2 | Deterministic & stochastic reaction kinetics: the common ground

Question 1

If 1000 molecules of a particular molecular species are present within a hepatocyte (a type of liver cell) with volume of 3.4⋅10−12
L, what is the molar concentration of this species in the considered hepatocyte?

Note: give your answer in nM (nano-molar), round your answer to one decimal digit and provide the decimal point as a comma, for example, if the answer was 5.472 nM, write 5,5

(ungraded quiz)

Answer 1

M = n/v = Number of molecules / (N</sub>A _{* V}hepatocyte

= ( 1000 / 6,022e23 ) / ( 3.4 / 10e12 )
= 1 / ( 6,022e8 * 3.4 )
= 4.88e10 M

= 0.5 nM

Concentration = Number of molecules / (N</sub>A _{* V}hepatocyte
= 1000 / (6,02e23mol</sup>-1 ^{* 3,4e−12L ≈ 0.488e9 mol/L = 0.488nM}

Question 2

Consider a reaction network involving the states (A,B,C,D)
(in this order). Give the state change vector corresponding to the reaction A + C → 2 ⋅ B

(ungraded quiz)

Answer 2

( -1, 2, -1, 0 )

Question 3

Select all second-order reactions among the following:

a.
A+B→C

b.
A→B+C

c.
2A→B

d.
∗→A+B

e.
A+B→C+D

(ungraded quiz)

Answer 3

c.
2A→B

e.
A+B→C+D

Question 4

The reaction system

NFκB_c ⟶ NFκB_n

NFκB_n + DNA ⇌ NFκB_n:DNA

IκB_n +NFκB_n ⇌ IκB_n:NFκB_n

IκB_n + NFkBκsub>n</sub>:DNA ⟶ IκB_n:NFκB_n + DNA

IκB_n:NFκB_n ⟶ IκB_c:NFκB_c

contains:
? zero-order reactions,
? first-order reactions and
? second-order reactions.

(ungraded quiz)

Answer 4

0 zero-order reactions,
4 first-order reactions and
3 second-order reactions.

Question 5

How many reactions of each order are present in the reaction
scheme?

∗ → A
A → 2B
B → ∗
C + A → C
C + B → B

? zero-order reactions
? first-order reactions
? second-order reactions

(2023_1, 2020_2)

Answer 5

1 zero-order reactions
2 first-order reactions
2 second-order reactions

Question 6

Constituents of reaction kinetics models?

Answer 6

• molecular species
• reaction compartments
• reactions
• reaction laws (det./stoch.)

Question 7

Model constituents: molecular species

✐2.1
A typical volume of an E. coli cell is 2 · 10^-15L. If a molecular species is present at a concentration of 1nM (nano-molar), what is the total number of molecules of this species in a single E. coli cell?

Answer 7

V = 2 * 10e-15 L
N_A = 6,022e23 / mol
C = 1 nM = 10e-9 mol/L

Molecules / L : N_A * C = …. = 6e14 / L

Molecules / E coli: n = N_A * V * C = … = 1.2 molecules

Question 8

Model constituents: molecular species

Avogadro constant?

Answer 8

N^A = 6,022 · 10²³ [1/mol]

Question 9

Model constituents: molecular species

Relationship between number of molecules and concentration?

Answer 9

Model constituents: molecular species

[number of molecules] = NA · V(t)· [molar concentration]

Question 10

Model constituents: molecular species

State of the system?

Answer 10

Different descriptions of the state of the system used:

in molar concentrations: x_i

in number of molecules: X_i

Question 11

Model constituents: molecular species

x and X?

relationship?

Answer 11

X_i : number of molecules

x_i : molar concentrations ( = [X_i] )

Relationship rewritten: x(t) = X(t) / ( N_A · V(t) )

Question 12

Model constituents: reactions

What is ν_µ ?

Answer 12

state change vector (= stoichiometric vector)

Question 13

Model constituents: reactions

What is r_µ ?

Answer 13

a coefficient for reactant species

Question 14

Model constituents: reactions

What is p_µ ?

Answer 14

a coefficient for a product species

Question 15

Model constituents: reactions

What is the state change vector for the reaction system below?

R1 : ∗ → IκB
R2 : IκB + NFκB → IκB:NFκB
R3 : IκB:NFκB → IκB + NFκB
R4 : IKK + IκB → IKK
R5 : IKK + IκB:NFκB → IKK + NFκB
R6 : IκB → ∗

X = (IκB, NFκB, IκB:NFκB, IKK)

Answer 15

= [ v₁, v₂, v₃, v₄, v₅, v₆]

R1 R2  R3  R4  R5  R6 
1   -1    1   -1    0   -1  (IκB) 
0   -1    1    0    1    0   (NFκB)
0    1   -1    0   -1    0   (IκB:NFκB) 
0    0    0    0    0    0   (IKK)

Question 16

Model constituents: reactions

Can different reactions have the same associated state
change vectors?

