Random fluctuations in gene regulatory networks are inevitable due to the probabilistic nature of... more Random fluctuations in gene regulatory networks are inevitable due to the probabilistic nature of chemical reactions and the small populations of proteins, mRNAs present inside cells. These fluctuations are usually reported in terms of the first and second order statistical moments of the protein populations. If the birth-death rates of the mRNAs or the proteins are nonlinear, then the dynamics of these moments generally do not form a closed system of differential equations, in the sense that their time-derivatives depends on moments of order higher than two. Recent work has developed techniques to obtain the two lowest-order moments by closing their dynamics, which involves approximating the higher order moments as nonlinear functions of the two lowest ones. This paper uses these moment closure techniques to quantify noise in several gene regulatory networks.
Gene network dynamics often involves processes that take place on widely differing time scalesfro... more Gene network dynamics often involves processes that take place on widely differing time scalesfrom the order of nanoseconds to the order of several days. Multiple time scales in mathematical models often lead to serious computational difficulties, such as numerical stiffness in the case of differential equations or excessively redundant Monte Carlo simulations in the case of stochastic processes. We present a method that takes advantage of multiple time scales and dramatically reduces the computational time for a broad class of problems arising in stochastic gene regulatory networks. We illustrate the efficiency of our method in two gene network examples, which describe two substantially different biological processes -cellular heat shock response and expression of the pap gene in Escherichia coli bacteria.
Stochasticity is well recognized to be of crucial importance in the analysis of gene regulatory p... more Stochasticity is well recognized to be of crucial importance in the analysis of gene regulatory problems. This importance stems from the fact that extremely rare but important regulatory molecules often cause a great amount of intrinsic noise within a cell. Such systems are frequently modeled at the mesoscopic level as jump Markov processes, whose probability distributions evolve according to the chemical master equation (CME). In this paper we review a number of attempts that have been made to solve the CME. These include various kinetic Monte Carlo approaches, such as the Stochastic Simulation Algorithm (SSA) and its deviates, as well as systems theory based analytical solutions to the CME, such as the Finite State Projection (FSP) method and various moment closure techniques.
In order to capture important subcellular dynamics, researchers in computational biology have beg... more In order to capture important subcellular dynamics, researchers in computational biology have begun to turn to mesoscopic models in which molecular interactions at the gene level behave as discrete stochastic events. While the trajectories of such models cannot be described with deterministic expressions, the probability distributions of these trajectories can be described by the set of linear ordinary differential equations known as the chemical master equation (CME). Until recently, it has been believed that the CME could only be solved analytically in the most trivial of problems, and the CME has been analyzed almost exclusively with Kinetic Monte Carlo (KMC) algorithms. However, concepts from linear systems theory have enabled the Finite State Projection (FSP) approach and have significantly enhanced our ability to solve the CME without resorting to KMC simulations. In this paper we review the FSP approach and introduce a variety of systems theory based modifications and enhancements to the FSP algorithm. Notions such as observability, controllability and minimal realizations enable large reductions and increase efficiency with little to no loss in accuracy. Model reduction techniques based upon linear perturbation theory allow for the systematic projection of multiple time scale dynamics onto a slowly varying manifold of much smaller dimension. We also present a powerful new reduction approach, in which we perform computations on a small subset of configuration grid points and then interpolate to find the distribution on the full set. The power of the FSP and its various reduction approaches is illustrated on few important models of genetic regulatory networks.
Many gene regulatory networks are modeled at the mesoscopic scale, where chemical populations cha... more Many gene regulatory networks are modeled at the mesoscopic scale, where chemical populations change according to a discrete state (jump) Markov process. The chemical master equation (CME) for such a process is typically infinite dimensional and unlikely to be computationally tractable without reduction. The recently proposed Finite State Projection (FSP) technique allows for a bulk reduction of the CME while explicitly keeping track of its own approximation error. In previous work, this error has been reduced in order to obtain more accurate CME solutions for many biological examples. In this paper, we show that this "error" has far more significance than simply the distance between the approximate and exact solutions of the CME. In particular, we show that this error term serves as an exact measure of the rate of first transition from one system region to another. We demonstrate how this term may 1 be used to (i) directly determine the statistical distributions for stochastic switch rates, escape times, trajectory periods, and trajectory bifurcations, and (ii) evaluate how likely it is that a system will express certain behaviors during certain intervals of time. We also present two systems-theory based FSP model reduction approaches that are particularly useful in such studies. We illustrate the benefits of this approach to analyze the switching behavior of a stochastic model of Gardner's genetic toggle switch.
