Sensitivity, robustness, and identifiability in stochastic chemical kinetics models

Abstract

We present a novel and simple method to numerically calculate Fisher information matrices for stochastic chemical kinetics models. The linear noise approximation is used to derive model equations and a likelihood function that leads to an efficient computational algorithm. Our approach reduces the problem of calculating the Fisher information matrix to solving a set of ordinary differential equations. This is the first method to compute Fisher information for stochastic chemical kinetics models without the need for Monte Carlo simulations. This methodology is then used to study sensitivity, robustness, and parameter identifiability in stochastic chemical kinetics models. We show that significant differences exist between stochastic and deterministic models as well as between stochastic models with time-series and time-point measurements. We demonstrate that these discrepancies arise from the variability in molecule numbers, correlations between species, and temporal correlations and show how this approach can be used in the analysis and design of experiments probing stochastic processes at the cellular level. The algorithm has been implemented as a Matlab package and is available from the authors upon request.

Fig. 1.

Determinant of FIM plotted against sampling frequency Δ (in hours). We used logarithms of four parameter sets (see Table S1). Sets 1 and 3 correspond to slow protein degradation (γ_p = 0.7); and sets 2 and 4 describe fast protein degradation (γ_p = 1.2). We assumed that 50 measurements (n = 50) of protein levels were taken from the stationary state. Observed maximum in information content results from the balance between independence and correlation of measurements.

Fig. 2.

Neutral spaces for TS and DT versions of the model of single gene expression for logs of parameters k_r and γ_p. (Left) Differences resulting from RNA, protein correlation: ρ_rp = 0.1 (Top) ρ_rp = 0.5 (Middle), ρ_rp = 0.9 (Bottom). Correlation 0.5 corresponds to parameter set 3 from Table S1 and was varied by equal-scaling of parameters k_p, γ_p. (Right) Differences resulting from temporal correlations. We assumed n = 50 and tuned correlation between observation by changing sampling frequency Δ = 0.3 h (Left) Δ = 3 h (Center) Δ = 30 h (Right). Set 3 of parameters was used (Table S1).

Fig. 3.

(Left) Diagonal elements of FIM for TS and TP versions of p53 model. Values of FIM for DT verison are not presented as they can not be compared with those for stochastic models. (Right) Sensitivity coefficients for TS, TP, DT version of p53 model. FIMs were calculated for parameters presented in Table S4.

Fig. 4.

Neutral spaces for TS, TP, and DT versions of p53 model for logs of two parameter pairs (a₀,a_k) and (b_x,a_y). The left column presents differences resulting form general variability, correlations between species and temporal correlation (comparison of TS and TP models). The right column shows differences due to variability and correlation between species (comparison of TS and TP models). The top row is an example of parameters for which differences are negligible, bottom row presents a parameter pair with substantial differences. FIM was calculated for 30 measurements of all model variables and Δ = 1 h.