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Exact on-lattice stochastic reaction-diffusion simulations using partial-propensity methods
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10.1063/1.3666988
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Affiliations:
1 MOSAIC Group, Institute of Theoretical Computer Science and Swiss Institute of Bioinformatics, ETH Zurich, CH–8092 Zürich, Switzerland
a) Electronic mail: rajeshr@ethz.ch.
b) Author to whom correspondence should be addressed. Electronic mail: ivos@ethz.ch.
J. Chem. Phys. 135, 244103 (2011)
/content/aip/journal/jcp/135/24/10.1063/1.3666988
http://aip.metastore.ingenta.com/content/aip/journal/jcp/135/24/10.1063/1.3666988

## Figures

FIG. 1.

Partitioning of the computation domain into subvolumes. (A–C) Different possibilities of a subdiving a box-shaped computational domain in one, two, and three dimensions, respectively. L x , L y , and L z are the edge lengths of the computational domain in each direction. K x , K y , and K z are the numbers of subvolumes of edge length h in each direction. (D) Diffusion is modeled as jump “reactions” to face-connected subvolumes. The same chemical in different subvolumes is treated as a different species. Unimolecular “diffusion reactions” convert species as shown.

FIG. 2.

Illustration of the binning of the total propensities of the subvolumes used for composition-rejection sampling of the next subvolume. The illustration shows a computational domain divided into 4 subvolumes. Points A and B refer to the example in main text used to explain rejection sampling.

FIG. 3.

The data structures in PSRD. The contents of the data structures shown corresponds to the example reaction network in Eq. (1) with 3 species and 4 reactions. We assume that the computational domain is divided into 4 subvolumes. In the illustration, , , and are the lumped specific probability rates of the “diffusion reactions” of species 1, 2, and 3 respectively. See main text for details.

FIG. 4.

Computational cost of PSRD and NSM for the aggregation model (Eq. (15)). (A) Computational cost Θ of PSRD (squares) and NSM (circles) as a function of the number of subvolumes N v with the size of the reaction network fixed to N = 10 (filled symbols) and N = 100 (empty symbols), respectively. The solid lines show the corresponding least-squares fits of the theoretical cost models: For N = 10, ΘPSRD ≈ 0.02861 log N v + 0.03925 N, ΘNSM ≈ 0.1095 log  N v + 0.00581 f r M + 0.00481(1 − f r)6N; for N = 100, ΘPSRD ≈ 0.04401 log N v + 0.003579 N, ΘNSM ≈ 0.288 log  N v + 0.001375 f r M + 0.001418(1 − f r)6N. We estimate for N = 10 and for N = 100. (B) Computational cost Θ of PSRD (squares) and NSM (circles) as a function of the number of species N in the reaction network with the number of subvolumes fixed to N v = 512 (filled symbols) and N v = 1000 (empty symbols), respectively. The solid lines show the corresponding least-squares fits of the theoretical cost models: For N v = 512, ΘPSRD ≈ 0.07559 log N v + 0.002258 N, ΘNSM ≈ 0.1356 log N v + 0.002784 f r(⌊N 2/2⌋ + N + 1) + 0.002633(1 − f r)6N; for N v = 1000, ΘPSRD ≈ 0.07205 log N v + 0.002777 N, ΘNSM ≈ 0.1198 log N v + 0.002762 f r(⌊N 2/2⌋ + N + 1) + 0.003163(1 − f r)6N. The fraction f r = 0.04 for N v = 512 and f r = 0.02 for N v = 1000.

FIG. 5.

Computational cost of PSRD and NSM for the linear chain model (Eq. (17)). (A) Computational cost Θ of PSRD (squares) and NSM (circles) as a function of the number of subvolumes N v with the size of the reaction network fixed to N = 10 (filled symbols) and N = 100 (empty symbols), respectively. The solid lines show the corresponding least-squares fits of the theoretical cost models: For N = 10, ΘPSRD ≈ 0.03312 log N v + 0.03703N, ΘNSM ≈ 0.08256 log N v + 0.02504 f r M + 0.002615(1 − f r)6N; for N = 100, ΘPSRD ≈ 0.04842 log N v + 0.003786N, ΘNSM ≈ 0.1428 log N v + 0.002934 f r M + 0.0001978(1 − f r)6N for N v ⪅ 512 and ΘPSRD ≈ 0.2923 log N v − 0.01199N, ΘNSM ≈ 0.5929 log N v + 0.0000008 f r M − 0.004924(1 − f r)6N for N v ⪆ 512. We estimate for N = 10 and for N = 100. (B) Computational cost Θ of PSRD (squares) and NSM (circles) as a function of the number of species N in the reaction network with the number of subvolumes fixed to N v = 512 (filled symbols) and N v = 1728 (empty symbols), respectively. The solid lines show the corresponding least-squares fits of the theoretical cost models: For N v = 512, ΘPSRD ≈ 0.03051 log N + 0.5291, ΘNSM ≈ 0.07885 log N + 0.5458; for N v = 1728, ΘPSRD ≈ 0.08479 log N + 0.5073, ΘNSM ≈ 0.1642 log N + 0.5561. The fraction f r = 0.06 for N v = 512 and f r = 0.03 for N v = 1728.

FIG. 6.

Normalized spatial concentration distribution of species S1 in the two-dimensional Gray-Scott reaction-diffusion system (Eq. (20)) for F = 0.04, k = 0.06, k 1 = 1, and D 1 = 2D 2 = 2 × 10−5 in a square computational domain of area 0.642, divided into N v = 642 subvolumes (or subareas) of edge length h = 0.01. The concentration in each subvolume is shown as a color ranging from blue (concentration zero) to red (concentration one). (A, B) Concentration distributions, normalized by u, obtained using PSRD for u = 106 (A) and u = 107 (B), respectively. (C) Normalized concentration distribution obtained from a deterministic simulation using the same parameters, simulated using second-order finite differences. All snapshots are taken at final time t f = 2000/(k 1 u 2).

FIG. 7.

Normalized spatial concentration distribution of species S1 in the three-dimensional Gray-Scott reaction-diffusion system for F = 0.04, k = 0.06, k 1 = 1, and D 1 = 2D 2 = 2 × 10−5 in a cubic computational domain of volume 0.643, divided into N v = 643 subvolumes of edge length h = 0.01. The concentration in each subvolume, normalized by u = 108, is shown as a color ranging from blue (concentration zero) to red (concentration one). The snapshot is taken at final time t f = 2000/(k 1 u 2).

## Tables

Table I.

The detailed algorithm of PSRD.

/content/aip/journal/jcp/135/24/10.1063/1.3666988
2011-12-23
2014-04-16

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