As preparation for discussing first-passage times, we briefly explain how steady-state probabilities are calculated in terms of the graph; see (Nam et al., 2022, §2) for more details. If we have a graph G, we noted in the Introduction that the steady state, u ^∞ (G), satisfies L ( G ) ⋅ u ∞ ( G ) = 0 , so that, in linear algebra terms, u ^∞ (G) lies by definition in the kernel of the Laplacian matrix: u ∞ ( G ) ∈ ker L ( G ) . If G is strongly connected—i.e., if, for any pair of distinct vertices i and j, there is a directed path of edges from i to j—then this kernel is one-dimensional (Gunawardena, 2012), dim   ker L G = 1 . (3) (The structure of ker L ( G ) is well understood for non-strongly connected graphs (Mirzaev and Gunawardena, 2013). We will not need this for steady states but we will encounter non-strong connectivity when discussing first-passage times in the next section.) Eq. 3 means that if z ∈ ker L ( G ) is any nonzero vector, then any other vector in the kernel, such as u ^∞ (G), is a scalar multiple of z: u ^∞ (G) = λz, for some number λ.

The classical Matrix-Tree theorem (MTT) yields a formula for a canonical basis vector, ρ ( G ) ∈ ker L ( G ) . We will describe this formula shortly but note first that, as just mentioned, u ^∞ (G) must be a scalar multiple of ρ(G), so that u i ∞ ( G ) = λ ρ i ( G ) for i = 1, …, N. Using the conservation law in Eq. 2 and recalling that u _tot = 1 for probabilities, λ may be removed by normalising, so that, u i ∞ G = ρ i G ρ 1 G + ⋯ + ρ N G . (4)

We need some terminology to explain how ρ(G) is determined from G. A spanning forest, F, of G is a subgraph that contains all vertices in G (“spanning”), lacks cycles when edge directions are ignored (“forest”), and has at most one outgoing edge from each vertex. The vertices with no outgoing edges are called the roots of F. If F has only one root, it is called a spanning tree. A forest consists of separate trees, although the forest is upside down, with each tree ascending to its root. Given any non-empty subset of vertices, ∅ ≠ U ⊆{1, …, N}, let Φ _U (G) denote the set of spanning forests of G that are rooted at U. Finally, given any subgraph H of G, let w(H) denote the product of all the edge labels in H: w(H) = ∏ _i→j∈H ℓ(i → j). As a matter of convention, if H has no edges, then w(H) = 1. Then, ρ _i (G) is obtained by summing w(F) over all spanning trees F of G that are rooted at i, ρ i G = ∑ F ∈ Φ i G w F . (5) ρ _i (G) is a manifestly positive polynomial in the edge labels, with each w(F) being a monomial with coefficient +1. The steady-state probabilities, u ^∞ (G), can be recovered from ρ _i (G) by using Eq. 4. Figure 2 illustrates this calculation for an example graph with five vertices and i = 5. Spanning trees are sufficient to calculate steady-state probabilities in Eq. 5 but spanning forests are also needed for the transient quantities considered below (Eqs. 6, 7).

Eq. 5 is a consequence of the classical MTT. The MTT is one of a family of theorems that describe the relationship between the minors of L ( G ) and spanning forests of G. The details of how Eq. 5 arises from the MTT, along with a statement and proof of the MTT itself, are given in Mirzaev and Gunawardena (2013).

Since a strongly connected graph contains at least one directed path from each vertex to every other vertex, there is always at least one spanning tree rooted at each vertex. Therefore, the right-hand side of Eq. 5 is never empty and has at least one term for any choice of i. However, the number of rooted spanning trees may depend on the vertex: in Figure 2, there are 20 spanning trees rooted at vertex 5 but the reader can check that there is only one spanning tree rooted at vertex 3. The size of ρ _i (G) can vary markedly with i, depending on the structure of G.

It follows from Eq. 4 that u ^∞ (G) is a manifestly positive rational function of the labels and is also always nonzero, irrespective of the values of the labels. It is well known in probability theory that the steady-state probabilities of a Markov process are always positive when the corresponding graph is strongly connected, and here we not only see why this is so but also how to calculate these probabilities in terms of the transition rates.

