On Parametric Stability of Gene Networks Controlling Ontogenetic Processes

The problem of evaluation of parametric stability of three models of pro- and eukaryotic gene networks controlling ontogenetic processes has been defined and solved. Experimental schemes of testing gene networks for parametric stability based on the method of generalized threshold models were developed and realized as a software application. We studied the sensitivity of the functioning modes to random variations of the parameters in three model systems: phage development control system, Arabidopsis thaliana flower morphogenesis control subsystem, and gene subnetwork controlling early ontogeny of Drosophila melanogaster. The parametric stability was quantitatively assessed for these models.     

Parametric stability evaluation in computer experiments on the mathematical model of Drosophila control gene subnetwork

Using the method of generalized threshold models, the problem is formulated and solved to evaluate the parametric stability of the model of a gene subnetwork controlling the early ontogenesis of the fruit fly Drosophila melanogaster. Computer experiments have been performed to test the parametric stability of the model. Quantitative evaluations have been obtained for parametric stability of the Drosophila gene subnetwork in nuclei along the embryo's anterior-posterior axis. The results of computer experiments have been compared with the previous research data on "sensitivity" of functioning regimes to random changes of the parameters in the models of prokaryotic and eukaryotic systems, namely the system controlling the lambda-phage development and the subsystem controlling the flower morphogenesis of Arabidopsis thaliana. The obtained results confirm high parametric stability of gene networks that control the development of organisms.     

Efficient parametric analysis of the chemical master equation through model order reduction

ABSTRACT: BACKGROUND: Stochastic biochemical reaction networks are commonly modelled by the chemical master equation, and can be simulated as first order linear differential equations through a finite state projection. Due to the very high state space dimension of these equations, numerical simulations are computationally expensive. This is a particular problem for analysis tasks requiring repeated simulations for different parameter values. Such tasks are computationally expensive to the point of infeasibility with the chemical master equation. RESULTS: In this article, we apply parametric model order reduction techniques in order to construct accurate low-dimensional parametric models of the chemical master equation. These surrogate models can be used in various parametric analysis task such as identifiability analysis, parameter estimation, or sensitivity analysis. As biological examples, we consider two models for gene regulation networks, a bistable switch and a network displaying stochastic oscillations. CONCLUSIONS: The results show that the parametric model reduction yields efficient models of stochastic biochemical reaction networks, and that these models can be useful for systems biology applications involving parametric analysis problems such as parameter exploration, optimization, estimation or sensitivity analysis.     

Adjusting phenotypes by noise control

Genetically identical cells can show phenotypic variability. This is often caused by stochastic events that originate from randomness in biochemical processes involving in gene expression and other extrinsic cellular processes. From an engineering perspective, there have been efforts focused on theory and experiments to control noise levels by perturbing and replacing gene network components. However, systematic methods for noise control are lacking mainly due to the intractable mathematical structure of noise propagation through reaction networks. Here, we provide a numerical analysis method by quantifying the parametric sensitivity of noise characteristics at the level of the linear noise approximation. Our analysis is readily applicable to various types of noise control and to different types of system; for example, we can orthogonally control the mean and noise levels and can control system dynamics such as noisy oscillations. As an illustration we applied our method to HIV and yeast gene expression systems and metabolic networks. The oscillatory signal control was applied to p53 oscillations from DNA damage. Furthermore, we showed that the efficiency of orthogonal control can be enhanced by applying extrinsic noise and feedback. Our noise control analysis can be applied to any stochastic model belonging to continuous time Markovian systems such as biological and chemical reaction systems, and even computer and social networks. We anticipate the proposed analysis to be a useful tool for designing and controlling synthetic gene networks.     

The incorporation of epigenetics in artificial gene regulatory networks

Artificial gene regulatory networks are computational models that draw inspiration from biological networks of gene regulation. Since their inception they have been used to infer knowledge about gene regulation and as methods of computation. These computational models have been shown to possess properties typically found in the biological world, such as robustness and self organisation. Recently, it has become apparent that epigenetic mechanisms play an important role in gene regulation. This paper describes a new model, the Artificial Epigenetic Regulatory Network (AERN) which builds upon existing models by adding an epigenetic control layer. Our results demonstrate that AERNs are more adept at controlling multiple opposing trajectories when applied to a chaos control task within a conservative dynamical system, suggesting that AERNs are an interesting area for further investigation.     

