeneration in the mdx mouse model of Duchenne muscular dystrophy. Our model suggests that the immune response contributes substantially to the muscle degeneration and regeneration processes. Furthermore, the analysis of the model predicts that the immune system response oscillates throughout the life of the mice, and the damaged fibers are never completely cleared.

An algebra-based method for inferring gene regulatory networks.

BACKGROUND: The inference of gene regulatory networks (GRNs) from experimental observations is at the heart of systems biology. This includes the inference of both the network topology and its dynamics. While there are many algorithms available to infer the network topology from experimental data, less emphasis has been placed on methods that infer network dynamics. Furthermore, since the network inference problem is typically underdetermined, it is essential to have the option of incorporating into the inference process, prior knowledge about the network, along with an effective description of the search space of dynamic models. Finally, it is also important to have an understanding of how a given inference method is affected by experimental and other noise in the data used. RESULTS: This paper contains a novel inference algorithm using the algebraic framework of Boolean polynomial dynamical systems (BPDS), meeting all these requirements. The algorithm takes as input time series data, including those from network perturbations, such as knock-out mutant strains and RNAi experiments. It allows for the incorporation of prior biological knowledge while being robust to significant levels of noise in the data used for inference. It uses an evolutionary algorithm for local optimization with an encoding of the mathematical models as BPDS. The BPDS framework allows an effective representation of the search space for algebraic dynamic models that improves computational performance. The algorithm is validated with both simulated and experimental microarray expression profile data. Robustness to noise is tested using a published mathematical model of the segment polarity gene network in Drosophila melanogaster. Benchmarking of the algorithm is done by comparison with a spectrum of state-of-the-art network inference methods on data from the synthetic IRMA network to demonstrate that our method has good precision and recall for the network reconstruction task, while also predicting several of the dynamic patterns present in the network. CONCLUSIONS: Boolean polynomial dynamical systems provide a powerful modeling framework for the reverse engineering of gene regulatory networks, that enables a rich mathematical structure on the model search space. A C++ implementation of the method, distributed under LPGL license, is available, together with the source code, at http://www.paola-vera-licona.net/Software/EARevEng/REACT.html.

Systems modeling of molecular mechanisms controlling cytokine-driven CD4+ T cell differentiation and phenotype plasticity.

Differentiation of CD4+ T cells into effector or regulatory phenotypes is tightly controlled by the cytokine milieu, complex intracellular signaling networks and numerous transcriptional regulators. We combined experimental approaches and computational modeling to investigate the mechanisms controlling differentiation and plasticity of CD4+ T cells in the gut of mice. Our computational model encompasses the major intracellular pathways involved in CD4+ T cell differentiation into T helper 1 (Th1), Th2, Th17 and induced regulatory T cells (iTreg). Our modeling efforts predicted a critical role for peroxisome proliferator-activated receptor gamma (PPARγ) in modulating plasticity between Th17 and iTreg cells. PPARγ regulates differentiation, activation and cytokine production, thereby controlling the induction of effector and regulatory responses, and is a promising therapeutic target for dysregulated immune responses and inflammation. Our modeling efforts predict that following PPARγ activation, Th17 cells undergo phenotype switch and become iTreg cells. This prediction was validated by results of adoptive transfer studies showing an increase of colonic iTreg and a decrease of Th17 cells in the gut mucosa of mice with colitis following pharmacological activation of PPARγ. Deletion of PPARγ in CD4+ T cells impaired mucosal iTreg and enhanced colitogenic Th17 responses in mice with CD4+ T cell-induced colitis. Thus, for the first time we provide novel molecular evidence in vivo demonstrating that PPARγ in addition to regulating CD4+ T cell differentiation also plays a major role controlling Th17 and iTreg plasticity in the gut mucosa.

Probabilistic polynomial dynamical systems for reverse engineering of gene regulatory networks.

Elucidating the structure and/or dynamics of gene regulatory networks from experimental data is a major goal of systems biology. Stochastic models have the potential to absorb noise, account for un-certainty, and help avoid data overfitting. Within the frame work of probabilistic polynomial dynamical systems, we present an algorithm for the reverse engineering of any gene regulatory network as a discrete, probabilistic polynomial dynamical system. The resulting stochastic model is assembled from all minimal models in the model space and the probability assignment is based on partitioning the model space according to the likeliness with which a minimal model explains the observed data. We used this method to identify stochastic models for two published synthetic network models. In both cases, the generated model retains the key features of the original model and compares favorably to the resulting models from other algorithms.

A mathematical framework for agent based models of complex biological networks.

