As cells divide, they need to transmit all the regulatory information that maintains homeostasis. This imperative is preserved across generations through a continuum of cell divisions in all organisms as evidenced by the similarity of form and function in successive generations. Such transmission of regulatory information across generations occurs in conjunction with the sequence information transmitted by replicating the genome during each cell division. The maximal information that can be transmitted using the genome sequence is proportional to its length (log₂[4].l=2l bits for l base pairs). To determine how the maximal information transmitted by interacting molecules increases with their number (Table 1), the regulatory architectures that can be formed by 1–4 entities were considered. Perpetual inheritance of such regulatory architectures requires sustained production of all the interacting molecules, that is every interactor must have regulatory input that promotes its production to overcome dilution at every cell division and other turnover mechanisms, if any. Indeed, this requirement was fundamental for conceiving the origin of life (Eigen, 1971; Gánti, 1975; Varela et al., 1974; Kauffman, 1993) and remains necessary for its persistence. Therefore, the minimal heritable regulatory architecture (HRA) is that formed by two molecules that mutually promote each other’s production (Figure 1, ‘A’), resulting in a positive feedback loop. However, not all positive feedback loops form HRAs. For example, positive feedback loops that promote the transient amplification of changes such as that formed by two molecules that mutually repress each other’s production (Mitrophanov and Groisman, 2008) are not compatible with perpetual inheritance because both molecules will be eventually lost by dilution or turnover.

Distinct architectures that can be formed by a set of interacting regulators can be represented as directed graphs that are non-isomorphic and weakly connected (Alon, 2020: 35, 68). Imposing the need for positive regulation for heritability reveals that only 7 of the 13 possible 3-node graphs formed by 3 interactors can be HRAs and only 125 of the 199 possible 4-node graphs can be HRAs (see Methods for computation). Including either positive or negative regulation for each interaction in a HRA and then selecting only architectures that include positive regulatory input for every interactor resulted in non-isomorphic weakly connected directed graphs that represent the distinct regulatory architectures that are heritable (Table 1): two entities form one HRA, three form 25, and four form 5604. Thus, with four interactors, the maximal information that can be transmitted using HRAs (log₂[5604] ≈12.45 bits) surpasses that transmitted by a four base-pair long genome (log₂[4].4 = 8 bits). The combinatorial growth in the numbers of HRAs with the number of interactors can thus provide vastly more capacity for storing information in larger HRAs compared to that afforded by the proportional growth in longer genomes.

To examine how each of the 26 simplest HRAs (Figure 1 and Figure 1—figure supplement 3) responds to a perturbation from steady state, ordinary differential equations that describe the rates of change of each entity in each HRA were developed (see Methods) and used to simulate steady states (Figure 2 and Figure 2—figure supplements 1–7). For each regulatory architecture, positive and negative regulatory interactions (green arrow and magenta bar, respectively, in Figure 1) are captured as linear functions (e.g. kxy.y-kxz.z if x is positively regulated by y and negatively regulated by z). To ensure the concentrations of all entities remain non-negative, as expected in real living systems, the equations for the rate of change are bounded to be applicable only when the values of the changing entity is greater than zero. At steady state, the concentrations of all interactors (x0,y0,z0) remain constant because the combination of all regulatory input, which must cumulatively promote the production of each entity (with rates kxy,kyz,kzy,etc), is equal to the turnover of that entity (with rates Tx,Ty,Tz). In principle, a genetic or non-genetic (i.e. epigenetic) perturbation could alter one or more of the following: the concentration of an entity, the strength of a regulatory link, the rate of turnover of each entity, and the polarity of an interaction. Of these, the most widely used perturbation that is easy to accomplish using current experimental techniques is reducing the concentration of an entity/sensor (e.g. using a loss-of-function mutation, knockdown of an mRNA, degradation of a protein, etc.). Indeed, the use of genome editing (Anzalone et al., 2020) for removal and RNA interference (RNAi) (Fire et al., 1998) for reduction of an entity/sensor are common during the experimental analysis of living systems. Therefore, the impact of permanent or transient loss of an entity was compared.

To simulate genetic change, the response after removal of each entity/sensor was examined in turn for each HRA (Supplementary file 1, left panels in Figure 2 and Figure 2—figure supplements 1–7). Deviations from unregulated turnover of the remaining entities (dotted lines, left panels in Figure 2 and Figure 2—figure supplements 1–7) reveal the residual regulation and are diagnostic of different regulatory architectures. When the removed interactor was an entity with no regulatory input into the other sensors, the remaining two sensors were unaffected (e.g. left panel, loss of z in Figure 2—figure supplement 1b, d and e). Residual promotion resulted in slower decay (e.g. left panels, y in Figure 2c, e and h) and residual inhibition resulted in more rapid decay (e.g. left panels, x in Figure 2g; y in Figure 2f and z in Figure 2c, d, e and h). In some cases, when the remaining architecture was composed of two sensors that promote each other’s production, there was continuous growth of both because their new rates of production exceed their rates of turnover (e.g. left panels, z in Figure 2b and Figure 2—figure supplements 2 and 3, Figure 2—figure supplement 5b-c, Figure 2—figure supplement 6b-d). In other cases, the remaining architecture resulted in slower decay of both because their new rates of production were insufficient to overcome turnover (e.g. left panels, z in Figure 2). Thus, genetic change can result in unrestrained growth or eventual decay of the remaining entities depending on the residual architecture (Figure 1—figure supplement 2c).

To examine scenarios where epigenetic perturbations could cause heritable changes, the threshold for observing a defect was set at half of the steady-state levels (dotted line, right panels in Figure 2 and Figure 2—figure supplements 1–7). RNAi can cause detectable defects that are heritable (Fire et al., 1998) and the conditions that promote or inhibit heritable epigenetic change after RNAi of a gene have been proposed to depend upon the regulatory architecture (Chey and Jose, 2022). To simulate RNAi of an entity/sensor, the response after a transient reduction of each entity/sensor to half of the threshold required for observing a defect was examined in turn for each HRA (right panels in Figure 2 and Figure 2—figure supplements 1–7). The responses after this transient epigenetic perturbation were different from that after genetic perturbation (compare left and right in Figure 2 and Figure 2—figure supplements 1–7), as expected. Many HRAs recovered the levels of all entities/sensors above the threshold required for detecting a defect (e.g. right panels in Figure 2b–e , and h). In some cases, this perturbation was sufficient to maintain the architecture but with a reduced steady-state level of all entities/sensors (e.g. reduction of x in Figure 2a, right; Figure 2—figure supplement 1b–e, right; Figure 2—figure supplement 2b, right; Figure 2—figure supplement 5, right). Notably, whether recovery occurs with the steady-state levels of each entity/sensor returning above or below the threshold for observing a defect depended not only on the architecture, but also on the identity of the perturbed entity/sensor (e.g. compare reduction of x vs. y in Figure 2—figure supplement 1a, right; x vs. y or z in Figure 2—figure supplement 1b, right; x vs. y or z in Figure 2—figure supplement 1c, right). Transient reduction of other entities/sensors below the threshold for observing defects was also observed in many cases (e.g. levels of z when x was perturbed in Figure 2c, right; of z when y was perturbed in Figure 2d, right; of z when x was perturbed in Figure 2e, right; of x and y when z was perturbed in Figure 2h, right). In some cases, recovery of original architectures occurred even after complete loss of one entity/sensor (e.g. after transient loss of z in Figure 2g, right; of y in Figure 2—figure supplement 6, right; of y in Figure 2—figure supplement 6, right). Such recovery from zero can be understood as re-establishment of the regulatory architecture within the duration of simulation (100 in Figure 2 and Figure 2—figure supplements 1–7) and is analogous to basal activity in the absence of inducers (e.g. leaky production of LacY permease from the lac operon in the absence of lactose (Robert et al., 2010), which allows the initial import of the lactose required for activating the lac operon). Alternatively, such recovery can also be understood as arising from the production of the missing entity/sensor as a byproduct when the activity of the upstream regulator increases beyond a threshold. Transient perturbations were also observed to induce different architectures that can persist for many generations (e.g. transient reduction of z resulting in loss of y and mutually promoted growth of x and z in Figure 2f, right; also see Figure 2—figure supplement 6 and Figure 2—figure supplement 7). Such continuous growth after an epigenetic change provides opportunities for achieving new steady states through dilution via cell divisions during development, potentially as part of a new cell type. Finally, some transient perturbations also led to the collapse of the entire architecture (e.g. transient perturbation of y in Figure 2—figure supplement 7, right). Thus, epigenetic change can result in unrestrained growth, eventual decay, or recovery at a new steady state level of the remaining entities depending on the residual architecture and the nature of the perturbation (Figure 1—figure supplement 2d).

