12 | Links between stochastic & deterministic approaches

Question 1

Why is Gillespie’s stochastic simulation algorithm not finishing in an acceptable computation time when using realistic parameter values? Explain how the hybrid algorithm can overcome this problem.
(2022_1)

Answer 1

chatgpt:

Gillespie’s SSA Limitations:
- High Reaction Rates: Frequent reactions require many small time steps.
- Large Populations: More molecules lead to more frequent reactions.
(- Stiffness: Systems with fast and slow reactions need small time steps for accuracy.)

Hybrid Algorithm Advantages:
- Timescale Separation: Partition of fast reactions –> det, slow reactions –> stoch.
- Reduced Stochastic Events: Fewer individual events to simulate.
- Adaptive Switching: Dynamically switches between stochastic and deterministic methods.
- Efficiency with Large Populations: Handles large populations more efficiently.

Summary:
The hybrid algorithm speeds up simulations by combining deterministic and stochastic methods, reducing computational load while maintaining accuracy.

Question 2

Let X_V denote the number of molecules in a stochastic reaction network with volume V, and x the concentration in the corresponding deterministic formulation. Which of the following statements about these two quantities is the thermodynamic limit?
(2023_1, 2020_2)

◦ If lim_V→∞X_V = x₀, then lim_V→∞X_V(t) = x(t)

◦ If lim_t→0X_V(t) / V = x₀, then lim_t→∞X_V(t) / V = x(t)

◦ If lim_V→∞X_V(0) / V = x₀, then lim_V→∞X_V(t) / V = x(t)

◦ If lim_t→0X_V(t) / V = x₀, then lim_V→∞X_V(t) / V = x(t)

◦ If lim_V→0X_V(0) / V = x₀, then lim_V→0X_V(t) / V = x(t)

◦ If lim_t→0X_V(t) = x₀, then lim_V→0X_V(t) = x(t)

Answer 2

◦ If lim_V→∞X_V(0) / V = x₀, then lim_V→∞X_V(t) / V = x(t)

Question 3

Mathematically state the thermodynamic limit
(2022_1)

Answer 3

The thermodynamic limit can be stated as follows. Given:

Deterministic setting:
concentration x^d(t) with x^d(0) = x^d₀

Stochastic setting:
number of molecules X^s_V(t) with lim_V→∞ X^s_V(0) / V = x^d₀
(V = volume)

Thermodynamic limit:
If lim_V→∞ X^s_V(0) / V = x^d₀,
then for each t > 0,
lim_V→∞ X^s_V(t) / V = x^d(t) (in probability)

More precisely, for each ε > 0:
lim_V→∞ P[|X^s_V(t) / V - x^d(t)| > ε] = 0

This statement describes how the stochastic behavior of a system approaches the deterministic behavior as the system size (volume V) approaches infinity, given that the initial conditions converge.

Question 4

Explain what the thermodynamic limit implies for the use of stochastic / deterministic models.
(2022_1)

Answer 4

• Thermodynamic limit bridges stochastic and deterministic descriptions of chemical reactions.
• States that as system size (volume) increases to infinity, while keeping conc constant, stochastic behavior approaches deterministic behavior.
• In practice means that for very small systems stochastic models maybe necessary to capture behavior accurately.
  For larger systems, deterministic models often suffice.

Stochastic Models - for small number of molecules
- Behavior: Significant random fluctuations (noise)
- Examples: Gene expression in a single cell

Deterministic Models - for large number of molecules
- Behavior: Random fluctuations average out, described by continuous differential equations
- Examples: Metabolic networks in a population of cell

Question 5

Thermodynamic limit in words

Answer 5

If lim_V→∞ X^s_V(0) / V = x^d₀,
initially, the concentration of molecules in the stochastic system matches the initial concentration 𝑥0 in the deterministic system when scaled by volume.

then for each t > 0,
lim_V→∞ X^s_V(t) / V = x^d(t) (in probability)
at any time t, the concentration of molecules in the stochastic system will match the concentration 𝑥(𝑡) in the deterministic system when scaled by volume.

Describes the relationship between the number of molecules in a stochastic reaction network and the concentration in the corresponding deterministic formulation as the volume
𝑉 approaches infinity.
–> the stochastic fluctuations average out, and the system’s behavior can be described deterministically.

Question 6

Let X_det / X_stoch denote the number of molecules in a deterministic / stochastic reaction network, respectively. For which of the following reaction schemes does the relationship X_det = E[X_stoch] hold?

a.
∗ → A
A → 2B
B → ∗

b.
∗ → A
2A → ∗

c.
A + B → A
∗ → A
∗ → B

d.
A → B
B → A

e.
∗ → A
A → B + C
B → ∗
C → ∗

Answer 6

a., d., e.

The relationship holds for linear reaction systems, where the propensity functions are linear in the number of molecules.

Question 7

Mean-field models for linear systems

Answer 7

Consider a linear reaction system (only zero- and first-order reactions).

Reaction propensities are
▶ either aµ(Xs(t)) = cµ
▶ or aµ(Xs(t)) = cµXs_i(t)
Expectation is linear ⇒2 E[aµ(Xs(t))] = aµ(E[Xs(t)]).

It can be shown that the “typical” trajectory (E[Xs(t)]) follows the deterministic model, i.e. E[Xs(t)] = Xd(t)!