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Epigenetic cell memory: The gene’s inner chromatin modification circuit
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.
doi: https://doi.org/10.1371/journal.pcbi.1009961.g003