In summary, genetic and epigenetic perturbations from steady state can cause a diversity of changes in HRAs that constrain the possible regulatory architectures consistent with experimental data obtained by perturbing them. HRAs that are nearly indistinguishable by genetic perturbation can be distinguished using epigenetic perturbations, underscoring the complementary nature of genetic and epigenetic perturbations.

Just as mutations in a DNA genome can persist through replication at each cell division, changes in HRAs can persist by the formation of new positive feedback loops or the liberation of previously inhibited positive feedback loops. Six types of changes in sequence can arise from the four bases in a DNA genome upon mutation (A‹–›T, A‹–›G, A‹–›C, T‹–›G, T‹–›C, G‹–›C, with density of the change matrix = 1 [6/6]). To determine the analogous types of changes in regulatory architectures, the capacity for each of the 26 simplest HRAs (A to Z in Figure 1) to change into other HRAs was considered (Figure 3). Single perturbations of any interactor (entity/sensor) could result in the loss of the interactor, loss of an interaction, or a change in the polarity of an interaction (e.g. Merdanovic et al., 2020). These perturbations could ‘mutate’ the HRA by either collapsing the entire architecture or stably changing it into a new HRA (Figure 3). Only changes that do not eliminate all positive regulatory inputs to an interactor can result in the persistence of a regulatory architecture rather than the eventual loss of one or more entities. Furthermore, since at steady state all gain of an entity/sensor is balanced by loss (via dilution at every cell division and/or other turnover mechanisms), any permanent reduction in the promotion of an entity/sensor will ultimately lead to its loss. Finally, if there is promotion of one sensor in a positive feedback loop and inhibition of another sensor in the same positive feedback loop then the net input can be positive or negative depending on the relative magnitudes of the inputs. With these considerations, enumeration of the changed HRAs that can result from a perturbation revealed that the 26 HRAs can be mutated to generate 61 different changes (24 through loss of interaction alone, 21 through change in polarity of interaction alone, and 16 through either change in regulation or though loss of an entity, with density of the change matrix ≈0.19 [61/325]). Thus, unlike changes in DNA sequence, not all changes are immediately accessible among HRAs (change matrix of 1 vs 0.19, respectively). Nevertheless, the heritable information transmitted using regulatory architectures is vast because even two or three interactors can form 26 heritable architectures that are collectively capable of 61 changes through single perturbations. This capacity is an underestimate because, single mutations can also result in the gain of new interactions that combine multiple HRAs into larger regulatory architectures with more interactors.

The constrained transition from one HRA at steady state to the next adjacent HRA through a single change (Figure 3) could skew the frequencies of different HRAs observed in nature and restrict the mechanisms available for development and/or evolution. For example, the HRA ‘A’ is accessible from 16 other HRAs but ‘C’ and ‘T’ are each accessible only from one other HRA (‘J’ and ‘U’, respectively). Furthermore, HRAs that rely on all components for their production (‘C’, ‘F’, ‘H’, ‘K’, and ‘T’) cannot change into any other HRAs from steady state without the addition of more positive regulation because any permanent loss in regulation without compensatory changes in turnover will result in the ultimate collapse of the entire architecture. These constraints can be overcome if change can occur through regulatory architectures that are not indefinitely heritable.

Deducing regulatory architectures from outcomes after perturbations is complicated by multiple HRAs resulting in the same HRA when perturbed (e.g. 16 HRAs can result in ‘A’ when perturbed, Figure 3). While measurement of dynamics after perturbations of each entity/sensor in turn can distinguish between all 26 architectures, the temporal resolution required is not obvious. This difficulty in accurate inference is apparent even for the simplest of perturbation experiments when inference relies only on end-point measurements, which are the most common in experimental biology. For example, a common experimental result is the loss of one regulator (x, say) leading to an increase in another (y, say), which is frequently interpreted to mean x inhibits y (Figure 1—figure supplement 2e). However, an alternative interpretation can be that y promotes x and itself via z, which competes with x (Figure 1—figure supplement 2e). In this scenario, removal of x leads to relatively more promotion of z, which leads to a relative increase in y. These equivalent outcomes upon loss or reduction of an entity in different architectures highlight the difficulty of inferring the underlying regulation after perturbation of processes with feedback loops. Therefore, simulations that enable exploration of outcomes when different interactors are perturbed at different times could enhance the understanding of underlying complexity, reduce biased inference, and better guide the next experiment.

The most commonly considered regulatory networks (Barabási and Oltvai, 2004) are either limited in scope and/or are not causal in nature. For example, gene regulatory networks (Levine and Davidson, 2005) consider transcription factors, promoter elements, and the proteins made as the key entities. Additional specialized networks include protein-protein interaction networks (e.g. Li et al., 2017), genetic interaction networks (e.g. Costanzo et al., 2019), and signaling networks (e.g. Azeloglu and Iyengar, 2015). However, regulation of any process can rely on changes in a variety of molecules within cells, ranging from small molecules such as steroid hormones to organelles such as mitochondria. Furthermore, experimental studies often seek to provide explanations of phenomena in terms of a diversity of interacting entities. A common expression for all possible regulatory networks that preserves both causation and heritability can be derived by parsing all the contents of the bottleneck stage between two generations (e.g. one-cell zygote in the nematode C. elegans) into entities, their sensors, and the sensed properties (Jose, 2020b). An additional advantage of such entity-sensor-property systems is that it is possible to consider entities that are sensed by a particular sensor even via unknown intermediate steps, allowing for simulation of regulatory networks despite incomplete knowledge of regulators.

For a given genome sequence, the number of distinguishable configurations of regulators represented as entities and sensors is given by the following equation (Jose, 2020b):(1)ctot=∑i=1bei(∑j=1sisj(∑k=1pjpk))

where, e is the measured entity (total b in the bottleneck stage between generations: nb in system, ob in environment), s is the measuring sensor (total s_i for i^th entity), which is itself a configuration of entities drawn from the total N per life cycle (i.e., f(Y), with each Y ⊆{e₁, e₂, …, e_N}), and p is the attainable and measurable values of the property measured by the sensor (total p_j for the j^th sensor of the i^th entity). An entity or configuration of entities is considered a sensor only if changes in its values can change the values of other entities in the system or in the environment at some later time.

The complex summation ∑iei∑jsj∑kpk can be simplified into a product of measurable property values of individual entities/sensors if the possible numbers of property values of every entity/sensor combination is independent:(2)ctot=∏i=1b ∏j=1sinij

where, nij=|{p1,p2,…pj}| is the number of property values as measured by the j^th sensor of the i^th entity. However, this upper limit is never reached in any system because regulatory interactions make multiple sensors/entities covary (Jose, 2020b). As a simple example, consider two sensors that activate each other in a Boolean network with no delay: the only possible values are {0,0} when both sensors are off and {1,1} when both sensors are on, because the mutual positive regulation precludes {0,1} and {1,0}.

Three simplifying assumptions were made to facilitate the simulation of regulatory networks with different architectures: (1) let the number of molecules be the only property measured by all sensors, (2) let each sensor be a single kind of molecule, and (3) let all entities be within the system. In these simple Entity-Sensor-Property (ESP) systems, the number of possible configurations is given by:(3)ctot=∑i=1Eei(∑j=1sinj) <∏i=1E ∏j=1sinij

where, e is the measured entity (total E in the system), s is the measuring sensor (total s_i for i^th entity), n_j is the attainable and measurable numbers of the i^th entity, and n_ij is the attainable and measurable numbers of the i^th entity as measured by the j^th sensor. In such simplified ESP systems, a sensor is simply an entity in the system that responds to changes in numbers of an entity by changing the numbers of that entity or another entity. Whether downstream changes occur depends on the sensitivity of the sensor and the step-size of changes in property value (i.e. number) of the downstream entity/sensor.

