Related work

Models of chromatin-mediated gene regulation, in which histone modifications, DNA methylation, and TF-mediated regulation are combined together, remain under-represented in the literature [20]. Most existing models that include chromatin modifications into gene expression regulation are based on phenomenological rules rather than on the molecular reactions that regulate chromatin state, and focus on specific biological processes, such as iPSC reprogramming [21–24] or epithelial-mesenchymal transition [25], and most of them are only suitable for computational simulations [21–23]. On the opposite side of the spectrum, highly detailed mechanistic models have appeared for histone modifications alone, in which each nucleosome within a gene is modeled and simulated as a distinct unit [26]. Different from these existing works, we explicitly address how the duration of memory of a chromatin state is modulated by the topological properties of the chromatin modification circuit, by relevant biochemical parameters, and by the interplay between TF-based regulation and chromatin dynamics.

Model of the gene’s inner chromatin modification circuit

In this paper, we focus on a circuit motif that captures the interactions among H3K4 methylation/acetylation, H3K9 methylation, and DNA methylation, and has the nucleosome as the basic unit that can carry these modifications (Fig 1A). H3K4 methylation (H3K4me3) typically co-exists with acetylation and is associated with active chromatin state [10](Chapter 3), [27], while H3K9 methylation (H3K9me2/3) and DNA methylation are a hallmark of repressed chromatin state and, as such, are found at pluripotency genes and at lineage specific genes in terminally differentiated cells [5].

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Fig 1. The gene’s inner chromatin modification circuit.

(A) Nucleosome modifications: D (unmodified nucleosome), D^A (nucleosome with a activating histone modification, H3K4me3/ac), (nucleosome with only DNA methylation, CpGme), (nucleosome with only a repressive histone modification, H3K9me3) and (nucleosome with both H3K9me3 and CpGme). (B) Pictorial representation of a gene with its nucleosomes carrying various modifications. The left side arrow represents the promoter. (C) Competitive interactions between opposing histone modifications (activating D^A and repressive ), wherein each modification recruits writers of itself and erasers of the opposing modification. (D) Complete chromatin modification circuit that includes all the interactions described in Section “Models” (see also Figs B and C in S1 File). The numbers shaded in gray on the arrows correspond to the reactions associated with the arrows, described in the main text and in Fig 2. In (C) and (D), enzymes that write (writers) and erase (erasers) each modification are represented as W₁, W₂, W₃ and E₁, E₂, E₃, respectively. The socket on each enzyme represents a domain that binds to protein readers of the indicated (by the dashed lines) modification, enabling the process by which each modification recruits writers or erasers to nearby histones. To distinguish the KMTs for H3K9 and H3K4, we define the writers for H3K9 methylation (Suv39H1) as KMT^R and the writers for H3K4 methylation (SETs and MLL1/2) as KMT^A. Similarly, to distinguish the HDMTs for H3K9 methylation and H3K4 methylation, we define the erasers for H3K9 methylation (JMJD2A) as HDMT^R and the erasers for H3K4 methylation (JARID) as HDMT^A. We use colored dotted lines to depict the recruitment process by H3K4me3/ac (green lines), H3K9me3 (red lines), and CpGme (orange lines) and we use dotted black arrows to depict the consequent effect on writing/erasing. The solid black arrow represents the nucleosome modification.

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Specifically, H3K4 and H3K9 methylation are typically mutually exclusive since the enzymes that methylate H3K4 do not do so if the neighboring K9 residues are already methylated [28] and viceversa [29]. We therefore assume that each nucleosome carries either H3K9 methylation or H3K4 methylation but not both. We also assume that H3K4 methylation and acetylation co-exist and we represent them by an activating “A” nucleosome modification (Fig 1A). Furthermore, for H3K9 methylation, we lump together the two methylation states (me2 and me3) for simplicity and call them “me3” because both of them are associated with gene repression. Also, although nucleosomes carry two copies of each histone, we consider them as one unit that can carry only one histone modification.

In addition to these histone modifications, DNA itself can be methylated in correspondence to the CpG dinucleotide (CpGme) [5, 10, 30]. DNA methylation is correlated with the absence of H3K4 methylation [10](Chapter 6), since the enzymes that methylate H3K4 recognize unmethylated DNA binding motifs [10](Chapter 1) and viceversa [10](Chapter 6), [31]. Therefore, we assume that each nucleosome can carry either one or the other but not both (Fig 1A). By contrast, DNA methylation can co-exist with H3K9 methylation [10](Chapters 6, 22) and hence we allow a nucleosome to carry both modifications (Fig 1A). As a result, each gene will include a number of nucleosomes with different modifications (Fig 1B).

In the next section, we introduce the molecular mechanisms and corresponding reactions by which these modifications can be written, erased, and copied to nearby nucleosomes, which will form the chromatin modification circuit (Fig 1C and 1D).

De novo establishment.