This article introduces the finite state projection Í‘FSPÍ’ method for use in the stochastic analys... more This article introduces the finite state projection Í‘FSPÍ’ method for use in the stochastic analysis of chemically reacting systems. One can describe the chemical populations of such systems with probability density vectors that evolve according to a set of linear ordinary differential equations known as the chemical master equation Í‘CMEÍ’. Unlike Monte Carlo methods such as the stochastic simulation algorithm Í‘SSAÍ’ or leaping, the FSP directly solves or approximates the solution of the CME. If the CME describes a system that has a finite number of distinct population vectors, the FSP method provides an exact analytical solution. When an infinite or extremely large number of population variations is possible, the state space can be truncated, and the FSP method provides a certificate of accuracy for how closely the truncated space approximation matches the true solution. The proposed FSP algorithm systematically increases the projection space in order to meet prespecified tolerance in the total probability density error. For any system in which a sufficiently accurate FSP exists, the FSP algorithm is shown to converge in a finite number of steps. The FSP is utilized to solve two examples taken from the field of systems biology, and comparisons are made between the FSP, the SSA, and leaping algorithms. In both examples, the FSP outperforms the SSA in terms of accuracy as well as computational efficiency. Furthermore, due to very small molecular counts in these particular examples, the FSP also performs far more effectively than leaping methods.
Ideas from System Theory lie behind many of the new powerful methods being developed in the burge... more Ideas from System Theory lie behind many of the new powerful methods being developed in the burgeoning field of Systems Biology. In this paper, we show two examples of this: one in the area of stochastic chemical kinetics, and the other in biological model invalidation. Stochastic chemical kinetics has gained a lot of attention in the last few years. In order to capture certain important dynamics in the subcellular environment, it is necessary to model molecular interactions at the gene level as discrete stochastic events. The dynamics of such processes is typically described by probability distributions, which evolve according to the set of linear ordinary differential equations known as the chemical master equation (CME). Until recently, it has been believed that the CME could not be solved analytically except in the most trivial of problems, and the CME has been analyzed almost exclusively with Monte Carlo (MC) algorithms. However, concepts from linear systems theory have enabled the Finite State Projection (FSP) approach and have significantly enhanced our ability to solve the CME without resorting to MC simulations. In this paper we review the FSP approach as well as a variety of systems theory based modifications to the FSP algorithm that dramatically improve the computational efficiency of the algorithm and expand the class of solvable problems. Notions such as observability, controllability and minimal realizations enable large reductions in the order of models and increase efficiency with little to no loss in accuracy. Model reduction techniques based upon linear perturbation theory allow for the systematic projection of multiple time scale dynamics onto a slowly varying manifold. Our second example shows the application of systems ideas in the area of biological model invalidation. As a specific case study, we use a dynamic model of the bacterial heat-shock response to demonstrate the approach. Using recent sum-of-squares techniques we show that the heat-shock model, when stripped from a certain protein-protein interaction that implements a certain feedback loop, cannot account for the input-output data regardless of the parameter choice for the model. In essence, such a deficient model is invalidated. Such conclusions are essential for pointing out the likelihood of missing components or interactions, thereby guiding new biological experiments.
IEEE Transactions on Circuits and Systems I-regular Papers, 2007
In order to capture important subcellular dynamics, researchers in computational biology have beg... more In order to capture important subcellular dynamics, researchers in computational biology have begun to turn to mesoscopic models in which molecular interactions at the gene level behave as discrete stochastic events. While the trajectories of such models cannot be described with deterministic expressions, the probability distributions of these trajectories can be described by the set of linear ordinary differential equations known as the chemical master equation (CME). Until recently, it has been believed that the CME could only be solved analytically in the most trivial of problems, and the CME has been analyzed almost exclusively with Kinetic Monte Carlo (KMC) algorithms. However, concepts from linear systems theory have enabled the Finite State Projection (FSP) approach and have significantly enhanced our ability to solve the CME without resorting to KMC simulations. In this paper we review the FSP approach and introduce a variety of systems theory based modifications and enhancements to the FSP algorithm. Notions such as observability, controllability and minimal realizations enable large reductions and increase efficiency with little to no loss in accuracy. Model reduction techniques based upon linear perturbation theory allow for the systematic projection of multiple time scale dynamics onto a slowly varying manifold of much smaller dimension. We also present a powerful new reduction approach, in which we perform computations on a small subset of configuration grid points and then interpolate to find the distribution on the full set. The power of the FSP and its various reduction approaches is illustrated on few important models of genetic regulatory networks.
In this tutorial the author introduce some of the concepts that frequently appear at the intersec... more In this tutorial the author introduce some of the concepts that frequently appear at the intersection of control theory and systems biology. Mustafa Khammash and Brian Munsky outline some of the key approaches for the modeling and analysis on cellular noise and the resulting fluctuations in the copy numbers of cellular constituents. Eduardo Sontagintroduces some tools for analyzing deterministic biochemical reaction networks. Pablo Iglesias looks at models of spatially varying chemical reactions, and in particular, how gradients and patterns are formed in cells. Finally, Domitilla Del Vecchio considers important control-theoretic problems in synthetic biology - that is, the synthesis of basic circuits inside cells using genetic regulatory components.