The Entities-Sensors-Property (ESP) framework is not entirely equivalent to a network, which is much less constrained. Nodes of a network can be parsed arbitrarily and the relationship between them can also be arbitrary. In contrast, entities and sensors are molecules or collections of molecules that are constrained such that the sensors respond to changes in particular properties of other entities and/or sensors. When considered as digraphs, sensors can be seen as vertices with positive indegree and outdegree. The ESP framework can be applied across any scale of organization in living systems and this specific way of parsing interactions also discretizes all changes in the values of any property of any entity (Figure 1—figure supplement 2f). In short, ESP systems are networks, but not all networks are ESP systems. Therefore, the results of network theory that remain applicable for ESP systems need further investigation.

To simulate ESP systems (e.g. Figure 4a) a hybrid approach with deterministic and stochastic aspects was used (see Methods for details). Specifically, the systems represented by equation (3) were simulated with random values for the numbers of each entity/sensor, the sensitivity of each sensor (i.e. the change in number needed to change a downstream entity/sensor), the step-size of changes in property (i.e. numbers changed per input from a sensor), and the active fraction (i.e. proportion that are available for interactions at any time). Only an arbitrary fraction of each entity (fixed over time) was simulated as active to account for processes such as folding, localization, diffusion, etc., that can limit regulatory interactions. Simulations were begun with a total of 500 molecules. A maximal increase of 5000 molecules was allowed per cell cycle before each cell division to account for depletion of precursors, reactants, or building blocks (which were not explicitly simulated). The simulation was ended if total number of simulated molecules increased beyond a maximal number (500,000 in Figure 4) to account for the limited capacity of a cell. Thus, with each time step, the numbers of all entities/sensors changed deterministically based upon the randomly established initial regulatory architecture with stochastic changes in the numbers of each entity arising from the random order of evaluation at each time step and the reduction by ~1/2 at the start of each cell cycle, which simulates experimentally observed noise (e.g. Wan et al., 2021). With these parameters, regulatory architectures were simulated and the relative concentration of each entity/sensor was plotted over time (Figure 4a). These profiles represent the change in ‘phenotype’ over time and are akin to measurements of relative RNA abundance using RNA-seq (Van den Berge et al., 2019) or relative protein abundance using proteomic approaches (Mund et al., 2022). Thus, ESP systems represent networks or graphs with weighted edges (because of the different activities of different sensors needed for the transmission of each regulatory change), complex nodes (because of the multiple properties of each entity/sensor), and transmission delays (because of the variation in the duration of each regulatory interaction). Notably, the relative concentrations of an entity can change over time (e.g. ‘c’ in Figure 4a) while the underlying regulatory architecture is preserved (see Figure 4—figure supplement 1, Video 1, and run ‘ESP_systems_single_system_explorer_v1.nlogo’ in NetLogo [Wilensky, 1999] for exploration).

To gain intuitions by exploring and perturbing simulated ESP systems, several interactive features were added to the ESP simulator (Figure 4—figure supplement 1, Video 1). These include parameters that control the setup and running of randomly generated ESP systems by specifying the probability of regulatory interactions in the system (link-chance, Figure 4—figure supplement 1), the probability of positive versus negative interactions (positive-interactions, Figure 4—figure supplement 1), the maximum number of molecules at the start of the simulation (max-molecules, Figure 4—figure supplement 1), the maximum number of molecules that will arrest growth until dilution through cell divisions to simulate depletion of raw materials or energy (stasis-level, Figure 4—figure supplement 1), maximum number of molecules in total reflecting the limited space occupied by living systems (max-ever-molecules, Figure 4—figure supplement 1), and duration of the cell cycle (cycle-time, Figure 4—figure supplement 1). Particular systems can be re-established and re-simulated by setting the random number seed that is used for controlling all stochastic steps (system-id, Figure 4—figure supplement 1) and by additionally specifying the above parameters along with the number of entities/sensors at the start of the simulation (molecule-kinds, Figure 4—figure supplement 1). For each such system, the number of entities that can increase or decrease at one time was set to be characteristic of each entity/sensor (unit change in property value, i.e. number) and the number of sensors needed to change one unit of each entity/sensor was set to be characteristic of each regulatory interaction (thickness of link increases with increasing sensitivity of regulation). Periodic loss-of-function or gain-of-function perturbations (perturb-kind, Figure 4—figure supplement 1) can be set up to begin in five different phases relative to the start of the simulation (perturb-phase [0, 1, 2, 3 or 4], Figure 4—figure supplement 1). Perturbations that can be made during the simulation include changing the number of molecules of any entity/sensor (change-a-node, Figure 4—figure supplement 1), adding an entity/sensor (add-a-node, Figure 4—figure supplement 1), removing an entity/sensor (remove-a-node, Figure 4—figure supplement 1), removing a particular regulatory interaction (remove-link-x-y, Figure 4—figure supplement 1), removing a random regulatory interaction (remove-a-link, Figure 4—figure supplement 1), and changing the strength of a regulatory input (link-hold, Figure 4—figure supplement 1). Finally, a reporter for any entity/sensor (add-a-reporter, Figure 4—figure supplement 1) can be set up that either perfectly or partially interacts with all its regulators (perfect?, Figure 4—figure supplement 1). A perfect reporter of an entity/sensor receives the same regulatory input as the entity/sensor of interest does. An imperfect reporter of an entity receives input from the same sensors as the entity/sensor of interest, but the polarity and strength of the input can vary. Regulatory outputs of the entity/sensor are not recreated for any reporter. As these changes are being made, both the regulatory architecture (Fig. Figure 4—figure supplement 1, top), which is re-drawn if the levels of any entity/sensor reaches zero, and the ‘phenotype’ as captured by the profile of relative concentrations of entities/sensors (Figure 4—figure supplement 1, bottom) can be observed.

Explorations of ESP systems revealed that some systems can be stable for a large number of cell divisions (or equivalently generations) before the level of an entity/sensor becomes zero (e.g. system-id 62795 was stable for 59,882.5 generations (Figure 4—figure supplement 2)). This long, yet finite duration of stability highlights the difficulty in claiming any architecture is heritable forever if one of its entities/sensors has low abundance and can be lost with a small probability. Systems responded to transient perturbations that reduce the levels of a randomly chosen entity/sensor in two major ways: Type I systems recovered relative levels of entities/sensors and maintained the same regulatory architecture; Type II systems changed by losing one or more entities, resulting in new relative levels of entities/sensors, and new regulatory architectures that persisted for many subsequent generations. These two types were observed even when the numbers of some entities/sensors were changed by just twofold for a few generations (Figure 4b versus Figure 4c).

For systematic analysis, architectures that could persist for ~50 generations without even a transient loss of any entity/sensor were considered HRAs. Each HRA was perturbed (loss-of-function or gain-of-function) after five different time intervals since the start of the simulation (i.e. phases). The response of each HRA to such perturbations were compared with that of the unperturbed HRA. For loss-of-function, the numbers of one randomly selected entity/sensor were held at half the minimal number of all entities/sensors for 2.5 generations every 50 generations. This perturbation is like the loss of transcripts through RNA interference. For gain-of-function, the numbers of one randomly selected entity/sensor were held at twice the maximal number of all entities/sensors for 2.5 generations every 50 generations. This perturbation is like overexpression of a particular mRNA or protein. Of 225,000 ESP systems thus simulated, 78,285 had heritable regulatory architectures that remained after 250 generations (system-ids and other details for exploration of individual systems are in Supplementary file 2). These persistent systems included entities/sensors with relative numbers that changed over time as well as those with nearly constant relative numbers. For each number of interactors considered (2–16), only a fraction of the regulatory architectures were stable, plateauing at ~50% of simulated systems with 10 or more entities/sensors (Figure 4—figure supplement 3, top left). This plateau is likely owing to the limit set for the maximal number of molecules in the system because most systems with many positive regulatory links quickly reached this limit. Although systems began with up to 16 entities/sensors, by the end of 250 generations, there was a maximum of ~8 entities/sensors and a median of ~4 entities/sensors in stable systems (Figure 4—figure supplement 3, bottom left). Furthermore, an excess of positive regulatory links was needed to sustain a system with negative regulatory links (Figure 4—figure supplement 3, right), with the minimal system that can sustain a negative regulatory interaction requiring three sensors, as expected. This bias reflects the inability of negative regulatory interactions alone to maintain regulatory architectures over time across cell divisions (Figure 1—figure supplement 2g).

