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

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[[Category:Lab Manuals]]
 
 
[[Category:20.309]]
 
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[[Category:Optical Microscopy Lab]]
 
[[Category:Optical Microscopy Lab]]
 
{{Template:20.309}}
 
{{Template:20.309}}
  
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==Introduction and Background==
+
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.
  
−
Many cellular functions such as migration, differentiation, and proliferation are regulated by the
+
==Contextual background==
−
mechanical properties of cells, specifically, their elasticity and viscosity. Rheology is the science of
+
===Background references===
−
measuring materials' mechanical properties. Microrheology is a subgroup of techniques that are
+
** R. Newburgh, [http://scitation.aip.org/content/aapt/journal/ajp/74/6/10.1119/1.2188962 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.
−
capable of measuring mechanical property from microscopic material volumes. Clearly, given the
+
** 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).
−
typical size of biological cells, microrheology is the technique needed to measure their elasticity and
+
** M. Haw, [http://stacks.iop.org/JPhysCM/14/7769 Colloidal suspensions, Brownian motion, molecular reality: a short history], J. Phys. Condens. Matter (2002).
−
viscosity.
+
** E. Frey and K. Kroy, [http://arxiv.org/PS_cache/cond-mat/pdf/0502/0502602v1.pdf Brownian motion: a paradigm of soft matter and biological physics], Ann. Phys. (2005).
−
The elastic and viscous properties of cells can be characterized by a complex-valued shear
+
** [http://www.youtube.com/watch?v=FAdxd2Iv-UA Random Force & Brownian Motion — 60 Symbols]
−
modulus (with units of Pa) <i>G*(ω) = G'(ω) + iG"(ω)</i>. The real part G'(ω), referred to as the
+
−
storage modulus, is a measure of cell elasticity, while the imaginary part G"(ω), the loss modulus,
+
−
is a measure of their viscosity. A generalized Hookian relationship can be written as:
+
  
−
[[File: Eq1.PNG]]
+
===Brownian motion===
 +
This section was adapted from http://labs.physics.berkeley.edu/mediawiki/index.php/Brownian_Motion_in_Cells.
  
−
where Δr<math>^2</math> is a generalized displacement, and F(ω) is a force linearly proportional to it via the
+
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.  
−
shear modulus. Therefore, we can measure the shear modulus if we can measure the deformation
+
−
of the cell under a known force. (Note that all these quantities are frequency-dependent).
+
−
Particle-tracking microrheometry is based on measuring the displacement of a particle with
+
−
radius a embedded in a cell driven by thermal forces (similarly to the vibrations of the cantilever in an AFM). One complication is that this relationship is frequency dependent - this is because in complex fluids, such as the cellular cytoskeleton, there are different energy dissipation mechanisms over different time scales.
+
−
To approach the derivation of the relevant formulas, it is more convenient to think in terms of
+
−
energy, rather than force. The relationship between stored energy and displacement has a familiar
+
−
form, similar to a spring-mass system (recall <math> KE \ \alpha \ k(\omega)^2</math>):
+
  
−
[[File: Eq2.PNG]]
+
Brown noted: ''[The movements] arose neither from currents in the fluid, nor from its gradual evaporation, but belonged to the particle itself''.  
  
−
What is the driving thermal energy U(ω)? Recall also from that thermal energy is "white,"
+
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!
−
i.e., it contains equal power at all frequencies and is equal to 0.5*<i>k<math>_B</math>T</i> for each degree of freedom in a
+
−
second-order system, where <i>k<math>_B</math></i> is Boltzmann's constant and <i>T</i> is the absolute temperature. From
+
−
this relationship (since we're observing motion in two dimensions), we have
+
  
−
[[File: Eq3.PNG]]
+
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.  
  
