CASE: Modelling Genetic Transcription

In a similar manner we shall also fashion an abstract model of genetic scription (Section 2.1.2), which is the second of our case studies. Genetic tran-scription is primarily a coordination scenario. It involves both the formation of coordination compounds and recurrent behaviour; this is modelled by recursive processes that coordinate within extruded scopes.

The resulting model has three components:

The genome is a collection of genes. Each gene repeatedly allows polymerase to attach at the promoter and then transcribe the gene by reading the nucleotides of the sequence one at a time. Hence, in the model we portray a gene as a recurrent processgene_iwhere each recurrence corresponds to a complete transcription and has three phases:

Initiation The gene supports the formation of coordination compounds by

gene1=

Table 3.6: Abstract model of genetic transcription.

offering scope extrusion of the private channels gi and bi on the public channel tr.

Elongation The gene supports a sequential ‘reading’ of its nucleotide sequence by offering a sequence of corresponding communications on the channel gi. At each step the gene is prepared for the coordination compound to be broken up via a message on the private channelbi.

Termination If all nucleotides are ‘read’ the gene breaks up the coordination compound by sending a message on the private channelb1.

Note that, in the concrete example, the genomeis a collection of only a single gene. As the BioAmbients syntax does not allow empty synchronising commu-nications we use thedi as dummy messages.

The cell provides a potentially unbounded supply of nucleoside tri-phosphates

that provides the nucleotides for the growing messenger RNA. In the model we capture this by replicating processes corresponding to each of the NTPs in question. Each replica offers just a single interaction on a public channel corresponding to the specific nucleotide. In the concrete model the nTP is simply an unbounded collection of ATP,GTP,CTP, andUTP.

Polymerase is an enzyme that makes RNA copies of DNA templates. It repeat-edly attaches to the promoter region of a (not specific) gene and then synthesises a corresponding RNA replica by processing one nucleotide at the time. Here we model the enzyme as a recurrent process,polymerase, where each major re-currence, recP.· · ·, corresponds to a complete transcription and each minor recurrence,recX.· · ·, corresponds to a single nucleotide. Thereby we achieve a generic polymerase that will transcribe a gene of any length in three phases:

Initiation The polymerase forms a coordination compound with a gene by engaging in interaction on the public channel tr. Here it receives two private channelsgeneandb that allows it to coordinate specifically with the gene in question.

Elongation The recursive (sub-)process,recX.· · ·, either handles the next nu-cleotide in two steps:

• First, the type of nucleotide is communicated from gene to poly-merase via the private channelgene.

• Then, if too long time passes before a corresponding NTP turns up, the polymerase breaks the coordination compound by sending a signal on the private channelb and then recurs to its initial state, (recP.· · ·). Otherwise, the polymerase reacts with the NTP and recurs to handle the next nucleotide (recX.· · ·).

Termination Or the genetic sequence (ACAC in the concrete case) is fully transcribed and the coordination compound is broken by a signal onb. The polymerase then recurs to its initial state (recP.· · ·) while, at the same time, releasing a piece of messenger RNA corresponding to the transcribed sequence.

Note that this model does not pay any attention to the cooperativity issues of activators, inhibitors, genes, and polymerases. These issues, that are particu-larly relevant to the initiation phase, have been treated in great detail by Kuttler and Niehren [KN06] and also by Blossey, Cardelli, and Phillips [BCP06].

Also note that the model rests on the assumption that transcription may termi-nate prematurely. This type of behaviour, however, does not seem to be frequent

in living systems; rather it seems to be connected with certain regulatory mech-anisms that we shall not treat in detail. For our purposes the assumption simply results in a model that is slightly more complex and, thus, constitutes a more interesting subject of analysis in later chapters.

3.5 Concluding Remarks

In this chapter we have introduced a variant of the BioAmbients calculus [Reg03, RPS⁺04] that incorporates general recursion in the manner of CCS [Mil80] and omits unrestricted choice in favour of guarded sum. The fundamental concepts of the calculus were introduced in three stages: First the fragment of simple reactive processes, which is strongly related to Milner’s CCS [Mil80]. Then the fragment of complex forming processes, which is strongly related to the π-calculus by Milner, Walker, and Parrow [MPW92, Mil99]. And, finally, com-partment forming processes in the form of the full BioAmbients calculus, which adds elements of Cardelli and Gordon’s Mobile Ambients [CG00], but in the style of Levi and Sangiorgi’s Mobile Safe Ambients [LS03] where all progress is a result of binary reactions rather than unary actions. One may observe that the resulting three classes of process expression correspond to the three classes of biological abstract machines identified by Cardelli [Car05a]:

The protein machine is concerned with Biochemical Networks, and seems to be adequately modelled by the notion of simple reactive processes. This is the view taken by Calder et al, who use the Performance Evaluation Process Algebra (PEPA), which is basically a stochastic extension of simple reactive processes, for the modelling and analysis of various signalling pathways [CGH06, CGH05, CDGH06, CGHV06], and, especially, Danos and Krivine in their work on formal biology in CCS [DK03]. Danos et al have made a few attempts at capturing special properties of the biological domain by addressing them at a more fundamental level: Reversible CCS aims to capture the concept of truly reversible reactions [DK03, DK04]. The κ-calculus focuses on the notion of complexation and its effect on the folding of proteins, in terms of exposure or hiding of binding sites, [DL03a, DL03b, DL04].

The Gene Machine is concerned with Gene Regulatory Networks, and seems to be adequately modelled by the notion of complex forming processes. This view was pioneered by Regev et al [RSS01], who worked with Priami [PRSS01]

in order to base their BioSPI stochastic simulation tool on Priami’s Stochastic version of the π-calculus [Pri96] coupled with Gillespie type rate estimations [Gil77, Gil76]. Technically this line of work has prospered at the hands of Phillips and Cardelli who have worked on graphical notations [PC05], simulation tools

[PC04], and modelling methodologies (Blossey et al) [BCP06]. The modelling efforts of Kuttler et al [KN06] have also led to technical extensions in the form of concurrent objects [Kut07]. Priami and Quaglia et al have fashionedβ-binders [PQ04] – a calculus that combines features from the κ-calculus [DL04], the stochastic π-calculus [Pri96], and the ambient calculus [CG00]. in a manner that goes some way towards relinquishing the deterministic key-lock assumption in favour of a more nuanced notion of affinity.

The Membrane Machine is concerned with Transport Networks, and seems to be adequately modelled by the notion of membrane forming (BioAmbients) pro-cesses – again a view that was pioneered by Regev et al [Reg03, RPS⁺04]. Other alternatives have emerged. Most notably Cardelli’s Brane calculus [Car05b]

that quite accurately models biomembranes as oriented 2-dimensional surfaces of bubbles that fuse and split. This model was simplified by Danos et al [DP04], but features for the modelling of large molecules and complexes have only been proposed quite recently [CP05]. A recent comparative analysis of BioAmbients and Brane calculus concludes that, from a meta point of view the two calculi are very similar [Ver07]. The β-binders [PQ04] also introduces a notion of boxed enclosures. These enclosures, however, do not nest and move, which renders the modelling of compartments less intuitive.