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35
Hybrid Modeling and Simulation of Biomolecular Networks
 Hybrid Systems: Computation and Control, LNCS 2034
, 2001
"... In a biological cell, cellular functions and the genetic regulatory apparatus are implemented and controlled by a network of chemical reactions in which regulatory proteins can control genes that produce other regulators, which in turn control other genes. Further, the feedback pathways appear t ..."
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Cited by 100 (7 self)
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In a biological cell, cellular functions and the genetic regulatory apparatus are implemented and controlled by a network of chemical reactions in which regulatory proteins can control genes that produce other regulators, which in turn control other genes. Further, the feedback pathways appear to incorporate switches that result in changes in the dynamic behavior of the cell. This paper describes a hybrid systems approach to modeling the intracellular network using continuous di#erential equations to model the feedback mechanisms and modeswitching to describe the changes in the underlying dynamics. We use two case studies to illustrate a modular approach to modeling such networks and describe the architectural and behavioral hierarchy in the underlying models. We describe these models using Charon [2], a language that allows formal description of hybrid systems. We provide preliminary simulation results that demonstrate how our approach can help biologists in their analysis of noisy genetic circuits. Finally we describe our agenda for future work that includes the development of models and simulation for stochastic hybrid systems.
Qualitative analysis and verification of hybrid models of genetic regulatory networks: Nutritional stress response in Escherichia coli
 in Hybrid Systems: Computation and Control
, 2005
"... Abstract. The switchlike character of the dynamics of genetic regulatory networks has attracted much attention from mathematical biologists and researchers on hybrid systems alike. We extend our previous work on a method for the qualitative analysis of hybrid models of genetic regulatory networks, ..."
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Cited by 27 (5 self)
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Abstract. The switchlike character of the dynamics of genetic regulatory networks has attracted much attention from mathematical biologists and researchers on hybrid systems alike. We extend our previous work on a method for the qualitative analysis of hybrid models of genetic regulatory networks, based on a class of piecewiseaffine differential equation (PADE) models, in two directions. First, we present a refinement of the method using a discrete or qualitative abstraction that preserves stronger properties of the dynamics of the PA systems, in particular the sign patterns of the derivatives of the concentration variables. The discrete transition system resulting from the abstraction is a conservative approximation of the dynamics of the PA system and can be computed symbolically. Second, we apply the refined method to a regulatory system whose functioning is not yet wellunderstood by biologists, the nutritional stress response in the bacterium Escherichia coli. 1
Parameter Space Structure of ContinuousTime Recurrent Neural Networks
, 2006
"... this article (see Figure 1). By transforming equation 2.1 to the output space defined by o #) and setting the time derivative to 0, we find that the SSIO curve of a neuron with selfweight w is given by I # 1 (o) w o. A single additive model neuron can exhibit either unistable or bista ..."
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Cited by 27 (3 self)
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this article (see Figure 1). By transforming equation 2.1 to the output space defined by o #) and setting the time derivative to 0, we find that the SSIO curve of a neuron with selfweight w is given by I # 1 (o) w o. A single additive model neuron can exhibit either unistable or bistable dynamics, depending on the strength of its selfweight and its net input (Cowan & Ermentrout, 1978). In a single CTRNN neuron, only unistable dynamics are possible when w<4 (see Figure 1A). When w>4, bistable dynamics occurs when I L (w) I R (w) (see 42 0 2 4 I+ 0 0.2 0.4 0.6 0.8 1 42 0 2 4 I+ 0 0.2 0.4 0.6 0.8 1 Figure 1: Representative steadystate inputoutput (SSIO) diagrams of a single CTRNN for (A) w 2 and (B) w 8. The solid line shows the output space location of the neuron's equilibrium points as a function of the net input I # . Note that the SSIO becomes folded for w>4, indicating the existence of three equilibrium points. When the SSIO is folded, the left and right edges of the fold are given by I L (w)andI R (w), respectively (black points in B). The ranges of synaptic inputs received from other neurons are indicated by gray rectangles. The lower (min and upper (max limits of this range play an important role in the analysis described in this article. In both plots, two synaptic input ranges are shown: one for which the neuron is saturated off (left rectangle) and one for which the neuron is saturated on (right rectangle). The dashed line in A shows the piecewise linear SSIO approximation used in section 4.2, which suggests using the intersections of the linear pieces (black points) as the analog of the fold edges in part B
Understanding the bacterial stringent response using reachability analysis of hybrid systems
 in Hybrid Systems: Computation and Control, ser. LNCS
, 2004
"... Abstract. In this paper we model coupled genetic and metabolic networks as hybrid systems. The vector fields are multi affine, i.e., have only product type nonlinearities to accommodate chemical reactions, and are defined in rectangular invariants, whose facets correspond to changes in the behavio ..."
