Lab 4 - lab 4 PDF

Preview

Summary

lab 4...

Description

Lab Report 4 Hypothesis: We hypothesize that our log vs log plot of predicted random-motion vesicles will have a slope of 1, indicating a proportional relationship between r² and t. For the log vs log plot of predicted directed-motion vesicles, we would expect the slope to be 2, indicating a proportional relationship between r² and t². Methods: Determining Random vs Direction Motion of Vesicles In order to identify vesicles that are moving randomly or directionally, the relationship between distance and time was evaluated--where r² would be proportional to t in the case of random motion, or proportional to t² in the case of directed motion. Looking at videos displaying the activity of vesicles in onion cells, both random and directed moving vesicles were identified and tracked. To achieve this, the video was imported into the Fiji software where the motion of onion vesicles could be further analyzed through the Manual Tracking plugin. The video was altered by enhancing the contrast, and adjusting the brightness and threshold through the features provided in the Fiji program, to make it easier to locate vesicles of interest. Utilizing the manual tracking plug-in, the X and Y positions for every 10 frames for each vesicle chosen, was determined. We would select 5 vesicles to represent random motion and 5 vesicles sto represent directed motion. The tracks were then used to determine the r² value for every frame, using the equation: r²=(X-X₀ )² + (Y-Y₀ )² with X and Y being from the specific time frame. Using the default scale, the distance the vesicles travelled was obtained in μm; however, the time as measured in frames was converted into seconds using 13.47 fps given to achieve velocity in terms of μm/sec. Following that, the r² value for every frame under random motion vesicles was averaged and graphed over time for the random motion graph. Similarly, the r² value for every frame under directional motion vesicles was averaged and graphed over time for the directional motion graph. In order to visualize the data and relationship between distance and time better, log-log graphs were generated by taking the log of r² and the log of t for their respective graphs. Observing the slope generated from the log-log graphs provides us information with the order of time which can then be used to consider whether our respective groups of vesicles are moving in a random or directed motion.

Determining Viscosity From Directed Motion Using the given experimental parameters, an estimate of vesicle size, the derivation of Wout produced in a single step, and knowledge from the free body diagram of the directed

motion vesicle, a value for viscosity was obtained. Step-by-step details on how an equation solving for viscosity was abstained can be found in the lab section titled “Model: Viscosity and Directed Motion” Determining Diffusion Constant from Random Motion Using the random motion vesicle data, a plot was created puting time on the x-axis and average R² on the y-axis. Next we rearranged the diffusion equation, = 4Dt into 4D=R²/T. The slope of our plot is then equated into 4D. In order to determine the diffusion constant, the slope is divided by 4. Model: Viscosity from Directed Motion Parameters Given: - Average Size of Myosin Motor Step: 10nm - Average Size of Kinesin Motor Step: 8nm - Ones step for a motor has EATP = 23kJ/mol - E′ATP = (23kJ/mol)*(1 mol/6.02x10^23molecules)*(1000J/1kJ) = 3.8x10^-20 J/molecule - Efficiency (e) = 60% Estimation of Vesicle Size: Using the 40x1024 calibration slide given, the pixels were converted to nm in order to find the diameter of an average vesicle. We assumed that the vesicles were all about the same size and spherical in shape. 1 mm on the 40x1024 calibration slide is equal to 42 pixels. Using this, the average vesicle was determined to be 0.265 mm in diameter. This was then converted to nm: 2.65x10^5 nm. Determining Average Velocity of Directed Motion Vesicle: To determine the terminal velocity of the directed motion vesicle, the square root of average R² values of directed motion vesicle data was taken. Then, an average R vs time graph was constructed with average R (distance traveled) on the y-axis and time on the x-axis. The slope of this graph would be used as the terminal velocity of the vesicle.

Figure i. Freebody Diagram for a vesicle attached to a motor, moving with some speed through fluid.

