Labels have been omitted for clarity. A A sequence of reversible edges, as considered by Ong et al. B A tree of reversible edges. A tree is characterised by having no cycle of reversible edges and is an example of a general graph structure that always satisfies detailed balance, irrespective of the kinds of edges in the graph and the labels on these edges Methods.

If the graph represents a system that is maintained away from thermodynamic equilibrium, then detailed balance may no longer hold. The graph may have irreversible edges and Equation 5 no longer works. This leads to the following procedure. Informally, a tree is a sub-graph with no cycles, it is spanning if it reaches every vertex and it is rooted at vertex i if i has no outgoing edges in the tree. Figure 4 B gives examples of rooted spanning trees.

It is not difficult to see that a graph is strongly connected if, and only if, it has a spanning tree rooted at each vertex and that a spanning tree always has one less edge than the number of vertices in G. Hence, by Equation 4 , each microstate has positive steady-state probability. The denominator in Equation 4 provides a non-equilibrium partition function.

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The SCCs inherit connections from the underlying graph but these connections can never form a cycle, for otherwise the SCCs would collapse into each other. It is therefore possible to identify terminal SCCs , from which there are no outgoing connections. The terminal SCCs yield steady states in the following way. Formation of an inherently bounded chromatin domain [ [ 47 ],[ 48 ] ].

A An array of nucleosomes is shown, with nucleation taking place at the right-hand end. White nucleosomes are unmarked, black nucleosomes are marked and grey nucleosomes are either marked or unmarked. B Directed graph for the model with three nucleosomes. Each microstate shows its marking pattern as a bit string with 0 denoting unmarked and 1 denoting marked. The microstates are enumerated by considering the bit string as a number in base 2 notation and adding 1. The edges correspond to nucleation, propagation and turnover, as above.

C On the left, an extension of the model to include mark stabilisation, with a stably marked nucleosome shown in magenta. A stabilised mark is no longer subject to turnover. This leads to the non-strongly connected graph shown on the right for an array of two nucleosomes, in which the digit 2 in the microstate description signifies a stabilised mark.

The strongly connected components SCCs are indicated by dotted outlines, with the two terminal SCCs identified by an asterisk. The dimension of the kernel is then t , the number of terminal SCCs. This average is given by. The same procedure is used for the examples studied here but the linear framework can accommodate the irreversible dynamics of mRNA polymerase initiation, open complex formation, promoter escape, elongation, pausing, etc. The dynamics of mRNAs and proteins can also be coupled to gene regulation within a graph-theoretic formalism [ 41 ]. However, this leads to infinite graphs because the number of mRNA or protein molecules may be unlimited.

Ong et al. They use ad hoc methods, which are independent of previous work on gene regulation. We show here how their analysis can be generalised and simplified within the linear framework. Recent work on steroid-hormone sensitive genes has revealed new co-regulators, such as the Ubiquitin conjugating enzyme, Ubc9, indicating the existence of multiple steps in addition to hormone-receptor binding to DNA [ 46 ]. Despite this additional complexity, gene-regulation functions [ 16 ], which describe how rates of gene expression depend on hormone concentration, are well fitted to Michaelis—Menten style functions, or first-order Hill dose—response curves FHDCs in the language of Ong et al.

They consider a sequence of reversible reactions Figure 5 A , representing the behaviour of the promoter of a hormone-sensitive gene.

## The equilibrium constant K

Such a sequence graph always satisfies detailed balance Methods. We consider the more general case of an arbitrary graph G of reversible edges that satisfies detailed balance. This might be, for instance, a tree graph Figure 5 B , which also always satisfies detailed balance Methods. If a general graph satisfies detailed balance it may not necessarily reach thermodynamic equilibrium and the edges of G may involve dissipative mechanisms. We assume that components R , U , Y 1 ,…, Y m are present and they can bind and unbind to form the microstates of G.

R and U are titratable components, which, crucially, are assumed to bind at most once in each microstate. The main result is that, provided gene expression only occurs from microstates in which both R and U are bound, the average rate of gene expression, g [ S ] , as given by Equation 10 , is also a FHDC Additional file 1 A ,. M G is evidently the average rate of gene expression at saturation i.

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Less obviously, K G is K R multiplied by the saturation probability of those microstates in which R is not bound. Additional file 1 A gives the details of the proof and shows how the formulas in Ong et al. It also discusses how Ong et al. The framework introduced here generalises and clarifies the work of Ong et al. The interpretation of the parameters in Equation 11 is new but emerges easily from our analysis Additional file 1 A. However, because detailed balance is assumed, the consequences of being away from equilibrium remain hidden, as we will see subsequently.