Modeling real eukaryotic control gene subnetworks based on generalized threshold models]

Mathematical and computational means are developed that take into consideration the specifics of control processes at the molecular level and allow one to obtain both qualitative and quantitative patterns of gene network dynamics. Using the method of generalized threshold models, models are constructed for the Arabidopsis thaliana flower morphogenesis control subsystem and gene subnetwork controlling the Drosophila melanogaster early ontogeny. The dynamics of these systems are investigated: kinetic curves are computed for molecular components (RNA, proteins), possible modes of functioning and steady states of the nets are revealed and biologically interpreted. The models are shown to be adequate to the real processes. The effectiveness of the generalized threshold model method is evaluated in the analysis of the actual eukaryotic gene networks.     

Inference of the Xenopus tropicalis embryonic regulatory network and spatial gene expression patterns

Background

During embryogenesis, signaling molecules produced by one cell population direct gene regulatory changes in neighboring cells and influence their developmental fates and spatial organization. One of the earliest events in the development of the vertebrate embryo is the establishment of three germ layers, consisting of the ectoderm, mesoderm and endoderm. Attempts to measure gene expression in vivo in different germ layers and cell types are typically complicated by the heterogeneity of cell types within biological samples (i.e., embryos), as the responses of individual cell types are intermingled into an aggregate observation of heterogeneous cell types. Here, we propose a novel method to elucidate gene regulatory circuits from these aggregate measurements in embryos of the frog Xenopus tropicalis using gene network inference algorithms and then test the ability of the inferred networks to predict spatial gene expression patterns.

Results

We use two inference models with different underlying assumptions that incorporate existing network information, an ODE model for steady-state data and a Markov model for time series data, and contrast the performance of the two models. We apply our method to both control and knockdown embryos at multiple time points to reconstruct the core mesoderm and endoderm regulatory circuits. Those inferred networks are then used in combination with known dorsal-ventral spatial expression patterns of a subset of genes to predict spatial expression patterns for other genes. Both models are able to predict spatial expression patterns for some of the core mesoderm and endoderm genes, but interestingly of different gene subsets, suggesting that neither model is sufficient to recapitulate all of the spatial patterns, yet they are complementary for the patterns that they do capture.

Conclusion

The presented methodology of gene network inference combined with spatial pattern prediction provides an additional layer of validation to elucidate the regulatory circuits controlling the spatial-temporal dynamics in embryonic development.     

Modeling of Actual Eukaryotic Control Gene Subnetworks Based on the Method of Generalized Threshold Models

Mathematical and computational means are developed that take into consideration the specifics of control processes at the molecular level and allow one to obtain both qualitative and quantitative patterns of gene network dynamics. Using the method of generalized threshold models, models are constructed for the Arabidopsis thalianaflower morphogenesis control subsystem and gene subnetwork controlling the Drosophila melanogasterearly ontogeny. The dynamics of these systems are investigated: kinetic curves are computed for molecular components (RNA, proteins), possible modes of functioning and steady states of the nets are revealed and biologically interpreted. The models are shown to be adequate to the real processes. The effectiveness of the generalized threshold model method is evaluated in the analysis of the actual eukaryotic gene networks.     

Parametric Sensitivity Analysis of Oscillatory Delay Systems with an Application to Gene Regulation

A parametric sensitivity analysis for periodic solutions of delay-differential equations is developed. Because phase shifts cause the sensitivity coefficients of a periodic orbit to diverge, we focus on sensitivities of the extrema, from which amplitude sensitivities are computed, and of the period. Delay-differential equations are often used to model gene expression networks. In these models, the parametric sensitivities of a particular genotype define the local geometry of the evolutionary landscape. Thus, sensitivities can be used to investigate directions of gradual evolutionary change. An oscillatory protein synthesis model whose properties are modulated by RNA interference is used as an example. This model consists of a set of coupled delay-differential equations involving three delays. Sensitivity analyses are carried out at several operating points. Comments on the evolutionary implications of the results are offered.     