The cellular environment is abuzz with noise originating from the inherent random motion of react... more The cellular environment is abuzz with noise originating from the inherent random motion of reacting molecules in the living cell. In this noisy environment, clonal cell populations exhibit cell-to-cell variability that can manifest significant prototypical differences. Noise induced stochastic fluctuations in cellular constituents can be measured and their statistics quantified using flow cytometry, single molecule fluorescence in situ hybridization, time lapse fluorescence microscopy and other single cell and single molecule measurement techniques. We show that these random fluctuations carry within them valuable information about the underlying genetic network. Far from being a nuisance, the ever-present cellular noise acts as a rich source of excitation that, when processed through a gene network, carries its distinctive fingerprint that encodes a wealth of information about that network. We demonstrate that in some cases the analysis of these random fluctuations enables the full identification of network parameters, including those that may otherwise be difficult to measure. We use theoretical investigations to establish experimental guidelines for the identification of gene regulatory networks, and we apply these guideline to experimentally identify predictive models for different regulatory mechanisms in bacteria and yeast.
Deterministic models fail to capture certain important dynamics in the subcellular environment du... more Deterministic models fail to capture certain important dynamics in the subcellular environment due to the discrete stochastic nature of the molecular interactions at the gene level. Such discrete stochastic interactions are exploited in the cell to implement stochastic switches whose state are predictable only in a statistical sense. This absence of determinism and the inherent variability resulting from it play an important role in creating biological diversity that improves the chance for survivability. This paper will use a simplified model of the Pap switch in E. coli in order to illustrate a variety of computational methodologies. It will be shown that continuous time discrete-state Markov chains are natural tools for modeling this switch, and a review of these approaches will be provided. Using the recently introduced Finite State Projection algorithm, it is shown that the probability of a given switch position can be computed within any a-priori tolerance without resorting to the Monte-Carlo simulations, which generally lack accuracy guarantees.
At the mesoscopic scale, chemical processes have probability distributions that evolve according ... more At the mesoscopic scale, chemical processes have probability distributions that evolve according to an infinite set of linear ordinary differential equations known as the chemical master equation (CME). Although only a few classes of CME problems are known to have exact and computationally tractable analytical solutions, the recently proposed Finite State Projection (FSP) technique provides a systematic reduction of the CME with guaranteed accuracy bounds. For many non-trivial systems, the original FSP technique has been shown to yield accurate approximations to the CME solution. Other systems may require a projection that is still too large to be solved efficiently; for these, the linearity of the FSP allows for many model reductions and computational techniques, which can increase the efficiency of the FSP method with little or no loss in accuracy. In this paper, we present a new approach for choosing and expanding the projection for the original FSP algorithm. Based upon his approach, we develop a new algorithm that exploits the linearity property of super-position. The new algorithm retains the full accuracy guarantees of the original FSP approach, but with significantly increased efficiency for some problems and a greater range of applicability. We illustrate the benefits of this algorithm on a simplified model of the heat shock mechanism in E. coli.
This article introduces the Observability Aggregated Finite State Projection (OAFSP) method for u... more This article introduces the Observability Aggregated Finite State Projection (OAFSP) method for use in the stochastic analysis of biological systems. The small chemical populations of such systems have probability distributions that evolve according to a set of linear, time-invariant, ordinary differential equations known as the Chemical Master Equation (CME). The original FSP algorithm directly approximates the full CME solution to within a prespecified error. However, one may be interested only in certain portions of the distribution or certain statistical quantities such as mean or variance, and the full FSP method may provide an excess of information. In these cases, one can define a linear output signal and extract only the reachable and observable regions from the full distribution state space. The unobservable regions of the distribution can be aggregated with no accuracy loss but with less computational cost. This paper presents the resulting OAFSP algorithm and illustrates its benefits on a simple chemical reaction.
Many gene regulatory networks are modeled at the mesoscopic scale, where chemical populations are... more Many gene regulatory networks are modeled at the mesoscopic scale, where chemical populations are assumed to change according a discrete state (jump) Markov process. The chemical master equation (CME) for such a process is typically infinite dimensional and is unlikely to be computationally tractable without further reduction. The recently proposed Finite State Projection (FSP) technique allows for a bulk reduction of the CME while explicitly keeping track of its own approximation error. In previous work, this error has been reduced in order to obtain more accurate CME solutions for many biological examples. In this paper, we show that this "error" has far more significance than simply the distance between the approximate and exact solutions of the CME. In particular, we show that apart from its use as a measure for the quality of approximation, this error term serves as an exact measure of the rate of first transition from one system region to another. We demonstrate how this term may be used to directly determine the statistical distributions for stochastic switch rates, escape times, trajectory periods, and trajectory bifurcations. We illustrate the benefits of this approach to analyze the switching behavior of a stochastic model of Gardner's genetic toggle switch.
