Figure 1.
From regulatory mechanism to Markov Chain.
(A) Schematics of two simplified models for initiation regulation (IR) and elongation regulation (ER). Transcription is represented in 4 steps: (1) naked DNA, (2) DNA-polymerase complex, (3) actively transcribing polymerase, and (4) completed mRNA. The enhancer is either (A) open or (B) bound. The enhancer must be bound (the permissive configuration) for the transcription chain to pass the gated step (), whose identity depends on the model (IR or ER). (B) The corresponding Markov chains for each regulation scheme. Colors of arrows denote the transition rates from (A). Note that one set of rate parameters determines all the numerical values for both chains, allowing for a direct test of the effects of topological change. (C) Distributions of log ratios of speed (
), variance of expression time (
), and transcript count variability (
) across 10,000 randomly chosen parameter vectors (as described in the text), showing that ER is faster, less variable, and produces less variability in transcript numbers over most possible combinations of rate parameters for this simple model.
Figure 2.
Each possible complex in the process is enumerated as a state of the promoter Markov chain. (see text for description of each complex) The promoter chain (states 1–8) is combined with the enhancer chain (states A and B) to make the full 16 state model of transcription. Transitions that in some scheme require an activated enhancer (state B) are indicated by a gate, . Forward rate transitions are in light font and backward transitions in dark font. The
transition is regulated in the IR scheme, and the
transition is regulated in the ER scheme.
Figure 3.
(A) Probability distributions for first passage times: Probability density functions of the time to first transcription, obtained by inversion of symbolically calculated Laplace transforms, using rate parameters computed in experimental studies of particular transcription systems. Rates inferred from Darzacq et al. [17] measurements of promoter binding and promoter escape rates (see text). (B) Distribution of total transcripts among a population of simulated cells during 600 minutes of transcription under the ER model with parameters as in (A) and a reinitiation probability of 0.8. (C) as in (B) but for the IR model. (D) Individual cell simulation (see text) showing of the expected results for an mRNA counting assay on the population of cells plotted in (B). Each mRNA transcript is represented by a red dot randomly positioned within the cell. Cells with less than two-thirds of mean mRNA concentration are shaded blue, cells with more than three-halves of mean mRNA concentration are shaded red. (E) as in (D) but for the IR scheme.
Figure 4.
(A) Comparison of log ratios of mean expression speed for the IR/ER schemes for 10,000 uniformly sampled rates. For all jump rates, the log ratio is positive (red line), indicating the ER scheme is always faster. Extreme values that would be off the edge of the graph are collected into the outermost bins. (B) Variance in timing of expression. (C) ratio of noise in transcript number, measured by the squared coefficient of variation between cells of total mRNA counts
up to time
:
– the ratio is approximately independent of
.
Figure 5.
Effect of scaffold stability for variation in transcript number.
(A) ratio of transcript variability,
, between the IR and ER model when all subsequent polymerases engage an assembled scaffold
. Extreme values that would be off the edge of the graph are collected into the outermost bins. (B) As in (A) when
, note the ER scheme is more often substantially more coordinated, though a few parameters still make the IR scheme the more coordinated by a smaller margin. (C)
. (D)
.
Figure 6.
Sensitivity analysis for mean expression time.
Histograms of the marginal distributions of relative sensitivities for both the ER and IR schemes, across uniform random samples from parameter space, as described in the text. The smallest bin of the histogram (values below ) is disproportionately large, and so is omitted; shown instead is the percent of parameter space on which the relative sensitivity is at least
. Note that often only a single parameter dominates (many sensitivities are near 1), that many parameters are almost never influential, and that ER and IR are similar except for the addition of sensitivity to
.