D.2 TPM data acquisition and analysis
D.2.4 Minimum number of trajectories and minimum observation time
and above the sticking-state threshold. For traces with well-separated and well-populated states, the looping probabilities calculated by this thresholding method were comparable to those calculated by the Gaussian fitting method; where they differed, we believe the thresholding method to better represent the behavior of the trajectory. Therefore all looping probabilities for the constructs in Fig. 4.1(F), Fig. 4.2, Fig. 4.4, and Chapter 6 were obtained by this thresholding method. As with the Gaussian fitting approach, where this thresholding method was used, it was done on a tether-by- tether basis and the mean looping probabilities and standard errors on these means for a population of tethers were reported.
0 0.2 0.4 0.6 0.8 1 0
0.2 0.4 0.6
Frequency
0 10 20 30 40
0 0.2 0.4 0.6 0.8 1
Number of Beads
Mean Looping Probability
0 10 20 30 40
0 0.1 0.2 0.3
Number of Beads
Standard Error
Looping Probability
(A) (B) (C)
Figure D.1: Distribution of looping probabilities obtained with Oid-E894-O1 at 10 pM repressor, as representative of the issues discussed in Sections D.2.4 and D.2.5. (A) Histogram of looping probabilities of all trajectories that lasted at least 3000 seconds, normalized by the total number of tethers included in the distribution. Black vertical dashed line is the mean of the distribution; solid black vertical line is the mean not including tethers that have zero looping probability (the “nonloopers” discussed in Section D.2.5). (B) Mean looping probability as a function of the number of tethers included in the mean, resampled with replacement 104 times per number included (that is, there are 104 blue points per x-value). Solid horizontal black line is the mean of the distribution that includes all 39 tethers in the distribution in (A) (so it is the same as the vertical dashed line in (A)); dashed lines here indicate the mean plus or minus the standard error of the distribution with all tethers. (C) Standard errors of the resampled distributions whose means are plotted in (B); that is, there is a red “x” for every blue circle. Horizontal dashed line indicates the standard error with all tethers included; green curve is the standard deviation of the 104blue points at each x-value tested (so it is a measure of the vertical spread of the blue points in (B)).
are used to calculate the mean, the mean fluctuates wildly between resamplings (indicated by the vertical spread in the blue points); but as the number of tethers included in the mean surpasses 20, the spread in the blue points remains constant, suggesting we need about 25 beads to accurately calculate the true mean. Similarly, as the number of trajectories included exceeds 20–25, the standard error of the distribution decreases until it reaches a constant value, which is the error associated with intrinsic tether-to-tether variation in looping probability and will not decrease with more data.
We found this number of 20–25 tethers to be consistent across repressor concentrations and DNA constructs, and so all reported means and standard errors in this work are obtained from sample sizes of at least 20 tethers.
As mentioned above, we asked not only how many trajectories were needed, but also how long each trajectory needed to be in order to be included in the analysis. As discussed in the next section, we considered several schemes for calculating the mean looping probability for a population of tethers. Most of these methods, and the analysis in [115], involved weighting each tether’s looping probability equally in the calculation of the population mean (the other strategy, discussed below, is to weight each tether’s looping probability by the observation time). For the schemes in which each tether was weighted equally, it is important to include in the population mean only those that
100-15 10-13 10-11 10-9 10-7 10-5 0.2
0.4 0.6 0.8 1
LacI Concentration (M)
Looping Probability
>1000 sec
>2000 sec
>3000 sec
>4000 sec
>5000 sec
(A)
100-15 10-13 10-11 10-9 10-7 10-5
0.2 0.4 0.6 0.8 1
LacI Concentration (M)
Looping Probability
>1000 sec
>2000 sec
>3000 sec
>4000 sec
>5000 sec
(B)
Figure D.2: Looping probability and observation time, method one. Means and standard errors of looping probability distributions as a function of minimum observation time for(A)Oid-E894-O1 and(B)Oid-TA94-O1. The legends indicate the minimum observation time for a bead to be included in the corresponding data point; for example, “>1000 seconds” means all trajectories at least 1000 seconds long. For the majority of these points it does not matter how strict the minimum observation time is—all of the means are within error of each other at all concentrations. Note that these data have not had “nonloopers” subtracted (Section D.2.5).
have been observed long enough to obtain an accurate measure of their looping probability.
We took two approaches to analyzing the minimum observation time. The first is analogous to the approach used to determine the minimum number of trajectories: the mean and standard error for sets of looping probabilities with successively stricter minimum observation requirements were calculated, to determine at what cutoff the mean and standard error of the distributions converges to a constant (Fig. D.2). We found these calculated means and standard errors to be surprisingly insensitive to minimum observation time, except at very low repressor concentrations where looping events are rare. We therefore considered a second approach, in which we considered each tether’s trajectory individually, and asked how its individual mean looping probability varied as data were removed from the end of the trajectory (Fig. D.3). Here again we found the looping probabilities for individual tethers to be surprisingly insensitive to observation time; where they did vary, an ideal minimal observation time (above which a tether’s looping probability generally remained constant) was about 3000 seconds. Therefore all tethers included in the analyses in this work lasted at least 3000 seconds.
0 2000 4000 6000 8000 10000 0
0.2 0.4 0.6 0.8 1
Time (seconds)
Looping Probability
10000 2000 3000 4000 5000 6000 7000 0.2
0.4 0.6 0.8 1
Time (seconds)
Looping Probability
(A) (B)
Figure D.3: Looping probability and observation time, method two. The looping probabilities of individual beads are shown as a function of how much of the trace is included in calculating the looping probability, for Oid-E894-O1 at (A)500 fM LacI and (B) 500 pM LacI. Each blue line corresponds to a different bead. Each bead’s looping probability was calculated by Gaussian fitting (Section D.2.3) including successively more 500 second intervals of its total trajectory, starting at 1000 seconds, then 1500, etc., until the bead breaks (blue points). Solid red line is the mean of all the beads’ looping probabilities at each time interval; dashed lines are the mean plus or minus the standard error. Note that in neither (A) nor (B) does the mean include more than 20 beads past about 5500 seconds, and is therefore unreliable past this point as an accurate estimate of the true looping probability (see previous section). Note also that to eliminate user oversight, the Gaussian fit parameters for each bead’s total trajectory were used as the starting point to this analysis, and at each other time point, only the amplitude of the Gaussian fit to each (looped or unlooped) state, not the mean value or the width, was refit; therefore in a small number of cases the re-fitting was suboptimal and the looping probability should be considered an estimate in all cases. There is no clear minimum observation time after which each bead’s looping probability remains constant, though it is clear that 1000–2000 seconds is too short a minimum, and it is also clear that lower repressor concentrations (e.g., (A)) require longer observation times before the majority of beads’ looping probabilities cease to change dramatically. Note that the red line, the mean looping probability over all beads, remains roughly constant, even at 1000–2000 seconds.