As described in [10](Chapter 21), sequence-specific transcriptional activators bind DNA and recruit HATs, such as the SAGA complex, to the promoter of a gene, which becomes acetylated. Examples of transcriptional activators that recruit HATs include Myc, GATA.1, and Gal4. The deposition of H3K4me3 then can occur co-transcriptionally as RNA Polymerase II recruits SETs, a KMT that methylates H3K4 [33], [10](Chapter 3), or through the recruitment of SETs and MLL1/2 (also KMTs for H3K4) to DNA by the CxxC binding domain that specifically recognizes unmethylated DNA [5](Chapter 7). Finally, MLLs can be recruited to specific promoters by transcriptional activators such as Oct4 [34]. H3K9me3 is established by the writer action of Suv39H1 (a KMT for H3K9), which can also be recruited to DNA by sequence-specific TFs. An example of this is the recruitment of this enzyme to GATA.1 targets by the PU.1 TF, as a means to silence GATA.1 targets and promote the myeloid lineage [35]. These modifications can be captured by the one-step enzymatic reactions ⓪, ①, ② and ③ in Fig 2. In the sequel, we will consider the k_W parameters as inputs to the chromatin modification circuit since they can be modulated by TFs external to the circuit (see Eqs (82), (83) in S1 File).

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Fig 2. Reactions associated with the chromatin modification circuit motifs.

Each reaction is associated with a number, which is referred to in the main text. Specifically, reactions ⓪, ①, ②, ③, ⑩ and ⑪ describe de novo establishment. Reactions ④ and ⑤ describe auto-catalysis, wherein a modification recruits writers of the same modification to nearby nucleosomes. Reactions ⑮ and ⑯ describe cross-catalysis, wherein DNA methylation recruits writers of repressive histone modifications and viceversa, respectively. Reactions ⑥, ⑦, and ⑫ represent basal erasure while reactions ⑧, ⑨, ⑬, and ⑭ represent recruited erasure, wherein competing modifications recruit erasers of each other. The different colored lines delimit the sets of reactions that take place for each of the circuit motifs shown in Fig 1 and Figs B and C in S1 File. Specifically, Fig B depicts the circuit between activating histone modifications and DNA methylation and Fig C depicts the circuit among repressive modifications. Here, we use one-step enzymatic reaction models [36] to capture histone and DNA modifications. In the bottom-right corner, we indicate how the reaction rate constants depend on the amount of writers and erasers, in which f(⋅) is a generic increasing function. Here, η and are between 0 and 1 and defined in Section “Models”. Specifically, η = 1 in the absence of DNMT1 and in the absence of MBD (see Section 1 in S1 File for the detailed form of the reaction constants and their derivation).

https://doi.org/10.1371/journal.pcbi.1009961.g002

Model of transcriptional regulation by chromatin state

Chromatin state affects transcription by modulating nucleosome compaction. DNA wrapped around highly compacted nucleosomes is difficult to access by TFs and therefore genes in such DNA regions are not transcribed. By contrast, DNA wrapped around less compacted nucleosomes is more likely to be accessible by TFs, and hence is susceptible to transcription. Repressive histone modifications and DNA methylation generally lead to increased compaction while activating histone modifications keep a more open chromatin structure [10](Chapter 3).

We therefore assume that transcription is allowed only by D^A, but not by D or any of , , and . In principle, nucleosome state D may allow some low basal transcription, through non-specific targeting by chromatin remodelers [32, 50], but we assume this is sufficiently low to be negligible. A more general model could include this basal rate, accounting for the fact that transcription by RNA Pol II of D occurs concurrently with the deposition of H3K4me3 and hence with the conversion of D to D^A [33]. Thus, by lumping together transcription and translation, we obtain the following gene expression model (Section 4 in S1 File): (1) in which X is the gene product, γ_x is the decay rate constant of X and α_x = F(A, R) is an increasing function of the concentration of TF activators A and a decreasing function of the concentration of TF repressors R. Here, for simplicity, we assume that α_x is a constant. The main conclusions of this paper are not affected by this assumption.

Time scale separation among constituent processes control bistability, hysteresis, and time to memory loss

We first analyze the ordinary differential equation (ODE) model of the chromatin modification circuit, focusing on how parameters affect bistability and hysteresis. Bistability is the co-existence of two stable steady states, specifically active and repressed chromatin states, in the absence of an input stimulus. Hysteresis occurs when the output of the system follows two different paths depending on whether the input is increased or decreased. In particular, the output remains in the proximity of the value reached for high input when this is removed that is, the system “remembers” a transient input stimulus. We then consider a stochastic model to address the question of temporal duration of this memory, given that noise intrinsic to molecular reactions can always displace the system’s state.

For our analysis, we let n^A, , , and n^D denote the number of nucleosomes with activating (D^A), repressive (, and ), and no modifications (D), respectively, within a gene composed of a total of D_tot nucleosomes. All analyses are performed considering the unitless normalized time with D_tot = D_tot/Ω and Ω the reaction volume. We define the following non-dimensional parameters, capturing the characteristic time scales of the circuit’s constituent processes: (2)

Here, α is the non-dimensional auto-catalysis rate parameter, while , and α′ are the non-dimensional cross-catalysis rate parameters, which we assume on the same order without loss of generality. We further let with b = O(1), implying that μ scales the ratio between the erasure rates of repressive and activating histone modifications. We similarly let with β = O(1), implying that μ′ scales the ratio between the erasure rates of DNA methylation and those of activating histone modifications. From these relationships, it also follows that and that . Therefore, the parameter ϵ scales the ratio between the basal rate at which all modifications are erased and the rate at which they are auto or cross-catalyzed. Finally, since and , it follows that ϵ′ scales the ratio between the rate at which modifications are removed through mutual recruitment of erasers and that at which they are auto or cross-catalyzed. These parameter definitions are pictorially conveyed in Fig 3A.