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Cited by 23 (5 self)
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Abstract. In this paper we model coupled genetic and metabolic networks as hybrid systems. The vector fields are multi affine, i.e., have only product type nonlinearities to accommodate chemical reactions, and are defined in rectangular invariants, whose facets correspond to changes in the behavior of a gene or enzyme. For such systems, we showed that reachability and safety verification problems can be formulated and solved (conservatively) in an elegant and computationally inexpensive way, based on the fact that multiaffine functions on rectangular regions of the space are determined at the vertices. Using these techniques, we study the stringent response system, which is the transition of bacterial organisms from growth phase to a metabolically suppressed phase when subjected to an environment with limited nutrients. 1
Analysis of lactose metabolism in E.coli using reachability analysis of hybrid systems
 IEE PROCEEDINGS  SYSTEMS BIOLOGY
, 2007
"... We propose an abstraction method for medium scale biomolecular networks, based on hybrid dynamical systems with continuous multiaffine dynamics. This abstraction method follows naturally from the notion of approximating nonlinear rate laws with continuous piecewise linear functions and can be easil ..."
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Cited by 14 (3 self)
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We propose an abstraction method for medium scale biomolecular networks, based on hybrid dynamical systems with continuous multiaffine dynamics. This abstraction method follows naturally from the notion of approximating nonlinear rate laws with continuous piecewise linear functions and can be easily automated. An efficient reachability algorithm is possible for the resulting class of hybrid systems. We construct an approximation for an ordinary differential equation model of the lac operon, and show that our abstraction passes the same experimental tests that were used to validate the original model. The wellstudied biological system exhibits bistability and switching behavior, arising from positive feedback in the expression mechanism of the lac operon. The switching property of the lac system is an example of the major qualitative features that are the building blocks of higher level, more coarsegrained descriptions. Our approach is useful in helping correctly identify such properties and in connecting them to the underlying molecular dynamical details. We use reachability analysis together with the knowledge of the steady state structure to identify ranges of parameter values for which the system maintains the bistable switching property.
V (2005) Computational techniques for analysis of genetic network dynamics
 Intl Journal of Robotics Research
"... In this paper we propose modeling and analysis techniques for genetic networks that provide biologists with insight into the dynamics of such systems. Central to our modeling approach is the framework of hybrid systems and our analysis tools are derived from formal analysis of such systems. Given a ..."
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Cited by 12 (4 self)
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In this paper we propose modeling and analysis techniques for genetic networks that provide biologists with insight into the dynamics of such systems. Central to our modeling approach is the framework of hybrid systems and our analysis tools are derived from formal analysis of such systems. Given a set of states characterizing a property of biological interestP, we present the MultiAffine Rectangular Partition (MARP) algorithm for the construction of a set of infeasible states I that will never reach P and the Rapidly Exploring Random Forest of Trees (RRFT) algorithm for the construction of a set of feasible statesF that will reach P. These techniques are scalable to high dimensions and can incorporate uncertainty (partial knowledge of kinetic parameters and state uncertainty). We apply these methods to understand the genetic interactions involved in the phenomenon of luminescence production in the marine bacterium V. fischeri. KEY WORDS—genetic networks, hybrid systems, formal analysis, rapidlyexploring random trees 1.
Identification of genetic network dynamics with unate structure
 in "Bioinformatics", 2010
"... Motivation: Modern experimental techniques for timecourse measurement of gene expression enable the identification of dynamical models of genetic regulatory networks. In general, identification involves fitting appropriate network structures and parameters to the data. For a given set of genes, exp ..."
