Allosteric conformational ensembles have unlimited capacity for integrating information

Integration of binding information by macromolecular entities is fundamental to cellular functionality. Recent work has shown that such integration cannot be explained by pairwise cooperativities, in which binding is modulated by binding at another site. Higher-order cooperativities (HOCs), in which binding is collectively modulated by multiple other binding events, appears to be necessary but an appropriate mechanism has been lacking. We show here that HOCs arise through allostery, in which effective cooperativity emerges indirectly from an ensemble of dynamically-interchanging conformations. Conformational ensembles play important roles in many cellular processes but their integrative capabilities remain poorly understood. We show that sufficiently complex ensembles can implement any form of information integration achievable without energy expenditure, including all HOCs. Our results provide a rigorous biophysical foundation for analysing the integration of binding information through allostery. We discuss the implications for eukaryotic gene regulation, where complex conformational dynamics accompanies widespread information integration.

Pairwise cooperativity by direct interaction on a target molecule (gray). As discussed in the text, the target could be any molecular entity. Left, target molecule with no ligands bound; numbers 1, · · · , 6 denote the binding sites. Right, target molecule after binding of blue ligand to site 2. B. Indirect long-distance pairwise cooperativity, which can arise "effectively" through allostery. C. Higher-order cooperativity, in which multiple bound sites, 2, 4 and 6, affect binding at site 5. and positive cooperativity for binding to site 3, given that site 2 is bound. 30 Studies of feedback inhibition in metabolic pathways revealed that information to modulate 31 binding could also be conveyed over long distances on a target molecule, beyond the reach of 32 direct interactions (Changeux, 1961, Gerhart, 2014 (Fig.1B). Monod and Jacob coined the term Plots of the binding function, whose shape reflects the interactions between binding sites, as described in the text. B. The MWC model with a population of dimers in two quaternary conformations, with each monomer having one binding site and ligand binding shown by a solid black disc. The two monomers are considered to be distinguishable, leading to four microstates. Directed arrows show transitions between microstates. This picture anticipates the graph-theoretic representation used in this paper, as shown in Fig.3. C. Schematic of the end points of the allosteric pathway between the tense, fully deoxygenated and the relaxed, fully oxygenated conformations of a single haemoglobin tetramer, α 1 α 2 β 1 β 2 , showing the tertiary and quaternary changes, based on (Perutz, 1970, Fig.4). Haem group (yellow); oxygen (cyan disc); salt bridge (positive, magenta disc; negative, blue bar); DPG is 2-3-diphosphoglycerate.
that is achievable at thermodynamic equilibrium. We work at the ensemble level (Fig.2B), using a graph-based representation of Markov processes developed previously (below). We introduce a 104 general method of "coarse graining", which is likely to be broadly useful for other studies. This 105 allows us to define the effective HOCs arising from any allosteric ensemble, no matter how complex. 106 The effective HOCs provide a quantitative language in which the integrative capabilities of an 107 ensemble can be specified. It is straightforward to determine the binding function from the effective 108 HOCs and we derive a generalised MWC formula for an arbitrary ensemble, which recovers the 109 functional perspective. Our results subsume and greatly generalise previous findings and clarify 110 issues which have remained confusing since the concept of allostery was introduced. Our graph-111 based approach further enables general theorems to be rigorously proved for any ensemble (below), 112 in contrast to calculation of specific models which has been the norm up to now. 113 Our analysis raises questions about how effective HOCs are implemented at the level of single- 114 molecules, similar to those answered by Perutz for haemoglobin and the MWC model (Fig.2C). Such  122 Wong and Gunawardena, 2020). Energy expenditure can bypass these barriers and substantially 123 enhance equilibrium capabilities. However, the study of non-equilibrium systems is more challenging 124 and we must defer analysis of this interesting problem to subsequent work (Discussion). 125 The integration of binding information through cooperativities leads to the integration of bio-126 logical function. Haemoglobin offers a vivid example of how allostery implements this relationship. 127 This one target molecule integrates two distinct functions, of taking up oxygen in the lungs and cooperativity causes the tense conformation to be dominant in the population, which thereby gives 133 up oxygen. Evolution may have used this integrative strategy more widely than just to transport 134 oxygen and we review in the Discussion some of the evidence for an analogy between functional 135 integration by haemoglobin and by gene regulation. 136