The following report uses a new stochastic model for the numerical study of the Pap pili epigenet... more The following report uses a new stochastic model for the numerical study of the Pap pili epigenetic switch in Escherichia Coli. The model focuses on the period immediately following DNA replication during which the cell's fate is decided as a result of a few critical stochastic chemical reactions. Gene methylation and protein-gene binding events are modeled as Markovian state transitions through which the Pap gene can reach any of 16 distinct epigenetic configurations; some of which allow for the production of the feedback regulator PapI. These patterns are composed with an infinitely variable PapI population level for an infinite number of possible system states. The resulting Chemical Master Equation is analytically approximated using the Finite State Projection method and compared with experimental data involving variations in the concentration of DNA adenine methylase. Cells with no PapI are considered to be OFF, and cells with PapI are considered to be ON. The model successfully captures all experimentally observed traits, and suggests further analysis and experimental testing.
The dynamics of chemical reaction networks often takes place on widely differing time scalesfrom ... more The dynamics of chemical reaction networks often takes place on widely differing time scalesfrom the order of nanoseconds to the order of several days. This is particularly true for gene regulatory networks, which are modeled by chemical kinetics. Multiple time scales in mathematical models often lead to serious computational difficulties, such as numerical stiffness in the case of differential equations or excessively redundant Monte Carlo simulations in the case of stochastic processes. We present a model reduction method for study of stochastic chemical kinetic systems that takes advantage of multiple time scales. The method applies to finite projections of the chemical master equation and allows for effective time scale separation of the system dynamics. We implement this method in a novel numerical algorithm that exploits the time scale separation to achieve model order reductions while enabling error checking and control. We illustrate the efficiency of our method in several examples motivated by recent developments in gene regulatory networks.
The cellular environment is abuzz with noise originating from the inherent random motion of react... more The cellular environment is abuzz with noise originating from the inherent random motion of reacting molecules in the living cell. In this noisy environment, clonal cell populations show cell-to-cell variability that can manifest significant phenotypic differences. Noise-induced stochastic fluctuations in cellular constituents can be measured and their statistics quantified. We show that these random fluctuations carry within them valuable information about the underlying genetic network. Far from being a nuisance, the ever-present cellular noise acts as a rich source of excitation that, when processed through a gene network, carries its distinctive fingerprint that encodes a wealth of information about that network. We show that in some cases the analysis of these random fluctuations enables the full identification of network parameters, including those that may otherwise be difficult to measure. This establishes a potentially powerful approach for the identification of gene networks and offers a new window into the workings of these networks.
Random fluctuations in gene regulatory networks are inevitable due to the probabilistic nature of... more Random fluctuations in gene regulatory networks are inevitable due to the probabilistic nature of chemical reactions and the small populations of proteins, mRNAs present inside cells. These fluctuations are usually reported in terms of the first and second order statistical moments of the protein populations. If the birth-death rates of the mRNAs or the proteins are nonlinear, then the dynamics of these moments generally do not form a closed system of differential equations, in the sense that their time-derivatives depends on moments of order higher than two. Recent work has developed techniques to obtain the two lowest-order moments by closing their dynamics, which involves approximating the higher order moments as nonlinear functions of the two lowest ones. This paper uses these moment closure techniques to quantify noise in several gene regulatory networks.
Gene network dynamics often involves processes that take place on widely differing time scalesfro... more Gene network dynamics often involves processes that take place on widely differing time scalesfrom the order of nanoseconds to the order of several days. Multiple time scales in mathematical models often lead to serious computational difficulties, such as numerical stiffness in the case of differential equations or excessively redundant Monte Carlo simulations in the case of stochastic processes. We present a method that takes advantage of multiple time scales and dramatically reduces the computational time for a broad class of problems arising in stochastic gene regulatory networks. We illustrate the efficiency of our method in two gene network examples, which describe two substantially different biological processes -cellular heat shock response and expression of the pap gene in Escherichia coli bacteria.
Stochasticity is well recognized to be of crucial importance in the analysis of gene regulatory p... more Stochasticity is well recognized to be of crucial importance in the analysis of gene regulatory problems. This importance stems from the fact that extremely rare but important regulatory molecules often cause a great amount of intrinsic noise within a cell. Such systems are frequently modeled at the mesoscopic level as jump Markov processes, whose probability distributions evolve according to the chemical master equation (CME). In this paper we review a number of attempts that have been made to solve the CME. These include various kinetic Monte Carlo approaches, such as the Stochastic Simulation Algorithm (SSA) and its deviates, as well as systems theory based analytical solutions to the CME, such as the Finite State Projection (FSP) method and various moment closure techniques.
