Heritable epigenetic changes are constrained by the dynamics of regulatory architectures

Reviewer #1 (Public Review):

The author studies a family of models for heritable epigenetic information, with a focus on enumerating and classifying different possible architectures. The key aspects of the paper are:

- Enumerate all 'heritable' architectures for up to 4 constituents.
- A study of whether permanent ("genetic") or transient ("epigenetic") perturbations lead to heritable changes.
- Enumerated the connectivity of the "sequence space" formed by these heritable architectures.
- Incorporating stochasticity, the authors explore stability to noise (transient perturbations).
- A connection is made with experimental results on C elegans.

The study is timely, as there has been a renewed interest in the last decade in non-genetic, heritable heterogeneity (e.g., from single-cell transcriptomics). Consequently, there is a need for a theoretical understanding of the constraints on such systems. There are some excellent aspects of this study: for instance:

- the attention paid to how one architecture "mutates" into another, establishing the analogue of a "sequence space" for network motifs (Fig 3).
- the distinction is drawn between permanent ("genetic") and transient ("epigenetic") perturbations that can lead to heritable changes.
- the interplay between development, generational timescales, and physiological time (as in Fig. 5).

The manuscript would be very interesting if it focused on explaining and expanding these results. Unfortunately, as a whole, it does not succeed in formalising nor addressing any particular open questions in the field. Aside from issues in presentation and modelling choices (detailed below), it would benefit greatly from a more systematic approach rather than the vignettes presented.

## Terminology
The author introduces a terminology for networks of interacting species in terms of "entities" and "sensors" -- the former being nodes of a graph, and the latter being those nodes that receive inputs from other nodes. In the language of directed graphs, "entities" would seem to correspond to vertices, and "sensors" those vertices with positive indegree and outdegree. Unfortunately, the added benefit of redefining accepted terminology from the study of graphs and networks is not clear.

## Heritability
The primary goal of the paper is to analyse the properties of those networks that constitute "heritable regulatory architectures". The definition of heritability is not clearly stated anywhere in the paper, but it appears to be that the steady-state of the network must have a non-zero expression of every entity. As this is the heart of the paper, it would be good to have the definition of heritable laid out clearly in either the main text or the SI.

## Model
As described in the supplementary, but not in the main text, the author first chooses to endow these networks with simple linear dynamics; something like $\partial_t \vec{x} = A x - T x$, where the vector $x$ is the expression level of each entity, $A$ has the structure of the adjacency matrix of the directed graph, and $T$ is a diagonal matrix with positive entries that determines the degradation or dilution rate of each entity. From a readability standpoint, it would greatly aid the reader if the long list of equations in the SI were replaced with the simple rule that takes one from a network diagram to a set of ODEs.

The implementation of negative regulation is manifestly unphysical if the "entities" represent the expression level of, say, gene products. For instance, in regulatory network E, the value of the variable z can go negative (for instance, if the system starts with z= and y=0, and x > 0).

The model seems to suddenly change from Figure 4 onwards. While the results presented here have at least some attempt at classification or statistical rigour (i.e. Fig 4 D), there are suddenly three values associated with each entity ("property step, active fraction, and number"). Furthermore, the system suddenly appears to be stochastic. The reader is left unsure of what has happened, especially after having made the effort to deduce the model as it was in Figs 1 through 3. No respite is to be found in the SI, either, where this new stochastic model should have been described in sufficient detail to allow one to reproduce the simulation.

## Perturbations
Inspired especially by experimental manipulations such as RNAi or mutagenesis, the author studies whether such perturbations can lead to a heritable change in network output. While this is naturally the case for permanent changes (such as mutagenesis), the author gives convincing examples of cases in which transient perturbations lead to heritable changes. Presumably, this is due the the underlying mutlistability of many networks, in which a perturbation can pop the system from one attractor to another.

Unfortunately, there appears to be no attempt at a systematic study of outcomes, nor a classification of when a particular behaviour is to be expected. Instead, there is a long and difficult-to-read description of numerical results that appear to have been sampled at random (in terms of both the architecture and parameter regime chosen). The main result here appears to be that "genetic" (permanent) and "epigenetic" (transient) perturbations can differ from each other -- and that architectures that share a response to genetic perturbation need not behave the same under an epigenetic one. This is neither surprising (in which case even illustrative evidence would have sufficed) nor is it explored with statistical or combinatorial rigour (e.g. how easy is it to mistake one architecture for another? What fraction share a response to a particular perturbation?)

