Analog epigenetic memory revealed by targeted chromatin editing

Summary

Cells store information by means of chromatin modifications that persist through cell divisions and can hold gene expression silenced over generations. However, how these modifications may maintain other gene expression states has remained unclear. This study shows that chromatin modifications can maintain a wide range of gene expression levels over time, thus uncovering analog epigenetic memory. By engineering a genomic reporter and epigenetic effectors, we tracked the gene expression dynamics following targeted perturbations to the chromatin state. We found that distinct grades of DNA methylation led to corresponding, persistent gene expression levels. Altering the DNA methylation grade, in turn, resulted in permanent loss of gene expression memory. Consistent with experiments, our chromatin modification model indicates that analog memory arises when the positive feedback between DNA methylation and repressive histone modifications is lacking. This discovery will lead to a deeper understanding of epigenetic memory and to new tools for synthetic biology.

Graphical abstract

Highlights

• •A system was engineered to edit chromatin marks and study gene expression memory
• •Gene expression can be memorized at a spectrum of levels, not just “on”' and “off”
• •DNA methylation grade is conserved and mediates gene expression level maintenance
• •A model indicates that analog memory emerges in the absence of DNAme-H3K9me3 feedback

Traditional models of epigenetic memory support binary memory. Targeted editing of the chromatin state, coupled with temporal tracking of gene expression with clonal resolution, reveals that epigenetic memory can be analog. DNA methylation serves as the “knob” of a molecular dimmer switch that sets and memorizes gene expression levels. A chromatin modification model capturing histone modifications and DNA methylation indicates that this is possible when positive feedback between DNA methylation and H3K9me3 is lacking.

Introduction

Epigenetic memory is the ability to maintain multiple gene expression states through cell divisions and time without differences in genetic sequence. It is required by many biological processes, including the maintenance of distinct cell identities,^1,2 innate and adaptive immunity,^3,4,5 memory formation in the brain,^6,7 and could become a powerful tool for mammalian cell engineering.⁸

Chromatin modifications, such as histone modifications and DNA methylation, are critical mediators of epigenetic memory,^{9,10,11,12,13} owing to their ability to affect gene expression by altering transcription^{14,15,16,17,18} and because they can self-propagate through DNA replication and cell division.^14,19,20 Because of these properties, chromatin modifications mediate memory of gene expression in a variety of biological contexts. DNA methylation and histone H3 lysine 9 tri-methylation (H3K9me3) enable the maintenance of cell identity by inducing and keeping stable silencing of alternate fate genes.^1,2 For example, somatic cell types maintain pluripotent genes such as Oct4 in a highly methylated, and hence silenced, state.²¹ In the context of immunology, the persistence of histone modifications, established after initial exposure to a pathogen, has been linked to the innate memory of the immune system.⁴ Finally, DNA methylation has been shown to be required for memory formation in the brain,⁶ although research is still unfolding to determine how that is established.

Despite the critical roles that chromatin modifications play in epigenetic memory across biological processes, the spectrum of gene expression states that these chemical modifications stably maintain remains uncharted.

Heterochromatin is a well-studied state of chromatin with concurrent DNA methylation and H3K9me3, in which gene expression is maintained silenced.¹⁹ The long-term temporal stability of this state was assessed in prior studies that performed time-resolved analysis of a reporter gene’s expression.^22,23 Related studies have shown that histone modifications mediate a bistable epigenetic switch where gene expression is either silenced or active.^24,25 Therefore, current models of epigenetic memory posit that memory is binary, wherein genes are maintained either “on” or “off” based on autocatalytic histone modifications that induce bistability²⁶ (Figure 1A). While informative, the abovementioned studies have been carried out in a small set of contexts limited to artificial chromosomes or to specific endogenous genes. Because the effect of chromatin modifications on gene expression is context dependent,^27,28,29,30 it remains unclear whether a model of binary memory generalizes across genetic contexts, particularly for endogenous loci.

Figure 1.

Binary versus analog epigenetic memory of gene expression

(A) A binary memory model posits that only silenced or active gene expression states are maintained long-term, but no other states. Each trajectory can be regarded as one cell. The emergent long-term distribution of gene expression across a cell population has two distinct peaks.

(B) In an analog memory model, any gene expression level is maintained. The resulting long-term distribution of gene expression levels can take any shape.

Here, we investigate whether chromatin modifications can maintain a broader spectrum of gene expression states and, hence, mediate non-binary memory. Additional forms of memory beyond binary include n-ary memory and analog memory. Memory that is n-ary extends the concept of binary memory to n $>2$ stable states and emerges from multi-stable networks.³¹ In the limit where the n stable states form a continuum of states, memory is analog. With analog memory, every possible gene expression state is stably maintained in time (Figure 1B). Elucidating the landscape of gene expression states that chromatin modifications maintain requires methods that initialize gene expression at different levels through targeted perturbations to the chromatin state. Although methods using targeted editing of reporters knocked into endogenous genes have been developed before, they can be subject to genetic context effects that are difficult to control and may influence the results.^{23,28,30,32,33} To mitigate the influence of context, prior studies have constructed reporter genes within chromatin insulators in artificial chromosomes²²; however, whether these findings generalize to natural chromosomes is unclear.

To overcome these difficulties, we developed a single-copy genomic reporter system in a natural chromosome at a specific site separated from endogenous genes. We also developed a set of chimeric proteins for targeted editing of the chromatin state and a chromatin modification model that, different from existing models, includes both histone and DNA methylation. We used these tools to map the spectrum of gene expression states that are memorized by chromatin modifications and determined the causal roles of histone modifications versus DNA methylation in long-term epigenetic memory. We show that memory can be analog and that, despite recent emphasis on histone modifications, DNA methylation is the causal determinant of analog epigenetic memory, with H3K9me3 only being an intermediate regulator of gene expression recruited via DNA methylation.

Results

An engineered genomic reporter system to study gene expression dynamics post targeted chromatin editing

We first set out to develop an experimental system to study epigenetic memory in a natural chromosome and the role that histone and DNA methylation play in its establishment. To this end, we constructed a single-copy reporter integrated site-specifically in an endogenous mammalian locus and engineered a set of proteins for targeted epigenetic editing (Figures 2A–2D, S1, S2A, and S2B; STAR Methods; and Tables S1 and S2). We constructed the genomic reporter in Chinese hamster ovary (CHO)-K1 cells, a common model cell line in mammalian synthetic biology. The gene comprises the mammalian elongation factor 1a promoter (EF1a) driving the expression of fluorescent reporter (EBFP2). This enables the study of changes in gene expression and protein level due to the correlation between fluorescent protein level and fluorescence intensity measured using flow cytometry.³⁴ We flanked the gene with cHS4 chromatin insulators for isolation of the reporter from other genes.³⁵ The reporter comprises five binding sites upstream of the promoter for each of the DNA-binding proteins PhlF and rTetR (Figures S1A and S1B and Table S2),³⁶ where the binding of PhlF and rTetR to DNA can be modulated using 2,4-diacetylphloroglucinol (DAPG) and doxycycline, respectively. Moreover, the region upstream of the promoter comprises five guide RNA (gRNA) target sites, which enable targeted chromatin editing using epigenetic effectors fused to catalytically dead Cas9 (dCas9) (Figures 2B and S1C and Tables S2 and S3). Epigenetic editing was performed by expressing epigenetic effectors fused to PhlF, rTetR, or dCas9 (Figures S1A–S1F and Table S2) using transient transfection. The epigenetic effectors were co-transfected with a fluorescent transfection marker (EYFP). Since the expression of two co-transfected vectors is highly correlated,³⁷ this enables sampling of cells with varying levels of epigenetic effector by sorting for cells with varying levels of EYFP using fluorescence-activated cell sorting (Figure 2E).

Figure 2.

The gene expression dynamics in DNMT3A-edited cells indicate a form of memory different from binary

(A) Transient epigenetic editing using KRAB, DNMT3A, or TET1 fused to dCas9, PhlF, or rTetR DNA-binding domains.

(B) Experimental system. A mammalian constitutive promoter (EF1a) drives expression of fluorescent protein EBFP2. Upstream binding sites enable targeted recruitment of epigenetic effectors fused to DNA-binding proteins rTetR, PhlF, or dCas9. The reporter gene is flanked by chromatin insulators (Figures S1, S2A, and S2B; STAR Methods; Tables S1 and S2).

(C) CpGs along the reporter (Table S1).

(D) Copy number of the genomic integration determined using qPCR. Control is the landing pad cell line (STAR Methods).

(E) Transient transfection into cells bearing the reporter, fluorescence-activated cell sorting based on transfection level, and time-course flow cytometry measurements.

(F) Gene expression dynamics following editing with DNMT3A-dCas9 according to the experimental timeline of (E). Shown are single-cell flow cytometry measurements (EBFP2) of DNMT3A-edited cells. DNMT3A-dCas9 was targeted to five target sites upstream of the promoter, and a scrambled gRNA target sequence was used as the control (Figures S2C and S2D; Table S3). Shaded yellow is the time that the transfection marker was detected (Figure S2D). The marker indicates geometric mean. Data shown are from a representative replicate from three independent replicates.

(G) MeDIP-qPCR and ChIP-qPCR analysis 14 days after transfection of DNMT3A-dCas9 and cell sorting for high levels of transfection (Table S4 and STAR Methods). Data are from three independent replicates. Reported are fold changes and their mean using the standard $\Delta \Delta C_t$ method with respect to the active state (STAR Methods). A non-targeting (NT) scrambled gRNA target sequence (gRNA NT) was used as the control. Error bars are SD. ∗p < 0.05, ∗∗p < 0.01, and ∗∗∗p < 0.001, unpaired two-tailed t test.

(H) Gene expression dynamics post PhlF-KRAB editing according to the experimental timeline of (E). Shown are flow cytometry measurements of the reporter gene (EBFP2) for single cells. The shaded yellow region indicates the time the transfection marker was detected during the time when DAPG was not applied. DAPG was applied in the PhlF-KRAB and PhlF conditions beginning at 6 days. The marker indicates geometric mean. A different independent replicate was measured on each day. Data shown are from three independent replicates.

(I) MeDIP-qPCR and ChIP-qPCR analysis 6 days after transfection of PhlF-KRAB and cell sorting for high levels of transfection (Table S4 and STAR Methods). Shown are fold changes and their mean with respect to the active state from three independent replicates. Fold changes were calculated using the standard ΔΔC_t method. Error bars are SD. ∗p < 0.05, ∗∗p < 0.01, and ∗∗∗p < 0.001, unpaired two-tailed t test.

(J) Chromatin modification circuit obtained when KRAB = 0 and TET1 = 0 (Figure S5C).

(K) Top: dose-response curve for the (CpGme and X) pair. Bottom: dose-response curve between the height of a DNMT3A pulse (Equation 7) and the DNA methylation grade (CpGme). See Figures S5D, S7, and S15.

(L) Stationary probability distribution of gene expression for the system in Tables S5 and S8, with parameter values in STAR Methods.

(M) Probability distributions of gene expression for the same system as in (L) after t = 28 days, with parameter values and initial conditions given in STAR Methods. For (L) and (M), the distributions were obtained computationally using the stochastic simulation algorithm (SSA)³⁸; STAR Methods. See Figures S5B and S14. Here, $\epsilon$$\left(\zeta \right)$ scales the ratio between the basal (recruited) erasure rate and the auto-catalysis rate of each modification. See Figures S5E, S7, and S15. A.U.F., arbitrary units of fluorescence.

Transient chromatin editing with DNMT3A leads to permanent changes in gene expression

We first investigated the gene expression dynamics following epigenetic editing by DNA methylation writer DNA methyltransferase 3 alpha (DNMT3A). To this end, we expressed a fusion of dCas9 and the catalytic domain of DNMT3A (DNMT3A-dCas9)³⁹ (Figure S1C and Table S2), via transient transfection targeted to five target sites upstream of the promoter in the reporter gene using a gRNA (Figures 2F, S2C, and S2D and Table S3). Subsequently, we utilized fluorescence-activated cell sorting to sample cell populations with various transfection levels and proceeded with time-course flow cytometry measurements of the resulting populations. We observed that gene expression was repressed, reaching maximal repression approximately 11 days after transfection (Figures 2F and S2C). After reaching maximal repression, the cell distributions remained stationary for all levels of DNMT3A-dCas9 (Figures 2F and S2C). This, along with no detection of the transfection marker of DNMT3A-dCas9 by day 8 (Figure S2D), indicates memory of the transient DNMT3A editing. No repression was observed for cells where a gRNA with a scrambled target sequence was co-transfected with DNMT3A-dCas9 or in control experiments where dCas9 was targeted to the gene (Figures S2E and S2F).

Although the epigenetic effector dosage increased the proportion of cells in the silenced state, all dosages of DNMT3A-dCas9 editing led to stationary distributions with cells expressing the reporter at all levels of gene expression, ranging from silenced to fully active (Figures 2F and S2C). Experiments where we transiently transfected the same DNMT3A catalytic domain fused to rTetR instead of dCas9 (rTetR-XTEN80-DNMT3A) (Figures S1D and S3 and Table S2) also led to stationary distributions of gene expression spanning the whole spectrum of levels, from silenced to fully active. These observations rule out binary memory (Figure 1). Since maximal repression is reached approximately 11 days post-transfection of DNMT3A-dCas9, we selected day 14 for analyzing DNA and histone methylation to ensure maximal repression has been reached. Methylated DNA immunoprecipitation (MeDIP) and chromatin immunoprecipitation (ChIP) followed by qPCR further showed an increase in DNA methylation and H3K9me3 levels, accompanied by a decrease in H3K4me3 levels in the reporter gene compared to the untransfected (active) cells (Figure 2G; STAR Methods; and Table S4). This is in accordance with reports showing the ability of methylated DNA to recruit writers for H3K9me3,⁴⁰ indicating that these two chromatin modifications play a role in the temporal maintenance of the gene expression states.

Transient chromatin editing with KRAB leads only to temporary changes in gene expression

In order to discriminate the roles of DNA methylation and H3K9me3 in the maintenance of gene expression states, we investigated the gene expression dynamics following epigenetic editing with writers of H3K9me3. To this end, we fused DNA-binding protein PhlF to KRAB (Figure S1E and Table S2),⁴¹ an epigenetic effector that promotes H3K9 trimethylation.^42,43 We thus expressed PhlF-KRAB via transient transfection in the cells bearing the reporter gene, sampled cell populations across different transfection levels using fluorescence-activated cell sorting, and performed time-course flow cytometry measurements (Figures 2H and S4A–S4C). We observed rapid repression of gene expression for cells transfected with PhlF-KRAB and no repression for cells transfected with PhlF. Irrespective of the dosages of PhlF-KRAB, gene expression reverted to the active gene state within 4 days after reaching maximal repression. This aligns with previous reports where KRAB-based epigenetic editing resulted in temporary repression while failing to establish permanent silencing.^23,39 Further experiments with transient expression of PhlF-KRAB also resulted in temporary repression of the gene while no long-term memory was observed (Figures S4D–S4F).

To determine what chromatin modifications emerged as a result of KRAB-mediated editing, we performed MeDIP and ChIP followed by qPCR on cells repressed with PhlF-KRAB (Figure 2I and STAR Methods). Since gene expression is maximally repressed around day 6 post-transfection and then reactivates, we selected day 6 for this analysis. We observed an increase in H3K9me3 levels in the reporter gene compared to untransfected (active) cells, while DNA methylation levels remained unchanged, and H3K4me3 levels decreased. Taken together, these data indicate that an increase in H3K9me3 in the absence of DNA methylation is unable to permanently downregulate gene expression. Therefore, DNA methylation, but not H3K9me3, emerges as a primary mediator of the temporal maintenance of gene expression states in the cells edited by DNMT3A (Figure 2F).

A model where DNA methylation drives H3K9me3 predicts analog memory

Since de novo DNA methylation leads to H3K9me3 (Figure 2G), we conclude that DNA methylation mediates the establishment of H3K9me3. This is consistent with studies reporting that methylated CpGs are bound by a reader protein (methyl-CpG-binding protein 2) that recruits H3K9me3 writers.⁴⁰ By contrast, H3K9me3 does not mediate the establishment of DNA methylation in our system (Figure 2I). Taken together, these data suggest a model in which DNA methylation drives H3K9me3. Furthermore, the permanent change in gene expression resulting from transient recruitment of DNA methylation writer (Figure 2F), but not from H3K9me3 writer (Figure 2H), suggests that DNA methylation’s decay is negligible, while H3K9me3 decay is relatively fast. This long-term stability of DNA methylation has been attributed to the fast action of maintenance enzyme DNMT1, which re-writes DNA methylation on the daughter DNA strand upon DNA replication.^14,44

We therefore propose a model in which the DNA methylation state remains constant in time in the absence of writer DNMT3A, drives the formation of H3K9me3 and the removal of H3K4me3.^12,45,46 These histone marks, in turn, are known to mutually inhibit each other by recruiting each other’s erasers⁴⁷ and are autocatalytic since they each recruit their own writers.^11,19,20,48 Furthermore, H3K4me3 but not H3K9me3 is associated with transcriptionally active promoters,²⁰ leading to our chromatin modification model (Figures 2J and S5; see STAR Methods; and Table S7 for the chemical reaction system).

This model predicts an inverse graded relationship between the DNA methylation grade in the gene, defined as the mean fraction of methylated CpGs, and gene expression level (Figure 2K [top] and STAR Methods). When the mutual inhibition between activating and repressive marks $\left(\zeta \right)$ is sufficiently strong, the map between the intensity of a DNMT3A pulse and the DNA methylation grade becomes ultrasensitive (Figures 2K [bottom] and S5D). This, in turn, with noise in the reactions and in the DNMT3A levels, leads to a bimodal distribution of gene expression in response to a transient DNMT3A input (Figures 2L and S5E), which matches well experimental observations (Figures S2C and S2D). The model also recapitulates the rest of the experimental observations (Figure S6).

Most importantly, the model predicts that any initially set DNA methylation grade remains approximately constant in time in the absence of DNMT3A, since the decay rate of DNA methylation is negligible. Thus, any gene expression level initially set with the corresponding equilibrium grade of DNA methylation persists in time, with some variability under the effect of noise (Figure 2M). This demonstrates analog memory. Concerning noise properties, the model also predicts that intermediate DNA methylation grades will result in broader gene expression distributions (compare the orange and yellow distributions to the gray and blue distributions in Figure 2M). This occurs when the basal erasure of histone modifications is sufficiently slow compared to their autocatalysis (small $ϵ$). In this regime, the histone modification circuit becomes bistable for intermediate DNA methylation grade (see STAR Methods and Figure S7A), which leads to increased gene expression variability under the influence of noise (Figure S7B). In summary, our data-educated model predicts that DNA methylation is primarily responsible for analog memory, with histone modifications following it to modulate the gene expression level and its variability.

Fluorescence-activated cell sorting reveals non-binary epigenetic memory

Following the model predictions, we hypothesized that epigenetic memory can be analog, thereby enabling cells to maintain any gene expression level. To initially test this hypothesis, we first sought to investigate whether cells with the gene in a state different from silenced or active within DNMT3A-edited populations exhibited long-term stability of gene expression. To this end, we performed fluorescence-activated cell sorting of the DNMT3A-edited cells from low, intermediate, and high gene expression levels (Figures 3A and 3B and STAR Methods). We then performed time-course flow cytometry measurements for 29 days to analyze the long-term gene expression trajectories (Figures 3C–3E). The cells with low and high levels of gene expression maintained an approximately stable distribution of gene expression levels after cell sorting, as expected by a binary memory model. Nonetheless, the cells with intermediate levels of expression, after an initial widening of the expression distribution, also kept a stable gene expression distribution across the population for the entire time course. This is incompatible with binary memory (Figure 1A). To characterize the epigenetic state of these cells, we performed ChIP followed by qPCR of the low, intermediate, and active cell populations (Figure 3F and STAR Methods). We found an increase of H3K9me3 in the low population compared to the intermediate and active cell populations. Accordingly, we observed a decrease in H3K4me3 in the low- and intermediate-expressing cells compared to the active cells. These results are consistent with the known role of H3K9me3 as a chromatin modification associated with repression and H3K4me3 as a chromatin modification associated with active transcription at promoters.⁴⁹

Figure 3.

Gene expression is stably maintained at multiple levels in DNMT3A-edited cells

(A) Experiment overview for (B–I).

(B) Fluorescence-activated cell sorting of DNMT3A-edited cells (DNMT3A-dCas9) according to low, intermediate, and high levels of gene expression (EBFP2). Single-cell gene expression measurements on the day of cell sorting using flow cytometry.

(C) Time-course flow cytometry data from cells obtained as shown in (B).

(D) Density of cells from flow cytometry measurements obtained on the last time point in (C) (29 days after cell sorting). (B–D) show a representative replicate from three independent replicates.