137
Construction of the allostery graph 138 Our approach uses the linear framework for timescale separation (Gunawardena, 2012), details of 139 which are provided in the Supplementary Information (Materials and methods) along with further 140 references. We briefly outline the approach here. 141 In the linear framework a suitable biochemical system is described by a finite directed graph with 142 labelled edges. In our context, graph vertices represent microstates of the target molecule, graph 143 edges represent transitions between microstates, for which the edge labels are the instantaneous 144 transition rates. A linear framework graph specifies a finite-state, continuous-time Markov process 145 and any reasonable such Markov process can be described by such a graph. We will be concerned 146 with the probabilities of microstates at steady state. These probabilities can be interpreted in two 147 ways, which reflect the ensemble and single-molecule viewpoints of Fig.2. From the ensemble per-148 spective, the probability is the proportion of target molecules which are in the specified microstate, 149 once the molecular population has reached steady state, considered in the limit of an infinite popu- interactions with the surrounding thermal bath. In mathematics, this approximation goes back 161 to the work of Wentzell and Freidlin on large deviation theory for stochastic differential equations 162 in the low noise limit (Ventsel' andFreidlin, 1970, Freidlin andWentzell, 2012). It has been 163 exploited more recently to sample energy landscapes in chemical physics (Wales, 2006) and in the 164 form of Markov state models arising from molecular dynamics simulations (Noé and Fischer, 2008, 165 Sengupta and Strodel, 2018). In this approximation, the microstates correspond to the minima 166 of the potential up to some energy cut-off, the edges correspond to appropriate limiting barrier 167 crossings and the labels correspond to transition rates over the barrier. 168 The linear framework graph, or the accompanying Markov process, describes the time-dependent 169 behaviour of the system. Our concern in the present paper is with systems graphs". These are also directed graphs with labelled edges but the edge labels no longer contain 176 dynamical information in the form of rates but rather ratios of forward to backward rates. These 177 ratios are the only parameters needed at equilibrium. From now on, in the main text, when we say 178 "graph", we will mean "equilibrium graph". 179 We explain such graphs using our main example. Fig.3 shows the graph, A, for an allosteric 180 ensemble, with multiple conformations c 1 , · · · , c N and multiple sites, 1, · · · , n, for binding of a all edges between them, also all have the same structure (Fig.3). 192 In an allostery graph, "conformation" is meant abstractly, as any state for which binding asso-193 ciation constants can be defined. It does not imply any particular atomic configuration of a target 194 molecule nor make any commitments as to how the pattern of binding changes. 195 The product-form structure of the allostery graph reflects the "conformational selection" view- in contrast to conformational selection (Koshland and Hamadani, 2002), but we will show below 203 that induced fit is not necessary for this and that negative HOCs arise naturally in our approach. 204 Accordingly, the product-form structure of our allostery graphs is both convenient and powerful. 205 The edge labels are the non-dimensional ratios of the forward transition rate to the reverse tran- The graph structure allows higher-order cooperativities (HOCs) between binding events to be 214 calculated, as suggested in the Introduction. We will define this first for the "intrinsic" HOCs which 215 arise in a given conformation and explain in the next section how "effective" HOCs are defined for 216 the ensemble. In conformation c k , the intrinsic HOC for binding to site i, given that the sites in S 217 are already bound, denoted ω c k ,i,S , is defined by normalising the corresponding association constant 218 to that for binding to site i when nothing else is bound, co nf or m at io ns binding COARSE-GRAINED GRAPH Figure 3: The allostery graph and coarse graining. A hypothetical allostery graph A (top) with three binding sites for a single ligand (blue discs) and conformations, c 1 , · · · , c N , shown as distinct gray shapes. Binding edges ("vertical" in the text) are black and edges for conformational transitions ("horizontal") are gray. Similar binding and conformational edges occur at each vertex but are suppressed for clarity. All vertical subgraphs, A c k , have the same structure, as seen for A c 1 (left) and A c N (right), and all horizontal subgraphs, A S , also have the same structure, shown schematically for A ∅ at the base. Example notation is given for vertices (blue font) and edge labels (red font), with x denoting ligand concentration and sites numbered as shown for vertices (c 1 , ∅) and (c N , ∅). The coarse graining procedure coalesces each horizontal subgraph, A S , into a new vertex and yields the coarse-grained graph, A φ (bottom right), which has the same structure as A c k for any k. Further details in the text and the Materials and methods.
pairwise cooperativity between sites i and j. There is positive or negative HOC if ω c k ,i,S > 1 or 221 ω c k ,i,S < 1, respectively, and independence if ω c k ,i,S = 1 (Fig.1A). 222 For any graph G, the steady-state probabilities of the vertices can be calculated from the edge 223 labels. For each vertex, v, in G, the probability, Pr v (G), is proportional to the quantity, µ v (G), 224 obtained by multiplying the edge labels along any directed path of edges from a fixed reference 225 vertex to v. It is a consequence of detailed balance that µ v (G) does not depend on the choice 226 of path in G. This implies algebraic relationships among the edge labels. These can be fully 227 determined from G and independent sets of parameters can be chosen (Materials and methods). 