More formally, each I/O-relation is a quadruple (χ, [x], [p], [y]), which we write in the form where [x] 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, Equation 1 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 H₂O, CO₂, 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.

Definition 1 The I/O-relation χ is traceable if and only if for each informational molecule y  [y] in the output and each letter y_k  y we can uniquely determine whether

The collection of traceable I/O-relations involving informational molecules represents the “linear” part of information metabolism. We suggest that it is not only well defined but also encompasses important processing steps in the “life-history” of a transcript, including: primary transcription, splicing, translation, insertion editing, cleavage, intein extraction, chemical modification, poly-adenylation, etc. Thus we can interpret as the subsystem of gene expression in a cell, organism, or ecosystem.

In a traceable reaction, we can determine, for any collection of sequence intervals in the output [y], the collection of all those sequence intervals in the input [x] that gave rise to the encoded letters in the output. This inverse map χ⁻¹ gives us a well-defined “footprint” of each stretch of output sequence on the input. The concatenation of such inverse maps is well defined and allows each sequence position in an informational molecules z to be traced back to its genomic source, the genomic footprint of z. This construction will provide us with the structural part of our gene concept. Any letter x_k of x that is not contained in the genomic footprint Γ(x) of x is identified as being inserted or appended at a particular stage in the production of z. The genomic “source” Γ(x) can be one of the cell’s genomes or, for instance, an intruding viral transcript.

In some cases, a product x that appears in the linear part of the information metabolism may have an empty genomic footprint Γ(x) =  This is the case e.g., for the so-called non-ribosomal peptides (NRPs), which are synthesized de novo without using the templating function of a messenger RNA (Walton 2006).

Formally, let us denote by param(χ) the set of parameters of the I/O-relation χ. This leads us to

Definition 2 The functionFct(z) of an object z  X the set of I/O-relationsWhile this looks somewhat contrived, it forms the basis of the official Nomenclature of Enzymes (see (NC-ICBMB and Webb 1992) and the annual supplements at http://www.chem.qmul.ac.uk/iubmb/enzyme/supplements/), in which enzymes are named for the chemical reactions that they catalyse: an alcohol dehydrogenase, for instance is per definitionem an enzyme that catalyzes the dehydrogenation of alcohols. In our setting, this would be represented by associating z with a set of I/O-relations Fct(z) that consists (largely) of chemical reactions χ describing the dehydrogenation of various alcohols. Also note that nothing precludes an object from having multiple disparate functions in this framework i.e., Fct(z) can be very large and contain several, semantically different, groups of I/O-relations.

Applying this notion of function, we identify the collection of functional informational molecules as those biopolymer sequences x for which Fct(x) ≠ . Note, again, that in this way we pin function only to physical objects that influence transformations (of other objects). “Being transformed”, on the other hand, by construction does not count as a function in itself. Furthermore, an informational molecule does not acquire a function, for example, by housing a cis-regulatory element that regulates its own subsequent processing step. Our model declares that the function of regulating/influencing subsequent processing step ξ is attributed to the parameter(s) of ξ, not to the argument of ξ. Intermediate processing steps thereby will not typically have a function. For instance, if x is the mRNA coding for a protein p, we will often observe that Fct(x) = , while the translation product p of x typically will have some function as an enzyme, signal molecule, or structural protein, such that Fct(p) ≠  Fig. 1. Not all proteins are necessarily functional. The precursor p of one or more small hormone peptides p₁′,...,p_r′, for instance, may not have any function alone (Dicou 2008). In this case, we have Fct(p_i′) ≠  for the hormone peptides p′, but Fct(p) =  for the prohormone protein, see c and d in Fig. 1.

In practise, Fct(z) is dependent upon the experimental and computational methodologies employed to determine the processes (I/O-relations) that are dependent upon z. Improved measurements thus have the potential to change our representation of Fct(z).

Definition 3 A gene on a given genome is the pair (Γ(z),z) consisting of a functional informational molecule z and its genomic footprint Γ(z), i.e., the collection of intervals on the genome that give rise to the encoded letters in the sequence z through a sequence of I/O-relations in Ξ.

In Definition 3, we require that there is at least one sequence of I/O-relations linking Γ(z) to the gene product z. Alternatively, we might want to include the specific sequence of I/O-relations in the definition of the gene. The distinction between these two alternative points of view is whether we would require that every gene has a unique way of being processed (each particular sequence of I/O-relations linking Γ(z) to z), or whether we allow that a gene (Γ(z),z) can be expressed in alternatives ways. At present, we lack sufficient evidence of alternative transcripts processed into the same functional “gene product”, to decide which version is biologically more useful.

In order to analyze the notion of a function in detail, we assume that there is a distance function D that allows us to measure how different two I/O-relations χ′ and χ′′ are. In the analysis of chemical reaction networks, distance functions between reactions are typically based on a notion of differences Δ among underlying objects (Maggiora and Shanmugasundaram 2004). Given a measure of dis-similarity of objects, one constructs a dis-similarity for input, output and parameter lists, which are finally combined into the desired measure D (Tohsato and Nishimura 2007). We can use D to cluster the elements of Fct(z) into distinct functional classes and to construct a measure D of the functional differences between two objects. The functional distance D between I/O-relations can be extended naturally to sets of I/O-relations and hence implies a notion of functional distance which is conceptually related to network distance (Forst et al. 2006).

In the case of information molecules, we can think of the object distance Δ as an edit or alignment distance that measures a quantity of sequence similarity. In contrast, which can be derived from Δ and the system of I/O-relations Ξ, measures functional dissimilarity. Homology-based gene annotation is based on the observation that Δ and are correlated in practice. Thus similar sequences (of functional information molecules, or of their genomic footprints) often—but not always—give rise to products with similar functions. Deviations from this rule exist in nature and indicate that either large changes in function are acquired by closely related sequences (small Δ, large ), or to horizontal replacement of a gene by a functionally equivalent one (small large Δ). An example of the first type is the imprinting-related ncRNA Xist in Mammalia, which originated by pseudogenization of the lnx3 transcript whose primary product is a PDZ-like ring finger protein (Duret et al. 2006). An example of the latter type are several unrelated “clans” of serine proteases that share only the common catalytic triad Ser-His-Asp (Krem and Di Cera 2001).

The concept of homology is the subject of intensive discussion, with several competing definitions, see e.g., Laubichler (2000), Brigandt and Griffiths (2007). In the context of evolutionary and molecular biology, one requires that homologous characters are linked by common descent. In the strict “phylogenetic” definition, this is the only requirement. In our framework, phylogenetic homology of (Γ(x),x) and (Γ(y),y) is naturally established by the existence of a common ancestor of the genomic footprints Γ(x) and Γ(y). In practice, evidence for homology of genes can be evaluated by comparative sequence analysis of Γ(x) and Γ(y), i.e., in terms of Δ, the same way as this done for protein, RNA, or DNA sequences. The only modification is a more precise recipe for delimiting the sequences that need to be compared.

The strictly functional notion that identifies functionally equivalent genes runs into an insurmountable problem because there is nothing to prevent it from identifying objects that do not share common descent. This would lead to a gene concept that is not compatible with (phylogenetic) homology.

Peter F. Stadler, Email: studla/at/bioinf.uni-leipzig.de.

Sonja J. Prohaska, Email: sonja/at/bioinf.uni-leipzig.de.

Christian V. Forst, Email: christian.forst/at/utsouthwestern.edu.

David C. Krakauer, Email: krakauer/at/santafe.edu.