Manifest positivity is what we would want for a formula that yields a steady-state probability. It is a striking fact that many well-known mathematical formulas of molecular biology, such as those of Michaelis–Menten and King–Altman in enzyme kinetics, Monod–Wyman–Changeux and Koshland–Némethy–Filmer in protein allostery and Ackers–Johnson–Shea in gene regulation, all have the structure of manifestly positive rational functions. However, they are typically derived in entirely different ways. In fact, all these rational functions can be shown to arise from Eqs. 4, 5 applied to appropriate linear framework graphs (Gunawardena, 2012; Wong et al., 2018; Nam et al., 2022), thereby revealing a surprising mathematical unity underlying the complexity of molecular biology.

We turn now from the steady state to the transient regime and specifically to first-passage times (FPTs) (Iyer-Biswas and Zilman, 2016). Given a graph G, the FPT from one vertex, i, to a distinct target vertex, j ≠ i, is the random variable for the time it takes the underlying Markov process, X(t), to reach j for the first time when starting from i. Formally, Θ i , j G = inf t > 0 : X t = j ∣ X 0 = i . Of interest are the mean and higher moments of the FPT distribution. Recurrence times for the process returning to i after leaving i can be treated similarly, as can FPTs for reaching a subset of target states from a distinct subset of initial states, but we will leave these refinements aside so as not to complicate the discussion.

For the kinds of stochastic molecular systems considered here, FPTs have been used to quantify several properties: the completion time of an enzymatic turnover (Fisher and Kolomeisky, 1999; Kou et al., 2005; Shaevitz et al., 2005; Kolomeisky and Fisher, 2007; Chemla et al., 2008; Garai et al., 2009; Bel et al., 2010; Moffitt et al., 2010; Cao, 2011; Moffitt and Bustamante, 2014); the speed with which an enzyme can discriminate between correct and incorrect substrates (Banerjee et al., 2017; Cui and Mehta, 2018; Mallory et al., 2019); the statistical structure of transcriptional bursting (Lammers et al., 2020); and the time by which a regulated molecule crosses an abundance threshold (Co et al., 2017; Ghusinga et al., 2017; Gupta et al., 2018). We briefly discuss two examples by way of motivation before proceeding to the technical details.

The development of single-molecule techniques for visualising transcription in live cells (Fukaya et al., 2016; Dufourt et al., 2018) has revealed that transcription is often characterised by transient “bursts” of mRNA expression interspersed by periods of inactivity. Efforts to explain how such bursting arises have focussed on stochastic transitions between transcriptionally active and inactive states in a Markovian setting (Peccoud and Ycart, 1995; Lammers et al., 2020). In active states, successive mRNAs are produced in a burst, which is terminated when the system makes a transition to an inactive state. The FPT to reach an active state from an inactive one provides an estimate of the time between bursts, which can be measured experimentally. As noted by Lammers et al. (2020), comparing the distributions of such FPTs offers a sensitive means to discriminate between different gene regulatory models.

FPTs have also been used to quantify the time at which a regulated molecule reaches a specific abundance threshold (Co et al., 2017; Ghusinga et al., 2017; Gupta et al., 2018). An example of this type of system is bacterial lysis by phage λ. Upon infecting Escherichia coli, phage λ expresses a protein, holin S105, that accumulates in the inner cell membrane until a threshold concentration is reached, at which point the holin molecules abruptly initiate lysis by puncturing the membrane with large irregular holes (White et al., 2010). Various other cellular processes, such as bacterial sporulation (Piggot and Hilbert, 2004), cell cycle progression (Liu et al., 2015) and cell migration during development (Gupta et al., 2018), rely on similar thresholding mechanisms. The FPT analysis undertaken by Ghusinga et al. (2017) shows the impact of different regulatory strategies on the variance in the FPT to reach the threshold and gives insight into the regulatory mechanism of bacterial lysis.