(E) Dynamics of gene expression exhibited by cells shown in (B–D). Shown is the mean of geometric means from three independent replicates. Error bars are the SD of the mean.

(F) ChIP-qPCR for H3K9me3 and H3K4me3 of the promoter in the reporter gene. Shown are fold changes and the corresponding mean from three independent replicates as determined by the $\Delta \Delta C_t$ method with respect to the active state (parental untransfected cells bearing the reporter gene) (STAR Methods). Error bars are SD. ∗p < 0.05, ∗∗p < 0.01, and ∗∗∗p < 0.001, unpaired two-tailed t test.

(G) Heatmap of targeted bisulfite sequencing of the reporter gene. Shown is the level of DNA methylation as the mean percentage of DNA methylation at each CpG from three independent replicates (STAR Methods).

(H) Probability distribution of the fraction of methylated CpGs (STAR Methods).

(I) Correlation plot between the mean fraction of CpGs methylated and the mean gene expression level. Shown are means and error bars indicating SD. The green line represents the best-fit $y=-0.5059x+0.5497$ obtained by linear regression. The p value associated with the slope coefficient is p$=0.038<0.05$ (STAR Methods).

(J) Experiment overview for (K–N).

(K) Fluorescence-activated cell sub-sorting of DNMT3A-edited cells (DNMT3A-dCas9) exhibiting intermediate levels of gene expression (EBFP2). Data are from gene expression measurements using flow cytometry on the day of cell sorting.

(L) Flow cytometry-based time-course analysis of the dynamics of gene expression exhibited by cells obtained in (K).

(M) Density of cells from flow cytometry measurements obtained on the last time point in (L) (14 days after cell sampling). (K–M) show a representative replicate from three independent replicates.

(N) Dynamics of gene expression exhibited by cells shown in (K–M). Shown is the mean of geometric means from three independent replicates. Error bars are the SD of the mean. A.U.F., arbitrary units of fluorescence.

Because we pinpointed DNA methylation as the critical mediator of the maintenance of gene expression states, we performed targeted bisulfite sequencing of the reporter gene for each of the three cell populations: low, intermediate, and active (Figure 3G and STAR Methods). While the active state is devoid of methylated CpGs, the probability of finding any given CpG methylated in the promoter and dCas9-rTetR binding region in the intermediate state is distinctly smaller than in the low state and higher than in the active state. From these data, we computed the probability distributions of the fraction of methylated CpGs in the promoter and dCas9-rTetR binding region, for each of the low, intermediate, and active states (Figure 3H and STAR Methods). These distributions reinforce that the intermediately expressing cells have intermediate levels of DNA methylation. In particular, the mean value of the fraction of methylated CpGs, which we define as the DNA methylation grade, is intermediate in cells with intermediate expression levels compared to that of the low and active cells and has a log-linear negative correlation (p value < 0.05) with the mean gene expression level (Figure 3I).

We additionally tested the hypothesis that epigenetic memory can operate in an analog manner by further subdividing the DNMT3A-edited cells according to the gene expression levels, followed by time-resolved tracking (Figures 3J–3N and STAR Methods). If epigenetic memory were analog, cells sampled from different gene expression levels should approximately maintain these distinct levels over time. Each of the four resulting cell populations maintained distinct distributions of gene expression levels according to the level on the day of cell sampling (Figures 3K–3N). We also sampled cells that had been edited with DNMT3A fused to rTetR (rTetR-XTEN80-DNMT3A) instead of dCas9, using fluorescence-activated cell sorting (Figure S8 and STAR Methods), and the same temporal stability was observed.

Overall, these observations support the hypothesis that the memory is analog (Figure 1B) in agreement with the model (Figures 2M and S9A–S9H; STAR Methods), and that DNA methylation grade is the primary mediator of the stable maintenance of gene expression levels.

Single-cell analysis reveals that a wide range of gene expression levels is maintained over time, along with the corresponding DNA methylation grade

To further test the hypothesis that epigenetic memory operates on an analog basis, we set out to characterize the long-term trajectories of gene expression with extensive sampling of single cells from DNMT3A-edited cell populations followed by monoclonal expansion and analysis (Figure 4). This approach provides a high-resolution evaluation of gene expression trajectories generated exclusively from one initial DNA methylation profile. We first performed highly dense sampling of single cells, using fluorescence-activated cell sorting, from three distinct cell populations that had been previously sampled from DNMT3A-edited cells (Figures 4A and 4B; STAR Methods). Specifically, we selected eight different consecutive regions of gene expression levels spanning across low (two regions), intermediate (four regions), and high (two regions) gene expression levels in DNMT3A-edited cells (STAR Methods). After sampling single cells from each of those regions, we performed clonal expansion, resulting in $\ge$ 10 monoclonal populations from each of the eight gene expression regions for a total of 93 monoclonal populations. Flow cytometry measurements were performed for each of the cell populations after 15 days of monoclonal expansion (Figure 4C). The monoclonal populations showed a continuum of gene expression levels ranging from background to active gene expression levels. To test whether the gene expression levels from the monoclonal populations could be maintained in time, we selected eight monoclonal populations (highlighted in Figure 4C), spanning the gene expression range from the 93 clones, and performed time-course flow cytometry measurements up to 161 days (> 5 months) since the beginning of monoclonal expansion (Figure 4D). In these measurements, we observed that the gene expression distributions remained approximately stable over time, validating the initial hypothesis that cells can maintain memory of any gene expression level, which defines analog memory.

Figure 4.

Epigenetic memory can be analog, and this is mediated by DNA methylation

(A) Experiment overview.

(B) Single cells were obtained after sampling of DNMT3A-edited cells with low, intermediate, and high levels of gene expression. After monoclonal expansion, the total number of monoclonal populations obtained was N = 93 (STAR Methods).

(C) Violin plots of the monoclonal populations (n = 93) measured 15 days after single-cell sampling from the cell populations shown in (B). Monoclonal populations were positioned on the x axis according to the geometric mean of gene expression. Clones indicated with numbers 1–8 were selected for further analysis. Colors represent the cell population of origin as shown in (B).

(D) Flow cytometry-based time course analysis of gene expression dynamics of the eight monoclonal populations selected in (C).

(E) Methylation level determined using bisulfite sequencing of the reporter gene in the eight monoclonal populations at two different time points (STAR Methods).

(F) Flow cytometry measurements of the reporter gene in the eight monoclonal populations at two different time points corresponding to those shown in (E).

(G) Probability distribution of the fraction of methylated CpGs (STAR Methods). The SD for each clone is: 0.1752, 0.4707, 0.9492, 1.2714, 1.3101, 1.262, 1.1683, and 0.8095, ordered from clone 1 to clone 8.

(H) Simulations of the probability distributions of the system in Tables S10 and S8. The distributions are obtained computationally using the SSA³⁸ (STAR Methods). The SD for each clone is: 0.1346, 0.5472, 1.2585, 1.1189, 1.0431, 1.1117, 0.8512, and 0.8259, ordered from clone 1 to clone 8. The parameter values used for these simulations are listed in STAR Methods.

(I) Correlation between the mean fraction of CpGs methylated and mean gene expression level at 42 days after single-cell sorting. Shown are means and error bars indicating SD. The green line represents the best linear fit $y=-0.3756x+0.4358$. The p value associated with the slope coefficient is p$=5.34·{10}^{-5}<0.05$ (STAR Methods).

(J) Correlation plot between the mean fraction of CpGs methylated and the mean gene expression level, x, obtained via simulations. Data shown are the mean and error bars indicating SD. The green line represents the best linear fit $y=-0.4673x+0.4580$. The p value associated with the slope coefficient is p$=2.18·{10}^{-5}<0.05$ (STAR Methods). A.U.F., arbitrary units of fluorescence.

We next investigated DNA methylation grade as a determinant of the maintenance of gene expression level. To this end, we performed targeted bisulfite sequencing of the reporter gene in each of the eight populations at two different time points (Figures 4E and 4F and STAR Methods). The data showed that clones (7 and 8) selected from cells in the high gene expression distribution are largely devoid of methylated CpGs. By contrast, clones (1 and 2) selected from cells within the low distribution have the majority of CpGs methylated. The remaining clones (3, 4, 5, and 6), selected from the intermediate distribution, display a graded decreasing number of CpGs that are methylated with high probability (Figure 4E). These trends inversely correlate with the gene expression levels in the clones (Figure 4F). To acquire a quantitative understanding of the graded nature of this variation, we computed from the bisulfite sequencing data the probability distribution of the fraction of methylated CpGs in the gene’s promoter and rTetR binding sites (STAR Methods). This confirmed a progressive and gradual shift of the distributions through the clonal populations (Figure 4G), in accordance with model predictions (Figure 4H). The correlation between the mean gene expression level of each clonal population and the mean number of methylated CpGs is log-linear (Figures 4I and 4J), confirming a gradual increase in the DNA methylation grade through the clonal populations from high to low gene expression levels. Furthermore, the distribution of the fraction of methylated CpGs remained approximately unchanged between the two time points where bisulfite sequencing was performed (Figure 4G), indicating that the DNA methylation grade is stably maintained.

Further analysis of the clonal distributions of the fraction of methylated CpGs revealed that the variability of the number of methylated CpGs about their mean value, measured by the standard deviation, is larger for intermediate DNA methylation grades compared to low and high methylation grades (Figure 4G). The mathematical model recapitulates this property (Figure 4H) when it includes the two following additional possible molecular interactions: (1) H3K9me3 helps recruit DNA methylation maintenance enzyme (DNMT1) to the gene⁵⁰ and (2) DNMT1 can cause de novo DNA methylation, although at a very low rate^51,52 (see STAR Methods for model derivation). With these interactions, the errors made by DNMT1 in maintaining methylation are less probable for a fully methylated gene due to a higher recruitment rate of DNMT1 via H3K9me3. Similarly, errors made by DNMT1 in de novo methylating additional CpGs are less probable for a gene with no methylation since DNMT1 recruitment is practically absent (see STAR Methods and Figure S10 for model analysis). Overall, these interactions can explain the tighter distributions observed for highly and lowly methylated gene states (Figure 4G).

The genome engineering system that we utilized in this study has been demonstrated to be highly precise.⁵³ Additionally, we have verified that the copy number of the chromosomal integration is one (Figure 2D). Nevertheless, we still sought to verify that the different clones had one genomic copy of the reporter in the correct location to rule out the possibility that the variation of gene expression observed across the clones is due to variability in the genomic integration. To this end, we verified using genomic PCRs that the integration was in the ROSA26 locus for all clones (Figure S11A and STAR Methods). Moreover, we performed copy-number variation analysis using digital PCR (dPCR), which showed that all of the clones have one copy (Figure S11B and STAR Methods). Taken together, these data establish that the DNA methylation grade is maintained by the clonal populations and that this grade fine-tunes the gene expression level.

Finally, we sought to rule out that the observed differences in gene expression could be due to differences in the genome of the cells bearing the chromosomal integration. To this end, we selected the active monoclonal population (monoclonal population 8), targeted DNMT3A-dCas9 to the reporter, and tracked gene expression over time (Figures S11C–S11E). Subsequently, we sorted eight populations with different gene expression levels for temporal tracking (Figures S11F and S11G). Since the monoclonal cell line is composed of identical cells, characterized in detail using dPCR, genomic PCRs, and bisulfite sequencing (Figures 4E, S11A, and S11B), any differences observed in gene expression are due to epigenetic changes. Consistent with all prior observations, the distribution of gene expression in the selected populations initially widened and then remained stable for each gene expression level (Figures S11F and S11G). Bisulfite sequencing data further confirmed grades of DNA methylation that change linearly with gene expression level across the eight populations, consistent with our prior results (Figures S11H–S11J).

DNA methylation drives the maintenance of analog gene expression states

Next, we sought to establish a causal link between the DNA methylation grade and the level of gene expression. To this end, we investigated whether altering the DNA methylation grade in cells maintaining an intermediate level of gene expression would impair this maintenance. To enhance DNA methylation, we utilized rTetR-XTEN80-DNMT3A, as we have shown that DNMT3A enhances DNA methylation (Figure 2G) in accordance with prior reports.³⁹ To remove DNA methylation, we engineered a fusion protein comprising rTetR fused to the catalytic domain of the ten eleven translocation protein 1 (TET1)^33,54 to construct rTetR-XTEN80-TET1 (Figure S1F and Table S2). This catalytic domain has been shown to enable targeted DNA demethylation of mammalian genes.³³ Moreover, we fused rTeR to KRAB to construct rTetR-KRAB (Figure S1G and Table S2) to evaluate changes to gene expression when enhancing H3K9me3 without changing DNA methylation.

We transiently transfected these epigenetic effectors in cells that had been sampled from intermediate levels of gene expression in DNMT3A-edited cells. As before, we utilized fluorescence-activated cell sorting to obtain transfected cells and subsequently performed time-course flow cytometry measurements (Figures 5 and S11K–S11M). By 10 days after transient transfection with rTetR-XTEN80-DNMT3A, the gene expression was repressed, remained silenced over time, and overlapped with the distribution of cells that had been previously sorted for low levels of gene expression post DNMT3A-editing (Figure 5A). Transient transfection with rTetR-XTEN80-TET1 resulted in nearly complete and permanent reactivation of gene expression (Figure 5B). By contrast, transient transfection with rTetR-KRAB did not result in any permanent change in gene expression. Specifically, 6 days after transfection, we observed levels of repression comparable to those of the low levels in DNMT3A-edited cells, but the expression reverted to the initial intermediate level and remained stable afterward (Figure 5C). These data indicate that transient recruitment of a DNA methylation writer or eraser to an intermediately methylated and partially active gene can permanently silence or reactivate the gene. By contrast, transient recruitment of an H3K9me3 writer does not exert a permanent change in gene expression. All of this is consistent with the model predictions (Figure S12).

Figure 5.

DNA methylation effectors (DNMT3A or TET1) impair the maintenance of analog gene expression states in DNMT3A-edited cells

Time-course flow cytometry measurements after transient editing using (A) rTetR-XTEN80-DNMT3A, (B) rTetR-XTEN80-TET1, or (C) rTetR-KRAB in DNMT3A-edited cells with stable intermediate levels of gene expression. The densities of cells shown are from one representative replicate from three independent replicates. The shaded blue region indicates the time the transfection marker was detected during the time that doxycycline was added (Figures S11K–S11M). Doxycycline was applied for 6 days. Shown is the mean of the geometric means from three independent replicates. Error bars are the SD of the mean. A.U.F., arbitrary units of fluorescence.

We thus hypothesized that distinct levels of gene expression are maintained because their corresponding DNA methylation grade is maintained over time. To test this hypothesis, we sought to inhibit DNMT1 activity since DNMT1 is the enzyme that maintains DNA methylation through cell division.^14,44 To this end, we selected clonal populations with intermediate levels of gene expression and DNA methylation from Figure 4D and treated them with DNMT1 inhibitor 5-azacytidine.⁵⁵ DNMT1 inhibition caused gene expression in these clones to completely reactivate within 4 days, indicating loss of gene expression memory (Figure S13).

Taken together, these results show that each gene expression level is memorized by the corresponding grade of DNA methylation through the activity of the DNA methylation maintenance enzyme DNMT1.

Discussion

Current models posit that epigenetic memory is binary, largely based on autocatalytic histone modifications that stabilize genes in either an “on” or “off” state.²⁶ Although experimental studies support this hypothesis,^22,24,25 the context-dependent nature of chromatin modifications^27,28,29,30 casts doubts on whether a binary memory model is broadly applicable. With our genomic reporter system and sequence-specific chromatin regulators, we have discovered that gene expression can be memorized at a continuum of levels, encoded by the grade of DNA methylation.

Unlike histone modifications, DNA methylation is not autocatalytic. This feature, when coupled with a negligible decay rate and a lack of positive feedback between DNA methylation and H3K9me3 (Figure 6A), makes any DNA methylation grade persist over time. Since DNA methylation modulates gene expression,^20,56 it confers long-term stability to the corresponding gene expression level. In our system, positive feedback between DNA methylation and H3K9me3 is lacking; although DNA methylation recruits writers of H3K9me3⁴⁰ (Figure 2G), H3K9me3 does not recruit writers of DNA methylation (Figure 2I).

Figure 6.

The interactions between DNA methylation and histone modifications explain when analog or binary epigenetic memory emerges

(A) Analog memory. DNA methylation is not under positive feedback.

(B) Binary memory. DNA methylation is under positive feedback because H3K9me3 recruits writer enzymes for DNA methylation (yellow dashed arrow marked by ${\alpha }^{\prime }$) and DNA methylation, in turn, mediates the recruitment of H3K9me3. Refer to Figures S5A and S5C for a complete diagram.

Because the crosstalk between DNA methylation and histone modifications is context-dependent,⁵⁷ different patterns of interactions likely appear in different cellular and genetic contexts. In situations where H3K9me3 also mediates the establishment of DNA methylation,^58,59 DNA methylation becomes subject to positive feedback (Figures 6B and S5). This feedback can push a partially methylated gene to be fully methylated. Likewise, the competitive interaction between DNA methylation and H3K4me3 can cause H3K4me3 to win over a low DNA methylation grade and to revert this to the unmethylated state. The result is binary memory (STAR Methods and Figure S14). The strength of the positive feedback (${\alpha }^{\prime }$ in Figure 6B), determines the speed at which intermediate states converge to the extreme ones (Figure S14). In practice, the cellular and genetic context will determine the strength of the positive feedback and thus the temporal duration of analog states of gene expression (Figure S14).

Although this study has placed a causal link between the grade of DNA methylation in a gene and the temporal maintenance of the expressed protein’s level, other factors can play a role in modulating this level. These include RNA stability, RNA modifications, and the possible crosstalk between DNA methylation and RNA methylation.^60,61,62 Investigating these interactions will enrich our understanding of how specific memory levels can be tuned. Accordingly, our mathematical model can be extended to include RNA stability modulation, such as by covalent modifications and interactions with effectors of the chromatin state.

We have provided a proof-of-concept study showing that analog epigenetic memory is possible and have identified graded, stably maintained DNA methylation as the enabling molecular mechanism. To achieve this, we have mitigated potential genetic context effects by insulating the reporter system from the surrounding gene loci through flanking chromatin insulators and by spacing it away from endogenous coding sequences (Figure 2B) Additional studies are required to investigate analog memory in more native and physiologic genetic contexts and to determine how it generalizes across different cell lines, promoters, and with varying promoter CpG contents. Our hypothesis is that analog memory will emerge in those contexts lacking the positive feedback between DNA methylation and repressive histone modifications (Figure 6).

Patterns of intermediate DNA methylation and gene expression level have been observed in a variety of biological contexts. For example, the hippocampus is composed of spatial gradients of cell types with varying degrees of gene expression⁶³ and DNA methylation.⁶⁴ Similarly, genome-wide studies have reported an inverse correlation between gene expression level and DNA methylation content in chronic lymphocytic leukemia cells,⁶⁵ embryonic stem cells,⁶⁶ and human placenta.⁶⁷ Intermediately methylated and partially active gene states that appear to be maintained in time were also found in a class of human stem/progenitor cells.⁶⁸ However, the temporal stability of gene expression and DNA methylation grade at the single cell and gene level will need to be addressed to demonstrate analog epigenetic memory in these contexts.

Analog epigenetic memory can be used as a tool for engineering mammalian cells that store analog information. Programming analog memory through the DNA methylation grade could allow us to create high-resolution spatial gradients of cell identities for future engineered tissues and organoids. To this end, additional research will be required to precisely control DNA methylation grade by user-defined inputs, such as small signaling molecules or drugs.

In summary, this study has revealed that chromatin modifications support analog epigenetic memory and has established tools that can be used for further biological discovery and future mammalian cell engineering applications.

Limitations of the study

We only used one promoter (EF1a), one cell line (CHO-K1), and one integration site. Further experiments using additional promoters, cell lines, and integration sites are required to determine the extent to which analog memory is broadly observed. We utilized single-site recombination for chromosomal integration, instead of recombinase-mediated cassette exchange, which may influence how chromatin regulators and methyl-CpG-binding proteins are recruited to the site due to the presence of the chromosomal integration backbone. We performed H3K9me3 editing indirectly through KRAB and not through the chromatin-modifying enzyme that KRAB recruits. We may find stronger recruitment of H3K9me3 in the latter case, which may enable us to reach a state where H3K9me3 is maintained on its own. We explored this with sequential transfections of PhlF-KRAB but still only observed a transient effect. This will not affect the presence of analog memory, which is established by using DNMT3A as the input.

Resource availability

Lead contact

Requests for reagents and resources should be directed to and will be fulfilled by the lead contact, Domitilla Del Vecchio (ddv@mit.edu).

Materials availability

All plasmids generated in this study are available from the corresponding author upon reasonable request and from Addgene. DNA sequences of all engineered proteins and the genomic reporters are included in the manuscript.