Putting these definitions together, in our model, ϵ and ϵ′ are non-dimensional parameters that quantify the time scales of basal and recruited erasure rates of all modifications relative to those of auto and cross-catalysis. Similarly, μ and μ′ are non-dimensional parameters that quantify the time scales of erasure rates (basal and recruited) of repressive histone modifications and DNA methylation, respectively, relative to those of activating histone modifications. Since these normalized parameters are functions of the biochemical reaction rate constants (Eq (2)) and, in turn, the reaction rate constants are known functions of the level of writer and eraser enzymes, TET enzymes, and MBD proteins (see Fig 2 and Section 1.6 in S1 File), it is possible to experimentally vary these parameters by over-expressing, inhibiting, or by recruiting these molecular players.

Deterministic behavior

We first analyze the ordinary differential equation (ODE) model of the chromatin modification circuit, focusing on how the above parameters affect dynamics. To this end, we assume that the number of total modifiable units D_tot is sufficiently large such that the variables n^A, , , and n^D can be considered real-valued and thus their temporal evolution can be described by ODEs. In particular, we describe the system in terms of fractions of modifiable units, that is , , , and . We further introduce the normalized inputs with and , with and , with and . With these definitions and letting , the ODEs describing the system in non-dimensional variables are given by (3) in which and the circled numbers indicate the corresponding reactions from Fig 2 (see Section 1.6 and Eqs (80) in S1 File for derivation).

In the absence of inputs (), system (3) has a unique stable steady state, if ϵ is not sufficiently small. This steady state has for small μ′. When ϵ is sufficiently small, the system is bistable, with both active () and repressed () co-existing stable chromatin states. But when μ′ is very small, even for small ϵ, the system returns monostable with a unique stable steady state corresponding to the repressed state (Fig 3C and 3D and Fig K in S1 File). Furthermore, as μ′ increases (DNA methylation erasure rate increases), the system approximates well the histone modification system (Fig 1C), wherein DNA methylation is not present. This is because when μ′ is sufficiently large, the DNA methylation is erased very quickly and then . As a consequence, when ϵ is sufficiently small the system is bistable for intermediate values of μ, monostable with when μ is large and monostable with when μ is small (Fig D in S1 File). Varying ϵ′ does not change these trends, except for ϵ′ ≪ ϵ, in which case the system returns monostable (Fig K in S1 File).

Fig 3E shows the input/output steady state characteristics of the chromatin modification circuit. Specifically, as u^A is increased starting from , the state switches to for u^A above a critical value, which increases as ϵ decreases. When u^A is decreased back to zero, the path that follows is different and it can keep even when u^A = 0 if ϵ is sufficiently small. In this case, we say that the state transition becomes irreversible and the system keeps memory of its input stimulus. From a biological point of view, this implies that an activating transient input stimulus allows to permanently re-activate a silenced gene. This memory vanishes as ϵ increases. When μ′ decreases, the value of u^A required to flip the system’s state to fully active increases until we reach a regime under which the memory is lost (Fig M in S1 File). A similar behavior is observed for the relationship between the repressing input u^R (with ) and (Fig N in S1 File). Biologically, this implies that a repressive transient input stimulus can permanently silence an initially active gene. However, as opposed to the characteristics, if we reduce μ′, the minimum value of u^R required to silence the system (, ) decreases (Fig 3F and Fig N in S1 File). This also shows that as μ′ is decreased, the active state becomes less robust to repressive input stimuli.

A comparison of the plots of Figs M and N in S1 File for small ϵ and μ = 1 reveals that, even when μ′ = 1, it takes smaller values of u^R to switch the state to than the values of u^A required to switch the state to . This fact highlights that it is the structure itself of the chromatin modification circuit, rather than the rate constants alone, to “bias” the system towards a repressed chromatin state. In fact, the two-layer topology of the repressive modifications confers increased robustness to the repressed state with respect to activating stimuli (Fig 3A). This is in contrast to what is observed for the system that does not include DNA methylation (Fig 1C), wherein once all parameters are equal to each other, the stability landscape is symmetric (Section 2.1 in S1 File). Varying ϵ′ does not significantly affect these trends (Figs M and N in S1 File), unless ϵ′ ≪ ϵ, in which case hysteresis is lost (Fig 3G).

In summary, when recruited erasure (ϵ′) and auto-catalysis/cross-catalysis () are sufficiently faster than basal erasure (ϵ), the circuit displays bistability and hysteresis, except if μ and/or μ′ are sufficiently small. In this case, the unique stable state is the fully repressed chromatin state, independent of all other parameters.

Stochastic behavior

We next evaluate how the stationary distribution of the chromatin state, the time to memory loss, and the ability to reactivate a repressed gene depend on ϵ, ϵ′, μ and μ′, and on the inputs u^A and u^R. To this end, we first perform a computational study by using Gillespie’s Stochastic Simulation Algorithm (SSA) [51]. We then complement this with a mathematical derivation of the approximate expressions of both the stationary distribution and the time to memory loss as a function of the parameters and input stimuli.