228 For the allostery graph, a convenient choice vertically is those association constants K c k ,i,S with i 229 less than all the sites in S, denoted i < S; horizontal choices are discussed in the Materials and 230 methods but are not needed for the main text. 231 Since probabilities must add up to 1, it follows that, 232 Pr . (1) Eq.1 yields the same result as equilibrium statistical mechanics, with the denominator being the 233 partition function for the thermodynamic grand canonical ensemble. HOCs provide a complete 234 parameterisation at thermodynamic equilibrium. They replace the parameterisation by free en-235 ergies of vertices, which is customary in physics, with a parameterisation by edge labels, at the 236 expense of the algebraic dependencies described above. Equilibrium thermodynamics has also been 237 used to define an overall measure of cooperativity for binding functions, for which higher-order 238 free energies are introduced (Martini, 2017). These differ from our HOCs in also being associated 239 to vertices rather than to edges. HOCs, higher-order free energies and conventional free energies 240 offer alternative parameterisations of an equilibrium system which are readily inter-converted. De-241 spite the lack of independence which arises from focussing on edges, HOCs are more effective for 242 analysing information integration, precisely because they are associated to binding, which is the 243 key mechanism through which information is conveyed to a target molecule. 244 Our specification of an allostery graph allows for arbitrary conformational complexity and ar-245 bitrary interacting ligands (we consider only one ligand here for simplicity), with the independent 246 association constants in each conformation being arbitrary and with arbitrary changes in these 247 parameters between conformations. Moreover, the abstract nature of "conformation", as described 248 above, permits substantial generality. Allostery graphs can be formulated to encompass the two The quantity µ S (A c k ) is calculated by multiplying labels over paths, as above, within the vertical We see that, although there is no intrinsic HOC in any conformation, effective HOC of each order 307 arises from the moments of K c k over the probability distribution on A ∅ . In particular, Eq.4 shows 308 that the effective pairwise cooperativity is ω φ In studies of G-protein coupled receptor (GPCR) allostery, Ehlert relates "empirical" to "ulti- This shows that ensembles with independent and identical sites, including the two-conformation 323 MWC model, can effectively implement high orders and high levels of positive cooperativity. Eq.5 324 is very informative and we return to it in the Discussion. 325 It is often suggested that negative cooperativity requires a different ensemble to those considered 326 here, such as one allowing KNF-style induced fit (Koshland and Hamadani, 2002). However, if two 327 sites are independent but not identical, so that K c k ,1,∅ = K c k ,2,∅ , then, with just two conformations, i,∅ , in arbitrary units of (concentration) −1 , and effective HOCs, ω φ i,S for i < S, in non-dimensional units; each example is coded by a colour (maroon, orange, red). Right panels show corresponding plots of average binding at each site (colours described in middle inset) and the binding function (black). Bottom left panels summarise the parameters of allostery graphs with N = 16 conformations, c 1 , · · · , c 16 , exhibiting the effective parameters to an accuracy of 0.01 (Materials and methods), showing the intrinsic bare association constants in the reference conformation, c 1 (left), and the probability distribution on the subgraph of empty conformations, A ∅ (right), colour coded as in the respective legends below. Intrinsic association constants for conformations other than c 1 are determined in the proof (Materials and methods). Numerical values are given in the Materials and methods. Calculations were undertaken in a Mathematica notebook, available on request. Background and references 527 The graphs described in the main text, like those in Fig.3, are "equilibrium graphs", which are 528 convenient for describing systems at thermodynamic equilibrium. Equilibrium graphs are derived 529 from linear framework graphs. The distinction between them is that the latter specifies a dynamics, 530 while the former specifies an equilibrium steady state. We first explain the latter and then describe 531 the former. Throughout this section we will use "graph" to mean "linear framework graph" and 532 "equilibrium graph" to mean the kind of graph used in the main text. The framework uses finite, directed graphs with labelled edges and no self-loops to analyse 541 biochemical systems under timescale separation. In a typical timescale separation, the vertices 542 represent "fast" components or states, which are assumed to reach steady state; the edges represent 543 reactions or transitions; and the edge labels represent rates with dimensions of (time) −1 . The labels 544 may include contributions from "slow" components, which are not represented by vertices but which 545 interact with them, such as binding ligands in the case of allostery. 546 Linear framework graphs and dynamics 547 Graphs will always be connected, so that they cannot be separated into sub-graphs between which 548 there are no edges. The set of vertices of a graph G will be denoted by ν(G). For a general graph, 549 the vertices will be indexed by numbers 1, · · · , N ∈ ν(G) and vertex 1 will be taken to be the 550 reference vertex. Particular kinds of graphs, such as the allostery graphs discussed in the Paper, 551 may use a different indexing. An edge from vertex i to vertex j will be denoted i → j and the 552 label on that edge by (i → j). A subscript, as in i → G j, may be used to specify which graph 553 is under discussion. When discussing graphs, we used the word "structure" to refer to properties 554 that depend on vertices and edges only, ignoring the labels. The dynamics in Eq.6 always tends to a steady state, at which du/dt = 0, and, under the funda-575 mental timescale separation, it is assumed to have reached a steady state. If the graph is strongly 576 connected, it has a unique steady state up to a scalar multiple, so that dim ker L(G) = 1. Strong 577 connectivity means that, given any two distinct vertices, i and j, there is a path of directed edges 578 from i to j, i = i 1 → i 2 → · · · → i k−1 → i k = j. Under strong connectivity, a representative steady 579 state for the dynamics, ρ(G) ∈ ker L(G), may be calculated in terms of the edge labels by the 580 Matrix Tree Theorem. We omit the corresponding expression, as it is not needed here, but it can 581 be found in any of the references given above. This expression holds whether or not the steady state 582 is one of thermodynamic equilibrium. However, at thermodynamic equilibrium, the description of 583 the steady state simplifies considerably because detailed balance holds. This means that the graph 584 is reversible, so that, if i → j, then also j → i, and each pair of such edges is independently in flux 585 balance, so that, This "microscopic reversibility" is a fundamental property of thermodynamic equilibrium. Note 587 that a reversible, connected graph is necessarily strongly connected. 588 Take any path of reversible edges from the reference vertex 1 to some vertex i, 1 = i 1 i 2

