Difference between revisions of "Optical Microscopy Part 4: Particle Tracking"

In this part of the lab, you will follow microscopic objects throughout a series of movie frames: first small, fluorescent microspheres diffusing in purely viscous solutions of glycerol-water, next small fluorescent bacteria cells swimming in salt water. Calculating the mean squared displacement of their motion as a function of time interval will allow you to characterize their physical environment and behavior, first in terms of diffusivity and viscosity coefficients of the glycerol-water mixtures, next recognizing diffusive vs. ballistic travel as a function of aerotactic stimuli.

Contextual background

Background references

• • R. Newburgh, Einstein, Perrin, and the reality of atoms: 1905 revisited, Am. J. Phys. (2006). A modern replication of Perrin's experiment. Has a good, concise appendix with both the Einstein and Langevin derivations.
  • A. Einstein, On the Motion of Small Particles Suspended in Liquids at Rest Required by the Molecular-Kinetic Theory of Heat, Annalen der Physik (1905).
  • M. Haw, Colloidal suspensions, Brownian motion, molecular reality: a short history, J. Phys. Condens. Matter (2002).
  • E. Frey and K. Kroy, Brownian motion: a paradigm of soft matter and biological physics, Ann. Phys. (2005).
  • Random Force & Brownian Motion — 60 Symbols

Brownian motion

This section was adapted from http://labs.physics.berkeley.edu/mediawiki/index.php/Brownian_Motion_in_Cells.

If you have ever looked at an aqueous sample through a microscope, you have probably noticed that every small particle you see wiggles about continuously. Robert Brown, a British botanist, was not the first person to observe these motions, but perhaps the first person to recognize the significance of this observation. Experiments quickly established the basic features of these movements. Among other things, the magnitude of the fluctuations depended on the size of the particle, and there was no difference between "live" objects, such as plant pollen, and things such as rock dust. Apparently, finely crushed pieces of an Egyptian mummy also displayed these fluctuations.

Brown noted: [The movements] arose neither from currents in the fluid, nor from its gradual evaporation, but belonged to the particle itself.

This effect may have remained a curiosity had it not been for A. Einstein and M. Smoluchowski. They realized that these particle movements made perfect sense in the context of the then developing kinetic theory of fluids. If matter is composed of atoms that collide frequently with other atoms, they reasoned, then even relatively large objects such as pollen grains would exhibit random movements. This last sentence contains the ingredients for several Nobel prizes!

Indeed, Einstein's interpretation of Brownian motion as the outcome of continuous bombardment by atoms immediately suggested a direct test of the atomic theory of matter. Perrin received the 1926 Nobel Prize for validating Einstein's predictions, thus confirming the atomic theory of matter.

Since then, the field has exploded, and a thorough understanding of Brownian motion is essential for everything from polymer physics to biophysics, aerodynamics, and statistical mechanics. One of the aims of this lab is to directly reproduce the experiments of J. Perrin that lead to his Nobel Prize. A translation of the key work is included in the reprints folder. Have a look – he used latex spheres, and we will use polystyrene spheres, but otherwise the experiments will be identical. In addition to reproducing Perrin's results, you will probe further by looking at the effect of varying solvent molecule size.

Diffusion coefficient of microspheres in suspension

According to theory,^[1][2][3][4] the mean squared displacement of a suspended particle is proportional to the time interval as: $ \left \langle {\left | \vec r(t+\tau)-\vec r(t) \right \vert}^2 \right \rangle=2Dd\tau $, where r(t) = position, d = number of dimensions, D = diffusion coefficient, and $ \tau $= time interval.

Instructions

Estimating the diffusion coefficient by tracking suspended microspheres

Imaging chamber for fluorescent microspheres diffusing in water:glycerol mixtures

1. Track some 0.84μm Nile Red Spherotech polystyrene beads in water-glycerin mixtures (Samples A, B and C contain 0%, 30% and 50% glycerin, respectively).

Notes: Fluorescent microspheres have been mixed for you by the instructors into water-glycerin solutions A, B, C, and D. (a) Vortex the stock Falcon tube, and then (b) transfer the bead suspension into its imaging chamber (consisting of a microscope slide, double-sided tape delimiting a 2-mm channel, and a 22x40mm No. 1.5 coverslip, and sealed at both ends nail polish).

Tip: Do not choose to monitor particles that remain stably in focus: these are likely to be 'sitting on the coverslip' and their motion will not be representative of diffusion in the viscous water-glycerol fluid.

2. Estimate the diffusion coefficient of these samples: $ \left \langle {\left | \vec r(t+\tau)-\vec r(t) \right \vert}^2 \right \rangle=2Dd\tau $, where r(t) = position, d = number of dimensions, D = diffusion coefficient, and $ \tau $= time interval. Use Sample A to verify that your algorithm correctly calculates the viscosity of water at the lab temperature (check the temperature on the clock on the wall or by other means).

• Consider how many particles you should track and for how long. What is the uncertainty in your estimate?
• From the viscosity calculation, estimate the glycerin/water weight ratio. (This chart is a useful reference.)
• See: this page for more discussion of Brownian motion and a Matlab simulation.

Report

Diffusion coefficient and viscosity

• • Estimate diffusion coefficient, viscosity for each water-glycerin mixture sample.
  • Comment on results, specifically how they are influenced by microscope stability and resolution.
  • Comment extensively on sources of error and approaches to minimize them, both utilized and proposed.
    • Categorize the sources of error as systematic, random, or just mistakes (so-called "illegitimate" errors).
  • Provide a bullet point outline of all calculations and data processing steps.

Optical microscopy lab

Code examples and simulations

• Converting Gaussian fit to Rayleigh resolution
• MATLAB: Estimating resolution from a PSF slide image
• Matlab: Scalebars
• Calculating MSD and Diffusion Coefficients

Background reading

• Geometrical optics and ray tracing
• Physical optics and resolution
• Optical aberrations
• Aperture and field stops
• Optical detectors, noise, and the limit of detection
• Manta G032 camera measurements
• Understanding log plots