Fig 4A and 4B shows how the stationary distribution is shaped by ϵ and μ′ (A) and by inputs u^R and u^A (B). Specifically, ϵ has a most critical role. As it decreases, with all other parameters remaining fixed, the peaks of the distribution become more concentrated about the fully repressed and fully active chromatin states. Highly concentrated peaks imply that the probability of finding the chromatin state outside of these configurations is very small. This property is connected to the time to memory loss, defined as the expected value of the earliest time the chromatin state reaches the active state starting from the repressed state and viceversa. In fact, as ϵ is decreased, the chromatin state starting in either the repressed or active state takes longer, on average, to reach the active or repressed state for the first time, respectively (Fig 4C and 4D). In contrast to what found for the histone modification model of Fig 1C (Fig FA in S1 File), decreasing ϵ also has an effect in increasing the height of the peak in correspondence of the repressed state to the detriment of the peak corresponding to the active state (Fig 4A). When μ′ is decreased, this bias increases, and the peak in correspondence of the repressed state becomes more concentrated (Fig 4A). Varying μ has a similar effect as varying μ′ (Fig S in S1 File). Inputs u^R and u^A each increase the height of the peak corresponding to the repressed and active state, respectively (Fig 4B). Consistent with the deterministic analysis (Fig M in S1 File), for smaller μ′, it is required a larger u^A to increase the probability of finding the state in the fully active chromatin state (Fig RB in S1 File). By contrast, ϵ′ does not significantly affect the trends observed for ϵ, μ′, μ and the inputs, although it modulates some aspects of the distribution (Section 3.5 and Figs Q and T in S1 File). Specifically, when ϵ′ decreases compared to ϵ, the peaks of the distribution become less concentrated and, in the extreme case where ϵ′ ≪ ϵ, the distribution becomes unimodal (Fig 4E).

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Fig 4. Time scale separation parameters ϵ, ϵ′ and μ′ control bimodality, time to memory loss, and reactivation time in the chromatin modification circuit.

(A)-(B) Stationary probability distributions π of (n^A, n^R) for the chromatin modification circuit in Fig 3A obtained by simulating the reactions listed in Fig 2 with the SSA. (A) Effect of ϵ and μ′ on the distribution: in the left side plots ϵ = 0.19, 0.02 and μ′ = 1. In the right side plots, μ′ = 1, 0.1 and ϵ = 0.19. (B) Effect of the input stimulus on the distribution: here, ϵ = 0.12 and μ′ = 0.8. In the top plots, u^A = 0, 1, and in the bottom plots and u^R = 0, 1 (Fig R shows different values of μ′ and ϵ). The parameter values for each panel are listed in Table A in S1 File. In all simulations, ϵ = 1 and μ = 1 (Figs S and T in S1 File show different values) and we decrease ϵ by decreasing (similar results can be obtained if we vary as shown in Fig Q in S1 File). (C) Time trajectories of n^R starting from a repressed chromatin state n^A = 5, , as ϵ and μ′ are varied. Simulations are stopped when n^R = 6 for the first time. (D) Time trajectories of n^A starting from an active chromatin state n^A = 45, , as ϵ and μ′ are varied. Simulations are stopped when n^A = 6 for the first time. In all plots of (C)-(D), time is normalized according to , ϵ = 0.36, 0.12, μ′ = 1, 0.5, μ = 1, and ϵ′ = 1 (Fig V in S1 File shows different values of ϵ′). The parameter values are listed in Table B in S1 File. (E) Effect of ϵ′ on the stationary probability distribution π. We set ϵ = 0.12, μ′ = 1, μ = 1 and ϵ′ = 1, 0.001 from left to right (Fig U in S1 File shows different values of ϵ′). The parameter values for each panel are listed in Table P in S1 File. (F) Time trajectories of the system starting from n^R = 45, n^A = 5 and applying an input u^A that, at steady state, leads to a unimodal distribution in the proximity of the active state n^A = D_tot. The parameter values are listed in Table C in S1 File. In particular, u^A = 3.2, μ′ = 0.1, ϵ = 0.24, 0.16, μ = 1 and ϵ′ = 1 (Fig W in S1 File shows different parameter values). In all plots, and simulations are obtained by implementing the set of reactions listed in Fig 2 with the SSA [51]. In all simulations, . In our model, parameters ϵ and ϵ′ quantify the time scales of basal and recruited erasure rates of all modifications relative to those of auto and cross-catalysis. Similarly, parameters μ and μ′ quantify the time scales of erasure rates (basal and recruited) of repressive histone modifications and DNA methylation, respectively, relative to those of activating histone modifications. Mathematical definitions are found in Eq (2). In all plots (n^A, n^R) ≈ (50, 0) corresponds to the active state and (n^A, n^R) ≈ (0, 50) corresponds to the repressed state. In each panel of (C), (D), and (F), the number of trajectories plotted is 10.