589
· · · i k−1 → i k = i, and let µ i (G) be the product of the label ratios along the path, It is straightforward to see from Eq.7 that µ i (G) does not depend on the chosen path and that 591 ρ i (G) = µ i (G)ρ 1 (G). The vector µ(G) is therefore a scalar multiple of ρ(G) and so also a steady 592 state for the dynamics. The detailed balance formula in Eq.7 also holds for µ in place of ρ. At 593 thermodynamic equilibrium, the only parameters needed to describe steady states are label ratios.

Equilibrium graphs and independent parameters
595 This observation about label ratios leads to the concept of an equilibrium graph. Suppose that G 596 is a linear framework graph which can reach thermodynamic equilibrium and is therefore reversible 597 (above). G gives rise to an equilibrium graph, E(G) as follows. The vertices and edges of E(G) are 598 the same as those of G but the edge labels in E(G), which we will refer to as "equilibrium edge 599 labels" and denote eq (i → j), are the label ratios in G. In other words, 600 eq (i → j) = (i → G j) (j → G i) .
Note that the equilibrium edge labels of E(G) are non-dimensional and that eq (j → i) = eq (i → j) −1 . The equilibrium edge labels are the essential parameters for describing a state of thermody-602 namic equilibrium. 603 These parameters are not independent because Eq.7 implies algebraic relationships among them. 604 Indeed, Eq.7 is equivalent to the following "cycle condition", which we formulate for E(G): given 605 any cycle of edges, i 1 → i 2 → · · · → i k−1 → i 1 , the product of the equilibrium edge labels along 606 the cycle is always 1, 607 eq (i 1 → i 2 ) × · · · × eq (i k−1 → i 1 ) = 1 .
This cycle condition is equivalent to the detailed balance condition in Eq.7 and either condition is 608 equivalent to G being at thermodynamic equilibrium. 609 There is a systematic procedure for choosing a set of equilibrium edge label parameters which 610 are both independent, so that there are no algebraic relationships among them, and also complete, 611 so that all other equilibrium edge labels can be algebraically calculated from them. Recall that a 612 spanning tree of G is a connected subgraph, T , which contains each vertex of G (spanning) and 613 which has no cycles when edge directions are ignored (tree). Any strongly connected graph has a 614 spanning tree and the number of edges in such a tree is one less than the number of vertices in 615 the graph. Since G and E(G) have the same vertices and edges, they have identical spanning trees. 616 The equilibrium edge labels eq (i → T j), taken over all edges i → j of T , form a complete and 617 independent set of parameters at thermodynamic equilibrium. In particular, if G has N vertices, 618 there are N − 1 independent parameters at thermodynamic equilibrium. 619 In the main text, we defined an equilibrium allostery graph, A (Fig.3), without specifying a 620 corresponding linear framework graph, G, for which E(G) = A. Because label ratios are used in an 621 equilibrium graph, there is no unique linear framework graph corresponding to it. However, some 622 choice of transition rates, (i → G j) and (j → G i), can always be made such that their ratio is 623 eq (i → E(G) j). Hence, some linear framework graph G can always be defined such that E(G) = A.

