Fig 1.
Polymerases Pi − 1, Pi, and Pi+1 in order on the DNA strand.
When Pi translocates, the DNA between Pi − 1 and Pi will over-twist and the DNA between Pi and Pi+1 will under-twist, increasing the elongation velocity of Pi − 1 and Pi+1.
Fig 2.
As a function of the initiation rate α ⋅ β, we present the average transcription time (A) and the average total delay per RNAP (B).
The ETAM model (blue triangles) and TASEP model (magenta) are both plotted for comparison. The dashed lines represent baseline simulations with no pauses. The error bars for the transcription time represent the standard deviation, with the standard deviation on the ETAM model decreasing as α increases.
Fig 3.
As a function of increasing initiation rates determined by α, the pause and collision results are presented for the ETAM model (blue triangles) and the TASEP model (magenta).
The number of pauses and collisions are computed as the average number per RNAP. The average number of pauses per RNAP is presented in (A) with the average pause duration in (B). Similarly the average number of collision per RNAP is given in (C) with the average collision duration in (D). RNAPs in the ETAM model experience significantly fewer collisions and shorter pause durations than their TASEP counterparts.
Fig 4.
Linear fit for the average number of collisions experienced per RNAP.
The collisions in the results for TASEP (magenta) and compared to the linear fit (magenta stars) and similarly for the ETAM results (blue and blue stars).
Table 1.
Results for ETAM model: percent of the strand covered by polymerases, average transcription time (s) per RNAP, average collision delay (s) per RNAP, average pause delay (s) per RNAP, and average total delay (s) per RNAP.
Table 2.
Results for TASEP model: Percent of the strand covered by polymerases, average transcription time (s) per RNAP, average collision delay (s) per RNAP, average pause delay (s) per RNAP, and average total delay (s) per RNAP.
Fig 5.
As a function of increasing initiation rates determined by α, the coefficient of variation (A), and the variance to mean ratio (B), are given for the transcription time. Both ETAM and TASEP models are presented as well as their baseline results (no pauses). The variance of the pause duration (C), and collision duration (D), are presented for the ETAM model (blue triangles) and the TASEP model (magenta).
Fig 6.
The portion of the cylinder that is a dashed line represents the original distance L0 and the current distance L. As the distances decreases from L0 to L, the total amount of twist added to the rod is ϕ. The small increment Δℓi represents a change in length due to RNAP motion by one nucleotide.
Table 3.
Parameters used in calculation of torque.
Fig 7.
This figure depicts the four different cases of RNAP Pi − 1, Pi, and Pi+1.
Fig 7A shows the three RNAPs with the original distances between them. Then there are four different configurations; Fig 7B, L0,i > Li and L0,i+1 > Li+1, Fig 7C, L0,i > Li and L0,i+1 < Li+1, Fig 7D, L0,i < Li and L0,i+1 > Li+1, and Fig 7E, L0,i < Li and L0,i+1 < Li+1.
Fig 8.
Sketch of the general functional behavior of the mean field velocity prediction of the torque model.
Fig 9.
The data published by Ma et. al. is presented (red dots) as well as the curve fit to the data (black).
The values that we choose based on biological information for very high and very low torque values are shown also (green dots). (A) shows the velocity as a function of torque, while pause frequency and pause duration as a function of torque are given by (B) and (C) respectively. The equation for each curve can be found in Eqs (19)–(21), respectively.
Fig 10.
The data published by Ma et. al. is presented (red dots) as well as the piecewise linear curve fit to the data (green).
The nonlinear polynomial fit is provided (blue) with our imposed values (green dots) for comparison purposes. As in Fig 9, the velocity is shown in (A), the pause frequency is presented in (B), and the pause duration is in (C). The equation for each piecewise linear curve can be found in Eqs (22)–(24), respectively.
Fig 11.
A flowchart for our simulation of the elongation process.
RNAPs that translocate follow the chart from elongation and continue in order depending on if that RNAP experiences a collision or a pause. If a neighboring RNAP translocates, updates for torque, velocity, and pause frequency are calculated for the affected RNAPs starting with that block of the flowchart.
Fig 12.
For the range of initiation rates α ⋅ β, we present our results for average transcription time (A) and average total delay (B) per RNAP in the ETAM model.
The nonlinear model (blue triangles) and piecewise linear model (green dots), detailed in the sections Incorporating Experimental Data into the Model and Linear Fit to the Data respectively, are both plotted for comparison.
Fig 13.
In addition to the nonlinear fit (blue triangles) and the piecewise linear fit (green dots), we also present our results for average transcription time (A) and average total delay (B) per RANP for a piecewise linear fit except for the pause duration on the interval [-10, 5], “Nonlinear Left” (red stars), or on the interval [5, 10], “Nonlinear Right” (black stars).
Fig 14.
The differences in the pause duration for the piecewise linear fit (green) and the nonlinear fit (blue) at low torque values is highlighted here.
This difference accounts for the very different results in average transcription time and average total delay depicted in Fig 12.
Fig 15.
Pause frequency function where the value at the end point, 11 pN⋅nm is raised.
The data (red dots) is fit with one quadratic function up to 7.5 pN⋅nm. A second quadratic function is used to fit the end point, with the exception of the lowest function which is the original quadratic for the entire interval.
Fig 16.
For the range of initiation rates α ⋅ β, these results show average transcription time (A) and average total delay (B) over different fits of the pause frequency function.
The legend labels the pause frequency at the highest torque value.
Table 4.
The average number of pauses per RNAP, average pause duration, and average pause delay per RNAP due to pauses experienced per RNAP for the different pause frequency functions.
Frequency represents the value of the pause frequency function when torque is 11 pN⋅nm. All of the data reported is for α = 0.0115.
Fig 17.
The percentage of times a torque value is calculated in a baseline (blue) simulation and a pause (red) simulation is presented here as a histogram.
The histogram bars at -10 and 10 represent the percentage of the total number of torque measurements within the simulation that those exteme values are computed. Each of the other bars represents the percentage total measurements that lie within (-10, -9], (-9, -8], etc., with the exception of the bars at the label <10. The histogram bars at <10 represent the percentage of torque values computed in (9, 10). (A) shows the torque values measured for α = 0.0001 which corresponds to the lowest initiation rate simulated, while (B) shows the torque values measured for α = 0.0115 which corresponds to the highest initiation rate simulated.