3 | Deterministic reaction kinetics: the law of mass action

Question 1

The (volume-dependent) in vivo infection rate constant of HIV infecting a CD4+ T-cell

HIV + CD4+T-cel l⟶ ( β^V _invivo) infected cell

has been estimated to be β^V_invivo = 8⋅10⁻¹² per day.

Assume that in vivo infection takes place in the plasma and interstitial lymph with average volume 41 ml per kg body weight and 121 ml per kg body weight, respectively, and that measurements were taken in a 70 kg adult.

What is the infection rate constant per day in an in vitro experiment with volume V_invitro = 200μL?

Note: give the answer in units 10⁻⁷ per day, round to one decimal digit and provide the decimal point as a comma, for example if the answer was 2.162⋅10⁻⁷ per day you would have to enter 2,2.

(ungraded quiz)

Answer 1

• We have a second-order reaction. Hence, β^V = β / N_AV
• We calculate: β^V_invitro = β / N_AV_invitro = β^V_invivo * ( V_invivo / V_invitro )
• According to the instructions, V_invivo= (0,041 + 0,121)L/kg⋅70kg = 11,34L.
• Doing the calculations, we obtain β^V_invitro = 4,536⋅10⁻⁷ per day.

–> The correct answer is: 4,5

Question 2

Consider the following reaction system:
R1: A+B→C | R2∗→B | R3: B+C→2A | R4: C→∗

What is the corresponding ODE system (in molar conc and ordered ABC)?

a.
x′1 = −k1x1x2 + 2k2x2x3
x′2 = −k1x1x2 − k2x2x3
x′3 = k1x1x2 − k2x2x3

b.
x′1 = −k1x1x2 + 2k3x2x3
x′2 = −k1x1x2 + k2 − k3x2x3
x′3 = k1x1x2 − k3x2x3 − k4x3

c.
x′1 = k1x1x2 − k3x2x3
x′2 = k1x1x2 + k2 + k3x2x3
x′3 = − k1x1x2 + k3x2x3 − k4x3

d.
x′1 = k1x1x2 + 2k3x2x3
x′2 = k1x1x2 + k2x2 − k3x2x3
x′3 = −k1x1x2 − k3x2x3 − k4

(ungraded quiz)

Answer 2

b.
x′1 = −k1x1x2 + 2k3x2x3
x′2 = −k1x1x2 + k2 − k3x2x3
x′3 = k1x1x2 − k3x2x3 − k4x3

Question 3

Assume that a molecular species is present within a cell at concentration 5nM, and that the cell is growing exponentially with growth rate kgrowth = 0.1/h.

Assuming that the molecular species is neither produced nor degraded or involved in any reaction: what is its concentration after three hours?

(ungraded quiz)

Answer 3

Since no reactions take place, we obtain the ODE

x′=−kgrowthx,

which can be solved to yield

x(t)=x0e−kgrowtht

Using the stated numbers we get

x(3 h)=5 nMe−0.3=3.704 nM

The correct answer is: 3.7

Question 4

Consider the reaction scheme for species (A_n, A_c), within reaction volumes V_n and V_c, respectively:

A_n → A_c
A_c → ∗

What is the ODE system corresponding to this scheme, stated in molar concentrations (and ordered as stated above)?

a.
x₁’ = -k₁ V_c/V_n x₁
x₂’ = k₁ x₁ - k₂ x₂

b.
x₁’ = -k₁ x₁
x₂’ = k₁ V_n/V_c x₁ - k₂ x₂

c.
x₁’ = -k₁ V_n/V_c x₁
x₂’ = k₁ x₁ - k₂ x₂

d.
x₁’ = -k₁ x₁
x₂’ = k₁ x₁ - k₂ V_c/V_n x₂

e.
x₁’ = -k₁ x₁
x₂’ = k₁ x₁ - k₂ V_n/V_c x₂

f.
x₁’ = -k₁ x₁
x₂’ = k₁ V_c/V_n x₁ - k₂ x₂

Answer 4

The correct answer is:
b.
x₁’ = -k₁ x₁
x₂’ = k₁ V_n/V_c x₁ - k₂ x₂

Explanation:
k₂ V_c/V_n x₂:
This accounts for the conversion of A_n to A_c considering the volume ratios.

Question 5

What is the law of mass action according to Guldberg and Waage (1864-1879)?

Answer 5

The law of mass action states:
For an elementary reaction R_μ in a spatially homogeneous reaction system with constant volume:
The reaction rate α_μ is proportional to the product of the molar concentrations of the reactant molecules involved.

Mathematically expressed as:
α_μ(x(t)) = k_μ · x₁(t)^r_μ,1 · … · x_N(t)^r_μ,N

Where:
k_μ is the reaction rate constant
x_i(t) are molar concentrations
r_μ,i are stoichiometric coefficients

fundamental in chemical kinetics for describing reaction rates.