Since the relative abundance of each entity/sensor is expected to trace a transgenerational waveform (Jose, 2020c) and to fluctuate with each time step, the relative timing of the perturbations (i.e. their phase) can impact the persistence of particular regulatory architectures. Specifically, entities/sensors with low numbers could be rescued from reaching zero by well-timed gain-of-function perturbations and those with high numbers could be rescued from arresting the simulation by well-timed loss-of-function perturbations. Therefore, the fractions of persistent ESP systems that showed heritable epigenetic change through loss of one or more entities were examined by starting with 2–16 molecules for each phase and type of perturbation (Figure 4d). As expected, only HRAs with a minimum of three entities/sensors at the start of the simulation showed heritable epigenetic changes. Fractions of HRAs showing heritable epigenetic changes were comparable across all perturbations for a given number of starting entities/sensors, with more such HRAs identified with increasing numbers of starting entities/sensors (compare Figure 4d, top, middle, and bottom). The variations in the numbers identified for different phases of perturbation when starting with a particular number of entities/sensors was comparable to the variation observed in systems that were not perturbed (compare Figure 4d, top with Figure 4d, middle and bottom), suggesting that no particular phase is more effective. Although some architectures were unaffected by all perturbations, many showed an altered response based on both the nature and phase of the perturbations. Consider the behavior of the illustrative example defined by system-id 46357 that begins with five entities/sensors (see Videos 2–9, Video 10, Video 11, Video 12). The unperturbed ESP system stabilizes with an architecture of three entities (of the type ‘E’ in Figure 1) until ~74 generations, when one of the entities is lost and the new architecture (of the type ‘A’ in Figure 1) remains stable until ~286.5 generations. However, perturbations yield a variety of different stabilities depending on phase and type of perturbation. Periodic loss-of-function perturbations with phase ‘0’, ‘3’, or ‘4’ resulted in stability of all 3 entities until collapse at ~130.5, ~181.5, or ~99.5 generations, respectively. But, such perturbations with phase ‘1’ resulted in an earlier change from type ‘E’ to type ‘A’ with collapse at ~193.5 generations and with phase ‘2’ resulted in a later change from ‘E’ to ‘A’ with collapse at ~184.5 generations. On the other hand, periodic gain-of-function perturbations with phase ‘0’, ‘1’, ‘2’, or ‘4’ prolonged the type ‘E’ architecture until ~306, ~253, ~212, or ~708 generations, respectively, after which the new type ‘A’ architecture persisted beyond 1000 generations, by when the simulation was ended. However, such perturbations with phase ‘3’ preserved the type ‘E’ architecture until collapse at ~311.5 generations.

Collectively, these results reveal that the heritability of regulatory architectures that are intrinsically unstable can be enhanced through interactions that alter the regulation of one or more sensors. Such external regulators could be part of the environment (e.g. periodic interactions such as circadian signals) or other cells (e.g. periodic interactions with somatic cells for regulatory architectures transmitted along a germline).

While for unicellular organisms, each cell division results in a new generation, for multicellular organisms, transmission of heritable information across generations occurs along a lineage of cells that can include many cell divisions with periods of quiescence and interactions with somatic cells. The nematode C. elegans is a well-characterized multicellular organism (Corsi, 2006) that has many features such as the early separation of the germline during development, sexual reproduction, and the generation of different somatic tissues (Figure 5a, top), that can be incorporated into simulations and are useful for generalization to other animals, including humans. For example, 14 cell divisions are necessary to go from the zygote of one generation to the zygote of the next (Supplementary file 3). The loading of oocytes with maternal molecules in multiple organisms makes one generation of delay in regulation (i.e. maternal regulation) common, but such ancestral effects could last longer in principle (e.g. grandparental effects are easily imagined in humans because oocytes begin developing within the female fetus of a pregnant woman). Indeed, studies on RNA silencing in C. elegans have revealed long delays in regulation within the germline. For example, parental rde-4 can enable RNA silencing in adult rde-4(-) progeny (Marré et al., 2016). Furthermore, loss of meg-3/–4 can result in persistent RNA silencing defects despite restoration of wild-type meg-3/–4 for multiple generations (Dodson and Kennedy, 2019; Lev et al., 2019; Ouyang et al., 2019). However, examining the impact of such long delays and of interactions with somatic cells (e.g. Abdu et al., 2016) is computationally expensive. Therefore, to simulate regulatory architectures that persist from one zygote to the next while satisfying some of the known constraints of C. elegans lineage and development, the ESP simulator was modified to incorporate the observed timings of cell division versus growth along the germline (Supplementary file 3, based on Oegema and Hyman, 2006; Bao et al., 2008; Giurumescu et al., 2012; Kimble and Crittenden, 2005; Jaramillo-Lambert et al., 2007; Hirsh et al., 1976) and allow a delay of up to two generations for the impact of a regulatory interaction (Figure 5b).

Since many genes expressed in the germline can be required for fertility or viability, the analysis of how they are regulated across generations poses a challenge. Following the behavior of a reporter across generations provides a proxy that can be used to understand transgenerational regulation in a relatively wild-type background (i.e. the tracer approach [Jose, 2018]). However, regulatory interactions that rely on the gene product will not be re-created by the reporter. For example, the mRNA sequence used to produce antisense small RNA of the gene will differ from that used for the reporter.

To simulate the regulation of a germline gene and its reporter, a modified ESP system is required. In addition to a minimal positive feedback loop required for heritability, positive and negative regulators that depend on cis-regulatory sequences shared by both the gene and its reporter as well as such regulators that depend on the different products (e.g. the mRNA and/or protein of gene versus reporter) need to be simulated. Of the 10,000 such ESP systems simulated, 11 maintained all entities/sensors for 10 generations (Supplementary file 4). A representative such system (Figure 5c) forms a HRA that incorporates regulatory delays ranging from 0 to 171 hr and can persist for hundreds of generations. Examining the levels of all entities/sensors over the first 10 generations (Figure 5d) reveals that despite starting at random values the HRA settles with a reproducible pattern during each generation (gen 2 onwards in Figure 5d). The transgenerational waveforms traced by the relative numbers of all entities/sensors reveal periods of increased expression/activity for some entities/sensors during development (red asterisks in Figure 5d), as observed for many genes expressed in the germline. Future studies that obtain data on key regulators with spatial and temporal resolution can be used to discriminate between different HRAs that drive the expression of different genes of interest.

