TY - JOUR

T1 - Variable-free exploration of stochastic models

T2 - A gene regulatory network example

AU - Erban, Radek

AU - Frewen, Thomas A.

AU - Wang, Xiao

AU - Elston, Timothy C.

AU - Coifman, Ronald

AU - Nadler, Boaz

AU - Kevrekidis, Ioannis G.

N1 - Funding Information:
This work was partially supported by DARPA [for four of the authors (T.A.F., R.C., I.G.K., and B.N.)], the Israel Science Foundation Grant No. 432/06 [to one of the authors (B.N.)], NIH Grant No. R01GM079271-01 [to two of the authors (T.C.E. and X.W.)], and the Biotechnology and Biological Sciences Research Council Grant No. BB/C508618/1 and Linacre College, University of Oxford [to one of the authors (R.E.)].

PY - 2007

Y1 - 2007

N2 - Finding coarse-grained, low-dimensional descriptions is an important task in the analysis of complex, stochastic models of gene regulatory networks. This task involves (a) identifying observables that best describe the state of these complex systems and (b) characterizing the dynamics of the observables. In a previous paper [R. Erban et al., J. Chem. Phys. 124, 084106 (2006)] the authors assumed that good observables were known a priori, and presented an equation-free approach to approximate coarse-grained quantities (i.e., effective drift and diffusion coefficients) that characterize the long-time behavior of the observables. Here we use diffusion maps [R. Coifman et al., Proc. Natl. Acad. Sci. U.S.A. 102, 7426 (2005)] to extract appropriate observables ("reduction coordinates") in an automated fashion; these involve the leading eigenvectors of a weighted Laplacian on a graph constructed from network simulation data. We present lifting and restriction procedures for translating between physical variables and these data-based observables. These procedures allow us to perform equation-free, coarse-grained computations characterizing the long-term dynamics through the design and processing of short bursts of stochastic simulation initialized at appropriate values of the data-based observables.

AB - Finding coarse-grained, low-dimensional descriptions is an important task in the analysis of complex, stochastic models of gene regulatory networks. This task involves (a) identifying observables that best describe the state of these complex systems and (b) characterizing the dynamics of the observables. In a previous paper [R. Erban et al., J. Chem. Phys. 124, 084106 (2006)] the authors assumed that good observables were known a priori, and presented an equation-free approach to approximate coarse-grained quantities (i.e., effective drift and diffusion coefficients) that characterize the long-time behavior of the observables. Here we use diffusion maps [R. Coifman et al., Proc. Natl. Acad. Sci. U.S.A. 102, 7426 (2005)] to extract appropriate observables ("reduction coordinates") in an automated fashion; these involve the leading eigenvectors of a weighted Laplacian on a graph constructed from network simulation data. We present lifting and restriction procedures for translating between physical variables and these data-based observables. These procedures allow us to perform equation-free, coarse-grained computations characterizing the long-term dynamics through the design and processing of short bursts of stochastic simulation initialized at appropriate values of the data-based observables.

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U2 - 10.1063/1.2718529

DO - 10.1063/1.2718529

M3 - Article

C2 - 17461667

AN - SCOPUS:34247367647

VL - 126

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 15

M1 - 155103

ER -