The activation-communication coarse graining. a) An example linear framework graph, H, representing a coarse-grained view of an enhancer. Each vertex contains a schematic of the enhancer (circle) and its target gene (black rectangle with the transcription start site marked with an arrow). The enhancer may be activated (filled red circle) or communicating (curved arrow to the target gene), encoded in the notation (i, j) used to denote vertices. The edge labels show that activation and communication take place independently of each other. b) A more detailed picture of the molecular complexity that may underlie the coarse-grained graph in panel a, as described further in the text. c) The example graph H in panel a is the graph product of two simpler 2-vertex graphs, Ka, which represents activation, and Kc, which represents communication. The product structure of H is equivalent to the independence of activation and communication.

Modeling mRNA production and degradation through copy-number graphs. a) The pipeline structure P represents the number of mRNA molecules and their production and loss. b) An example regulatory graph, G, and the resulting copy-number graph G ⋉ P. In this example G has two production states, X and Y, with corresponding mRNA production rates rx and ry respectively. We note that G ⋉ P is a sub-structure of G × P ; it has the same vertices but lacks the edges corresponding to a production rate of zero. G and P also do not operate independently in G ⋉ P because the mRNA production rates depend on the regulatory state. c) A compact way to represent G ⋉ P. The production states are outlined in purple with corresponding mRNA production rates. The degradation rate, δ, is shown above the arrow from purple squiggles (mRNA) to the empty set ∅. d) The compact representation of the graph H ⋉ P, where H is given in Fig.1a. The only production state is the state (1, 1), in which the enhancer is both activated and communicating, which has production rate r.

Two examples a and b of the default model construction. The copy-number graphs are depicted in compact format, as shown in Fig.2c but omitting the degradation symbols for clarity. Production states are outlined in bold purple with corresponding production rates in purple text. The model in b is adapted from Figure 9 of [26].

The Independent-Activation-Communication (IAC) model. A gene described by the IAC model follows Assumptions 1-6 for the component graphs H1, …, HN. The ordered pair of binary digits for the vertices in each Hi represent the activation and communication status, respectively, of each enhancer. Each Hi has the same structure as the graph in Fig.2d, but different labels, and represents the independence of activation and communication within each enhancer. Each Hi is assumed to have the same degradation rate which is omitted for clarity. See also Fig.S1.

A model of non-independence in activation between enhancers. (a) The graph A♯ represents the activation components of each enhancer. A♯ has the structure of Ka,1 × Ka,2. Labels on the unmarked edges can be arbitrary. (b) The graph C = Kc,1 ⊗ Kc,2 represents the communication components of each enhancer.

IAC model for N = 2 enhancers as a product graph of product graphs. (a) Graphs describing the activation and communication status of enhancer 1. (b) Graphs describing the activation and communication status of enhancer 2. (c) The graph H1 whose underlying regulatory graph is Ka,1 ⊗ Kc,1. (d) The graph H2 whose underlying regulatory graph is Ka,2 ⊗ Kc,2. (e) The graph H1 ⊛ H2 satisfying the default model Assumptions 1-4 with components H1 and H2. The regulatory graph of H1 ⊛ H2 is given by (Ka,1 ⊗ Kc,1) ⊗ (Ka,2 ⊗ Kc,2). Each vertex of this graph corresponds to the activation and communication statuses of both enhancers. For the vertices along the top of the graph, the product-graph binary notation is also provided using the coordinate system ((a1, c1), (a2, c2)). Reverse edges and labels of most edges are omitted for clarity. Production states are highlighted in purple with corresponding production rates also in purple font.