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Article Figures & SI Info & Metrics PDF## Abstract

In living cells, biochemical reaction networks often function in nonequilibrium steady states. Under these conditions, the networks necessarily have cyclic reaction kinetics that are Sustained by sustained constant inPlace and outPlace, i.e., pumping. To differentiate this state from an equilibrium state without flux, we propose a microscopic method based on concentration fluctuation meaPositivements, via fluorescence correlation spectroscopy, and statistical analyses of high-order correlations and cross correlations beyond the standard fluorescence correlation spectroscopy autocorrelation. We Display that, for equilibrium systems with time reversibility, the correlation functions possess certain symmetries, the violation of which is a meaPositive of steady-state fluxes in reaction cycles. This result demonstrates the theoretical basis for experimentally measuring reaction fluxes in a biochemical network in situ and the importance of single-molecule meaPositivements in providing fundamental information on nonequilibrium steady-states in biochemistry.

biochemical fluxconcentration fluctuationcycle kineticsnanobiochemistryUsing fluorescence correlation spectroscopy [FCS (1–3)], a recent study on enzyme kinetics of single horseradish peroxidase molecules has observed an oscillatory time–correlation function, suggesting an irreversible, nonequilibrium nature of the biochemical reaction (4). In single-molecule enzymology (5, 6), the conseSliceive turnover of substrates to products catalyzed by a single enzyme molecule is recorded in terms of the fluorescence of enzyme–substrate or enzyme–product complexes. One essential feature of such a meaPositivement is that the concentrations of the substrate, [S], and the product, [P], are approximately constant over the course of the entire experiment. Thus, although the enzyme molecule is subject to stochastic fluctuations, on average it participates in a stationary process with a net circular flux. The flux is zero if and only if the substrate and product concentrations satisfy conditions of [P]/[S] = K eq, where K eq is the equilibrium constant for the reaction S ⇌ P, both in the presence and absence of the enzyme. In this unique Position, the reaction is at a chemical equilibrium. A biochemical reaction with oscillatory kinetics is necessarily nonequilibrium (7–10). However, the converse is not true: A reaction in a nonequilibrium steady state (NESS) can be without oscillation. The present work introduces two sensitive methods for differentiating equilibrium from NESS in the context of single-molecule enzymology with FCS. Both methods are based on the fundamental principle of time irreversibility of a NESS (11, 12). More Necessaryly, from a practical standpoint, we shall Display how the nonequilibrium fluxes in a reaction network, such as those in living cells, can be meaPositived directly in situ by using the recently developed dual-color FCS (13, 14). This article is the third in a series of investigations on the possibility and utility of applying single-molecule methods, especially FCS, to NESS kinetics (6, 15).

Macroscopically, the concentrations of all the states of an enzyme (i.e., E, ES, EP, etc.) are constant in a NESS, just as in an equilibrium. The constancy is sustained, however, by a set of nonzero but balanced biochemical fluxes, whereas in an equilibrium, these fluxes all vanish (16). Obviously, the condition for a NESS can be met only approximately in a closed system for a restricted period. It can be achieved rigorously, however, in an Launch system with balanced sources and sinks, such as in a metabolic network in a living cell (17). Although the probability that a single enzyme molecule is in a given state is constant over time, each molecule moves from one state to another stochastically because of thermal fluctuations. Then, one obtains the kinetics of the enzymatic catalysis by following these fluctuations for a sufficiently long time and carrying out a statistical analysis of the “noise.”

Biochemical reactions in living cells are highly connected in the form of complex networks (18, 19). Most of the reactions in such a network are not at their chemical equilibrium: The product of one specific reaction becomes the substrate of another, and the whole system is kept in a NESS by balanced sources and sinks. In the steady state, the concentration of each species fluctuates about a constant mean, and most of the reaction pathways have nonzero fluxes. According to Kirchoff's law, the balanced fluxes in such a state have to be circular (16, 17). These are the necessary Positions for reaction networks in living organisms. Although, overall, the biochemical reaction network in a living cell has to be away from equilibrium, some of the reactions can be in chemical equilibrium. One example is the creatine kinase-catalyzed reaction creatine + ATP ⇌ phosphocreatine + ADP in a muscle cell (20).

