Difference between revisions of "Spring 2020 Assignment 9"

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Unfortunately, we won't be collecting our own data in the lab this semester, but it's still important to have a feel for what the raw data looks like, and what ''signal'' we are measuring. Download the data file named <tt>'fall2019_StudentData_3.mat'</tt> from the course dropbox folder. This is raw data that was collected by 20.309 students in the Fall of 2019. Load the file into your MATLAB workspace, and you should see a variable called <tt>yeastOsmoticShockData</tt>. This is a struct which contains the movie data, along with some other relevant experimental parameters:
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<pre>
 +
>> yeastOsmoticShockData
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yeastOsmoticShockData =
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  struct with fields:
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                        Movie: [544×728×2×32 uint16]
 +
                          Time: [32×2 double]
 +
                    ValveState: [32×1 logical]
 +
        ValveOscillationPeriod: 480
 +
    BlueCameraGainAndExposure: [3 5000000]
 +
    GreenCameraGainAndExposure: [15 5000000]
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</pre>
 +
 +
Notice that the movie contains two colors (the third dimension of the movie has a length of 2). The movie matrix is constructed so that the first color represents the GFP-Hog1 signal, and the second color represents the nuclear signal (tagged with RFP).
 +
 +
Use <tt>implay</tt> to watch each color of the movie:
 +
<pre>
 +
implay(double(yeastOsmoticShockData.Movie(:,:,1,:))/4095);
 +
implay(double(yeastOsmoticShockData.Movie(:,:,2,:))/4095);
 +
</pre>
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Do they behave as you would expect?
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{{Template:Assignment Turn In|message =
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# Step through the frames of the movie using the implay controls. Identify a frame where the signal is "high" - in other words, where the Hog1 signal is localized in the nucleus. Turn in the frame number that you've identified and a screen shot of the GFP-Hog1 movie at that frame number.
 +
# Repeat for a frame where the signal is "low" - in other words, where the Hog1 signal is uniformly distributed throughout the cell. Turn in the frame number that you've identified and a screen shot of the GFP-Hog1 movie at that frame number.
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}}
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{{Template:20.309 bottom}}
 
{{Template:20.309 bottom}}

Revision as of 14:25, 28 April 2020

20.309: Biological Instrumentation and Measurement

ImageBar 774.jpg


Signals and systems


Pencil.png

Fill out the table below. Match each system function with its Bode magnitude and phase plot, step response, and pole zero diagram. (Write one letter A-E in each box below.) In the row labeled “Description,” write a descriptive name of each system, such as “low-pass filter” or “overdamped second-order system.”


System function $ \frac{1}{s+1} $ $ \frac{s}{s+1} $ $ \frac{s}{s^2+2s+1} $ $ \frac{s}{s^2+0.1s+1} $ $ \frac{1}{s^2+10s+1} $
Magnitude plot
Phase plot
Step response
Pole/zero plot
Description

Magnitude Plots

Transfer function matching magnitude plots.png

Phase Plots

Transfer function matching phase plots.png

Step Response Plots

Transfer function matching step response plots.png

Pole Zero Plots

Transfer function matching pole zero plots.png


Pencil.png
  • Use graphical methods to find the Fourier transform of the half-cosine pulse function x(t) plotted below, which consists of the positive half of a 1 Hz cosine, repeated forever at a rate of 1 Hz.
  • What is the lowest frequency component of x(t), not counting $ \hat{X}(0) $?


Cosine pulse function.png One way to create x(t) using functions that appear on the transform table is:

  1. multiply a cosine by a rectangle, and then
  2. convolve the result with the comb function $ \mathrm{III(}t)=\sum\limits_{n=-∞}^{∞} \delta(t-nT) $.

Use the diagram below to help you find the answer. The left column of shows signals in the time domain, and the right column shows the magnitude of the Fourier transform of each signal. The top right plot is filled in for you, plus a little hint that might help you make an accurate plot.

(The phase of the transforms in this problem is zero at all frequencies, so it is not plotted.)

Cosine pulse transform worksheet.png

Feedback systems


Pencil.png
  • Find the transfer function H ̂(s)=(V ̂_out (s))/(V ̂_in (s)) of the circuit shown below, assuming L = 1 H and R = 1Ω.
  • Plot the poles and zeros of H ̂(s) on a set of axes using x’s for poles and o’s for zeroes.
  • The circuit from Figure 1 is placed in a feedback system, as shown in the block diagram below. The triangle represents an amplifier with gain G that does not depend on frequency. Find the transfer function of the feedback system $ \hat{F}(s)=(\hat{Y}(s))/(\hat{X}(s)). $
  • Plot the poles and zeros of $ \hat{F}(s) $ for G=1,9,and 19. Label the gain value for each point.</math>


LR Low Pass Filter for S20 Assignment 9.png Feedback System Block Diagram for S20 Assignment 9.png


Pencil.png
  1. Find the transfer function of the circuit below for L = 1 H and R = 1 Ω and C=1 F.
  2. The circuit is placed in the same feedback system shown in the previous question. Plot the poles for gains of 1/4, 3/4, 5/4, 10/4, and 17/4. You may generate your plot by hand or use MATLAB.


