2015 exam

Question 1

Write the stochiometric matrix for a biochemical network with fluxes according to:
-> A -> 2B -> C ->

Assume that the flux from A to B in a) is described by a non-reversible Hill equation with Hill
coefficient 2, and that all other fluxes are described by non-reversible MM-eqations. Write the
equations for all the fluxes.

Explain/describe the prerequisites (assumptions) for the MM-equations.

Answer 1

d/dt (A,B,C) = N*(j_in, j1, j2, j_ut
N = (1 -1 0 0
0 2 -2 0
0 0 1 -1)

Note, we do not know we are at steady-state. Also note that info on
sensitivity (Hill etc) does not really affect fluxes per se.
dA/dt = 1*j_in-1*j1
dB/dt = 2*j1-2*j_2
dC/dt = 1*j2-1*j_ut
or 
j_in = -k_0*A
j_1 = k_1*A - k_-1*B^2
j_2 = k_2*B^2 - k_-2*C
j_out = k_3*C

no spontaneous conversion between substrate to product
the catalytic step is irreversible
the product does not rebind to the enzyme
the disociation step is very fast compared to the catalytic step
(and in addition the solution should be well mixed and at steady-state)

Question 2

Below follows a number of case descriptions of genetic networks. For each case, what kind of
model would you choose, and what in the description made you make this choise? If you think
that a description does not fit into any of the model types (including the assumptions they build
upon) you should label the case as ”Difficult case” and explain the problem.

a) A fungus(family) is studied. A very large number of genes have been observed. All reactants
exist in large numbers during the study, and the reactions are saturating. Transitions between
expressed and not expressed genes are in all cases rapid.

b) Sampling from a crime scene. The collected volume is very small and genes exist in quite
small numbers. Measurements show clear signs of stochasticity in the reactions.

c) A very small bacteria that lives deep in earths crust is studied. The reaction system that is
studied is quite small, about 0.1*10-15 litres. There are observations that at least 2 of the reactions
have an extremely rapid kinetics even at the low twmperature of +4 degrees centigrade at which
the experiment is conducted at.

d) A large volume is used in an industrial process and all genes and all reactants exist in great
numbers. The temperature is not controlled but is at room temperature, about 20 degrees
centigrade. The system shows good signal to noise ratio during measurements. The signal
pathways that are found seem unique and free from randomness.

Answer 2

a)
binary model
-large number of reactants / large volume gives large N, points to a differential eq might be used (which also is required for a boolean model)
-large number of genes points to probles using a differential eq model as it does not scale well with number of components
-saturating reactions, needed for a boolean model
-rapid transitions means a binary representation is ok

b)
stochastic model
-reactants are in small numbers / the volume is small, points to a stochastic model
-stochasticity in the reactions points to use of a stochastic model

c)
Difficult case.
-A very small volume points to a stochastic model, but as there are processes with extremely rapid dynamics, the system is not well mixed. The fact that temperature is low makes diffusion slower, which adds to the problem.

d)
continious differential eq model
-large number of reactants / large volume gives large N, e.g. continious representation ok
-good signal to noise relationship points to small fluctuations
-signal pathways are unique and free from randomness points to a deterministic system

Question 3

Give the equation that describes the current through a cloride leak channel. Leak channels are
linear channels, that is they have a purely resistive/fixed voltage characteristic. Also make a
figure of the electrical equivalent circuit for a very small cell that only has this channel.

Answer 3

I = Gcl * (V - Vcl)

in

   ---------------
   |        |       |
   |        -      Gcl	  
 Gm   -Cm   |
   |        |       - Vcl
   |        |      ---
   |        |       |
   ---------------
           |
out    ---
          -

Question 4

For the giant squid axon, concentrations inside are the following: [K+] = 400, [Na+] = 50, [Cl-] =
40. At the outside, concentrations are: [K+] = 10, [Na+] = 460, [Cl-] = 540. Consider a case at
temperature 20 degrees C, axon diameter 1mm and axon length 45 mm. At resting potential the
ratio of the permeabilities PK:PNa:PCl are 1:0.03:0.1. During the action potential the ratio of the
permeabilities are 1:15:0.1. Calculate the ”resting” potential during the action potential.

Answer 4

Vm=58 log [ (1(10) + 15(460) + 0.1(40)) / (1(400) + 15(50) + 0.1(540))] = +44mV

Question 5

Assume you have a long dendrite of constant radius. Assume a constant density of calcium
channels and calcium pumps in the membrane. Describe the type of model to use to model the
calcium inside this dendrite

Answer 5

I use a “onion” type model with iso-concentration shells. Cylindrical
symmetry gives cylindrical shells oriented aling the symmetry
axis. Diffusion occurrs only along the radial direction. Use shells of
thickness delta_r. If the innermost shell (which is not a shell but a
cylinder, gets a radius r

Question 6

a model with no spatial coordinate can be simulated as a ______ model

Answer 6

point

Question 7

models use simulation as a calcium pore through the _______ and calcium decay due to ______

Answer 7

channels

buffers

Question 8

If a model is thick you need a ________ model because theres _____________

Answer 8

onion

thickness along the core

Question 9

long dendrite of constant radius =

Answer 9

cylinder, constant density of calcium

Question 10

diffusion occurs along

Answer 10

the radial direction