20.309: Biological Instrumentation and Measurement
Signals and systems

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 AE in each box below.) In the row labeled “Description,” write a descriptive name of each system, such as “lowpass filter” or “overdamped secondorder 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
Phase Plots
Step Response Plots
Pole Zero Plots
Feedback systems

 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 the axes below 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. $

The Frequency Dependence of OsmoAdaptation in Saccharomyces cerevisiae
Read The Frequency Dependence of OsmoAdaptation 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.

Answer the following questions about The Frequency Dependence of OsmoAdaptation in S. cerevisiae:
 What is the primary mechanism by which S. cerevisiae recovers from hyperosmotic shock?
 What model did Mettetal, et. al. use for Hog1 activation in response to a hyperosmotic shock?
 Mettetal, et. al. found that that the hyperosmotic shock response of wildtype yeast was (choose one): underdamped, critically damped, or overdamped.
 The response of the mutant (low Pbs) yeast was (choose one): underdamped, critically damped, or overdamped.
 In the plot below, indicate which step response corresponds to the wildtype strain and the mutant strain.
f) Indicate which of the pole zero diagrams below corresponds to the wildtype strain and the mutant strain.
g) Indicate which curve in the Bode plots below corresponds to the wildtype strain and the mutant strain.
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Our goal for assignment 10 is to reproduce the bode plot in the paper (Figure 2 B and C), and fit it to a model secondorder system. We will only measure the 'wild type' yeast strain, since measuring the mutant would take too much time.
 What mathematical model did Mettetal, et. al. use for the yeast response network? Express the model in the following forms: transfer function (TF), poles and zeros (ZPK), single differential equation (SDE), and 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 $.
 Find an expression for the step response and plot it over a range of values of $ \omega_0 $ and $ \zeta $. A handdrawn plot is fine, but you should probably look into MATLAB's step function.
 Plot the frequency response (i.e. make a Bode plot) of the system over a range of $ \omega_0 $ and $ \zeta $ values that includes over damped, critically damped, and under damped.
 What are two questions that you have about the paper's methodology or how we're going to implement the experiment in 20.309?


Turn in all your MATLAB code in pdf format. No need to include functions that you used but did not modify.

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