Data and code availability

Code for mathematical modeling is available at https://zenodo.org/records/15345727 (Zenodo, DOI: 15345727) and GitHub at https://github.com/simonbruno100/AnalogEpigeneticMemoryPaper2025/tree/v1.0. Targeted bisulfite sequencing data are available online (NCBI SRA: PRJNA1271686). All original data are available from the lead contact upon reasonable request.

Acknowledgments

We thank Dr. James J. Collins for providing scientific comments and suggestions on the link between the model predictions and the experimental data, Dr. Lacramioara Bintu for support with DNA methylation assays, Kalon Overholt for providing useful references on DNA methylation and providing feedback on some of the figures, and the Biological Engineering Comm Lab for helping us refine some of the diagrams. This study was supported by the National Science Foundation, by NSF-MODULUS (award no. 2027949), and partially by the Vannevar-Bush Faculty Fellowship (Office of Naval Research award no. N00014-25-1-2053).

Author contributions

S.P., S.B., and D.D.V. conceptualized the study. D.D.V. directed and supervised the study. S.P., S.B., R.W., E.S., A.K., and D.D.V. contributed to experimental design and/or data analysis. S.P., E.S., I.G.-M., A.K., and K.I. conducted or assisted in experimental investigation. S.P. performed computational protein structure modeling and analysis. S.B. performed mathematical modeling and data analysis. S.P., S.B., and D.D.V. wrote the manuscript.

Declaration of interests

The authors declare no competing interests.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Antibodies
Rabbit polyclonal anti-H3K4me3	Diagenode	C15410003; AB_2924768
Rabbit polyclonal anti-H3K9me3	Diagenode	C15410056; AB_3674193
Mouse monoclonal anti-5mC	Diagenode	C15200081; 2572207
Chemicals, peptides, and recombinant proteins
Doxycycline	Sigma-Aldrich	D9891
2,4-diacetylphloroglucinol (DAPG)	Santa Cruz	c-206518
Penicillin-Streptomycin	Sigma	P4333
Puromycin	InvivoGen	ant-pr-1
Formaldehyde	Thermo Fisher	28906
Critical commercial assays
DNeasy Blood and Tissue Kit	Qiagen	69504
Q5 High-Fidelity 2X Master Mix	NEB	M0492
Power SYBR™ Green PCR Master Mix	Thermo Fisher	4368577
MagMeDIP Kit	Diagenode	C02010021
MicroChIP-seq kit	Diagenode	C01010132
Deposited data
Targeted bisulfite sequencing original data	This paper	NCBI SRA: PRJNA1271686
Experimental models: Cell lines
Hamster: CHO-K1 cell line	ATCC	CCL-61
Hamster: CHO-K1 landing pad cell line	Gaidukov et al.54	sLP20
Oligonucleotides
Primers qPCRs, see Table S4E	This paper	N/A
Primers for targeted bisulfite sequencing, see Table S4E	This paper	N/A
Primers of PCRs, see Table S4E	This paper	N/A
Recombinant DNA
Sequences for genomic reporter, see Table S1E	This paper	N/A
Sequences for epigenetic effectors, see Table S2E	This paper	N/A
Sequences for gRNAs, see Table S3E	This paper	N/A
DNMT3A-dCas9	O’Geen et al.40	Addgene 100090
Software and algorithms
Computational model code	This paper	Zenodo, DOI: 1534572. https://github.com/simonbruno100/AnalogEpigeneticMemorPaper2025/tree/v1.0
MATLAB	MathWorks	https://www.mathworks.com/products/matlab.html
Bismark	Babraham Institute	http://www.bioinformatics.babraham.ac.uk/projects/bismark/
Trim Galore	Babraham Institute	https://www.bioinformatics.babraham.ac.uk/projects/trim_galore
FastQC	Babraham Institute	https://www.bioinformatics.babraham.ac.uk/projects/fastqc/
Cytoflow	MIT	https://cytoflow.readthedocs.io/en/stable/
LightCycler 96 Application Software	Roche	https://lifescience.roche.com/global/en/products/product-category/lightcycler.html#4
Benching	Benchling	https://www.benchling.com/
AlphaFold 2 (ColabFold)	Mirdita et al.69	https://colab.research.google.com/github/sokrypton/ColabFold/blob/main/AlphaFold2.ipynb
QIAcuity Software Suite	Qiagen	https://www.qiagen.com/us/resources/resourcedetail?id=def90e09-0c22-4dad-ba8d-820fedb5ec0d&lang=en
QuantStudio Real-Time PCR Software	Thermo Fisher	https://www.thermofisher.com/us/en/home/global/forms/life-science/quantstudio-6-7-flex-software.html
Python	Python Software Foundation	https://www.python.org
Prism	Graphpad	https://www.graphpad.com/

Experimental model and study participant details

Cell engineering

The reporter gene was integrated using BxB1 site-specific chromosomal integration into an endogenous mammalian locus (Rosa26) in monoclonal CHO-K1 cells (female) bearing a monoallelic landing pad⁵³ (Figure S2A). We selected this method because it is highly efficient and precise in mammalian cells.^53,70 This locus had been previously identified by a homology-based computational search (BLAST) using the murine Rosa26 locus as reference. Each insulator consists of two tandem copies of the 5′ 250bp core sequence of the 1.2kb cHS4 insulator sequence.³⁵ The core sequence isolates chromosomal integrations via both enhancer-blocking and barrier activity.^30,35,71,72 EYFP is expressed in the cells exclusively before site-specific chromosomal integration into the Rosa26 locus (Figures S2A and S2B). The cells become puromycin resistant exclusively after correct chromosomal integration (Rosa26 locus)(Figures S2A and S2B). For integration, 500ng of the chromosomal integration vector and 500ng of the Bxb1 recombinase expression vector were transfected into the cells. After 24 h, routine cell culture was commenced with the supplementation of growth media with puromycin (8 $\mu$g/ml) for 2 weeks. Subsequently, a fluorescence-activated cell sorter (BD Aria) was utilized to obtain cells that were positive for EBFP2 expression and negative for EYFP expression. Real-time PCR (qPCR) was used to confirm 1 genomic reporter per genome (Figure 2D). Genomic PCRs were used to confirm integration into the Rosa26 locus (Figure S2B). The cells were also assayed and confirmed to test negative for mycoplasma contamination (MIT’s Koch Institute Preclinical Modeling Core).

Cell culture conditions

Cells were maintained in tissue culture plates (Corning) with growth media comprised of Dulbecco’s Modified Eagle Medium (DMEM) (Corning) supplemented with 10% FBS (Corning), 1X non-essential amino acids (Gibco), 100 U/ml penicillin (Sigma), and 100 $\mu$g/ml streptomycin (Sigma) at 37°C in a 5% CO2 incubator. The growth media was changed every 2–4 days. For routine passaging, the growth media was removed, followed by the addition of 0.25% trypsin (Corning) and incubation at 37°C in a 5% CO2 incubator for 7 min. Trypsin was then neutralized with growth media at room temperature. The cells were then centrifuged for 3 min at 200g, the supernatant was discarded, and the cell pellet was resuspended in fresh growth media. The cells were then diluted (ratio between 1 and 20) and reseeded into new tissue culture plates for subsequent cell culture.

Copy number determination using real-time PCR

To determine the copy number of the integration in our engineered cell line (Figure 2D), genomic DNA extraction was performed on the cells bearing the chromosomal integration and the cells before chromosomal integration (landing pad cell line) as control.⁵³ The extraction was performed using the DNeasy Blood & Tissue Kit per the manufacturer’s instructions (Qiagen) with approximately 1x ${10}^6$ cells. After genomic extraction, the DNA samples were submitted to Azenta Life Sciences for performing the rest of the assay. TaqMan probes and primers were designed by Azenta Life Sciences for the chromosomal integration and the endogenous COG1 gene as reference. Real-Time PCRs were performed in triplicates on a QuantStudio Real-Time PCR system and analyzed using QuantStudio Real-Time PCR Software. An amplicon-based standard curve was developed to quantify copies per reaction. The COG1 gene was used as reference gene for normalization to obtain copies per genome (assuming 2 COG1 copies per genome).

Copy number determination using digital PCR

To determine the copy number of the integration in each of the 8 clones (Figure S11B), genomic DNA extraction was performed with the DNeasy Blood & Tissue Kit (Qiagen) using approximately 1x ${10}^6$ cells, according to the manufacturer’s instructions. Samples were then submitted to Azenta Life Sciences for conducting the rest of the assay. A NanoDrop spectrophotometer was first used to assess sample concentration and confirm purity of the samples. The samples were then loaded in triplicate for analysis on a QIAcuity 8.5k 96-Well Nanoplate using the Qiagen QIAcuity digital PCR system, with primes and probes targeting EBFP2 and COG1. Copy number calculation was conducted using the QIAcuity Software Suite (version 2.2.0.26). The COG1 gene was used as reference gene for normalization (assuming 2 COG1 copies per genome). Each sample was compared to a no-template control to verify the absence of external contamination.

Genomic PCRs

The DNeasy Blood & Tissue Kit (Qiagen) was used to extract genomic DNA according with the manufacturer’s instructions and using approximately 1x ${10}^6$ cells per sample. PCRs were performed using the Q5 High-Fidelity 2X Master Mix (NEB) for 25–35 cycles using the primers in Table S4.

Method details

Transfections

Cells were cultured in growth media at 37°C in a 5% CO2 incubator. The cells were then treated with 0.25% trypsin (Corning) and incubated for 7 min at 37°C in a 5% CO2 incubator. The cell suspension was subsequently neutralized with growth media at room temperature. Cells were centrifuged for 3 min at 200g followed by removal of the supernatant and cell resuspension in fresh growth media. Cells were then seeded in 12-well tissue culture plates (Corning) at 300,000 cells/well and cultured for 24 h at 37°C in a 5% CO2 incubator. Growth media was replaced with growth media without antibiotics. Transfection was then performed using Lipofectamine LTX (Invitrogen) according to the manufacturer’s instructions. After 24 h of incubation at 37°C in a 5% CO2 incubator, replacement for routine growth media supplemented with penicillin and streptomycin was performed. All transfections with rTetR-XTEN80-DNMT3A, rTetR-XTEN80-DNMT3A, rTetR-XTEN80-TET1 and rTetR-KRAB were performed using 500ng of the fusion protein and 300ng of trasfection marker (EYFP). Transfections with PhlF-KRAB that involved cell sorting were performed using 450ng of either PhlF-KRAB or PhlF, and 450ng of the transfection marker (450ng). Experiments of PhlF-KRAB without cell sorting were perfomed using 450ng of PhlF-KRAB and 450ng of the transfection marker (1:1 stochiometry), 45ng of PhlF-KRAB and 450ng of the transfection marker (0.1:1 stochiometry), or 4.5ng of PhlF-KRAB and 450ng of the transfection marker (0.01:1 stochiometry). An empty vector was used to maintain the DNA amounts constant for the various stoichiometries. Transfections of DNMT3A-dCas9 that involved cell sorting were performed using 500ng of DNMT3A-dCas9, 600ng of the gRNA expression vector, and 300ng of the trasfection marker (EYFP). Transfections of DNMT3A-dCas9 in Figures S11D and S11E used 190ng of DNMT3A-dCas9, 600ng of the gRNA expression vector, and 300ng of the trasfection marker (EYFP). In experiments using dCas9, 450ng of the gRNA expression vector, 300ng of either DNMT3A-dCas9 or dCas9, and 300ng of the transfection marker (EYFP) were used. All the experiments presented use 2 $\mu$M of dox and 30 $\mu$M of DAPG.

DNA construction

DNA cloning was performed using a combination of Golden Gate Assembly, Gibson Assembly, restriction cloning, and PCR cloning. Expression vectors for the epigenetic effectors were cloned using Golden Gate Assembly. The cloning was performed in a single reaction with the plasmids comprising a destination vector, cHS4 chromatin insulator, EF1a promoter, 5′ UTR, gene sequence, 3′ UTR, and polyadenylation signal. Gene sequences for rTetR-TET1, rTetR-XTEN80-TET1, rTeTR-DNMT3A, rTeTR-XTEN80-DNMT3A, PhlF-KRAB, rTetR-KRAB, and PhlF were synthesized and cloned into a plasmid with compatible overhangs for Golden Gate Assembly (Azenta Life Sciences). The expression vectors for TET1 and rTetR were cloned by performing PCR cloning using the rTetR-XTEN80-TET1 expression vector as a template. For the reporter gene, the region comprising the PhlF and rTetR binding sites was synthesized (Azenta Life Sciences) and subsequently cloned upstream of a vector containing the EF1a promoter using restriction cloning. Golden Gate Assembly was then performed in a reaction comprising the EF1a promoter with upstream DNA-binding sites, destination vector, upstream cHS4 chromatin insulator, 5′ UTR, EBFP2, 3′ UTR, and polyadenylation signal. The resulting vector was cloned along with the downstream cHS4 chromatin insulator into the chromosomal integration vector using Gibson Assembly. All sequences were sequenced verified via Sanger sequencing (Azenta Life Sciences) or whole plasmid sequencing (Primordium).

Flow cytometry

Cells were trypsinized using 0.25% trypsin (Corning) and incubation at 37°C in a 5% CO2 incubator for 7 min. Following this, trypsin was neutralized by adding growth media at room temperature. Afterward, the cells were centrifuged at 200g for 3 min. The supernatant was removed, and the resulting cell pellet was then resuspended in fresh growth media. The cell suspension was filtered through a filter cap into flow cytometry tubes (Falcon). Flow cytometry data was then collected using a flow cytometer (BD LSRFortessa). EBFP2 was measured using a 405nm laser and a 450/50 filter with 259 V, EYFP was measured using a 488nm laser and 530/30 filter with 190 V, mKO2 was measured using a 561nm laser and 610/20 filter with 250 V, iRFP was measured using a 637nm laser and 670/30 filter with 326 V. For flow cytometry in Figure S11, data was collected on a BD LSRFortessa with EBFP2 measured using a 405nm laser and a 450/40 filter with 332 V, and EYFP was measured using a 488nm laser and a 530/30 filter with 310 V. Cytoflow (version 1.2) was used for analyzing all flow cytometry data and obtaining cell densities, means, standard deviations, and percentages of flow cytometry measurements. Measurements reported in the study refer to arbitrary units of fluorescence. For analysis, measurements were gated for single cells based on forward and side scatter (Figure S13B), and only gated measurements were used for subsequent analysis. CHO-K1 cells (WT) were used to assess background fluorescence, which is comparable to the maximal silencing we observed when the genomic reporter was edited with DNMT3A or KRAB (Figure S13C). URCP-100-2H calibration particles (SPHERO) were used to ensure instrument linearity and no day-to-day variation within experiments (Figure S13D). After processing the flow cytometry data using Cytoflow, the data was plotted using either Cytoflow, GraphPad Prism or Matplotlib. Graphpad Prism was used for generating bar plots. For the correlation plot between mean EBFP2 fluorescence and DNA methylation, Cytoflow’s gated measurements were imported into MATLAB, where the correlation was performed using the polyfit function.

Time-course flow cytometry

A fluorescence-activated cell sorter (BD Aria) was used to obtain cells with specific levels of transfection (EYFP) or gene expression (EBFP2). For transfection experiments that involved cell sorting, cell sorting was performed between 42 and 48 h after transfection. When flow cytometry was performed on the day of cell sorting, a fraction of the cells that were sorted was utilized for flow cytometry measurements (BD LSRFortessa), and the rest of the cells were reseeded on tissue culture plates. For flow cytometry measurements, cells were trypsinized using 0.25% trypsin (Corning) followed by incubation at 37°C in a 5% CO2 incubator. Trypsin neutralization was performed with growth media at room temperature. The cells were then centrifuged for 3 min at 200g. The supernatant was removed followed by resuspension of the cells in fresh in growth media. A fraction of the cells was utilized for the acquisition of flow cytometry measurements (BD LSRFortessa). The rest of the cells were diluted (ratio between 1 and 20) and reseeded into new tissue culture plates for subsequent measurements.

Cell sorting

Cells were trypsinized using 0.25% trypsin (Corning) followed by incubation at 37°C in a 5% CO2 incubator for 7 min. Subsequently, trypsin was neutralized with growth media at room temperature. The cells were then centrifuged at 200g for 3 min. The supernatant was discarded and the cell pellet was resuspended in growth media supplemented with 2.5g/ml BSA and corresponding small molecules according to the experiment. The cell suspension was then filtered through a filter cap into flow cytometry tubes. Cell sorting was then performed using a fluorescence-activated cell sorter (BD Aria) approximately 42–48 h after transfection, and cells were collected in growth media supplemented with penicillin/streptomycin and corresponding small molecules according to the experiment. EBFP2 was measured using a 405nm laser and a 450/50 filter with 332 V. EYFP was measured using a 488nm laser and 515/20 filter with 200 V. For cell sorting in Figure S11, a BD Aria was used with EBFP2 measured using a 405nm laser and a 450/50 filter with 225 V, and EYFP was measured using a 488nm laser and a 530/30 filter with 186 V. Cells were sorted either for transfection level (EYFP) or gene expression level (EBFP2). During cell sorting for EBFP2, a gate was also applied to sort exclusively for EYFP negative cells. Cell sorting to study the gene expression dynamics after epigenetic editing using DNMT3A-dCas9 (Figures 2F, S2C and S2D) used ranges 100–200, 222–445, 500–1000 and 1150–2300 EYFP A.U.F. Figure 2F shows the 222–445 range. Cell sorting to study the gene expression dynamics after epigenetic editing using PhlF-KRAB (Figures 2H and S4A–S4C) used ranges 25–100, 100–1000, 1000–10000, and 10000–100000 corresponding to EYFP A.U.F. Figure 2H shows the 100–1000 range.

Cell sorting to study histone and DNA methylation changes using MeDIP-qPCR and ChIP-qPCR (Figures 2G and 2I) used ranges 3000–100000 EYFP A.U.F. Cell sorting to demonstrate the stability of DNMT3A-edited (DNMT3A-dCas9) cells expressing intermediate levels of gene expression (Figure 3B) was performed according to <100 A.U.F for low expression, 200–500 A.U.F for intermediate expression, and >1000 A.U.F for high expression. Cells had been edited with transfection of DNMT3A-dCas9 targeted to the upstream binding sites of the promoter (Tables S1 and S3), cell sorting for 7000–100000 EYFP A.U.F., and were maintained in culture for $>$ 2 weeks. Cell sorting to subdivide cell populations from stable intermediate levels of gene expression (Figure 3K) was performed according to <100 A.U.F for bin 1, 179–224 for bin 2, 400–500 for bin 3, >1000 A.U.F for bin 4. Cells expressing intermediate levels of gene expression had been sorted from DNMT3A-edited cells (DNMT3A-dCas9) from the range 300–400 EBFP A.U.F. and maintained in culture for $>$ 2 weeks. Cells had been edited with transfection of DNMT3A-dCas9 targeted to the upstream binding sites (Tables S1 and S3), cell sorting for 3000–100000 EYFP A.U.F. (transfection marker), and maintained in culture for $>$ 2 weeks. Cell sorting for rTetR-XTEN80-DNMT3A (Figure 5A) and rTetR-KRAB (Figure 5C) transfections in cells with stable intermediate levels of expression used 3000–100000 EYFP A.U.F. Cell sorting for rTetR-XTEN80-TET1 (Figure 5B) transfections in cells with stable intermediate levels of expression used 410–2800 EYFP A.U.F. Cells expressing low and intermediate levels of gene expression had been sorted from DNMT3A-edited cells (DNMT3A-dCas9) using <100 A.U.F for low expression and 200–500 A.U.F for intermediate expression and maintained in culture for $>$ 2 weeks. Cells had been edited with transfection of DNMT3A-dCas9 targeted to the upstream binding sites (Tables S1 and S3), cell sorting for 3000–100000 EYFP A.U.F., and maintained in culture for $>$ 2 weeks. Cell sorting to study the gene expression dynamics after epigenetic editing using rTetR-XTEN80-DNMT3A (Figures S3A–S3C) used ranges 60–445, 500–7000, and 7000–100000 EYFP A.U.F. Figure S3A shows the 60–445 range. Cell sorting to demonstrate the stability of DNMT3A-edited (rTetR-XTEN80-DNMT3A) cells expressing intermediate levels of gene expression (Figure S8) was performed according to <80 EBFP2 A.U.F for low expression, and ranges 304–380, 400–500, and 530–662 EBFP2 A.U.F for intermediate expression. The cells had been edited with transfection of rTetR-XTEN80-DNMT3A, supplementation of dox for 8 days, cell sorting for 60–445 EYFP A.U.F., and maintained in culture for $>$ 2 weeks. Cell sorting to study the effect of DNMT3A-dCas9 on monoclonal population 8 used ranges 100–200, 210–450, 500–1000, 1100–2200 EYFP A.U.F. (Figures S11C–S11E). The cells from the 1100–2200 EYFP A.U.F. range were used for subsequent subdivision after maintenance in culture for $>$ 2 weeks. Cell sorting in Figure S11F used ranges 100–180, 250–310, 480–600, 800–1000, 1610–2000, 3210–4000, 5500–6850, 11000–13700 EBFP2 A.U.F.