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Cited by 11 (2 self)
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Motivation: Modern experimental techniques for timecourse measurement of gene expression enable the identification of dynamical models of genetic regulatory networks. In general, identification involves fitting appropriate network structures and parameters to the data. For a given set of genes, exploring all possible network structures is clearly prohibitive. Modelling and identification methods for the a priori selection of network structures compatible with biological knowledge and experimental data are necessary to make the identification problem tractable. Results: We propose a differential equation modelling framework where the regulatory interactions among genes are expressed in terms of unate functions, a class of gene activation rules commonly encountered in Boolean network modelling. We establish analytical properties of the models in the class and exploit them to devise a twostep procedure for gene network reconstruction from product concentration and synthesis rate time series. The first step isolates a family of model structures compatible with the data from a set of most relevant biological hypotheses. The second step explores this family and returns a pool of best fitting models along with estimates of their parameters. The method is tested on a simulated network and compared to stateoftheart network inference methods on the benchmark synthetic network IRMA. Contact:
Learning cyclelinear hybrid automata for excitable cells
 In Proc. of HSCC’07, the 10th International Conference on Hybrid Systems: Computation and Control, volume 4416 of LNCS
, 2007
"... Abstract. We show how to automatically learn the class of Hybrid Automata called CycleLinear Hybrid Automata (CLHA) in order to model the behavior of excitable cells. Such cells, whose main purpose is to amplify and propagate an electrical signal known as the action potential (AP), serve as the “bi ..."
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Abstract. We show how to automatically learn the class of Hybrid Automata called CycleLinear Hybrid Automata (CLHA) in order to model the behavior of excitable cells. Such cells, whose main purpose is to amplify and propagate an electrical signal known as the action potential (AP), serve as the “biologic transistors ” of living organisms. The learning algorithm we propose comprises the following three phases: (1) Geometric analysis of the APs in the training set is used to identify, for each AP, the modes and switching logic of the corresponding Linear Hybrid Automata. (2) For each mode, the modified Prony’s method is used to learn the coefficients of the associated linear flows. (3) The modified Prony’s method is used again to learn the functions that adjust, on a percycle basis, the mode dynamics and switching logic of the Linear Hybrid Automata obtained in the first two phases. Our results show that the learned CLHA is able to successfully capture AP morphology and other important excitablecell properties, such as refractoriness and restitution, up to a prescribed approximation error. Our approach is fully implemented in MATLAB and, to the best of our knowledge, provides the most accurate approximation model for ECs to date. 1
Modeling the molecular network controlling adhesion between human endothelial cells: Inference and simulation using constraint logic programming
 In CMSB’04: Proceedings of the 20 international conference on Computational Methods in Systems Biology
, 2004
"... Abstract. Cellcell adhesion plays a critical role in the formation of tissues and organs. Adhesion between endothelial cells is also involved in the control of leukocyte migration across the endothelium of blood vessels. The most important players in this process are probably identified and the o ..."
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Cited by 8 (3 self)
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Abstract. Cellcell adhesion plays a critical role in the formation of tissues and organs. Adhesion between endothelial cells is also involved in the control of leukocyte migration across the endothelium of blood vessels. The most important players in this process are probably identified and the overall organization of the biochemical network can be drawn, but knowledge about connectivity is still incomplete, and the numerical values of kinetic parameters are unknown. This calls for qualitative modeling methods. Our aim in this paper is twofold: (i) to integrate in a unified model the biochemical network and the genetic circuitry. For this purpose we transform our system into a system of piecewise linear differential equations and then use Thomas theory of discrete networks. (ii) to show how constraints can be used to infer ranges of parameter values from observations and, with the same model, perform qualitative simulations. 1
Subtilin production by Bacillus subtilis: Stochastic hybrid models and parameter identification
 IEEE Transactions on Circuits and Systems I – IEEE Transactions on Automatic Control
, 2008
"... This paper presents methods for the parameter identification of a model of subtilin production by Bacillus subtilis. Based on a stochastic hybrid model, identification is split in two subproblems: estimation of the genetic network regulating subtilin production from gene expression data, and estima ..."
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Cited by 7 (1 self)
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This paper presents methods for the parameter identification of a model of subtilin production by Bacillus subtilis. Based on a stochastic hybrid model, identification is split in two subproblems: estimation of the genetic network regulating subtilin production from gene expression data, and estimation of population dynamics based on nutrient and population level data. Techniques for identification of switching dynamics from sparse and irregularly sampled observations are developed and applied to simulated data. Numerical results are provided to show the effectiveness of our methods.