The Thermodynamic Uncertainty Relation in Biochemical Oscillations

Robert Marsland III,1 Wenping Cui,1, 2 and Jordan M. Horowitz3, 4, 5

1Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, MA 02215∗2Department of Physics, Boston College, 140 Commonwealth Avenue, Chestnut Hill, MA 02467

3Physics of Living Systems Group, Department of Physics,Massachusetts Institute of Technology, 400 Technology Square, Cambridge, MA 02139

4Department of Biophysics, University of Michigan, Ann Arbor, MI, 481095Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48104

(Dated: August 19, 2020)

Living systems regulate many aspects of their behavior through periodic oscillations of molecularconcentrations, which function as “biochemical clocks.” The chemical reactions that drive theseclocks are intrinsically stochastic at the molecular level, so that the duration of a full oscillationcycle is subject to random fluctuations. Their success in carrying out their biological function isthought to depend on the degree to which these fluctuations in the cycle period can be suppressed.Biochemical oscillators also require a constant supply of free energy in order to break detailedbalance and maintain their cyclic dynamics. For a given free energy budget, the recently discovered‘thermodynamic uncertainty relation’ yields the magnitude of period fluctuations in the most preciseconceivable free-running clock. In this paper, we show that computational models of real biochemicalclocks severely underperform this optimum, with fluctuations several orders of magnitude larger thanthe theoretical minimum. We argue that this suboptimal performance is due to the small number ofinternal states per molecule in these models, combined with the high level of thermodynamic forcerequired to maintain the system in the oscillatory phase. We introduce a new model with a tunablenumber of internal states per molecule, and confirm that it approaches the optimal precision as thisnumber increases.

Many living systems regulate their behavior using aninternal “clock,” synchronized to the daily cycles of lightand darkness. In the past 15 years, the isolation of thekey components of several bacterial circadian clocks hasopened the door to systematic and quantitative study ofthis phenomenon. In particular, a set of three proteinspurified from the bacterium Synechococcus elongatus arecapable of executing sustained periodic oscillations invitro when supplied with ATP [53]. One of the proteins,KaiC, executes a cycle in a space of four possible phos-phorylation states, as illustrated in figure 1. This cycle iscoupled to the periodic association and dissociation fromthe other two proteins, KaiA and KaiB.

Steady oscillations break detailed balance, and mustbe powered by a chemical potential gradient or other freeenergy source. In this system and in related experimentsand simulations, it is commonly observed that the oscil-lator precision decreases as this thermodynamic drivingforce is reduced [38]. At the same time, recent theoreticalwork indicates that the precision of a generic biochemi-cal clock is bounded from above by a number that alsodecreases with decreasing entropy production per cycle[35–37, 44, 45, 47, 60, 68]. This has led to speculationthat this universal bound may provide valuable informa-tion about the design principles behind real biochemicalclocks.

So far, most discussion of this connection has focusedon models with cyclic dynamics hard-wired into the dy-namical rules [37, 50, 68]. But real biochemical oscilla-

∗ Email: marsland@bu.edu

tors operate in a high-dimensional state space of concen-tration profiles, and the cyclic behavior is an emergent,collective phenomenon [37, 38, 54]. These oscillators typ-ically exhibit a nonequilibrium phase transition at a finitevalue of entropy production per cycle. As this thresholdis approached from above, the oscillations become morenoisy due to critical fluctuations [46, 49, 54, 64]. Belowthe threshold, the system relaxes to a single fixed pointin concentration space, with no coherent oscillations atall. In some systems, the precision may still be well be-low the theoretical bound as the system approaches thisthreshold. In these cases, the precision will never comeclose to the bound, for any size driving force.

As we show in Section II, computational models ofreal chemical oscillations typically fall into this regime,never approaching to within an order of magnitude ofthe bound. Macroscopic in vitro experiments on the Ka-iABC system perform even worse, remaining many or-ders of magnitude below the bound. Previous theoreticalwork suggests that the performance could be improvedby increasing the number of reactions per cycle at fixedentropy production, and by making the reaction ratesmore uniform [35, 61, 68]. In Section I, we elaborate onthese ideas, introducing an effective number of states percycle and showing how the relationship of this quantityto the location of the phase transition threshold controlsthe minimum distance to the bound. In Section III, weintroduce a new model based on these design principles,with nearly uniform transition rates in the steady stateand with a tunable number of reactions per cycle. Weshow that this model approaches the optimal precisionas the number of reaction steps per cycle grows.