Despite their broad usefulness in biology, FPTs have often been calculated by numerical simulations (Lammers et al., 2020) or by analytical methods that rely on the special structure of the model (Ghusinga et al., 2017). We describe here a systematic graph-theoretic scheme, similar to that in Eq. 5, by which the moments of the FPT distribution can be expressed as rational functions of the edge labels.

Since Θ _i,j (G) measures the time taken by X(t) to reach j from i for the first time, the distribution of Θ _i,j (G) does not depend on the outgoing edges from j or their labels. Therefore, one can remove from G the edges leaving j without affecting the distribution of Θ _i,j (G). For example, the distribution of Θ _i,5(G) is the same for the strongly connected graph in Figure 2A and for the graph in Figure 3A, which is formed by removing the edges leaving 5 from the graph in Figure 2A. In consequence, it is convenient when working with FPTs to deal with graphs that may not be strongly connected, for which some additional terminology is helpful.

A graph G always has a unique decomposition into strongly connected components (SCCs), which can be thought of as the maximal strongly connected subgraphs; see Mirzaev and Gunawardena (2013) for the full details. The directed edges which leave these SCCs give rise to a partial order on the set of SCCs. Those SCCs which are maximal in the partial order are called terminal. For example, the graph in Figure 2A is strongly connected and therefore has only a single SCC, but if the edge 5 → 1 is removed, to yield the graph in Figure 3A, this graph has 3 SCCs in the partial order {1, 2, 3}⪯{4}⪯{5}. Let us consider the special case where G has a unique terminal SCC that contains just one vertex, say, q ∈ {1, …, N}, like the graph in Figure 3A. This is what happens upon removal of the edges leaving a vertex, q, in a strongly connected graph, as in Figure 2A: q forms a unique terminal SCC, {q}, with only one vertex. If the underlying Markov process X(t) starts from any other vertex, say i, then the probability that X(t) eventually reaches q is 1. There may, of course, be trajectories of the process along which q is never reached but these form a set of probability zero.

We need just a bit more notation. The quantities we want to calculate are the kth moments of the probability distribution of the FPT from i to q, τ i , q k G = ⟨ Θ i , q G k ⟩ , where ⟨ − ⟩ denotes the average over the underlying sample space of trajectories. Let I denote the subset of non-terminal vertices, I = { 1 , … , N } \ { q } . Given any non-empty subset of vertices, ∅ ≠ U ⊂ {1, …, N}, and vertices j ∈ {1, …, N} and r ∈ U, let Φ _U:j⇝r (G) denote the set of spanning forests of G that are rooted at U and contain a directed path of edges from j to the root r, specified by j ⇝ r. By convention, there is always a (trivial) directed path from any vertex to itself, so that r ⇝ r. Then, for the mean FPT, we have (Nam and Gunawardena, 2023), τ i , q 1 G = ∑ j ∈ I ∑ F ∈ Φ j , q : i ⇝ j G w F ∑ F ∈ Φ q G w F . (6) The numerator in Eq. 6 runs over all doubly-rooted spanning forests of G in which q is one root and there is a directed path of edges from i to the other root. Figure 3B demonstrates this calculation for the graph in Figure 3A. The denominator in Eq. 6 runs over all spanning trees of G rooted at q and is similar in that respect to the right-hand side of Eq. 5.

The combinatorics become more complicated for the higher moments of Θ _i,q (G). Choose k-tuples of non-terminal vertices, j 1 , … , j k ∈ I × ⋯ × I ⏟ k  times , and set j ₀ = i. Then, for the kth moment, we have (Nam and Gunawardena, 2023), τ i , q k G = k ! ∑ j 1 , … , j k ∏ u = 1 k ∑ F ∈ Φ j u , q : j u − 1 ⇝ j u G w F ∑ F ∈ Φ q G w F k . (7) The product in the numerator of Eq. 7 again involves doubly-rooted spanning forests, in which q is one of the roots and the other root shifts along the k-tuple from j ₁ to j _k , with j _u−1 having a directed path to j _u as u runs from 1 to k. Eq. 7 reduces to Eq. 6 when k = 1.