The simple fact that organisms resemble their parents in most respects provides evidence for the homeostatic preservation of form and function across generations. Yet, this ‘transgenerational homeostasis’ (Jose, 2018) is overcome in some cases such that epigenetic changes persist for many generations (reviewed in Fitz-James and Cavalli, 2022). Indeed, DNA methylation patterns of unknown origin are thought to have persisted for millions of years in the fungus C. neoformans (Catania et al., 2020). Studies on RNA silencing in C. elegans (reviewed in Frolows and Ashe, 2021) provide strong evidence for heritable epigenetic changes in the expression of particular genes, facilitating analysis. When a gene expressed in the germline is silenced using double-stranded RNA of matching sequence and/or germline small RNAs called piRNAs, the silencing can last from one or two generations to hundreds of generations (Devanapally et al., 2021; Shukla et al., 2021; Priyadarshini et al., 2022). The maintenance of RNA silencing across generations is thought to require a positive feedback loop formed by antisense small RNAs called 22G RNAs that are bound to the Argonaute HRDE-1 (Buckley et al., 2012) and sense mRNA fragments processed into poly-UG RNAs (pUG RNAs) (Shukla et al., 2020) that can act as templates for RNA-dependent RNA polymerases that synthesize the 22G RNAs. However, this mechanism does not explain the variety of effects that can arise when such genes with long-term silencing are exposed to other genes of matching sequence (Seth et al., 2018; Devanapally et al., 2021). For example, when a gene silenced by disrupting RNA regulation within the germline (iT, a transgene with silenced mCherry sequences) is exposed to genes with matching sequences (Figure 6, Devanapally et al., 2021), different outcomes are possible. After initial silencing in trans, the newly exposed genes can recover from silencing (mCherry and mCherry∆pi in Figure 6a) when separated from the source of silencing signals, but can be either continually silenced (mCherry/iT in Figure 6a) or become resistant to silencing (mCherry∆pi/iT in Figure 6a) depending on the presence of intact piRNA-binding sequences despite the continued presence of the source. While this observation suggests that recognition of the target mRNA by piRNAs prolongs RNA silencing, loss of the Argonaute PRG-1 that binds piRNAs and regulates more than 3000 genes in the germline (e.g. Reed et al., 2020) also prolongs the duration of heritable RNA silencing (Shukla et al., 2021). Although the release of shared regulators upon loss of piRNA-mediated regulation in animals lacking PRG-1 could be adequate to explain enhanced HRDE-1-dependent transgenerational silencing initiated by dsRNA in prg-1(-) animals, such a competition model alone cannot explain the observed alternatives of susceptibility, recovery and resistance (Figure 6a). Recent considerations of such competition for regulatory resources in populations of genes that are being silenced suggest explanations for some observations on RNA silencing in C. elegans (Karin et al., 2023). Specifically, based on Little’s law of queueing, with a pool of M genes silenced for an average duration of T, new silenced genes arise at a rate λ that is given by M = λT. However, this theory cannot predict which gene is silenced at any given time, why some genes are initially susceptible to silencing but subsequently become resistant (e.g. mCherry∆pi in Figure 6a), and why the silencing of some genes can last for hundreds of generations. Thus, there is a need for understanding the origins of gene-specific differences in the dynamics of heritable epigenetic changes.

22G RNAs and pUG RNAs are experimentally measurable molecular markers whose levels are thought to be proportional to the extent of gene silencing (e.g. Gu et al., 2009; Pak et al., 2012; Shirayama et al., 2012), although formally gene-specific regulatory features could influence this proportionality (Knudsen et al., 2023). To understand how the activity of the underlying positive feedback loop that maintains the levels of these RNAs could relate to the extent of observed silencing, the HRDE-1-dependent loop was abstracted into a minimal positive feedback loop with 22G RNAs and pUG RNAs promoting each other’s production (Figure 6b, top). Ordinary differential equations (Figure 6b, bottom) were developed for interdependent change in both RNAs by considering their rates of promotion (kxy for 22G and kyx for pUG) and turnover (Tx for 22G and Ty for pUG). All molecules and chemical modifications that are necessary to maintain a positive feedback loop are sensors because they need to transmit the change from an ‘upstream’ regulator to a ‘downstream’ regulator. This 22G-pUG-positive feedback loop thus represents mutual promotion by two sensors (i.e. the HRA ‘A’ in Figure 1). When any changed molecule or chemical modification is thus viewed as one component of a heritable regulatory architecture driven by a positive feedback loop, two criteria that impact the duration of epigenetic changes through the reduction of particular sensors are immediately suggested: (1) for every permanent change that is observed, all sensors that participate in the regulatory loop must be reduced to a level below that required for observing the change; and (2) for eventual recovery from a change, at least one sensor should be above the threshold required to drive the increase of all other sensors above their respective levels for observing the change. Consistently, a weak and brief reduction of 22G RNAs from steady state levels results in the eventual recovery of both 22G RNA and pUG RNA levels above the threshold required for them to be effective for silencing (Figure 6c, left). Stronger (Figure 6c, middle) or longer (Figure 6c, right) reductions can result in new steady-state levels for both RNAs that are below the threshold required for silencing.

To obtain an analytic expression for how long heritable RNg needs to be inhibited for eventual recovery, the impact of transient reduction in the activity of the 22G-pUG positive feedback loop from steady state ([22G]0 and [pUG]0) was considered. Let dx.[22G]0 or lower be insufficient for silencing and let 22G RNAs be transiently perturbed to p.dx.[22G]0≠0, where dx<1andp<1. The critical duration (t22G) of such a perturbation for permanent reduction of 22G RNA below the level required for silencing is given by(4)t22G>1Tyln[1dx.p−1(1p−1)(TyTx+1)]

where Tx and Ty are the rates of turnover for 22G RNAs and pUG RNAs, respectively. Analogously, the critical duration (tpUG) of such a perturbation for permanent reduction of pUG RNA below the level required for silencing is given by(5)tpUG>1Txln[1dy.p−1(1p−1)(TxTy+1)]

where dy.[pUG]0 or lower is insufficient for silencing. Derivation of the general case for these equations (HRA ‘A’) is presented in Methods. These equations suggest that depending on the parameters of the architecture, different sensors may be more easily perturbed to cause heritable epigenetic changes. For example, for the same critical threshold below steady state (dx=dy=0.5) and the same extent of perturbation (p=0.8), an architecture with [22G]0 = 10, [pUG]0=7.14, Tx=0.05, Ty=0.1, kxy=0.07, kyx=0.0714, is more quickly inhibited by reducing 22G RNAs than by reducing pUG RNAs (6.93 vs 27.72 units of time).

Combining these considerations with the observations that both piRNA binding to target mRNAs (Figure 6a) and loss of the piRNA-binding Argonaute PRG-1 (Shukla et al., 2021; Priyadarshini et al., 2022) prolong the duration of heritable RNA silencing suggests a unified mechanism that sets gene-specific durations of heritable RNA silencing. Specifically, the dynamics of recovery from silencing depends on the strength of an inhibitory feedback that can act across generations to reduce the HRDE-1-dependent positive feedback loop. This transgenerational inhibition relies on sensor(s) that are regulated by PRG-1 and is opposed by piRNA binding to the silenced mRNA (Figure 6d).

This proposed mechanism implies the existence of sensor(s) that respond to the activity of the HRDE-1-dependent positive feedback loop by recognizing one or more of the molecules and/or chemical modifications generated. Consistently, a chromodomain protein HERI-1 has been reported to be recruited to genes undergoing heritable RNA silencing and is required to limit the duration of the silencing (Perales et al., 2018). Additional sensors are likely among the >3000 genes mis-regulated in animals lacking PRG-1 (Reed et al., 2020; Shukla et al., 2021). The levels or activities of these sensor(s) could either increase or decrease in response to the activity of the HRDE-1-dependent loop depending on which of the multiple equivalent configurations of the negative feedback are present at different genes (expected to decrease in Figure 6—figure supplement 1, left and right, but increase in Figure 6—figure supplement 1, middle). However, in every case, the net result is a reduction in the activity of the HRDE-1-dependent loop (Figure 6—figure supplement 1). Therefore, genes encoding such sensors could be among those that show increased mRNA levels (e.g. 2517 genes in Reed et al., 2020) and/or that show decreased mRNA levels (e.g., 968 genes in Reed et al., 2020) upon loss of PRG-1. Another set of genes that could encode similar sensors are those identified using repeated RNAi as modifiers of transgenerational epigenetic kinetics (Houri-Ze’evi et al., 2016). Regardless of the identities of such sensors, differences in the transgenerational feedback that reduces some component(s) of the 22G-pUG positive feedback loop can explain the persistence of, recovery from, and resistance to heritable RNA silencing when different genes are targeted for silencing.

In summary, experimental evidence and theoretical considerations suggest that the HRDE-1-dependent positive feedback loop that generates 22G RNAs and pUG RNAs is tuned by negative feedback that acts across generations to cause different durations of heritable RNA silencing. Such tuning can explain silencing for a few generations followed by recovery from silencing as well as resistance to silencing. Future studies are required for testing the quantitative predictions on the impact of reducing 22G RNA or pUG RNA levels (Equations (4) and (5)) and for identifying the PRG-1-dependent genes that could have roles in the transgenerational inhibition of heritable RNA silencing (Figure 6d).