624
In some of the constructions below, we will work with the linear framework graph, G, rather than 625 with the equilibrium graph A and will then show that the construction does not depend on the 626 choice of G. The steady-state probability of vertex i, Pr i (G), can be calculated from the steady state of the 629 dynamics by normalising, so that, 630 Pr where the first formula holds for any strongly-connected graph and the second formula also holds if 631 the graph is at thermodynamic equilibrium. In the latter case, Eq.7 holds and µ(G) can be defined 632 by Eq.8. The second formula in Eq.10 corresponds to Eq.1. If the graph is at thermodynamic 633 equilibrium, the equilibrium edge labels may be interpreted thermodynamically, where ∆Φ is the free energy difference between vertex i and vertex j, k B is Boltzmann's constant 635 and T is the absolute temperature. If Eq.11 is used to expand the second formula in Eq.10, it gives 636 the specification of equilibrium statistical mechanics for the grand canonical ensemble, with the 637 denominator being the partition function. 638 It will be helpful to let Π(G) and Ψ(G) denote the corresponding denominators in Eq.10, so that, 639 Π(G) = ρ 1 (G) + · · · + ρ N (G) for any strongly-connected graph and Ψ(G) = µ 1 (G) + · · · + µ N (G) 640 for a graph which is at thermodynamic equilibrium. We will refer to Π(G) and Ψ(G) as partition 641 functions. It follows from Eq.10 that, 642 Pr depending on the context. subgraphs having the same structure as each other (Fig.3). Scheme 1 below illustrates this further. 657 As for the labels, the vertical binding edges have equilibrium labels, To see this, let us calculate the steady-state, µ (c k ,S) (A), using Eq.8. Taking the reference vertex 664 in A to be (c 1 , ∅), we can always find a path to any given vertex (c k , S) of A, by first moving 665 horizontally within A ∅ from (c 1 , ∅) to (c k , ∅) and then moving vertically within A c k from (c k , ∅) to 666 (c k , S). According to Eq.8, the steady state is given by the product of the equilibrium labels along 667 this path, so that, Now consider any horizontal edge in A, (c k , S) → (c l , S). Since A is at thermodynamic equilibrium, 669 it follows from Eq.7, using µ in place of ρ, and Eq.14 that, 670 eq ((c k , S) → A (c l , S)) = µ (c l ,S) (A) Applying Eq.7 to A ∅ , with µ in place of ρ, we see that Hence, it follows that, Accordingly, all the labels in A are determined by the vertical labels in Eq.13, from which µ S (A c k ) Coarse graining a linear framework graph and Eq.2 690 We will describe the coarse-graining procedure for an arbitrary reversible linear framework graph, 691 G, and then explain how this can be adapted to an equilibrium graph, as described for the allostery 692 graph A in the main text. 693 We will say that a graph G is in-uniform if, given any vertex j ∈ ν(G), then for all edges i → j, 694 (i → j) does not depend on the source vertex i.

695
Lemma 1 Suppose that G is reversible and in-uniform. Then, G is at thermodynamic equilibrium 696 and the vector θ given by θ j = (i → j), which is well-defined by hypothesis, is a basis element in 697 ker L(G) and a steady state for the dynamics.

698
Proof: If i 1 i 2 · · · i i−1 i k is any path of reversible edges in G, then the product of the 699 label ratios along the path satisfies, because the intermediate terms cancel out by the in-uniform hypothesis. If the path is a cycle, so 701 that i k = i 1 , then, again because of the in-uniform hypothesis, the right-hand side of Eq. 15 is 1. 702 Hence, G satisfies the cycle condition in Eq.9 and is therefore at thermodynamic equilibrium. For 703 the last statement, assume that i 1 is the reference vertex 1 and that i k = j, for any vertex j. Using 704 Eq.8, we see that µ j (G) = θ j /θ 1 . Since θ 1 is a scalar multiple, the last statement follows.

