We formally characterize a set of causality-based properties of metabolic networks. This set of properties aims at making precise several notions on the production of metabolites, which are familiar in the biologists' terminology. From a theoretical point of view, biochemical reactions are abstractly represented as causal implications and the produced metabolites as causal consequences of the implication representing the corresponding reaction. The fact that a reactant is produced is represented by means of the chain of reactions that have made it exist. Such representation abstracts away from quantities, stoichiometric and thermodynamic parameters and constitutes the basis for the characterization of our properties. Moreover, we propose an effective method for verifying our properties based on an abstract model of system dynamics. This consists of a new abstract semantics for the system seen as a concurrent network and expressed using the Chemical Ground Form calculus. We illustrate an application of this framework to a portion of a real metabolic pathway.
Understanding the relationships amongst the elements of biological interaction networks is a relevant problem in Systems Biology. In the words of [23], "diagrams of interconnections represent a sort of static roadmaps, but what we really seek to know are the traffic patterns, why such patterns emerge, and how we can control them". Formal descriptions of interconnections and methodologies for performing traffic simulations in silico can orientate in vitro experimentation.
We focus here on metabolic networks, i.e. the set of the cellular biochemical pathways involved in energy management and in the synthesis of structural components. Biochemical pathways are typically composed of chains of enzymatically catalyzed chemical reactions and are interconnected in a complex way. This makes difficult to understand the overall emerging behaviour of a network, starting from the detailed knowledge of the single reactions.
An interesting issue is the identification of the parts of a network whose integrity is crucial for certain functionalities. These “hot points” represent candidate drug targets for repressing undesired metabolic functions involved in pathological states, such as infectious diseases and cancer [9,15]. Several properties characterizing different aspects of the network functionalities have been introduced in the biological literature, often with slightly different versions for the same property. What formal methods can offer is a way to make precise and classify properties, too often expressed only at an intuitive level.
Since, broadly speaking, causality plays a key role in finding chains of reactions that connect the parts of a network, we base our understanding of properties in terms of causality relations. Following the approach in [5], in order to give a formal characterization of causality-based properties, we interpret biochemical reactions as “logical consequences”, where the source metabolites can cause, i.e. produce the target ones. Furthermore, we adopt the notion of explanation of a certain metabolite. Given a set of reactions and initial conditions, an explanation represents the chain of reactions, causally dependent one from the another, that leads to the metabolite. Our approach therefore models the biochemical dynamics, capturing causal dependencies, while abstracting away from other aspects, like quantities, stoichiometric and thermodynamic parameters.
On top of the causality notion, we formalize several properties from a potentially longer list. Beyond the relevance of their biological meaning, these properties show how the few and simple ingredients we propose are sufficiently expressive to make precise several common notions, often intuitively used in biology. Specifically, the set of properties we present concerns the role and the relations of metabolites and reactions within a metabolic network.
We propose an effective method for verifying the formalized causal properties, based on the construction once for all of an abstract representation of the dynamics of the biological system. The system is specified as a concurrent network in terms of the Chemical Ground Form calculus [7]. We opted for the CGC for its extreme simplicity and well established theories and techniques, while it is, at the same time, sufficiently concrete to capture our properties. For our verification purposes, we have defined a slightly different semantics from the one in [7,10]. It is worth pointing out that our choice mainly strives for simplicity. Other specification languages suitable for biological networks, e.g. [29,34,33,8,21], could have been adopted as well, some perhaps even more expressive, but generally requiring higher costs for model construction and verification procedures.
Overall, we are interested in efficiently evaluating the impact of changes on working hypotheses, such as the variation of the initial conditions and of the sets of reactions, according to a what-if strategy. The method we propose is meant to be exploited as a sort of preliminary in silico screening, aiming at determining the most promising experiments to be carried out in vitro. Finally, we believe that our framework should be palatable to biologists, since it is very close to the biochemical intuition of causality and to the spirit of many informal notions currently in use.
Related Work. Due to recent progress of wet-lab techniques, many metabolic networks are structurally well characterized and can be reconstructed for many organisms up to the genome-scale level (see e.g. [30]). However, approaches grounded on dynamical modeling, e.g. Metabolic Control Analysis or Metabolic Flux Analysis (see [16]), may encounter difficulties, mainly because part of the needed kinetic parameters are not known. In contrast, structure oriented analysis only requires information about the topology of the investigated networks, which is often known. Even though this kind of approach may not provide a detailed knowledge of the dynamics under
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