Metabolic Pathway Analysis in the Presence of Biological Constraints

Metabolic pathway analysis is a key method to study a metabolism in its steady state, and the concept of elementary fluxes (EFs) plays a major role in the analysis of a network in terms of non-decomposable pathways. The supports of the EFs contain in particular those of the elementary flux modes (EFMs), which are the support-minimal pathways, and EFs coincide with EFMs when the only flux constraints are given by the irreversibility of certain reactions. Practical use of both EFMs and EFs has been hampered by the combinatorial explosion of their number in large, genome-scale systems. The EFs give the possible pathways in a steady state but the real pathways are limited by biological constraints, such as thermodynamic or, more generally, kinetic constraints and regulatory constraints from the genetic network. We provide results on the mathematical structure and geometrical characterization of the solution space in the presence of such biological constraints (which is no longer a convex polyhedral cone or a convex polyhedron) and revisit the concept of EFMs and EFs in this framework. We show that most of the results depend only on very general properties of compatibility of constraints with vector signs: either sign-invariance, satisfied by regulatory constraints, or sign-monotonicity (a stronger property), satisfied by thermodynamic and kinetic constraints. We show in particular that the solution space for sign-monotone constraints is a union of particular faces of the original polyhedral cone or polyhedron and that EFs still coincide with EFMs and are just those of the original EFs that satisfy the constraint, and we show how to integrate their computation efficiently in the double description method, the most widely used method in the tools dedicated to EFs computation. We show that, for sign-invariant constraints, the situation is more complex: the solution space is a disjoint union of particular semi-open faces (i.e., without some of their own faces of lesser dimension) of the original polyhedral cone or polyhedron and, if EFs are still those of the original EFs that satisfy the constraint, their computation cannot be incrementally integrated into the double description method, and the result is not true for EFMs, that are in general strictly more numerous than those of the original EFMs that satisfy the constraint.

Metabolic pathway analysis in presence of biological constraints

Metabolic pathway analysis is a key method to study metabolism at steady state and the elementary fluxes (EFs) is one major concept allowing one to analyze the network in terms of non-decomposable pathways. The supports of the EFs contain in particular the supports of the elementary flux modes (EFMs), which are the support-minimal pathways, and EFs coincide with EFMs when the only flux constraints are given by the irreversibility of certain reactions. Practical use of both EFMs and EFs has been hampered by the combinatorial explosion of their number in large, genome-scale, systems. The EFs give the possible pathways at steady state but the real pathways are limited by biological constraints, such as thermodynamic or, more generally, kinetic constraints and regulatory constraints from the genetic network. We provide results about the mathematical structure and geometrical characterization of the solutions space in presence of such biological constraints and revisit the concept of EFMs and EFs in this framework. We show that most of those results depend only on a very general property of compatibility of the constraints with sign: either sign-based for regulatory constraints or sign-monotone (a stronger property) for thermodynamic and kinetic constraints. We show in particular that EFs for sign-monotone constraints are just those original EFs that satisfy the constraint and show how to efficiently integrate their computation in the double description method, the most widely used method in the tools dedicated to EFMs computation.

Enterprise in a Market Environment: A Double Tetrad Model

System-based economic theory is a natural platform for the duality concept application in modeling and studying of economic systems. Aside from revealing the nature of economic events and showing regular connections between them, the analysis of duality also allows to develop methods to efficiently manage economic processes. This paper reexamines modeling methods of economic system performance. New approach uses double description method, via dual tetrads, one illustrative of intrasystem processes and the other exhibiting external interactions of the system. It is suggested that interactions between business and market are representable as production and implementation cycles. During the former, the initial batch of raw and producing materials are sequenced through object, environment, process and project subsystems of the enterprise. Then finished product enters implementation cycle and passes through object, environment, process and project subsystems of the enterprise’s external environment. Within this cycle the object subsystem represent the enterprise as a manufacturer; environment subsystem represents market environment for sales; process subsystem represents sales process; project subsystem represents delivery of products to the customer and transfer of revenue to the manufacturer. Production flow therefore loops through the stages of these cycles as a going concern, ensuring continuous operation of the enterprise conditioned upon coordination with market. The dual tetrad model allows to determine the roles of administration (with internal content being a controlled object) and marketing (where external environment is a controlled object) in the structure of enterprise management. This paper also analyses general patterns for identification of duality phenomena in system-based economic theory and mathematical economic models of business administration.This article is an expanded report presented at the plenary session of the XX “Corporate Planning and business Development” Anniversary Symposium (April 9–10, 2019, Moscow).