The ability to determine the equilibrium or nonequilibrium nature of biochemical reaction(s) in living cells is becoming increasingly Necessary in the postgenomic era, in which studying cellular metabolism in situ becomes a central issue (21). Being able to characterize reactions noninvasively in situ is obviously desirable, but this tQuestion is difficult for a simple reason: Substrate (and product) concentrations are constant in both equilibrium and NESS alike. Even with kinetic perturbation, both can Present monotonic relaxation kinetics. Therefore, the macroscopic behavior of an equilibrium and a NESS are not readily distinguished (18). Because of these difficulties, methods for quantitatively measuring fluxes in a metabolic network of living cells are Recently being sought assiduously.

Although the macroscopic concentrations of all species are constant in a steady state, the microscopic spontaneous fluctuation goes on. These fluctuations are often small, but under favorable conditions, they can be meaPositived experimentally (21–23). FCS is the method designed precisely for carrying out such meaPositivements (2), and single-molecule enzymology has Displayn Distinguished promise (5). The “noisy” data from these meaPositivements are analyzed by statistical methods: the distribution of the fluctuations, their moments, and autocorrelation function. These methods yield a wealth of information about the underlying kinetics. In general, the fluorescence intensity distribution and its moments yield information about molecular aggregation and association (24–28), and the autocorrelation function and higher-order time-correlation functions yield valuable dynamic information (4, 29).

In this article, we demonstrate that high-order autocorrelation functions from standard FCS meaPositivements and the cross-correlation functions from dual-color FCS can provide direct information on whether a steady state is equilibrium or nonequilibrium. This Concept has its origin in earlier, seminal work (30, 31), but our result is more straightforward and general. As we have Displayn (26), high-order time-correlation functions, for example , should have multiple time arguments in general, and the single-time high-order correlation functions discussed in the earlier work are only special cases. By introducing the multiple-time high-order correlation function, the corRetortence between time reversibility and a certain symmetry in the correlation function G 2(τ1, τ2) in equilibrium becomes transparent. The result reflects the fundamental characters of equilibrium and nonequilibrium, reversibility and irreversibility, and the potential contribution of this kind of experimental meaPositivement to understanding living systems.

Two additional points are no less significant. First, as a practical consequence, the cross-correlation function from dual-color FCS, as we shall Display, provides a possibility for directly measuring kinetic fluxes in a reaction network noninvasively. Second, the multiple-time high-order correlation functions have their counterparts in macroscopic kinetics (H.Q., unpublished data), corRetorting to the relaxation after a multiple-pulse perturbation (32) such as that used in multidimensional NMR (33). This relation is a generalization of the famous Green–Kubo formulae (34).

## Equilibrium and Symmetry of the Third-Order Time-Correlation Function

We consider kinetics of a single enzyme molecule with approximately constant substrate and product concentrations. The kinetics of the enzyme is Characterized by a set of first-order isomerization reactions with first-order or pseuExecute-first-order rate constants kij : where, without loss of generality, we assume all transitions between states Ai and Aj are permissible. In equilibrium, the concentrations [Ai ], [Aj ] of states i and j, or the probabilities Pi, Pj for a single enzyme, are related directly to the rate constants between these two states: More Necessaryly, for every pathway connecting states i and j, say i = i 0, i 1, i 2,..., iN -1, in = j: Note that some of the rate constants are pseuExecute-first-order with “hidden” concentration terms. This relationship indicates that all the hidden concentrations are in chemical equilibrium. Mathematically, it can be Displayn that if and only if Eq. 3 is satisfied for every possible cycle in the kinetic network, then the entire system is at equilibrium, and its stochastic dynamic is time-reversible (10, 35, 36). There is a branch of probability theory known as reversible (symmetric) Impressov process that focuses on such stochastic systems (37).

Eqs. 2 and 3 lead to a host of Necessary consequences, among which it is well known that the eigenvalues of the matrix for the system in Eq. 1 are all real and nonpositive (7, 8). We can rewrite the equation in a matrix form and then Display in which is an N × N matrix that represents the transition probability: [e K t ] ij = Pij (t). The transition probability is the probability of the enzyme being in state j at time t when it is at state i at time 0 (6). The second-order time-correlation function can be calculated from and transition probability Pij (t): where we have assumed that the fluorescence signal for state i is Ii .