Agent-based modeling and simulation is a useful method to study biological phenomena in a wide range of fields, from molecular biology to ecology. Since there is currently no agreed-upon standard way to specify such models, it is not always easy to use published models. Also, since model descriptions are not usually given in mathematical terms, it is difficult to bring mathematical analysis tools to bear, so that models are typically studied through simulation. In order to address this issue, Grimm et al. proposed a protocol for model specification, the so-called ODD protocol, which provides a standard way to describe models. This paper proposes an addition to the ODD protocol which allows the description of an agent-based model as a dynamical system, which provides access to computational and theoretical tools for its analysis. The mathematical framework is that of algebraic models, that is, time-discrete dynamical systems with algebraic structure. It is shown by way of several examples how this mathematical specification can help with model analysis. This mathematical framework can also accommodate other model types such as Boolean networks and the more general logical models, as well as Petri nets.

Polynomial algebra of discrete models in systems biology.

MOTIVATION: An increasing number of discrete mathematical models are being published in Systems Biology, ranging from Boolean network models to logical models and Petri nets. They are used to model a variety of biochemical networks, such as metabolic networks, gene regulatory networks and signal transduction networks. There is increasing evidence that such models can capture key dynamic features of biological networks and can be used successfully for hypothesis generation. RESULTS: This article provides a unified framework that can aid the mathematical analysis of Boolean network models, logical models and Petri nets. They can be represented as polynomial dynamical systems, which allows the use of a variety of mathematical tools from computer algebra for their analysis. Algorithms are presented for the translation into polynomial dynamical systems. Examples are given of how polynomial algebra can be used for the model analysis. CONTACT: alanavc@vt.edu SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

The dynamics of conjunctive and disjunctive Boolean network models.

For many biological networks, the topology of the network constrains its dynamics. In particular, feedback loops play a crucial role. The results in this paper quantify the constraints that (unsigned) feedback loops exert on the dynamics of a class of discrete models for gene regulatory networks. Conjunctive (resp. disjunctive) Boolean networks, obtained by using only the AND (resp. OR) operator, comprise a subclass of networks that consist of canalyzing functions, used to describe many published gene regulation mechanisms. For the study of feedback loops, it is common to decompose the wiring diagram into linked components each of which is strongly connected. It is shown that for conjunctive Boolean networks with strongly connected wiring diagram, the feedback loop structure completely determines the long-term dynamics of the network. A formula is established for the precise number of limit cycles of a given length, and it is determined which limit cycle lengths can appear. For general wiring diagrams, the situation is much more complicated, as feedback loops in one strongly connected component can influence the feedback loops in other components. This paper provides a sharp lower bound and an upper bound on the number of limit cycles of a given length, in terms of properties of the partially ordered set of strongly connected components.

Algebraic models of biochemical networks.

With the rise of systems biology as an important paradigm in the life sciences and the availability and increasingly good quality of high-throughput molecular data, the role of mathematical models has become central in the understanding of the relationship between structure and function of organisms. This chapter focuses on a particular type of models, so-called algebraic models, which are generalizations of Boolean networks. It provides examples of such models and discusses several available methods to construct such models from high-throughput time course data. One specific such method, Polynome, is discussed in detail.

A systems biology view of cancer.

In order to understand how a cancer cell is functionally different from a normal cell it is necessary to assess the complex network of pathways involving gene regulation, signaling, and cell metabolism, and the alterations in its dynamics caused by the several different types of mutations leading to malignancy. Since the network is typically complex, with multiple connections between pathways and important feedback loops, it is crucial to represent it in the form of a computational model that can be used for a rigorous analysis. This is the approach of systems biology, made possible by new -omics data generation technologies. The goal of this review is to illustrate this approach and its utility for our understanding of cancer. After a discussion of recent progress using a network-centric approach, three case studies related to diagnostics, therapy, and drug development are presented in detail. They focus on breast cancer, B-cell lymphomas, and colorectal cancer. The discussion is centered on key mathematical and computational tools common to a systems biology approach.

The effect of negative feedback loops on the dynamics of boolean networks.

Feedback loops play an important role in determining the dynamics of biological networks. To study the role of negative feedback loops, this article introduces the notion of distance-to-positive-feedback which, in essence, captures the number of independent negative feedback loops in the network, a property inherent in the network topology. Through a computational study using Boolean networks, it is shown that distance-to-positive-feedback has a strong influence on network dynamics and correlates very well with the number and length of limit cycles in the phase space of the network. To be precise, it is shown that, as the number of independent negative feedback loops increases, the number (length) of limit cycles tends to decrease (increase). These conclusions are consistent with the fact that certain natural biological networks exhibit generally regular behavior and have fewer negative feedback loops than randomized networks with the same number of nodes and same connectivity.

A virtual look at Epstein-Barr virus infection: biological interpretations.

