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A pathwise derivative approach to the computation of parameter sensitivities in discrete stochastic chemical systems
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10.1063/1.3677230
/content/aip/journal/jcp/136/3/10.1063/1.3677230
http://aip.metastore.ingenta.com/content/aip/journal/jcp/136/3/10.1063/1.3677230

Figures

Image of FIG. 1.
FIG. 1.

Monte Carlo approaches to computing sensitivity in stochastic models of chemical kinetics.

Image of FIG. 2.
FIG. 2.

Comparison of sensitivity estimates for E[X(T)|x 0 = 0] for the birth-death process with parameters c 1 = 10, c 2 = 0.5 as the final time (T) is varied, with the method parameters chosen for RPD and CRP to yield a fixed bias of 0.1%. (a) and (b) The sensitivity estimates from each method are compared with the analytical solutions (dashed line). (c) and (d) The variance of the sensitivity estimators at each time. (e) and (f) The algorithm parameter values (blue triangles: RPD, red x: CRP) used in each computation. All estimates were computed independently using N = 104 sample paths.

Image of FIG. 3.
FIG. 3.

Sensitivity analysis for E(1 {aX(T) ⩽ b}) for the birth-death process with parameters T = 10, c 1 = 10, c 2 = 0.5, and x 0 = 0. (a) The sensitivity estimates are shown for the methods indicated by color and compared with the analytical result (black circle). Error bars indicate 95% confidence intervals of the estimates. (b) The variance of the sensitivity estimates is shown, with the vertical axis in log scale. All estimates were obtained using N = 104 sample paths. For RPD and CRP, w and h were selected to achieve a 0.1% bias relative to the exact sensitivities (w = 0.978, h = 0.0069 for 15 ⩽ X(T) ⩽ 16; w = 0.876, h = 0.0098 for 10 ⩽ X(T) ⩽ 20). For 1 ⩽ X(T) ⩽ 100 w = 0.876 and h = 0.0024 were used.

Image of FIG. 4.
FIG. 4.

Comparison of sensitivity estimates for E[X(T)|x 0] for the birth-death process with parameters T = 3, c 1 = 10, c 2 = 0.5 as the initial state x 0 is varied. (a) and (b) The sensitivity estimates from each method are compared with the analytical solution (dashed line). (c) and (d) Variance of the sensitivity estimates from each method. The window lengths (w 1 = 0.289, w 2 = 0.309) and perturbation sizes (h 1 = 9.0, h 2 = 0.041) were chosen for RPD and CRP, respectively, to yield a fixed bias of 0.1% of the exact sensitivity with x 0 = 0. All estimates were computed using N = 104 sample paths.

Image of FIG. 5.
FIG. 5.

Reversible isomerization process with parameters c 1 = 0.3, c 2 = 0.2, X 1(0) = 50, X 2(0) = 0. (a) The evolution of the expected value for each species (dashed line) is shown and compared with a representative sample path (solid lines) generated using SSA. (b) Variance and bias of sensitivity estimates for E[X 1(T)] with respect to c 1 at T = 3 (solid) and T = 40 (dashed) obtained from N = 104 independent samples for all methods. Note the Girsanov estimator (green) is unbiased, and its variance from one ensemble of N = 104 samples is shown for reference only. The RPD estimates (triangles) and CRP estimates (x) were computed using parameters w and h, respectively, which achieved the identical relative bias indicated on the horizontal axis. Representative w and h values are labeled.

Image of FIG. 6.
FIG. 6.

Sensitivity analysis for E[A(T)] for the genetic oscillator at T = 5 using N = 104 samples at parameter values listed in Table I. (a) The sensitivity estimates are shown for the RPD method (w = 0.1) and the Girsanov method. Error bars indicate the 95% confidence intervals of the sensitivity estimates. (b) The estimator variance is shown for the respective methods, with the same coloring as in (a).

Image of FIG. 7.
FIG. 7.

Sensitivity analysis for the toggle switch example, at T = 10 using N = 105 samples and parameter values listed in the text. (a) Sensitivity estimates computed from the RPD, Girsanov, and CRP methods are compared with the exact result (1.1897), calculated using the finite state projection. Colored bands indicate the 95% confidence intervals of the point estimates for each method. RPD was computed using windows of varying widths between w = 10−4 and 100. A perturbation of size h = 0.01 was used for the CRP estimates. (b) Variance of the estimates.

Tables

Generic image for table
Table I.

Model reactions, propensity functions, and parameters for the genetic oscillator example.

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/content/aip/journal/jcp/136/3/10.1063/1.3677230
2012-01-20
2014-04-23
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: A pathwise derivative approach to the computation of parameter sensitivities in discrete stochastic chemical systems
http://aip.metastore.ingenta.com/content/aip/journal/jcp/136/3/10.1063/1.3677230
10.1063/1.3677230
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