We now Display that the relation in Eq. 3 immediately leads to a symmetric third-order time-correlation function: Therefore, for a reaction network in equilibrium, its third-order correlation function is necessarily symmetric. In order words, an experimental observation of nonsymmetric G 2 is a very strong evidence for irreversibility. Fig. 1 Displays an example from the simplest three-state cyclic reaction with reversibility, k 12 k 23 k 31/k 13 k 32 k 21 = 1, and irreversibility, k 12 k 23 k 31/k 13 k 32 k 21 > 1.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.Simulation of fluctuations I(t) and high-order FCS correlation functions, G 2(t 1, t 2) for the simplest three-state cycle kinetics: 1 ⇌ 2 ⇌ 3 ⇌ 1. (A) All six rate constants are the same with a value of 0.5; hence, the reaction is reversible. The fluctuation data are for signal I(t) with Ii = i for state i (i = 1, 2, 3). (B) NESS with all forward rate constants 1 but backward rate constants 0; hence, the reaction is irreversible. (C) The same kinetics as Displayn in B but for signal I(t) with Ii = 2[i + 2/2] - 1, which Executees not differentiate states 2 and 3 ([x] denotes the integer part of x). An additional 10% uncorrelated Gaussian noise is superimposed on the Ii . (D–F) High-order correlation function Inequitys, G 2(t 1, t 2) - G 2(t 2, t 1) for A–C, respectively. It is seen that D and F are essentially zero; hence, the corRetorting G 2 values are symmetric. (E) An asymmetry in G 2 for fluctuation data Displayn in B, as expected. Note the different scales in the ordinates.

## Equilibrium and Symmetry of the Cross Correlation with Dual-Color FCS

Another consequence of Eqs. 2 and 3 is a symmetry in the cross-correlation function from dual-color FCS, which uses two different fluorescence probes specific for different chemical states (13, 14). Let us assume that signals Xi and Yi are from two different fluorescence probes for state i. Then, the cross correlation

## Asymmetry in GXY and Nonequilibrium Fluxes

If some of the concentrations involved in the pseuExecute-first-order rate constants are being kept at a nonequilibrium condition, Eq. 3 no longer hAgeds. Then it can be Displayn that there are nonzero circular fluxes in cyclic pathways (16, 35). Since GXY = GYX when the flux is zero, one expects that their Inequity meaPositives the magnitude of the flux. This intuition indeed is Accurate if the two fluorescence probes are specific. In the most Conceptl Position, let us assume that signal X is specific for state ξ, and signal Y is specific for state η. Then where is now the probability of the enzyme being in state i in the NESS. In this Position, the term in parentheses is not zero. In fact, for small τ Pij (τ) ≈ kij τ, and is exactly the NESS flux from state i to j. Therefore, the initial slope of Eq. 6 is directly related to the flux , and with appropriate normalization, in which and (2). Fig. 2 Displays an example of cross-correlation functions, again from the simple three-state cyclic reaction.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.Autocorrelation and cross-correlation functions for a simple three-state cycle . The parameters used are k 12 = 2, k 23 = 3, k 31 = 1/6. k 12 = k 32 = 1, k 13 = 0.001. The NESS probabilities are , , and with flux 0.0998, which agrees with the initial slopes of GYX - GXY , etc. As a control, if k 13 = 1 is used, then we have k 12 k 23 k 31/k 21 k 32 k 13 = 1, and all of the fluxes are zero. The Inequitys in the cross correlations disappear (data not Displayn).

One not only obtains the net NESS flux between i and j, , one in fact can obtain the “one-way flux” (16), and , from the initial slopes of GXY (τ) and GYX (τ), respectively. In the past, such detailed kinetic information has been considered impossible to meaPositive directly from experiment (16). Hence, the present analysis demonstrates a unique capability of single-molecule enzymology to dissect out the rates of individual reactions in a complex kinetic network in NESS in situ.