Random fluctuations in gene regulatory networks are inevitable due to the probabilistic nature of... more Random fluctuations in gene regulatory networks are inevitable due to the probabilistic nature of chemical reactions and the small populations of proteins, mRNAs present inside cells. These fluctuations are usually reported in terms of the first and second order statistical moments of the protein populations. If the birth-death rates of the mRNAs or the proteins are nonlinear, then the dynamics of these moments generally do not form a closed system of differential equations, in the sense that their time-derivatives depends on moments of order higher than two. Recent work has developed techniques to obtain the two lowest-order moments by closing their dynamics, which involves approximating the higher order moments as nonlinear functions of the two lowest ones. This paper uses these moment closure techniques to quantify noise in several gene regulatory networks.
Gene network dynamics often involves processes that take place on widely differing time scalesfro... more Gene network dynamics often involves processes that take place on widely differing time scalesfrom the order of nanoseconds to the order of several days. Multiple time scales in mathematical models often lead to serious computational difficulties, such as numerical stiffness in the case of differential equations or excessively redundant Monte Carlo simulations in the case of stochastic processes. We present a method that takes advantage of multiple time scales and dramatically reduces the computational time for a broad class of problems arising in stochastic gene regulatory networks. We illustrate the efficiency of our method in two gene network examples, which describe two substantially different biological processes -cellular heat shock response and expression of the pap gene in Escherichia coli bacteria.
Stochasticity is well recognized to be of crucial importance in the analysis of gene regulatory p... more Stochasticity is well recognized to be of crucial importance in the analysis of gene regulatory problems. This importance stems from the fact that extremely rare but important regulatory molecules often cause a great amount of intrinsic noise within a cell. Such systems are frequently modeled at the mesoscopic level as jump Markov processes, whose probability distributions evolve according to the chemical master equation (CME). In this paper we review a number of attempts that have been made to solve the CME. These include various kinetic Monte Carlo approaches, such as the Stochastic Simulation Algorithm (SSA) and its deviates, as well as systems theory based analytical solutions to the CME, such as the Finite State Projection (FSP) method and various moment closure techniques.
In order to capture important subcellular dynamics, researchers in computational biology have beg... more In order to capture important subcellular dynamics, researchers in computational biology have begun to turn to mesoscopic models in which molecular interactions at the gene level behave as discrete stochastic events. While the trajectories of such models cannot be described with deterministic expressions, the probability distributions of these trajectories can be described by the set of linear ordinary differential equations known as the chemical master equation (CME). Until recently, it has been believed that the CME could only be solved analytically in the most trivial of problems, and the CME has been analyzed almost exclusively with Kinetic Monte Carlo (KMC) algorithms. However, concepts from linear systems theory have enabled the Finite State Projection (FSP) approach and have significantly enhanced our ability to solve the CME without resorting to KMC simulations. In this paper we review the FSP approach and introduce a variety of systems theory based modifications and enhancements to the FSP algorithm. Notions such as observability, controllability and minimal realizations enable large reductions and increase efficiency with little to no loss in accuracy. Model reduction techniques based upon linear perturbation theory allow for the systematic projection of multiple time scale dynamics onto a slowly varying manifold of much smaller dimension. We also present a powerful new reduction approach, in which we perform computations on a small subset of configuration grid points and then interpolate to find the distribution on the full set. The power of the FSP and its various reduction approaches is illustrated on few important models of genetic regulatory networks.
Many gene regulatory networks are modeled at the mesoscopic scale, where chemical populations cha... more Many gene regulatory networks are modeled at the mesoscopic scale, where chemical populations change according to a discrete state (jump) Markov process. The chemical master equation (CME) for such a process is typically infinite dimensional and unlikely to be computationally tractable without reduction. The recently proposed Finite State Projection (FSP) technique allows for a bulk reduction of the CME while explicitly keeping track of its own approximation error. In previous work, this error has been reduced in order to obtain more accurate CME solutions for many biological examples. In this paper, we show that this "error" has far more significance than simply the distance between the approximate and exact solutions of the CME. In particular, we show that this error term serves as an exact measure of the rate of first transition from one system region to another. We demonstrate how this term may 1 be used to (i) directly determine the statistical distributions for stochastic switch rates, escape times, trajectory periods, and trajectory bifurcations, and (ii) evaluate how likely it is that a system will express certain behaviors during certain intervals of time. We also present two systems-theory based FSP model reduction approaches that are particularly useful in such studies. We illustrate the benefits of this approach to analyze the switching behavior of a stochastic model of Gardner's genetic toggle switch.
