Figure 1.
A framework for combining intrinsic and extrinsic noise.
(A) Distribution prediction starts with predicting distributions based on intrinsic noise only and then adds in extrinsic noise. (B) A schematic for analysis of molecule number distributions in biochemical networks. Predicted distributions based on a model can be compared to experimental data, and information about parameters can be inferred.
Figure 2.
A modified Gibbs sampling method in comparison to previous methods.
(A) The Hartree approximation can distort the joint distribution for multimodal distributions by generating false peaks. (B) Gibbs sampling method: sampling of each molecule number is based on current values of the other molecule numbers rather than on mean values, to avoid this distortion. This method can result in samples being “stuck” in one peak of the probability distribution. The blue arrows in (B) and (C) indicate sampling directions, which are used sequentially. (C) A modified Gibbs sampling method based on coordinate changes can avoid the sampling bias.
Figure 3.
Calculating steady-state distributions for a simple birth-death process.
Calculating the steady-state distribution of a (A) simple birth-death process; A is expressed in terms of molecule number for all distributions. (B) Simulated distribution using the Gillespie algorithm (histogram) as compared with the analytical solution (blue line) from the CME, which is a Poisson distribution. (C) Final distributions from this systems with extrinsic noise were generated by taking ks/kd∼Γ(20,1) (green in C) or ks/kd∼Γ(10,2) (green in D); best-fit distributions based on convolution with a Gaussian (red; standard deviations 4.4 for C and 6.1 for D) and the intrinsic-noise-only distribution (blue) are shown as well. (The shift to lower molecule numbers arising from extrinsic noise in the parameter-distribution representation is equivalent to changing the base parameter set in the convolution representation; the “base” parameter set is less well defined in the parameter-distribution representation).
Figure 4.
Calculating steady-state distributions for a toggle switch.
(A) Circuit diagram. (B) The probability distribution, based only on intrinsic noise, for parameters Ku = Kv = 1, ru = rv = 10, β = γ = 2, du = dv = 1. This was calculated by solving the CME directly. U and V are expressed in molecule numbers for each distribution. (C) The same distribution except with extrinsic noise added (σ1 = σ2 = 2); peak spreading is evident. (D) The intrinsic-noise-only distribution except with ru = rv = 1000; calculated using scaling; peak focusing is evident. (E) A sample (10000 points) from this distribution (intrinsic noise only), using modified Gibbs sampling using the same parameters as in (B). (F) Another 10,000-point sample, generated the same way except with extrinsic noise (σ1 = σ2 = 1).
Table 1.
Comparison of MGS and direct CME solution for the toggle switch.
Figure 5.
Toggle switch: comparison with experiments.
(A) A typical workflow for analyzing experimental data based on the framework for noise calculations presented in this study. Modes of the distribution(s) are determined and used with the deterministic model to estimate some reaction parameters (i.e. intrinsic noise parameters); prior knowledge may also be included in these estimates. Then, using the calculation methods (e.g. convolution for extrinsic noise) presented here, one can obtain a best fit for the full parameter set, which describes both intrinsic and extrinsic noise. (B) Theoretical (blue) and experimental (green) fraction of ON cells as a function of [NFX]. Inset shows the perturbation to the circuit. (C) Experimental distribution of GFP fluorescence in cells with 250 ng/mL NFX. (D) Predicted distribution of U molecule numbers (proportional to fluorescence) with 250 ng/mL NFX. Note that the distribution in (C) is used in (A) to illustrate the general computational procedure.
Figure 6.
(A) T7 RNAP enhances its own transcription. In addition, T7 RNAP expression slows down cell growth, which dilutes T7 RNAP. (B) Probability distribution with M = 10, k0 = 0.001, kf = 0.01, kb = 0.1, k1 = 0.01, dx0 = 0.003, μ = 0.01, and θ = 1; shown with (green) and without (blue) extrinsic noise (σ = 3). (C) Comparison of experiment and modeling on the perturbations to growth and T7 RNAP synthesis rates. Contours denote computed fractions of ON cells, by varying parameters k1 and μ. X's denote experimental data points with experimental parameters (IPTG concentration and OD) as labeled; the k1 and μ values for each data point were determined by fitting the fractions of ON cells.
Figure 7.
(A) A network diagram of Myc/Rb/E2F at the G1-S cell cycle checkpoint. (B) Comparison of the MGS predicted probability distribution (red) and the SDEs predicted distribution (green) of selected molecular species of the repressed and the activated state of the network. The solid lines represent Gaussian distributions fitted over all random samples corresponding to each method. (C) Convolution of extrinsic noise and combination of the repressed and the activated modes based on empirical data derived from SDE simulations, with the variance of the shift distribution separately optimized. Shown in red are MGS predicted distributions and green are SDEs predicted distributions of E2F with either low or high level of extrinsic noise as defined in the SDE framework. The SDE model and the associated parameters are described in detail in Lee et al [51].