706
Now let G be an arbitrary reversible graph, which need not satisfy detailed balance. Let 707 G 1 , · · · , G m be any partition of the vertices of G, so that G i ⊆ ν(G), G 1 ∪ · · · ∪ G m = ν(G) 708 and G i ∩ G j = ∅ when i = j. Let C(G) be the labelled directed graph with ν(C(G)) = {1, · · · , m} 709 and let u → C(G) v if, and only if, there exists i ∈ G u and j ∈ G v such that i → G j. Finally, let the 710 edge labels of C(G) be given by, The quantity Q in Eq.16 is chosen arbitrarily so that the dimension of (u → v) is (time) −1 , as 712 required for an edge label. This is necessary because, by the Matrix Tree Theorem, the dimension 713 of ρ i (G) is (time) 1−N , where N is the number of vertices in G. However, Q plays no role in the 714 analysis which follows because the coarse graining applies only to the steady state of C(G), not its 715 transient dynamics, and, as we will see, C(G) is always at thermodynamic equilibrium, so that Q 716 disappears when equilibrium edge labels are considered. 717 Note that C(G) inherits reversibility from G and that C(G) is in-uniform. Hence, by Lemma 1, 718 C(G) is at thermodynamic equilibrium and, 719 λµ where λ is a scalar that does not depend on v ∈ ν(C(G)). Since G 1 , · · · , G m is a partition of the 720 vertices of G, it follows from Eq.17 that Eqs.12 and 17 then show that both λ and Q cancel in the ratio for the steady-state probabilities, 722 so that, 723 Pr Eq.18 is the coarse-graining equation, as given in Eq.2.

724
Coarse graining an equilibrium graph 725 The coarse graining procedure described above can be applied to any reversible graph, which need 726 not be at thermodynamic equilibrium. However, the coarse-graining described in the Paper was 727 for an equilibrium graph. It is not difficult to see that the construction above can be undertaken 728 consistently for any equilibrium graph. It is helpful to first establish a more general observation. 729 The choice of edge labels for C(G), as given in Eq.16, is not the only one for which Eq.18 holds, as 730 the appearance of the factor Q indicates. However, the label ratios in C(G) are uniquely determined 731 by the labels of G.

732
Suppose that G is a reversible graph with a vertex partition G 1 , · · · , G m , as above. G need not 733 be at thermodynamic equilibrium. Suppose that C is a graph which is isomorphic to C(G) as a 734 directed graph ("structurally isomorphic"), in the sense that it has identical vertices and edges but 735 may have different edge labels. (Technically speaking, an "isomorphism" allows for the vertices 736 of C to have an alternative indexing to those of C(G), as long as the two indexings can be inter-737 converted so as to preserve the edges. For simplicity of exposition, we assume that the indexing is, 738 in fact, identical. No loss of generality arises from doing this.) 739 Lemma 2 Suppose that C is at thermodynamic equilibrium and the coarse-graining equation (Eq.18) 740 holds for C, so that Pr u (C) = i∈Gu Pr i (G). If u C v is any reversible edge, then its equilibrium 741 label depends only on G, , and C and C(G) are isomorphic as equilibrium graphs, so that identical edges have identical equi-743 librium labels.

744
Proof: It follows from Eq.12 that Pr i (G) = ρ i (G)/Π(G) and, since C is at thermodynamic equi-745 librium, Pr u (C) = µ u (C)/Ψ(C). Using the coarse-graining equation for Pr u (C), we see that Since C is at thermodynamic equilibrium, Eq.7, with µ in place of ρ, implies that, Substituting with Eq.19, the partition functions cancel out to give the formula above. Since C(G) 748 satisfies the same assumptions as C, it has the same equilibrium labels. Hence, C and C(G) must 749 be isomorphic as equilibrium graphs.

751
Corollary 1 Suppose that A is an equilibrium graph and that G is any graph for which E(G) = A, 752 as described above. If any coarse graining of G is undertaken to yield the coarse-grained graph 753 C(G), which must be at thermodynamic equilibrium, then 754 and E(C(G)) depends only on A and not on the choice of G.

