Fig 1.
The gene’s inner chromatin modification circuit.
(A) Nucleosome modifications: D (unmodified nucleosome), DA (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 DA 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 KMTR and the writers for H3K4 methylation (SETs and MLL1/2) as KMTA. Similarly, to distinguish the HDMTs for H3K9 methylation and H3K4 methylation, we define the erasers for H3K9 methylation (JMJD2A) as HDMTR and the erasers for H3K4 methylation (JARID) as HDMTA. 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.
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).
Fig 3.
Time scale parameters ϵ, ϵ′, μ and μ′ control bistability and hysteresis in the chromatin modification circuit.
(A) Diagram of the gene’s inner chromatin modification circuit in which, compared to Fig 1D, we removed the lines and arrows indicating recruitment and catalysis. The labels on each arrow specify the processes enabling that nucleosome modification as indicated below the reaction diagram. We use purple labels for repressive modifications and blue labels for activating modifications. A visual representation of the relationships between the rates of these processes and the parameters ϵ, ϵ′, μ and μ′ is provided. In particular, 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, 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. For the mathematical definition, refer to Eq (2) and the related text. (B) Block diagram corresponding to the chromatin modification circuit. Here, nA, ,
and
denote the numbers of modified nucleosomes DA,
,
, and
within the gene,
and
. The pair
are the inputs and (nR, nA) are the outputs. (C) Steady states of the system as a function of ϵ, μ and μ′. Here,
and
are the fractions of nucleosomes with activating or repressive modifications within the gene with a total of Dtot nucleosomes. Plots are obtained from system (3) with
. The solid lines represent stable steady states, the dashed lines represent unstable steady states and the black circle represents the bifurcation point (saddle-node bifurcation). In these plots
and all the other parameters are set equal to 1 (Fig K in S1 File shows different values). (D) Chart depicting the (ϵ, μ′) combinations that result in a monostable (red, green or white) or bistable (yellow) system for μ = 10 (Fig L in S1 File shows different values of μ). Here, ϵ = 1. (E) Input/output steady state characteristics displaying hysteresis for the
and
pairs, with
, for different values of ϵ obtained from simulations of system (3). We consider
as initial conditions and we set uR = 0,
,
, ϵ = 1, μ = 1 and μ′ = 0.8 (Fig M in S1 File shows different values of μ, μ′ and ϵ′). (F) Input/output steady state characteristics for the
pair, for different values of μ′ obtained from simulations of system (3). We consider
as initial conditions and we set uA = 0, ϵ = 0.07,
,
, and all the other parameters equal to 1 (Fig OA in S1 File shows the
steady state characteristics for the same parameter values). (G) Input/output steady state characteristics for the
pair, for different values of ϵ′ obtained from simulations of system (3). We consider
as initial conditions and we set uR = 0, ϵ = 0.15, μ = 1, μ′ = 1,
and all the other parameters equal to 1 (Fig OB in S1 File shows the
steady state characteristics for different values of ϵ′). In all plots
,
for i ϵ {1, 2},
corresponds to the active state and
corresponds to the repressed state. In the figure, we use green and red, respectively, to indicate the activating and repressive modifications and related quantities.
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 (nA, nR) 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, uA = 0, 1, and in the bottom plots and uR = 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 nR starting from a repressed chromatin state nA = 5,
, as ϵ and μ′ are varied. Simulations are stopped when nR = 6 for the first time. (D) Time trajectories of nA starting from an active chromatin state nA = 45,
, as ϵ and μ′ are varied. Simulations are stopped when nA = 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 nR = 45, nA = 5 and applying an input uA that, at steady state, leads to a unimodal distribution in the proximity of the active state nA = Dtot. The parameter values are listed in Table C in S1 File. In particular, uA = 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 (nA, nR) ≈ (50, 0) corresponds to the active state and (nA, nR) ≈ (0, 50) corresponds to the repressed state. In each panel of (C), (D), and (F), the number of trajectories plotted is 10.
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 DA. 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 nX, 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 px 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, px = 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 nA = 45,
and nX = px nA. Simulations are stopped the first time at which nA = 6. In all plots, time is normalized according to
, ϵ = 0.36, μ′ = 0.5, μ = 1, ϵ′ = 1, and px = 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 (nA = 45, nD = 5 and nX = 45). At the indicated times, nX is reset artificially to zero (dashed lines). The parameter values for each simulation are listed in Table D in S2 File. In particular, px = 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.
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, nX and nZ 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 nA, ℓ with ℓ = X, Z represents the number of nucleosomes in each gene with activating histone modifications. In (C), px = pz = 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 nA, X and nA, Z starting from nA, X = Dtot and nA, Z = 0, in which nX and nZ 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 (nA, ℓ, nR, ℓ) ≈ (50, 0) and (nA, ℓ, nR, 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 DA,X and DA,Z and related quantities.