Single-cell sampling and expansion

Single cell sorting was performed using a fluorescence-activated cell sorter (BD Aria) for various levels of EBFP2 fluorescence. For the cells exhibiting low levels of fluorescence, single cells were obtained from range 21–30 A.U.F. and range 50–60 A.U.F. For the cells exhibiting intermediate levels of fluorescence, single cells were obtained from range 200–240 A.U.F, range 254–305 A.U.F, range 320–384 A.U.F, and range 417–500 A.U.F. For the cells exhibiting high levels of fluorescence, single cells were obtained from range 1000–1200 A.U.F and range 1750–2100 A.U.F. From each range, 90 single cells well obtained for a total of 720 single cells, each seeded on a separate well of a 96-well tissue culture plate (Corning) with growth media supplemented with penicillin/streptomycin. Single cells were then cultured under standard cell culture conditions at 37°C in a 5% CO2 incubator. Growth media was replaced independently for each well every four days. After 14 days of cell culture, 93 monoclonal populations were observed, representing $\ge$ 10 monoclonal populations for each of the 8 ranges sampled. The cells in Figures 3B–3E were used for single-cell sampling at day 29.

DNMT1 inhibition

Cells were seeded and cultured in growth media at 37°C in a 5% CO2 incubator for 24 h. Subsequently, DNMT1 inhibition was initiated by growth media supplementation with 10μM of 5-azacytidine. Beginning a day after supplementation, the cells were measured by flow cytometry every day, diluted 1:2, and reseeded in fresh media supplemented with 10μM 5-azacytidine. The intermediately DNMT3A-silenced clonal cells had been cultured after single-cell sorting for 232 days.

MeDIP-qPCR and ChIP-qPCR

MeDIP-qPCR was performed using the MagMeDIP-qPCR kit (Diagenode) according to the manufacturer’s instructions. For MeDIP-qPCR, genomic DNA was extracted using the DNeasy Blood & Tissue Kit (Qiagen) according to the manufacture’s instructions using approximately 0.5x ${10}^6$ cells, followed by dilution to 100ng/ $\mu$ l. ChIP-qPCR was performed using the True MicroChIP-seq kit (C01010132, Diagenode) according to the manufacturer’s instructions using approximately 1x ${10}^5$ cells per immunoprecipitation reaction, which were cross-linked for 8 min at room temperature with methanol-free formaldehyde (Thermo Fisher) at a concentration of 1%. Sonication was performed using a Bioruptor Pico (Diagenode). We utilized the anti-5mC antibody in the MagMeDIP-qPCR kit (Diagenode) for DNA methylation. We utilized antibodies for H3K4me3 (C15410003, Diagenode) and H3K9me3 (C15410056, Diagenode) for histone methylation. The qPCRs were performed using Power SYBR Green PCR Master Mix (Applied Biosystems) and primers in Table S4 using a LightCycler 96 (Roche). Melt curves were used to confirm a single amplification product and the LightCycler 96 Application Software (Roche) was used to obtain the $C_t$ values. For the reporter gene, the promoter region was amplified. $\beta$-Actin (H3K4me3) and IgF2 (H3K9me3 and DNA methylation) were used as reference endogenous genes.²²

Targeted bisulfite sequencing

Genomic DNA was extracted using the DNeasy Blood & Tissue Kit (Qiagen) according to the manufacture’s instructions. Samples were then submitted to Zymo Research for conducting the rest of the assay. Assays targeting CpG sites along the chromosomal integration region comprising the continuous sequence covering all DNA binding sites, promoter and EBFP2 were developed using the Rosefinch tool from Zymo Research (Table S4). Primers were used to amplify bisulfite-converted DNA from the samples, using the EZ DNA Methylation-Lightning Kit for conversion. Amplified products were pooled, barcoded, and prepared for sequencing on an Illumina MiSeq system with a V2 300bp kit, following paired-end sequencing. Trim Galore was used to trim low quality nucleotides and adapter sequences. FastQC was used to evaluate sequence quality. Sequencing reads were aligned to the chromosomal integration sequence using Bismark. The estimation of methylation levels for each sampled cytosine was calculated by dividing the count of reads identifying a cytosine (C) by the sum of reads identifying either a cytosine (C) or a thymine (T).

Design of gRNA sequences

Sequences for gRNAs were designed in Benchiling using the chromosomal integration sequence and the CHO-K1 genome sequence (National Library of Medicine (NCBI) RefSeq assembly GCF_000223135.1).

Protein structure predictions

Protein structure predictions were performed with AlphaFold2⁷³ using MMseqs2 on the Google Colab Platform (ColabFold v1.5.5)⁷⁴ on GPUs. AlphaFold2 was used to perform structural predictions using 5 models for each protein. The top ranking predicted structure (according to plDDT score for single chains and pTM score for protein complexes) was then relaxed using Assisted Model Building with Energy Refinement (AMBER) relaxation. The protein structures of PhlF-KRAB, rTetR-KRAB, PhlF, and rTetR were predicted as multimers (homodimers). The protein structures of DNMT3A-dCas9, rTetR-XTEN80-DNMT3A, rTetR-XTEN80-TET1 were predicted as single chains. The structures of rTetR-XTEN80-TET1 and rTetR-XTEN80-DNMT3A were then aligned using the predicted homodimer of rTetR as a template in PyMOL (2.5.2). Minimal rotations ($\le$ 1 per linker) were performed at the flexible linkers in PyMOL.

Mathematical model and computational analysis

The chromatin modification model of Figure 2J is comprised of the chemical reactions listed below:

$$①\mathrm{D}\stackrel{k_W^1}{\to }{\mathrm{Y}}_1,②\mathrm{D}\stackrel{k_W^A}{\to }{\mathrm{D}}^{\mathrm{A}},③{\mathrm{D}}^{\mathrm{A}}\stackrel{{\overline{k}}_E^A}{\to }\mathrm{D},④{\mathrm{D}}^{\mathrm{A}}\stackrel{\delta }{\to }\mathrm{D},⑤\mathrm{D}+{\mathrm{D}}^{\mathrm{A}}\stackrel{k_M^A}{\to }{\mathrm{D}}^{\mathrm{A}}+{\mathrm{D}}^{\mathrm{A}}\text{,}$$

$$⑥{\mathrm{D}}^{\mathrm{A}}+{\mathrm{D}}_2^{\mathrm{R}}\stackrel{k_E^A}{\to }\mathrm{D}+{\mathrm{D}}_2^{\mathrm{R}},⑦{\mathrm{D}}^{\mathrm{A}}+{\mathrm{Y}}_1\stackrel{k_E^A}{\to }\mathrm{D}+{\mathrm{Y}}_1,⑧\mathrm{D}\stackrel{k_W^2}{\to }{\mathrm{D}}_2^{\mathrm{R}},⑨{\mathrm{D}}_2^{\mathrm{R}}\stackrel{{\overline{k}}_E^R}{\to }\mathrm{D}\text{,}$$

$$⑩{\mathrm{D}}_2^{\mathrm{R}}\stackrel{\delta }{\to }\mathrm{D},⑪\mathrm{D}+{\mathrm{D}}_2^{\mathrm{R}}\stackrel{k_M}{\to }{\mathrm{D}}_2^{\mathrm{R}}+{\mathrm{D}}_2^{\mathrm{R}},⑫\mathrm{D}+{\mathrm{Y}}_1\stackrel{{\overline{k}}_M}{\to }{\mathrm{D}}_2^{\mathrm{R}}+{\mathrm{Y}}_1,⑬{\mathrm{D}}_2^{\mathrm{R}}+{\mathrm{D}}^{\mathrm{A}}\stackrel{k_E^R}{\to }\mathrm{D}+{\mathrm{D}}^{\mathrm{A}}\text{,}$$

$$⑭{\mathrm{D}}^{\mathrm{A}}\stackrel{{\beta }_m^A}{\to }{\mathrm{D}}^{\mathrm{A}}+\mathrm{m},⑮\mathrm{D}\stackrel{{\beta }_m}{\to }\mathrm{D}+\mathrm{m},⑯{\mathrm{D}}_2^{\mathrm{R}}\stackrel{{\beta }_m}{\to }{\mathrm{D}}_2^{\mathrm{R}}+\mathrm{m},⑰\mathrm{m}\stackrel{\beta }{\to }\mathrm{m}+\mathrm{X}$$

$$⑱\mathrm{m}\stackrel{{\gamma }_m}{\to }\varnothing ,⑲\mathrm{X}\stackrel{\delta }{\to }\varnothing ,$$

in which ${\text{Y}}_1$ represents nucleosomes with methylated DNA, and the other species are defined in the STAR Methods subsection “Model of chromatin modification circuit.” The chromatin modification circuit model of Figure 2J is a reduction of the full chromatin modification circuit depicted in Figure S5, which was previously published in Bruno et al.⁷⁵ This circuit includes both DNA methylation and histone modifications, the positive feedback loop between DNA methylation and H3K9me3, without making any assumptions on the relative time scales between DNA methylation and histone modifications. The chromatin modification circuit model of Figure 2J can be obtained from this circuit by removing the recruitment of DNA methylation by H3K9me3, thereby breaking the positive feedback loop between H3K9me3 and DNA methylation, and by removing the decay rate of DNA methylation, assumed slow compared to that of histone modifications. See STAR Methods subsection “Derivation of model (11)” for the derivation.

For the plots obtained in Figure 2K, we used the reaction rate equation model corresponding to the reactions listed above, using the law of mass action.⁷⁶ This is the deterministic system of ordinary differential equations (ODEs) given in Equation 11 with gene expression dynamics ODE model given in Equation 10 with $D_1^R=D_{12}^R=0$. In order to obtain the dose-response curves show in Figure 2K, the system of ODEs was simulated using MATLAB and endpoint simulation quantities were plotted.

The plots of Figures 2L and 2M were obtained by simulating the chemical reaction system in Tables S1 and S4 using Gillespie’s Stochastic Simulation Algorithm (SSA).³⁸ Specifically, we obtained $N=1000$ different sample temporal paths using the SSA starting at the indicated initial condition and with parameter values listed at the end of the STAR Methods, and then computed the distributions by obtaining histograms of the final values of the species abundance across the different sample paths. The SSA is a Monte Carlo method that generates temporal trajectories of a chemically reacting system while accounting for the randomness of chemical reactions. We used Gillespie’s first reaction method³⁸ and the MATLAB codes have been deposited on GitHub and archived on Zenodo (DOI: 15345726).

Derivation and characterization of mathematical models

Model of chromatin modification circuit

The chromatin modification circuit reaction model considered in our study, shown in Figure S5A, was constructed starting from the one in Bruno et al.⁷⁵ and it is based on known molecular interactions from the literature. Here, we give a brief description of the reaction model (see Bruno et al.⁷⁵ for additional details). The chromatin modifications considered in the circuit are H3K9 methylation (H3K9me3), H3K4 methylation (H3K4me3), and DNA methylation. The basic unit of the model is the nucleosome with DNA wrapped around it, D, that can be modified with H3K4me3, ${\mathrm{D}}^{\mathrm{A}}$, DNA methylation, ${\mathrm{D}}_1^{\mathrm{R}}$, H3K9me3, ${\mathrm{D}}_2^{\mathrm{R}}$, or both H3K9me3 and DNA methylation, ${\mathrm{D}}_{12}^{\mathrm{R}}$. The reaction model can be graphically represented by the circuit of Figure S5A, whose associated reactions are given in Table S5. More precisely, writer enzymes have the ability to de novo establish chromatin marks. Additionally, histone modifications recruit writer enzymes for the same modification to nearby modifiable nucleosomes (auto-catalysis),^11,19,20,48 and DNA methylation and repressive histone modification cooperate by recruiting each other’s writer enzymes (cross-catalysis).⁴⁰ Ultimately, these modifications can be passively removed through dilution during DNA replication or by the action of eraser enzymes (basal erasure). Both activating and repressive modifications are involved in the recruitment of each other’s eraser enzymes (recruited erasure).^12,45,46,47

Now, defining the number of ${\mathrm{D}}^{\mathrm{A}},{\mathrm{D}}_1^{\mathrm{R}},{\mathrm{D}}_2^{\mathrm{R}},{\mathrm{D}}_{12}^{\mathrm{R}},\mathrm{D}$ as $n^A$, $n_1^R$, $n_2^R$, $n_{12}^R$, and $n^D$, let us derive the related ordinary differential equation (ODE) model in terms of the fractions ${\overline{D}}^A=n^A/{\mathrm{D}}_{\text{tot}}$, ${\overline{D}}_1^R=n_{D_1^R}/{\mathrm{D}}_{\text{tot}}$, ${\overline{D}}_2^R=n_{D_2^R}/{\mathrm{D}}_{\text{tot}}$, ${\overline{D}}_{12}^R=n_{D_{12}^R}/{\mathrm{D}}_{\text{tot}}$ and $\overline{D}=n_D/{\mathrm{D}}_{\text{tot}}$, with ${\mathrm{D}}_{\text{tot}}$ the total number of nucleosomes in a gene of interest. This can be done by assuming that ${\mathrm{D}}_{\text{tot}}$ is sufficiently large so that ${\overline{D}}^A$, ${\overline{D}}_1^R$, ${\overline{D}}_2^R$, ${\overline{D}}_{12}^R$ and $\overline{D}$ can be treated as real number. Now, let us introduce $D_{tot}={\mathrm{D}}_{\text{tot}}/\mathrm{\Omega }$ where $\mathrm{\Omega }$ is the reaction volume, and define the normalized inputs: $u_{10}^R=k_{W0}^1/k_M^A{D}_{tot}$, $u_1^R=k_W^1/\left(k_M^A{D}_{tot}\right)$, $u_{20}^R=k_{W0}^2/\left(k_M^A{D}_{tot}\right)$, $u_2^R=k_W^2/\left(k_M^A{D}_{tot}\right)$, $u_0^A=k_{W0}^A/\left(k_M^A{D}_{tot}\right)$, and $u^A=k_W^A/\left(k_M^A{D}_{tot}\right)$. Furthermore, let us introduce the following dimensionless parameters:

$$\alpha =\frac{k_M}{k_M^A}\overline{\alpha }=\frac{{\overline{k}}_M}{k_M^A},{\alpha }^{\prime }=\frac{k_M^{\prime }}{k_M^A}\text{,}$$ (Equation 1)

in which $\alpha$ ($\overline{\alpha }$, ${\alpha }^{\prime }$) is the non-dimensional rate constant associated with auto-catalysis (cross-catalysis). Let

$$\epsilon =\frac{\delta +{\overline{k}}_E^A}{k_M^A{D}_{tot}},\zeta =\frac{k_E^A}{k_M^A},\mu =\frac{k_E^R}{k_E^A},{\mu }^{\prime }=\frac{k_T^{\prime \ast }}{k_E^A}\text{,}$$ (Equation 2)

with $b=O\left(1\right)$ such that $\left(\delta +{\overline{k}}_E^R\right)/\left(\delta +{\overline{k}}_E^A\right)=b\mu$ and $\beta =O\left(1\right)$ such that $\left({\delta }^{\prime }+k_T^{\prime }\right)/\left(\delta +{\overline{k}}_E^A\right)=\beta {\mu }^{\prime }$. Then, $\mu$ $\left({\mu }^{\prime }\right)$ encapsulates the asymmetry between the erasure rate of repressive histone modifications (the DNA demethylation rate) and the erasure rate of activating marks. Furthermore, given that $\left(\delta +{\overline{k}}_E^R\right)/\left(k_M^A{D}_{tot}\right)=b\epsilon \mu$, $\left({\delta }^{\prime }+k_T^{\prime }\right)/\left(k_M^A{D}_{tot}\right)=\beta \epsilon {\mu }^{\prime }$, $k_E^R/k_M^A=\mu \zeta$ and $k_T^{\prime \ast }/k_M^A={\mu }^{\prime }\zeta$, $\epsilon$ $\left(\zeta \right)$ is a parameter that scales the ratio between the basal (recruited) erasure rate and the auto-catalysis rate of each modification. Considering the normalized time $\tau =t{k}_M^A{D}_{tot}$, the ODEs associated with the chromatin modification circuit are

$$\begin{array}{l}\frac{d{\overline{D}}_1^R}{d\tau }=\left(u_{10}^R+u_1^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R\right)\right)\overline{D}+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{12}^R-\left(u_{20}^R+\alpha \left({\overline{D}}_2^R+{\overline{D}}_{12}^R\right)+\overline{\alpha }\left({\overline{D}}_1^R+{\overline{D}}_{12}^R\right)\right){\overline{D}}_1^R\\ & -{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)){\overline{D}}_1^R\\ \frac{d{\overline{D}}_{12}^R}{d\tau }=\left(u_{10}^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R\right)\right){\overline{D}}_2^R+\left(u_{20}^R+\alpha \left({\overline{D}}_2^R+{\overline{D}}_{12}^R\right)+\overline{\alpha }\left({\overline{D}}_1^R+{\overline{D}}_{12}^R\right)\right){\overline{D}}_1^R\\ & -\left({\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_{12}^R\\ \frac{d{\overline{D}}_2^R}{d\tau }=\left(u_{20}^R+u_2^R+\alpha \left({\overline{D}}_2^R+{\overline{D}}_{12}^R\right)+\overline{\alpha }\left({\overline{D}}_1^R+{\overline{D}}_{12}^R\right)\right)\overline{D}+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{12}^R-\left(u_{10}^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R\right)\right){\overline{D}}_2^R\\ & -\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)){\overline{D}}_2^R\\ \frac{d{\overline{D}}^A}{d\tau }=\left(u_0^A+u^A+{\overline{D}}^A\right)\overline{D}-\left(\epsilon +\zeta \left({\overline{D}}_2^R+{\overline{D}}_{12}^R\right)+\zeta \left({\overline{D}}_1^R+{\overline{D}}_{12}^R\right)\right){\overline{D}}^A\text{,}\end{array}$$ (Equation 3)

with $\overline{D}=1-{\overline{D}}_1^R-{\overline{D}}_{12}^R-{\overline{D}}_2^R-{\overline{D}}^A$ (see Bruno et al.⁷⁵ for full derivation). It is important to note that in our reporter gene, the expression of fluorescent protein EBFP2 is driven by a constitutive promoter. This means that the promoter remains active continuously under normal conditions, without the need for external activators. This is the reason why, for ${\mathrm{D}}^{\mathrm{A}}$, we introduce a constant non-basal $de-novo$ establishment term $\left(u^A>0\right)$. As for the other species, we will describe in the following section how external inputs, such as KRAB, DNMT3A, and TET1, modulate the parameters of the chromatin modification circuit model.

How KRAB, DNMT3A, and TET1 modulate model parameters

In our study, we exploit KRAB, DNMT3A, and TET1 to modulate chromatin modification state and they affect the model parameters as follows. KRAB is a well-known epigenetic effector that mediates the recruitment of H3K9me3 writers.⁴² As we introduce KRAB through transient transfection, its concentration will gradually decrease due to dilution until it eventually dies out. Furthermore, KRAB is fused to the DNA binding domain (DBD) PhlF. This fusion not only allows us to obtain sequence-specific recruitment, but also to regulate the binding and unbinding of KRAB through DAPG, as DAPG inhibits the ability of PhlF-KRAB to bind to DNA. To incorporate these aspects into our model, we write the expression for $k_W^2$, i.e., reaction rate constant for H3K9me3 establishment (see reactions in Table S5), as follows:

$$k_W^2={\tilde{k}}_W^2\frac{1}{1+\frac{DAPG}{K_2}}{W}_2{e}^{-\delta t}\text{,}$$ (Equation 4)

in which $W_2$ is the total amount of PhlF-KRAB, ${\tilde{k}}_W^2$ is a parameter independent of PhlF-KRAB, $K_2$ is the dissociation constant related to the binding - unbinding reactions between DAPG and PhlF-KRAB, and $\delta$ is the dilution rate constant. Then, the normalized input $u_2^R$ in the ODE model (3) can be written as

$$u_2^R=\frac{k_W^2}{k_M^A{D}_{tot}}=\frac{{\tilde{k}}_W^2}{k_M^A}\frac{1}{1+\frac{DAPG}{K_2}}\frac{W_2}{D_{tot}}{e}^{-\frac{\delta }{k_M^A{D}_{tot}}t{k}_M^A{D}_{tot}}={\tilde{u}}_2^R\frac{1}{1+\frac{DAPG}{K_2}}{\overline{W}}_2{e}^{-\epsilon \tau }\text{,}$$ (Equation 5)

in which we define ${\tilde{u}}_2^R={\tilde{k}}_W^2/k_M^A$ and ${\overline{W}}_2=W_2/D_{tot}$.

