11 | Thermodynamic constraints

Question 1

Gibbs free energy is a measure of ______ energy.
All ______ systems (and thus, ______ , too) tend towards states of ______ ______ ______ ______ .
The chemical energy of a reaction is affected by two driving forces:
- The change in ______ , ______ , ie change in the ______ content of the reaction
- The change in ______ , ______, ie change in the ______ of the system

Answer 1

Gibbs free energy is a measure of chemical energy.
All chemical systems (and thus, reactions, too) tend towards states of minimum Gibbs free energy.
The chemical energy of a reaction is affected by two driving forces:
- The change in enthalpy, ∆𝐻, ie change in the heat content of the reaction
- The change in entropy, ∆𝑆, ie change in the disorder of the system

Question 2

What is the equation for Gibbs free energy using enthalpy and entropy?
What can be said about the conditions for measuring this?

Answer 2

• ∆G = ∆H − T∆S
• Can be measured at any set of conditions
• However, if the data are collected under standard-state conditions, we talk about standard-state free energy of reactions ∆G^o

Question 3

What is the equation for Gibbs free energy using gas constant, temperature, reaction quotient?

Answer 3

ΔG = ΔG⁰ + RT ln Q
* ΔG: Gibbs free energy change under actual conditions
* ΔG^o: standard Gibbs free energy change
* R: gas constant
* T: temperature in Kelvin
* Q: reaction quotient

Question 4

What’s the difference between ΔG and ΔG⁰ and how can these be related?

Answer 4

• ΔG^o: standard Gibbs free energy
• ΔG: actual free energy under cellular conditions
• Related by ΔG = ΔG⁰ + RT ln Q

Question 5

What does it mean if ΔG < 0?

Answer 5

ΔG < 0 → reaction proceeds forward

Question 6

What does it mean if ΔG = 0?

Answer 6

ΔG = 0 → equilibrium

Question 7

What does it mean if ΔG > 0?

Answer 7

ΔG > 0 → reaction proceeds in reverse

Question 8

What is an exergonic reaction?

Answer 8

• Favorable, or spontaneous reaction
• ∆G^{o < 0}

Question 9

What is an endergonic reaction?

Answer 9

• Unfavorable, or non-spontaneous reaction
• ∆G^{o > 0}

Question 10

∆G^o = ∆H^o − T∆S^o
Complete the table:
∆H^o / ∆S^o / ∆G^o / Reaction

Answer 10

∆H^o / ∆S^o / ∆G^o / Reaction
exothermic(–) / increase(+) / - / product-favored
endothermic(+) / decrease(-) / + / reactant-favored
exothermic(–) / decrease(-) / ? / T dependent
endothermic(+) / increase(+) / ? / T dependent

Question 11

What is ΔG⁰ and what two methods are there for calculating it?

Answer 11

Standard Gibbs free energy (under 1M, 298K, 1 atm)

Two methods of calculating ∆G^o
1. Determine ∆H^o and ∆S^o and use the Gibbs equation.
2. Use tabulated values of standard-state free energies of formation, ∆G_f^o.

Given by the difference between the free energy of the substance and the free energies of its elements in their thermodynamically most stable states at 1 atm.

(see google doc for example)

Question 12

How can ΔG⁰ be estimated if we have no information on ∆G_f^o, the standard state free energy of formation?

Answer 12

Mavrovouiniotis (1990) Biotechnology & Bioengineering proposed the group contribution method
Assumes molecules are built from additive functional groups
Basic principle:
Decompose a molecule into groups and identify higher order substructures
Have access to ∆G_f^o of individual groups

Question 13

What is the group contribution method?

Answer 13

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* Estimates ΔG⁰_f for metabolites
* Each molecule = sum of free energy contributions of its subgroups
* ΔG⁰_rxn = Σ ΔG⁰_f(products) – Σ ΔG⁰_f(reactants)

Question 14

What is the group contribution method used for?

Answer 14

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* To estimate ΔG⁰_f for compounds
* Allows ΔG⁰_rxn to be calculated from sub-structure data

Question 15

How does the group contribution method work?

