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Steady-state simulation of metastable stochastic chemical systems
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10.1063/1.4804191
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Affiliations:
1 Automatic Control Lab, ETH Zurich, Physikstrasse 3, 8092 Zurich, Switzerland
a) Electronic mail: milias@control.ee.ethz.ch
J. Chem. Phys. 138, 184109 (2013)
/content/aip/journal/jcp/138/18/10.1063/1.4804191
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/18/10.1063/1.4804191

## Figures

FIG. 1.

A simulated trajectory for set 1, demonstrating the metastable behavior of the system. Blue line corresponds to protein X, and green to protein Y. Notice that when X is high, Y is low and vice versa, and that the system spends very little time in the region where the proteins have relatively the same abundance.

FIG. 2.

Logarithm of the invariant distribution for the first parameter set computed from a large finite truncation.

FIG. 3.

Approximation error for the invariant distribution for the first parameter set, obtained by SSA and complement-based simulation, plotted against the simulation of length time units for the censored chain. The results are based on 50 simulation runs. Empty circles and horizontal bands inside boxes denote means and medians, respectively, upper and lower ends of boxes correspond to 75th and 25th percentiles, and whiskers extend to extreme data points.

FIG. 4.

Logarithm of the invariant distribution for set 2 computed from a large finite truncation.

FIG. 5.

Approximation error for the invariant distribution of the second parameter set, obtained by complement-based simulation plotted against the simulation length of t.u. for the censored chain. The results are based on 20 simulation runs. Empty circles and horizontal bands inside boxes denote means and medians, respectively, upper and lower ends of boxes correspond to 75th and 25th percentiles, and box whiskers extend to approximately 95% percentiles. Outliers are denoted by dots.

FIG. 6.

Deviation from equality for the stationary probabilities of the two metastable regions, plotted against the simulation length for the censored chain. The results are based on 50 simulation runs. Empty circles and horizontal bands inside boxes denote means and medians, respectively, upper and lower ends of boxes correspond to 75th and 25th percentiles, and box whiskers extend to approximately 95% percentiles. Outliers are denoted by dots.

FIG. 7.

Logarithm of the steady-state marginal probability distribution of the total numbers of and molecules in the system, approximated from a simulation run of 5 × 10 t.u. for the censored chain. To produce the simulated trajectory that yielded this approximation, a total computation time of about 15 min (including matrix calculations) was needed.

## Tables

Table I.

The two largest eigenvalues of and , along with the average time spent in each metastable region prior to the first exit.

Table II.

Chemical reactions for the toggle switch model. denotes the common regulatory region of both genes, which can either be bound to ( ) or ( ). The reaction rates are taken from Ref. and are equal for the corresponding reactions of the two proteins, leading to a symmetric bimodal stationary distribution.

/content/aip/journal/jcp/138/18/10.1063/1.4804191
2013-05-13
2014-04-19

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