Mechanisms of noise-resistance in genetic oscillatorsJose´M. G. Vilar*†, Hao Yuan Kueh*, Naama Barkai‡, and Stanislas Leibler*†§*Howard Hughes Medical Institute, Departments of Molecular Biology and Physics, Princeton University, Princeton, NJ 08544;†The Rockefeller University,1230 York Avenue, New York, NY 10021; and‡Departments of Molecular Genetics and Physics of Complex Systems, Weizmann Institute of Science,Rehovot 76100, IsraelCommunicated by Michael E. Fisher, University of Maryland, College Park, MD, March 7, 2002 (received for review November 6, 2001)A wide range of organisms use circadian clocks to keep internalsense of daily time and regulate their behavior accordingly. Mostof these clocks use intracellular genetic networks based on positiveand negative regulatory elements. The integration of these ‘‘cir-cuits’’ at the cellular level imposes strong constraints on theirfunctioning and design. Here, we study a recently proposed model[Barkai, N. & Leibler, S. (2000) Nature (London), 403, 267–268] thatincorporates just the essential elements found experimentally. Weshow that this type of oscillator is driven mainly by two elements:the concentration of a repressor protein and the dynamics of anactivator protein forming an inactive complex with the repressor.Thus, the clock does not need to rely on mRNA dynamics tooscillate, which makes it especially resistant to fluctuations. Oscil-lations can be present even when the time average of the numberof mRNA molecules goes below one. Under some conditions, thisoscillator is not only resistant to but, paradoxically, also enhancedby the intrinsic biochemical noise.The environment changes in a highly periodic manner. Thereare, among other changes, daily cycles of light and dark aswell as annual cycles of changing climates and physical condi-tions. Such environmental periodicity may create the necessityfor organisms to develop internal time-keeping mechanisms toaccurately anticipate these external changes and modify theirstate accordingly (1). In particular, a wide range of organisms,as diverse as cyanobacteria and mammals, have evolved circa-dian rhythms—biological clocks with a period of about 24 h thatevoke and regulate physiological and biochemical changes tobest suit different times of the day.Recent findings show that the molecular mechanisms uponwhich these clocks rely share many common features amongspecies (2). The main characteristic is the presence of intracel-lular transcription regulation networks with a set of clockelements that give rise to stable oscillations in gene expression.A positive element activates genes coupled to the circadian clock.It simultaneously promotes the expression of a negative element,which in turn represses the positive element. The cycle completesitself upon degradation of the negative element and re-expression of the positive element.A crucial feature of circadian clocks is the ability to maintaina constant period over a wide range of internal and externalfluctuations (1). Such robustness ensures that the clock runsaccurately and triggers the expression of clock-dependent genesat the appropriate time of the day. For instance, fluctuations intemperature affect chemical reaction rates and may perturboscillatory behavior. Another source of fluctuations may be thepresence of internal noise caused by the stochastic nature ofchemical reactions (3). Low numbers of molecules may beresponsible for random fluctuations that can destabilize theoscillatory behavior of the biochemical network (4). Yet,circadian clocks maintain a fairly constant period amidst suchfluctuations.Description of the Model. To study possible strategies, or princi-ples, that biological systems may use to minimize the effect ofstochastic noise on circadian clocks, we examined a minimalmodel based on the common positive and negative controlelements found experimentally (3). This model is described inFig. 1. It involves two genes, an activator A and a repressor R,which are transcribed into mRNA and subsequently translatedinto protein. The activator A binds to the A and R promoters,which increases their transcription rate. Thus, A acts as thepositive element in transcription, whereas R acts as the negativeelement by sequestering the activator.The deterministic dynamics of the model is given by the set ofreaction rate equationsdDA/dt ⫽␪AD⬘A⫺␥ADAAdDR/dt ⫽␪RD⬘R⫺␥RDRAdD⬘A/dt ⫽␥ADAA ⫺␪AD⬘AdD⬘R/dt ⫽␥RDRA ⫺␪RD⬘RdMA/dt ⫽␣⬘AD⬘A⫹␣ADA⫺␦MAMAdA/dt ⫽␤AMA⫹␪AD⬘A⫹␪RD⬘R⫺ A共␥ADA⫹␥RDR⫹␥CR ⫹␦A兲dMR/dt ⫽␣⬘RD⬘R⫹␣RDR⫺␦MRMRdR/dt ⫽␤RMR⫺␥CAR ⫹␦AC ⫺␦RRdC/dt ⫽␥CAR ⫺␦AC,[1]where the variables and constants are as described in the captionfor Fig. 1. This simple model is not intended to reproduce theparticular details of each organism but to grasp the propertiesthat the core design confers. As in any general model, theparameters of the values we use are typical ones. For instance,the rates for bimolecular reactions are all in the range ofdiffusion limited reactions.The preceding equations would be strictly valid in a well stirredmacroscopic reactor. At the cellular level, a more realisticapproach has to consider the intrinsic stochasticity of chemicalreactions (5), which can be done by transforming the reactionrates into probability transition rates and concentrations intonumbers of molecules. One then obtains the so-called masterequation, which gives the time evolution of the probability ofhaving a given number of molecules. There is no generalprocedure to solve this type of equation analytically, but it is thestarting point to simulate the stochastic behavior of the system.The basic idea behind such simulations is to perform a randomwalk through the possible states of the system, which are definedby the numbers of molecules of the different reacting species.Starting from a state with given numbers of molecules, theprobability of jumping to other states with different numbers ofmolecules (i.e., the probability for an elementary reaction tohappen) can be computed from the master equation. One canpick up a state and the jumping time according to that probabilitydistribution, consider the resulting state as a new initial state,§To whom reprint requests should be addressed at: The Rockefeller University, Box 34, 1230York Avenue, New York, NY 10021.The publication costs of this article were defrayed in part by