A diagram technique for cumulant equations in biomolecular reaction networks with mass-action kinetics

We present a diagram technique for the derivation of cumulant equations with zero-cumulant closure for the approximate description of biomolecular reaction networks. Our approach is applicable both to the chemical master equation and to the Kramers-Moyal expansion. It allows one to “see” the structure of the equations. This facilitates the investigation of the properties of the approximation and the comparison with other approximation methods. As an example of this, we derive a transparent relation between the cumulant equations for the chemical master equation and for the Kramers-Moyal expansion.