-

-

Applied force is driving the vesicle as the viscous force is pulling it back; we will assume constant velocity in the directed motion vesicles, and thus, the velocity value experimentally obtained will be representative of terminal velocity. F applied = F viscosity F applied = 6πμrv

Wout Produced in a Single Step - Derivation: - e = Wout/Ein - W = Fd - e = W/E′atp - E′atp*e = Fd - e*Eatp=Fapp*s (s = step size) Knowing Fapp = 6πμrv and e*Eatp=Fapp*s, we can obtain the equation e*E′atp = 6πμrvs. - When we rearrange the equation to solve for viscosity, we obtain the equation, μ = (e*E ′atp)/(6πrvs)

Results: Random Motion vs Directed Motion of Vesicles

Figure 1: For the hypothesized random motion of the cell, the r² was calculated. Using the X and Y positions of the vesicle at a specific frame, the r² was determined using pythagorean theorem and averaged for each frame. Then, they were plotted against the change in time (top graph). The log-log plot was created to analyze the relationship between r² and time. The log-log plot has a slope of 0.75 (μm²/sec) with an R² of 0.5272 (bottom graph). Table 1: The LINEST function for the Log-Log plot was calculated. The slope was determined to have an error of 0.75 (μm²/sec) ±0.20 with an R² of 0.527.

Linest Function Slope: 0.75 (μm²/sec)

Y-intercept: 2.92 (μm)

±0.20

±0.21

R²: 0.527188

s(y): 0.249405

Figure 2: For the Hypothesized Directed Motion, the r² was calculated. Using the X and Y positions of the vesicle at a specific frame, the r² was determined using pythagorean theorem. The average for each frame was determined and then plotted against the change in time (top graph). The log-log plot was then created to look at the relationship between r² and time. The log-log plot has a slope of 2.1061 (μm²/sec) with an R² of 0.992 (bottom graph). Table 2: The LINEST function for the Log-Log plot was calculated. The Slope was determined to have an error of 2.11 (μm²/sec) ±0.08 with an r² of 0.992.

Linest Function Slope: 2.11 μm²/sec

Y intercept: 2.93 μm

±0.08

±0.06

R²: 0.992003

s(y): 0.062396

Determining Viscosity Coefficient from Directed Motion Derivation of the viscosity coefficient equation: Fapplied=6 πμrv W =F∗d e E ' ATP=F∗D W e= E' ATP e E ' ATP =F applied∗step e E ' ATP μ= 6 πrsv Solution: Parameters given: e

0.6

E’

3.8x10⁻ ²⁰ J/molecule

Step size

9 nm

Radius Estimate

1.33x10⁵nm

Step Size = avg of Myosin and Kinesin motor steps = (8nm + 10nm)/2 = 9nm = 9x10⁻⁹m Radius of Vesicle Estimate: - 2.65x10⁵nm in diameter, and diameter/2 = radius - Radius of vesicle = 1.33x10⁵nm = .000133m

Figure 3: The velocity of the directed motion was determined by looking at the change in average r (μm) over time (sec). This was plotted to determine a slope of 33.155 μm/sec with an R² of 0.9982. Table 3: The LINEST function of the velocity graph was taken to determine the error in the slope to 33.155 ± 0.49 μm/sec.

Slope: 33.15 μm/sec

Y intercept: 0 μm

±0.491551

-

r²: 0.998245

s(y): 10.42428

Terminal Velocity of Vesicle: 33.155 μm/sec = .000033155 m/sec (0.6 )(3.8∗1 0−20 J /mole) μ= −9 6 π (9∗1 0 m)(0.000133 m)(0.000033155 m/sec) μ = 3.05 x 10⁻⁵ N*s*m⁻²

Diffusion Constant from Random Motion

Figure 4: To determine the diffusion constant of random motion, the r² average was plotted over time to give a slope of 522.08 μm²/sec with an R² of 0.86. Table 4: The LINEST function was used to determine the error of the slope to be 522.08 ± 55.36 μm²/sec.

Slope: 522.08 μm²/sec

Y intercept: 0

±55.36

-

r^2: 0.85568458

s(y): 2894.416

Calculation for Diffusion Coefficient: 2 4 D =R /T μm ² / sec 4 D =522.1 μm ²/ sec D=130.5 μm ² / sec

Data Interpretation: Random motion: log-log plot shows a linear relationship (slope ~1) because r² is proportional to t. Directed motion: log-log plot shows a squared relationship (slope ~2) because r² is proportional t². The random motion log-log plot was found to have a slope of 0.75± 0.20 μm²/sec. This suggests that when the vesicles exhibited random motion, they had an average r² (μm²) that was

linear in relationship to time (sec) (Figure 1). While the value was slightly lower than 1, we still feel justified in this linear association as there may have been some error in the manual tracking of the vesicles due to their fast movement and sometimes low contrast. This is seen as the r² value is around 0.6 suggesting that the data is slightly variable. When 2 outliers are removed from the data, the graphs slope and r² values significantly increased supporting our hypothesis (Figure 4).