Distinctive topologies of partner-switching signaling networks correlate with their physiological roles

Regulatory networks controlling bacterial gene expression often evolve from common origins and share homologous proteins and similar network motifs. However, when functioning in different physiological contexts, these motifs may be re-arranged with different topologies that significantly affect network performance. Here we analyze two related signaling networks in the bacterium Bacillus subtilis in order to assess the consequences of their different topologies, with the aim of formulating design principles applicable to other systems. These two networks control the activities of the general stress response factor sigma(B) and the first sporulation-specific factor sigma(F). Both networks have at their core a "partner-switching" mechanism, in which an anti-sigma factor forms alternate complexes either with the sigma factor, holding it inactive, or with an anti-anti-sigma factor, thereby freeing sigma. However, clear differences in network structure are apparent: the anti-sigma factor for sigma(F) forms a long-lived, "dead-end" complex with its anti-anti-sigma factor and ADP, whereas the genes encoding sigma(B) and its network partners lie in a sigma(B)-controlled operon, resulting in positive and negative feedback loops. We constructed mathematical models of both networks and examined which features were critical for the performance of each design. The sigma(F) model predicts that the self-enhancing formation of the dead-end complex transforms the network into a largely irreversible hysteretic switch; the simulations reported here also demonstrate that hysteresis and slow turn off kinetics are the only two system properties associated with this complex formation. By contrast, the sigma(B) model predicts that the positive and negative feedback loops produce graded, reversible behavior with high regulatory capacity and fast response time. Our models demonstrate how alterations in network design result in different system properties that correlate with regulatory demands. These design principles agree with the known or suspected roles of similar networks in diverse bacteria.     

Bioprocess iterative batch-to-batch optimization based on hybrid parametric/nonparametric models

This paper presents a novel method for iterative batch-to-batch dynamic optimization of bioprocesses. The relationship between process performance and control inputs is established by means of hybrid grey-box models combining parametric and nonparametric structures. The bioreactor dynamics are defined by material balance equations, whereas the cell population subsystem is represented by an adjustable mixture of nonparametric and parametric models. Thus optimizations are possible without detailed mechanistic knowledge concerning the biological system. A clustering technique is used to supervise the reliability of the nonparametric subsystem during the optimization. Whenever the nonparametric outputs are unreliable, the objective function is penalized. The technique was evaluated with three simulation case studies. The overall results suggest that the convergence to the optimal process performance may be achieved after a small number of batches. The model unreliability risk constraint along with sampling scheduling are crucial to minimize the experimental effort required to attain a given process performance. In general terms, it may be concluded that the proposed method broadens the application of the hybrid parametric/nonparametric modeling technique to "newer" processes with higher potential for optimization.     

A linear theory for global electrocortical activity and its control by the lateral hypothalamus

A linear model for electrocortical waves and their control by the lateral hypothalamus is proposed. It is argued that such a linear model is not in contradition to non-linearity of neural elements on the microscopic scale. Telencephalic structures are treated as a mass of linked oscillators generating activity with a number of resonant modes. The lateral hypothalamus is regarded as controlling damping of activity in the telencephalic mass, and therefore exerting a specific parametric control over all signal processing in the cortical networks. An initial test is proposed to assess the constancy of telencephalic natural frequencies, with variation in lateral hypothalamic damping.     

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.     