−
Our argument is clearly very rough but a complete (and much more di±cult) derivation results
+
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.
−
in the following equation (see <ref>Mason </ref> for details):
+
  
−
==Experiment Details==
+
===Diffusion coefficient of microspheres in suspension===
−
===Stability and Setup===
+
According to theory,<ref>A. Einstein, [http://www.math.princeton.edu/~mcmillen/molbio/papers/Einstein_diffusion1905.pdf On the Motion of Small Particles Suspended in Liquids at Rest Required by the Molecular-Kinetic Theory of Heat], Annalen der Physik (1905).</ref><ref>E. Frey and K. Kroy, [http://www3.interscience.wiley.com/cgi-bin/abstract/109884431/ 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.</ref><ref>R. Newburgh, [http://scitation.aip.org/journals/doc/AJPIAS-ft/vol_74/iss_6/478_1.html 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.</ref><ref>M. Haw, [http://stacks.iop.org/JPhysCM/14/7769 Colloidal suspensions, Brownian motion, molecular reality: a short history], J. Phys. Condens. Matter (2002).</ref> the mean squared displacement of a suspended particle is proportional to the time interval as: <math>\left \langle {\left | \vec r(t+\tau)-\vec r(t) \right \vert}^2 \right \rangle=2Dd\tau</math>, where <i>r</i>(<i>t</i>) = position, <i>d</i> = number of dimensions, <i>D</i> = diffusion coefficient, and <math>\tau</math>= time interval.
−
The major challenge of particle tracking microrheometery is the small scale of the thermal forces and
+
−
the associated nanometer scale displacements. A few things you can do to ensure the experiment
+
−
works:
+
−
Be sure all the microscope components are rigidly assembled and firmly tightened. Poorly built
+
−
scopes shake. It is also vital that when you perform this experiment that the optical tables are
+
−
floating so that building noise is isolated. Of course, avoid touching the optical table and the
+
−
microscope during the measurement. There are also cardboard boxes available that you can put
+
−
over your microscope to isolate it from air currents. Finally, make sure that you and the people
+
−
around you are not talking too loudly during the experiment, because acoustic noise is significant.
+
  
 +
==Instructions==
  
 +
===Estimating the diffusion coefficient by tracking suspended microspheres===
  
−
[[File:Camera_settings.PNG|center|thumb|350px|Proper camera gain settings for particle tracking: Choose "Properties..." under the "Device"
+
[[Image: 20.309_130924_GlycerolChamber.png|right|thumb|200px|Imaging chamber for fluorescent microspheres diffusing in water:glycerol mixtures]]
−
menu and set the gain and brightness level to zero. Change the exposure time until the intensity of the fluorescent beads is just high enough to achieve good contrast, but do not saturate the intensity value to 255.]]
+
1. Track some 0.84&mu;m Nile Red Spherotech polystyrene beads in water-glycerin mixtures (Samples A, B and C contain 0%, 30% and 50% glycerin, respectively).
  
−
===System Verification===
+
:''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).
−
To verify that your system is suficiently rigid/stable, first measure a specimen containing red
+
−
fluorescent beads (Molecular Probes) dried in a cell dish. Chose a field of view in which you can
+
−
see at least 3-4 beads. Using a 40x objective record an .avi movie for about 3 min. at a frame
+
−
rate of 30 frames/sec.
+
  
−
From your experience with image processing, you already know how to import .avi movie data
+
:''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.
−
into MATLAB. To improve signal to noise ratio, sum every 30 frames together, which will make your
+
−
sequence have a temporal interval of 1 sec.
+
  
−
Use the bead tracking processing algorithm on two beads to calculate two trajectories. To
+
2. Estimate the diffusion coefficient of these samples: MSD = <math>\left \langle {\left | \vec r(t+\tau)-\vec r(t) \right \vert}^2 \right \rangle=2Dd\tau</math>, where <i>r</i>(<i>t</i>) = position, <i>d</i> = number of dimensions, <i>D</i> = diffusion coefficient, and <math>\tau</math>= 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).
−
further reduce common-mode motion from room vibrations, calculate the differential trajectory
+
:* '''Consider how many particles you should track and for how long. What is the uncertainty in your estimate?'''
−
from the individual trajectories of these two beads. Calculate the MSD <<math>\Delta r^2</math>> from this differential
+
:* '''From the viscosity calculation, estimate the glycerin/water weight ratio.''' (This [https://dl.dropboxusercontent.com/u/12957607/Viscosity%20of%20Aqueous%20Glycerine%20Solutions.pdf chart] is a useful reference.)
−
trajectory. Your MSD should start out less than 10 nm<math>^2</math> at <math>\tau</math> = 1 sec and still be less than 100 nm<math>^2</math> for <math>\tau</math> = 180 sec. If you don't get this, do not proceed further and ask for help.
+
:* See: [http://labs.physics.berkeley.edu/mediawiki/index.php/Simulating_Brownian_Motion this page] for more discussion of Brownian motion and a Matlab simulation.
  
