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The equivalence of the chemical master equation and Kolmogorov differential equations can now be easily seen by interpreting i N as the number assigned to state x X , that is, pi(t) = p (t) (x), qij = m (x) if j is the number assigned to state x + vm , and qj i = m (x vm ) if j is the number assigned to state x vm
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1123 Model structure and rare event issues
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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
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present, stochastic simulation approaches to estimating unknown reaction rates or rates of reactive pathways between metastable regions almost exclusively appear in chemical physics However, since the models are similar these methods are applicable in systems biology Many of the algorithms in use are variations around importance sampling or closely related ideas; see [16, 28, 31] For advanced, well-worked-out approaches we refer the interested reader in particular to transition path sampling [10, 30] and forward ux sampling [1, 2, 50, 76] Presumably, sooner or later they will become more widespread and make their way into systems biology just like methods for simulating stiff models with known reaction rates Our major focus is on ef cient simulation techniques for stiff models with all rates known Rather than aiming to estimate unknown reaction rates or unknown probabilities of certain trajectories or sets of states, we consider accelerated simulation in the presence of rare events with known probabilities or rates, respectively Some such techniques are established in systems biology and have been more or less extensively applied After giving some background on stochastic simulation of biological systems in Section 113, we discuss in Section 114 the applicability of importance sampling Section 115 outlines partitioning-based techniques as one important class of approximation approaches that are well suited for model reduction as well as for hybrid analysis Section 116 provides a quite comprehensive description of adaptive tau-leaping, an approximate multistep method that currently appears to be the most mature approach to accelerated simulation of stiff biochemically reacting systems
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