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 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.
Often a few elementary events of binding and covalent modification [31] in a biomolecular reaction system give rise to a combinatorial number of non-isomorphic reachable species or complexes [17,18]. Instances of such systems are signaling pathway, polymerizations involved in cytoskeleton maintainance, the formation of transcription factor complexes in gene-regulation.
For such biomolecular systems, traditional chemical kinetics face fundamental limitations, that are related to the question of how biomolecular events are represented and translated into a mathematical model [24]. More specifically, chemical reactions can only operate on a collection of fully specified molecular species and each such species gives rise to one differential equation, describing the rate of change of that species’ concentration. Many combinatorial systems do not permit the enumeration of all molecular species and thus render their traditional differential description prohibitive. However, even if one could enumerate them, it remains questionable whether chemical reactions are the appropriate way to represent and to reason about such systems.
As the dynamics of a biomolecular reaction mixture comes about through the repeated execution of a few elementary events one may wonder about the effective degrees of freedom of the reaction mixture’s dynamics. If the velocity of all events -or their probabilities to occur per time-unit per instance -are different for all complexes (w.r.t. modification) and pairs of complexes (w.r.t. binding) to which the events can apply to, then the degrees of freedom would equal to the number of molecular species. However, due to the local nature of physical forces underlying molecular dynamics, the kinetics of most events appear to be ignorant with respect to the global configuration of the complexes they are operating on. More provocatively, one may say that even if there would be variations of kinetics of an event from one context to another, experimental biology does not -and most likely never will -have the means to discern between all different contexts. For instance, fluorescence resonance energy transfer (FRET), may report on a specific protein-binding event and even its velocity, however we have no means to determine whether the binding partners are already part of a protein complex -not to speak of the composition and modification state of these complexes. To this end, molecular species remain elusive and appear to be inappropriate entities of descriptions.
To align with the mentioned experimental insufficiencies and with the underlying biophysical locality, rule-based or agent-based descriptions were introduced as a framework to encode such reaction mixtures succinctly and to enable their mathematical analysis [8,3]. Rules exploit the limited context on which most elementary events are conditioned. They just enumerate that part of a molecular species that is relevant for a rule to be applicable. Thus, in contrast to chemical reactions, rules can operate on a collection of partially specified molecular species. Recently, attempts have been made to identify the set of those partially specified species that allow for a self-consistent description of the rule-set’s dynamics [11,6]. Naturally, as partially specified species -or fragments -in general encompass many fully specified species, the cardinality of that set is less than of the set of molecular species. These approaches aim to obtain a self-consistent fragment dynamics based on ordinary differential equations. It represents the dynamics in the thermodynamic limit of stochastic kinetics when scaling species multiplicities to infinity while maintaining a constant concentration (multiplicity per unit volume) [22]. In many applications in cellular biology this limiting dynamics is an inappropriate model due to the low multiplicities of some molecular species -think of transcription factor -DNA binding events. We recently showed that the obtained differential fragments cannot be used to describe the finite volume case of stochastic kinetics [12]. Exploiting statistical independence of events we instead derived stochastic fragments that represent the effective degrees of freedom in the stochastic case. Conceptually, the procedure replaces the Cartesian product by a Cartesian sum for statistically independent states. In contrast to the differential case, stochastic fragments have the important property that the sample paths of molecular species can be recovered from that of partially specified species.
We believe that interdisciplinary fields, such as systems biology, can move forward quickly enough only if the well-established knowledge in each of the disciplines involved is exploited to its maximum, i.e. if the standardized, well-established theories are recognized and re-used. For that reason, in this paper we translate our abstraction method ( [12]) into the language of well-established contexts of abstraction for probabilistic systems -lum
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