Note that a spanning forest, or the special case of a spanning tree, that has q as a root cannot include any outgoing edge from q. Hence, the spanning forests or trees with q = 5 as a root are the same for the strongly connected graph in Figure 2A as for the graph in Figure 3A, in which {q} has become the unique terminal SCC by removing the edges that leave q. Accordingly, both the numerator and denominator in Eqs. 6, 7 give the same result for q = 5 in either graph. This is the graph-theoretic consequence of the fact, mentioned above, that the probability distribution of Θ _i,5(G) is the same for the graphs in Figure 2A and Figure 3A.

Eq. 7 and, by specialisation, Eq. 6 can be derived, after some manipulations, from the All-Minors Matrix-Tree theorem, a more recent generalisation of the classical MTT (Nam and Gunawardena, 2023).

As a sanity check on Eq. 7, we note that if G has N vertices, then any spanning forest with r roots has N − r edges, as can be checked for the examples in Figure 2B and Figure 3B. It follows from Eq. 7 that τ i , q ( k ) ( G ) has dimensions of (time) ^k , as expected for the kth moment of an FPT.

Let us see what Eq. 7 tells us for the graph G consisting of just two vertices, 1 and 2, with ℓ(1 → 2) = a and ℓ(2 → 1) = b. If we consider τ 1,2 ( k ) ( G ) , then, for the denominator of Eq. 7, we need the spanning trees rooted at 2, given by Φ_{2}(G). There is only one such tree F, for which w(F) = a. As for the numerator, we need the spanning forests rooted at j _u and 2, given by Φ { j u , 2 } : j u − 1 ⇝ j u ( G ) . Since the roots have to be distinct, the only possibility is that j _u = 1. But then the only forest, F, with these roots has just these vertices and no edges. Recalling the convention for what happens when there are no edges, we find that w(F) = 1. It follows that Eq. 7 collapses to the simple conclusion that τ 1,2 k G = k ! a k . In particular, the mean FPT is 1/a and the variance, which is τ 1,2 ( 2 ) ( G ) − ( τ 1,2 ( 1 ) ( G ) ) 2 , is 1/a ². Only the rate a is relevant, as we would expect, since the rate b is the label on an edge that leaves the target vertex. Because this example is so simple, the moments of the FPT distribution can be readily calculated without the paraphernalia of Eq. 7. The case of a longer pipeline of vertices is more demanding, as we will see below (Figure 5).

Eq. 7 gives a general and systematic method to calculate FPTs from the linear framework graph associated with a Markov process. It can be used to calculate exact formulas in simple graphs and to avoid estimating FPT moments by cumbersome numerical simulations of the Markov process. The combinatorics rapidly become formidable as the graph becomes larger or less symmetric, as is perhaps already evident in Figure 2B and Figure 3B. The broader value of Eq. 7 is that it reveals the mathematical structure of the FPT moments as manifestly positive rational functions of the edge labels. This can often be informative in its own right, as we will see in discussing enzyme kinetics below. We will say more about ways of dealing with the combinatorial complexity in the Discussion.

In the previous section, we considered the FPT distribution from a given vertex i to a single target vertex. It is, however, often the case that there are several target vertices and one wants to know the probability of reaching a particular target vertex or the FPT to that vertex conditioned on the Markov process actually reaching it. (If target vertices lie in different SCCs that are not related in the partial order, then a trajectory that reaches one target can never reach any other target, so that the mean FPT to each target becomes infinite. Conditioning on reaching the target is therefore essential.) Let us suppose, therefore, that G is a graph with one or more terminal SCCs, each of which consists of a single vertex. Let T ⊂ { 1 , … , N } be the subset consisting of these terminal vertices. Given i ∈ {1, …, N} and q ∈ T , define the splitting probability from i to q, denoted π _i,q (G), to be the probability that the underlying Markov process, when started from i, eventually reaches q, as opposed to any other terminal vertex. Then we have (Nam and Gunawardena, 2023), π i , q G = ∑ F ∈ Φ T : i ⇝ q G w F ∑ F ∈ Φ T G w F . (8) The denominator in Eq. 8 runs over all spanning forests of G rooted at T , and the numerator runs over the subset of those spanning forests in which there is a directed path of edges from i to the root q. Accordingly, the right-hand side of Eq. 8 must lie between 0 and 1, as expected for a probability. If i ∈ T and i ≠ q, then there is no directed path from i to q and so Eq. 8 gives 0, while if i = q, then every spanning forest has a (trivial) path of directed edges from i to q and so Eq. 8 gives 1. If G contains only one terminal vertex, then every spanning forest of G rooted at T = { q } has a path of directed edges from i to q, and so Eq. 8 again gives 1. Figure 4 illustrates the calculation of the splitting probability from i = 1 to q = 5 on a six-vertex graph with two terminal vertices, 5 and 6.