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In the regime where both ϵ′ ≪ 1 and ϵ ≪ 1, in which the system displays a bimodal distribution (Figs Q-T in S1 File), we can analytically derive a one-dimensional Markov chain approximation that allows analytical computation of the stationary distribution and of the time to memory loss as functions of the parameters (Sections 3.2–3.4 in S1 File). Biophysically, in this regime, the reactions represented by the labels BE_i and RE_i in Fig 3A become slower compared to those represented by the label C_i. In particular, letting with x ϵ [0, D_tot], we have that, when ϵ ≪ 1, the stationary probability distribution π(x) can be approximated by (4) with (5) in which , and , defined in Eqs (161) and (171) in S1 File, independent of ϵ, μ, μ′ and ϵ′ (see Section 3.3 in S1 File for the mathematical derivations). From this expression, we conclude that if ϵ ≪ 1, π(x) ≈ 0 for all x except for x = D_tot (fully repressed state) and x = 0 (fully active state), implying that the probability of finding the system in one of the intermediate states is about zero. This is in accordance with the computational results of Fig 4A, which show higher concentration around the extreme states, and with Fig 4C and 4D, which show smaller likelihood of transitioning out of those extreme states as ϵ is decreased. When ϵ goes to zero, P → ∞, which implies that π(D_tot) ≈ 1. This is consistent with the structural bias of the chromatin modification circuit towards the repressed chromatin state (Fig 3A) and the computational results of Fig RA in S1 File. This bias is enhanced as μμ′ is decreased since P → ∞ as μμ′ → 0, leading to π(D_tot) ≈ 1. Furthermore, in agreement with what observed in Fig 4B, an increase of u^A decreases P and hence increases π(0). By contrast, an increase of u^R increases P, which, in turn, increases π(D_tot). Also, larger u^A values are required to increase the probability of the active state to the same level when μ′ is smaller.

Similarly, the normalized time to memory loss of the repressed state can be approximated as (Section 3.4 in S1 File) (6) with an increasing function, , and K_R and functions independent of ϵ, μ′ and μ, while the normalized time to memory loss of the active state can be approximated as (7) with an increasing function and , and K_A and functions independent of ϵ, μ′ and μ. In the limiting condition where ϵ → 0, we have that both and . Therefore, decreased ϵ is the driver for increasing the extent of epigenetic memory of both the active and repressed chromatin states. Furthermore, because of the structural asymmetry of the chromatin modification circuit, the power of ϵ is different in the two above expressions, indicating that decreased ϵ increases the extent of memory much more for the repressed state than for the active state. This is in contrast with what obtained for the histone modification circuit, in which both and are O(1/ϵ) (Eqs (138) and (140) in S1 File). Finally, decreased μ and μ′ lead to larger but to lower . These trends are in agreement with the simulations in Fig 4C and 4D. In particular, in each panel, we plotted a number of trajectories equal to 10 to provide a comprehensive representation of the parameter effects on the trajectories and, at the same time, a clear reading of the figure.

We next determine via simulation how the parameters ϵ, μ′ and ϵ′ affect the time to reactivation, i.e., the time it takes to re-activate an initially repressed chromatin state by applying a sufficiently large activating input stimulus u^A (Fig 4F). The time trajectories show a switch-like behavior, which is more prominent for ϵ ≪ 1, μ′ ≪ 1, and ϵ′ > 1 (Fig W in S1 File). Also, the latency, i.e., the time the trajectory takes to switch after an activating input is applied, is highly variable and its variability is mostly dictated by small ϵ and small μ′, which, together, make gene reactivation a low probability event. Furthermore, the time to reactivation decreases by increasing μ′ and/or increasing ϵ, consistent with results from the deterministic model (Fig X in S1 File).

Taken together, our results indicate that basal erasure rate sufficiently smaller than recruited erasure rate (ϵ sufficiently small and ϵ′ sufficiently larger than ϵ) allows a bimodal distribution such that both active and repressed chromatin states each have high probabilities compared to intermediate states. As ϵ decreases, this distribution is more concentrated about the fully active and repressed states and the time to memory loss of either state increases. Even when there is no asymmetry between activating and repressive modification erasure rates (μ = 1 and μ′ = 1), the time to memory loss is larger for the repressed chromatin state. This asymmetry is accentuated by μ′ < 1.

Relationship with published data

In [52], the authors study the kinetics of silencing and reactivation of chromatin modifications, including H3K9me3 and DNA methylation, at the single-cell level. In particular, they show that silencing and reactivation has a all-or-none behavior. Our model predicts that this all-or-none response is observed when the underlying chromatin modification circuit has a bimodal stationary distribution, which we proved requires ϵ ≪ 1 and ϵ′ sufficiently larger than ϵ (Fig W in S1 File). This suggests that in practice we should expect ϵ ≪ 1 and ϵ′ not too small. Furthermore, data on temporal dynamics of DNA de-methylation in vivo indicate that μ′ ≪ 1 (Section 3.1 in S1 File). Early elegant experimental studies performed by [53] further show that the latency of reactivation of a silenced gene is highly variable. In our model, a highly variable latency of reactivation events is predicted under the parameter regime where both ϵ and μ′ are small (Fig 4F and Fig W in S1 File), suggesting that μ′ ≪ 1 may in fact be responsible for the experimentally observed latency of reactivation [53]. Furthermore, in [53] it is shown that increased proliferation rate leads to a faster reactivation kinetics. In our model, increased proliferation is captured by increased ϵ and in fact higher ϵ leads to faster kinetics of reactivation events (Fig YA-YC in S1 File), in agreement with the observations in [53].