Network-Based Inference of Cancer Progression from Microarray Data

Cancer cells exhibit a common phenotype of uncontrolled cell growth, but this phenotype may arise from many different combinations of mutations. By inferring how cells evolve in individual tumors, a process called cancer progression, we may be able to identify important mutational events for different tumor types, potentially leading to new therapeutics and diagnostics. Prior work has shown that it is possible to infer frequent progression pathways by using gene expression profiles to estimate ldquodistancesrdquo between tumors. Here, we apply gene network models to improve these estimates of evolutionary distance by controlling for correlations among coregulated genes. We test three variants of this approach: one using an optimized best-fit network, another using sampling to infer a high-confidence subnetwork, and one using a modular network inferred from clusters of similarly expressed genes. Application to lung cancer and breast cancer microarray data sets shows small improvements in phylogenies when correcting from the optimized network and more substantial improvements when correcting from the sampled or modular networks. Our results suggest that a network correction approach improves estimates of tumor similarity, but sophisticated network models are needed to control for the large hypothesis space and sparse data currently available.     

Robustness and evolvability in genetic regulatory networks

Living organisms are robust to a great variety of genetic changes. Gene regulation networks and metabolic pathways self-organize and reaccommodate to make the organism perform with stability and reliability under many point mutations, gene duplications and gene deletions. At the same time, living organisms are evolvable, which means that these kind of genetic perturbations can eventually make the organism acquire new functions and adapt to new environments. It is still an open problem to determine how robustness and evolvability blend together at the genetic level to produce stable organisms that yet can change and evolve. Here we address this problem by studying the robustness and evolvability of the attractor landscape of genetic regulatory network models under the process of gene duplication followed by divergence. We show that an intrinsic property of this kind of networks is that, after the divergence of the parent and duplicate genes, with a high probability the previous phenotypes, encoded in the attractor landscape of the network, are preserved and new ones might appear. The above is true in a variety of network topologies and even for the case of extreme divergence in which the duplicate gene bears almost no relation with its parent. Our results indicate that networks operating close to the so-called "critical regime" exhibit the maximum robustness and evolvability simultaneously.     

Inferring models of gene expression dynamics

We study the problem of identifying genetic networks in which expression dynamics are modeled by a differential equation that uses logical rules to specify time derivatives. We make three main contributions. First, we describe computationally efficient procedures for identifying the structure and dynamics of such networks from expression time series. Second, we derive predictions for the expected amount of data needed to identify randomly generated networks. Third, if expression values are available for only some of the genes, we show that the structure of the network for these "visible" genes can be identified and that the size and overall complexity of the network can be estimated. We validate these procedures and predictions using simulation experiments based on randomly generated networks with up to 30,000 genes and 17 distinct regulators per gene and on a network that models floral morphogenesis in Arabidopsis thaliana.     

Comparing different ODE modelling approaches for gene regulatory networks

A fundamental step in synthetic biology and systems biology is to derive appropriate mathematical models for the purposes of analysis and design. For example, to synthesize a gene regulatory network, the derivation of a mathematical model is important in order to carry out in silico investigations of the network dynamics and to investigate parameter variations and robustness issues. Different mathematical frameworks have been proposed to derive such models. In particular, the use of sets of nonlinear ordinary differential equations (ODEs) has been proposed to model the dynamics of the concentrations of mRNAs and proteins. These models are usually characterized by the presence of highly nonlinear Hill function terms. A typical simplification is to reduce the number of equations by means of a quasi-steady-state assumption on the mRNA concentrations. This yields a class of simplified ODE models. A radically different approach is to replace the Hill functions by piecewise-linear approximations [Casey, R., de Jong, H., Gouz, J.-L., 2006. Piecewise-linear models of genetic regulatory networks: equilibria and their stability. J. Math. Biol. 52 (1), 27-56]. A further modelling approach is the use of discrete-time maps [Coutinho, R., Fernandez, B., Lima, R., Meyroneinc, A., 2006. Discrete time piecewise affine models of genetic regulatory networks. J. Math. Biol. 52, 524-570] where the evolution of the system is modelled in discrete, rather than continuous, time. The aim of this paper is to discuss and compare these different modelling approaches, using a representative gene regulatory network. We will show that different models often lead to conflicting conclusions concerning the existence and stability of equilibria and stable oscillatory behaviours. Moreover, we shall discuss, where possible, the viability of making certain modelling approximations (e.g. quasi-steady-state mRNA dynamics or piecewise-linear approximations of Hill functions) and their effects on the overall system dynamics.