This article introduces the finite state projection Í‘FSPÍ’ method for use in the stochastic analys... more This article introduces the finite state projection Í‘FSPÍ’ method for use in the stochastic analysis of chemically reacting systems. One can describe the chemical populations of such systems with probability density vectors that evolve according to a set of linear ordinary differential equations known as the chemical master equation Í‘CMEÍ’. Unlike Monte Carlo methods such as the stochastic simulation algorithm Í‘SSAÍ’ or leaping, the FSP directly solves or approximates the solution of the CME. If the CME describes a system that has a finite number of distinct population vectors, the FSP method provides an exact analytical solution. When an infinite or extremely large number of population variations is possible, the state space can be truncated, and the FSP method provides a certificate of accuracy for how closely the truncated space approximation matches the true solution. The proposed FSP algorithm systematically increases the projection space in order to meet prespecified tolerance in the total probability density error. For any system in which a sufficiently accurate FSP exists, the FSP algorithm is shown to converge in a finite number of steps. The FSP is utilized to solve two examples taken from the field of systems biology, and comparisons are made between the FSP, the SSA, and leaping algorithms. In both examples, the FSP outperforms the SSA in terms of accuracy as well as computational efficiency. Furthermore, due to very small molecular counts in these particular examples, the FSP also performs far more effectively than leaping methods.
Ideas from System Theory lie behind many of the new powerful methods being developed in the burge... more Ideas from System Theory lie behind many of the new powerful methods being developed in the burgeoning field of Systems Biology. In this paper, we show two examples of this: one in the area of stochastic chemical kinetics, and the other in biological model invalidation. Stochastic chemical kinetics has gained a lot of attention in the last few years. In order to capture certain important dynamics in the subcellular environment, it is necessary to model molecular interactions at the gene level as discrete stochastic events. The dynamics of such processes is typically described by probability distributions, which evolve according to the set of linear ordinary differential equations known as the chemical master equation (CME). Until recently, it has been believed that the CME could not be solved analytically except in the most trivial of problems, and the CME has been analyzed almost exclusively with Monte Carlo (MC) algorithms. However, concepts from linear systems theory have enabled the Finite State Projection (FSP) approach and have significantly enhanced our ability to solve the CME without resorting to MC simulations. In this paper we review the FSP approach as well as a variety of systems theory based modifications to the FSP algorithm that dramatically improve the computational efficiency of the algorithm and expand the class of solvable problems. Notions such as observability, controllability and minimal realizations enable large reductions in the order of models and increase efficiency with little to no loss in accuracy. Model reduction techniques based upon linear perturbation theory allow for the systematic projection of multiple time scale dynamics onto a slowly varying manifold. Our second example shows the application of systems ideas in the area of biological model invalidation. As a specific case study, we use a dynamic model of the bacterial heat-shock response to demonstrate the approach. Using recent sum-of-squares techniques we show that the heat-shock model, when stripped from a certain protein-protein interaction that implements a certain feedback loop, cannot account for the input-output data regardless of the parameter choice for the model. In essence, such a deficient model is invalidated. Such conclusions are essential for pointing out the likelihood of missing components or interactions, thereby guiding new biological experiments.
IEEE Transactions on Circuits and Systems I-regular Papers, 2007
In order to capture important subcellular dynamics, researchers in computational biology have beg... more In order to capture important subcellular dynamics, researchers in computational biology have begun to turn to mesoscopic models in which molecular interactions at the gene level behave as discrete stochastic events. While the trajectories of such models cannot be described with deterministic expressions, the probability distributions of these trajectories can be described by the set of linear ordinary differential equations known as the chemical master equation (CME). Until recently, it has been believed that the CME could only be solved analytically in the most trivial of problems, and the CME has been analyzed almost exclusively with Kinetic Monte Carlo (KMC) algorithms. However, concepts from linear systems theory have enabled the Finite State Projection (FSP) approach and have significantly enhanced our ability to solve the CME without resorting to KMC simulations. In this paper we review the FSP approach and introduce a variety of systems theory based modifications and enhancements to the FSP algorithm. Notions such as observability, controllability and minimal realizations enable large reductions and increase efficiency with little to no loss in accuracy. Model reduction techniques based upon linear perturbation theory allow for the systematic projection of multiple time scale dynamics onto a slowly varying manifold of much smaller dimension. We also present a powerful new reduction approach, in which we perform computations on a small subset of configuration grid points and then interpolate to find the distribution on the full set. The power of the FSP and its various reduction approaches is illustrated on few important models of genetic regulatory networks.
In this tutorial the author introduce some of the concepts that frequently appear at the intersec... more In this tutorial the author introduce some of the concepts that frequently appear at the intersection of control theory and systems biology. Mustafa Khammash and Brian Munsky outline some of the key approaches for the modeling and analysis on cellular noise and the resulting fluctuations in the copy numbers of cellular constituents. Eduardo Sontagintroduces some tools for analyzing deterministic biochemical reaction networks. Pablo Iglesias looks at models of spatially varying chemical reactions, and in particular, how gradients and patterns are formed in cells. Finally, Domitilla Del Vecchio considers important control-theoretic problems in synthetic biology - that is, the synthesis of basic circuits inside cells using genetic regulatory components.