Let us turn now to the conditional FPT for reaching a particular target vertex, q ∈ T , from the vertex i ∈ I , where, as before, I is the subset of non-terminal vertices, I = { 1 , … , N } \ T . For the mean conditional FPT from i ∈ I to q ∈ T , denoted by χ i , q ( 1 ) ( G ) , we find that (Nam and Gunawardena, 2023), χ i , q 1 G = ∑ j ∈ I ∑ F ∈ Φ T ∪ j : i ⇝ j G w F ∑ F ∈ Φ T : j ⇝ q G w F ∑ F ∈ Φ T : i ⇝ q G w F ∑ F ∈ Φ T G w F . (9) If there is only one terminal vertex, so that T = { q } , then the mean conditional FPT, χ i , q ( 1 ) ( G ) , as given by Eq. 9, is equal to the mean FPT, τ i , q ( 1 ) ( G ) , as given by Eq. 6. Formulas for the higher moments of the conditional FPT can be obtained in a similar way.

Evidently, the unconditional mean FPT to reach any terminal vertex in T from i, denoted ψ i ( 1 ) ( G ) , is now given by, ψ i 1 G = ∑ p ∈ T π i , p G χ i , p 1 G . Combining Eqs. 8, 9, we can show that this mean FPT can also be expressed in terms of the spanning forests of G, as ψ i 1 G = ∑ j ∈ I ∑ F ∈ Φ T ∪ j : i ⇝ j G w F ∑ F ∈ Φ T G w F , (10) which specialises to Eq. 6 when there is only a single terminal vertex.

Splitting probabilities and conditional FPTs have not been as widely used as have the unconditional FPTs described in the previous section. This reflects the relatively simple models that have been formulated so far in the literature. However, as we have shown here, there is no greater difficulty in dealing with these more complex quantities, at least within the graph-theoretic approach that we have outlined here. All the quantities we have considered are manifestly positive rational functions of the edge labels. This mathematical accessibility should allow deeper analysis of transient stochastic properties.

Single-molecule experimental methods have given unprecedented access to the stochastic kinetics of individual enzymes and have stimulated the development of theoretical models to account for the resulting data. This literature offers a convenient setting to illustrate the ideas introduced above.

A frequently used model in enzyme kinetics corresponds to a pipeline graph (Figure 5) (Fisher and Kolomeisky, 1999; Kou et al., 2005; Kolomeisky and Fisher, 2007; Chemla et al., 2008; Garai et al., 2009; Moffitt et al., 2010; Moffitt and Bustamante, 2014). Such a graph consists of vertices 1, …, N, representing different conformations of the enzyme, with nearest-neighbour transitions, i → i + 1 or i → i − 1. Substrate may bind at any forward transition, i → i + 1, so that ℓ(i → i + 1) incurs a concentration term that we will denote by x, and binding is assumed to be reversible, so that i + 1 → i. The final transition, N − 1 → N, is usually treated as an irreversible catalytic step, with the enzyme returning to its initial conformation, so that vertex N corresponds to vertex 1 in the next enzymatic cycle. A pipeline may be thought of as partitioned into reversible “blocks” that are separated by sequences of irreversible transitions. Figures 5A, C show pipeline graphs with 1 and 3 reversible blocks, respectively.