Finally, a basic DNA methylation model that does not include MBD proteins cannot reflect the low effective active erasure rates of DNA methylation through the action of TET enzymes, k_T, encountered in vivo (reactions (45) in S1 File). In our model, the ability of TET enzymes to convert methylated DNA to hydorxilmethylated DNA is hampered by the binding of MBD proteins to methylated DNA (Section 1.3 in S1 File). As a result, the effective active erasure rate constant, , derived in Eq (62) in S1 File, can be written as in which k_T is a constant proportional to the level of TET enzyme, where if MBD proteins are highly abundant, and if MBD proteins are absent (Eq (62) in S1 File). This implies that the parameter is proportional to the amount of TET enzymes and decreases with the amount of MBD proteins. Lower levels of MBD proteins result in higher μ′ and in our model this leads to faster reactivation kinetics and decreased variability of reactivation time (Fig YD-YF in S1 File). This is consistent with the experimental observations of [54], which show that knocking down of MBD proteins not only makes the reprogramming process faster, with almost the entire cell population converted into iPS cells within 7 days, but makes it also almost deterministic.

Positive TF-enabled autoregulation extends memory of active chromatin state robustly to perturbations

The study of the chromatin modification circuit reported in the previous section shows that the two-layer topology of the repressive modifications biases the system towards a repressed chromatin state, and that this bias becomes even more pronounced for low values of μ′. It follows that, when the chromatin state is in the active configuration, without any activating input stimulus, even a small perturbation can cause a quick silencing of the gene. For an engineered gene expression system, this silencing will disrupt the intended engineered function of the cell line, as observed in experiments with different cell types [16–19]. Here, we study how TF-enabled positive autoregulation can alleviate this problem by restoring memory of the active gene state and by making it more robust to endogenous silencing. A possible form of positive autoregulation is when the gene product X acts as a TF activator that recruits writers of activating histone modifications to the gene. This is the case of many TFs involved in cell fate determination, such as Oct4 and Nanog [55]. The analysis of this design thus not only provides a potential engineering solution to the silencing problem of chromosomally integrated gene expression systems, but it also allows for a different understanding of the role of positive autoregulation in natural networks.

In this case, the input of the chromatin modification circuit is a monotonically increasing function of n^X (Section 1.7 in S1 File). The positive autoregulation model is thus given by combining the chromatin modification circuit reactions (Fig 2) with the gene expression reactions (1), and the expression for (Eq (82) in S1 File). The corresponding interaction diagram is shown in Fig 5A and 5B shows the block diagram representation with input and output n^X.

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Fig 5. Positive autoregulation allows robust temporal extension of memory of the active chromatin state.

(A) Diagram of a positively autoregulated gene where the gene’s product X recruits writers of D^A. A simplified representation of the chromatin modification circuit is introduced, in which the the labels are not represented. (B) Block diagram corresponding to the circuit in panel (A). Here, . Furthermore, is the input and n^X, the number of molecules of X, is the output, which feeds back on by increasing its value. (C) Input/output steady state characteristics for the pair, with , for different values of p_x obtained from simulations of system Eqs (1) in S2 File with as initial conditions. The parameter values are listed in Table A in S2 File. In particular, ϵ′ = 1, μ = 1, μ′ = 0.7, ϵ = 0.1. (D) Stationary probability distribution π obtained by simulating the reactions listed in Table B in S2 File with the SSA. As before, . The parameter values of each plot are listed in Table B in S2 File. In particular, p_x = 0, 0.1, 10, ϵ′ = 1, μ = 1, μ′ = 0.5, ϵ = 0.12 (Figs B and C in S2 File show different parameter values). (E) Time trajectories obtained by simulating the reactions listed in Table C in S2 File with the SSA with no inputs and starting with initial conditions n^A = 45, and n^X = p_x n^A. Simulations are stopped the first time at which n^A = 6. In all plots, time is normalized according to , ϵ = 0.36, μ′ = 0.5, μ = 1, ϵ′ = 1, and p_x = 0, 0.2, 5. The parameter values of each panel are listed in Table C in S2 File. In each panel, the number of trajectories plotted is 10. (F) Time trajectories of system of reactions listed in Table D in S2 File with the SSA starting from the active chromatin state (n^A = 45, n^D = 5 and n^X = 45). At the indicated times, n^X is reset artificially to zero (dashed lines). The parameter values for each simulation are listed in Table D in S2 File. In particular, p_x = 1, μ′ = 0.1, μ = 1, ϵ′ = 1 and ϵ = 0.4, 0.02. For both values of ϵ, the system is bistable. In our model, ϵ, defined in Eq (2), is a non-dimensional parameter that quantifies the time scales of basal erasure rate of all modifications relative to those of auto and cross-catalysis. In the figure, we use green to indicate the activating modification and related quantities.

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Deterministic behavior

Considering again the normalized time , and letting , with n^X the number of molecules of X, , and , , the ODE associated with gene expression in non-dimensional variables can be written as (8)

By assuming molecular counts sufficiently high to use ODEs and by combining the ODEs of the chromatin modification circuit (3) with those of gene expression (8), we obtain the ODE model of the positive autoregulation system (Eqs (1) in S2 File). We first realize bifurcation plots with p_x = α_x/γ_x the bifurcation parameter (Fig A in S2 File). As p_x increases, the system transitions from monostable with one stable equilibrium point corresponding to the fully repressed chromatin state to bistable in which also a fully active chromatin state is a stable equilibrium. For lower values of μ′, higher values of p_x are required to make the fully active state appear as a stable equilibrium. Therefore, increased p_x can compensate, to some extent, for the asymmetry of the system, thus restoring stability of the active state. Varying μ has a similar effect as varying μ′. In order to gain a qualitative understanding of the way positive autoregulation affects the stability of the system, we also conducted a mathematical analysis assuming that protein dynamics are much faster than chromatin modification dynamics. This analysis shows that positive autoregulation has an effect equivalent to increasing the auto-catalysis rate of the activating histone modification. Based on the bifurcation plots realized for the chromatin modification circuit (Fig K in S1 File), a higher auto-catalysis rate constant can restore the active state stability, consistent with the results in Fig A in S2 File.

