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16
Sliding window abstraction for infinite Markov chains
 In Proc. CAV, volume 5643 of LNCS
, 2009
"... Abstract. We present an onthefly abstraction technique for infinitestate continuoustime Markov chains. We consider Markov chains that are specified by a finite set of transition classes. Such models naturally represent biochemical reactions and therefore play an important role in the stochastic ..."
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Cited by 21 (8 self)
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Abstract. We present an onthefly abstraction technique for infinitestate continuoustime Markov chains. We consider Markov chains that are specified by a finite set of transition classes. Such models naturally represent biochemical reactions and therefore play an important role in the stochastic modeling of biological systems. We approximate the transient probability distributions at various time instances by solving a sequence of dynamically constructed abstract models, each depending on the previous one. Each abstract model is a finite Markov chain that represents the behavior of the original, infinite chain during a specific time interval. Our approach provides complete information about probability distributions, not just about individual parameters like the mean. The error of each abstraction can be computed, and the precision of the abstraction refined when desired. We implemented the algorithm and demonstrate its usefulness and efficiency on several case studies from systems biology. 1
Formalisms for Specifying Markovian Population Models
"... We compare several languages for specifying Markovian population models such as queuing networks and chemical reaction networks. These languages —matrix descriptions, stochastic Petri nets, stochastic process algebras, stoichiometric equations, and guarded command models — all describe continuoust ..."
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Cited by 11 (3 self)
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We compare several languages for specifying Markovian population models such as queuing networks and chemical reaction networks. These languages —matrix descriptions, stochastic Petri nets, stochastic process algebras, stoichiometric equations, and guarded command models — all describe continuoustime Markov chains, but they differ according to important properties, such as compositionality, expressiveness and succinctness, executability, ease of use, and the support they provide for checking the wellformedness of a model and for analyzing a model.
Approximation of event probabilities in noisy cellular processes
 In Proc. of CMSB
, 2009
"... Abstract. Molecular noise, which arises from the randomness of the discrete events in the cell, significantly influences fundamental biological processes. Discretestate continuoustime stochastic models (CTMC) can be used to describe such effects, but the calculation of the probabilities of certain ..."
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Cited by 9 (4 self)
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Abstract. Molecular noise, which arises from the randomness of the discrete events in the cell, significantly influences fundamental biological processes. Discretestate continuoustime stochastic models (CTMC) can be used to describe such effects, but the calculation of the probabilities of certain events is computationally expensive. We present a comparison of two analysis approaches for CTMC. On one hand, we estimate the probabilities of interest using repeated Gillespie simulation and determine the statistical accuracy that we obtain. On the other hand, we apply a numerical reachability analysis that approximates the probability distributions of the system at several time instances. We use examples of cellular processes to demonstrate the superiority of the reachability analysis if accurate results are required. 1
Analysis of spike statistics in neuronal systems with continuous attractors or multiple, discrete attractor states.
, 2005
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Bidirectional classical stochastic processes with measurements and feedback
, 2006
"... A measurement on a quantum system is said to cause the “collapse” of the quantum state vector or density matrix. An analogous collapse occurs with measurements on a classical stochastic process. This paper addresses the question of describing the response of a classical stochastic process when there ..."
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A measurement on a quantum system is said to cause the “collapse” of the quantum state vector or density matrix. An analogous collapse occurs with measurements on a classical stochastic process. This paper addresses the question of describing the response of a classical stochastic process when there is feedback from the output of a measurement to the input, and is intended to give a simplified model for quantummechanical processes that occur along a spacelike reaction coordinate. The classical system can be thought of in physical terms as two counterflowing probability streams, which stochastically exchange probability currents in a way that the net probability current, and hence the overall probability, suitably interpreted, is conserved. The proposed formalism extends the mathematics of those stochastic processes describable with linear, singlestep, unidirectional transition probabilities, known as Markov chains and stochastic matrices. It is shown that a certain rearrangement and combination of the input and output of two stochastic matrices of the same order yields another matrix of the same type. Each measurement causes the partial collapse of the probability current distribution in the midst of such a process, giving rise to calculable, but nonMarkov, values for the ensuing modification of the system’s output probability distribution. The paper concludes with an analysis of a simple classical probabilistic version of a socalled grandfather paradox. 1
Multivalent RandomWalkers — A Model for Deoxyribozyme Walkers
"... Abstract. We propose a stochastic model for molecular transport at the nanoscale that describes the motion of twodimensional molecular assemblies called multivalent random walkers (MVRWs). This walker model is an abstract description of the motion of multipedal molecular assemblies, called molecul ..."
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Abstract. We propose a stochastic model for molecular transport at the nanoscale that describes the motion of twodimensional molecular assemblies called multivalent random walkers (MVRWs). This walker model is an abstract description of the motion of multipedal molecular assemblies, called molecular spiders, which use deoxyribozyme legs to move over a surface covered with substrate DNA molecules, cleaving them to produce shorter product DNA molecules as they go. In this model a walker has a rigid inert body and several flexible enzymatic legs. A walker moves over a surface of fixed chemical sites. Each site has one of several molecular species displayed, and walker legs can bind to and unbind from these sites to move over the surface. Additionally, the enzymatic activity of the legs allows them to catalyze irreversible chemical changes to the sites, thereby permanently modifying the state of the surface. We describe a MVRW system as a continuoustime Markov process, where all state transitions in the process correspond to chemical reactions of the legs with the sites. We model the kinetics of the leg reactions by considering the constrained diffusion of the walker body and unattached leg. Through kinetic Monte Carlo simulations, we show that the irreversibility of the enzymatic action of the legs can bias the motion of walkers and cause them to move superdiffusively over significant distances. 1