The simple ESP systems simulated here (Figure 4) can be extended to include a wide variety of properties to explore heritable epigenetic changes that can occur in cell/organelle geometry, phase separation, protein folding, etc. In general, each kind of molecule or entity can have multiple properties that are sensed by different sensors. For example, concentration, folded structure, primary sequence, and subcellular localization of a protein could each be measured by different sensors that respond by causing different downstream effects. If a protein (x) is regulated by three different regulators that each change its concentration (C), subcellular localization (L), or folded structure (F), then the protein x could be considered as having C, L, and F as values for its regulated properties. If the protein x in turn acts as a regulator that changes another entity y, then the activity of x regulating y could be simulated as a combined function of its concentration, localization and folded structure (i.e. activity of x=f(C, L, F)). Such simulations preserve both the independent regulation of different properties of a protein along with the potential equivalence in the activity of a higher concentration of partially folded proteins and a lower concentration of well-folded proteins. Thus, appropriate mapping onto an ESP system would enable the explanation of many phenomena in terms of regulatory architectures formed by any set of interactors while rigorously considering heritability.

While no entity can promote changes in all its properties by itself, some entities can promote changes in one of their properties through self-regulatory interactions under some conditions. For example, prions can act as replicating stores of information that template changes in the conformation of other proteins with the same sequence (Scheckel and Aguzzi, 2018), although other properties of prions such as their concentration, subcellular localization, rate of turnover, etc. are determined through interactions with other entities/sensors. Such self-regulatory interactions for the control of some properties can be considered by allowing self-referential loops. Specifically, if the protein x above were a prion, then its properties will include C, L, and F as above, with the value of F changing with time as a function of both concentration and prior proportion folded (i.e. F(t+1)=g(C, F(t))). Similar considerations underscore that any one positive feedback loop can promote only one property of an entity and not all of its properties. For example, consider small RNAs that are associated with gene silencing in C. elegans. The targeting of a gene by small RNAs could be preserved in every generation through a regulatory loop whereby recognition of mRNA by antisense small RNAs results in the production of additional small RNAs by RNA-dependent RNA Polymerases. However, the mere existence of this feedback loop cannot explain the different concentrations of small RNAs targeting different genes (Gu et al., 2009) or the different durations of persistent small RNA production when initiated experimentally (Devanapally et al., 2021). Similar considerations apply for chromatin modifications, DNA modifications, RNA modifications, and all other ‘epigenetic marks’.

Inducing heritable changes in epigenetic information is more challenging than inducing similar changes in genetic information (Jose, 2020c). Chemically altering a single molecule (typically DNA) is sufficient for inducing a genetic change, however, similarly altering one entity of a regulatory architecture (say, a protein) to induce an epigenetic change requires simultaneously altering the many copies of that entity without altering DNA sequence. All DNA bases with induced chemical changes are deleted or converted into one of the other bases by the replication and repair pathways. Consequently, only 6 different base exchanges are possible in DNA sequence through a single mutation, but even when only up to three interactors are considered, 61 different HRA changes are possible through a single mutation (Figure 3). Importantly, after a heritable change, the DNA sequence remains similarly mutable, but the impact of a change (genetic or epigenetic) on subsequent mutability of a regulatory architecture could increase or decrease. Consider a change that incorporates a new transcription factor into a regulatory architecture. If it is an activator, it could promote the expression of many genes leading to the incorporation of more RNAs and proteins into the regulatory architecture. Conversely, if it is a repressor, it could repress the expression of many genes leading to the removal of RNAs and proteins from the regulatory architecture. Either of these consequences could make the entire architecture more robust such that drastic perturbation is needed to cause observable change. When such robust architectures occur during development, the identity of a cell could become relatively fixed and heritable through cell divisions and yet remain compatible with specific natural (Jarriault et al., 2008) and/or induced (Shi et al., 2017) cell fate transformations. Thus, such cell fate determination reflects the acquisition of different robust states, providing the appearance of a cell ‘rolling down an epigenetic landscape’ (Waddington, 1957).

Despite imposing heritability, regulated non-isomorphic directed graphs soon become much more numerous than unregulated non-isomorphic directed graphs as the number of interactors increase (125 vs 5604 for 4 interactors, Table 1). With just 10 interactors, there are >3 × 10²⁰ unregulated non-isomorphic directed graphs (Sloane, 2021) and HRAs are expected to be more numerous. This tremendous variety highlights the vast amount of information that a complex regulatory architecture can represent and the large number of changes that are possible despite sparsity of the change matrix (Figure 3). This number is potentially a measure of epigenetic evolvability – the ability to adapt and survive through regulatory change without genetic mutations. However, architectures made up of numerous interactors that are all necessary for the positive feedback loop(s) required for transmitting information across generations are more vulnerable to collapse (e.g., ‘C’ and ‘T’ in Figure 1). Furthermore, spatial constraints of the bottleneck stage (one cell in most cases) present a challenge for the robust transmission of regulatory information. A speculative possibility is that complex architectures are compressed into multiple smaller positive feedback loops for transmission between generations with the larger HRAs being re-established through interactions between the positive feedback loops in every generation. Examples of such numerous but small positive feedback loops include small RNA-mediated, chromatin-mediated, and prion-mediated loops that specify the identity of genes for particular forms of regulation in the next generation. However, determining how the rest of the regulatory information is transmitted in each case and whether such compression with redundancy is a general principle of heredity require further study.

Individual entities transmitted across generations can be considered as carrying part of the heritable information if such information is always seen in the context of the sensors that interact with the entity. While the maximal information that can be carried by DNA sequence is a product of its length (l) and the number of bits contributed by each base (2l = log₂[4].l), the maximal number of relevant bits contributed by any entity - including the genome - depends on the number of states sensed by all interacting sensors (for i^th entity = log₂(∑j=1sisj∑k=1pjpk) from Equation 1; Jose, 2020b). The genome likely interacts with the largest number of sensors per molecule, making it the entity with the most information density. When a gene sequence is transcribed and translated, sequence information is transmitted from one molecule (DNA) to many (RNA(s) and/or protein(s)), thereby reducing the density of sequence information per entity (e.g. RNA or protein of a particular sequence). However, information is also added to these entities through the process of RNA folding and protein folding, which depend on interactions with the surrounding chemical and physical context (e.g. Porter and Looger, 2018). The resultant structural (and potentially catalytic) specialization of RNAs and proteins provides them with additional properties that are relevant for other sensors, thereby increasing their information content beyond that in the corresponding genome sequence. Importantly, these proteins and RNAs can interact to create molecular complexes and organelles that again concentrate information in higher-order entities present as fewer copies within cells (e.g. centrosomes). Such complexes and organelles therefore are entities with high information density. In this view, the information density throughout a bottleneck stage connecting two generations (e.g. the single-cell zygote) is non-uniform and ranges from entities with very high density (e.g. the genome) to those with very low density (e.g. water). Beginning with the simplest of regulatory architectures (Figure 1), progressive acquisition of entities and their interacting sensors would lead to the incorporation of regulators with increasing information density. An entity that is connected to numerous sensors that each respond to changes in a fraction of its properties can appear to be the chief ‘information carrier’ of the living system (e.g. DNA in cells with a genome). Thus, as heritable regulatory architectures evolve and become more complex, entities with a large number of sensed properties can appear central or controlling (see Dyson, 1982; Noble, 2008 for similar ideas). This pathway for increasing complexity through interactions since before the origin of life suggests that when making synthetic life, any form of high-density information storage that interacts with heritable regulatory architectures can act as the ‘genome’ analogous to DNA.

All calculations were performed by hand or using custom programs in Python (v. 3.8.5), and/or R (v. 3.6.3). Simulations and analyses were performed using NetLogo (v. 6.1.1), Python (v. 3.8.5), and/or R (v. 3.6.3). Transitions between HRAs were depicted using the circular layout plugin in Gephi (v. 0.10.1 202301172018). All programs used in this study are available at https://github.com/AntonyJose-Lab/Jose_2023, copy archived at Jose, 2023.

