4,919 research outputs found
Analysis of Petri Net Models through Stochastic Differential Equations
It is well known, mainly because of the work of Kurtz, that density dependent
Markov chains can be approximated by sets of ordinary differential equations
(ODEs) when their indexing parameter grows very large. This approximation
cannot capture the stochastic nature of the process and, consequently, it can
provide an erroneous view of the behavior of the Markov chain if the indexing
parameter is not sufficiently high. Important phenomena that cannot be revealed
include non-negligible variance and bi-modal population distributions. A
less-known approximation proposed by Kurtz applies stochastic differential
equations (SDEs) and provides information about the stochastic nature of the
process. In this paper we apply and extend this diffusion approximation to
study stochastic Petri nets. We identify a class of nets whose underlying
stochastic process is a density dependent Markov chain whose indexing parameter
is a multiplicative constant which identifies the population level expressed by
the initial marking and we provide means to automatically construct the
associated set of SDEs. Since the diffusion approximation of Kurtz considers
the process only up to the time when it first exits an open interval, we extend
the approximation by a machinery that mimics the behavior of the Markov chain
at the boundary and allows thus to apply the approach to a wider set of
problems. The resulting process is of the jump-diffusion type. We illustrate by
examples that the jump-diffusion approximation which extends to bounded domains
can be much more informative than that based on ODEs as it can provide accurate
quantity distributions even when they are multi-modal and even for relatively
small population levels. Moreover, we show that the method is faster than
simulating the original Markov chain
Bisimulation Relations Between Automata, Stochastic Differential Equations and Petri Nets
Two formal stochastic models are said to be bisimilar if their solutions as a
stochastic process are probabilistically equivalent. Bisimilarity between two
stochastic model formalisms means that the strengths of one stochastic model
formalism can be used by the other stochastic model formalism. The aim of this
paper is to explain bisimilarity relations between stochastic hybrid automata,
stochastic differential equations on hybrid space and stochastic hybrid Petri
nets. These bisimilarity relations make it possible to combine the formal
verification power of automata with the analysis power of stochastic
differential equations and the compositional specification power of Petri nets.
The relations and their combined strengths are illustrated for an air traffic
example.Comment: 15 pages, 4 figures, Workshop on Formal Methods for Aerospace (FMA),
EPTCS 20m 201
Petri nets for systems and synthetic biology
We give a description of a Petri net-based framework for
modelling and analysing biochemical pathways, which uni¯es the qualita-
tive, stochastic and continuous paradigms. Each perspective adds its con-
tribution to the understanding of the system, thus the three approaches
do not compete, but complement each other. We illustrate our approach
by applying it to an extended model of the three stage cascade, which
forms the core of the ERK signal transduction pathway. Consequently
our focus is on transient behaviour analysis. We demonstrate how quali-
tative descriptions are abstractions over stochastic or continuous descrip-
tions, and show that the stochastic and continuous models approximate
each other. Although our framework is based on Petri nets, it can be
applied more widely to other formalisms which are used to model and
analyse biochemical networks
Computational Modeling for the Activation Cycle of G-proteins by G-protein-coupled Receptors
In this paper, we survey five different computational modeling methods. For
comparison, we use the activation cycle of G-proteins that regulate cellular
signaling events downstream of G-protein-coupled receptors (GPCRs) as a driving
example. Starting from an existing Ordinary Differential Equations (ODEs)
model, we implement the G-protein cycle in the stochastic Pi-calculus using
SPiM, as Petri-nets using Cell Illustrator, in the Kappa Language using
Cellucidate, and in Bio-PEPA using the Bio-PEPA eclipse plug in. We also
provide a high-level notation to abstract away from communication primitives
that may be unfamiliar to the average biologist, and we show how to translate
high-level programs into stochastic Pi-calculus processes and chemical
reactions.Comment: In Proceedings MeCBIC 2010, arXiv:1011.005
A structured approach for the engineering of biochemical network models, illustrated for signalling pathways
http://dx.doi.org/10.1093/bib/bbn026Quantitative models of biochemical networks (signal transduction cascades, metabolic pathways, gene regulatory circuits) are a central component of modern systems biology. Building and managing these complex models is a major challenge that can benefit from the application of formal methods adopted from theoretical computing science. Here we provide a general introduction to the field of formal modelling, which emphasizes the intuitive biochemical basis of the modelling process, but is also accessible for an audience with a background in computing science and/or model engineering. We show how signal transduction cascades can be modelled in a modular fashion, using both a qualitative approach { Qualitative Petri nets, and quantitative approaches { Continuous Petri Nets and Ordinary Differential Equations. We review the major elementary building blocks of a cellular signalling model, discuss which critical design decisions have to be made during model building, and present ..