755
Proof. A acquires from G the same coarse graining, with the partition A 1 , · · · , A m of ν(A), where 756 A i = G i ⊆ {1, · · · m}. By hypothesis, G is at thermodynamic equilibrium, so that ρ i (G) = λµ i (G) 757 for some scalar multiple λ. Also, since E(G) = A, µ i (G) = µ i (A). Substituting in the formula in 758 Lemma 2 yields the formula above. The equilibrium labels of C(G) therefore depend only on the 759 equilibrium labels of A, as required.
Since A is at thermodynamic equilibrium, we can make use of the formula in Corollary 1 to rewrite 775 this as, .
Eqs.8 and 13 tell us that µ (c k ,S∪{i}) (A) = xK c k ,i,S µ (c k ,S) (A), so that, after rearranging, HORIZONTAL SUBGRAPH OF EMPTY CONFORMATIONS ( ) COARSE-GRAINED GRAPH ( ) Scheme 1: Coarse graining and effective association constants. At top left is an example allostery graph, with binding of a single ligand to n = 2 sites for N = 3 conformations. Vertices indicate a bound site with a solid black dot and an unbound site with a black dash and binding subsets are colour coded: both sites unbound, black; only site 1 bound, magenta; only site 2 bound, cyan; both sites bound, blue. Some vertices are annotated and some edge labels are shown, with x denoting ligand concentration. Example calculations of µ S based on Eq.8 are shown for the vertical subgraph A c 3 . At bottom is the horizontal subgraph A ∅ along with the calculation of its steady-state probability distribution in terms of the equilibrium labels, l 1 , l 2 and the quantities µ c k . At top right is the coarse-grained allostery graph, A φ , with vertices colour coded as for the binding subsets of the allostery graph. Eq.24 for the effective association constants is illustrated below A φ .
We can now appeal to Eqs.12 and 14 to rewrite the term in brackets on the right as, At this point, it will be helpful to introduce the following notation. If G is any equilibrium graph 779 and u : ν(G) → R is any real-valued function defined on the vertices of G, let u denote the 780 average of u over the steady-state probability distribution of G, rearranging, we obtain a formula for the effective association constant as a ratio of averages, Here, the "dot" signifies a product to make the formula easier to read. Scheme 1 demonstrates 786 this calculation. Recall from the main text that higher-order cooperativities (HOCs) are defined by 787 normalising to the empty binding subset, so that ω φ i,S = K φ i,S /K φ i,∅ . Furthermore, since the reference 788 vertex of the vertical subgraphs, A c k , is taken to be the empty binding subset, µ ∅ (A c k ) = 1. It 789 follows that the effective HOCs are given by, The probability, Pr S (A φ ), can be calculated using Eq.10, which requires the quantities µ S (A φ ). 808 These can in turn be calculated by the path formula in Eq.8. We can choose the path in A φ to use 809 the independent parameters introduced above. Let S = {i 1 , · · · , i s }, where i 1 < · · · < i s . Making 810 use of Eq.20, we see that Eq.27 can be rewritten in terms of the non-dimensional effective HOCs but it is simpler for our 812 purposes to use instead the effective association constants, K φ i,S . The dependence on x in Eq.27 813 shows that average binding is given by the logarithmic derivative of the partition function, Ψ(A φ ), 814 so the fractional saturation can be written, With this in mind, Eq.27 shows that the partition function can be written as a polynomial in x, It follows from Eq.8 that µ S (A c k ) = (K c k ) s , where s = #(S). Eq.25 then tells us that ω φ i,S also 828 depends only on s, so that we can write it as ω φ s , and Eq.25 simplifies to, which gives Eq.4. 830 If we consider the effective association constant instead of the effective HOC, then, with the 831 same assumptions as above, Eq.24 tells us that, Suppose that only two conformations, R and T , are present. Let eq (c R → c T ) = L and write K c T 833 and K c R as K T and K R , respectively. Then, for any random variable on conformations, X c k , the 834 average is given by, X c k = (X c R + X c T L)/(1 + L). Hence, which is the formula for the (s + 1)-th "intrinsic binding constant" given by Gruber in which the right-hand side depends only on s = #(S). Collecting together subsets of the same 842 size, the partition function of A φ may be written as, It then follows from Eq.28 that the fractional saturation is given by, If we set α = xK R and cα = xK T this gives, for the fractional saturation, which recovers the classical Monod-Wyman-Changeux formula in the notation of (Monod et al., 846 1965, Eq.2). 847 Proof of Eq.5

848
The following result is unlikely not to be known in other contexts but we have not been able to 849 find mention of it. 850 Lemma 3 Suppose that X is a positive random variable, X > 0, over a finite probability distribu-851 tion. If s ≥ 1, the following moment inequality holds, with equality if, and only if, X is constant over the distribution.
The quantity α s = X s+1 X s−1 − X s 2 can then be written as, Collecting together terms in p i p j , we can rewrite this as Note that the terms corresponding to i = j yield (X s+1 i X s−1 i − X s i X s i )p 2 i = 0 and so do not 858 contribute to Eq.33. Choose any pair 1 ≤ i ≤ m and i < j ≤ m and let X j = µX i . Then, the 859 coefficient of p i p j in Eq.33 becomes, Now, 1 − 2µ + µ 2 = (µ − 1) 2 ≥ 0 for µ ∈ R, with equality if, and only if, µ = 1. Since X > 0 by 861 hypothesis, µ > 0, so the coefficient of p i p j is positive unless µ = 1. Hence, α s > 0 unless X i = X j 862 whenever 1 ≤ i ≤ m and i < j ≤ m, which means that X is constant over the distribution. Of 863 course, if X is constant, then clearly α s = 0 for all s ≥ 1. The result follows.

