The basis for our construction is an abstract computational model of regulation. We start with the observation that cellular processes can be described as chemical reactions. This includes the interconversions of metabolites, the interactions of regulators, the aggregation of supermolecular structures, and the transport of molecules. There will be no need to operate at the level of individual molecules. It is more practical to employ coarse-grained representations. For instance, transcription could be viewed as an input/output (I/O)-relation that takes genomic DNA and a set of transcription factors as input, and results in a specific output transcript. In this way, we emphasize the computational aspects of bulk chemical reactions.
More formally, each I/O-relation is a quadruple (χ, [x
]), which we write in the form
] is a list of material components (inputs) transformed into a list of material outputs [y
] by means of a process χ that depends on a list [p
] of additional influences. We call [x
] the arguments and [p
] the parameters of χ. Equation 1
is an abstract, and arbitrarily coarse-grained representation of a chemical reaction. In chemical notation, we could write it in the form,
can also represent transport “reactions”, where input and output describe the same object(s) in different spatial locations or compartments, as well as other high-level aggregate processes including replication, transcription, translation, or the production of biomass (if one chooses not to model such parts of the system in detail). In contrast to an implementation at the finest level, that of elementary chemical reactions, the I/O-relations are not required to satisfy conservation of mass or atom types. We are able, for instance, to ignore ubiquitous chemical species (such as H2
, or coenzyme A) and energy and redox currencies ATP and NADH, if we choose. Our framework is consistent with, but will be more coarse-grained than, a full-fledged representation of all chemical reactions. This is a common coarse graining in Systems Biology models (Palsson 2006
). For our purposes, it will be convenient to model transcription and translation as I/O-relations that “produce” primary transcripts from a DNA template and a polypeptide from an RNA template. Equation 1
may also include compartment/spatial information and thus can describe cellular processes of more than one cell or organism, including a complete microbial community or even entire ecologies with complex predator–prey dynamics. Note that some or all elements of the output list [y
] of χ will typically appear as inputs [x
] and/or parameters [p
] of other I/O-relations ξ.
A system Ξ of I/O-relations over a given domain of “objects” X
has a natural interpretation as a model of computations on X
(Berry and Boudol 1992
; Taylor 1998
). This gives us considerable freedom in implementing a model of cellular processes in the form of Eq. 1
depending on: (1) the level of aggregation or abstraction beyond elementary chemical reactions; and (2) the effect that a parameter p
must have on the outcome of χ to be considered relevant. For example, we may define p
to be relevant to a particular I/O-relation χ if the absence of p
makes the transformation χ impossible. Alternatively, we could consider p
a relevant influence whenever it affects the reaction rate.
Before proceeding, a formal issue requires attention. Each process χ links a particular triplet of input, output, and parameter lists. Hence, transformations utilizing the same input [x
] to produce different outputs [y
′] ≠ [y
] are necessarily two distinct reactions χ and χ′. Here, we admit only physical objects as elements of the input and output lists [x
] and [y
]. The parameters [p
], on the other hand, may be either objects or physical quantities such as temperature or pH. The parameter list may be empty, [p
, e.g., in spontaneous chemical reactions or transport by diffusion.
If an object a
appears both as an argument, a
], and as a parameter, a p
, in the same I/O-relation χ, this implies an autocatalytic mechanism. The argument and the parameter are necessarily two different instantiations of the object type a
. The simplistic distinction between arguments and parameters in the formalism is akin to the notions of cis
action in molecular biology. Note, however, that the concepts are not equivalent in all cases.