The possible weakly connected non-isomorphic graphs that form regulatory architectures capable of indefinite persistence and the maximal information that can be stored using them (log₂N) were calculated. Systems of ordinary differential equations (ODEs) were used to describe the rate of change for each interacting entity (x,y,z) in each of the 26 heritable regulatory architectures (A to Z) with the relative amounts of all entities at any time defined as the ‘phenotype’ at that time. Each regulatory architecture is characterized by a maximum of 9 parameters in addition to the relative amounts of each entity: a rate of turnover for each entity (3 total; Tx,Ty,Tz) and rates for each regulatory interaction between entities (6 total; e.g. kxy is the regulatory input to x from y, kyz is the regulatory input to y from z, etc.). Relative amounts of each interacting entity for each architecture at steady state (x0,y0,z0) were determined by setting all rate equations to zero, which results in constraints on the variables for each architecture (e.g. for A: y0=x0.Txkxy=x0.kyxTy ; for B: y0=x0.Txkxy=x0.kyxTy ; z0=x0.Txkxy.kzyTz , etc.). These constraints were used to obtain a total of 128,015 parameter sets (Tx,Ty,Tz,kxy,kyz, etc.) that are compatible with steady state for each architecture (Supplementary file 1). Particular parameter sets were then used to illustrate the consequences of genetic and epigenetic changes in all architectures (Figure 2 and Figure 2—figure supplements 1–7). The simpler expressions for steady-state levels for all regulatory architectures when there are no turnover mechanisms for any entities and the only ‘turnover’ occurs by dilution upon cell division (T=Tx=Ty=Tz) were derived. Expressions for steady state values when all entities are lost at a constant rate to complex formation (γ) and for changes upon complete loss of an entity were also derived.

The impacts of transient perturbations from steady state were examined for each architecture to identify conditions for heritable epigenetic change by defining thresholds (dx,dy,dz) for observing a defect for each entity (e.g. for the function of x, dx.x0 is not sufficient, when dx<1, or dx.x0 is in excess, when dx>1). Responses after perturbations are illustrated for all architectures (Figure 2—figure supplement 1) and cases where genetic and epigenetic perturbations result in distinct responses are highlighted (Figure 2). The duration and extent of perturbation needed for heritable epigenetic changes were explored using numerical solutions of the equations describing the dynamics of each regulatory architecture. Explorations of 22G-pUG positive feedback loop were similarly performed by simulating a type ‘A’ HRA using ODEs (Figure 6). Analytical expressions for conditions that enable heritable epigenetic change were derived for the simplest heritable regulatory architecture where two entities (x and y) mutually promote each other’s production. Transitions between HRAs (Figure 3) were worked out manually by considering the consequence of each change from steady state for each HRA.

The dynamics of the 26 heritable regulatory architecture (HRAs in Figure 1) can be described by systems of ordinary differential equations. The rate of change of each entity (x, y, or z for each HRA in Figure 1) can be written by aggregating positive and negative contributions from the other sensors (e.g. kxy.y-kxz.z if x is positively regulated by y and negatively regulated by z) with loss of each entity described with a turnover term (e.g. -Tx.x for x). To ensure the concentrations of all entities remain non-negative, as expected in real living systems, the equations are bounded to be applicable only when the values of the changing entity is greater than zero. These equations can then be used to derive other equations and inequalities of interest.

At steady state, each architecture results in relative amounts of each entity (x0,y0,z0) that could define a ‘phenotype’. These relative concentrations can be derived by setting the rate of change of all entities to zero.

For the heritable regulatory architecture A,dxdt=kxy.y−Tx.x,∀(x+dx)>0,else(x+dx)=0dydt=kyx.x−Ty.y,∀(y+dy)>0,else(y+dy)=0

where, kxy is the rate constant for the production of x promoted by y; kyx is the rate constant for the production of y promoted by x; Tx is the rate of turnover of x; and Ty is the rate of turnover of y.

i.e., X˙=A.X, where X˙ = x˙y˙ ; A = -Txkxykyx-Ty ; X=xy

At steady state,-Tx.x+kxy.y=0kyx.x-Ty.y=0

that is, A.X=0, where A = [−Txkxykyx−Ty] ; X=x0y0 , which has solutions that satisfy:x0y0=kxyTx=Tykyx

Equations for steady state and the resulting solutions for the other HRAs can be similarly derived (see previous version of the paper for details).

A common way in which entities change in living systems is through the formation of intermolecular complexes that then interact with different entities to perform different functions. If the same number of molecules per unit time (γ) are lost for all entities (e.g. through incorporation into a 1:1:1 stoichiometric complex), then each entity needs to grow at the same rate (γ) to maintain steady state (x0,y0,z0).

For the heritable regulatory architecture A,kxy.y-Tx.x=γ-Ty.y+kyx.x=γ

that is A.X=B, where A = -Txkxykyx-Ty ; X=x0y0 ; B=γγ

The solution is given by X=A-1.B

(For A = abcd , A-1=1ad-bcd-b-ca)X=1Tx.Ty−kxy.kyx[−Ty−kxy−kyx−Tx].[γγ]i.e.,[x0y0]=γkxy.kyx−Tx.Ty[Ty+kxyTx+kyx]

To similarly identify the rates of growth for the other heritable regulatory architectures (B to Z) under a constant rate of loss for all entities, inverses for the 3x3 matrices can be used (see previous version of the paper for details). If all molecules are diluted through cell division (typically one cell dividing to give two), then for maintaining steady state on average, each molecule needs to accumulate to twice the average steady-state value per cell cycle (See Figure 4 for simulations that include cell divisions).

Loss of an entity (typically the RNA or protein product of a gene) through a genetic mutation is a common perturbation used for analyzing living systems.

For the heritable regulatory architecture A,

When x is lost,y˙=-Ty.y

Which has the solution,

y=y0.e-Tyt , that is the concentration of y undergoes exponential decay through turnover from its steady-state value (y0).

Similarly, when y is lost,x=x0.e-Txt

For all other architectures, loss of one entity can result in different dynamics of the other two entities (α and β, say) depending on their regulatory interactions. The equations for their dynamics is given by a pair of differential equations that can be coupled.

that is X˙=A.X, where X˙ = α˙β˙ ; A = abcd ; X=αβ .

The exact equations that result upon loss of each entity in each regulatory architecture can be derived for each HRA (see previous version of the paper for details) and potentially be used to distinguish the different architectures through genetic experiments (e.g. knockout of individual genes using genome editing). Since for the steady state of each architecture, there are a maximum of three equations, a maximum of three variables among rates of production (kxy,kyx etc.), rates of turnover (Tx,Ty,Tz), and the steady-state concentrations (x0, y0,z0) are constrained. Changes in all entities after each loss can be determined for all architectures through simulations by choosing random values for the unconstrained parameters (Figure 2 and Figure 2—figure supplements 1–7, left).

Analytic expressions for heritable epigenetic change after reducing the levels of a sensor from steady state are derived below for the simplest of heritable regulatory architectures ‘A’ (Figure 1).

The dynamics of two entities (x and y) that mutually promote each other’s production is given by a pair of differential equations that are coupled.

that is X˙=A.X, where X˙ = x˙y˙ ; A = abcd ; X=xy and a=-Tx , b=kxy , c=kyx , d=−Ty

The general solution for the concentrations x(t) and y(t) are:x(t)=K12m(e0.5t(a+d−m)(etm(a−d+m)+m+d−a))−bK2m(e0.5t(a+d−m)−e0.5t(a+d+m))y(t)=K22m(e0.5t(a+d−m)(etm(d−a+m)+m−d+a))−cK1m(e0.5t(a+d−m)−e0.5t(a+d+m))

where, m=a2-2ad+4bc+d2 and, K1 and K2 are constants.

Since the system was already at steady state before the perturbation, TxTy=kxykyx ,m=Tx2−2TxTy+4kxykyx+Ty2=Tx+Ty

Substituting the value of m in the equations above and simplifying yields, x(t)=K1.Ty+K2.kxyTx+Ty+K1.Tx−K2.kxyTx+Ty.e−(Tx+Ty)t

andy(t)=K2.Tx+K1.kyxTx+Ty+K2.Ty−K1.kyxTx+Ty.e−(Tx+Ty)t

Let dx be the reduction in x (reduction-of-function) needed to observe a defect when x0 is the steady-state value before perturbation. That is, dx.x0 is not sufficient for the function of x in a living system, where dx<1. Let x be perturbed to xp=p.dx.x0≠0 from t=0 until t=tp , where p<1. For heritable epigenetic changes using reduction-of-function perturbations (dx<1 and/or dy<1), which preserve the architecture at a new steady state: xps < dx.x0 and yps < dy.y0 .

