2.3 Results and Discussion
2.3.1 End-grafted DNA Molecules in the Absence of Shear
Center-of-mass position
The measured center-of-mass position for a particular DNA molecule is shown as a function of time in the x and y directions in Figure 2.22. The uctuations of the molecule around its grafted point are caused exclusively by Brownian motion of the molecule. The random walk of this tethered molecule is shown in Figure 2.23 by plotting yCM versus xCM with time t as a parameter.
0 5 10 15 20 25 30
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1 1.2
Time [s]
x CM, y CM [µm]
xCM yCM
Figure 2.22: Fluctuation of the center-of-mass position of a particular DNA molecule around its grafted point in the x and y directions as a function of time (0 < t < 28s).
−1 −0.5 0 0.5 1
Figure 2.23: Random walk of the molecule around its grafted point caused by Brownian motion.
Representing the exemplary data shown in Figure 2.22 and 2.23 in the form of histograms reveals the probability distributions of the x and y positions. As can be seen from Figure 2.24 these histograms for xCM and yCM approximate Gaussian distributions. This is as expected for a ran-dom walk.
Mean square uctuation of the center-of-mass position of a sin-gle DNA molecule
The mean square uctuation of the center-of-mass position of the molecule is given by
RCM = p
hx− hxii2 + hy − hyii2 (2.91) where the averages are performed over all images of a particular molecule in a given movie. Repeating this procedure for 15 dierent molecules a value
RCM = (0.49±0.11)µm (2.92) was obtained, where the error indicates the standard deviation between the molecules. As discussed in chapter 2.1.3 for a semiexible polymer RCM can be calculated using equation 2.29. Assuming a polymer with
−1 −0.5 0 0.5 1
Figure 2.24: Histograms of the center-of-mass position for a particular DNA molecule in the x and y directions in the absence of shear ow approximate Gaussian distribu-tions for both direcdistribu-tions.
length L = 21µm [13] and lp = 0.05µm [39, 40] as its persistence length, this parameter should be equal to 0.68µm . This value is larger than the measured value given above, about two standard deviations from the measured value. It may be that analysis of a larger number of molecules would reduce the discrepancy. On the other hand, neither the Gaussian chain model nor the Kratky-Porod model may actually fail to reproduce faithfully the behavior of an end-grafted DNA molecule.
Radius of gyration of a single DNA molecule
The uorescence intensity of uorescently labeled DNA molecule was averaged over a time of ' 40 sec, corresponding to 1000 images. The upper left image in Figure 2.25 shows the uorescence image of a DNA carpet after averaging. The upper right image shows a zoom into the image of one individual DNA molecule from the DNA carpet. The prole of this averaged intensity, averaged in addition over the azimuthal angle, is shown in the graph at the bottom of Figure 2.25.
The widths obtained directly from the averaged images do not show the
−3 −2 −1 0 1 2 3
−0.2 0 0.2 0.4 0.6 0.8 1 1.2
Radial Distance [µm]
Normalized Intensity
Figure 2.25: upper left image: Fluorescence image of an end-grafted DNA carpet, averaged over a time span of 40 seconds. upper right image: Zoom into an image of one of the end-grafted DNA molecules. bottom graph: Azimuthally averaged prole of the uorescence intensity of the chosen molecule and a t to equation 2.94 (compare the text)
real sizes of the molecule because of the diraction-limited resolution of the imaging by the objective lens (compare the discussion in section 2.2.3). In order to get the real size of the molecule it is necessary to deconvolute the concentration of the DNA with the point spread function (see Figure 2.16). In contrast to a previous study by R. Lehner [37]
using confocal optical microscopy, which was primarily interested in the density of the segmental concentration prole as a function of distance z from the surface and averaged over the directions parallel to the
surface, the present study uses non-confocal uorescence microscopy of single DNA molecules and therefore cannot resolve the z dependence of the monomer density because of overlapping out-of-focus background.
Imaging by non-confocal uorescence microscopy provides a signal that is proportional to the convolution of the PSF of the non-confocal optical system with the monomer concentration.
