The simplest heritable regulatory architectures.

Of the 99 possible regulatory architectures with fewer than four entities (see Fig. S2), only 26 can be indefinitely heritable (A through Z with x, y, and z entities/sensors). Entities that act as sensors (black circles) or that do not provide any regulatory input (blue circles), positive (green arrows) and negative (magenta bar) regulatory interactions are indicated.

Capacity of heritable regulatory architectures to store information.

The number of heritable, regulated, and heritably regulated architectures, and the information they can store were calculated using a program that enumerates non-isomorphic weakly connected graphs that satisfy specified criteria (‘Heritable_Regulatory_Architectures_1-4_entities.py’).

Epigenetic and genetic changes can provide complementary information about heritable regulatory architectures.

(A to H) In each panel, cases where specific perturbations of architectures (top left) characterized by sets of parameters that support a steady state (bottom left) result in different outcomes for permanent or genetic (middle) versus transient or epigenetic (right) changes are illustrated. Relative concentrations of each entity during periods of steady state (thick grey line), the point of genetic change (red arrow), periods of epigenetic reduction (red bar, for a duration tp = 5 (a.u); with the threshold for observing a defect d = 0.5; and an extent of perturbation beyond the threshold p = 0.5), and periods of recovery after perturbation (thin grey line) are shown. Architectures are depicted as in Figure 1 (A, B, C, D, E, F, G, and H depict the heritable regulatory architectures A, G, P, R, W, X, Y and Z, respectively) with transient reductions in an entity or sensor and associated interactions depicted using lighter shades. Dotted lines indicate unregulated turnover (in middle) or thresholds for observing defects upon reduction in levels of an entity/sensor (in right).

Possible conversions between the simplest heritable regulatory architectures.

Table summarizing the possible changes in regulatory architecture observed after a single perturbation from steady state (blue, loss of a regulatory interaction; orange, change in the polarity of a regulatory interaction; black, either change in regulatory interaction and/or loss of an entity). For example, Z can arise from P through the loss of a regulatory interaction or from W through a change in the polarity of a regulatory interaction. Bottom left, Network diagram summarizing possible changes arranged clockwise by frequency of change to the HRA (color-matched numbers). Edges (black, blue, or orange) are colored as in table and nodes are colored according to number of adjacent HRAs.

Regulatory architectures can be simulated as entity-sensor-property systems to examine how they persist or change in response to transient perturbations.

(A) An ESP system illustrating the stability of a regulatory architecture despite changes in the relative numbers of the interactors (entities/sensors) over time. Left, Simulation of an ESP system showing how interacting molecules create regulatory architectures. This system consists of four entities (a, b, c, d), where ‘d’ and ‘a’ are also sensors. Each sensor (red) sends regulatory input (grey, positive or black, negative) to increase or decrease another sensor or entity (blue). Numbers of each entity (i.e., its property value) change in fixed steps per unit time. The number of sensors needed to cause one unit of change in property differs for each regulatory input (lower number = thicker line, representing lower threshold for downstream change). Each entity is depicted with property step, active fraction, and number at the start of the first generation (gen 1) and at the end of the third generation (gen 3). Right, The relative numbers of the entities, which can be together considered as ‘phenotype’, can change over time. Note that relative amounts of ‘a’, ‘b’, or ‘d’ remain fairly constant, but that of ‘c’ changes over time. (B and C) ESP systems can differ in their response to epigenetic change. Top, ESP systems are depicted as in A. Bottom, Relative abundance of each entity/sensor (different colors) or ‘phenotype’ across generations. Blue bars = times of epigenetic perturbation (reduction by two fold). In response to epigenetic perturbation that lasts for a few generations, Type I systems recover without complete loss of any entity/sensor (B) and Type II systems change through loss of an entity/sensor (C). (D) ESP systems of varying complexity can show heritable epigenetic changes, depending on when the system is perturbed. The numbers of randomly chosen entities were unperturbed (none, top), reduced to half the minimum (loss of function), or increased to twice the maximum (gain of function, bottom) every 50 generations for 2.5 generations and the number of systems responding with a new stable regulatory architecture that lasts for >25 generations were determined. Perturbations were introduced at each of five different time points with respect to the starting generation (phase – 0,1,2,3,4). Of the 78,285 stable systems, 14,180 showed heritable epigenetic change.

ESP systems that incorporate the timings of cell division during C. elegans development and temporal delays in regulatory interactions can recreate periods of increased expression in every generation.

