Fig 1.
Extension of the canonical TASEP to include nucleosomes as dynamic defects.
(A) Depiction of the canonical TASEP. Particles are injected into the first lattice site at rate α and advance with rate q = 1 when unobstructed by other particles. Particles are removed from the final lattice site at rate ß. (B) The canonical TASEP phase diagram in the αß-plane with the fundamental relationships for J, ρ, and v summarized for each phase. (C) Simulated density profiles for each phase are shown by black lines, and dashed red lines indicate the mean-field theory prediction obtained for an infinitely long lattice. (D) Depiction of the proposed dynamic defect TASEP (ddTASEP) where periodically spaced nucleosomes function as extended-body dynamic defects. RNAPII bind the first lattice site (transcription start site, TSS) at rate α, RNAPII advances at rate q when unobstructed by other polymerases or nucleosomes. RNAPII is then released from the gene at the end of transcription at rate ß, which is set equal to q (ß = q). DNA in the nucleosome associated regions (containing 3 subsequent sites) unwraps from the nucleosome to the open conformation at rate ho and returns to the wrapped/closed state at rate hc when there are no polymerase present on the nucleosome associated sites. All lattice sites are 50 bp long.
Fig 2.
Qualitative behavior of the ddTASEP over a range of representative parameters.
All sub-figures are composed of 9 subplots. For all plots, the transcription initiation rate α was set to 0.05, while ho is held constant along rows (increasing from 0.001→0.01→0.1 from top to bottom) and while hc is held constant along columns (increasing from 0.001→0.01→0.1 from left to right). The top right subplot deviates the furthest from the canonical TASEP behavior while the bottom left most closely resembles ideal TASEP behavior. Simulations shown in panels A, C, D and F utilized 4.5×105 Monte Carlo Steps with 50 replicates. (A) Number of completed mRNAs as a function of time. Notice that the simulation times for the top, middle, and bottom rows are 60, 8, and 2 hours, respectively. (B) Kymographs plotting the position of each RNAPII on the gene (x-axis) as time advances (y-axis). (C) Time averaged RNAPII site density ρi as a function of position. The wavy density profiles are caused by the fact that the linker sites have a higher average RNAPII occupancy than the neighboring nucleosome sites. (D) Time averaged nucleosome site density ρN,i (black line) as a function of position (ignoring linker sites). The bulk nucleosome density is anti-correlated with the polymerase density, reaching a maximum at the end points where polymerases are rapidly ejected. The dotted red lines indicate the resting-state nucleosome density prior to transcriptional initiation. (E) Inter-site correlation heatmaps with colors bars ranging from -0.2 to 1. (F) Average inter-site correlation profile (rΔ, black line) as a function of relative inter-site distance Δ with the dotted red line indicating zero.
Fig 3.
Deriving and validating the mean and variance of first passage time equations.
(A) A state transition diagram utilized to calculate the mean (Ee) and variance (Ve) of first passage time to enter a nucleosome. Each arrow is labeled with the rate constant for the transition between the neighboring states. (B) Log-log plot of simulated mean first passage time (Eq 22) against the predicted analytical expression for mean first passage time. The black line represents the theoretical prediction. The dashed red line represents the Bare DNA limit obtained by simulations with hc = 0. (C) Log-log plot of simulated variance of first passage time against the predicted analytical expression for VFPT (Eq 26) with the Bare DNA limit represented by the dashed red line. (D) Plot of index of dispersion of the waiting time to enter a nucleosome De = Ve/Ee against the resting state binding probability of the nucleosomes (). hc is set to be 0.001 (blue), 0.01 (green), 0.1 (orange), and 1 (red). ho was adjusted to achieve γ∈{0.002, 0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 0.99}. Solid lines indicate the theoretical prediction given by Eq 27. (E) Log-log plot of the analytical expression for the index of dispersion of the waiting time to enter a nucleosome De (Eq 27, analytical) as a function of the expected waiting time for nucleosome opening (1/ho). The dashed black line shows the index of dispersion in the limit of ho→0.
Fig 4.
The ddTASEP phase diagram in the αγ-plane.
(A) Phase diagram in the αγ-plane (orthogonal to the original αß-plane in Fig 1B at ß = q = 1). The critical line (Eq 37) divides the phase diagram into two regions: the initiation-limited and the nucleosome-limited For γ→1, the ddTASEP’s dynamics converge to the classical Max Flux Limit shown in Fig 1B. (B) The theoretical predictions (top row) for the bulk elongation rate v (Eq 38), bulk density ρ (Eq 39), and bulk transcription flux J (Eq 40) are qualitatively compared to simulated results for hc = 0.1, 0.01, and 0.001 (2nd through 4th rows) Mean first passage rate, γ was varied from between 0 and 1 by specifying hc (for given ho) to achieve the desired value. 1.2 million simulation steps were used for each heat map point.
Fig 5.
Dependence of the elongation rate v, polymerase density ρ, and transcription flux J on the initiation rate α and the mean first passage rate γ.