Our next application is to a model of chromatin organisation, with no explicit gene regulation.

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Hathaway et al. To explain the dynamics of such domains, they developed a mathematical model based on a linear array of nucleosomes [ 47 ],[ 48 ]. This model is readily translated into our framework. We considered nucleosome arrays with varying numbers of sites n.

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We placed the nucleation site at the right-hand end of our array Figure 6 A. The microstates correspond to array marking patterns, of which there are 2 n , while the edges correspond to mark nucleation, propagation and turnover Figure 6 A,B. This irreversibility reflects the dissipative mechanism of histone marking and the non-equilibrium nature of the model. The graph does not satisfy detailed balance but is strongly connected.

Monte Carlo simulation is an efficient method for studying very large graphs: an array of nucleosome has a graph with approximately 10 77 microstates.

However, the linear framework provides mathematical access to the steady-state probabilities for any array size and this yields insights that are not easily found by simulation. The importance of the ratio becomes readily apparent within our framework. More significantly, Hathaway et al.

They imposed a stabilisation of the nucleosome mark through a transition to a hypothetical new marked state, whose turnover was inhibited Figure 6 C, left. Because turnover is prevented by the stabilised mark, the graph is no longer strongly connected. If nucleation is stopped, as was done in the simulation, then the resulting graph has two terminal SCCs, each consisting of a single extreme microstate, one in which the entire nucleosome array is unmarked and the other in which the entire array is stably marked.

According to Equation 9 , all other microstates have zero steady-state probability. Which of the two extreme microstates is reached in a simulated trajectory depends on the microstate in which nucleation is stopped. If some nucleosome has become stably marked in that microstate, then it cannot become unmarked, so the trajectory can only reach the completely stably marked microstate.

In their simulation, Hathaway et al. They concluded that the inherently bounded domain was stably maintained in the absence of the initial nucleating stimulus. Our analysis shows that this conclusion is incorrect. Once nucleation is stopped, the bounded domain becomes a transient phenomenon, which eventually expands to fill the whole array. It is conceivable that a bound on the domain size is maintained for sufficiently long to still be biologically relevant.

Such fine-tuning of rate constants is inherently fragile and we think it is more likely that other mechanisms are at work to ensure stable inheritance of the inherently bounded domain. Our framework allows these conclusions to be reached by elementary mathematical deductions, without the need for the numerical simulations undertaken by Hathaway et al.

We now turn back to gene regulation and to one of the very few models in which a non-equilibrium mechanism has been rigorously analysed without assuming detailed balance. Pho5 is an acid phosphatase in Saccharomyces cerevisiae that is expressed under phosphate-starvation conditions. Regulation of yeast PHO5 , adapted from Figures one and four b of [ [ 52 ] ].

A Schematic of the experimental set-up. The TATA box is occluded by nucleosome Labels d k remod and e k reass correspond to disassembly and assembly, respectively, of nucleosomes Figure 3 F , which introduce the non-equilibrium and irreversible features of the graph. Nucleosome -3 has been ignored in the graph.

They pointed out that the nucleosomal transitions were dissipative and in some cases irreversible under their assumptions, so that detailed balance could not be assumed. They assumed that the binding of Pho4 saturates according to a Hill function, which can be accommodated in a similar way to Figure 3 B. The non-binding reactions correspond to unbinding of Pho4 Figure 3 C , or to nucleosomal assembly or disassembly Figure 3 F.

We used our own software written in the programming language Python to enumerate the spanning trees by a fast algorithm and then used the polynomial algebra capabilities of Mathematica to calculate the microstate probabilities and the gene-regulation function Methods. This strongly suggests that what can be done for the yeast PHO5 gene can be systematically undertaken for other genes with non-equilibrium features, with the solution now being understood explicitly through Equation 7 , without recourse to MATLAB.

They used their synthetic construct Figure 7 A, with details in the caption to measure the PHO5 gene-regulation function. In response to doxycycline, individual cells expressed Pho4-YFP, which was treated as the input to the gene-regulation function, and this induced the expression of CFP from the Pho4-responsive promoter in the construct. CFP was treated as the output as a proxy for Pho5. By using different doses of doxycycline to cover a range of Pho4-YFP expression levels, the gene-regulation function was assembled from single-cell measurements.