Figure 4: The log-log plot of the random motion was replotted with the removal of 2 outliers to determine if the data evaluation of a linear relationship between r² average (μm²) and time (sec) remained true.

The Directed motion log-log plot was found to have a slope of 2.11 ± 0.08 μm²/sec suggesting a squared relationship. This supported our claim that the vesicles selected where exhibiting directed motion as their r² averages were proportional to r² (sec) (Figure 2). Given the r² value of 0.99 and the low levels of error (±0.08) in the slope, this evidence is statistically significant. To then determine the viscosity coefficient of the onion cell, an equation was derived using our assumptions, given parameters, and experimental model. It was determined that the '

e E ATP . The viscosity coefficient for directed 6 πrsv motion was determined to be 3.05*10⁻⁵ N*s*m⁻ ². With regards to viscosity, error can come from two sources: terminal velocity calculated and our estimate. Our estimate does not have a numerical error margin, but terminal velocity did: ±0.491551 μm/sec. Given the fact that terminal velocity was converted into units of m/sec, this error has no significant effect on our viscosity outcome. There is also evidence for accuracy on the directed motion velocity track through its high r² value: 0.9982. To determine the diffusion coefficient of the vesicles under random motion, the average r² (μm²) was plotted against time (sec) and the slope was used in the equation 4D=r²/T to calculate viscosity coefficient would be equivalent to

μ=

a diffusion coefficient of 130.5 ±55.36 μm²/sec (Figure 4). The high level of error within the slope suggests that some data points may be outliers. Given the random motion of the vesicles was difficult to track, we considered this to still be accurate as the R² was 0.86. Evaluation: https://docs.google.com/document/d/1QlRja6o-jWatMaiXhs5hnbRib8iqEftxQOuPTKkeIo/edit?usp=sharing Based on the results, our data did match our hypothesis. The log-log r² vs t random motion graph had a slope that was close to one. This means that there was a proportional relationship between r² and t. Looking at the log-log plot of the r² vs t² directed motion graph showed the slope being close to 2 indicating that there was a proportional relationship between r² and t². In comparing our results to other groups there were some slight differences. For example, a Group 2’s log-log random motion graph had a slope that was a bit higher than ours. Their slope was 0.9 compared to our which was 0.7478. Thier log-log directed motion graph has a slope that was lower with a slope of 1.43 and our data had a slope of 2.1061. In addition, their diffusion constant was very different then one we calculated. Group 2 calculated a diffusion constant of 0.00545 μm²/sec where our group calculated a Diffusion constant of 130.5 μm²/sec. Although Group 4 had a similar value for the diffusion coefficient, 120.92 μm²/sec. This could also indicate that different vesicles tracked could impact the diffusion coefficient based on the vesicles movement over time. There were also major differences in velocity. Group 2 calculated an average velocity of 1.6045 μm/sec. We got an average velocity of 33.155 μm/sec. Some of these stark differences in data could have been due to mistakes in the data collection. We tracked the vesicles in the video using a manual tracker. This left room to lose track of the vesicales and could have resulted in a lot of discrepancies in the data. In addition, because it was difficult to track the vesicles this also led to us only being able to collect a small amount of data. If we were to do this experiment again it would be best to figure out how to use an automated tracker that could feasibly follow the vesicles. In addition, each person collected various parts of the data individually. This could mean that there could have been overlap in tracking the vesicles or at the very least could introduce a lot of discrepancies into the results. It would have been optimal if only one person was tracking the data. This experiment focused heavily on the concept of diffusion and movement of particles. This concept is seen throughout various disciplines and many other sciences. For example, the diffusion of molecules is extremely important to the biological processes in the body. How fast molecules like oxygen and CO2 can diffuse in water and across membranes is essential to knowing how our body operates. In terms of the movement of molecules, whether it be directed or random, this is also an important aspect of biological science. As seen in this experiment we measure vesicles moving in an onion. This process of moving vesicles in a directed motions across cells through the cytoplasm is important in the functioning of a cell. Oftentimes vesicles are instrumental in transporting proteins and various other materials outside of the cell. The use of onion cells to observe random motion (like in diffusion) and directional movement (like with

myosin and and kinesin motors), serves as a simple model in understanding bigger and more complex organisms....