The enzyme DNMT3A is a DNA methyltransferase that catalyzes de novo DNA methylation (Allis et al.,¹¹ Chapter 5) and, as a consequence, it modulates the rate of de novo DNA methylation establishment $k_W^1$ (see reactions in Table S5). As for KRAB, we introduce DNMT3 through transient transfection, and therefore its concentration will gradually decrease. To incorporate this aspect into our model, we can modify the expression for $k_W^1$ as follows:

$$k_W^1={\tilde{k}}_W^1{W}_1{e}^{-\delta t}\text{,}$$ (Equation 6)

in which $W_1$ is the total amount of DNMT3A and ${\tilde{k}}_W^1$ is a parameter independent of DNMT3A. Then, the normalized input $u_1^R$ in the ODE model (3) can be written as

$$u_1^R=\frac{k_W^1}{k_M^A{D}_{tot}}=\frac{{\tilde{k}}_W^1}{k_M^A}\frac{W_1}{D_{tot}}{e}^{-\frac{\delta }{k_M^A{D}_{tot}}t{k}_M^A{D}_{tot}}={\tilde{u}}_1^R{\overline{W}}_1{e}^{-\epsilon \tau }\text{,}$$ (Equation 7)

in which we define ${\tilde{u}}_1^R={\tilde{k}}_W^1/k_M^A$ and ${\overline{W}}_1=W_1/D_{tot}$.

Finally, TET1 is an enzyme that catalyzes the conversion of DNA methylation into modifications that are not recognized by the maintenance enzyme DNMT1, and hence are passively removed through dilution.^12,45,69 This enzyme then modulates the constants $k_T^{\prime }$ and $k_T^{\prime \ast }$ in the reactions of Table S5. As for KRAB, we introduce TET1 through transient transfection and we fuse it to a DBD. More specifically, we employ rTetR, a DBD that binds DNA exclusively in the presence of doxycycline (dox). To incorporate these aspects into our model, we can modify the expressions for $k_T^{\prime }$ and $k_T^{\prime \ast }$ as follows:

$$k_T^{\prime }=k_{T1}^{\prime }+{\tilde{k}}_T^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}T{e}^{-\delta t},k_T^{\prime \ast }=k_{T1}^{\prime \ast }+{\tilde{k}}_T^{\prime \ast }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}T{e}^{-\delta t}\text{,}$$ (Equation 8)

in which $k_{T1}^{\prime }$ $\left(k_T^{\prime \ast }1\right)$ is the component of the rate coefficient that does not depend on the external rTetR-TET1 transiently transfected, $T$ is the total amount of rTetR-TET1 transiently transfected, ${\tilde{k}}_T^{\prime }$ and ${\tilde{k}}_T^{\prime \ast }$ are parameters independent of rTetR-TET1, and $K_1$ is the dissociation constant related to the binding - unbinding reactions between dox and rTetR-TET1. Then, the non-dimensional parameter ${\mu }^{\prime }$ in the ODE model (3) can be written as

$${\mu }^{\prime }=\frac{k_T^{\prime \ast }}{k_E^A}=\frac{k_{T1}^{\prime \ast }}{k_E^A}+\frac{{\tilde{k}}_{T{D}_{tot}}^{\prime \ast }}{k_E^A}\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\frac{T}{D_{tot}}{e}^{-\frac{\delta }{k_M^A{D}_{tot}}t{k}_M^A{D}_{tot}}={\mu }_1^{\prime }+{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\text{,}$$ (Equation 9)

in which we define ${\mu }_1^{\prime }=\frac{k_{T1}^{\prime \ast }}{k_E^A}$, ${\tilde{\mu }}^{\prime }$, and $\overline{T}=T/D_{tot}$. From 9, it is possible to note that, without transfection of rTetR-TET1, ${\mu }^{\prime }={\mu }_1^{\prime }$, indicating that ${\mu }^{\prime }$ is non-zero. This is because of the presence of passive DNA demethylation. However, previous experimental studies suggest that passive DNA demethylation, although present, is a slow process.^22,44

Model of transcriptional regulation

The gene expression process involves two main steps. The first step is transcription, during which the genetic information is transcribed from DNA into mRNA (m). The second step is translation, during which the mRNA is translated into the gene product (X). Chromatin state affects transcription by modulating nucleosome compaction and then gene expression.^11,77 Thus, we assume that transcription is predominantly allowed by ${\mathrm{D}}^{\mathrm{A}}$, while allowing a low level of transcription to D and to all the species characterized by repressive chromatin modifications. Furthermore, mRNA m is subject to dilution due to cell division and degradation, while the gene product X is subject to dilution only. This is because fluorescent reporters are highly stable and thus not degraded but only diluted.⁷⁶ The reactions associated with the gene expression model can be found in Table S8. Considering the normalized time $\tau =t{k}_M^A{D}_{tot}$ and introducing the non-dimensional parameters ${\overline{\beta }}_m^A={\beta }_m^A/k_M^A{D}_{tot}$, ${\overline{\beta }}_m={\beta }_m/k_M^A{D}_{tot}$, $\overline{\beta }=\beta /k_M^A{D}_{tot}$, ${\overline{\gamma }}_m={\gamma }_m/k_M^A{D}_{tot}$, $\overline{\delta }=\delta /k_M^A{D}_{tot}$, the ODE model for X and m can be written as

$$\frac{d\overline{m}}{d\tau }={\overline{\beta }}_m^A{\overline{D}}^A+{\overline{\beta }}_m\left(\overline{D}+{\overline{D}}_1^R+{\overline{D}}_2^R+{\overline{D}}_{12}^R\right)-{\overline{\gamma }}_m\overline{m},\frac{d\overline{X}}{d\tau }=\overline{\beta }\overline{m}-\overline{\delta }\overline{X}\text{.}$$ (Equation 10)

In this paper, we will refer to the model that combines the complete chromatin modification circuit model (Equation 3) along with the transcriptional regulation model outlined in this section as the “4D + X model”.

Effect of ${\alpha }^{\prime }$ and ${\mu }^{\prime }$ on the behavior of the system

In this section, we analyze the stochastic behavior of the 4D + X model with a specific focus on understanding how the parameters ${\alpha }^{\prime }$ (normalized rate of DNA methylation establishment through repressive histone modifications) and ${\mu }^{\prime }$ (ratio between the DNA demethylation rate and the activating histone modification erasure rate) affect the probability distribution of gene expression levels and thus the type of memory that can be achieved.

When the rate of DNA demethylation is 0 $\left({\mu }^{\prime }=0\right)$, achieving analog memory is possible only when repressive histone modification (H3K9me3) does not catalyze the de novo establishment of DNA methylation $\left({\alpha }^{\prime }=0\right)$ (Figure S5B). For ${\alpha }^{\prime }>0$, the gene expression level shifts either to a low or high level. Furthermore, the higher ${\alpha }^{\prime }$, the more the distribution tends to shift toward a low gene expression level (Figure S14 - top panel). When ${\mu }^{\prime }$ is different from 0, only binary memory can be achieved (Figure S14 - intermediate and bottom panels). Moreover, the higher ${\mu }^{\prime }$ is, the more the gene expression level shifts toward a high level (Figure S14 - bottom panel). Overall, these results suggest that analog memory can be achieved only when DNA demethylation rate is sufficiently small compared to histone modification erasure rate (that is, ${\mu }^{\prime }$ can be approximated as zero on the time scales of interest) and DNA methylation is not catalyzed by repressive histone modifications $\left({\alpha }^{\prime }=0\right)$.

If these conditions are verified, and external inputs (KRAB, DNMT3A, and TET1) are not applied or are applied transiently (transient transfection), then the number of nucleosomes with methylated DNA stabilizes at a constant value. Introducing the variable ${\overline{Y}}_1={\overline{D}}_1^R+{\overline{D}}_{12}^R$, that is, the fraction of nucleosomes with methylated DNA in the gene, the behavior of the original model (3) can be captured by a reduced 2D ordinary differential equation (ODE) model below:

$$\frac{d{\overline{D}}_2^R}{d\tau }=\left(\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_1\right)\overline{D}-\mu \left(b\epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_2^R,\frac{d{\overline{D}}^A}{d\tau }=\left(u^A+{\overline{D}}^A\right)\overline{D}-\left(\epsilon +\zeta \left({\overline{D}}_2^R+{\overline{Y}}_1\right)\right){\overline{D}}^A\text{,}$$ (Equation 11)

with $\overline{D}=1-{\overline{D}}^A-{\overline{D}}_2^R-{\overline{Y}}_1$ and ${\overline{Y}}_1=constant$ (full derivation below). A representative diagram of the reduced system is depicted in Figure S5C. It is important to point out that in the original model of chromatin modification circuit, it is assumed that the basic unit is a nucleosome with DNA wrapped around it and with only one modifiable CpG.⁷⁵ There are cases in which this assumption is not verified. However, it is possible to demonstrate that removing this assumption yields a similar reduced model. This indicates that the assumption does not impact the qualitative results regarding the influence of the parameters ${\alpha }^{\prime }$ and ${\mu }^{\prime }$ on the probability distribution of gene expression levels (see STAR Methods subsection “Modification to the chromatin modification circuit model to include more than one CpG per nucleosome”).

Derivation of model (11)

Let us first merge the rates linked to the enhancement of H3K9me3 establishment by ${\overline{D}}_{12}^R$ into a single rate. We will assume that this rate is identical to the one associated with the enhancement of H3K9me3 establishment by ${\overline{D}}_1^R$. Similarly, let us merge the rates linked to the recruited erasure of H3K4me3 by ${\overline{D}}_{12}^R$ into a single rate and assume that this rate is identical to the one associated with the recruited erasure of H3K4me3 by ${\overline{D}}_1^R$. These simplifying assumptions do not affect the qualitative results related to the effect of the cooperative and competitive interactions among chromatin modifications on epigenetic cell memory. The ODE system (3) can then be rewritten as

$$\begin{array}{l}\frac{d{\overline{D}}_1^R}{d\tau }=\left(u_{10}^R+u_1^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R\right)\right)\overline{D}+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{12}^R-\left(u_{20}^R+\alpha \left({\overline{D}}_2^R+{\overline{D}}_{12}^R\right)+\overline{\alpha }\left({\overline{D}}_1^R+{\overline{D}}_{12}^R\right)\right){\overline{D}}_1^R\\ & -{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)){\overline{D}}_1^R\\ \frac{d{\overline{D}}_{12}^R}{d\tau }=\left(u_{10}^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R\right)\right){\overline{D}}_2^R+\left(u_{20}^R+\alpha {\overline{D}}_2^R+\overline{\alpha }\left({\overline{D}}_1^R+{\overline{D}}_{12}^R\right)\right){\overline{D}}_1^R\\ & -\left({\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_{12}^R\\ \frac{d{\overline{D}}_2^R}{d\tau }=\left(u_{20}^R+u_2^R+\alpha {\overline{D}}_2^R+\overline{\alpha }\left({\overline{D}}_1^R+{\overline{D}}_{12}^R\right)\right)\overline{D}+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{12}^R-\left(u_{10}^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R\right)\right){\overline{D}}_2^R\\ & -\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)){\overline{D}}_2^R\\ \frac{d{\overline{D}}^A}{d\tau }=\left(u_0^A+u^A+{\overline{D}}^A\right)\overline{D}-\left(\epsilon +\zeta \left({\overline{D}}_2^R\right)+\zeta \left({\overline{D}}_1^R+{\overline{D}}_{12}^R\right)\right){\overline{D}}^A\text{,}\end{array}$$ (Equation 12)

Now, let us rewrite system (12) by assuming negligible basal de novo establishment $\left(u_{10}^R=u_{20}^R=u_0^A=0\right)$ and introducing the variable ${\overline{Y}}_1={\overline{D}}_1^R+{\overline{D}}_{12}^R$, which corresponds to the fraction of nucleosomes with methylated DNA. We then obtain

$$\begin{array}{l}\frac{d{\overline{Y}}_1}{d\tau }=\left(u_1^R\right)\overline{D}+\left({\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R\right)\right)\left(\overline{D}+{\overline{D}}_2^R\right)-{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{Y}}_1\\ \frac{d{\overline{D}}_{12}^R}{d\tau }=\left({\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R\right)\right){\overline{D}}_2^R+\left(\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_1\right){\overline{D}}_1^R-\left({\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_{12}^R\\ \frac{d{\overline{D}}_2^R}{d\tau }=\left(u_2^R+\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_1\right)\overline{D}+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{12}^R-\left({\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R\right)\right){\overline{D}}_2^R-\left(\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_2^R\\ \frac{d{\overline{D}}^A}{d\tau }=\left(u^A+{\overline{D}}^A\right)\overline{D}-\left(\epsilon +\zeta {\overline{D}}_2^R+\zeta {\overline{Y}}_1\right){\overline{D}}^A\text{,}\end{array}$$ (Equation 13)

with $\overline{D}=1-{\overline{Y}}_1-{\overline{D}}_2^R-{\overline{D}}^A$. Based on previous experimental results²² and our experimental findings (see Figure 2I), we can further simplify our model by accounting for the fact that the enhancement of DNA methylation establishment by H3K9me3 is negligible, i.e., ${\alpha }^{\prime }=0$. The ODE system (13) can then be rewritten as

$$\begin{array}{l}\frac{d{\overline{Y}}_1}{d\tau }=\left(u_1^R\right)\overline{D}-{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{Y}}_1\\ \frac{d{\overline{D}}_{12}^R}{d\tau }=\left(\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_1\right){\overline{D}}_1^R-\left({\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_{12}^R\\ \frac{d{\overline{D}}_2^R}{d\tau }=\left(u_2^R+\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_1\right)\overline{D}+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{12}^R-\mu \left(b\epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_2^R\\ \frac{d{\overline{D}}^A}{d\tau }=\left(u^A+{\overline{D}}^A\right)\overline{D}-\left(\epsilon +\zeta {\overline{D}}_2^R+\zeta {\overline{Y}}_1\right){\overline{D}}^A\text{,}\end{array}$$ (Equation 14)

with $\overline{D}=1-{\overline{Y}}_1-{\overline{D}}_2^R-{\overline{D}}^A$. Now, let us rewrite the ODE system (14) introducing the expressions for $u_2^R$, $u_1^R$, and ${\mu }^{\prime }$ derived in the previous subsection (Expressions (5), (7), and (9)):

$$\begin{array}{l}\frac{d{\overline{Y}}_1}{d\tau }={\tilde{u}}_1^R{\overline{W}}_1{e}^{-\epsilon \tau }\overline{D}-\left({\mu }_1^{\prime }+{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\right)\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{Y}}_1\\ \frac{d{\overline{D}}_{12}^R}{d\tau }=\left(\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_1\right){\overline{D}}_1^R-\left(\left({\mu }_1^{\prime }+{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\right)\left(\beta \epsilon +\zeta {\overline{D}}^A\right)+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_{12}^R\\ \frac{d{\overline{D}}_2^R}{d\tau }=\left({\tilde{u}}_2^R\frac{1}{1+\frac{DAPG}{K_2}}{\overline{W}}_2{e}^{-\epsilon \tau }+\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_1\right)\overline{D}+\left({\mu }_1^{\prime }+{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\right)\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{12}^R\\ & -\mu \left(b\epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_2^R\\ \frac{d{\overline{D}}^A}{d\tau }=\left(u^A+{\overline{D}}^A\right)\overline{D}-\left(\epsilon +\zeta {\overline{D}}_2^R+\zeta {\overline{Y}}_1\right){\overline{D}}^A\text{,}\end{array}$$ (Equation 15)

with $\overline{D}=1-{\overline{Y}}_1-{\overline{D}}_2^R-{\overline{D}}^A$ and ${\overline{D}}_1^R={\overline{Y}}_1-{\overline{D}}_{12}^R$. Let us now consider the parameter regime in which the rate of DNA demethylation, without transfection of rTetR-TET1, is sufficiently low compared to histone modification dynamics, i.e., ${\mu }_1^{\prime }\approx 0$. This assumption is consistent with experimental data that suggest that the passive DNA demethylation is a slow process.^22,44 The system (15) can then be rewritten as follows:

$$\begin{array}{l}\frac{d{\overline{Y}}_1}{d\tau }={\tilde{u}}_1^R{\overline{W}}_1{e}^{-\epsilon \tau }\overline{D}-{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{Y}}_1\\ \frac{d{\overline{D}}_{12}^R}{d\tau }=\left(\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_1\right){\overline{D}}_1^R-\left({\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_{12}^R\\ \frac{d{\overline{D}}_2^R}{d\tau }=\left({\tilde{u}}_2^R\frac{1}{1+\frac{DAPG}{K_2}}{\overline{W}}_2{e}^{-\epsilon \tau }+\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_1\right)\overline{D}+{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{12}^R\\ & -\mu \left(b\epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_2^R\\ \frac{d{\overline{D}}^A}{d\tau }=\left(u^A+{\overline{D}}^A\right)\overline{D}-\left(\epsilon +\zeta {\overline{D}}_2^R+\zeta {\overline{Y}}_1\right){\overline{D}}^A\text{,}\end{array}$$ (Equation 16)

with $\overline{D}=1-{\overline{Y}}_1-{\overline{D}}_2^R-{\overline{D}}^A$ and ${\overline{D}}_1^R={\overline{Y}}_1-{\overline{D}}_{12}^R$. From 16, it follows that, after a temporary phase during which the externally transfected inputs decrease, $e^{-\epsilon \tau }\approx 0$ and then $d{\overline{Y}}_1/d\tau \approx 0$, and the last two ODEs in 16 depend only on ${\overline{D}}_2^R$, ${\overline{D}}^A$, and ${\overline{Y}}_1$. This implies that, after a transient change, ${\overline{Y}}_1$ stabilizes at a constant value and the behavior of the original model can be captured by the reduced two-dimensional ODE model given by the last two equations in 16, that corresponds to the ODE model in 11.

Effect of $\epsilon$ and $\zeta$ on the behavior of the system

In this section, we analyze the deterministic and stochastic behavior of our system, with a specific focus on understanding the effect of parameters $\epsilon$ (parameter that scales the ratio between the basal erasure rate and the auto-catalysis rate of each modification) and $\zeta$ (parameter that scales the ratio between the recruited erasure rate and the auto-catalysis rate of each modification) on the probability distribution of gene expression levels.

We first conduct a deterministic analysis by studying the ODE model associated with the simplified chromatin modification circuit, i.e., Equation 11, and determine how $\epsilon$ and $\zeta$ affect the value of ${\overline{D}}^A$ at the equilibrium for different fractions of methylated CpGs in the gene, i.e., ${\overline{Y}}_1$ (Figure S7A). When $\epsilon$ is large, the system consistently exhibits a unique stable steady state characterized by low ${\overline{D}}^A$ (Figure S7A). The value of ${\overline{D}}^A$ is smaller for higher ${\overline{Y}}_1$. As $\epsilon$ decreases, the steady-state value of $D^A$ increases, particularly for low values of ${\overline{Y}}_1$, where ${\overline{D}}^A\approx 1$ (Figure S7A). Further reduction in $\epsilon$ results in the system becoming bistable for intermediate values of ${\overline{Y}}_1$ (Figure S7A). Changes in $\zeta$ do not significantly impact these trends, except for when $\epsilon$ is small. In such cases, larger values of $\zeta$ reduce the range of ${\overline{Y}}_1$ for bistability and diminish the difference in the values of ${\overline{D}}^A$ between the two steady states (Figure S7A). We next study the impact of $\epsilon$ and $\zeta$ on the number of methylated CpGs at equilibrium for various initial levels of DNMT3A, represented by ${\overline{W}}_1$ as defined in Equation 6 (Figure S15A). In general, ${\overline{Y}}_1$ increases with higher ${\overline{W}}_1$. However, for large values of $\epsilon$, ${\overline{Y}}_1$ remains low over an extended range of ${\overline{W}}_1$ values. As $\epsilon$ decreases, ${\overline{Y}}_1$ reaches higher values. In the case of small $\epsilon$, an increase in $\zeta$ results in a more ultrasensitive curve.

Overall, this analysis suggests that high fractions of nucleosomes with H3K4me3, and then a high level of gene expression, is achievable only when $\epsilon$ is sufficiently small (Figure S7A). In this parameter regime, when $\zeta$ is sufficiently high, ${\overline{Y}}_1$ has an ultrasensitive response to DNMT3A transient dosage and then different ranges of DNMT3A pulse height would lead to either high or low gene expression levels only (Figures S5D and S15A). To validate these findings, we use the Stochastic Simulation Algorithm (SSA)³⁸ to perform a computational study on the 4D + X model, that is, the model that incorporates the full chromatin modification circuit, whose reactions are listed in Tables S5 and S8 (Figures S5E and S15B). Here, we consider several ranges of the initial amount of DNMT3A transfected $\left({\overline{W}}_1\right)$, which are uniformly distributed.