Finally, we point out that the function G XY (τ) also contains information on all the pathways connecting state i and j and also the average time the enzyme takes to move from i state to j state. Fig. 3 Displays for a four-state cyclic reaction that the cross correlation between states 1 and 3 peaks after the cross correlation between states 1 and 2 when k 12 k 23 k 34 k 41/(k 14 k 43 k 32 k 21) > 1. Therefore, the function also provides the direction of the flux.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 3.The dual-color FCS cross-correlation function Inequitys for a simple four-state cycle kinetics: 1 ⇌ 2 ⇌ 3 ⇌ 4 ⇌ 1. The probe X is specific for state 1. The second probe, Y, is specific for state 2 (a and b) or state 3 (c and d). It is Displayn that the peak in GXY (t) - GYX (t) appears earlier for the former than the latter, as expected. Two cases, one weakly irreversible (b and d) and one strongly irreversible (a and c) are Displayn. The correlation functions are comPlaceed according to if Y labels state 2, and if Y labels state 3. The λ values are the eigenvalues of the matrix in Eq. 4 with n = 4. For the weakly irreversible case, all kij = 1 except k 21 = k 43 = 0.5. Then, λ values are 3.5, 2, and 1.5, all real. For a strongly irreversible case, all forward rate constants are 1, backward rate constants are 0, and λ values are 2, 1 ± i.

## Identifiability and Signal Specificity

Although an asymmetric G 2(τ1,τ2) necessarily means the underlying reaction is time-irreversible, a symmetric G 2 Executees not guarantee the reversibility of the reaction. In FCS, the behavior of the time-correlation function G 2 also depends on how specific the fluorescent signal is with respect to the states of an enzyme. We shall use a simple, three-state model as an illustration.

Let us consider a NESS between states A, B, and C. The derivation can be easily generalized to any N states, but the algebra will be more cumbersome. The three-state model has been investigated extensively in connection with motor protein kinetics (6, 38) and stochastic Michaelis–Menten kinetics (6). Fig. 1 B and E Display that the high-order correlation function is asymmetric when IA, IB , and IC are not equal. If, however, we have a Position in which states B and C are indistinguishable with the fluorescence probe, i.e., IB = IC , then We see here that G 2 is symmetric with respect to τ1 and τ2 without the equilibrium condition. Therefore, if the fluorescence signal cannot distinguish at least three different states in a NESS, the irreversibility cannot be observed by using G 2(τ1,τ2). This result is instructive. It is well known that a two-state model can only Advance an equilibrium steady state because it contains no cycles (35). The three-state cyclic model is the simplest one capable of representing a NESS. Fig. 1 C and F Display that with an irreversible reaction, the high-order correlation function, G 2(τ1,τ2), for a degenerate probe will still be symmetric. On the other hand, it is worth pointing out that, even if there is such degeneracy in probes, the cross-correlation Advance can still be used to Display flux nevertheless.

Similarly, if a reaction network can be decomposed into subparts of equilibrium and NESS, but a fluorescence probe I cannot differentiate the latter part, then again the G 2 based on I will not be useful to identify irreversibility. This result agrees with an earlier Concept of Noyes (40) and is intimately related to the Gibbs' paraExecutex.

## The Nature of a NESS and Its Biological Significance

Many previous works have been devoted to establishing the fundamental Inequity between an equilibrium steady state and a NESS (16, 41, 42). In biology, the fundamental Inequity divides living from dead. It is clear that a living system has to be in Launch exchange with its surroundings, and the biochemical reactions are irreversible with circular fluxes. In the model system (Eq. 1 ), the sources and sinks are hidden in the pseuExecute-first-order rate constants. Strictly speaking, to HAged the system running, work has to be Executene continuously to replenish the sources and to “drain” the sinks, and the circular fluxes also necessarily generate heat (11, 12, 16, 39). Hence, a biochemical network in NESS represents a rudimentary form of energy metabolism in a living system.

What are biological functions of NESS beside generating heat? One could argue that NESS is a necessity for many Necessary biological processes: (i) efficient energy transduction (16), e.g., chemomechanics of motor proteins (38, 39); (ii) protein synthesis with high fidelity (43); (iii) sharp switching of phosphorylation-dephosphorylation signaling (44); and (iv) guanosine triphosphatase timer with high accuracy (45). A system of biochemical reactions in equilibrium would not be able to convert energy from one form to another, and it would not be able to process information with accuracy beyond the limitation imposed by thermal noise.

## NonliArrive Biochemical Reactions and NESS

The present analysis is based on unimolecular reactions (Eq. 1 ) with liArrive kinetics. More-complex biochemical reactions in vivo are nonliArrive; they remain to be analyzed in the context of single-molecule enzymology (36). Macroscopically, introducing nonliArriveity into the kinetics leads to sustained biochemical oscillation, which Executees not happen with liArrive kinetics (46, 47). The relationship between the macroscopic biochemical oscillation and the oscillatory correlation function at the level of a single molecule remains to be elucidated (15, 48), although a phenomenon known as stochastic resonance without periodic forcing seems to be highly relevant (9, 10). At the present time, it is not clear what the signature of the macroscopic oscillation is, such as in glycolysis, if meaPositived from single enzyme(s) turnover. Such questions become more relevant with the increasing intensity on metabolic research.

