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PNAS 107 (18): 8171-8176

Copyright © 2010 by the National Academy of Sciences.

BIOLOGICAL SCIENCES / PHYSICAL SCIENCES / BIOPHYSICS AND COMPUTATIONAL BIOLOGY / APPLIED MATHEMATICS

Costs and constraints from time-delayed feedback in small gene regulatory motifs

Andreas Grönlunda, Per Lötstedtb, and Johan Elfc,1

aDepartment of Mathematics, Uppsala University, SE-751 06 Uppsala, Sweden; bDivision of Scientific Computing, Department of Information Technology, Uppsala University, SE-751 05 Uppsala, Sweden; and cDepartment for Cell and Molecular Biology, Uppsala University, SE-751 24 Uppsala, Sweden.

Edited by Peter G Wolynes, University of California, San Diego, La Jolla, CA, and approved March 16, 2010 (received for review November 17, 2009)

Abstract: The multistep character of transcription, translation, and protein modification inevitably leads to time delays between sensing gene regulatory signals and responding with changed concentrations of functional proteins. However, the interplay between the time-delayed and the stochastic nature of gene regulation has been poorly investigated. Here we present an extension of the linear noise approximation which makes it possible to estimate second moments—variances and covariances—of fluctuations around stationary states in time-delayed systems. The usefulness of the method is exemplified by analyzing two ubiquitous regulatory motifs. In the first system, we show that there is an optimal combination of transcriptional repression and direct product inhibition in determining the activity of an enzyme system. In particular, we demonstrate that direct product inhibition is necessary to avoid deleterious fluctuations in a system when the gene regulatory response is delayed. The second system is an anabolic motif where the substrate fluxes are balanced by time-delayed regulation responding to the substrate concentrations. The extended linear noise approximation makes it possible to show analytically that increased association rate between the substrates leads to a lower product flux because of increasing unbalance in substrate pools.

Key Words: biochemical networks • intrinsic noise

Author contributions: A.G., P.L., and J.E. designed research, performed research, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.0913317107/-/DCSupplemental.

1To whom correspondence should be addressed. E-mail: johan.elf{at}icm.uu.se.

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