Optical Microscopy Part 4: Particle Tracking
In this part of the lab, you will follow microscopic objects throughout a series of movie frames: small, fluorescent microspheres first diffusing in purely viscous solutions of glycerol-water, and next moving in fibroblast cells after endocytosis.
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 other material or transport properties in fibroblast cells.
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
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: MSD = $ \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).
Live cell particle tracking of endocytosed beads
We can also use particle tracking to probe cell samples. 0.84 μm diameter red fluorescent microspheres were mixed with the growth medium and added to the plated cells for a period of 12 to 24 hours for bead endocytosis.
You will be given two plates of cells for these experiments:
- Dish 1 will be used to monitor particles in untreated cells, while
- Dish 2 will be reserved to track microspheres after adding CytoD.
- Pre-warm your DMEM++ and CytoD to 37°C
- Carefully pipet out the medium from Dish 1. Gently rinse with 1mL of medium 2X to remove beads that were not endocytosed. Then, place 2 mL of fresh medium in dish.
- Choose cells in Dish 1 with 3 or 4 particles embedded in them and capture movies of the samples. Take as many movies as you can with about 3-5 cells in the field of view in each movie. Make sure to do this quickly, as the cells become unhealthy without the temperature and carbon dioxide regulation.
- Next, carefully pipet out the medium in Dish 2. Gently rinse with 1mL of medium 2X to remove beads that were not endocytosed.
- Treat the cells in Dish 2 with the cytoskeleton-modifying CytoD: Pipet out remaining medium, add 1 mL pre-warmed CytoD solution at 10 μM (pre-mixed for you) to the dish, and incubate for 20 minutes at 37°C. It's a good idea to check on your cells after 15 minutes: sometimes they are in bad shape at that point but sometimes they still look very healthy. Wash 2X with 2 mL of pre-warmed DMEM++, leaving 2 mL in the dish when imaging.
- Perform and repeat the particle tracking measurements again in Dish 2 as quickly as you are able. It would be good to image the beads in only one cell at a time, since different cells may have different degrees of cytoskeletal disruption. Take as many videos as you can before the cells become sad. The cells' physiology has now been significantly disrupted by the toxin CytoD, and they will die within a couple of hours.
Report
Find and follow all guidelines on the Microscopy report outline wiki page. Remember that all numerical quantities must be reported with an associated level of uncertainty or significance.
- Viscosity
- Procedure
- Document the samples you prepared and used and how you captured images (camera settings including frame acquisition rate, number of frames, number of particles in the region of interest, choice of sample plane, etc)
- Data
- Include a snapshot of the 0.84 μm fluorescent beads monitored.
- Plot two or more example bead trajectories for each of the glycerin samples. (Hint: If you subtract the initial position from each trajectory, then you can plot multiple trajectories on a single set of axes.)
- Analysis and Results
- Plot the average MSD vs τ results for all glycerin samples (A, B, C, and D); use log-log axes. Use the minimum number of axes that can convey your results clearly.
- Include a table of the diffusion coefficient, viscosity and glycerin/water ratio for each of the samples (A, B, C, and D).
- Provide a bullet point outline of all calculations and data processing steps.
- Discussion
- How do your viscosity calculations compare to your expectations? (This chart is a useful reference.)
- Include a thorough discussion of error sources and the approaches to minimize them. It may be helpful to list out the error sources in a table, including a category for the error source, type of error (random, systematic, fundamental, technical, etc.), the magnitude of the error, and a description and way to minimize each one.
- Procedure
- Particle Tracking in Cells
- Procedure
- Document the samples you prepared and used and how you captured images (camera settings including frame acquisition rate, number of frames, number of particles in the region of interest, choice of sample plane, etc)
- Data
- Include a snapshot of the 0.84 μm fluorescent beads monitored.
- Plot two or more example bead trajectories for each of the samples. (Hint: If you subtract the initial position from each trajectory, then you can plot multiple trajectories on a single set of axes.)
- Analysis and Results
- Combine your data with others from the class to increase your sample size.
- Plot the average MSD for untreated and Cyto D treated cells on a single set of log-log axes.
- Discussion
- What kind of motion do you see described by your MSD vs τ results?
- What differences do you see between the untreated and Cyto D treated MSD curves?
- Please suggest an interpretation of the behavior of your cells based on your data.
- Include a discussion of your error sources.
- Procedure
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
References
- ↑ 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).
- ↑ E. Frey and K. Kroy, Brownian motion: a paradigm of soft matter and biological physics, Ann. Phys. (2005). Published on the 100th anniversary of Einstein’s paper, this reference chronicles the history of Brownian motion from 1905 to the present.
- ↑ 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.
- ↑ M. Haw, Colloidal suspensions, Brownian motion, molecular reality: a short history, J. Phys. Condens. Matter (2002).