For different ranges of initial DNMT3A transfection levels, a bimodal probability distribution of gene expression levels emerges when $\zeta$ is sufficiently large (Figure S15B, left-hand side plots). As $\zeta$ decreases, the stationary distribution becomes unimodal (Figure S15B, left-hand side plots). When considering ranges that include higher values of ${\overline{W}}_1$, it becomes more evident how the unique peak of the distribution shifts toward lower gene expression levels for sufficiently high values of ${\overline{W}}_1$ (Figure S5E, left-hand side plots; Figure S15B, right-hand side plots).

We then conduct a similar computational analysis to assess the impact of $\epsilon$ and $\zeta$ on number of methylated CpGs and gene expression level across various initial levels of TET1 enzyme $\left(\overline{T}\right)$ (Figures S15C and S15D). The obtained results mirror those described in the previous analysis (Figures S15A and S15B).

By looking at our experimental results involving transfection of KRAB (Figure 2H), it is possible to note that the probability distribution of gene expression levels is bimodal. Previous computational study suggests that this behavior can be obtained only for sufficiently small values of $\epsilon$^75,78. Furthermore, our experimental results involving transfection of DNMT3A (Figures 2F and S2C) also show a bimodal probability distribution for gene expression levels. Based on our mathematical analysis (Figures S15A and S15B), these outcomes can be observed only when $\epsilon$ is sufficiently small and $\zeta$ is sufficiently large. We validate these hypotheses by successfully replicating our experimental data via simulations considering this parameter regime, i.e., small $\epsilon$, large $\zeta$ (Figure S6).

Refined chromatin modification circuit model to replicate the bisulfite sequencing data from the monoclonal experiment (Figure 4)

In the study conducted in the previous sections, we observed that in the absence of external stimuli and when ${\alpha }^{\prime }$ and ${\mu }^{\prime }$ can be assumed equal to zero, the chromatin modification circuit model, whose associated reactions are listed in Table S5, predicts a “frozen” level of DNA methylation, i.e., ${\dot{\overline{Y}}}_1=0$. However, this model does not perfectly align with the experimental results obtained from the bisulfite sequencing data of the synthetic reporter system in the eight different clones, each once associated with a distinct EBFP2 expression level (Figures 4A, 4B and 4D–4G). In fact, for each clone, we do not observe a single value for methylated CpGs across the entire cell population. Instead, we observe a distribution with mean and standard deviation that remain approximately constant throughout the observed time points (Figure 4G). Notably, the distribution of methylated CpGs appears more spread for clones associated with an intermediate level of methylated CpGs.

In order to recapitulate this observed behavior, we incorporated in the full model known molecular mechanisms involved in the DNA methylation maintenance process that were omitted before. Specifically, in the original full model we have

$${\mathrm{D}}_1^{\mathrm{R}}\stackrel{{\delta }^{\text{'}}}{\to }\mathrm{D}\text{,}$$ (Equation 17)

in which ${\delta }^{\prime }$ represents the net passive demethylation rate constant obtained from the balance between the dilution and the DNMT1 maintenance process. In order to recapitulate the experimental results, we added the following reactions:

$$\mathrm{D}\stackrel{c_0}{\to }{\mathrm{D}}_1^{\mathrm{R}},{\mathrm{D}}_1^{\mathrm{R}}\stackrel{{\tilde{\delta }}^{\prime }·\left(\frac{1}{1+D_2^R/K}\right)}{\to }\mathrm{D},\mathrm{D}\stackrel{c·D_2^R}{\to }{\mathrm{D}}_1^{\mathrm{R}}\text{.}$$ (Equation 18)

It is important to point out that, for the sake of simplicity in the description of the DNA methylation maintenance model above, we used, with abuse of notation, ${\mathrm{D}}_1^{\mathrm{R}}$ to denote any species with DNA methylation and ${\mathrm{D}}_2^{\mathrm{R}}$ to denote any species with H3K9me3. The first reaction in 18 represents the possibility that the DNA methylation maintenance enzyme DNMT1 can cause de novo DNA methylation, although at a very low rate (low $c_0$).^51,52 The second reaction represents the fact that H3K9me3 helps recruit DNMT1 to the gene and then reinforces DNMT1-mediated maintenance.⁵⁰ Finally, we introduce the third reaction since H3K9me3 helps recruit DNMT1 and DNMT1 can, by mistake, establish DNA methylation on previously unmethylated sites.

Considering the normalized time $\tau =t{k}_M^A{D}_{tot}$ and introducing the non-dimensional parameters ${\overline{\delta }}^{\prime }={\delta }^{\prime }/k_M^A{D}_{tot}$, ${\overline{\tilde{\delta }}}^{\prime }={\tilde{\delta }}^{\prime }/k_M^A{D}_{tot}$, ${\overline{c}}_0=c_0/k_M^A{D}_{tot}$, $\overline{c}=c/k_M^A$, the ODE describing the dynamics of fraction of nucleosomes with methylated DNA, that is, ${\overline{Y}}_1$, can be then rewritten as

$$\frac{d{\overline{Y}}_1}{d\tau }=\left({\overline{c}}_0+\overline{c}{\overline{D}}_2^R\right)\overline{D}-\left({\overline{\delta }}^{\prime }+{\overline{\tilde{\delta }}}^{\prime }\frac{1}{1+{\overline{D}}_2^R/\overline{K}}\right){\overline{D}}_1^R\text{.}$$ (Equation 19)

Then, the average level of DNA methylation remains approximately constant in time when $d{\overline{Y}}_1/d\tau \approx 0$, that is, if all the reaction rate constants are sufficiently small $\left({\overline{c}}_0,\overline{c},{\overline{\delta }}^{\prime },{\overline{\tilde{\delta }}}^{\prime }\ll 1\right)$.

This is confirmed by the computational study conducted by simulating the reactions of the refined chromatin modification circuit model that incorporates the DNA methylation maintenance model described above, whose reactions are listed in Table S10, using the SSA (Figure S10A). Additionally, this analysis shows that the reactions encapsulating the recruitment mechanism of DNMT1 by H3K9me3 (the second and the third in 18) are crucial for obtaining broader distributions in correspondence of intermediate levels of methylated CpGs (Figure S10B). In fact, as illustrated in the figure, when the rate constants associated with these reactions are low, all distributions have approximately the same height and width. However, higher values of these rates result in broader distributions at intermediate levels of methylated CpGs.

4D + X model: Reactions

Chromatin modification circuit

In this reaction model, it is assumed that the rate constants of the establishment, auto and cross-catalytic, and erasure processes of H3K9me3 (DNA methylation) do not change if the other repressive mark is present on the same modifiable unit. The chromatin modification circuit reaction model can then be written as in Table S5.

When DNA methylation is not included in the system, the only species included in the model are $\mathrm{D}$, ${\mathrm{D}}^{\mathrm{A}}$, and ${\mathrm{D}}_2^{\mathrm{R}}$, and the reactions included in the model become only the ones listed in Table S6.

Finally, the model represented in Figure 2J can be obtained starting from the full chromatin modification circuit model (Table S5) and assuming negligible basal de novo establishment $\left(u_{10}^R=u_{20}^R=u_0^A=0\right)$, merging the rates linked to the enhancement of H3K9me3 establishment by ${\overline{D}}_{12}^R$ into a single rate, merging the rates linked to the recruited erasure of H3K4me3 by ${\overline{D}}_{12}^R$ into a single rate, assuming negligible DNA methylation establishment by H3K9me3, and assuming a DNA demethylation rate, without transfection of rTetR-TET1, sufficiently low compared to histone modification dynamics. The reaction list associated with this model can be written as in Table S7 and the corresponding ODE model can be found in 16.

Transcriptional regulation

The reactions associated with the gene expression model described in the STAR Methods subsection “Model of transcriptional regulation” are listed in Table S8. Here, m is the mRNA, X is the gene product, ${\gamma }_m$ is the decay rate constant of m due to dilution and degradation, $\delta$ is the decay rate constant of X due to dilution, ${\beta }_m^A,{\beta }_m$ are the transcription rate constants, with ${\beta }_m<{\beta }_m^A$, and $\beta$ is the translation rate constant.

When DNA methylation, and then species ${\mathrm{D}}_1^{\mathrm{R}}$ and ${\mathrm{D}}_{12}^{\mathrm{R}}$ are not included in the chromatin modification circuit, then the gene expression model can be simplified as in Table S9.

DNA methylation maintenance and refined chromatin modification circuit

To recapitulate the experimental results shown in Figure 4G, modifications need to be made to the reaction system that models the DNA methylation maintenance process, as explained in STAR Methods subsection “Refined chromatin modification circuit model to replicate the bisulfite sequencing data from the monoclonal experiment (Figure 4)”. The reaction model for the chromatin modification circuit can then be revised and expressed as detailed in Table S10.

Modification to the chromatin modification circuit model to include more than one CpG per nucleosome

In the original model of the chromatin modification circuit, it is assumed that the basic unit, that is a nucleosome with DNA wrapped around, can have only one modifiable CpG.⁷⁵ There are cases in which this assumption is not verified. However, it is possible to show that this assumption does not affect the qualitative results related to the effect of the parameters ${\alpha }^{\prime }$ (normalized rate of DNA methylation establishment through repressive histone modifications) and ${\mu }^{\prime }$ (ratio between the DNA demethylation rate and the activating histone modification erasure rate) on the probability distribution of gene expression levels.

To this end, let us assume that the basic unit of the model, D, is the nucleosome with DNA wrapped around it with two modifiable CpGs. The conclusion obtained in this section would not change if we consider the DNA wrapped around the nucleosome having $n\in {\mathbb{Z}}^{+}$ modifiable CpGs. The basic unit of the model can then be modified with H3K4me3, ${\mathrm{D}}^{\mathrm{A}}$, one DNA methylation, ${\mathrm{D}}_1^{\mathrm{R}}$, H3K9me3, ${\mathrm{D}}_2^{\mathrm{R}}$, one DNA methylation and H3K9me3, ${\mathrm{D}}_{12}^{\mathrm{R}}$, two DNA methylations, ${\mathrm{D}}_{11}^{\mathrm{R}}$, and two DNA methylation and H3K9me3, ${\mathrm{D}}_{112}^{\mathrm{R}}$. Now, defining the number of ${\mathrm{D}}^{\mathrm{A}},{\mathrm{D}}_1^{\mathrm{R}},{\mathrm{D}}_2^{\mathrm{R}},{\mathrm{D}}_{12}^{\mathrm{R}},{\mathrm{D}}_{11}^{\mathrm{R}},{\mathrm{D}}_{112}^{\mathrm{R}},\mathrm{D}$ as $n^A$, $n_1^R$, $n_2^R$, $n_{12}^R$, $n_{D_{11}^R}$, $n_{112}^R$ and $n^D$, let us derive the related ordinary differential equation (ODE) model in terms of the fractions ${\overline{D}}^A=n^A/{\mathrm{D}}_{\text{tot}}$, ${\overline{D}}_1^R=n_{D_1^R}/{\mathrm{D}}_{\text{tot}}$, ${\overline{D}}_2^R=n_{D_2^R}/{\mathrm{D}}_{\text{tot}}$, ${\overline{D}}_{12}^R=n_{D_{12}^R}/{\mathrm{D}}_{\text{tot}}$, ${\overline{D}}_{11}^R=n_{D_{11}^R}/{\mathrm{D}}_{\text{tot}}$, ${\overline{D}}_{112}^R=n_{D_{112}^R}/{\mathrm{D}}_{\text{tot}}$ and $\overline{D}=n_D/{\mathrm{D}}_{\text{tot}}$, with ${\mathrm{D}}_{\text{tot}}$ the total number of nucleosomes on a gene of interest. This can be done by assuming that ${\mathrm{D}}_{\text{tot}}$ is sufficiently large so that ${\overline{D}}^A$, ${\overline{D}}_1^R$, ${\overline{D}}_2^R$, ${\overline{D}}_{12}^R$ and $\overline{D}$ can be treated as real number. Now, considering the dimensionless parameters and normalized inputs introduced in the STAR Methods subsection “Model of chromatin modification circuit” and the normalized time $\tau =t{k}_M^A{D}_{tot}$, the ODEs associated with the chromatin modification circuit are

$$\begin{array}{l}\frac{d{\overline{D}}_1^R}{d\tau }=\left(u_{10}^R+u_1^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)\right)\overline{D}+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{12}^R+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{11}^R-\left(u_{20}^R+\alpha \left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)+\overline{\alpha }\left({\overline{D}}_1^R+{\overline{D}}_{12}^R+2{\overline{D}}_{11}^R+2{\overline{D}}_{112}^R\right)\right){\overline{D}}_1^R-\left(u_{10}^R+u_1^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_1^R\\ \frac{d{\overline{D}}_{12}^R}{d\tau }=\left(u_{10}^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)\right){\overline{D}}_2^R+\left(u_{20}^R+\alpha \left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)+\overline{\alpha }\left({\overline{D}}_1^R+{\overline{D}}_{12}^R+2{\overline{D}}_{11}^R+2{\overline{D}}_{112}^R\right)\right){\overline{D}}_1^R+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{112}^R-\left({\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)+u_{10}^R+u_1^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)\right){\overline{D}}_{12}^R\\ \frac{d{\overline{D}}_{11}^R}{d\tau }=\left(u_{10}^R+u_1^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)\right){\overline{D}}_1^R+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{112}^R-\left(u_{20}^R+\alpha \left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)+\overline{\alpha }\left({\overline{D}}_1^R+{\overline{D}}_{12}^R+2{\overline{D}}_{11}^R+2{\overline{D}}_{112}^R\right)+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_1^R\\ \frac{d{\overline{D}}_{112}^R}{d\tau }=\left(u_{20}^R+\alpha \left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)+\overline{\alpha }\left({\overline{D}}_1^R+{\overline{D}}_{12}^R+2{\overline{D}}_{11}^R+2{\overline{D}}_{112}^R\right)\right){\overline{D}}_{11}^R+\left(u_{10}^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)\right){\overline{D}}_{12}^R-\left({\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_{112}^R\\ \frac{d{\overline{D}}_2^R}{d\tau }=\left(u_{20}^R+u_2^R+\alpha \left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)+\overline{\alpha }\left({\overline{D}}_1^R+{\overline{D}}_{12}^R+2{\overline{D}}_{11}^R+2{\overline{D}}_{112}^R\right)\right)\overline{D}+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{12}^R-\left(u_{10}^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_2^R\\ \frac{d{\overline{D}}^A}{d\tau }=\left(u_0^A+u^A+{\overline{D}}^A\right)\overline{D}-\left(\epsilon +\zeta \left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)+\zeta \left({\overline{D}}_1^R+{\overline{D}}_{12}^R+2{\overline{D}}_{11}^R+2{\overline{D}}_{112}^R\right)\right){\overline{D}}^A\text{,}\end{array}$$ (Equation 20)

with $\overline{D}=1-{\overline{D}}_1^R-{\overline{D}}_{12}^R-{\overline{D}}_{11}^R-{\overline{D}}_{112}^R-{\overline{D}}_2^R-{\overline{D}}^A$.

As we did for the original ODE system (3), let us merge the rates linked to the enhancement of H3K9me3 establishment by ${\overline{D}}_{12}^R$, ${\overline{D}}_{11}^R$, and ${\overline{D}}_{112}^R$ into a single rate. We will assume that this rate is identical to the one associated with the enhancement of H3K9me3 establishment by ${\overline{D}}_1^R$. Similarly, let us merge the rates linked to the recruited erasure of H3K4me3 by ${\overline{D}}_{12}^R$, ${\overline{D}}_{11}^R$, and ${\overline{D}}_{112}^R$ into a single rate and assume that this rate is identical to the one associated with the recruited erasure of H3K4me3 by ${\overline{D}}_1^R$. The ODE system (20) can then be rewritten as

$$\begin{array}{l}\frac{d{\overline{D}}_1^R}{d\tau }=\left(u_{10}^R+u_1^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)\right)\overline{D}+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{12}^R+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{11}^R-\left(u_{20}^R+\alpha {\overline{D}}_2^R+\overline{\alpha }\left({\overline{D}}_1^R+{\overline{D}}_{12}^R+{\overline{D}}_{11}^R+{\overline{D}}_{112}^R\right)\right){\overline{D}}_1^R-\left(u_{10}^R+u_1^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_1^R\\ \frac{d{\overline{D}}_{12}^R}{d\tau }=\left(u_{10}^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)\right){\overline{D}}_2^R+\left(u_{20}^R+\alpha {\overline{D}}_2^R+\overline{\alpha }\left({\overline{D}}_1^R+{\overline{D}}_{12}^R+{\overline{D}}_{11}^R+{\overline{D}}_{112}^R\right)\right){\overline{D}}_1^R+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{112}^R-\left({\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)+u_{10}^R+u_1^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)\right){\overline{D}}_{12}^R\\ \frac{d{\overline{D}}_{11}^R}{d\tau }=\left(u_{10}^R+u_1^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)\right){\overline{D}}_1^R+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{112}^R-\left(u_{20}^R+\alpha {\overline{D}}_2^R+\overline{\alpha }\left({\overline{D}}_1^R+{\overline{D}}_{12}^R+{\overline{D}}_{11}^R+{\overline{D}}_{112}^R\right)+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_1^R\\ \frac{d{\overline{D}}_{112}^R}{d\tau }=\left(u_{20}^R+\alpha {\overline{D}}_2^R+\overline{\alpha }\left({\overline{D}}_1^R+{\overline{D}}_{12}^R+{\overline{D}}_{11}^R+{\overline{D}}_{112}^R\right)\right){\overline{D}}_{11}^R+\left(u_{10}^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)\right){\overline{D}}_{12}^R-\left({\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_{112}^R\\ \frac{d{\overline{D}}_2^R}{d\tau }=\left(u_{20}^R+u_2^R+\alpha {\overline{D}}_2^R+\overline{\alpha }\left({\overline{D}}_1^R+{\overline{D}}_{12}^R+{\overline{D}}_{11}^R+{\overline{D}}_{112}^R\right)\right)\overline{D}+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{12}^R-\left(u_{10}^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_2^R\\ \frac{d{\overline{D}}^A}{d\tau }=\left(u_0^A+u^A+{\overline{D}}^A\right)\overline{D}-\left(\epsilon +\zeta \left({\overline{D}}_2^R+{\overline{D}}_1^R+{\overline{D}}_{12}^R+{\overline{D}}_{11}^R+{\overline{D}}_{112}^R\right)\right){\overline{D}}^A\text{,}\end{array}$$ (Equation 21)