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(d)τ1 τ

2 τ3 τ

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FIG. 1. Coherent cycles in a biochemical oscillator. (a) Schematic of the KaiC biochemical oscillator: This simplifieddiagram shows four different internal states of the KaiC molecule, labeled U, T, S and ST (unphosphorylated, phosphorylatedon threonine, phosphorylated on serine, and phosphorylated on both residues). The molecules execute cycles around thesefour states in the indicated direction. They interact with each other via additional molecular components, in such a way thatmolecules in state S slow down the forward reaction rate for other molecules in state U. (b) Simulated trajectory from a detailedkinetic model of the KaiC system (adapted from [56]) with 360 interacting KaiC hexamers. The state space of this system isdescribed by the list of copy numbers of all the molecular subspecies. Here we have projected the state onto a two-dimensionalplane, spanned by the copy numbers of monomers in states T and S. The inset gives a magnified view of a small portion of theplot, showing the discrete reaction steps caused by single phosphorylation and dephosphorylation events. (c) Time-evolutionof the fraction fT of monomers in state T. (d) Histograms of the time τn required for 1,200 independent sets of 360 interactingKaiC hexamers to complete n collective cycles, for n = 1 through 10. (e) Variances var(τn) of the histograms as a function ofn. Error bars are bootstrapped 95% confidence intervals. Black line is a linear fit, with slope D = 2.05± 0.05 hours2.

I. EFFECTIVE NUMBER OF STATES ANDCRITICAL ENTROPY PRODUCTION CONTROL

DISTANCE TO THERMODYNAMIC BOUND

As illustrated in figure 1(a), a KaiC monomer has twophosphorylation sites, one on a threonine residue (T) andone on a serine (S), giving rise to four possible phospho-rylation states [53]. The monomer also has two ATP-binding pockets, and forms hexamers that collectivelytransition between two conformational states [58, 67].All these features are important for the dynamics of thesystem, and have been incorporated into a thermody-namically consistent computational model that correctlyreproduces the results of experiments performed with pu-rified components [56]. In particular, the ATP hydroly-sis rate in one of the binding pockets has been shownto be essential for determining the period of the cir-cadian rhythms [58, 66, 67]. Although our simulationswill use the detailed model just mentioned, the simpli-fied schematic in figure 1 highlights only the features ofthe model that we will explicitly discuss: the fact that

each molecule can execute a directed cycle among severalinternal states (pictured as black arrows), and the factthat the state of one molecule affects transition rates ofthe others (suggestively represented as a red inhibitionsymbol).

Fluctuations in the time required for a simplified modelof a single KaiC hexamer to traverse the reaction cy-cle have recently been studied in [37]. But the biolog-ical function of this clock demands more than preciseoscillations of isolated molecules; rather, it has evolvedto generate oscillations in the concentrations of variouschemical species. The concentrations are global vari-ables, which simultaneously affect processes throughoutthe entire cell volume. These global oscillations can stillbe described by a Markov process on a set of discretestates, but with a very different topology from the uni-cyclic network of an isolated monomer. For a well-mixedsystem, each state can be labeled by a list of copy num-bers of all molecules in the reaction volume as shown infigure 1(b), with each distinct internal state counted as adifferent kind of molecule.

In the KaiC system, molecules in one of the phospho-

3

rylation states can suppress further phosphorylation ofother molecules, by sequestering the enzyme (KaiA) re-quired to catalyze the phosphorylation. This mechanismcan stably synchronize the progress of all the moleculesaround the phosphorylation cycle, slowing down the onesthat happen to run too far ahead of the rest. This is cru-cial for the maintenance of sustained oscillations in theconcentration of free KaiA and of each of the four formsof KaiC. Figure 1(c) shows a sample trajectory of theconcentration of one of the KaiC phosphorylation statesin the detailed computational model mentioned above[56].

Unlike the cycles of an idealized mechanical clock, theperiod τ1 of these oscillations is subject to random fluc-tuations, due to the stochastic nature of the underly-ing chemical reactions. The precision can be quantifiedby considering an ensemble of identical reaction volumesthat are initially synchronized. The histogram of timesτn for each molecule to complete n cycles will widen as nincreases and the clocks lose their initial synchronization,as illustrated in figure 1(d). When the width exceeds themean period T ≡ 〈τ1〉, the clocks are totally desynchro-nized. This leads to a natural measure of the precisionof the clock in terms of the number of coherent cycles Nthat take place before the synchronization is destroyed.