LRC circuit for SP20 assignment 9.png

The Frequency Dependence of Osmo-Adaptation in Saccharomyces cerevisiae


Pencil.png

Read The Frequency Dependence of Osmo-Adaptation in Saccharomyces cerevisiae and the supporting information.. This paper will be the focus of exam 2. We will discuss the paper and the supporting information on Thursday and Friday (4/30 and 5/1).

Answer the following questions about The Frequency Dependence of Osmo-Adaptation in S. cerevisiae:

  1. What is the primary mechanism by which S. cerevisiae recovers from hyperosmotic shock?
  2. What mathematical model did Mettetal, et. al. use for Hog1 activation in response to a hyperosmotic shock? Express the model in the following forms
    • transfer function (TF)
    • poles and zeros (ZPK)
    • single differential equation (SDE)
    • coupled differential equations (CDE)
    Express the TF, SDE, and ZPK models in terms of the undamped natural frequency, $ \omega_0 $, damping ratio $ \zeta $, and/or damped natural frequency $ \omega_D $.
  3. What mathematical model did Mettetal, et. al. use to account for nonlinearities in the system?
  4. Plot the frequency response (i.e. make a Bode plot) of the model over a range of $ \omega_0 $ and $ \zeta $ values that includes over damped, critically damped, and under damped.
  5. Find an expression for the step response and plot it over a range of values of $ \omega_0 $ and $ \zeta $. A hand-drawn plot is fine, but you should probably look into MATLAB's step function.
  6. Mettetal, et. al. found that the hyperosmotic shock response of wild-type yeast was (choose one): underdamped, critically damped, or overdamped.
  7. The response of the mutant (low Pbs) yeast was (choose one): underdamped, critically damped, or overdamped.
  8. Which of the step responses below corresponds to Mettetal's model for the wild-type strain and the mutant strain (neglecting the nonlinear element)?
  9. Which of the Bode plots below corresponds to Mettetal's model for the wild-type strain and the mutant strain?
  10. Which of the pole zero diagrams below corresponds to Mettetal's model for the wild-type strain and the mutant strain?
  11. What are two questions that you have about the paper's methodology?

    12. </div>


Mettetal yeast model step response.png Mettetal yeast model Bode plots.png Mettetal yeast model pole zero diagrams.png

Unfortunately, we won't be collecting our own data in the lab this semester, but it's still important to have a feel for what the raw data looks like, and what signal we are measuring. Download the data file named 'fall2019_StudentData_3.mat' from the course dropbox folder. This is raw data that was collected by 20.309 students in the Fall of 2019. Load the file into your MATLAB workspace, and you should see a variable called yeastOsmoticShockData. This is a struct which contains the movie data, along with some other relevant experimental parameters: 

>> yeastOsmoticShockData

yeastOsmoticShockData = 

  struct with fields:

                         Movie: [544×728×2×32 uint16]
                          Time: [32×2 double]
                    ValveState: [32×1 logical]
        ValveOscillationPeriod: 480
     BlueCameraGainAndExposure: [3 5000000]
    GreenCameraGainAndExposure: [15 5000000]

Notice that the movie contains two colors (the third dimension of the movie has a length of 2). The movie matrix is constructed so that the first color represents the GFP-Hog1 signal, and the second color represents the nuclear signal (tagged with RFP).

Use implay to watch each color of the movie: 

implay(double(yeastOsmoticShockData.Movie(:,:,1,:))/4095);
implay(double(yeastOsmoticShockData.Movie(:,:,2,:))/4095);

Do they behave as you would expect?


Pencil.png
  1. Step through the frames of the movie using the implay controls. Identify a frame where the signal is "high" - in other words, where the Hog1 signal is localized in the nucleus. Turn in the frame number that you've identified and a screen shot of the GFP-Hog1 movie at that frame number.
  2. Repeat for a frame where the signal is "low" - in other words, where the Hog1 signal is uniformly distributed throughout the cell. Turn in the frame number that you've identified and a screen shot of the GFP-Hog1 movie at that frame number.


</div>
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