The left panels A, B and C show the dependence of v (Eq 38), ρ (Eq 39), and J (Eq 40) on γ. γ is varied from 0.001 to 0.99 by adjusting ho for fixed values of hc equal to 0.001 (blue), 0.01 (green), 0.1 (orange), and 1 (red). For each subpanel of A, B, and C, a different α values is assumed: 0.1 (top left), 0.25 (top right), 0.5 (bottom left), 1 (bottom right). The right panels D, E and F show the dependence of v (Eq 38), ρ (Eq 39), and J (Eq 40) on α. α is varied from 0.001 to 1 for fixed values of γ equal to 0.1 (blue), 0.2 (green), 0.4 (orange), and 0.7 (red) under the constraint that ho = hc. Data points represent the average simulation results from 50 simulations with 2×106 Monte Carlo steps each, and solid lines represent theoretical estimates.
Fig 6.
Evaluation of the effects of gene length and nucleosome spacing on bulk transcriptional properties.
Panels A (v), B (ρ), and C (J) show the results of a parameter sweep with respect to gene length from 2,000 to 20,000 bp with α = 0.1 (top left), 0.25 (top right), 0.5 (bottom left), and 1 (bottom right), and γ = 0.1 (blue), 0.2, (green), 0.4 (orange), and 0.75 (red). Panels D (v), E (ρ), and F (J) show the results of a parameter sweep with respect to the number of evenly spaced nucleosomes (by varying number of linker sites) giving NNuc∈{1, 4, 8, 16, 20, 25, 40, 50, 80, 100} with α = 0.1 (top left), 0.25 (top right), 0.5 (bottom left), and 1 (bottom right). Since γ depends on geometry, ho and hc were set equal to 1 (blue), 0.1 (green), and 0.01 (orange). Both sweeps were performed with 2×106 Monte Carlo steps and 50 replicates.
Fig 7.
Investigation of the ddTASEP Inter-Burst Interval and Average Burst Size.
(A) Representative smoothed kernel density estimates of log10(tw) weighted by log10(tw) using same parameter sets as Fig 2. (B) Empirical cumulative distribution functions of waiting times (black line) from ddTASEPs with the same parameters as in Fig 2. The time-headway cumulative distribution function is overlaid for each plot (Eq 42). The red dashed time headway distributions have robust bursting with an effective density ρeff = 0.5. In contrast, the green dashed lines indicate initiation limited cases where unimodality was observed in Fig 7A, and ρ is given by Eq 39. (C) shows the average inter-burst interval plotted as a function of Eh = 4/γ. (D) shows the average inter-burst interval
plotted as a function of index of dispersion of the first passage time to enter a nucleosome De. (E) shows burst size
plotted as a function of mean first passage rate γ with the solid lines proportional to
. Panels (C), (D), and (E) were generated with Fig 5A–5C. γ is varied from 0.001 to 0.99 by adjusting ho for fixed values of hc equal to 0.001 (blue), 0.01 (green), and 0.1 (orange) omitting cases that failed to show bimodality.
Fig 8.
Bursting in the low initiation rate regime.
(A) shows the average inter-burst interval as a function of α with the headway distribution prediction for the average waiting time
given by a dashed line. (B) shows the average burst size
plotted as a function of α. Panels (A) and (B) were generated with Fig 5D–5F with α∈[0.001, 1] with hc = ho∈{0.0143, 0.033} to achieve γ∈{0.1, 0.2}. (C) shows that the variance of burst size
(simulated) is equal to the variance of a geometric distribution given by Eq 46 and is colored by −log10 α. (D) compares the mean inter-burst interval and the variance of the inter-burst intervals to establish that the waiting time distribution is exponentially distributed using the data from Fig 5A–5C and is colored by −log10 γ. (E) and (F) show representative burst size probability mass functions generated from geometric distributions with means given by
= 3.62 and 1.49 associated with the parameter sets (α = 0.01, γ = 0.2) and (α = 0.002, γ = 0.2).
Fig 9.
Effects of static defects and nucleosome rate constant variability on RNAPII density profiles.
All subpanels have hc increasing from 0.001 to 0.1 from left to right and ho increasing from 0.001 to 1 from top to bottom as in Fig 2. 50 replicates of NMCS = 2×106 Monte Carlo steps are performed for all cases. All plots list the simulated flux J, the theory prediction for the unperturbed system JNorm, and the theory prediction based on the slowest site Jslow. The left column (A, C) contains simulations at a low initiation rate of α = 0.05 while the right column (B, D) contains simulations at α = 0.25 which is saturating for the slow sites. The top row (A, B) shows density profiles from a gene with static defects representing a CpG Island (32 sites) on the promoter and 7 exons (8 sites) with a static defect advance rate of qi = 0.5q on these sites with additional GC content heterogeneity given by Eq 48 leading to a minimum advance rate qs = 0.43. The bottom row (C, D) shows density profiles from the same gene, but the variability is introduced through the rate constants ho,m and hc,m. GC content heterogeneity is introduced via Eqs 49 and 50. On the CpG Island and exon sites, hc,m is increased by 25%, and ho,m is decreased by 25%. With the addition of GC content heterogeneity, the maximum observed max(hc,m) was 27% higher than the average, and the minimum observed min(ho,m) was 26% lower than the average.