To measure N in a systematic way, we first note thatthe variance var(τn) = Dn, for some constant of propor-tionality D, as illustrated in figure 1(e). This is exactlytrue in a renewal process [63], such as the isolated KaiCmonomer, where each period is an independent randomvariable (cf. [68]). It remains asymptotically valid forarbitrarily complex models in the large n limit, as longas the correlation time is finite. The number of cyclesrequired for the width

√var(τn) of the distribution to

reach the average period T is therefore given by

N ≡ T 2

D. (1)

Any chemical oscillator must be powered by a detailed-balance-breaking thermodynamic driving force that gen-erates a positive average rate of entropy production S.The number of coherent cycles is subject to a universalupper bound as a function of S, holding for arbitrarilycomplex architectures [35–37, 41, 45, 47, 60]. The boundsays thatN is never greater than half the entropy produc-tion per cycle ∆S ≡ ST (setting Boltzmann’s constantkB = 1 from here on) [44]:

N ≤ ∆S

2. (2)

The validity of this bound depends on the proper defi-nition of N , which in our formulation also depends onthe definition of τn. Determining τn is a subtle matterfor systems of interacting molecules. Our solution is pre-sented in detail in the Appendix, but it always roughlycorresponds to the peak-to-peak distance in figure 1(c).

As ∆S →∞, Equation (2) says that N is also allowedto become arbitrarily large. But as the entropy released

2000 2250 2500 2750 3000Entropy Production per Cycle S/M

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0.6

0.8

1.0

1.2

Num

ber o

f Coh

eren

t Cyc

les

/M

KaiABC ModelNeff/M = 1.1

FIG. 2. Number of coherent oscillations saturates as∆S → ∞. The number of coherent cycles N is plotted asa function of the entropy production per cycle ∆S for theKaiC model discussed in figure 1 above. Both axes are scaledby the system size M , so that all quantities are molecular-scale values. Error bars are ±1 standard deviation, estimatedwith the bootstrap procedure described in the Appendix. Theblack dotted line is the estimated ∆S → ∞ limit N = Neff .See Appendix and [56] for model parameters.

in the reactions coupled to the driving force increases,detailed balance implies that the reverse reaction ratestend towards zero. Once the reverse rates are negligi-ble compared to the other time scales of the problem,these reactions can be ignored, and further changes in∆S produce no effect. In any given biochemical model,therefore, N approaches some finite value as ∆S → ∞(as was already noted in [38]), which depends on the net-work topology and the rest of the reaction rates. We cansee this in our detailed computational model in figure 2.For unicyclic networks in particular, where the topologyis a single closed cycle like the isolated KaiC monomer,the maximum possible value for this asymptote is thenumber of states N [40, 50]. By analogy, we will referto the ∆S → ∞ limit of N for any model as the effec-tive number of states per cycle Neff ≡ lim∆S→∞N . Foran oscillator built from coupled cycles of internal states,such as the KaiC system, Neff reaches its maximum valuewhen the dynamics constrain the oscillations to a singlepath through concentration space, and when all reactionrates along this path are equal. In this case, the dynam-ics are equivalent to a single ring of NM states, where Nis the number of internal states per molecule and M isthe number of molecules. This upper bound on Neff canbe easily computed for any model or experiment from abasic knowledge of the component parts.

In all five models we will analyze below, N monotoni-cally increases as a function of ∆S. The existence of thefinite ∆S → ∞ limit thus implies that N can only ap-proach the thermodynamic bound of Equation (2) when

4

∆S < ∆Sb ≡ 2Neff . But the collective oscillations ofthese models also exhibit a nonequilibrium phase transi-tion as a function of ∆S, whose critical behavior has re-cently been studied [38, 55]. In the thermodynamic limit,the inverse precision 1/N diverges as ∆S approaches acritical value ∆Sc from above, in a way that dependson the architecture of the reaction network. Below ∆Sc,there are no collective oscillations, and the concentra-tions relax to a single fixed point. Since the oscillationscease to exist below ∆Sc, the bound is only relevant for∆S > ∆Sc. Combining these two observations, we seethat models with ∆Sb < ∆Sc can never approach thethermodynamic bound.