The strength of positive autoregulation p_x also affects the robustness of the active gene state’s memory to endogenous silencing. Specifically, we introduce in our model possible perturbations that can lead to silencing, due to for example chromatin spreading from surrounding gene loci, by considering . The input/output steady state characteristics () with , shows persistence of the active state to increasing values of u^R as p_x is increased (Fig 5C). These results imply enhanced robustness of the active state to repressive input perturbations.

Stochastic behavior

Here, we determine how positive autoregulation modulates the stationary probability distribution of the chromatin state and the time to memory loss of the active state. To this end, we first perform a computational study by simulating the full set of reactions (Table B in S2 File) with the SSA [51]. This study shows that, increasing p_x, the peak in correspondence to the active state becomes more concentrated and its height increases to the detriment of the peak corresponding to the repressed state (Fig 5D). However, for lower values of μ′, the height of the peak in correspondence to the active state is also lower, implying that a higher value of p_x is required to increase the peak to the same level (Fig C in S2 File). Reducing μ has a similar effect as reducing μ′ (Fig C in S2 File). These results are in agreement with the temporal trajectories of n^A (Fig 5E), which indicate that the time to memory loss of the active state increases when p_x is increased. The trends with which p_x affects the time to memory loss of the active state do not depend on ϵ′ (Fig D in S2 File).

To gain an analytical understanding of how p_x affects the stationary distribution and the time to memory loss, we also computed analytically these quantities as a function of the system parameters in the regime ϵ′ ≪ 1 and ϵ ≪ 1 and by assuming that protein dynamics are fast compared to chromatin dynamics. The latter assumption may be satisfied if the protein is quickly degrading due to proteolysis [56]. In this regime, we obtain a one-dimensional Markov chain reduction of the system (Section 1.3 in S2 File), which leads to the stationary probability distribution for ϵ ≪ 1 as (9) with (10) in which x = n^R, , and defined in Eq (161) in S1 File. From these expressions, we conclude that by increasing p_x, P decreases and, in turn, the height of the peak in correspondence to x = 0 (active chromatin state) increases to the detriment of the height of the peak at x = D_tot (repressed chromatin state). Furthermore, the smaller μ and μ′ are, the higher p_x has to be in order to raise the peak corresponding to the active state. These results are in agreement with the computational results (Fig 5D and Fig C in S2 File).

Using the same model reduction, we derive the formula of the normalized time to memory loss of the active chromatin state, which, for ϵ ≪ 1, can be approximated as (11) in which f₂ and are increasing functions of their arguments, , and are functions independent of ϵ, μ′, μ and p_x (Section 1.4 in S2 File). From this expression, it follows that if p_x increases, increases. This confirms that positive autoregulation helps extend the memory of the active state.

Taken together, these findings indicate that positive TF-enabled autoregulation contributes to extend the duration of memory of an active chromatin state, thus re-balancing the asymmetry that naturally biases the chromatin modification circuit towards a repressed state. This is also manifested in the effect of positive autoregulation on the time to reactivation of a gene, wherein the time to reactivation decreases by increasing p_x (Fig E in S2 File).

We also determined how the chromatin modification circuit contributes to the robustness of the active state’s memory to perturbations that reduce the availability of X at the gene (Fig 5F). For low ϵ the robustness is improved, implying that the system is able to keep the active chromatin state even when X is transiently removed. Then, small ϵ may be a way in which the chromatin modification circuit enables robustness of the active state’s memory to disruption of the binding of the TF X to DNA, such as due to DNA replication and cell division.

Wiring positively autoregulated chromatin modification circuits allows long-term persistence of gene expression patterns

Long-term and reconfigurable memory of multiple gene expression patterns is desirable for a number of applications, such as programmable cell therapies and in vivo transdifferentiation, wherein TFs need to be activated in a specific temporal sequence and in an exclusive fashion [57, 58]. We thus analyze a candidate circuit to implement this function: two positively autoregulated chromatin modification circuits that mutually recruit writers of repressive modifications to each other (Fig 6A). This is also a common motif in GRNs involved in cell fate determination, such as the PU.1/GATA.1 antagonism [35, 59], the Nanog/GATA interaction [6], and the Cdx2/Oct4 circuit motif [6, 60].

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Fig 6. Mutual repression circuit: Robust memory of multiple co-existing gene expression patterns.

