The splicing of precursor messenger RNA (pre-mRNA) is an essential process in the expression of most eukaryotic genes. The five small nuclear ribonucleoproteins (snRNPs) and the many non-snRNP-associated proteins that constitute the splicing machinery, assemble anew on each precursor RNA to form the spliceosome complex that catalyses the two chemical reactions of splicing 
. Both the spliceosome components and the spliceosome assembly process are largely conserved between human and yeast. The complexity of the spliceosome is indicated by the
170 proteins that are associated with it 
. Adding to the complexity, splicing may occur partly, or entirely, concurrently with transcription. In eukaryotes, the interaction of the spliceosome with the precursor RNA can be considered to be an allosteric cascade in which early recognition steps induce conformational changes required for subsequent steps and for catalytic activation (reviewed by 
). However, the wealth of knowledge of molecular interactions, obtained mainly through extensive biochemical and genetic analyses, has yet to be formalised as a systems model of transcription and splicing.
Spliceosome assembly is thought to occur via a series of events with many points of regulation 
. In the first step, U1 snRNP binds to the 5′ splice site (5′SS), followed by the U2 snRNP at the branchsite. The U4, U5 and U6 snRNPs join as a tri-snRNP complex and, after the association of other, non-snRNP proteins, the spliceosome complex is activated for the first chemical step of splicing. The 5′ splice site is cleaved and, simultaneously, the 5′ end of the intron becomes covalently attached to the branchsite to form a branched, lariat structure. In the second step, the 3′ splice site (3′SS) is cleaved, which excises the intron, and the exons are joined to produce the mature mRNA. Between the two steps of splicing, a conformational change is required in the catalytic centre of the spliceosome 
, and at several stages during the cycle of spliceosome assembly, splicing and spliceosome dissociation, proofreading mechanisms are thought to operate 
. Nascent transcripts also have to be matured at their 3′ end, by cleavage and polyadenylation. illustrates spliceosome assembly and the two steps of splicing for a pre-mRNA with one intron that has already been polyadenylated and released from the DNA template.
Post-transcriptional and co-transcriptional splicing.
Splicing can also occur co-transcriptionally, prior to 3′ end maturation (), and there is considerable experimental evidence for functional coupling of transcription, splicing and 3′ end maturation in vivo 
. However, little is known about the impact of coupling on kinetic rates. Splicing has been modelled, but not to the same level of detail as transcription, and models of transcription have yet to fully incorporate the splicing reaction. Quantifying the dynamics of these processes remains a challenge 
, and modelling may have an important role to play in distinguishing functional dependencies from coincidental and contemporaneous effects, and in identifying and characterising the interactions that effect coupling.
Existing models of splicing have allowed splicing efficiency to be defined 
, and have shown that transcription by RNA polymerase II (Pol II) greatly increases splicing efficiency in comparison with transcription by T7 polymerase 
. A correlation between splicing efficiency and the pausing of Pol II on short terminal exons has been reported 
. Splicing has been represented as a single irreversible reaction that creates the product mRNA from pre-mRNA 
, and as a single irreversible reaction that creates mRNA from the pre-mRNA+spliceosome complex 
. To-date, steps one and two of splicing have not been modelled as separate reactions, nor have the co- and post-transcriptional splicing pathways been distinguished. Further insights into splicing can be expected by more detailed modelling and analysis.
As noted above, splicing can occur during messenger RNA transcription. Transcription begins with the assembly of the pre-initiation complex at the promoter. This complex includes Pol II, which, after initiation, begins the transcript elongation process that transcribes DNA into RNA. Early in elongation, the pre-mRNA is capped at its 5′ end by the capping enzymes. Elongation involves a sequence of many hundreds of individual polymerisation reactions, and hence the time required to complete the elongation of a transcript is predicted to have less variability than a single-step process with an equivalent rate 
. The mature 3′ end of the RNA is formed by an endonucleolytic cleavage at the so-called poly A site and the newly formed 3′ end is extended by polyadenylation (reviewed by 
). The elongation process and the 3′ end formation steps can also be accounted for when modelling transcription 
The recruitment of Pol II enzymes and spliceosomal proteins are important steps in transcription and splicing, but are not believed to be rate limiting under normal conditions. Kinetic studies of Pol II complexes indicate that a minority of them are actively involved in transcription at any given time. The remainder move by diffusion through the nucleus 
, as do the product mRNA molecules 
. Three kinetically distinct populations of Pol II have been identified at the site of transcription; those bound to the promoter, those initiating transcription, and those engaged in elongation 
. The movement of the spliceosomal proteins that catalyse the splicing reactions can be modelled as Brownian diffusion 
: these RNPs move continuously throughout the nucleus independently of transcription and splicing.
We have developed a stochastic model that represents splicing in the context of transcript elongation and RNA 3′ end maturation, as shown diagrammatically in . (All pathway models are provided as files in Dataset S1
.) A stochastic formulation allows the effects of small numbers of molecules to be explored, and simulations of the model can be averaged in order to obtain the population mean over time. Experimental values for the model species (population averages in copies/cell), including fully-spliced mRNA (see Materials and Methods
) and two precursor species in both 3′ uncleaved and cleaved/polyadenylated forms, have been obtained by a rapid sampling protocol that is capable of capturing transient species 
. We first describe the structure of the pathway, then present the data, and subsequently discuss alternative representations of the steps in the RNA pathway in the light of the data. The simplest description that might be adopted for the elongation, 3′ end formation and splicing steps is a single irreversible reaction. However, we find this provides a poor fit to the available data, and consequently a number of alternative representations for these reactions are considered. The extent to which the alternative pathways fit the data is assessed by the Akaike information criterion (AIC) for optimal parameter choices.