To determine the concentration of y at the end of the perturbation (yp) the equation y˙=xp.kyx−Ty.y can be solved using y(t)=y0 at t=0. The general solution of the equation is given by,y(t)=xp.kyxTy+C1.e−Tyt

Substituting for y0=y0 at t=0, and rearranging gives C1=y0.Ty−xp.kyxTy . Thus, at the end of the perturbation (i.e., at tp),y(tp)=yp=xp.kyxTy+(y0.Ty−xp.kyxTy).e−Tytp

The new steady states (xpsandyps) will be reached from the initial concentrations of xp and yp .

Therefore, to determine the new steady state, the initial values of xp and yp can be used at new t=0 to get the values for the constants K1andK2 .xp=K1.Ty+K2.kxyTx+Ty+K1.Tx-K2.kxyTx+Ty.e-Tx+Tyt=0yp=K2.Tx+K1.kyxTx+Ty+K2.Ty-K1.kyxTx+Ty.e-(Tx+Ty)t=0

Which simplifies to,xp=K1.Ty+K2.kxyTx+Ty+K1.Tx-K2.kxyTx+Tyyp=K2.Tx+K1.kyxTx+Ty+K2.Ty-K1.kyxTx+Ty

Solving for each,K1 =xpK2 =yp

To obtain the new steady state value xps , set t=∞ in the equation using the above constants.xps=xp.Ty+yp.kxyTx+Tyyps=yp.Tx+xp.kyxTx+Ty

Consider the equality that is the threshold for observing heritable epigenetic effects,xps=xp.Ty+yp.kxyTx+Ty=dx.x0

Substituting for xp and simplifying yields,yp.kxy=dx.x0.Tx+dx.x0.Ty(1−p)

Substituting for yp(xp.kyxTy+(y0.Ty−xp.kyxTy).e−Tytp).kxy=dx.x0.Tx+dx.x0.Ty(1−p)

Collecting exponential terms,e−Tytp(y0.kxy.Ty−p.dx.x0.kyx.kxyTy)=dx.x0.Tx+dx.x0.Ty(1−p)−p.dx.x0.kyx.kxyTye−Tytp=dx.x0.Tx.Ty+dx.x0.Ty.Ty.(1−p)−p.dx.x0.kyx.kxyy0.kxy.Ty.Ty−p.dx.x0.kyx.kxye−Tytp=dx.x0.Tx.Ty+dx.x0.Ty.Ty−p.dx.x0.(kyx.kxy+Ty.Ty) y0.kxy.Ty−p.dx.x0.kyx.kxy

Dividing numerator and denominator with Ty ,e−Tytp=dx.x0.Tx+dx.x0.Ty−p.dx.x0.(kyx.kyxTy+Ty)y0.kxy−p.dx.x0.kyx.kyxTy

At steady state, the ratio x(t)y(t) will be independent of the concentrations of x and y. That is, xpsyps=x0y0=kxyTx=Tykyx . Therefore, these equalities can be used to simplify the above equations. Substituting kyx.kyxTy=Tx ,e−Tytp=dx.x0.Tx +dx.x0.Ty−p.dx.x0.(Tx+Ty)y0.kxy−p.dx.x0.Tx

Substituting ,y0.kxy=x0.Txe−Tytp=dx.x0.Tx+dx.x0.Ty−p.dx.x0.(Tx+Ty)x0.Tx−p.dx.x0.Tx

Dividing numerator and denominator by x0.Txe−Tytp=dx+dx.TyTx−p.dx.(1+TyTx)1−p.dx

Simplifying,e−Tytp=(1+TyTx).(1−p).dx1−p.dx

Dividing numerator and denominator by dxe−Tytp=(1+TyTx).(1−p)(1dx−p)

Taking the log_e on both sides,−Tytp=ln[(1+TyTx).(1−p)1dx−p]

that istp=1Tyln[1dx−p(1+TyTx).(1−p)]

Dividing numerator and denominator within the antilogarithm by p,tp=1Tyln[1dx.p−1(1p−1)(1+TyTx)]

This equation relates the duration of a perturbation (tp) and the extent of the perturbation (p<1 for loss-of-function) beyond the threshold that causes a defect in the function of x (i.e. dx). Increasing the duration of the perturbation beyond tp for a given extent of perturbation (p) will result in heritable epigenetic change where the steady-state levels of both interactors are insufficient for appropriate function.

Similarly, the minimal duration of perturbation for heritable epigenetic changes through a defect in the function of y is given by,tp>1Txln[1dy.p−1(1p−1)(1+TxTy)]

These inequalities were verified using numerical simulations (see ‘HRA_A_ tp_analytical_expression_check.py’) and additional HRAs were similarly simulated to gain intuitions about the consequences of epigenetic reduction in the levels of entities (Figure 2 and Figure 2—figure supplements 1–7).

Simple ESP systems were simulated using custom programs in NetLogo (Wilensky, 1999)(details are available within the ‘code’ and ‘info’ tabs of each program). Results from systematic explorations obtained by running NetLogo programs from the command line (i.e. ‘headless’) were analyzed using R (e.g. Figure 4d, Figure 4—figure supplement 3). The developmental timings of cell divisions in C. elegans were curated manually from the literature (Supplementary file 3) and used to simulate ESP systems that incorporate developmental timings and temporal delays in regulation (Figure 5). Videos (Videos 1–13) were made by recording screen captures of NetLogo runs using QuickTime Player.

A model was created in NetLogo to simulate entity-sensor-property systems and their evolution across generations for exploring regulatory architectures (ESP_systems _explorer_v1.nlogo). A variety of regulatory architectures were simulated using this model to identify ones with some architecture that persisted for 250 generations with or without perturbations (e.g. 78285 out of 225000 tested using the experiment ‘ESP_origins_2–16_mols’ described under behaviorspace). Each stable ESP system can be further analyzed in detail using the related ESP_systems_single_system_explorer_v1. All randomly chosen values for parameters in the regulatory architectures that lead to stability, that is heritable for many generations, can be recreated because the random-seed is set for each run using the behaviorspace-run-number. The behaviorspace-run-number serves as the ‘system-id’ in the related ESP_systems_single_system_explorer_v1.nlogo.

For each system, entities/sensors were defined as ‘turtles’ with 3 variables that stored its attributes:

The regulatory interactions in each system were specified using ‘links’ that were weighted to indicate the threshold required for the regulatory interaction and colored to indicate the nature of the regulation. Specifically, the links have two variables:

Parameters that were varied in the exploration of ESP systems were:

Additional parameters, which were not varied in the exploration of ESP systems were:

Monitors reporting behaviorspace-run-number (system-id), generation number, total molecules, the number of generations of stability for considering a regulatory architecture stable (stability gen), stable generations since last instability, the value of the perturbed entity (perturb-value), the phase of the perturbation (perturb-phase), the duration of each perturbation (perturb-time), the frequency of the perturbations (perturb-freq) and the identity of the perturbed entity (perturbed node) were included in the interface.

For simulating changes over time, this model used a combination of deterministic and stochastic functions. The values of each entity/sensor changes at each tick using a deterministic equation: val @ t+1 = val @ t + sum of inputs from all sensors. The change in value contributed by each sensor for a given entity = round((property of entity) x (value of sensor) x (1 - inactive-fraction of sensor) / (weight of regulatory link)). In other words, change = round(k x (value of sensor)), where k is a different constant for each sensor of each entity and round indicates rounding to the nearest integer. For positive regulators (link color grey), this change in value was added and for negative regulators (link color black), it was subtracted. The order of operation on the entities/sensors varies with every tick. After every two ticks, only about half the number of each entity/sensor was kept using a random number generator to simulate dilution and random partitioning upon cell division.

The value of each entity/sensor was plotted relative to the most abundant entity/sensor. This profile at each time point can be considered as the ‘phenotype’ of the system. These scaled values are also used to depict each entity/sensor in the regulatory architecture at each time point.