D.2 TPM data acquisition and analysis
D.2.2 Particle tracking and calculation of the root-mean-squared motion of the bead 163
As in [115], beads were tracked with custom Matlab code by cross-correlating each frame with the initial frame in a time series on a bead-by-bead basis. This results in “raw” x and y positions of each bead, relative to the tethering point, as a function of time. Drift was removed from these raw data as in [115], by subtracting the results of a low-pass first-order Butterworth filter with a cutoff frequency fcB = 0.05 Hz for the 0.49 µm beads or fcB = 0.07 Hz for the 0.27 µm beads. The root-mean-square motion was obtained by applying a Gaussian filter with a -3 dB frequency offcG
= 0.0326 Hz for the 0.49µm beads orfcG= 0.461 Hz for the 0.27 µm beads, corresponding to a 4 second or 2.8 second standard deviation of the filter, to the mean-squared displacement of the data (that is, to the quantity (~x2+~y2)). The root-mean-squared (RMS) motion of the bead is then the square root of the result of the convolution of (~x2+~y2) and the Gaussian filter.
More precisely, filters were applied in Fourier space (so that convolutions become simple mul- tiplication). This means for both the drift-correction and the Gaussian filter smoothing, we first Fourier transformed the data (which in the case of the Butterworth, applied separately to thexand y position coordinates, means we Fourier transformed the raw ~xand ~y position data; whereas the Gaussian filter is applied to the transform of (~x2+~y2)), multiplied by the appropriate filter, and then inverse Fourier transformed the result. In frequency space the Butterworth filter takes the form
B(f) = 1
(1 + (f /cfB)2n), (D.1)
wheref is frequency,n= 1 so that this is a first-order filter (which determines the sharpness of the transition at the cutoff frequency), andcfB is a rescaled cutoff frequency of the filter based on how we define our frequency axis. We choose to establish our frequency axis from −num. frames/2 to
+num. frames/2, where “num. frames” is the number of image frames in a trajectory. If we want a cutoff frequency for the filter atfcB= 0.05 Hz, then we must define
cfB= num. frames f ps×fc−1
, (D.2)
wheref psis the frame rate of the camera (30 Hz in our case). This is essentially a unit conversion, since our frequency axis is unitless but fc is in Hz. Note that now the units work out, because the Matlab command that Fourier transforms the data (the fft function) returns a vector the same length as the input vector, in frequencies from 0 tof ps/2.
In the case of the Gaussian filter, the filter has the form in Fourier space of
G(f) =e−0.3466(f /cfG)2. (D.3)
The factor of 0.3466 in the exponent is chosen to give 3 dB of attenuation at the cutoff frequency [158]. That is, whenf =cfG, the attenuation is half (3 dB corresponds to a change in power ratio of a factor of 2). For this to be the case, we must have a pre factor in the exponent of ln 2/2 = 0.3466.
As with the Butterworth filter, here we also have the problem of needing a unit conversion between fcG, which is in Hz, andcfG. This conversion is the same as in Eq. (D.2) because we define thef axis for the Gaussian filter in the same manner as for the Butterworth.
Finally we note that the Fourier transform of a Gaussian is a Gaussian, so in time space the Gaussian filter defined in Eq. (D.3) becomes
g(t) = 1
√2πσge−
t2 2σ2
g, (D.4)
whereσg defines the width of the filter and is related tofcGby [158, 120]
σg=
√ ln 2
2πfcG. (D.5)
Therefore the 0.0326 Hz Gaussian filter we apply to most of our data corresponds to a Gaussian- shaped smoothing profile with a 4 second standard deviation in time space, and the 0.461 Hz filter applied to the smaller beads results in a 2.8 second smoothing window. The choice ofσg is directly related to the temporal resolution of our analysis, as discussed in Section 5.A.2.
D.2.3 Determining the looping probability for each trajectory
For the constructs of Chapter 3 and Fig. 4.1(D–E) in Chapter 4 (the constructs with 94 bp loops, which have primarily only the “middle” looped state; or in Chapter 3, the PUC306 construct), data for each tether were histogrammed separately and fit to one (all looped or unlooped), two (unlooped and one looped state), or three (two looped states and an unlooped state) Gaussians.
The looping probability was determined as the area under Gaussian(s) corresponding to the looped state(s) divided by the sum of the areas under all the Gaussians. This 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.
However, for the predominantly three-state DNAs in Fig. 4.1(F), Fig. 4.2, Fig. 4.4, and Chapter 6 (excluding the three-operator constructs), the two looped states were often not well described by Gaussians (see Figs. E.1 and E.2 for examples). We therefore investigated a thresholding approach to calculating each bead’s looping probability [158, 113, 109, 104, 111]. Threshholding was performed subsequent to the Gaussian-fitting method described above. The intersection points of the fitted Gaussians were used to identify initial threshold values, which were adjusted manually as needed, such that the thresholds split the trajectories into the unlooped state and any looped state(s). A threshold above which data were excluded was set to the mean RMS of the tether in the absence of repressor plus three times the standard deviation of the no-repressor RMS; data above this point were usually due to tracking errors or free beads in solution temporarily entering the field of view.
An empirically determined lower bound was set at 80 nm to exclude sticking events. The looping probability was then determined as the number of data points between the thresholds that delineated looped state(s), divided by the total number of points in the trajectory below the topmost threshold
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.