MCB 195:
Tuesday 22nd
Problem Set #2
1.
System 1:
dx
x
dt
dy
 y
dt
a.
Given that at t=0 x=xo and y=yo find x(t) and y(t).
b.
Hand draw a phase portrait of this system.
c.
Find the fixed point.
d.
Write the Jacobian of this system.
e.
Find the eigenvalues and determine their corresponding eigenvectors (compare
this to 1a and b).
f.
What type of fixed point is this?
dx
 y 1
dt
Repeat 1b-f. (Don’t resolve a!)
dy
 x  2
dt
dx
 2 y  2
dt
Repeat 1b-f. (Don’t resolve a!)
dy
 x  2
dt
System 2:
g-k.
System 3:
l-p.
2.
Nutrient Uptake
Cells need to uptake nutrients (carbon, nitrogen, phosphorus, magnesium, etc.) from their
environment. Typically there are specific permeases which allow the passage of each
needed nutrient (glucose, fructose, ammonia, arginine, etc.) into the cell. In order to
ensure proper nutrient uptake from a potentially changing environment, regulation of
these transporters are need. In problem 3 and 4 we will examine two general ways in
which this regulation is achieved.
It will be useful to have a relationship for the percent occupancy of a permease (R) given
the external nutrient concentration (Nex) and the binding affinity of the nutrient to the
permease.
k on

k trans
Nex   R  koff Nex R  
R   N

R binds Nex at a rate kon to form NexR complex. This complex falls apart at a rate koff.
The complex transports the nutrient into the cell at a rate ktrans.
a.
Assuming that the transporter nutrient complex is at steady state write a
differential equation for the concentration of NexR.
b.
Solve for the concentration of NexR at steady-state in terms of [Nex], [R], kon, koff,
and ktrans.
N ex R
k trans  k off
write an equation for the percent occupancy
of
N ex R  R
k on
R in terms of Km and Nex.
c.
Using Km=
*Note In biological problems concentrations cannot be negative!
*
For problems 3 and 4 only list the biologically obtainable fixed points.
3.
In yeast, phophate is taken up from the environment (Pext) through a permease (R).
Intracellular phosphate (P) inhibits production of new transporter.
 Pext 
dP
 R - uP
 v
dt
K

P
 m ext 
 Ks 
dR
 - bR
 a
dt
 Ks  P 
u is the rate at which cells use phosphate.
b is the rate at which permease is degraded.
v is the maximal rate at which a single permease can transport phosphate
a is the maximal synthesis rate of permease.
a.
b.
c.
d.
e.
f.
h.
Draw a schematic of this system of equations (of the same general style as that in
Q2) showing all the species and arrows (labeling with the appropriate rate
constants).
 Pext 
 Pext 
 Ks 
 , Km, v
 R ; 
 , Ks and
In words, what do 
K

P
K

P
K

P
 m ext 
 s

 m ext 
 Ks 
 represent.
a
 Ks  P 
Solve for the fixed point(s) of this system
 Pext 
 with c.
Feel free to replace v
 K m  Pext 
Write the Jacobian for this system.
Assume all the constants are equal to 1 except Ks=0.5.
What type of fixed point(s) would we have?
Assume all the constants are equal to 1 except u=9 and a=4 .
What type of fixed point(s) would we have?
Hand draw one of the phase portraits (label).
4.
Lactose is taken up by bacteria from the environment (Lext) through a permease (R).
Intracellular lactose (L) stimulates production of new transporter. Assume that there is
no basal synthesis.
a.
b.
c.
d.
e.
f.
g.
h.
i.
Draw a schematic of this system of equations (of the same general style as that in
Q2) showing all the species and arrows (labeling with the appropriate rate
constants).
Rewrite the differential equations for 3 to describe this regulation.
Solve for the fixed point(s) of this system
 Lext 
 with c.
Feel free to replace v
K

L
ext 
 m
Write the Jacobian for this system.
Evaluate the Jacobian at one of the two fixed point(s).
Evaluate the stability of the fixed point(s).
Assume all the constants are equal to 1 except Ks are equal to 0.5. Evaluate both
fixed points. What type of fixed points are these?
Calculate the eignenvalues and their corresponding eigenvectors at each fixed
point using the values specified in g.
Hand draw one phase portrait (label).
5.
What would happen without feedback? Assume nutrient is taken up by bacteria from the
environment (Next) through a permease (R). Intracellular nutrient (N) does not affect the
production of new transporter.
a.
b.
c.
d.
e.
f.
g.
h.
6.
a.
b.
c.
7.
Draw a schematic of this system of equations (of the same general style as that in
Q2) showing all the species and arrows (labeling with the appropriate rate
constants).
Rewrite the differential equations for 3 to describe this regulation.
Solve for the fixed point(s) of this system
 Lext 
 with c.
Feel free to replace v
 K m  Lext 
Write the Jacobian for this system.
Evaluate the Jacobian at the fixed point(s).
Evaluate the stability of the fixed point(s).
Assume all the constants are equal to 1 except Ks are equal to 0.5. Evaluate the
fixed point(s)s. What type of fixed point(s) is this?
Hand draw one phase portrait (label).
What type of feedback occurs in problems 3, 4, and 5 respectively?
How does feedback affect the relationship between the external nutrient
concentration and the internal nutrient concentration?
Propose conditions under which each feedback system could be advantageous.
Read for section:
Ferrell JE Jr, and Machleder EM
The biochemical basis of an all-or-none cell fate switch in Xenopus oocytes.
Science. 1998 May 8;280(5365):895-8.