The cellular environment is abuzz with noise originating from the inherent random motion of react... more The cellular environment is abuzz with noise originating from the inherent random motion of reacting molecules in the living cell. In this noisy environment, clonal cell populations exhibit cell-to-cell variability that can manifest significant prototypical differences. Noise induced stochastic fluctuations in cellular constituents can be measured and their statistics quantified using flow cytometry, single molecule fluorescence in situ hybridization, time lapse fluorescence microscopy and other single cell and single molecule measurement techniques. We show that these random fluctuations carry within them valuable information about the underlying genetic network. Far from being a nuisance, the ever-present cellular noise acts as a rich source of excitation that, when processed through a gene network, carries its distinctive fingerprint that encodes a wealth of information about that network. We demonstrate that in some cases the analysis of these random fluctuations enables the full identification of network parameters, including those that may otherwise be difficult to measure. We use theoretical investigations to establish experimental guidelines for the identification of gene regulatory networks, and we apply these guideline to experimentally identify predictive models for different regulatory mechanisms in bacteria and yeast.
Deterministic models fail to capture certain important dynamics in the subcellular environment du... more Deterministic models fail to capture certain important dynamics in the subcellular environment due to the discrete stochastic nature of the molecular interactions at the gene level. Such discrete stochastic interactions are exploited in the cell to implement stochastic switches whose state are predictable only in a statistical sense. This absence of determinism and the inherent variability resulting from it play an important role in creating biological diversity that improves the chance for survivability. This paper will use a simplified model of the Pap switch in E. coli in order to illustrate a variety of computational methodologies. It will be shown that continuous time discrete-state Markov chains are natural tools for modeling this switch, and a review of these approaches will be provided. Using the recently introduced Finite State Projection algorithm, it is shown that the probability of a given switch position can be computed within any a-priori tolerance without resorting to the Monte-Carlo simulations, which generally lack accuracy guarantees.
At the mesoscopic scale, chemical processes have probability distributions that evolve according ... more At the mesoscopic scale, chemical processes have probability distributions that evolve according to an infinite set of linear ordinary differential equations known as the chemical master equation (CME). Although only a few classes of CME problems are known to have exact and computationally tractable analytical solutions, the recently proposed Finite State Projection (FSP) technique provides a systematic reduction of the CME with guaranteed accuracy bounds. For many non-trivial systems, the original FSP technique has been shown to yield accurate approximations to the CME solution. Other systems may require a projection that is still too large to be solved efficiently; for these, the linearity of the FSP allows for many model reductions and computational techniques, which can increase the efficiency of the FSP method with little or no loss in accuracy. In this paper, we present a new approach for choosing and expanding the projection for the original FSP algorithm. Based upon his approach, we develop a new algorithm that exploits the linearity property of super-position. The new algorithm retains the full accuracy guarantees of the original FSP approach, but with significantly increased efficiency for some problems and a greater range of applicability. We illustrate the benefits of this algorithm on a simplified model of the heat shock mechanism in E. coli.
This article introduces the Observability Aggregated Finite State Projection (OAFSP) method for u... more This article introduces the Observability Aggregated Finite State Projection (OAFSP) method for use in the stochastic analysis of biological systems. The small chemical populations of such systems have probability distributions that evolve according to a set of linear, time-invariant, ordinary differential equations known as the Chemical Master Equation (CME). The original FSP algorithm directly approximates the full CME solution to within a prespecified error. However, one may be interested only in certain portions of the distribution or certain statistical quantities such as mean or variance, and the full FSP method may provide an excess of information. In these cases, one can define a linear output signal and extract only the reachable and observable regions from the full distribution state space. The unobservable regions of the distribution can be aggregated with no accuracy loss but with less computational cost. This paper presents the resulting OAFSP algorithm and illustrates its benefits on a simple chemical reaction.
Many gene regulatory networks are modeled at the mesoscopic scale, where chemical populations are... more Many gene regulatory networks are modeled at the mesoscopic scale, where chemical populations are assumed to change according a discrete state (jump) Markov process. The chemical master equation (CME) for such a process is typically infinite dimensional and is unlikely to be computationally tractable without further reduction. The recently proposed Finite State Projection (FSP) technique allows for a bulk reduction of the CME while explicitly keeping track of its own approximation error. In previous work, this error has been reduced in order to obtain more accurate CME solutions for many biological examples. In this paper, we show that this "error" has far more significance than simply the distance between the approximate and exact solutions of the CME. In particular, we show that apart from its use as a measure for the quality of approximation, this error term serves as an exact measure of the rate of first transition from one system region to another. We demonstrate how this term may be used to directly determine the statistical distributions for stochastic switch rates, escape times, trajectory periods, and trajectory bifurcations. We illustrate the benefits of this approach to analyze the switching behavior of a stochastic model of Gardner's genetic toggle switch.
