Chemical Langevin equation: A path-integral view of Gillespie's derivation

Abstract

In 2000, Gillespie rehabilitated the chemical Langevin equation (CLE) by describing two conditions that must be satisfied for it yield a valid approximation of the chemical master equation (CME). In this work, we construct an original path integral description of the CME, and show how applying Gillespie's two conditions to it directly leads to a path integral equivalent to the CLE.We compare this approach to the path integral equivalent of a large system size derivation, and show that they are qualitatively different. In particular, both approaches involve converting many sums into many integrals, and the difference between the two methods is essentially the difference between using the Euler-Maclaurin formula and using Riemann sums. Our results shed light on how path integrals can be used to conceptualize coarse-graining biochemical systems, and are readily generalizable.

Interpretable and tractable models of transcriptional noise for the rational design of single-molecule quantification experiments

“…We can encode these fluctuations by defining a multi-state promoter activated by : where a is the production rate and κ is the degradation rate. If the number of regulator molecules r ( t ) is very large, we can accurately approximate regulator birth and death dynamics as a real-valued stochastic process using the framework associated with the chemical Langevin equation (CLE) 39 , 59 . Under the assumptions of rapid, weak binding, the effective transcription rate K ( t ) ≔ θ r ( t ) satisfies the SDE where ξ ( t ) is a Gaussian white noise term and θ = k ini k on / k off (see Section 3.3.1 in the Supplementary Note) .…”

“…The CIR model is solved using a state space path integral representation of P ( x N , x M , K , t ) which combines a path integral representation of the CME 59 with a more conventional continuous state space path integral. The Γ-OU model can also be solved using this method, along with a plethora of other discrete-continuous hybrid models.…”

Interpretable and tractable models of transcriptional noise for the rational design of single-molecule quantification experiments

“…The following reaction list (𝒩: nascent RNA, ℛ: regulator) crudely models this idea: where a is the ℛ production rate, κ is the ℛ degradation rate, and θ is the ‘gain’ relating the number of regulator molecules to the rate of transcription. If the number of regulator molecules r ( t ) is very large, we can accurately approximate regulator dynamics as a continuous stochastic process using the framework associated with the chemical Langevin equation (38, 55). The effective transcription rate K ( t ) := θ r ( t ) satisfies the SDE where ξ ( t ) is a Gaussian white noise term (see Section S3.3.1).…”

“…This implies that the joint distribution of the downstream species coincides with the system driven by Γ-OU transcription. The generating function of SDE-driven system can be computed using the solution of the bursty system, reported in Equation 11, where U 0 ( s ; u N , u M ) = A 0 e −κs + A 1 e −βs + A 2 e −γs can be computed by solving Equation 12: The CIR model is solved using a state space path integral representation of P ( x N , x M , K, t ) which combines the path integral representation of the CME from (55) with a more conventional continuous state space path integral. The Γ-OU model can also be solved using this method, along with a plethora of other discrete-continuous hybrid models.…”

Studying stochastic systems biology of the cell with single-cell genomics data

“…This PDE, for a large class of biologically relevant systems, can then be solved using the method of characteristics 98 , which converts the problem of solving a PDE into integrating a system of ordinary differential equations (ODEs). This is mathematically equivalent to using certain path integral methods 17,20,99 . Define the generating functions (GFs)…”