865
Corollary 2 If A is an MWC-like allostery graph, its effective HOCs satisfy, with equality at any stage if, and only if, K c k is constant over A ∅ .

867
Proof: It follows from Eq.29 that we can rewrite the effective HOCs recursively as, Since ω φ 0 = 1, the result follows by recursively applying Lemma 3 to X = K c k > 0. Eq.34 gives 869 Eq.5.
The left-hand side factors to give, We see that negative cooperativity arises if, and only if, the sites have opposite patterns of associ-884 ation constants in the two conformations.

886
The integrative flexibility theorem 887 Some preliminary notation is needed. Recall that if X is a finite set-typically, a subset of 888 {1, · · · , n}-then #(X) will denote the number of elements in X. If X and Y are sets, then 889 X\Y will denote the complement of Y in X, X\Y = {i ∈ X , i ∈ Y }. Recall also the "little o" Given any ε > 0 and δ > 0, there exists an allosteric conformational ensemble, which has no 896 intrinsic HOC in any conformation, such that for all corresponding values of i and S.

898
Proof: We will construct an allostery graph A whose conformations are indexed by subsets T ⊆ 899 {1, · · · , n} and denoted c T (Scheme 2). Both conformations and binding subsets will then be indexed 900 by subsets of {1, · · · , n}. They should not be confused with each other. The reference vertex of A is 901 r = (c ∅ , ∅). For the horizontal subgraph of empty conformations, A ∅ , let λ T = µ c T (A ∅ ). It follows 902 from Eq.8, using µ in place of ρ, that the λ c T determine the equilibrium labels of A ∅ . Keeping in 903 mind that λ ∅ = 1, the λ T form a set of 2 n − 1 independent parameters for A ∅ , as explained above. 904 The partition function of A ∅ is then given by Ψ(A ∅ ) = T λ T and the steady-state probabilities 905 are given by Pr c T (A ∅ ) = λ T /Ψ(A ∅ ) (Eq.12). 906 The other parameters will be n quantities from which we will construct the association constants 907 in each conformation. Let κ 1 , · · · , κ n > 0 be positive quantities whose values we will subsequently 908 choose. We assume that all intrinsic HOCs are 1 and, for any binding microstate S ⊆ {1, · · · , n}, 909 we set, Scheme 2 illustrates Eq.36 for the case n = 2. If c T is a conformation and S ⊆ {1, · · · , n} is a 911 binding microstate, it follows from Eq.36 that, After coarse-graining, we can calculate effective association constants and effective HOCs using 913 the formulas in Eqs.24 and 25. It follows from Eq.24 and Eq.36 that the effective bare association 914 constants are, For the last step, we have used the trick mentioned above of letting ε → 0 to determine the term 916 independent of ε. We have written {i} ⊆ T in place of the equivalent i ∈ T for consistency with 917 formulas which will appear below. Note that the association constants in Eq.38 are linear in the 918 κ i , whose values are therefore readily determined once the λ T have been chosen. Now let S be a 919 binding microstate and i ∈ S. Using Eq.24 and Eqs.36 and 37, 920 .
Letting ε → 0, we can use the trick above to rewrite this as, Note that Eq.39 simplifies to Eq.38 when S = ∅. It follows from Eq.38 and Eq.39, using the same 922 trick to reorganise the terms which are O ε (1), that the effective HOCs are, We see that the effective HOCs are independent of the quantities κ i and depend only on the 924 parameters, λ T , of the horizontal subgraph A ∅ . 925 We can now specify the λ T . Of course, λ ∅ = 1. If T = {i 1 , · · · , i k }, where i 1 < i 2 < · · · < i k , we 926 set, where each of the α quantities is given by hypothesis. In particular, if T = {j}, then λ T = δ. Note It follows from Eq.41 that, given any X ⊆ {1, · · · , n}, 931 X⊆T λ T = λ X (1 + O δ (1)) .
It follows from Eq.41 that, when i < S, λ S∪{i} = α i,S λ S δ, so using the trick above for reorganising 933 the O δ (1) terms, we can rewrite Eq.42 as α i,S + O δ (1). Substituting back into Eq.40, we see that, 934 when i < S, With the choice of λ T given by Eq.41, we can return to Eq.38 and define, 936 Substituting back into Eq.38, we see that, The result follows from Eqs.43 and 44.