Time-dependent product-form Poisson distributions for reaction networks with higher order complexes

Area of Science:

• Stochastic reaction networks
• Chemical kinetics
• Probability theory

Background:

• Complex balanced reaction networks exhibit stationary Poisson product distributions.
• The question of preserving this product-form distribution from an initial Poisson state is explored.

Purpose of the Study:

• To determine the necessary and sufficient conditions for a stochastically modeled reaction network's Poisson distribution to remain constant over time.
• To introduce a new condition governing the dynamics of such systems.

Main Methods:

• Drawing inspiration from Crispin Gardiner's "Poisson representation" for the chemical master equation.
• Analyzing the conditions under which an initial Poisson distribution is preserved.

Main Results:

• A necessary and sufficient condition, termed "dynamical and restricted complex balance" (DR), is derived.
• This condition applies specifically to reaction complexes with multiplicity greater than or equal to two, affecting non-linear dynamics.

Conclusions:

• The DR condition ensures the long-term preservation of Poisson product distributions in stochastic reaction networks.
• This finding offers new insights into the dynamics of complex chemical systems.