(A) Top, Schematic of cell divisions between two successive generations of C. elegans. Cells that maintain the intergenerational continuity through cell divisions (magenta, germline), cells that cannot contribute to the next generation through cell divisions (white, soma) but arise in each generation (gen x and gen x+1) from the bottleneck stage, and the interactions between these two cell types (red line) are depicted. Bottom, Experimentally determined timing of cell division (1) versus growth (0) from one zygote to the next in C. elegans in 15 minute intervals (= 1 time step in simulations), which give a generation time of ∼91.25 hours (= 365 time steps). See Table S3 for the relative timing of cell divisions based on past studies. (B) Key control features for simulating HRAs that incorporate organismal timing of cell divisions and temporal delays in regulation. In addition to controls used in the single system explorer (Fig. S10A), the following sliders were added: one to set the number of generations of ancestors that can contribute regulation (ancestral-effect-generations, e.g., 2 for parental effects), one to set the probability of the regulatory origin for each interaction from one of the two sensors that form the positive feedback loop required for heritability (cyc1-vs-cyc2), and one to set the probability of the gene of interest being a sensor providing regulatory input into the positive feedback loop instead of an entity (gene-is-sensor). Monitors that show the current generation and the total number of molecules, and an input to set the system-id were also added. (C) Representative simulated HRA that incorporates temporal delays and the characteristic timings of cell divisions in C. elegans. Different types of positive (+) and negative (-) regulators (red) that depend on cis-regulatory sequences (+s and –s, e.g., transcription factors), and that depend on the gene product (+p and –p for gene and reporter, e.g., small RNAs made using mRNA template, chaperones that promote the folding of the protein, etc.) are depicted with color coded arrows (+, grey and –, black). Different relative delays in regulation (hours on arrows, maximum of 2x generation time to allow for the widely observed parental regulation) are also depicted. The unknown components of the core positive feedback loops required for heredity were simulated as two sensors that promote each other’s production in addition to the production of all other entities/sensors. (D) Relative concentrations of entities/sensors regulated by the HRA in (C) over 10 generations showing transgenerational waveforms. Properties, active fractions, relative numbers, and regulatory interactions were considered and relative numbers of each entity/sensor depicted as in Fig. 4 with colors as in (C). Although the simulation began with random numbers for all entities/sensors, the HRA settles into a reproducible pattern within two generations with periods of increased relative concentrations for some entities/sensors in every generation (red asterisks). Also see Movie S13.

Regulation of a positive feedback loop can explain the magnitude and duration of experimentally observed heritable RNA silencing.

(A) Experimental evidence from C. elegans for susceptibility to, recovery from, and resistance to trans silencing by a silenced gene (adapted from [42]). Left, Schematic of experiment showing a gene silenced for hundreds of generations by mating-induced silencing (iT = mex-5p::mCherry::h2b::tbb-2 3’ utr::gpd-2 operon::gfp::h2b::cye-1 3’ utr) exposed to genes with matching sequences (mCherry and mCherryΔpi, i.e., mCherry without piRNA binding sites) to initiate trans silencing. Right, Dynamics of heritable RNA silencing showing the initial exposure to trans silencing by iT (F1 generation), subsequent recovery after separation from iT (‘mCherry since F2’ and ‘mCherryΔpi since F2’), resistance to silencing by iT (iT/mCherryΔpi), or persistence of silencing by iT (iT/mCherry). Fractions of animals that recover mCherry or mCherryΔpi expression (fraction unsilenced) are depicted with error bars eliminated for simplicity. (B) Abstraction of the HRDE-1-dependent positive feedback loop required for the persistence of RNA silencing. Top, Representation of the mutual production of RNA intermediates (22G and pUG) with rates of production (kyxand kxy) and turnover (Tx and Ty). Bottom, Ordinary differential equations for the rates of change of pUG RNAs (pUG) and 22G RNAs (22G). See text for details. (C) Impact of transient epigenetic perturbations on subsequent activity of a positive feedback loop. Left, response to a brief and weak reduction in the levels of one sensor (22G) of the positive feedback loop. The steady-state levels after recovery were above the threshold required for a silencing effect (dotted lines). Steady states ([22G]0 and [pUG]0), perturbation level (p. dx. [22G]0), and levels required for silencing (dx. [22G]0 and dy. [pUG]0) are indicated. Middle and Right, Stronger (middle) or longer (right) reduction can result in steady-state levels after recovery being below the threshold required for a silencing effect (dotted lines). (D) Deduced regulatory architecture that explains data shown in (A) by including enhancement of silencing by piRNA binding on target mRNA and a gene-specific inhibitory loop that can act across generations through as yet unidentified sensor(s). Prolonged silencing in prg-1(-) animals [43] suggests that these sensor(s) are among the genes mis-regulated in prg-1(-) animals (e.g., [48]). See Fig. S13 for depictions of additional equivalent architectures.