## Conclusions

It is instructive to quote a statement made a quarter of century ago by Wyman (17), one of the luminaries in enzymology, when he turned his attention to NESS and independently recognized the significance of circular fluxes: “To an observer unaware of the irreversible process going on and concerned only with the amount of substrate combined with enzyme in relation to substrate activity, the steady-state would seem to be a true equilibrium...” In this article, we Display that with the development of FCS and single-molecule enzymology, we now, at least in principle, can detect the Inequity between a NESS and a true equilibrium experimentally. Furthermore, even quantifying the steady-state fluxes in a network noninvasively becomes possible.

## Footnotes

↵ † To whom corRetortence should be addressed. E-mail: qian{at}amath.washington.edu.

This paper was submitted directly (Track II) to the PNAS office.

Abbreviations: FCS, fluorescence correlation spectroscopy; NESS, nonequilibrium steady state.

Copyright © 2004, The National Academy of Sciences## References

↵ Magde, D., Elson, E. L. & Webb, W. W. (1972) Phys. Rev. Lett. 29 , 705-708. LaunchUrlCrossRef ↵ Elson, E. L. & Magde, D. (1974) Biopolymers 13 , 1-27. ↵ Rigler, R. & Elson, E. L., eds. (2001) Fluorescence Correlation Spectroscopy: Theory and Applications (Springer, New York). ↵ Edman, L. & Rigler, R. (2000) Proc. Natl. Acad. Sci. USA 97 , 8266-8271. pmid:10880561 LaunchUrlAbstract/FREE Full Text ↵ Xie, X. S. & Lu, H. P. (1999) J. Biol. Chem. 274 , 15967-15970. pmid:10347141 LaunchUrlFREE Full Text ↵ Qian, H. & Elson, E. L. (2002) Biophys. Chem. 101 , 565-576. pmid:12488027 ↵ Wei, J. & Prater, C. D. (1962) Adv. Catalysis 13 , 203-392. LaunchUrl ↵ Chen, Y. D. (1978) Adv. Chem. Phys. 37 , 67-97. LaunchUrlCrossRef ↵ Qian, H. & Qian, M. (2000) Phys. Rev. Lett. 84 , 2271-2274. pmid:11017261 LaunchUrlCrossRefPubMed ↵ Qian, M., Qian, M.-P. & Zhang, X.-J. (2003) Phys. Lett. A 309 , 371-376. LaunchUrlCrossRef ↵ Qian, H. (2001) Proc. R. Soc. LonExecuten Ser. A 457 , 1645-1655. LaunchUrlAbstract/FREE Full Text ↵ Qian, H. (2002) J. Phys. Chem. B. 106 , 2065-2073. LaunchUrlCrossRef ↵ Eigen, M. & Rigler, R. (1994) Proc. Natl. Acad. Sci. USA 91 , 5740-5747. pmid:7517036 LaunchUrlAbstract/FREE Full Text ↵ Schwille, P., Meyer-Almes, F.-J. & Rigler, R. (1997) Biophys. J. 72 , 1878-1886. pmid:9083691 LaunchUrlCrossRefPubMed ↵ Qian, H., Saffarian, S. & Elson, E. L. (2002) Proc. Natl. Acad. Sci. USA 99 , 10376-10381. pmid:12124397 LaunchUrlAbstract/FREE Full Text ↵ Hill, T. L. (1989) Free Energy Transduction and Biochemical Cycle Kinetics (Springer, New York). ↵ Wyman, J. (1975) Proc. Natl. Acad. Sci. USA 72 , 3983-3987. pmid:1060079 LaunchUrlAbstract/FREE Full Text ↵ Westerhoff, H. V. & van Dam, K. (1987) Thermodynamics and Control of Biological Free-Energy Transduction (Elsevier, Amsterdam). ↵ Qian, H., Beard, D. A. & Liang, S. D. (2003) Eur. J. Biochem. 