II. MODELS OF REAL CHEMICALOSCILLATORS SEVERELY UNDERPERFORM

THE BOUND

Cao et. al. recently measured N as a function of ∆Sin computational models of four representative chemi-cal clock architectures: activator-inhibitor, repressilator,Brusselator, and the glycolysis network [38]. The datafor all four models produced an acceptable fit to a four-parameter phenomenological equation, which is repro-duced in the Appendix along with the parameter valuesobtained by Cao et. al. for each model. In figure 3, weplot these phenomenological curves and the thermody-namic bound of Equation (2). We also obtained N and∆S for a detailed model of the KaiC system based on[56] as described in the Appendix, with the parametersobtained in that paper by extensive comparison with ex-perimental data, for twenty values of the ATP/ADP ra-tio.

The values of Neff , ∆Sb and ∆Sc can be estimated di-rectly from figure 3, by noting where each curve saturatesand where it drops to zero. Both axes are scaled by thesystem size M , which equals the number of KaiC hexam-ers for the KaiC model, and the number of kinases in theactivator-inhibitor model. The other three models lacka direct physical interpretation of M , since there are noconserved molecular species, but it still defines a genericmolecular scale. For any physically reasonable model, Nis expected to be an extensive parameter, proportionalto M , as is ∆S. This has been confirmed numerically fora number of models, and appears to break down signifi-cantly only in the immediate vicinity of the critical point[38, 49, 55]. The models plotted here have Neff/M ≈ 2,which is reasonable for molecules that only have a fewinternal states and highly non-uniform reaction rates.

But Neff/M ≈ 2 implies that ∆Sb/M ≈ 4, whichmeans that the entropy production per molecule per cy-cle must be less than 4 for the thermodynamic bound tobecome relevant. This is a very small number even bybiochemical standards, equal to the entropy change fromforming four hydrogen bonds between protein residuesin solution. The activator-inhibitor, Brusselator, andglycolysis models have phase transitions at ∆Sc/M val-

101 102 103

S/M

100

101

/M

Thermodynamic BoundRepressilatorGlycolysisActivator-InhibitorBrusselatorKaiABC Model

FIG. 3. Models of collective oscillations comparedwith thermodynamic bound. Same as figure 2, but in-cluding all four models studied in [38]. The black dotted lineis the thermodynamic bound N = ∆S/2. Curves for the firstfour models are phenomenological fits obtained in [38].

ues of 360, 100.4 and 80.5, respectively, under the pa-rameter choices of [38]. They all exceed ∆Sb/M by atleast an order of magnitude, guaranteeing that the preci-sion can never come close to the thermodynamic bound.The KaiC model appears to have ∆Sc/M ∼ 1, 000 andNeff/M = 1.1, so that ∆Sc exceeds ∆Sb by more thantwo orders of magnitude. The only model with ∆Sb >∆Sc is the repressilator model, where ∆Sc/M = 1.75 and∆Sb ≈ 4. But even here, the critical fluctuations beginto severely degrade the precision when ∆S is still muchgreater than ∆Sb.

Estimates of N can also be extracted directly from ex-periments, as shown by Cao et. al. for an in vitro recon-stitution of the KaiC system with purified componentsin a macroscopic reaction volume [38]. They analyzedtimeseries data from a set of experiments at differentATP/ADP ratios, and fit their phenomenological equa-tion to describe N as a function of this ratio. As we showin the Appendix, this fit implies that ∆Sc/M ∼ 1, 000,consistent with our model results. The ∆S →∞ asymp-tote of the fit, however, gives Neff/M ∼ 10−11, whichis astronomically small compared to the model predic-tion Neff/M ≈ 1. This surprising result reflects the factthat the dominant sources of uncertainty in these macro-scopic experiments are fluctuations in temperature andother environmental perturbations, rather than the in-trinsic stochasticity of the reaction kinetics. Since thesefluctuations are independent of the system size, their ef-fect is inflated when we divide by the number of hexamersM ∼ 1014 in a 100 µL reaction volume at 1 µM concen-tration. The only way to observe the effect of intrinsicstochasticity in such a noisy environment is to decrease

5

(a) (b)