I = P SF ⊗C(r, z) (2.93)
Here, the monomer concentration C(r,z) is given by equation 2.33. There-fore .Figure 2.25 shows this convolution for a tted radius of gyration of Rg = (0.85±0.05)µm. This should be compared to the predictions of the theory explained in section 2.1.3. Using equations 2.16 and 2.21 we obtain, within the random walk model Rg = √23RCM. From the experimental result obtained above RCM = (0.49±0.11)µm we obtain
Rg(Random W alk) =
√3
2 RCM = 0.42±0.10µm (2.95) which is far outside of the error of the experimental result for Rg. Using equation 2.28, corresponding to the Kratky-Porod model, on the other hand, we obtain
Rg(Kratky−P orod) = q
Llp/3 = 0.59µm (2.96) (with L=21 µm, lp=0.05 µm), which is closer to the measured value than the value expected from a simple random walk model, but still too small.
Considering nally the Zimm model for a good solvent, equation 2.50, we obtain
Rg(Zimm model) = Rg = 0.203 kBT
√6ηD = 0.81µm (2.97)
(with T = 298 K, η = 8.9×10−4Pas, D = 0.47 µm2/s [43]). This is within the error of the experimental result and strongly hints at the validity of the Zimm model for a good solvent for tethered λ-phage DNA. The value found should be compared to the value Rg ≈ 0.7µm for λ-phage DNA in a bulk solvent [43, 69, 70]. It was stated previously that the radius of gyration for end-grafted λ DNA in a good solvent should be similar to Rg for a free chain [38].
Longest relaxation time of end-grafted DNA molecule
The relaxation times of the autocorrelation functions hxCM(0)xCM(t)i,hyCM(0)yCM(t)i were determined by extracting the center-of-mass positions xCM(t), yCM(t) of individual molecules, as described above, from movies taken in the absence of shear ow, calcu-lating the corresponding autocorrelation functions, and averaging them over 31 molecules. Figure 2.26 shows the autocorrelation function of the center-of-mass movement of tethered DNA, in the absence of shear ow, in the x and y directions, respectively. Exponential ts to these data, weighted by their standard deviations, yield Ax=(1.34±0.04) s−1, Ay=(1.18±0.04) s−1 for the exponential decay rates. Because, in the absence of shear ow, the movement in both directions is independent and should give the same result, we obtain the longest relaxation time from the average of these values:
τ = (0.79±0.03)s (2.98)
Fourier analysis of the movement of a tethered DNA molecule in the absence of shear ow
In order to determine the spectrum of uctuations of a tethered DNA molecule in the absence of shear ow Fast Fourier Transforms (FFT) were calculated of the x and y components of the movement for 5 dierent molecules. The magnitudes of these 10 Fourier transforms were then averaged and plotted on a double logarithmic scale. Figure 2.27 shows the result of this procedure. In the low frequency range at frequencies smaller than about 0.3 Hz the spectrum does not show reliable data, because of
0 0.5 1 1.5 2 2.5 3 3.5 4
−0.2 0 0.2 0.4 0.6 0.8 1
Time [s]
Normalized autocorrelation − x direction
0 0.5 1 1.5 2 2.5 3 3.5 4
−0.2 0 0.2 0.4 0.6 0.8 1
Time [s]
Normalized autocorrelation − y direction
Figure 2.26: Upper part: Average of the autocorrelation functions of the x component of the center-of-mass positions of 31 DNA molecules. The error bars represent the standard deviations of the individual autocorrelation functions of the 31 molecules.
Solid blue line: Exponential t to the data weighted with the reciprocal of the error.
Fitted decay constant Ax = (1.34±0.04) s−1. Lower part: Average of the autocorrela-tion funcautocorrela-tions of the y component of the center-of-mass movement of the same 31 DNA molecules. Solid red line: Exponential t to the data weighted with the reciprocal of the error. Fitted decay constant Ay = (1.18±0.04) s−1
insucient length of the movies. In the frequency range between about 1 Hz and 6 Hz an approximate power law is observed with an exponent of -1.4. At still larger frequencies the spectrum levels o. High frequency uctuations are therefore stronger than expected from such a power law.
10−3 10−2 10−1 100 101
10−3 10−2 10−1
Frequency [Hz]
FFT amplitude [µm]
Figure 2.27: Spectrum of uctuations of tethered DNA molecules. Average over the magnitude of the FFT of the movements of 5 dierent molecules in the x and y directions, respectively, that is over 10 FFT (compare the text). In the frequency range ν = (1-6) Hz the autocorrelation follows an approximate power law with an exponent -1.4, as indicated by the dashed red line.
2.3.2 End-grafted DNA Molecule under Oscillatory Shear Flow