Error vs runtime for the spectral method and stochastic simulation. Error is the Jensen-Shannon divergence [] between

*p*_{nm} obtained using the |

*n,m*〉 basis (via iterative solution of the original master equation) and that obtained using the |

*j,k*〉 basis [circles; cf. ], the |

*j,k*_{j}〉 basis [triangles; cf. ], the |

*n,k*_{n}〉 basis [squares; cf. ], the |

*j,k*_{n}〉 basis [diamonds; cf. ], or stochastic simulation [] (dots). Runtimes are scaled by that of the iterative solution, 150 s (in

matlab). Spectral basis data is obtained by varying

*K*, the cutoff in the eigenmode number

*k* of the second gene; simulation data is obtained by varying the integration time. The input distribution

[from which

*g*_{n} is calculated via ] is a mixture of two Poisson distributions with

*λ*_{1}=0.5,

*λ*_{2}=15, and

*π*_{1}=0.5. The regulation function

is a Hill function with

*q*_{−}=1,

*q*_{+}=11,

*n*_{0}=7, and

*ν*=2. The gauge choices used (cf. ) are

*g‾*=Σ

_{n} *p*_{n}g_{n}, q‾=Σ

_{n} *p*_{n}q_{n}, q‾_{n}=

*q*_{n}, and

*q‾*_{j}=Σ

_{n}〈

*j*|

*n*〉

*q*_{n}〈

*n*|

*j*〉. The cutoffs used are

*J*=80 for the eigenmode number

*j* of the first gene and

*N*=50 for the protein numbers

*n* and

*m*. Inset: the joint probability distribution

*p*_{nm}. The peak at low protein number extends to

*p*_{00}≈0.1.

## PubMed Commons