Now, let us rewrite system (21) by assuming negligible basal establishment $\left(u_{10}^R=u_{20}^R=u_0^A=0\right)$ and introducing the variable ${\overline{Y}}_1={\overline{D}}_1^R+{\overline{D}}_{12}^R$, which corresponds to the fraction of nucleosomes with one methylated CpG, and the variable ${\overline{Y}}_{1tot}={\overline{D}}_1^R+{\overline{D}}_{12}^R+{\overline{D}}_{11}^R+{\overline{D}}_{112}^R$, which corresponds to the fraction of nucleosomes with at least one methylated CpG. We then obtain

$$\begin{array}{l}\frac{d{\overline{D}}_1^R}{d\tau }=\left(u_1^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{Y}}_{1tot}-{\overline{D}}_1^R-{\overline{D}}_{11}^R\right)\right)\overline{D}+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\left({\overline{Y}}_1-{\overline{D}}_1^R\right)+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{11}^R\\ -\left(\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_{tot}+u_1^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{Y}}_{1tot}-{\overline{D}}_1^R-{\overline{D}}_{11}^R\right)+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_1^R\\ \frac{d{\overline{D}}_{11}^R}{d\tau }=\left(u_1^R+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{Y}}_{1tot}-{\overline{D}}_1^R-{\overline{D}}_{11}^R\right)\right){\overline{D}}_1^R+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\left({\overline{Y}}_{1tot}-{\overline{Y}}_1-{\overline{D}}_{11}^R\right)\\ -\left(\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_{1tot}+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_1^R\\ \frac{d{\overline{Y}}_1}{d\tau }=u_1^R\left(\overline{D}-{\overline{D}}_1^R\right)+\left({\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{Y}}_{1tot}-{\overline{D}}_1^R-{\overline{D}}_{11}^R\right)\right)\left(\overline{D}+{\overline{D}}_2^R\right)\\ +{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)\left({\overline{Y}}_{1tot}-{\overline{Y}}_1\right)-\left({\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)+{\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{Y}}_{1tot}-{\overline{D}}_1^R-{\overline{D}}_{11}^R\right)\right){\overline{Y}}_1\\ \frac{d{\overline{Y}}_{1tot}}{d\tau }=u_1^R\overline{D}+\left({\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{Y}}_{1tot}-{\overline{D}}_1^R-{\overline{D}}_{11}^R\right)\right)\left(\overline{D}+{\overline{D}}_2^R\right)-{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{Y}}_1\\ \frac{d{\overline{D}}_2^R}{d\tau }=\left(u_2^R+\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_{1tot}\right)\overline{D}+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{12}^R-\left({\alpha }^{\prime }\left({\overline{D}}_2^R+{\overline{D}}_{12}^R+{\overline{D}}_{112}^R\right)+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_2^R\\ \frac{d{\overline{D}}^A}{d\tau }=\left(u^A+{\overline{D}}^A\right)\overline{D}-\left(\epsilon +\zeta \left({\overline{D}}_2^R+{\overline{Y}}_{1tot}\right)\right){\overline{D}}^A\text{,}\end{array}$$ (Equation 22)

with $\overline{D}=1-{\overline{Y}}_{1tot}-{\overline{D}}_2^R-{\overline{D}}^A$. As we did for the ODE system associated with our original chromatin modification circuit in STAR Methods subsection “Model of chromatin modification circuit”, we can further simplify our model by assuming that the enhancement of DNA methylation establishment by H3K9me3 is negligible, i.e., ${\alpha }^{\prime }=0$. The ODE system (22) can then be rewritten as

$$\begin{array}{l}\frac{d{\overline{D}}_1^R}{d\tau }=\left(u_1^R\right)\overline{D}+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\left({\overline{Y}}_1-{\overline{D}}_1^R\right)+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{11}^R\\ & -\left(\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_{tot}+u_1^R+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_1^R\\ \frac{d{\overline{D}}_{11}^R}{d\tau }=\left(u_1^R\right){\overline{D}}_1^R+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\left({\overline{Y}}_{1tot}-{\overline{Y}}_1-{\overline{D}}_{11}^R\right)-\left(\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_{1tot}+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_1^R\\ \frac{d{\overline{Y}}_1}{d\tau }=u_1^R\left(\overline{D}-{\overline{D}}_1^R\right)+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)\left({\overline{Y}}_{1tot}-{\overline{Y}}_1\right)-{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{Y}}_1\\ \frac{d{\overline{Y}}_{1tot}}{d\tau }=u_1^R\overline{D}-{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{Y}}_1\\ \frac{d{\overline{D}}_2^R}{d\tau }=\left(u_2^R+\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_{1tot}\right)\overline{D}+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{12}^R-\mu \left(b\epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_2^R\\ \frac{d{\overline{D}}^A}{d\tau }=\left(u^A+{\overline{D}}^A\right)\overline{D}-\left(\epsilon +\zeta \left({\overline{D}}_2^R+{\overline{Y}}_{1tot}\right)\right){\overline{D}}^A\text{,}\end{array}$$ (Equation 23)

with $\overline{D}=1-{\overline{Y}}_{1tot}-{\overline{D}}_2^R-{\overline{D}}^A$. Now, introducing in 23 the expressions for $u_2^R$, $u_1^R$, and ${\mu }^{\prime }$ previously derived (Expressions (5), (7), and (9)), we obtain:

$$\begin{array}{l}\frac{d{\overline{D}}_1^R}{d\tau }={\tilde{u}}_1^R{\overline{W}}_1{e}^{-\epsilon \tau }\overline{D}+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\left({\overline{Y}}_1-{\overline{D}}_1^R\right)+\left({\mu }_1^{\prime }+{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\right)\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{11}^R-\left(\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_{tot}+{\tilde{u}}_1^R{\overline{W}}_1{e}^{-\epsilon \tau }+\left({\mu }_1^{\prime }+{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\right)\left(\beta \epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_1^R\\ \frac{d{\overline{D}}_{11}^R}{d\tau }={\tilde{u}}_1^R{\overline{W}}_1{e}^{-\epsilon \tau }{\overline{D}}_1^R+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\left({\overline{Y}}_{1tot}-{\overline{Y}}_1-{\overline{D}}_{11}^R\right)-\left(\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_{1tot}+\left({\mu }_1^{\prime }+{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\right)\left(\beta \epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_1^R\\ \frac{d{\overline{Y}}_1}{d\tau }={\tilde{u}}_1^R{\overline{W}}_1{e}^{-\epsilon \tau }\left(\overline{D}-{\overline{D}}_1^R\right)+\left({\mu }_1^{\prime }+{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\right)\left(\beta \epsilon +\zeta {\overline{D}}^A\right)\left({\overline{Y}}_{1tot}-{\overline{Y}}_1\right)-\left({\mu }_1^{\prime }+{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\right)\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{Y}}_1\\ \frac{d{\overline{Y}}_{1tot}}{d\tau }={\tilde{u}}_1^R{\overline{W}}_1{e}^{-\epsilon \tau }\overline{D}-\left({\mu }_1^{\prime }+{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\right)\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{Y}}_1\\ \frac{d{\overline{D}}_2^R}{d\tau }=\left({\tilde{u}}_2^R\frac{1}{1+\frac{DAPG}{K_2}}{\overline{W}}_2{e}^{-\epsilon \tau }+\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_{1tot}\right)\overline{D}+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{12}^R-\mu \left(b\epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_2^R\\ \frac{d{\overline{D}}^A}{d\tau }=\left(u^A+{\overline{D}}^A\right)\overline{D}-\left(\epsilon +\zeta \left({\overline{D}}_2^R+{\overline{Y}}_{1tot}\right)\right){\overline{D}}^A\text{,}\end{array}$$ (Equation 24)

with $\overline{D}=1-{\overline{Y}}_{1tot}-{\overline{D}}_2^R-{\overline{D}}^A$. Furthermore, if we consider the parameter regime in which the rate of DNA demethylation, without transfection of rTetR-TET1, is sufficiently low compared to histone modification dynamics,i.e., ${\mu }_1^{\prime }\to 0$, the system (24) can then be rewritten as follows:

$$\begin{array}{l}\frac{d{\overline{D}}_1^R}{d\tau }={\tilde{u}}_1^R{\overline{W}}_1{e}^{-\epsilon \tau }\overline{D}+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\left({\overline{Y}}_1-{\overline{D}}_1^R\right)+{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{11}^R\\ -\left(\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_{tot}+{\tilde{u}}_1^R{\overline{W}}_1{e}^{-\epsilon \tau }+{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_1^R\\ \frac{d{\overline{D}}_{11}^R}{d\tau }={\tilde{u}}_1^R{\overline{W}}_1{e}^{-\epsilon \tau }{\overline{D}}_1^R+\mu \left(b\epsilon +\zeta {\overline{D}}^A\right)\left({\overline{Y}}_{1tot}-{\overline{Y}}_1-{\overline{D}}_{11}^R\right)-\left(\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_{1tot}+{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)\right){\overline{D}}_1^R\\ \frac{d{\overline{Y}}_1}{d\tau }={\tilde{u}}_1^R{\overline{W}}_1{e}^{-\epsilon \tau }\left(\overline{D}-{\overline{D}}_1^R\right)+{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\left(\beta \epsilon +\zeta {\overline{D}}^A\right)\left({\overline{Y}}_{1tot}-{\overline{Y}}_1\right)-{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{Y}}_1\\ \frac{d{\overline{Y}}_{1tot}}{d\tau }={\tilde{u}}_1^R{\overline{W}}_1{e}^{-\epsilon \tau }\overline{D}-{\tilde{\mu }}^{\prime }\frac{\frac{dox}{K_1}}{1+\frac{dox}{K_1}}\overline{T}{e}^{-\epsilon \tau }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{Y}}_1\\ \frac{d{\overline{D}}_2^R}{d\tau }=\left({\tilde{u}}_2^R\frac{1}{1+\frac{DAPG}{K_2}}{\overline{W}}_2{e}^{-\epsilon \tau }+\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_{1tot}\right)\overline{D}+{\mu }^{\prime }\left(\beta \epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_{12}^R-\mu \left(b\epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_2^R\\ \frac{d{\overline{D}}^A}{d\tau }=\left(u^A+{\overline{D}}^A\right)\overline{D}-\left(\epsilon +\zeta \left({\overline{D}}_2^R+{\overline{Y}}_{1tot}\right)\right){\overline{D}}^A\text{,}\end{array}$$ (Equation 25)

with $\overline{D}=1-{\overline{Y}}_1-{\overline{D}}_2^R-{\overline{D}}^A$. From the fourth equation in 25, it is possible to notice that, following a temporary phase during which the externally transfected inputs decrease, $e^{-\epsilon \tau }\approx 0$, and then $d{\overline{Y}}_{1tot}/d\tau \approx 0$. This implies that, once the transient phase concludes, ${\overline{Y}}_{1tot}$ stabilizes at a constant value and the behavior of the original model can be captured by a reduced 2D model described by the subsequent ODEs:

$$\frac{d{\overline{D}}_2^R}{d\tau }=\left(\alpha {\overline{D}}_2^R+\overline{\alpha }{\overline{Y}}_{1tot}\right)\overline{D}-\mu \left(b\epsilon +\zeta {\overline{D}}^A\right){\overline{D}}_2^R,\frac{d{\overline{D}}^A}{d\tau }=\left(u^A+{\overline{D}}^A\right)\overline{D}-\left(\epsilon +\zeta \left({\overline{D}}_2^R+{\overline{Y}}_{1tot}\right)\right){\overline{D}}^A\text{,}$$ (Equation 26)

with $\overline{D}=1-{\overline{D}}^A-{\overline{D}}_2^R-{\overline{Y}}_{1tot}$ and ${\overline{Y}}_{1tot}=constant$. It is evident that the ODE model (26) closely resembles the original model, in which we assume to have one CpG per nucleosome, i.e., (11), with the only distinction being the presence of ${\overline{Y}}_{1tot}$ instead of ${\overline{Y}}_1$. That is, in this revised model, the representation of the fraction of the number of methylated CpGs in the gene is denoted by ${\overline{Y}}_{1tot}$. Then, the introduction of the simplifying assumption, having one CpG per nucleosome, does not affect the qualitative results of the analysis that focuses on identifying the impact of the parameters ${\alpha }^{\prime }$ (normalized rate of DNA methylation establishment through repressive histone modifications) and ${\mu }^{\prime }$ (ratio between the DNA demethylation rate and the activating histone modification erasure rate) on the probability distribution of gene expression levels.

Values of parameters and rationale for parameter selections used in generating figures in the main paper and SI

In our model, we defined the total number of nucleosomes within a gene of interest as ${\mathrm{D}}_{\text{tot}}$. Assuming approximately one nucleosome per 200 bp⁷⁹ and given that the length of our gene is $\approx 3000$ bp, then in our computational study we set the value of ${\mathrm{D}}_{\text{tot}}$ to 15.

With regard to the effect of dilution due to cell growth, we use a standard approach by modeling it as a first-order decay reaction.⁸⁰ The decay rate constant can then be expressed as $\delta =\ln \left(2\right)/T$, where $T$ corresponds to the cell cycle length. Given that in our experiments we use CHO cells, whose cycle length has been estimated to be $\approx$ 20 h,²² in the simulations aimed at replicating our experimental data, we set $\delta =0.035$ (corresponding to $T=19.8$ h).

Additionally, our mathematical study in Methods subsection “Effect of $\epsilon$ and $\zeta$ on the behavior of the system” suggests that our 4D + X model can replicate the experimental data related to the transfection of KRAB and DNMT3A (Figure 2) only within a parameter regime characterized by small $\epsilon$ and large $\zeta$. The expressions defining these parameters are given in Equation 2. Therefore, we set $k_M^A/\mathrm{\Omega }=0.0347$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0176$ ${\mathrm{h}}^{-1}$, and $k_E^A/\mathrm{\Omega }=0.8675$ ${\mathrm{h}}^{-1}$, since these parameter values, along with the previously listed values of $\delta$ and ${\mathrm{D}}_{\text{tot}}$, allow us to be within the desired parameter regime and accurately replicate our experimental results.

Regarding the other parameters, we considered multiple values and selected those that allow us to accurately replicate all experimental results presented in this paper.

Parameter values used to realize the plots in Figure 2K - top panel

$\alpha =1$, $\overline{\alpha }=1$, $\mu =0.1$, $b=1$, $\epsilon =0.1$, $\zeta =25$, $u^A=15$, ${\overline{\beta }}_m^A=0.4911$, ${\overline{\beta }}_m=0.004$, $\overline{\beta }=4.84$, ${\overline{\gamma }}_m=1$, As initial conditions, we set $\left({\overline{Y}}_1,{\overline{D}}_2^R,{\overline{D}}^A,\overline{m},\overline{X}\right)=\left(k,0,1-k,\frac{{\overline{\beta }}_m^A}{{\overline{\gamma }}_m}\left(1-k\right),\frac{\overline{\beta }}{\overline{\delta }}\frac{{\overline{\beta }}_m^A}{{\overline{\gamma }}_m}\left(1-k\right)\right)$, with $0\le k\le 1$.

Parameter values used to realize the plots in Figure 2K - bottom panel

${\tilde{u}}_2^R=0$, ${\tilde{\mu }}^{\prime }=0$, $\beta =1$, ${\tilde{u}}_1^R=1$, ${\overline{W}}_1=\left[0,4\right]\alpha =1$, $\overline{\alpha }=1$, $\mu =0.1$, $b=1$, $\epsilon =0.1$, $\zeta =25$, $u^A=15$. As initial conditions, we set $\left({\overline{Y}}_1,{\overline{D}}_2^R,{\overline{D}}^A\right)=\left(0,0,1\right)$.

Parameter values used to realize the plots in Figure 2L

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0315$ ${\mathrm{h}}^{-1}$, $\delta =0.035$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.174$ ${\mathrm{h}}^{-1}$, $k_{W0}^1=0$, $k_W^1\in \left[0.2602,0.6246\right]e^{-\delta t},\left[0.5205,0.8848\right]e^{-\delta t},\left[0.7808,1.1451\right]e^{-\delta t}$ ${\mathrm{h}}^{-1}$, $k_T^{\prime }=0$, ${\delta }^{\prime }=0$, $\frac{k_M^{\prime }}{\mathrm{\Omega }}=0$, $\frac{k_T^{\prime \ast }}{\mathrm{\Omega }}=0$, $k_{W0}^2=0$, $k_W^2=0$, ${\overline{k}}_E^R=0.0032$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0174$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. As initial condition, we consider $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (0,0,0,15) and the initial values of $n^X$ and $n^m$ at time $t=0$ were set to their steady states of the ODEs.

Parameter values used to realize the plots in Figure 2M

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0118$ ${\mathrm{h}}^{-1}$, $\delta =0.0291$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.8675$ ${\mathrm{h}}^{-1}$, $k_{W0}^1=0$, $k_W^1=0$, $k_T^{\prime }=0$, ${\delta }^{\prime }=0$, $\frac{k_M^{\prime }}{\mathrm{\Omega }}=0$, $\frac{k_T^{\prime \ast }}{\mathrm{\Omega }}=0$, $k_{W0}^2=0$, $k_W^2=0$, ${\overline{k}}_E^R=0.0012$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. We consider four initial conditions: $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (14,0,1,0) (blue), $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (6,0,8,1) (red), $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (4,0,5,6) (yellow), $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (1,0,1,13) (purple). For each case, the initial values of $n^X$ and $n^m$ at time $t=0$ were set to their steady states of the ODEs.

Parameter values used to realize the plots in Figures 4H and 4J

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0176$ ${\mathrm{h}}^{-1}$, $\delta =0.035$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.8675$ ${\mathrm{h}}^{-1}$, $k_{W0}^1=0$, $k_W^1=0$, $c=1.11·{10}^{-6}$ ${\mathrm{h}}^{-1}$, $c_0=8.37·{10}^{-5}$ ${\mathrm{h}}^{-1}$, ${\delta }^{\prime }=2.09·{10}^{-5}$ ${\mathrm{h}}^{-1}$, ${\tilde{\delta }}^{\prime }=4.15·{10}^{-5}$ ${\mathrm{h}}^{-1}$, $K=1$, $\frac{k_M^{\prime }}{\mathrm{\Omega }}=0$, $\frac{k_T^{\prime \ast }}{\mathrm{\Omega }}=$ ${\mathrm{h}}^{-1}$, $k_{W0}^2=0$, $k_W^2=0$, ${\overline{k}}_E^R=0.0018$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. We consider eight initial conditions: $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (13,0,0,2) (Bin 1), (12,0,0,3) (Bin 2), (8,0,0,7) (Bin 3), (6,0,0,9) (Bin 4), (5,0,0,10) (Bin 5), (4,0,0,11) (Bin 6), (1,0,0,14) (Bin 7), (0,0,0,15) (Bin 8). For each case, the initial values of $n^X$ and $n^m$ at time $t=0$ were set to their steady states of the ODEs.

Parameter values used to realize the plots in Figure S5B

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0118$ ${\mathrm{h}}^{-1}$, $\delta =0.0291$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.8675$ ${\mathrm{h}}^{-1}$, $k_{W0}^1=0$, $k_W^1=0$, $k_T^{\prime }=0$, ${\delta }^{\prime }=0$, $\frac{k_M^{\prime }}{\mathrm{\Omega }}=0,0.00347$ ${\mathrm{h}}^{-1}$, $\frac{k_T^{\prime \ast }}{\mathrm{\Omega }}=0$, $k_{W0}^2=0$, $k_W^2=0$, ${\overline{k}}_E^R=0.0012$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. We consider four initial conditions: $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (14,0,1,0) (blue), $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (6,0,8,1) (red), $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (4,0,5,6) (yellow), $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (1,0,1,13) (purple). For each case, the initial values of $n^X$ and $n^m$ at time $t=0$ were set to their steady states of the ODEs.

Parameter values used to realize the plots in Figure S5D - left hand-side panel

$\alpha =1$, $\overline{\alpha }=1$, $\mu =0.1$, $b=1$, $\epsilon =0.1$, $\zeta =1,25$, $u^A=15$. As initial conditions, we set $\left({\overline{Y}}_1,{\overline{D}}_2^R,{\overline{D}}^A\right)=\left(k,0,1-k\right)$, with $0\le k\le 1$.

Parameter values used to realize the plots in Figure S5D - right hand-side panel

${\tilde{u}}_2^R=0$, ${\tilde{\mu }}^{\prime }=0$, $\beta =1$, ${\tilde{u}}_1^R=1$, ${\overline{W}}_1=\left[0,4\right]\alpha =1$, $\overline{\alpha }=1$, $\mu =0.1$, $b=1$, $\epsilon =0.1$, $\zeta =1,25$, $u^A=15$. As initial conditions, we set $\left({\overline{Y}}_1,{\overline{D}}_2^R,{\overline{D}}^A\right)=\left(0,0,1\right)$.

Parameter values used to realize the plots in Figure S5E

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0315$ ${\mathrm{h}}^{-1}$, $\delta =0.035$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.0520,0.8675$ ${\mathrm{h}}^{-1}$, $k_{W0}^1=0$, $k_W^1\in \left[0,0.3643\right]e^{-\delta t},\left[0.2602,0.6246\right]e^{-\delta t},\left[0.5205,0.8848\right]e^{-\delta t}$ ${\mathrm{h}}^{-1}$, $k_T^{\prime }=0$, ${\delta }^{\prime }=0$, $\frac{k_M^{\prime }}{\mathrm{\Omega }}=0$, $\frac{k_T^{\prime \ast }}{\mathrm{\Omega }}=0$, $k_{W0}^2=0$, $k_W^2=0$, ${\overline{k}}_E^R=0.0032$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0052,0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. As initial condition, we consider $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (0,0,0,15) and the initial values of $n^X$ and $n^m$ at time $t=0$ were set to their steady states of the ODEs.

Parameter values used to realize the plots in Figure S6C

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0176$ ${\mathrm{h}}^{-1}$, $\delta =0.035$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.8675$ ${\mathrm{h}}^{-1}$, $k_{W0}^2=0$, $k_W^2\in \left[0.78,3.90\right]e^{-\delta t}$, $\left[10.41,52.05\right]e^{-\delta t},\left[23.42,117.11\right]e^{-\delta t},\left[2082,10410\right]e^{-\delta t}$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^R=0.0018$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. As initial condition, we consider $\left(n_2^R,n^A\right)$ = (0,15) and the initial values of $n^X$ and $n^m$ at time $t=0$ were set to their steady states of the ODEs.

Parameter values used to realize the plots in Figure S6D

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0176$ ${\mathrm{h}}^{-1}$, $\delta =0.035$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.8675$ ${\mathrm{h}}^{-1}$, $k_{W0}^1=0$, $k_W^1\in \left[0.10,0.36\right]e^{-\delta t}$,

$\left[0.18,0.64\right]e^{-\delta t},\left[0.31,1.09\right]e^{-\delta t},\left[0.47,1.61\right]e^{-\delta t}$ ${\mathrm{h}}^{-1}$, $k_T^{\prime }=0$, ${\delta }^{\prime }=0$, $\frac{k_M^{\prime }}{\mathrm{\Omega }}=0$, $\frac{k_T^{\prime \ast }}{\mathrm{\Omega }}=0$, $k_{W0}^2=0$, $k_W^2=0{\overline{k}}_E^R=0.0018$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. As initial condition, we consider $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (0,0,0,15) and the initial values of $n^X$ and $n^m$ at time $t=0$ were set to their steady states of the ODEs.