The mean FPT for reaching vertex N from vertex 1 is a measure of the enzyme’s completion time. Bustamante and colleagues have emphasised how the substrate dependence of τ 1 , N ( 1 ) ( G ) and τ 1 , N ( 2 ) ( G ) contains information about the enzyme mechanism, and they have built on previous studies (Derrida, 1983) to analyse this theoretically (Moffitt et al., 2010). This amounts to studying τ 1 , N ( 1 ) ( G ) and τ 1 , N ( 2 ) ( G ) as functions of x, which falls directly into the scope of the results described above. We will show how the graph-theoretic methods introduced here provide a straightforward way to recover some of these previous findings. We do not intend to be exhaustive and there is much more of interest in the cited references. We hope, rather, to show the advantages of the graph-theoretic approach over the variety of approaches used previously, such as recursive solution of the master equation (Derrida, 1983) or Fourier transformation and determinants (Chemla et al., 2008).

Consider first a pipeline graph, G, with a single reversible block consisting of the vertices 1, …, N − 1 and recall Eq. 6 for the mean FPT, where the terminal vertex is q = N. An example is shown in Figure 5A with the notation that we will use for the edge labels, ℓ(i → i + 1) = p _i and ℓ(i + 1 → i) = q _i . It is evident that there is only a single spanning tree, T ∈ Φ_{N}(G), consisting of all the forward edges, so that w(T) = p ₁⋯p _N−1. This gives the denominator of τ 1 , N ( 1 ) ( G ) . As for the doubly-rooted spanning forests of Φ_{j,N}(G) in the numerator, they can be indexed as F(j, k, N), where j < k ≤ N and k is the vertex with the smallest index that has a directed path to the root N (Figure 5B). Furthermore, each such forest has a directed path from 1 to the root j, so that Φ_{j,N}:1⇝j (G) = Φ_{j,N}(G). We see from the labels in Figure 5B that w F j , k , N = p 1 ⋯ p j − 1 q j ⋯ q k − 2 p k ⋯ p N − 1 , (11) where the “missing” label, between vertices k − 1 and k, corresponds to the gap between the tree rooted at j and the tree rooted at N in the forest. If we divide by the denominator, we see that each spanning forest F(j, k, N) contributes a rational function of the labels that we may write in the form, w F j , k , N w T = 1 p j ∏ u = j k − 2 q u p u + 1 . The spanning forests in Φ_{j,N}(G) therefore contribute the sum, ∑ F ∈ Φ j , N : 1 ⇝ j G w F w T = Δ j , N p j , where, Δ j , N = ∑ k = j + 1 N ∏ u = j k − 2 q u p u + 1 . (12) Note that, in Eq. 12, the empty product for k = j + 1 is by convention taken to be 1. It follows from Eq. 6 that the enzyme completion time is given by, τ 1 , N 1 G = ∑ j = 1 N − 1 Δ j , N p j . (13) With some notational translation, Eq. 13 can be seen to be the same as (Moffitt et al., 2010, Eq. S2). The quantity Δ(j, N) in Eq. 12 first appears in Derrida’s derivation of the velocity and diffusion constant of a Markov particle on a periodic pipeline (Derrida, 1983, Eq. 24); Δ(j, N) = Γ(j + 1, N − 1), where Γ is the quantity defined in Eq. S3 of Moffitt et al. (2010). The calculation above, using the general formula for the mean FPT in Eq. 6, is hopefully more transparent.

Suppose now that substrate binds at s forward transitions in the pipeline graph, with concentration x. We will refer to terms other than x in the edge labels as “kinetic parameters,” which thereby include both simple rates and on-rates. Since we can exclude the final catalytic transition from substrate binding, it follows that 1 ≤ s ≤ N − 2. Eq. 11 then shows that the enzyme completion time has the following structure as a rational algebraic function of x, τ 1 , N 1 G = a 0 + a 1 x + ⋯ + a s x s b x s . (14) Here, the coefficients a ₀, …, a _s and b are all manifestly positive polynomials in the kinetic parameters. In particular, the forest F(N − 1, N, N) includes all the substrate-binding transitions, which confirms that a _s > 0. If the substrate-binding transitions are specified, these polynomials may be explicitly calculated using Eq. 11. Eq. 14 already provides some insight. In the limit of low substrate, the completion time diverges at an order, 1/x ^s , that depends on the number of substrate-binding transitions. In contrast, in the limit of high substrate, the completion time asymptotes to the positive value a _s /b. If substrate binds at only one transition in the pipeline, so that s = 1, then the completion time exhibits a reciprocal Michaelis–Menten form (Kou et al., 2005; Garai et al., 2009; Moffitt et al., 2010; Moffitt and Bustamante, 2014) (Discussion), τ 1 , N 1 G = a 0 + a 1 x b x . (15)

