Fluorescence Microscopy

Fluorescence microscopy has been used for almost a century to observe uorescent specimen using excitation by ultraviolet light. More recently, substantial progress like labeling of antibodies with uorescent dyes or the genetic manipulation of uorescent proteins has been achieved making uorescence microscopy nowadays one of the most important tools of investigation in biology. In addition, the sensitivity of uorescence microscopy has been pushed down to the single (dye) molecule level. In a standard uorescence microscope the uorescence is separated from the excitation light, usually provided by a mercury lamp, using suitable short-pass excitation and long-pass uorescence lters and dichroic beam splitters. In the present thesis the technique is used to monitor the dynamics of single multiply labeled DNA molecules tethered to a surface. This aims at providing insight into the microscopic mechanisms of externally driven polymer molecular dynamics.

Spatial resolution

Fluorescence imaging by an optical system

A central tool of this thesis is optical imaging of individual uorescent DNA molecules. The radius of gyration of these molecules is below 1 micron (see chapter 2.1.3). Therefore, the imaging properties of the

optical microscope are important and the nite resolution, which is limited due to diraction, can not be neglected. In the following these properties are very briey reviewed.

In the case of transmission or reection microscopy coherent superposition of all contributions in the object plane to the image has to be considered.

In the case of uorescence microscopy, on the other hand, coherence is broken and an incoherent superposition in the image plane is obtained, given by the integral over all intensity contributions from points in the object plane. The intensity distribution near the image plane caused by a uorescent point source in the object plane and produced by the optical system is then called the point spread function. This intensity point spread function is given by:

P SF(x, y, z) =|u(mx, my, mz)|² (2.63) where the factor m takes into account the magnication of the imaging system and u(x, y, z) is the diraction integral [65]

u(x, y, z) =−i

Here, s is the path from a point in the aperture of the lens to the point of observation, k = 2π/λ,λ the wavelength of observation, f the focal length of the lens, and the integral extends over the aperture of the lens. The formula is a good approximation when the numerical aperture of the lens is small compared to one. In this case also the scalar approximation is justied where the amplitude of the wave is assumed to be scalar and the vector character of the electromagnetic wave is neglected. Both approxi-mations typically work well even for relatively large numerical apertures.

Furthermore, a constant amplitude of the illumination is assumed every-where within the aperture.

The intensity point spread function in the image plane resulting from Equation (2.64) follows a

dependence, where the axis in the image plane. This intensity distribution exhibits rings of zero intensity in the focal plane, the so-called Airy rings. From Equation (2.65) it can be concluded that the full width at half maximum of the main maximum in the image plane corresponds to an apparent width in the object plane of

dF W HM = 0.51λ

NA (2.67)

Here, NA is the numerical aperture of the objective lens. This determines the diraction limit in the image plane. The axial dependence of the point spread function is given by

sin(u/4)

Equation 2.68 predicts zeros of intensity along the propagation axis. The maxima of the side lobes reach few percent of the main maximum. In reality the zeros of intensity and the side lobes are often smeared out and the symmetry for positive and negative defocus is destroyed due to aberrations occurring in the lens. Because the radius of gyration of a tethered DNA molecule is below 1 µm, we are interested, however, only in the region of the central maximum of the point spread function. We will therefore choose below a simple approximation of the intensity point spread function consisting in a Gaussian function with a width that in-creases quadratically with defocus. The quality of this approximation will be demonstrated experimentally and the necessary parameters will be deduced from the measurement. As described above the imaging of an extended incoherently emitting object with an intensity distribution O(x, y) can be described as summation of the images of all individual object points. The image is therefore given by multiplying the value of

O(mx, my, mz) by the function P SF(x−x⁰, y−y⁰, z−z⁰) and summing the result in the form of a convolution integral (compare Figure 2.16).

The displacement in the argument of the point spread function is thereby

Object Intensity Function O(x´, y´)

Point Spread Function E(x´, y´)

Image Intensity Function I(x´, y´)

Convolution

Deconvolution

Figure 2.16: The image intensity is given by the convolution of the PSF and the object intensity. Therefore the real size of an object can be derived from a deconvolution of the image intensity with the PSF [66].

due to the dierent positions of the individual object points and the cor-responding shift of their images in the image plane.

I(x⁰, y⁰, z⁰) =

Z Z Z

O(mx, my, mz)P SF(x−x⁰, y−y⁰, z −z⁰)dxdydz (2.70) Measurement of the point-spread function

The point-spread function of the microscope for a given microscope ob-jective lens was determined experimentally by measuring the uorescence intensity distribution in the image plane of 100 nm diameter red uores-cent latex beads (A.1.32) (excitation / emission maxima at 488/605 nm).

For such spheres it can be easily shown that the nite contribution of

the size of the object can be neglected within a very good approximation compared to the width of the instrumental point spread function. For a diraction-limited microscope (like the Nikon Eclipse TE 200 inverted mi-croscope used in the experiments) the latter is given by equation (2.67).

A droplet of diluted beads was deposited on a cover slip and allowed

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−0.5 0 0.5 1 1.5 2 2.5x 10⁴

Radial distance [µm ]

Normalized Intensity

Defocus H=0 Defocus H=0.8µm Defocus H=1.6µm Gaussian fit Gaussian fit Gaussian fit

Figure 2.17: Upper part: Images of a 100 nm uorescent latex bead at dierent defocus z. Experimentally obtained point spread function at dierent defocus. Lower part:

Fit of the azimuthally averaged intensity distributions to Gaussian functions.

to dry in order to x the beads to the surface. For observation with the microscope a droplet of water was added slowly to the dried beads and images of isolated beads on the surface were taken by means of a 100x/1.3 NA oil immersion lens, the same that was used for the experiment. Suc-cessive camera images were taken with the bead rst in focus and then at increasing defocus in steps of 0.8 µm until the bead was not visible any

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Figure 2.18: Fitted denominators in equation 2.71 are plotted versus defocus together with a quadratic t to the data. Resulting coecients from the t: a = (0.16± 0.009)µm⁻¹,b = (−0.04±0.01)µm,c= (0.24±0.016)µm.

more on the camera image (compare Figure 2.17). The same procedure was repeated for correspondingly negative defocus values. The intensity distributions of all images were tted (compare Figure 2.17) to Gaussian functions of normalized area whose widths were assumed to depend on defocus approximately as a quadratic function. The tted widths were correspondingly tted to such a quadratic function of defocus (compare Figure 2.18). The coecients, corresponding to equation 2.71 obtained in this way are given with their standard deviations by a = (0.16±0.009)µm⁻¹, b = (−0.04±0.01)µm, c = (0.24±0.016)µm. This result will be used for the calculation of the radius of gyration (see chapter 2.3).

The minimum full width at half maximum (FWHM), corresponding to zero defocus z=0 in equation 2.71, is given by

d_{F W HM} = 2c√

2ln2 (2.72)

Using c=0.24µm we obtain dFWHM = 0.57µm, which should be compared with the diraction limit FWHM = 0.22µm calculated from equation 2.67 with an average uorescence wavelength of 0.56µm and NA = 1.3.

Whereas this discrepancy has not been investigated in detail, it appears most likely that it is due to mechanical instabilities of the system.