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Observation of Molecular Diffusion Through Fluctuation Correlation Spectroscopy
Samantha Schmitt and Anna Andrick
University of Minnesota, Twin Cities
Methods of Experimental Physics - Spring 2019
Abstract
A fluctuation correlation spectrometer was made to observe the diffusion of Rhodamine B in an aqueous solution. This spectrometer was assembled using a 2 mW 543.5 nm laser, a series of lenses, mirrors, and pinholes, and a microscope objective. It was possible to differentiate between different concentrations of Rhodamine but no autocorrelation curve could be seen, due to the signal being overwhelmed by noise.
Introduction
Fluctuation correlation spectroscopy examines fluctuations in the intensity of fluorescence emitted by a fluorophore. These fluctuations can be caused by Brownian motion, the random movement of particles in water caused by collisions with the water molecules, or by chemical reactions. In this experiment, Rhodamine B was used, with fluctuations only through Brownian motion. Fluorescent particles move in and out of a small observation volume, and the intensity of fluorescence is dependent on how many molecules are in the viewing area.
Fluctuation correlation spectroscopy, or FCS, is used in biophysics and environmental sciences, as it can be used to investigate small channels without causing obstructions. Fluorophores are small, and FCS uses low concentrations, so it can be used to probe flow through microchannels, and even to observe cellular structure and behavior. Using FCS, we can learn about the diffusion of a fluorophore in a solution. Thus, we can find the viscosity of nucleoplasm, or observe binding affinities of DNA.
The goal of this experiment was to build and calibrate a fluctuation correlation spectrometer. Several intrinsic properties of the spectrometer must be found experimentally by using a known concentration of a fluorophore with a known diffusion coefficient.
Theory
Figure 1 shows the intensity of the fluorescence in the form of photons observed over a time-step as a function of time. The intensity of fluorescence is compared to the average fluorescence through an autocorrelation function (Equation 1):
where T = t + τ and
From Fick's Law (Equation 2),
Figure 1 shows fluctuations in fluorescence as a function of time. where δc(r,t) gives concentration fluctuations and D is the diffusion coefficient of a particle defined by the
This is from a 1 μM sample with a sampling frequency of 1MHz. Stokes-Einstein relation (Equation 3),
with R as the hydrodynamic radius of the particle, T is the temperature, and η is the viscosity of the solvent, we can derive a non-linear least squares fit (Equation 4):
By fitting this to the autocorrelation given by Equation 1, the fit parameters N, τD, ro, and zo can be obtained. N = CVeff is the average number of particles in the observation volume, where C is the concentration and Veff = π3/2ro2zo. We define ro and zo are intrinsic properties of the spectrometer defined as the radial and axial distances for the fluorescence intensity to decay by e-2. The correlation time for translational diffusion τD, or the time for a particle to cross the observation volume, is given by (Equation 5)
For Rhodamine B, D = 3.36*10-10 m2/s in water at room temperature, so ro and zo can be obtained to calibrate the spectrometer easily.
Experimental Set-Up
A confocal geometry is crucial to getting meaningful results from FCS. The pinholes remove interference and allow us to get a cleaner output from a small sample. Looking at a small sample area is necessary as FCS examines deviations from the mean. With a large area, deviations will be too small to be meaningful.
Lenses are used to control the width of the laser beam. In our set-up, two pinholes (P3 and P5) are placed at the focal distance associated with their respective lens. These pinholes are set to the size of the first Airy disk, the brightest central circle in an Airy pattern. This removes diffracted light from the system while keeping as much light as possible. This pinhole diameter is found using (Equation 6)
where y is the radius of the pinhole, f is the focal distance of the lens, λ is the wavelength of the laser, and d is the beam width of the laser. Rhodamine B has an excitation maximum at ~544 nm, which is the wavelength of the laser we used, and an emission maximum of ~565 nm.
Figure 2 shows a schematic of the apparatus. Through the use of lenses and pinholes the beam is
widened to the size of the back aperture of the objective and unwanted light contamination is reduced.
