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Although we are concerned with Markovian models, the speci c structure of biological systems as well as the practically relevant topics largely disable direct one-to-one mappings from other domains to systems biology Markovian models of biological and genetic networks are almost always potentially in nite in either case the state space is multidimensional and extremely huge They are typically not ergodic and contain absorbing states or absorbing classes since some molecular species might get exhausted, which often implies that no more reactions are possible at all or only reactions that do not lead out of a certain set of states Absorption times can be important such as the time to apoptosis (cell death) or the time to viral infection Stationary distributions are less relevant than in many other domains They are relevant, for example, in the case of multiple absorbing classes, meta- or bistable systems such as bistable genetic switches, but transient phenomena are often of primary interest In general, transient analysis tends to be much more dif cult than steady-state (limiting to in nite time) analysis Moreover, in biological and genetic network models, except for zero-order reactions, we are always concerned with state-dependent rates given by the propensity functions This is a signi cant difference from, for example, the vast majority of queuing models where constant arrival and service rates are rather usual Inherent to biochemically reacting systems are multiple time scales There are fast and slow reactions with rates differing by many orders of magnitude such that the system is stiff, which makes numerical analysis cumbersome both in the domain of ODEs and in the domain of CTMCs Stiffness also causes serious problems in stochastic simulation Consequently, ef cient simulation of stiff Markovian models is a major challenge in stochastic modeling and analysis of biological systems Obviously, reactions with very small rates are rare events Within the particular framework of huge and stiff Markovian models, some transition rates might be unknown In practice, models are built according to real-world observations which in the context of biological systems often correspond to experiences and conclusions drawn from lab experiments If the rates of speci c reactions are unknown but empirical data is available, rare event simulation can be suitable for estimating small reaction rates Therefore, estimating unknown rare event probabilities or rates is one topic of potentially high relevance Even if the actual goal of the study is not the determination of the reaction rates, these rates are required for an appropriate model Then rare event simulation becomes part of model building Almost all stochastic simulation techniques that have been applied in systems biology originate from chemical physics, which is quite natural against the background of molecular reactions driving biological and genetic networks At. java ean 13 reader Java Barcode Reader Tutorial to scan, read linear, 2d barcodes in ...
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