FIG. 4. Symmetric toy model compared with ther-modynamic bound. (a) Schematic of toy model inspiredby the KaiC system. Each protein has N distinct conforma-tions whose transitions are arranged in a ring topology witha net clockwise drift. Each protein suppresses the transitionrate for proteins further along in the circulation around thering. (b) N/M versus ∆S/M in this model for three differentvalues of the number of internal states N . Error bars are ±1standard deviation, estimated with the same bootstrap pro-cedure used in figure 2. See Appendix for model details andparameters.

the reaction volume. Assuming that the minimum con-tribution of the external noise to N remains fixed at thevalue of 500 found in the experiments, and that the in-trinsic contribution is of order Neff ≈M as given by themodel, we find that M = 500 hexamers is the system sizeat which the intrinsic fluctuations become detectable. At1 µM concentration of hexamers, the corresponding reac-tion volume is about 1µm3, the size of a typical bacterialcell.

Because of this separation of scales, the apparent di-vergence of 1/N at a critical value of the ATP/ADP ratioin the experiments is probably due to the expected diver-gence of susceptibility at the critical point, which makesthe oscillation period increasingly sensitive to environ-mental fluctuations as the ATP/ADP ratio is reduced. Inany case, this analysis suggests that an important designconsideration for oscillators in living systems is robust-ness against external perturbations, as recently exploredin [42, 51, 62].

III. TOY MODEL WITH VARIABLE NUMBEROF STATES CAN SATURATE THE BOUND

The failure of all five of these models to approach thethermodynamic bound raises the question of whether it ispossible in principle for any chemical oscillator to do so.Put another way, whether it is possible to simultaneouslyachieve a large enoughNeff/M and small enough ∆Sc/M .In a simple unicyclic network of N states, Neff = N whenall the reaction rates in the forward direction are equal,and so one can always approach arbitrarily close to thebound by increasing N . But a chemical oscillator can-not have uniform rates, since the transition rates of eachmolecule have to change based on the states of the othersin order to achieve collective oscillations. Furthermore,

it is not known how changing the number of internalstates affects ∆Sc/M , and so it is not obvious whether∆Sc < ∆Sb is achievable at all.

To answer this question, we devised a new modelloosely inspired by the KaiC model of figure 1, with Minteracting molecules each containing N distinct inter-nal states. Molecules in any one of these states suppressthe transition rates for other molecules that are furtherahead in the cycle, as illustrated in figure 4 and describedin detail in the Appendix. All internal states have thesame energy, and each reaction carries the same fractionof the total thermodynamic force.

In this highly symmetrized model, ∆Sc/M can easilybe reduced to between 2 and 3 by choosing a sufficientlyhigh coupling strength, as shown in the Appendix. Atthe same time Neff/M scales linearly with N , and can bemade arbitrarily large by increasing this parameter. Infigure 4, we plot N/M versus ∆S/M for three differentvalues ofN , and show that the data does indeed approachthe thermodynamic bound as N increases. This extendsthe validity of design principles obtained for unicyclicnetworks in various context to these collective dynamics:the rates should be made as uniform as possible, whilethe number of internal states is made as large as possibleat fixed thermodynamic driving force [37, 48, 68].

IV. DISCUSSION

The thermodynamic uncertainty relation is a powerfulresult with impressive universality. It has been widelyassumed that the relation should have some relevancefor the evolution of biochemical oscillators. Based ondata from experiments and extensive simulations withrealistic parameters, we have argued that these oscillatorstypically underperform the bound by at least an order ofmagnitude. From a thermodynamic perspective, they arefree to evolve higher precision without increasing theirdissipation rate.

We have also derived a simple criterion for estimatinghow closely a given oscillator can approach the thermo-dynamic bound, in terms of an effective number of statesNeff and the entropy production per cycle ∆Sc at theonset of oscillatory behavior. For an oscillator composedof M identical molecules with N internal states, glob-ally coupled through mass-action kinetics, we noted thatNeff ≤ NM , with equality only when all the cycles areperfectly synchronized, and when all reactions that actu-ally occur have identical rates. Assuming that the num-ber of coherent periods N is monotonic in the entropyproduction per cycle ∆S, the thermodynamic bound canonly be approached when NM ≥ Neff � ∆Sc.