Answer 15

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Each molecule is a sum of known functional groups
ΔG⁰_f = sum of group energies
ΔG⁰_rxn = Σ ΔG⁰_f(products) − Σ ΔG⁰_f(reactants)

Question 16

What does the reaction quotient Q represent?
Eg for reaction: aA + bB ⇌ cC + dD

Answer 16

Q = ([C]^c [D]^d) / ([A]^a [B]^b)
Describes ratio of product to substrate concentrations at a given time

Question 17

How do we define the reaction quotient Q_r(t) for a reaction:
∑_i=1ⁿ α_i S_i → ∑_i=1ⁿ β_i S_i

Answer 17

The reaction quotient is:
Q_r(t) = (∏_i=1ⁿ x_i(t)^β_i) / (∏_i=1ⁿ x_i(t)^α_i)
x_i(t): concentration of metabolite S_i at time t

Question 18

What is the role of Q_r(t) in thermodynamics?

Answer 18

It modifies ΔG⁰ to give ΔG
Used in: ΔG = ΔG⁰ + RT ln Q_r(t)
Accounts for actual concentrations in the system

Question 19

The reaction quotient ______ as a reaction proceeds.
The ______ of the ______ is governed by the Gibbs free energy ∆𝐺 = 𝑅𝑇𝑙𝑛(𝑄/𝐾_𝑒𝑞).
Where 𝐾_𝑒𝑞 is the ______ ______, independent of initial ______.

Answer 19

The reaction quotient changes as a reaction proceeds.
The direction of the change is governed by the Gibbs free energy ∆𝐺 = 𝑅𝑇𝑙𝑛(𝑄/𝐾_𝑒𝑞).
Where 𝐾_𝑒𝑞 is the equilibrium constant, independent of initial composition.

Question 20

The reaction proceeds in the ______ directions if ∆𝐺 < 0 (towards ______ values of 𝑄).

Answer 20

The reaction proceeds in the forward directions if ∆𝐺 < 0 (towards larger values of 𝑄).

Question 21

The reaction proceeds in the ______ direction if ∆𝐺 > 0 (towards ______ values of 𝑄).

Answer 21

The reaction proceeds in the reverse direction if ∆𝐺 > 0 (towards smaller values of 𝑄).

Question 22

How can thermodynamically infeasible stoichiometrically balanced cycles (SBCs) be eliminated using TMFA?

Answer 22

How can thermodynamically infeasible stoichiometrically balanced cycles (SBCs) be eliminated using TMFA?

Use an extended TMFA formulation as a MILP:

max_v,G,a c^Tv

s.t.

• Nv = 0
• ∀i, 1 ≤ i ≤ m₁ (internal):
  v_i,min(1 − a_i) ≤ v_i ≤ v_i,maxa_i
  G_i,mina_i + (1 − a_i) ≤ G_i ≤ −a_i + G_i,max(1 − a_i)
• P_int·G = 0
• ∀j, m₁ + 1 ≤ j ≤ m (exchange):
  v_j,min ≤ v_j ≤ v_j,max
• a_i ∈ {0,1}, G_i ∈ ℝ for i ∈ R_internal

The solution v is a thermodynamically feasible flux distribution without internal loops.

Question 23

How can the standard Gibbs free energy change ΔG° for all reactions be computed from standard formation energies of metabolites?

Answer 23

• Let ΔG°_f be a vector of standard formation Gibbs energies (per metabolite)
• Let S be the stoichiometric matrix
• Then: ΔG° = ΔG°_f^T · S
• This computes ΔG° for each reaction based on: ΔG° = ∑ (β_i − α_i) · ΔG°_f(S_i)

🧪 For a single reaction:

You have a reaction of the form:

∑_i=1ⁿ α_iS_i → ∑_i=1ⁿ β_iS_i

• α_i, β_i: Stoichiometric coefficients of substrates and products.
• S_i: Metabolites (reactants/products).

Then, the standard Gibbs energy change of the reaction is:

ΔG° = ∑_i=1ⁿ (β_i − α_i) · ΔG°_f(S_i)

This says: take the difference in coefficients (products − reactants), and weight that by the standard formation Gibbs free energy of each metabolite.

🧮 Matrix Form for Entire Network:
Let:

• ΔG°_f be a vector of standard formation Gibbs free energies — one for each metabolite.
• S be the stoichiometric matrix (size: metabolites × reactions)

Then the standard Gibbs free energy change of each reaction across the whole network can be written compactly as:

💡 ΔG° = ΔG°_f^T · S

This gives a vector ΔG° with one entry per reaction, computed using the weighted sum of formation energies based on stoichiometry.