As an additional comment, many of the results here are presented as depending on the topology of the network. However, each network is specified by many kinetic constants, and there is no attempt to consider the robustness of results to changes in parameters.

## DNA analogy
At two points, the author makes a comparison between genetic information (i.e. DNA) and epigenetic information as determined by these heritable regulatory architectures. The two claims the author makes are that (i) heritable architectures are capable of transmitting "more heritable information" than genetic sequences, and (ii) that, unlike DNA, the connectivity (in the sense of mutations) between heritable architectures is sparse and uneven (i.e. some architectures are better connected than others).

In both cases, the claim is somewhat tenuous -- in essence, it seems an unfair comparison to consider the basic epigenetic unit to be an "entity" (e.g., an entire transcription factor gene product, or an organelle), while the basic genetic unit is taken to be a single base-pair. The situation is somewhat different if the relevant comparison was the typical size of a gene (e.g., 1 kb).

Reviewer #2 (Public Review):

Summary:
This manuscript uses an interesting abstraction of epigenetic inheritance systems as partially stable states in biological networks. This follows on previous review/commentary articles by the author. Most of the molecular epigenetic inheritance literature in multicellular organisms implies some kind of templating or copying mechanisms (DNA or histone methylation, small RNA amplification) and does not focus on stability from a systems biology perspective. By contrast, theoretical and experimental work on the stability of biological networks has focused on unicellular systems (bacteria), and neglects development. The larger part of the present manuscript (Figures 1-4) deals with such networks that could exist in bacteria. The author classifies and simulates networks of interacting entities, and (unsurprisingly) concludes that positive feedback is important for stability. This part is an interesting exercise but would need to be assessed by another reviewer for comprehensiveness and for originality in the systems biology literature. There is much literature on "epigenetic" memory in networks, with several stable states and I do not see here anything strikingly new.

An interesting part is then to discuss such networks in the framework of a multicellular organism rather than dividing unicellular organisms, and Figure 5 includes development in the picture. Finally, Figure 6 makes a model of the feedback loops in small RNA inheritance in C. elegans to explain differences in the length of inheritance of silencing in different contexts and for different genes and their sensitivity to perturbations. The proposed model for the memory length is distinct from a previously published model by Karin et al. (ref 49).

Strengths:
A key strength of the manuscript is to reflect on conditions for epigenetic inheritance and its variable duration from the perspective of network stability.

Weaknesses:
- I found confusing the distinction between the architecture of the network and the state in which it is. Many network components (proteins and RNAs) are coded in the genome, so a node may not disappear forever.

- From the Supplementary methods, the relationship between two nodes seems to be all in the form of dx/dt = Kxy . Y, which is just one way to model biological reactions. The generality of the results on network architectures that are heritable and robust/sensitive to change is unclear. Other interactions can have sigmoidal effects, for example. Is there no systems biology study that has addressed (meta)stability of networks before in a more general manner?

- Why is auto-regulation neglected? As this is a clear cause of metastable states that can be inherited, I was surprised not to find this among the networks.

- I did not understand the point of using the term "entity-sensor-property". Are they the same networks as above, now simulated in a computer environment step by step (thus allowing delays)?

- The final part applies the network modeling framework from above to small RNA inheritance in C. elegans. Given the positive feedback, what requires explanation is how fast the system STOPs small RNA inheritance. A previous model (Karin et al., ref. 49) builds on the fact that factors involved in inheritance are in finite quantity hence the different small RNAs "compete" for amplification and those targeting a given gene may eventually become extinct.

The present model relies on a simple positive feedback that in principle can be modulated, and this modulation remains outside the model. A possibility is to add negative regulation by factors such as HERI-1, that are known to limit the duration of the silencing.

The duration of silencing differs between genes. To explain this, the author introduces again outside the model the possibility of piRNAs acting on the mRNA, which may provide a difference in the stability of the system for different transcripts.
At the end, I do not understand the point of modeling the positive feedback.

- From the initial analysis of abstract networks that do not rely on templating, I expected a discussion of possible examples from non-templated systems and was a little surprised by the end of the manuscript on small RNAs.