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Problem 10.26PP
Excitation-Inhibition Model from Systems Biology {Yang and Iglesias, 2005): In Dictyostelium 
cells, the activation of key signaling molecules involved in chemoattractant sensing can be 
modeled by the following third order linearized model. The external disturbance to the output 
transfer function is:
w d ) ' ' ( I + o ) ( j + 1)(J + y ) '
where, w is the external disturbance signal proportional to chemoattractant concentration, and y 
is the output which Is the fraction of active response regulators. Show that there is an alternate 
representation of the system with the “plant” transfer function 
____________O jlf ! ) __________
j2 + ( l + o + y )« + (a + y + o K ) ’
and the “feedback regulator"
j^ + ( l+ o + K)5 + (« + y + a y )
and the “feedback regulator"
It is known that a # 1 for this version of the model. Draw the feedback block diagram of the 
system showing the locations of the disturbance Input and the output. What is the significance of 
this particular representation of the system? What hidden system property does it reveal? Is the 
disturbance rejection a robust property for this system? Assume the system parameter values 
are a = 0.5 and y = 0.2, then plot the disturbance rejection response of the system for a unit step 
disturbance input.
Step-by-step solution
step 1 of 5
The external disturbance to the ou^ut transfer function is given by 
w M ( s + a ) ( s + l) ( s + y )
Where w is the external disturbance signal proportional to chemo attractant 
concentration, and^ is the output which is the fraction of active response regulators. 
It is known that a ^ Ifor this version of the model.
Step 2 of 5
Sketch the Feedback loop representatioa
Step 3 of 5 ^
Find the Lr^lace transform of the given equation
___________ 0 - “ )___________
^ ( ^ ) _ + (1 + Of + y)s + {a + y + ay)
1+ ay
+ (1 + Of + y)s + ( a + y + ay)
W(s) (s + a ) (« + l) ( f f + y)
Step 4 of 5
The significance of this particular representation is that it reveals the internal model, 
namely the pure integrator. Hence the system is Type I with respect to disturbance 
rejection. It rejects constant disturbances in a robust &shion
FindG(s) fo ra= 0.5 and y = 0.2.
(1 -0 .5 )b
0 ( s ) =
s" + (1 + 0.5 + 0.2)s + (0.5 + 0.2 + 0.5 X 0.2)
0 ( . ) =
0.5s
s’ + 1.7s + 0.8
Step 5 of 5 ^
The disturbance response is shown in the following figure.
Step Response
Time (sec)
Thus, the disturbance rejection response of the system is sketched.