(A) Interaction diagram of two mutually repressing and positively autoregulated genes, wherein the product of each gene recruits writers of repressive chromatin modifications to the other gene. Here, X and Z represent the products of the two genes. (B) Block diagram corresponding to the circuit in panel (A). Here, n^X and n^Z correspond to the number of molecules of X and Z, respectively. (C) Stationary probability distribution π of the system obtained by simulating the reactions listed in Tables E and F in S2 File with the SSA, in which n^{A, ℓ} with ℓ = X, Z represents the number of nucleosomes in each gene with activating histone modifications. In (C), p_x = p_z = p with p = 0, 0.1, 10 and ϵ = 0.48, 0.2. The parameter values of each plot are listed in Tables E and F in S2 File. For all simulations we have μ = 1, μ′ = 0.6, and ϵ′ = 1 (Figs G-I in S2 File show different parameter values). (D) Time trajectories of n^{A, X} and n^{A, Z} starting from n^{A, X} = D_tot and n^{A, Z} = 0, in which n^X and n^Z are reset to zero at the indicated times (dashed line). Time is normalized with respect to . The parameter values for each panel are listed in Tables G and H in S2 File. In particular, p = 0.15, μ′ = 0.6, μ = 1, ϵ = 1 and ϵ as indicated. In all plots, we assume equal parameters for both chromatin modification circuits. In our model, ϵ, defined in Eq (2), is a non-dimensional parameter that quantifies the time scales of basal erasure rate of all modifications relative to those of auto and cross-catalysis. In all plots (n^{A, ℓ}, n^{R, ℓ}) ≈ (50, 0) and (n^{A, ℓ}, n^{R, ℓ}) ≈ (0, 50) correspond to the active and repressed state of gene ℓ, with ℓ = X, Z. In the figure, we use green and purple, respectively, to indicate D^A,X and D^A,Z and related quantities.

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For each gene, the chromatin modification circuit is described by the reactions in Fig 2 and the gene expression reactions are listed in (1). In each gene’s chromatin modification circuit, is a monotonically increasing function of the other gene’s product level and is a monotonically increasing function of the gene’s self product level (Section 2.1, Eqs (10) in S2 File). The system is then obtained by input/output connecting two chromatin modification circuits through the TFs that they are expressing (Fig 6B).

Deterministic behavior

By combining the ODEs of the chromatin modification circuit and those of the gene expression circuit (Eqs (3) and (8), respectively) for each gene, by defining X ≔ n^X/D_tot, Z ≔ n^Z/D_tot and by properly setting the inputs according to Eqs (13) in S2 File, we obtain the ODEs of the mutual repression system (Eqs (14) in S2 File). To determine the number and stability of equilibria, we exploit the results obtained for the positive autoregulation circuit. In fact, the block diagram in Fig 6B makes it explicit that the mutual repression circuit is the input/output composition of two positively autoregulated genes, in which the output of one gene, n^X or n^Z, is used as an input to the other gene by increasing its . This analysis, described in Section 2.3 in S2 File, shows that, for low p values and large ϵ, the system has a unique stable steady state about the origin, where both genes are “off”. By increasing p, the system acquires two stable steady states, in which one gene is “on” and the other is “off”. That is, one equilibrium with and the other equilibrium with . When ϵ is small, also the steady state about the origin is stable, while when ϵ is large, it is not. Thus, when p is large (high expression rate) and ϵ is small, the system is tri-stable. When μ′ is increased, the system can have four co-existing stable steady states, acquiring a new stable steady state in which both genes are “on” (Fig F in S2 File). Since we expect μ′ to be much smaller than one, this last scenario may not be observed in practice.

Stochastic behavior

Here, denoting the number of nucleosomes in each gene with activating histone modifications as n^{A, X} and n^{A, Z}, we first determine how p_x, p_z, and ϵ shape the stationary distribution of (n^{A, X}, n^{A, Z}) and hence the admissible gene expression patterns. Specifically, we perform a computational study of the distribution by simulating the full set of reactions (Tables E and F in S2 File) with the SSA [51]. If the production rate constant p of both proteins is low, independent of ϵ, the distribution has one peak only in correspondence to both genes being “off”, consistent with the deterministic analysis. Therefore, the only possible gene expression pattern in this regime is both genes “off”. When p is sufficiently increased, the peak in correspondence of both genes “off” disappears and two peaks arise in correspondence of one gene “on” and the other “off”, thus enabling two symmetric gene expression patterns. There is an intermediate protein production rate constant regime, in which a third gene expression pattern with both genes “off” is also possible. In all cases, decreased ϵ makes the distribution more concentrated about each of the peaks, thereby also reducing the probability of transitioning out of that expression pattern (Fig 6C). The admissible gene expression patterns are largely shaped by the choice of the two protein production rate constants (Fig I in S2 File). Reducing μ′, as before, biases the distribution towards the repressed chromatin state (n^{A, X}, n^{A, Z}) ≈ (0, 0) (Fig H in S2 File) while varying ϵ′ does not change the trends with which the key parameters affect the distribution (Fig G in S2 File). The computational results of Fig 6C are in agreement with those obtained from the analytical study (Section 2.4 in S2 File).

Furthermore, we determine computationally how ϵ affects the robustness of the memory of an admissible gene expression pattern with respect to resetting the amount of the gene’s protein products X and Z to 0 (Fig 6D). Lower ϵ values result into enhanced robustness of the pattern’s memory to resetting n^X and n^Z to zero, indicating that the chromatin modification circuit with lower ϵ helps memory of co-existing gene expression patterns persist through changes in the availability of TFs to the promoters. Finally, we analyzed whether overexpression of TF can transition the system from one gene expression pattern to the other one. Indeed, even when a gene expression pattern displays high robustness to TFs variability (low ϵ), the system allows a transition from one admissible expression pattern to the other one by TF overexpression (Fig J in S2 File). The speed of transition increases with the level of overexpression and decreases with ϵ.