−
===Live Cell Measurements===
 
  
−
Now that your system is su±ciently stable, you can run the experiment on cell samples. A key
+
===Live cell particle tracking of endocytosed beads===
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technique to keep in mind when working with live cells - to avoid shocking them with "cold" at
+
−
20°C, be sure that any solutions you add are pre-warmed to 37°C. We will keep a warm-water bath
+
−
running on the hotplate for this purpose, in which we will keep the various media.
+
  
−
You are provided with NIH 3T3 ibroblasts, which were prepared as follows:
+
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.  
−
Cells were cultured at 37°C in 5% CO<math>_2</math> in standard 100 mm x 20 mm cell culture dishes (Corning) in a medium referred to as DMEM++. This consists of DMEM (Cellgro) supplemented with 10% fetal bovine serum (FBS - from Invitrogen) and 1% penicillin-streptomycin (Invitrogen). The day prior to the microrheology experiments, fibroblasts were plated on 35 mm glass-bottom
+
−
cell culture dishes (MatTek). On the day of the experiments, the cell confluency should reach
+
−
about 60%. 1 μm diameter orange fluorescent microspheres (Molecular Probes) were mixed with
+
−
the growth medium (at a concentration of 5 x 10^5</math> beads/mL) and added to the plated cells for a
+
−
period of 12 to 24 hours for bead endocytosis. Choose cells with 3 or 4 particles embedded in them
+
−
and take a movie as before. Take movies of about 3-5 cells.
+
  
−
Now treat the cell with the cytoskeleton-modifying chemical cytochalasin D (CytoD). Pipet
+
You will be given two plates of cells for these experiments:
−
out the buffer, add 1 mL CytoD solution at 10 μM (pre-mixed for you) to the dish, and wait for
+
* <u>Dish 1</u> will be used to monitor particles in untreated cells, while
−
20 min. It's a good idea to check on your cells after 20 min.: sometimes they are in bad shape
+
* <u>Dish 2</u> will be reserved to track microspheres after adding CytoD.
−
at that point but sometimes they still look very healthy. Wash and replace with buffer twice with
+
−
2 mL pre-warmed DMEM++.
+
  
−
Repeat the particle tracking measurements again for 3-5 cells as quickly as you are able, since
+
# Pre-warm your DMEM++ and CytoD to 37&deg;C
−
their physiology has now been significantly disrupted and they will die within a couple of hours.
+
# 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.
−
It's very unlikely that you'll be able to find the exact same cells you've already tracked; however
+
# 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.
−
it's very much advisable to use the same dish for the "before" and "after" so you're aren't also
+
# Next, carefully pipet out the medium in Dish 2. Gently rinse with 1mL of medium 2X to remove beads that were not endocytosed.
−
comparing between different cell populations.
+
# 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&deg;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.
  
−
==References==
+
==Report==
  
−
<References/>
+
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.
 +
{{:Optical Microscopy: Part 4 Report Outline}}
 +
 
 +
{{:Optical microscopy lab wiki pages}}
  
 
{{Template:20.309 bottom}}
 
{{Template:20.309 bottom}}
 +
 +
==References==
 +
<references/>

Latest revision as of 01:01, 10 March 2016

20.309: Biological Instrumentation and Measurement

ImageBar 774.jpg


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

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: 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).

  • 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.


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.
  1. Pre-warm your DMEM++ and CytoD to 37°C
  2. 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.
  3. 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.
  4. Next, carefully pipet out the medium in Dish 2. Gently rinse with 1mL of medium 2X to remove beads that were not endocytosed.
  5. 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.
  6. 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.

  1. Viscosity
    1. 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)
    2. 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.)
    3. 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.
    4. 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.
  1. Particle Tracking in Cells
    1. 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)
    2. 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.)
    3. 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.
    4. 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.

Optical microscopy lab

Code examples and simulations

Background reading

References

  1. ↑ 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).
  2. ↑ 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.
  3. ↑ 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.
  4. ↑ M. Haw, Colloidal suspensions, Brownian motion, molecular reality: a short history, J. Phys. Condens. Matter (2002).
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