The spectrometer we made follows the schematic in Figure 2. A 543.5 nm 2 mW laser was used to excite a sample of RB. The intensity of the emitted fluorescence was measured with a Hamamatsu h6240-20 photomultiplier tube (PMT). The laser used was a Melles-Griot Helium Neon (HeNe) laser, and some red light escaped along with the wavelength we wanted. Pinholes P1 and P2 were in place to reduce that and other external light sources. The width of the laser beam was controlled by lens L1, with pinhole P3 at the focal distance of L1 to filter out scattered light. The diameter of the beam was set to fill the back aperture plane of the objective, thus achieving a minimal spot size in the sample to get the most precise measurements. The beam passed an excitation filter (MF542-20) to try to further block unwanted light, then reached a dichroic mirror (DM) ( MD568) and was directed through a Leitz Wetzlar 100x 1.30NA, focal length 170 mm oil immersion objective(Ob).
as one can see in the transmission profile in Figure 3. This makes it quite possible that laser light transmitting through the dichroic could overpower the fluorescence being observed, which is one explanation that the results Figure 3 shows the transmission profile of the dichroic, from the
obtained were nothing but noise. seller's website. Approximately 15% of the ~565 nm emission was transmitted.
Results and Analysis
One problem with the components used is that the dichroic only transmitted ~20% of the fluorescence,
The microscope objective was a 100x oil immersion objective with a focal length of 170 mm. Thus, it was crucial to position P3 at that
distance, and L2 at that distance plus its own focal distance. The entire set-up was covered to block external light, and all trials were taken with as little light in the room as possible to decrease interference. The
rhodamine was diluted in water and pipetted into a cell counting slide.
Here the laser excited the RB, which emits light at ~565 nm. This fluorescence was transmitted through the dichroic and a barrier filter (MF56
5-24) to try to filter out laser light that the dichroic transmitted. It was then focused through lens L2 to pass through the confocal pinhole P5 to be measured with a PMT.
The intensity of fluorescence was recorded by a LabView program connected to the PMT with a DAQ.
Figure 4 displays the two forms of autocorrelation observed in this experiment. Both were obtained from the same 10 μM sample at a
frequency of 1 MHz, and were taken five minutes apart with no changes. Both were dominated by noise of different sorts.
In this experiment, five concentrations of RB were used, at factors of 10 from 1 nM to 10 μM. It was possible to differentiate between the three highest concentrations by observing photons/second, but the three lowest concentrations were indistinguishable from each other. They were all significantly greater than the thermal noise of the PMT, so we can confirm some fluorescence was being detected. Figure 4 shows the two forms observed autocorrelations took, and Figure 5 shows an expected autocorrelation curve.
The plot on the left of Figure 4 is the same shape as is obtained with no slide used, and thus is believed to be an example of purely random noise. The plot on the right is dominated by an unidentified noise. It always appears with roughly the same frequency, ~2 kHz. This could be random electrical noise, or it could be an interference pattern caused by laser light reflecting from the table.
Figure 5 shows an autocorrelation fit of simulated data, based off
the results of Rieger and Roecker's paper.
It is believed that the fluorescent signal was drowned out by outside light sources and excess laser light. To account for this, the set-up was enclosed in a cardboard box and covered with black cloth, and all external light sources were blocked or turned off. To try to minimize interfering laser light, it is proposed to invert the spectrometer, so the PMT and the sample switch places. In doing so, any light passing through the sample and reflecting back through from the table below can instead be redirected.
The alignment of lenses and pinholes is also not perfect, but should be close enough where a messy curve could be seen. From observing this curve, it would be possible to make minor adjustments to clean it up as much as possible. Another trick to get the cleanest possible signal would be to cool the PMT with dry ice, to minimize thermal noise.
References
Rieger and Roecker, Fluctuation Correlation Spectroscopy for the Advanced Physics Laboratory, https://aapt.scitation.org/doi/10.1119/1.2074047
https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=2990&pn=MF542-20 (Dichroic Transmission Profile)
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