Article

# Lumpability Abstractions of Rule-based Systems

(Impact Factor: 0.66). 05/2012; 431:137--164. DOI: 10.1016/j.tcs.2011.12.059
Source: OAI

ABSTRACT

The induction of a signaling pathway is characterized by transient complex formation and mutual posttranslational modification of proteins. To faithfully capture this combinatorial process in a mathematical model is an important challenge in systems biology. Exploiting the limited context on which most binding and modification events are conditioned, attempts have been made to reduce the combinatorial complexity by quotienting the reachable set of molecular species, into species aggregates while preserving the deterministic semantics of the thermodynamic limit. Recently we proposed a quotienting that also preserves the stochastic semantics and that is complete in the sense that the semantics of individual species can be recovered from the aggregate semantics.
In this paper we prove that this quotienting yields a sufficient condition for \emph{weak lumpability} and that it gives rise to a backward Markov bisimulation between the original and aggregated transition system. We illustrate the framework on a case study of the EGF/insulin receptor crosstalk.

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Available from: Tatjana Petrov,
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• "Other reaction network model reduction methods are restricted to deterministic models [32] [33] including a reduction from 42 to 29 molecular species [34]. The most comparable method in terms of problem size may be [35] which like GCCD applies to rule-based reaction networks. We will quantitatively compare degrees of model reduction of these methods to GCCD in section 3.4. "
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ABSTRACT: A stochastic reaction network model of Ca(2+) dynamics in synapses (Pepke et al PLoS Comput. Biol. 6 e1000675) is expressed and simulated using rule-based reaction modeling notation in dynamical grammars and in MCell. The model tracks the response of calmodulin and CaMKII to calcium influx in synapses. Data from numerically intensive simulations is used to train a reduced model that, out of sample, correctly predicts the evolution of interaction parameters characterizing the instantaneous probability distribution over molecular states in the much larger fine-scale models. The novel model reduction method, 'graph-constrained correlation dynamics', requires a graph of plausible state variables and interactions as input. It parametrically optimizes a set of constant coefficients appearing in differential equations governing the time-varying interaction parameters that determine all correlations between variables in the reduced model at any time slice.
Physical Biology 06/2015; 12(4):045005. DOI:10.1088/1478-3975/12/4/045005 · 2.54 Impact Factor
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• "A simple example of a Kappa model is presented in Fig. 1. For the stochastic semantics of Kappa, that is a continuoustime Markov chain (CTMC) assigned to a rule-based model, we refer to [12] or [17]. Intuitively, any rule-based system can be expanded to an equivalent reaction system (with potentially infinitely many species and reactions). "
##### Article: Efficient reduction of Kappa models by static inspection of the rule-set
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ABSTRACT: When designing genetic circuits, the typical primitives used in major existing modelling formalisms are gene interaction graphs, where edges between genes denote either an activation or inhibition relation. However, when designing experiments, it is important to be precise about the low-level mechanistic details as to how each such relation is implemented. The rule-based modelling language Kappa allows to unambiguously specify mechanistic details such as DNA binding sites, dimerisation of transcription factors, or co-operative interactions. However, such a detailed description comes with complexity and computationally costly execution. We propose a general method for automatically transforming a rule-based program, by eliminating intermediate species and adjusting the rate constants accordingly. Our method consists of searching for those interaction patterns known to be amenable to equilibrium approximations (e.g. Michaelis-Menten scheme). The reduced model is efficiently obtained by static inspection over the rule-set, and it represents a particular theoretical limit of the original model. The Bhattacharyya distance is proposed as a metric to estimate the reduction error for a given observable. The tool is tested on a detailed rule-based model of a $\lambda$-phage switch, which lists $96$ rules and $16$ agents. The reduced model has $11$ rules and $5$ agents, and provides a dramatic reduction in simulation time of several orders of magnitude.
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• "Reduction of stochastic rule-based models, based on a weakened version of the exact lumpability criterion, has been proposed by Feret et al. (2012) to define abstract species or stochasticfragments that can be further used in simplified calculations. More generally, rule-based models alow to overcome combinatorial complexity in stochastic simulations (Danos et al., 2007b). "
##### Article: Reduction of dynamical biochemical reaction networks in computational biology
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ABSTRACT: Biochemical networks are used in computational biology, to model mechanistic details of systems involved in cell signaling, metabolism, and regulation of gene expression. Parametric and structural uncertainty, as well as combinatorial explosion are strong obstacles against analyzing the dynamics of large models of this type. Multiscaleness, an important property of these networks, can be used to get past some of these obstacles. Networks with many well separated time scales, can be reduced to simpler models, in a way that depends only on the orders of magnitude and not on the exact values of the kinetic parameters. The main idea used for such robust simplifications of networks is the concept of dominance among model elements, allowing hierarchical organization of these elements according to their effects on the network dynamics. This concept finds a natural formulation in tropical geometry. We revisit, in the light of these new ideas, the main approaches to model reduction of reaction networks, such as quasi-steady state (QSS) and quasi-equilibrium approximations (QE), and provide practical recipes for model reduction of linear and non-linear networks. We also discuss the application of model reduction to the problem of parameter identification, via backward pruning machine learning techniques.
Frontiers in Genetics 05/2012; 3:131. DOI:10.3389/fgene.2012.00131