To show that this criterion can in principle be satisfiedby emergent oscillations of molecular concentrations, wedevised a toy model that oscillates with less than 3 kBTof free energy per molecule per cycle and can contain anarbitrary number of internal states per molecule. Butit is hard to imagine a biochemically plausible mecha-

6

nism for sustained oscillations powered by the free en-ergy equivalent of three hydrogen bonds per moleculeper cycle. Certainly this could not be a phosphorylationcycle, since cleaving a phosphate group from ATP andreleasing it into the cytosol dissipates about 20 kBT un-der physiological conditions [52]. Furthermore, we notedthat the five models we analyzed all have an effectivenumber of internal states per molecule Neff/M of around2. This number may be constrained by a trade-off withthe complexity of the oscillator. For KaiC, the inter-molecular coupling required for collective oscillations ismediated by sequestration of KaiA at a particular pointin the cycle. Implementing the symmetric N -state cycleof figure 4(a) in this way would require each transitionto be catalyzed by a different molecule, and for each ofthese molecules to be selectively sequestered by the statea quarter-cycle behind the transition. Finally, biochemi-cal oscillators are subject to other performance demandsthat may be more important than the number of coher-ent cycles of the free-running system. In particular, anoscillator’s ability to match its phase to an external sig-nal (e.g., the day/night cycle of illumination intensity)is often essential to its biological function, placing an

independent set of constraints on the system’s architec-ture [39, 62]. Entrainment efficiency has recently beenshown to increase with dissipation rate in some modelseven when the free-running precision has saturated, pro-viding an impetus for increasing the entropy productionper cycle beyond what is required to achieve N = Neff

[43].

ACKNOWLEDGMENTS

We thank J. Paijmans for his help with adjustments tothe KaiC simulation, and Y. Cao for valuable discussionson the technical details of reference [38]. RM acknowl-edges Government support through NIH NIGMS grant1R35GM119461. JMH is supported by the Gordon andBetty Moore Foundation as a Physics of Living SystemsFellow through Grant No. GBMF4513. The computa-tional work reported on in this paper was performed onthe Shared Computing Cluster which is administered byBoston Universitys Research Computing Services.

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APPENDIX

A. Measuring the stochastic period

The definition of the number of coherent cycles N de-pends on a prior notion of the n-cycle completion timeτn. In a unicyclic transition network, this time can bestraightforwardly defined in terms of the integrated cur-rent J through an arbitrarily chosen transition in thenetwork. Each time the transition is executed in theforward direction, J increases by 1, and each time it isexecuted in the reverse direction, J decreases by 1. Then-cycle completion time τn is then naturally defined asthe time when the system first reaches J = n, given thatit was initialized in the state immediately adjacent to themeasured transition in the positive direction [44, 63, 68].

For a chemical oscillator, the definition is not so clear.One common approach is to fit the autocorrelation func-tion of some observable to a sine wave with exponentiallydecaying amplitude. If the autocorrelation function ex-actly fits this functional form, then N can be obtainedfrom the ratio of the decay time to oscillation period viaa numerical conversion factor [38]. One can also evaluatethe ratio of imaginary to real parts of the leading eigen-value of the rate matrix for the Master equation of thedynamics, which gives the same result when all the othereigenvalues are much smaller in amplitude [37, 54, 64].While this ratio is conjectured to be bounded by thethermodynamic driving force powering the oscillations,it is not the approach we study here [37]. Instead, we

(b)

(a)

Inte

grat

ed C

urre

nt (J

)

+1

-1

FIG. 5. Measuring the n-cycle completion time τn (a)Trajectories of the toy model of figure 4 with N = 6 for twodifferent values of the thermodynamic driving force, projectedonto their first two principal components. A cut from theorigin along the positive vertical axis provides the criterionfor cycle completion. (b) Integrated current (net number ofcompleted cycles) J as a function of time for the same twotrajectories. The first-passage time for a net increase n in theintegrated current defines the n-cycle completion time τn.

note that the value of N generated by these precedingprocedures only satisfies the hypotheses of the thermo-dynamic uncertainty relation under the specific condi-tions of a perfectly sinusoidal autocorrelation function(cf. [37]). For our analysis of the KaiC model and ournew toy model, we instead utilize a definition of τn thattreats the oscillations in the full concentration space asone large cycle.