The possibility of using computer simulation and mathematical modeling to gain insight into biological and other complex systems is receiving increased attention. However, it is as yet unclear to what extent these techniques will provide useful biological insights or even what the best approach is. Epstein-Barr virus (EBV) provides a good candidate to address these issues. It persistently infects most humans and is associated with several important diseases. In addition, a detailed biological model has been developed that provides an intricate understanding of EBV infection in the naturally infected human host and accounts for most of the virus' diverse and peculiar properties. We have developed an agent-based computer model/simulation (PathSim, Pathogen Simulation) of this biological model. The simulation is performed on a virtual grid that represents the anatomy of the tonsils of the nasopharyngeal cavity (Waldeyer ring) and the peripheral circulation--the sites of EBV infection and persistence. The simulation is presented via a user friendly visual interface and reproduces quantitative and qualitative aspects of acute and persistent EBV infection. The simulation also had predictive power in validation experiments involving certain aspects of viral infection dynamics. Moreover, it allows us to identify switch points in the infection process that direct the disease course towards the end points of persistence, clearance, or death. Lastly, we were able to identify parameter sets that reproduced aspects of EBV-associated diseases. These investigations indicate that such simulations, combined with laboratory and clinical studies and animal models, will provide a powerful approach to investigating and controlling EBV infection, including the design of targeted anti-viral therapies.

Simulating Epstein-Barr virus infection with C-ImmSim.

MOTIVATION: Epstein-Barr virus (EBV) infects greater than 90% of humans benignly for life but can be associated with tumors. It is a uniquely human pathogen that is amenable to quantitative analysis; however, there is no applicable animal model. Computer models may provide a virtual environment to perform experiments not possible in human volunteers. RESULTS: We report the application of a relatively simple stochastic cellular automaton (C-ImmSim) to the modeling of EBV infection. Infected B-cell dynamics in the acute and chronic phases of infection correspond well to clinical data including the establishment of a long term persistent infection (up to 10 years) that is absolutely dependent on access of latently infected B cells to the peripheral pool where they are not subject to immunosurveillance. In the absence of this compartment the infection is cleared. AVAILABILITY: The latest version 6 of C-ImmSim is available under the GNU General Public License and is downloadable from www.iac.cnr.it/~filippo/cimmsim.html

Nested Canalyzing, Unate Cascade, and Polynomial Functions.

This paper focuses on the study of certain classes of Boolean functions that have appeared in several different contexts. Nested canalyzing functions have been studied recently in the context of Boolean network models of gene regulatory networks. In the same context, polynomial functions over finite fields have been used to develop network inference methods for gene regulatory networks. Finally, unate cascade functions have been studied in the design of logic circuits and binary decision diagrams. This paper shows that the class of nested canalyzing functions is equal to that of unate cascade functions. Furthermore, it provides a description of nested canalyzing functions as a certain type of Boolean polynomial function. Using the polynomial framework one can show that the class of nested canalyzing functions, or, equivalently, the class of unate cascade functions, forms an algebraic variety which makes their analysis amenable to the use of techniques from algebraic geometry and computational algebra. As a corollary of the functional equivalence derived here, a formula in the literature for the number of unate cascade functions provides such a formula for the number of nested canalyzing functions.

Cryopreservation of canine spermatozoa: theoretical prediction of optimal cooling rates in the presence and absence of cryoprotective agents.

In the present study a shape independent differential scanning calorimeter (DSC) technique was used to measure the dehydration response during freezing of ejaculated canine sperm cells. Volumetric shrinkage during freezing of canine sperm cell suspensions was obtained at cooling rates of 5 and 10 degrees C/min in the presence of extracellular ice and CPAs (6 different combinations of freezing media were used, ranging from a media with no CPAs, and those with 0.5%, 3%, and 6% glycerol and with 0.5% and 3% Me(2)SO). Using previously published data, the canine sperm cell was modeled as a cylinder of length 105.7mum and a radius of 0.32mum with an osmotically inactive cell volume, V(b), of 0.6 V(o), where V(o) is the isotonic cell volume. By fitting a model of water transport to the experimentally obtained volumetric shrinkage data the best fit membrane permeability parameters (L(pg) and E(Lp)) were determined. The "combined best fit" membrane permeability parameters at 5 and 10 degrees C/min for canine sperm cells in the absence of CPAs are: L(pg)=0.52x10(-15)m(3)/Ns (0.0029mum/min-atm) and E(Lp)=64.0kJ/mol (15.3kcal/mol) (R(2)=0.99); and the corresponding parameters in the presence of CPAs ranged from L(pg)[cpa]=0.46 to 0.53x10(-15) m(3)/Ns (0.0027-0.0031mum/min-atm) and E(Lp)[cpa]=46.4-56.0kJ/mol (11.1-13.4kcal/mol). These parameters are significantly different than previously published parameters for canine and other mammalian sperm obtained at suprazero temperatures and at subzero temperatures in the absence of extracellular ice. The parameters obtained in this study also suggest that optimal rates of freezing canine sperm cells ranges from 10 to 30 degrees C/min; these theoretical cooling rates are found to be in close conformity with previously published but empirically determined optimal cooling rates.