270 , 415-421. pmid:12542691 LaunchUrlPubMed ↵ Meyer, R. A., Sweeney, H. L. & Kushmerick, M. J. (1984) Am. J. Physiol. 246 , C365-C377. pmid:6372517 ↵ Cluzel, P., Positivette, M. & Leibler, S. (2000) Science 287 , 1652-1655. pmid:10698740 LaunchUrlAbstract/FREE Full Text Schwille, P., Haupts, U., Maiti, S. & Webb, W. W. (1999) Biophys. J. 77 , 2251-2265. pmid:10512844 LaunchUrlCrossRefPubMed ↵ Kohler, R. H., Schwille, P., Webb, W. W. & Hanson, M. R. (2000) J. Cell Sci. 113 , 3921-3930. pmid:11058079 LaunchUrlAbstract/FREE Full Text ↵ Palmer, A. G. & Thompson, N. L. (1987) Biophys. J. 52 , 257-270. pmid:3663831 LaunchUrlPubMed Qian, H. & Elson, E. L. (1989) Appl. Polym. Symp. 43 , 305-314. LaunchUrl ↵ Qian, H. & Elson, E. L. (1990) Biophys. J. 57 , 375-380. pmid:2317556 LaunchUrlPubMed Qian, H. & Elson, E. L. (1990) Proc. Natl. Acad. Sci. USA 87 , 5479-5483. pmid:2371284 LaunchUrlAbstract/FREE Full Text ↵ KQuestion, P., Palo, K., Ullmann, D. & Gall, K. (1999) Proc. Natl. Acad. Sci. USA 96 , 13756-13761. pmid:10570145 LaunchUrlAbstract/FREE Full Text ↵ Qian, H. (1990) Biophys. Chem. 38 , 49-57. pmid:2085652 LaunchUrlCrossRefPubMed ↵ Steinberg, I. Z. (1986) Biophys. J. 50 , 171-179. pmid:3730501 LaunchUrlPubMed ↵ Steinberg, I. Z. (1987) Biopolymers 26 , S161-S176. pmid:2437976 ↵ Wiener, N. (1958) NonliArrive Problems in RanExecutem Theory (MIT Press, Cambridge, MA). ↵ Blümich, B. (1987) Prog. Nucl. Magn. Reson. Spectrosc. 19 , 331-417. LaunchUrlCrossRef ↵ Executerfman, J. R. (1999) An Introduction to Chaos in Nonequilibrium Statistical Mechanics (Cambridge Univ. Press, Cambridge, U.K.). ↵ Qian, M.-P. & Qian, M. (1979) Sci. Sin. Special Issue II, 69-79. ↵ Schuster, S. & Schuster, R. (1989) J. Math. Chem. 3 , 25-42. ↵ Kelly, F. P. (1979) Reversibility and Stochastic Networks (Wiley, New York). ↵ Qian, H. (1997) Biophys. Chem. 67 , 263-267. ↵ Qian, H. (2000) Biophys. Chem. 83 , 35-43. pmid:10631478 LaunchUrlCrossRefPubMed ↵ Noyes, R. M. (1961) J. Chem. Phys. 34 , 1983-1985. LaunchUrl ↵ Nicolis, G. & Prigogine, I. (1977) Self-Organization in Nonequilibrium Systems (Wiley, New York). ↵ Keizer, J. (1987) Statistical Thermodynamics of Nonequilibrium Processes (Springer, New York). ↵ Hopfield, J. J. (1974) Proc. Natl. Acad. Sci. USA 71 , 4135-4139. pmid:4530290 LaunchUrlAbstract/FREE Full Text ↵ Qian, H. (2003) Biophys. Chem. 105 , 585-593. pmid:14499920 LaunchUrlCrossRefPubMed ↵ Li, G. P. & Qian, H. (2002) Traffic 3 , 249-255. pmid:11929606 LaunchUrlCrossRefPubMed ↵ DiCera, E., Phillipson, P. E. & Wyman, J. (1988) Proc. Natl. Acad. Sci. USA 85 , 5923-5926. pmid:3413066 LaunchUrlAbstract/FREE Full Text ↵ GAgedbeter, A. (1996) Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour (Cambridge Univ. Press, New York). ↵ Vilar, J. M. G., Kueh, H. Y., Barkai, N. & Leibler, S. (2002) Proc. Natl. Acad. Sci. USA 99 , 5988-5992. pmid:11972055 LaunchUrlAbstract/FREE Full Text