The following report uses a new stochastic model for the numerical study of the Pap pili epigenet... more The following report uses a new stochastic model for the numerical study of the Pap pili epigenetic switch in Escherichia Coli. The model focuses on the period immediately following DNA replication during which the cell's fate is decided as a result of a few critical stochastic chemical reactions. Gene methylation and protein-gene binding events are modeled as Markovian state transitions through which the Pap gene can reach any of 16 distinct epigenetic configurations; some of which allow for the production of the feedback regulator PapI. These patterns are composed with an infinitely variable PapI population level for an infinite number of possible system states. The resulting Chemical Master Equation is analytically approximated using the Finite State Projection method and compared with experimental data involving variations in the concentration of DNA adenine methylase. Cells with no PapI are considered to be OFF, and cells with PapI are considered to be ON. The model successfully captures all experimentally observed traits, and suggests further analysis and experimental testing.
The dynamics of chemical reaction networks often takes place on widely differing time scalesfrom ... more The dynamics of chemical reaction networks often takes place on widely differing time scalesfrom the order of nanoseconds to the order of several days. This is particularly true for gene regulatory networks, which are modeled by chemical kinetics. Multiple time scales in mathematical models often lead to serious computational difficulties, such as numerical stiffness in the case of differential equations or excessively redundant Monte Carlo simulations in the case of stochastic processes. We present a model reduction method for study of stochastic chemical kinetic systems that takes advantage of multiple time scales. The method applies to finite projections of the chemical master equation and allows for effective time scale separation of the system dynamics. We implement this method in a novel numerical algorithm that exploits the time scale separation to achieve model order reductions while enabling error checking and control. We illustrate the efficiency of our method in several examples motivated by recent developments in gene regulatory networks.
The cellular environment is abuzz with noise originating from the inherent random motion of react... more The cellular environment is abuzz with noise originating from the inherent random motion of reacting molecules in the living cell. In this noisy environment, clonal cell populations show cell-to-cell variability that can manifest significant phenotypic differences. Noise-induced stochastic fluctuations in cellular constituents can be measured and their statistics quantified. We show that these random fluctuations carry within them valuable information about the underlying genetic network. Far from being a nuisance, the ever-present cellular noise acts as a rich source of excitation that, when processed through a gene network, carries its distinctive fingerprint that encodes a wealth of information about that network. We show that in some cases the analysis of these random fluctuations enables the full identification of network parameters, including those that may otherwise be difficult to measure. This establishes a potentially powerful approach for the identification of gene networks and offers a new window into the workings of these networks.
Random fluctuations in gene regulatory networks are inevitable due to the probabilistic nature of... more Random fluctuations in gene regulatory networks are inevitable due to the probabilistic nature of chemical reactions and the small populations of proteins, mRNAs present inside cells. These fluctuations are usually reported in terms of the first and second order statistical moments of the protein populations. If the birth-death rates of the mRNAs or the proteins are nonlinear, then the dynamics of these moments generally do not form a closed system of differential equations, in the sense that their time-derivatives depends on moments of order higher than two. Recent work has developed techniques to obtain the two lowest-order moments by closing their dynamics, which involves approximating the higher order moments as nonlinear functions of the two lowest ones. This paper uses these moment closure techniques to quantify noise in several gene regulatory networks.
Gene network dynamics often involves processes that take place on widely differing time scalesfro... more Gene network dynamics often involves processes that take place on widely differing time scalesfrom the order of nanoseconds to the order of several days. Multiple time scales in mathematical models often lead to serious computational difficulties, such as numerical stiffness in the case of differential equations or excessively redundant Monte Carlo simulations in the case of stochastic processes. We present a method that takes advantage of multiple time scales and dramatically reduces the computational time for a broad class of problems arising in stochastic gene regulatory networks. We illustrate the efficiency of our method in two gene network examples, which describe two substantially different biological processes -cellular heat shock response and expression of the pap gene in Escherichia coli bacteria.
Stochasticity is well recognized to be of crucial importance in the analysis of gene regulatory p... more Stochasticity is well recognized to be of crucial importance in the analysis of gene regulatory problems. This importance stems from the fact that extremely rare but important regulatory molecules often cause a great amount of intrinsic noise within a cell. Such systems are frequently modeled at the mesoscopic level as jump Markov processes, whose probability distributions evolve according to the chemical master equation (CME). In this paper we review a number of attempts that have been made to solve the CME. These include various kinetic Monte Carlo approaches, such as the Stochastic Simulation Algorithm (SSA) and its deviates, as well as systems theory based analytical solutions to the CME, such as the Finite State Projection (FSP) method and various moment closure techniques.
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Papers by Brian Munsky