Answer 16

Yes, eg in NFkB system: degradation of IkB with or without IKK enzyme.

Question 17

Model constituents: reactions

complex and elementary reactions?

Answer 17

If reactant molecules react through intermediate states that can be
identified as separate molecules the reaction is called complex,
otherwise elementary

Question 18

Model constituents: reactions

✐2.3 Why is the complex reaction less probable than the several elementary reactions?

Answer 18

requires meeting of three species simultaneously under the right conditions (e.g. orientation, speed), vs. meeting of two species under right conditions

more difficult that (3) Molecules of the complex come to the right place/time

Question 19

Model constituents: reactions

Definition of reaction order?

Answer 19

For a reaction:

R_µ : r_µ1S₁ + . . . + r_µNS_N → product species

with stoichiometric coefficients r_µk ∈ ℕ₀, the order of reaction R_µ is defined as

ord(R_µ) = r_µ1 + . . . + r_N

A reaction R_µ is of k-th order, if ord(R_µ) = k.

Question 20

Model constituents: reaction laws (det./stoch.)

What is a reaction counter?

Answer 20

ζ_µ

number of times that reaction Rµ occurred in [0,t]

Question 21

Model constituents: reaction laws (det./stoch.)

What is ζ_µ ?

Answer 21

Reaction counter

number of times that reaction Rµ occurred in [0,t]

Question 22

Model constituents: reaction laws (det./stoch.)

ζ_µ for deterministic?

Answer 22

ζ_µ deterministic process related to the reaction rate

In the large volume/numbers of molecules limit, the deterministic
model is an approximation to the stochastic model

Question 23

Model constituents: reaction laws (det./stoch.)

when is stoch more appropriate / needed?

Answer 23

with low numbers of molecules, deterministic
and reaction kinetics differ a lot. Stochastic
framework is more appropriate.
E.g. infection probability with low initial infectious molecule exposure

Question 24

Model constituents: reaction laws (det./stoch.)

ζ_µ for stochastic?

Answer 24

ζ_µ stochastic process based on
collision probabilities of reactant molecules

Question 25

Conservation laws

how to find?

Answer 25

A (linear) conservation law is a linear combination of states that is
constant over time:

c1X1(t) + . . . + cNXN(t) = const.

⇒ find linear combinations of rows of ν adding up to (0, . . . , 0).

= either 1 row is all 0s, or find two rows that add up to all 0s.

Question 26

Conservation laws

dimensions of stoichiometric matrix?

Answer 26

the matrix ν has dimension
(number of species) × (number of reactions)

Question 27

Classify the following ODE: (dy(t)/dt) = e^t⋅y(t)

Is it homogeneous?
Is it linear?
Is it autonomous?

(stack quiz)

Answer 27

For a linear ODE to be homogeneous, all terms must be on one side of the equation, with zero on the other side. This equation can be rewritten as:
(d/dt)y(t) - e^t * y(t) = 0
Therefore, it is homogeneous.

Is it linear?
A linear ODE has the dependent variable and its derivatives appearing only to the first power, and not as arguments of other functions. In this case, y(t) and its derivative appear only to the first power, and e^t is a function of the independent variable t only.
Therefore, it is linear.

Is it autonomous?
An autonomous ODE does not explicitly depend on the independent variable (in this case, t). However, this equation contains e^t, which is an explicit function of t.
Therefore, it is not autonomous.

Question 28

Classify the following ODE: (dy(t)/dt) = y(t)⋅t³ / (1+y)

(stack quiz)

Answer 28

An ODE is linear if the dependent variable y(t) and its derivatives appear to the power of one and are not multiplied or divided by each other.
–> nonlinear

An ODE is autonomous if it does not explicitly depend on the independent variable t.
–> nonautonomous

Question 29

Rate constant units

What does the reaction rate need to be ?

Answer 29

[conc/time]

eg [µM/s]

Question 30

Rate constant units

If there are 0 reactants?

example?

Answer 30

[conc/time]

synthesis from * –> S
k_syn has units [µM/s]

Question 31

Rate constant units

If there is 1 reactant (unimolecular reaction)?

example?

Answer 31

[1/time] eg [µM/s]

catalysis from E:S –> E + P
k_cat has units [1/s]

Question 32

Rate constant units

If there are 2 reactants (bimolecular reaction)?

example?

Answer 32

[1/(conc*time)]

forming a complex from E + S –> E
k_on has units [1/(µM*s)]