Parameter values used to realize the plots in Figure S7A

$\alpha =1$, $\overline{\alpha }=1$, $\mu =0.1$, $b=1$, $\epsilon =0.0001,0.1,1,50$, $\zeta =1,5,25$, $u^A=15$. As initial conditions, we set $\left({\overline{Y}}_1,{\overline{D}}_2^R,{\overline{D}}^A\right)=\left(k,0,1-k\right)$, with $0\le k\le 1$.

Parameter values used to realize the plots in Figure S7B - top panels

$k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0315$ ${\mathrm{h}}^{-1}$, $\delta =0.035$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.0520,0.8675$ ${\mathrm{h}}^{-1}$, $k_W^1=0$, $k_W^2=0$, ${\overline{k}}_E^R=0.0032$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0052,0.0868$ ${\mathrm{h}}^{-1}$. As initial condition, we consider $\left(n_2^R,n^A\right)$ = (0,15 - $n^{Y_1}$).

Parameter values used to realize the plots in Figure S7B - bottom panels

$k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0315$ ${\mathrm{h}}^{-1}$, $\delta =0.035$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.0520,0.8675$ ${\mathrm{h}}^{-1}$, $k_W^1=0$, $k_W^2=0$, ${\overline{k}}_E^R=0.0032$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0052,0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. As initial condition, we consider $\left(n_2^R,n^A\right)$ = (0,15 - $n^{Y_1}$) and the initial values of $n^X$ and $n^m$ at time $t=0$ were set to their steady states of the ODEs.

Parameter values used to realize the plots in Figure S9B

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0176$ ${\mathrm{h}}^{-1}$, $\delta =0.035$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.8675$ ${\mathrm{h}}^{-1}$, $k_{W0}^1=0$, $k_W^1\in \left[0.31,1.09\right]e^{-\delta t}$ ${\mathrm{h}}^{-1}$, $k_T^{\prime }=0$, ${\delta }^{\prime }=0$, $\frac{k_M^{\prime }}{\mathrm{\Omega }}=0$, $\frac{k_T^{\prime \ast }}{\mathrm{\Omega }}=0$, $k_{W0}^2=0$, $k_W^2=0{\overline{k}}_E^R=0.0018$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. As initial condition, we consider $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (0,0,0,15) and the initial values of $n^X$ and $n^m$ at time $t=0$ were set to their steady states of the ODEs. The distribution shown is after 30 days.

Parameter values used to realize the plots in Figure S9C

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0176$ ${\mathrm{h}}^{-1}$, $\delta =0.035$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.8675$ ${\mathrm{h}}^{-1}$, $k_{W0}^1=0$, $k_W^{1}0$, $k_T^{\prime }=0$, ${\delta }^{\prime }=0$, $\frac{k_M^{\prime }}{\mathrm{\Omega }}=0$, $\frac{k_T^{\prime \ast }}{\mathrm{\Omega }}=0$, $k_{W0}^2=0$, $k_W^2=0{\overline{k}}_E^R=0.0018$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. As initial condition, we consider values of $\left(n_{12}^R,n_1^R,n_2^R,n^A,n^X,n^m\right)$ from the simulations shown in Figure S9B and divide them with respect to their value of $n^X$, following the following intervals: [0.100] (red), [200,800] (orange), [900,10000] (blue).

Parameter values used to realize the plots in Figure S9G

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0176$ ${\mathrm{h}}^{-1}$, $\delta =0.035$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.8675$ ${\mathrm{h}}^{-1}$, $k_{W0}^1=0$, $k_W^{1}0$, $k_T^{\prime }=0$, ${\delta }^{\prime }=0$, $\frac{k_M^{\prime }}{\mathrm{\Omega }}=0$, $\frac{k_T^{\prime \ast }}{\mathrm{\Omega }}=0$, $k_{W0}^2=0$, $k_W^2=0{\overline{k}}_E^R=0.0018$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. As initial condition, we consider values of $\left(n_{12}^R,n_1^R,n_2^R,n^A,n^X,n^m\right)$ from the simulations associated with the orange distribution in Figure S9D and divide them with respect to their value of $n^X$, following the following intervals: [0,80] (Bin 1), [90,150] (Bin 2), [250,400] Bin 3, [500,10000] (Bin 4).

Parameter values used to realize the plots in Figure S10A

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0176$ ${\mathrm{h}}^{-1}$, $\delta =0.035$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.8675$ ${\mathrm{h}}^{-1}$, $k_{W0}^1=0$, $k_W^1=0$, $k_T^{\prime }=0$, $c=1.74·{10}^{-6}$, $3.47·{10}^{-6}$, $3.47·{10}^{-5}$ ${\mathrm{h}}^{-1}$, $c_0=1.74·{10}^{-6}$, $3.47·{10}^{-6}$, $3.47·{10}^{-5}$ ${\mathrm{h}}^{-1}$, ${\delta }^{\prime }=1.74·{10}^{-6}$, $3.47·{10}^{-6}$, $3.47·{10}^{-5}$ ${\mathrm{h}}^{-1}$, ${\tilde{\delta }}^{\prime }=1.74·{10}^{-6}$, $3.47·{10}^{-6}$, $3.47·{10}^{-5}$ ${\mathrm{h}}^{-1}$, $K=1$, $\frac{k_M^{\prime }}{\mathrm{\Omega }}=0$, $\frac{k_T^{\prime \ast }}{\mathrm{\Omega }}=0$, $k_{W0}^2=0$, $k_W^2=0$, ${\overline{k}}_E^R=0.0018$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. We consider four initial conditions: $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (14,0,0,1) (blue), $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (6,0,0,9) (red), $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (4,0,0,11) (yellow), $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (0,0,0,15) (purple). For each case, the initial values of $n^X$ and $n^m$ at time $t=0$ were set to their steady states.

Parameter values used to realize the plots in Figure S10B

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0176$ ${\mathrm{h}}^{-1}$, $\delta =0.035$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.8675$ ${\mathrm{h}}^{-1}$, $k_{W0}^1=0$, $k_W^1=0$, $k_T^{\prime }=0$, $c=3.47·{10}^{-7}$, $1.04·{10}^{-6}$, $3.47·{10}^{-6}$ ${\mathrm{h}}^{-1}$, $c_0=3.47·{10}^{-6}$ ${\mathrm{h}}^{-1}$, ${\delta }^{\prime }=3.47·{10}^{-6}$ ${\mathrm{h}}^{-1}$, ${\tilde{\delta }}^{\prime }=3.47·{10}^{-7}$, $1.04·{10}^{-6}$, $3.47·{10}^{-6}$ ${\mathrm{h}}^{-1}$, $K=1$, $\frac{k_M^{\prime }}{\mathrm{\Omega }}=0$, $\frac{k_T^{\prime \ast }}{\mathrm{\Omega }}=0$, $k_{W0}^2=0$, $k_W^2=0$, ${\overline{k}}_E^R=0.0018$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. We consider four initial conditions: $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (14,0,0,1) (blue), $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (6,0,0,9) (red), $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (4,0,0,11) (yellow), $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (0,0,0,15) (purple). For each case, the initial values of $n^X$ and $n^m$ at time $t=0$ were set to their steady states of the ODEs.

Parameter values used to realize the plots in Figure S12A

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0176$ ${\mathrm{h}}^{-1}$, $\delta =0.035$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.8675$ ${\mathrm{h}}^{-1}$, $k_{W0}^1=0$, $k_W^1=0$, $k_T^{\prime }=0$, ${\delta }^{\prime }=0$, $\frac{k_M^{\prime }}{\mathrm{\Omega }}=0$, $\frac{k_T^{\prime \ast }}{\mathrm{\Omega }}=0$, $k_{W0}^2=0$, $k_W^2\in \left[26.02,130.13\right]e^{-\delta t}$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^R=0.0018$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. As initial condition, we consider values of $\left(n_{12}^R,n_1^R,n_2^R,n^A,n^X,n^m\right)$ from the simulations associated with the orange distribution in Figure S9C.

Parameter values used to realize the plots in Figure S12B

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0176$ ${\mathrm{h}}^{-1}$, $\delta =0.035$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.8675$ ${\mathrm{h}}^{-1}$, $k_{W0}^1=0$, $k_W^1\in \left[0.1041,0.36\right]e^{-\delta t}$ ${\mathrm{h}}^{-1}$, $k_T^{\prime }=0$, ${\delta }^{\prime }=0$, $\frac{k_M^{\prime }}{\mathrm{\Omega }}=0$, $\frac{k_T^{\prime \ast }}{\mathrm{\Omega }}=0$, $k_{W0}^2=0$, $k_W^2=0$, ${\overline{k}}_E^R=0.0018$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. As initial condition, we consider values of $\left(n_{12}^R,n_1^R,n_2^R,n^A,n^X,n^m\right)$ from the simulations associated with the orange distribution in Figure S9C.

Parameter values used to realize the plots in Figure S12C

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0176$ ${\mathrm{h}}^{-1}$, $\delta =0.035$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.8675$ ${\mathrm{h}}^{-1}$, $k_{W0}^1=0$, $k_W^1=0$, $k_T^{\prime }\in \left[0.7·{10}^{-4},0.24·{10}^{-3}\right]e^{-\delta t}$ ${\mathrm{h}}^{-1}$, ${\delta }^{\prime }=0$, $\frac{k_M^{\prime }}{\mathrm{\Omega }}=0$, $\frac{k_T^{\prime \ast }}{\mathrm{\Omega }}\in \left[3.5·{10}^{-3},1.2·{10}^{-2}\right]e^{-\delta t}$ ${\mathrm{h}}^{-1}$, $k_{W0}^2=0$, $k_W^2=0$, ${\overline{k}}_E^R=0.0018$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. As initial condition, we consider values of $\left(n_{12}^R,n_1^R,n_2^R,n^A,n^X,n^m\right)$ from the simulations associated with the orange distribution in Figure S9C.

Parameter values used to realize the plots in Figure S14

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0118$ ${\mathrm{h}}^{-1}$, $\delta =0.0291$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.8675$ ${\mathrm{h}}^{-1}$, $k_{W0}^1=0$, $k_W^1=0$, $k_T^{\prime }=0,0.00012,0.0012$ ${\mathrm{h}}^{-1}$, ${\delta }^{\prime }=0$, $\frac{k_M^{\prime }}{\mathrm{\Omega }}=0,3.47·{10}^{-4},3.47·{10}^{-3}$ ${\mathrm{h}}^{-1}$, $\frac{k_T^{\prime \ast }}{\mathrm{\Omega }}=0,0.0087,0.087$ ${\mathrm{h}}^{-1}$, $k_{W0}^2=0$, $k_W^2=0$, ${\overline{k}}_E^R=0.0012$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. We consider four initial conditions: $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (14,0,1,0) (blue), $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (6,0,8,1) (red), $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (4,0,5,6) (yellow), $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (1,0,1,13) (purple). For each case, the initial values of $n^X$ and $n^m$ at time $t=0$ were set to their steady states of the ODEs.

Parameter values used to realize the plots in Figure S15A

${\tilde{u}}_2^R=0$, ${\tilde{\mu }}^{\prime }=0$, $\beta =1$, ${\tilde{u}}_1^R=1$, ${\overline{W}}_1=\left[0,4\right]\alpha =1$, $\overline{\alpha }=1$, $\mu =0.1$, $b=1$, $\epsilon =0.0001,0.1,1,50$, $\zeta =1,5,25$, $u^A=15$. As initial conditions, we set $\left({\overline{Y}}_1,{\overline{D}}_2^R,{\overline{D}}^A\right)=\left(0,0,1\right)$.

Parameter values used to realize the plots in Figure S15B

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0315$ ${\mathrm{h}}^{-1}$, $\delta =0.035$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.0520,0.8675$ ${\mathrm{h}}^{-1}$, $k_{W0}^1=0$, $k_W^1\in \left[0,0.3643\right]e^{-\delta t},\left[0.2602,0.6246\right]e^{-\delta t},\left[0.5205,0.8848\right]e^{-\delta t}$ ${\mathrm{h}}^{-1}$ (left hand-side panel) and $k_W^1\in \left[0,0.2082\right]e^{-\delta t},\left[1.2492,1.4574\right]e^{-\delta t},\left[1.7697,1.9779\right]e^{-\delta t}$ ${\mathrm{h}}^{-1}$ (right hand-side panel), $k_T^{\prime }=0$, ${\delta }^{\prime }=0$, $\frac{k_M^{\prime }}{\mathrm{\Omega }}=0$, $\frac{k_T^{\prime \ast }}{\mathrm{\Omega }}=0$, $k_{W0}^2=0$, $k_W^2=0$, ${\overline{k}}_E^R=0.0032$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0052,0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. As initial condition, we consider $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (0,0,0,15) and the initial values of $n^X$ and $n^m$ at time $t=0$ were set to their steady states of the ODEs.

Parameter values used to realize the plots in Figure S15C

${\tilde{u}}_2^R=0$, ${\tilde{\mu }}^{\prime }=1$, $\beta =1$, ${\tilde{u}}_1^R=0$, $\overline{T}=\left[0,4\right]\alpha =1$, $\overline{\alpha }=1$, $\mu =0.1$, $b=1$, $\epsilon =0.0001,0.1$, $\zeta =1,5,25$, $u^A=15$. As initial conditions, we set $\left({\overline{Y}}_1,{\overline{D}}_{12}^R,{\overline{D}}_2^R,{\overline{D}}^A\right)=\left(1,1,0,0\right)$.

Parameter values used to realize the plots in Figure S15D

$k_{W0}^A=0$, $k_W^A=7.8075$ ${\mathrm{h}}^{-1}$, ${\overline{k}}_E^A=0.0315$ ${\mathrm{h}}^{-1}$, $\delta =0.035$ ${\mathrm{h}}^{-1}$, $\frac{k_M^A}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^A}{\mathrm{\Omega }}=0.0520,0.8675$ ${\mathrm{h}}^{-1}$, $k_{W0}^1=0$, $k_W^1=0$, $k_T^{\prime }\in \left[0,0.0016\right]e^{-\delta t},\left[0.0076,0.0088\right]e^{-\delta t},\left[0.0113,0.0126\right]e^{-\delta t}$ ${\mathrm{h}}^{-1}$, ${\delta }^{\prime }=0$, $\frac{k_M^{\prime }}{\mathrm{\Omega }}=0$, $\frac{k_T^{\prime \ast }}{\mathrm{\Omega }}\in \left[0,0.0434\right]e^{-\delta t},\left[0.2082,0.2429\right]e^{-\delta t},\left[0.3123,0.3470\right]e^{-\delta t}$ ${\mathrm{h}}^{-1}$, $k_{W0}^2=0$, $k_W^2=0$, ${\overline{k}}_E^R=0.0032$ ${\mathrm{h}}^{-1}$, $\frac{k_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{{\overline{k}}_M}{\mathrm{\Omega }}=0.0347$ ${\mathrm{h}}^{-1}$, $\frac{k_E^R}{\mathrm{\Omega }}=0.0052,0.0868$ ${\mathrm{h}}^{-1}$, ${\beta }_m=0.0021$ ${\mathrm{h}}^{-1}$, ${\beta }_m^A=0.2556$ ${\mathrm{h}}^{-1}$, $\beta =2.52$ ${\mathrm{h}}^{-1}$, ${\gamma }_m=0.24$ ${\mathrm{h}}^{-1}$. As initial condition, we consider $\left(n_{12}^R,n_1^R,n_2^R,n^A\right)$ = (15,0,0,0) and the initial values of $n^X$ and $n^m$ at time $t=0$ were set to their steady states of the ODEs.

Quantification and statistical analysis

DNA methylation analysis

To determine the probability of a specific fraction of CpGs being methylated based on bisulfite sequencing data, which offers the probability of a specific CpG being methylated for the cell population, we employ the following method. This approach is applied to both bisulfite sequencing datasets presented in Figures 3G and 4E, resulting in the corresponding plots in Figures 3H and 4H, respectively.

Let us assume that the gene of interest has a total of $N$ CpGs. Then, for any $0\le j\le N$, having $j$ CpGs methylated corresponds to the event where precisely $j$ CpGs are methylated, and all other CpGs (i.e., $N-j$) are unmethylated. Now, let us define the probability of the described event as $P_j$, where $0\le j\le N$. Additionally, let the probability of CpG $i$ being methylated be $p_i$, and the probability of CpG $i$ being unmethylated be $q_i$, where $1\le i\le N$. The bisulfite sequencing data provide us with $p_i$, and since the event “CpG unmethylated” is the complement of “CpG methylated”, $q_i$ is known and can be expressed as $q_i=1-p_i$. Then, the probability that exactly $j$ CpGs are methylated out of a total of $N$ CpGs, where each CpG has its own probability of being methylated $\left(p_i\right)$ or not being methylated $\left(q_i\right)$, can be calculated by using the following formula:

$$P\left(\text{Exactly}j\text{CpGs}\text{are}\text{methylated}\right)=\sum _{\begin{array}{c}S\subseteq \left\{1,2,\dots ,N\right\}\\ \left|S\right|=j\end{array}}\left(\prod _{i\in S}{p}_i·\prod _{k\in \left\{1,2,\dots ,N\right\}\setminus S}{q}_k\right)\text{.}$$ (Equation 27)

More precisely, the sum $(\sum )$ is taken over all possible subsets $S$ of events with exactly $j$ elements. For each subset $S$, we calculate the product of the probabilities of the events in $S$ occurring $\left({\prod }_{i\in S}{p}_i\right)$. We then calculate the product of the probabilities of the events not in $S$ occurring $\left({\prod }_{k\in \left\{1,2,\dots ,N\right\}\setminus S}{q}_k\right)$. The overall probability is the sum of these products for all subsets $S$. In simpler terms, the formula considers all possible combinations of $j$ events occurring out of $N$ events, multiplying the probabilities of those events occurring and the probabilities of the remaining events not occurring in each combination. The sum aggregates these probabilities for all valid combinations, providing the probability that exactly $j$ events happen.

As an illustrative example, let us consider the case in which the total number of CpG in the gene of interest are $N=3$. Then, defining the probability of CpG $i$ to be and not to be methylated as $p_i$ and $q_i=1-p_i$, with $i=1,2,3$, we then can calculate the probability of having $j$ CpGs methylated, i.e., $P_j$, with $j=0,1,2,3$, using 27, obtaining $P_0=q_1{q}_2{q}_3$, $P_1=p_1{q}_2{q}_3+q_1{p}_2{q}_3+q_1{q}_2{p}_3$, $P_2=p_1{p}_2{q}_3+p_1{q}_2{p}_3+q_1{p}_2{p}_3$, and $P_3=p_1{p}_2{p}_3$. The probability distribution of the fraction of CpGs methyalated is then obtained by plotting $\left(j,P_j\right)$ for $j=0,2,\dots ,N$.

Linear regression analysis

Concerning the best-fit linear regression lines shown in Figures 3I, 4I, 4J and S11J they were obtained using MATLAB polyfit function. Furthermore, the p-value associated with the slope coefficient was determined through the MATLAB fitlm function.

Fold change calculation

Fold changes from qPCRs in the study were calculated using a standard $\Delta \Delta C_t$ method as described below. When calculating $\Delta \Delta C_t$, the mean of $\Delta C_{t,\text{Active}\text{State}}$ across independent replicates was used.

$$\begin{array}{l}\Delta C_{t,\text{Epigenetically}\text{Edited}}=C_{t,\text{Epigenetically}\text{Edited},\text{Reporter}\text{Gene}}-C_{t,\text{Epigenetically}\text{Edited},\text{Endogenous}\text{Reference}\text{Gene}}\\ \Delta C_{t,\text{Active}\text{State}}=C_{t,\text{Active}\text{State},\text{Reporter}\text{Gene}}-C_{t,\text{Active}\text{State},\text{Endogenous}\text{Reference}\text{Gene}}\\ \Delta \Delta C_t=\Delta C_{t,\text{Epigenetically}\text{Edited}}-\Delta C_{t,\text{Active}\text{State}}\\ \text{Fold}\text{change}=2^{-\Delta \Delta C_t}\end{array}$$

Statistical analysis for fold changes and percentages

For statistical tests comparing percentages calculated from flow cytometry measurements or fold changes from MeDIP-qPCR and ChIP-qPCR, unpaired two sample t-tests were performed. Significance was defined using ∗p < 0.05, ∗∗p < 0.01, ∗∗∗p < 0.001.

Published: September 9, 2025