6 | Model reduction: the quasi steady state approximation

Question 1

What is the QSS assumption for the following model (Michaelis-Menten)?

2S + E _koff⇌^k_on C →^kcat E + P

Answer 1

S’ = - V_max * S / ( K_m + S)

= rate of change of substrate S

(Briggs/Haldane)

Question 1

What are the conservation laws for the system below?

2S + E _koff⇌^k_on C →^kcat E + P

And what is the central idea behind this?

Answer 1

E + C = E_tot
S + C + P = S₀

Central idea: Substrate conversion to product much slower than complex formation → C is in quasi steady state with respect to S

Question 2

In quasi steady state assumption, one takes the ODE of which species and sets it to 0?

Answer 2

’ The fast species ‘

Question 3

We want to make a quasi-steady state assumption on X with respect to Y.

Which ODE do we set equal to 0?

Answer 3

X

Question 4

Which statements on a quasi-steady state approximation (QSSA) are true?

a. The reduced model aims to faithfully represent both the fast and the slow dynamics.
b. The reduced model can show properties of the system not directly visible in the original model.
c. Reactions are separated into slow and fast reactions.
d. The reduced model aims to faithfully represent the slow dynamics, but not the fast dynamics.
e. If one reaction rate constant is much larger than all others, this ensures the conditions for a QSSA.
f. Molecular species are separated into slow and fast species.

(ungraded quiz)

Answer 4

b. The reduced model can show properties of the system not directly visible in the original model.
f. Molecular species are separated into slow and fast species.
d. The reduced model aims to faithfully represent the slow dynamics, but not the fast dynamics.

Question 5

What is the relationship between quasi-steady state approximation (QSSA) and total quasi-steady state approximation (tQSSA)?

a. When the assumption of small initial perturbations is dropped in a QSSA, the resulting approximation is called tQSSA.
b. In tQSSA, all concentrations are replaced by total concentrations.
c. In tQSSA, a QSSA is made in a transformed set of ODEs.
d. In tQSSA, the set of ODEs is transformed after making a QSSA.

(ungraded quiz)

Answer 5

In tQSSA, a QSSA is made in a transformed set of ODEs.

Question 6

Consider the reaction system A →(k1) B →(k2) C, with xA(0)>0
and xB(0)=xC(0)=0. Assume k1»k2

Which of the following statements on model reduction for the reaction network is true?

a. There is a conservation law which allows to reduce the number of ODEs
b. All assumptions for a quasi-steady state assumption on B with respect to C are met
c. A total quasi-steady state assumption on A+B with respect to C is possible
d. The timescale condition for a quasi-steady state assumption on B with respect to C is violated
e. The small perturbation condition for a quasi-steady state assumption on B with respect to C is violated

(ungraded quiz)

Answer 6

a. There is a conservation law which allows to reduce the number of ODEs
c. A total quasi-steady state assumption on A+B with respect to C is possible
e. The small perturbation condition for a quasi-steady state assumption on B with respect to C is violated

Question 7

In the following questions, we consider the reaction scheme
∗ → X
X → ∗,
X + Y → ∗
given by the ODEs (in molar concentrations):
dx/dt = k1 − k2x − k3xy,
dy/dt =−k3xy
We want to make a quasi-steady state assumption on X (fast species) with respect to Y (slow species).

What is the quasi-steady state in this reaction system?
What is the reduced system in the example?
What is the response time τqss?

(ungraded quiz)

Answer 7

dy/dt = − k1k3y / (k2+k3y)

Question 8

Steps for QSS approximation?

Answer 8

The QSSA approximation involves four steps:
* Partition the system into slow and fast variables.
* Approximate fast variables by their quasi-steady state
* Determine approximate slow kinetics by substituting fast variables by their QSS.
* Check validity

Question 9

Validity of QSS approximation
Which two points?

Answer 9

• Time Scale Separation: 𝜏_qss ≪ 𝜏_red
• relative change / small perturbation of initial condition

Question 10

Validity of QSS approximation
Explain the condition of time scale separation

Answer 10

Condition:

τ_qss≪τ_appr

Explanation: This condition states that there must be a clear separation between the time scales of the fast and slow kinetics.

Specifically, the response time for the system to reach the quasi-steady state (τ_qss) must be much shorter than the response time of the reduced model (τ_appr).

This ensures that the fast dynamics settle quickly, allowing the slow dynamics to be accurately approximated by the reduced model.

Question 11

Validity of QSS approximation
Explain the condition of relative change

Answer 11

Condition:

∣ Δx_slow / x_slow,0 ∣ ≪ 1
(Eg 1 order of magnitude less so under 0.1)

Explanation: This condition ensures that the relative change in the slow variable during the initial fast transient is very small = Negligible relative change for slow variables during fast initial transient.

Here, x_slow,0 is the initial value of the slow variable, and x_slow(τ^∗_qss) is the value of the slow variable after the fast transient period (i.e., after the system has reached the quasi-steady state).

If the relative change is small, it means that the slow variable does not change significantly during the fast transient, validating the use of the quasi-steady state approximation.

Question 12

K_m = ?

Answer 12

Michaelis constant !
(K_off + k_cat) / k_on

Question 13

V_max= ?

Answer 13

= k_catE_tot

Question 14

Total quasi steady state assumption?

Answer 14

T’ = - V_max * S_qss / ( K_m + S_qss)

T = total amount of substrate = S + C

Approximation holds more generally than MM.

(Borghans et al)