Figure 5 illustrates our procedure. We started by pro-jecting the state of the system from the high-dimensionalconcentration space to two dimensions. We projectedonto the plane that captured the largest percentage ofthe total variation in system state over a cycle, usinga Principal Component Analysis (PCA) of a trajectorycontaining at least one full cycle (using the Python pack-age scikit-learn [57]). In this plane, the oscillating tra-jectories describe a noisy ring, as shown in Figure 5a.Because the ring has a finite width, we cannot select asingle transition to count the integrated current J . In-stead, we draw a half-line starting from the middle of

9

the ring, representing a half-hyperplane in the full statespace, and include all the transitions cut by this hyper-plane in the current. Each time the line is crossed in theclockwise direction, J increases by 1, and each time itis crossed in the counterclockwise direction, J decreasesby 1. Sample traces of J(t) are plotted in figure 5 (b).The n-cycle completion time τn can now be defined as be-fore, measuring the first-passage time for reaching J = n.These definitions fulfill the hypotheses of the thermo-dynamic uncertainty relations for currents and for firstpassage times [35, 44, 45]. In the notation of [44, 45],they correspond to setting d(y, z) = 1 for all transitionsfrom y to z cut by the hyperplane, and d(y, z) = 0 forall other transitions. Note that the uncertainty relationsare obtained in the J →∞ limit, where initial conditionsare irrelevant, but for our numerical analysis we chosespecial initial conditions that gave rapid convergence tothe asymptotic form. Specifically, we employed a con-ditional steady-state distribution over states adjacent tothe hyperplane. This was achieved by running the simu-lation longer than the relaxation time, and then startingthe counter at J = 0 the next time the hyperplane wascrossed.

B. Phenomenological fits from reference [38]

The extensive numerical simulations performed by Caoet. al. on four different models of chemical oscillators canbe summarized by the parameters of a phenomenologicalfitting function

NM

=

[A+B

(∆S −∆Sc

M

)α]−1

(3)

with four parameters A,B,∆Sc and α. The exponent αis always negative, so N/M goes to zero as ∆S → ∆Sc.(To convert from their notation to ours, use V → M ,W0 → B, C → A, Wc → ∆Sc/M , D/T → N−1). Theparameters for these fits are given in the figure captionsof [38], and are reproduced in the following table:

Model A B ∆Sc/M α

Activator-Inhibitor 0.6 380 360 -0.99

Repressilator 0.4 25.9 1.75 -1.1

Brusselator 0.5 846 100.4 -1.0

Glycolysis 0.5 151.4 80.5 -1.1

Note that A controls the ∆S → ∞ asymptote, and isequal to N−1

eff .

C. Analysis of experimental data

Cao et. al. also analyze experimental data on the KaiCsystem, extracting the ratio of decay time to oscillationperiod from fits to the autocorrelation function at differ-ent values of the ATP/ADP ratio. They fit Equation (3)

above to the data, but using ln([ATP]/[ADP]) instead of∆S/M , and without converting the autocorrelation ratioto N or dividing by volume.

To compare these results to the thermodynamicbound, we first had to convert from the logarithmof the ATP/ADP ratio to entropy production per cy-cle. In the text, they estimate that the critical valueln([ATP]/[ADP])c = −1.4 obtained from the fit corre-sponds to an entropy production per ATP hydrolysisevent of 10.6. Combining this with the measurementfrom [66] of 16 hydrolysis events per cycle per KaiCmonomer, we find a critical entropy production per cycleper hexamer of ∆Sc/M ≈ 10.6× 16× 6 ≈ 1, 020.

Next, we had to convert the observed ratio of decaytime/period to N/M . Since the autocorrelation functionwas well fit by an exponentially decaying sinusoid, weapplied the corresponding conversion factor N = 2π2 τ

Twhere T is the period and τ is the decay time [38, Eq.2]. We then estimated the number of hexamers M ≈3 × 1013 using the KaiC monomer concentration of 3.4µM reported with the original publication of the data[59, 65], and the typical reaction volume in a 96-well plateof 100 µL. With these two conversions, we found that the∆S →∞ value of N/M was 2× 10−11.