CSL model checking of Deterministic and Stochastic Petri Nets
Deterministic and Stochastic Petri Nets (DSPNs) are a widely used high-level formalism for modeling discrete-event systems where events may occur either without consuming time, after a deterministic time, or after an exponentially distributed time. The underlying process dened by DSPNs, under certain restrictions, corresponds to a class of Markov Regenerative Stochastic Processes (MRGP). In this paper, we investigate the use of CSL (Continuous Stochastic Logic) to express probabilistic properties, such a time-bounded until and time-bounded next, at the DSPN level. The verication of such properties requires the solution of the steady-state and transient probabilities of the underlying MRGP. We also address a number of semantic issues regarding the application of CSL on MRGP and provide numerical model checking algorithms for this logic. A prototype model checker, based on SPNica, is also described
Flux Analysis in Process Models via Causality
We present an approach for flux analysis in process algebra models of
biological systems. We perceive flux as the flow of resources in stochastic
simulations. We resort to an established correspondence between event
structures, a broadly recognised model of concurrency, and state transitions of
process models, seen as Petri nets. We show that we can this way extract the
causal resource dependencies in simulations between individual state
transitions as partial orders of events. We propose transformations on the
partial orders that provide means for further analysis, and introduce a
software tool, which implements these ideas. By means of an example of a
published model of the Rho GTP-binding proteins, we argue that this approach
can provide the substitute for flux analysis techniques on ordinary
differential equation models within the stochastic setting of process algebras
Exploring the concept of interaction computing through the discrete algebraic analysis of the Belousov–Zhabotinsky reaction
Interaction computing (IC) aims to map the properties of integrable low-dimensional non-linear dynamical systems to the discrete domain of finite-state automata in an attempt to reproduce in software the self-organizing and dynamically stable properties of sub-cellular biochemical systems. As the work reported in this paper is still at the early stages of theory development it focuses on the analysis of a particularly simple chemical oscillator, the Belousov-Zhabotinsky (BZ) reaction. After retracing the rationale for IC developed over the past several years from the physical, biological, mathematical, and computer science points of view, the paper presents an elementary discussion of the Krohn-Rhodes decomposition of finite-state automata, including the holonomy decomposition of a simple automaton, and of its interpretation as an abstract positional number system. The method is then applied to the analysis of the algebraic properties of discrete finite-state automata derived from a simplified Petri net model of the BZ reaction. In the simplest possible and symmetrical case the corresponding automaton is, not surprisingly, found to contain exclusively cyclic groups. In a second, asymmetrical case, the decomposition is much more complex and includes five different simple non-abelian groups whose potential relevance arises from their ability to encode functionally complete algebras. The possible computational relevance of these findings is discussed and possible conclusions are drawn
Computational models for inferring biochemical networks
Biochemical networks are of great practical importance. The interaction of biological compounds in cells has been enforced to a proper understanding by the numerous bioinformatics projects, which contributed to a vast amount of biological information. The construction of biochemical systems (systems of chemical reactions), which include both topology and kinetic constants of the chemical reactions, is NP-hard and is a well-studied system biology problem. In this paper, we propose a hybrid architecture, which combines genetic programming and simulated annealing in order to generate and optimize both the topology (the network) and the reaction rates of a biochemical system. Simulations and analysis of an artificial model and three real models (two models and the noisy version of one of them) show promising results for the proposed method.The Romanian National Authority for Scientific Research, CNDI–UEFISCDI,
Project No. PN-II-PT-PCCA-2011-3.2-0917
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