Measuring Viscosity of Artificial Cytoplasm using Brownian Motion

Blake Antos, Michael Rush

University of Minnesota - School of Physics and Astronomy

Minneapolis, MN 55455

Abstract

The viscosity of an artificial cytoplasm was measured by monitoring the Brownian motion of suspended spherical microparticles. A value of 1.609±.029 mPa•s was obtained. The change in viscosity over time was then measured after backteriophage DNA was added to the cytoplasm and allowed to replicate. No change in viscosity was measured, as DNA was observed to flocculate.

Goal

Measure the diffusion coefficient and viscosity of an artificial cytoplasm solution using the Brownian motion of spherical particles.

Introduce replicating bacteriophage DNA and measure how the viscosity changes over time.

Introduction & Theory

Diffusion is a ubiquitous process that describes how molecules are able to move across various membranes due to the presence of a concentration gradient. The diffusivity is a measure of how quickly this diffusion occurs, defined for a pair of substances. In this experiment, this pair is artifical cytoplasm and spherical polystyrene microparticles. Such microparticles undergo Brownian motion when suspended in a liquid, wherein they travel short distances in random directions and random velocities due to collision with the particles that make up the liquid (on the order of 10²¹ per second). By measuring the rate of this motion, one can calculate the diffusivity, and from that, the viscosity of the liquid. By introducing bacteriophage DNA and allowing it to replicate in the cytoplasm, the viscosity ay be increased due to the polymeric nature of the DNA.

In order to measure the diffusivity, we must measure the mean-square displacement:

Since Brownian motion is random, we expect the distribution of <latex> \Delta L </latex> to be normal. Einstein relates the MS displacement to the diffusion coefficient, generalized to 3 dimensions, by

where D is the diffusivity and <latex> t_s </latex> is the length of the time step [1]. The time step must be chosen such that it is long enough that the motions of the particle between frames are independent, but short enough that the particles does not undergo several changes in direction during the step. The diffusivity is then related to the viscosity <latex> \eta </latex> using the Stokes-Einsten relation:

where T is the temperature and r is the radius of the spherical particles being diffused [1]. Considering the fact that the motion of the particles occurs in 3 dimensions, while we only measure 2 and assuming the MS displacements in each direction are equal:

, the MS displacement is then

Thus we obtain an expression for the viscosity:

It is assumed through this derivation, that the suspended particles are large compared to those that make up the medium in which the particle is suspended (1nm ~ 10 μm) , that they are of comparable density to the fluid, and that they are not interacting with eachother.

Adding DNA and allowing it to replicate will create large, dense polymers that may impede the motion of the microspheres, increasing the viscosity of the cytoplasm. The viscosity should increase steadily as DNA replicates until the resources needed to continue to replicate become scarce, causing the viscosity to stabilize. Though it is possible that the larger size of the DNA molecules will prevent true Brownian motion of the particles, making this measurement inconclusive.

Experimental Apparatus & Procedure

The apparatus consists of a microscope using a 100x objective lens equipped with a digital camera (shown in figure 1) and glass slides containing the cytoplasm and microbeads. The polystyrene latex microbeads have radii of 1.445μm and a density of 1.05<latex> \frac{g}{cm^3} </latex>. The artificial cytoplasm consists of E. coli cytoplasm extract and ~10 mg/mL of polyethylene glycol, maltose, magnesium, water-soluble proteins, ribosomes, and a mixture of amino- and nucleic acids. Found by trial and error, the approximate concentration of the beads in the cytoplasm was .0025% in 5μL of solution sealed between the glass slides. The spacing between these slides was measured to be 180±3.7μm, allowing ample space for the particles to undergo Brownian motion without interacting with the slides. Isolated beads were viewed under the microscope and pictures taken every .5 seconds for 5 minutes.

Figure 1. Picture of the microscope setup. The camera is located on the lower left side of the microscope.

Each image was then processed with matlab to remove the background and enhance the contrast of the image. Figure 2 below is an example of a pre-processed image showing particles circled in green, and the 2 kinds of background circled in blue and red.

Figure 2. A typical image taken by the apparatus described above. In green are the particles of interest, which appear either as a donut shape or a set of concentric rings. In blue is the mattled, low gradient background, and in red is the dust or particle-like background.

Due to the static nature of the background, it can be removed by subtracting an image taken without a slide under the microscope. The images were then processed using the MTrackJ plugin for ImageJ. This method requires the user to click on the particle to limit the area over which the plugin's algorithm will look for a bright or dark centroid. A processed image can be seen in figure 3 below.

Figure 3. One example of a processed image. Left: a single tracked particle. Right: full-sized image showing the 3 tracked particles. This image is from a slide with DNA.

The program returns an x and y position for each particle in the image, allowing multiple particles to be tracked independently.

Data Analysis

This procedure was followed on three 600 image data sets for each of 3 slides without DNA and eight 300 image data sets for each of 2 slides with DNA. The data was analyzed for any trends in the trajectories of the particles. A good example is shown in figure 4 below.

Figure 4. Example of position vs. time plots and an absolute position plot. This particle appears to be undergoing Brownian motion, judging by its lack of preferential direction. Error bars are shown on the time plots. These data were taken from a particle on a slide without DNA.

To calculate diffusion coefficient, use the rms deviations:

where D is the diffusion coefficient, t is the time step, and delta x is the movement during that time step.

To calculate viscosity from diffusion coefficient:D=(kT)/6πηr

where k is Boltzmann's constant, T is temperature, η is the viscosity, and r is the radius of the beads.

Apparatus:

Image Analysis: Used Matlab's Image Processing capabilities to turn images like this:

Into images like this:

Conclusions

Overall, we cannot say with any degree of certainty how the viscosity of an artificial cytoplasm depends on the time DNA has been allowed to replicate within it. It was difficult to eliminate the effects of flow within the solution; the source of this problem is still not known. In addition, the sheer amount of images required to produce this type of data renders the acquisition and analysis an unwieldy and tedious process that consumes much computational time. The majority of the semester was spent experimenting with Matlab’s Image Toolbox trying to write a code that would consistently track the positions of beads. Although it was a good exercise to learn how to use Matlab’s image processing capabilities, it would be much simpler in the future to use software that already exists for this purpose. If viscosity data for the replicating DNA solution continues to be sought, groups should use software like Image-J's "Spot tracker" plug-in. Another potential improvement in this experiment is to assign some sort of visible chemical change to the DNA replication reaction so that it can be verified that replication is indeed taking place.