The higher moments of the FPT, as specified by Eq. 7, are more complicated to calculate but the doubly-rooted spanning forests that are needed for the numerator, which are contained in Φ { j u , N } : j u − 1 ⇝ j u ( G ) , have already been enumerated by the forests F(j, k, N) introduced above (Figure 5B). It seems reasonable to conclude from Eq. 7 that τ 1 , N ( k ) ( G ) has a similar rational algebraic structure as shown in Eq. 14 but with a degree of ks for both the numerator and the denominator. In particular, if substrate binds at only one transition, so that s = 1, the second moment of the FPT is a quadratic rational function (Moffitt et al., 2010).

In their study of the packaging motor for the φ29 bacteriophage, Bustamante and colleagues consider a more general pipeline graph, G, that consists of multiple reversible blocks separated by single irreversible transitions (Figure 5C) (Moffitt et al., 2010). The packaging motor is a pentameric ring of identical ATPase units that compacts the φ29 double-stranded DNA into the assembling viral capsid. It has been found to do this in a burst of four ATP-consuming steps per cycle. ATP hydrolysis during the catalytic step is typically irreversible under physiological conditions and a pipeline with 4 reversible blocks serves as a model for the motor (Moffitt et al., 2010, Figure 4A).

If the Markov process takes an irreversible transition in G, it cannot subsequently visit the preceding reversible blocks. Also, every irreversible transition must be taken to reach N. Hence, any trajectory that begins at 1 and reaches N must take each irreversible transition exactly once. It follows from this that the FPT from 1 to N is just the sum of the FPTs for each reversible block considered separately and these FPTs are all independent of each other. Suppose there are m reversible blocks which start at the vertices e ₀, e ₁, …, e _m−1, where 1 = e ₀ < e ₁ < e ₂ < ⋯ < e _m−1 < N. Let G _i be the subgraph consisting of the vertices from e _i−1 to e _i , which includes the ith reversible block and the immediately following irreversible transition. It follows that, τ 1 , N k G = τ 1 , e 1 k G 1 + τ e 1 , e 2 k G 2 + ⋯ + τ e m − 1 , N k G m . (16)

If substrate binds at the same number of transitions in each reversible block, then Eq. 7 shows that the τ e i − 1 , e i ( k ) ( G i ) all have the same rational algebraic structure with the same degrees in both the numerator and the denominator. It follows from Eq. 16 that τ 1 , N ( k ) ( G ) must also have this same rational algebraic structure. For the case of the φ29 packaging motor, ATP binds at only one transition in each reversible block, so the completion time has the reciprocal Michaelis–Menten form of Eq. 15 and the resulting curve may be fitted to the experimental data (Moffitt et al., 2010, Figure 3A). Bustamante and colleagues make use of the reciprocal of the coefficient of variation, n min = τ 1 , N 1 G 2 τ 1 , N 2 G − τ 1 , N 1 G 2 , which is readily seen from the discussion above to be a quadratic rational function of x, and they also fit this curve to the experimental data (Moffitt et al., 2010, Figure 3B). A theorem due to Aldous and Shepp (1987), which is of independent interest, tells us that, for an arbitrary graph with N vertices, n _min < N.

An interesting question arises as to whether n _min itself is also manifestly positive, as might be expected of a coefficient of variation, given that this is true for both τ 1 , N ( 1 ) ( G ) and τ 1 , N ( 2 ) ( G ) . A further point made by Moffitt et al. (2010) is that the quadratic structure of n _min may not be limited to pipeline graphs but may be true also for some graphs with branches and parallel pathways. If so, the graph-theoretic methods described here offer a way to generalise their findings.