D. Thermodynamically consistent KaiC model

Paijmans et. al. have recently developed a mechanis-tically explicit computational model of the KaiC oscil-lator [56]. This model is particularly interesting fromthe theoretical point of view because it captures the ex-tremely large dimensionality characteristic of real bio-chemical systems. Each KaiC hexamer contains six KaiCproteins, which each contain two nucleotide binding sitesand two possible phosphorylation sites (“S” and “T”from figure 1). Each of these sites can be in one of twopossible states (ATP-bound/ADP-bound, or phosphory-lated/unphosphorylated). Furthermore, the whole hex-amer can be in an “active” or an “inactive” conformation.Thus each hexamer has (2 · 2 · 2 · 2)6 · 2 = 225 possible in-ternal states. As noted in the main text, the state spacefor a well-mixed chemical system is the vector of con-centrations of all molecular types. For the Paijmans et.al. KaiC model, this vector therefore lives in a space ofdimension 225 = 3.4× 107.

The original implementation of this model in [56]lacked the reverse hydrolysis reaction ADP + P → ATP,which never spontaneously happens in practice underphysiological conditions. To obtain full thermodynamicconsistency, we added this reaction to the model with theassistance of one of the original authors. This required in-troducing a new parameter ∆G0, the free energy changeof the hydrolysis reaction at standard concentrations. Forall the simulations analyzed here, we chose ∆G0 and the

10

concentration of inorganic phosphate [Pi] such that

[Pi]0[Pi]

e−∆G0 = 108. (4)

In other words, the entropy generated during a singlehydrolysis reaction when nucleotide concentrations areequal ([ATP] = [ADP]) is ∆Shyd = ln 108 ≈ 18.4.

Since the steady-state supply of free energy in thismodel comes entirely from the fixed nonequilibrium con-centrations of ATP and ADP, we can measure the aver-age rate of entropy production S by simply counting howmany ATP molecules are hydrolyzed over the course of along simulation, multiplying by ∆Shyd, and dividing bythe total time elapsed in the simulation.

All parameters other than ∆G0 are described andtabulated in the original publication [56], and theonly parameter altered during our simulations was theATP/ADP ratio.

The revised C code and scripts for generating datacan be found on GitHub at https://github.com/robertvsiii/kaic-analysis.

E. Symmetric toy model

We also developed our own abstract toy model to iso-late the operating principles of the KaiC oscillator, andto check whether the thermodynamic bound could be sat-urated by a collective oscillator with the right design.

Consider a molecule with N states, as sketched in fig-ure 6 (a). Transitions are allowed from each state to twoother states, such that the network of transitions has thetopology of a ring. The rates for “clockwise” and “coun-terclockwise” transitions around this ring are k+ = Nkand k− = Nke−A/N , respectively, where A is the cycleaffinity. Under these definitions, the total entropy pro-duced when one ring executes a full cycle is always equalto A.

Now consider M copies of this molecule in the samewell-mixed solution with Mi copies at position i, wherei increases in the “clockwise” direction from 1 to N . Wecan couple their dynamics together by allowing the barerate k to vary around the ring. Specifically, we make thebare rate ki for transitions between states i and i + 1depend on the occupancy fractions fj = Mj/M of all Nstates:

ki = exp

−C∑j

fj sin[2π(i− j)/N ]

(5)

The constant C controls the strength of the coupling,and the rate for a uniform distribution over states is 1.This dependence of the rates on the fi mimics the ef-fect of KaiA sequestration in the KaiC system. Recallthat high occupancy of the inactive conformation of KaiCcauses KaiA to be sequestered, slowing down nucleotideexchange in other hexamers, as illustrated in figure 1. In

i i+1

i+2

Nki

Nkie-A/N

i+3

i+4

N

1

i+5

(a) (b)

FIG. 6. Toy model with variable number of inter-nal states (a) Transition rates for a single molecule with Ninternal states. Multiple copies of the molecule are coupledtogether kinetically, by making ki depend on the fraction ofmolecules fi in each state. (b) Dependence of critical affinityAc on N for different values of the coupling C.

this toy model, high occupancy of any one state slowsdown transition rates ahead of that state in the cycle,by up to a factor of e−C for transitions a quarter-cycleahead. Due to the symmetry of our model, high occu-pancy of a given state also speeds up transition ratesbehind that state.

The data for figure 4 was obtained with C = 5, for 18values of A from 1 to 30. Note that ∆S/M ≈ A, sinceall M molecules execute approximately one cycle duringa given oscillation period.

We simulated this model using a Gillespie algorithmwith the reaction rates specified above. The Python codecan be found in the GitHub repository https://github.com/robertvsiii/kaic-analysis.

In the limit of infinite system size, the dynamics be-come deterministic, and are described by the followingset of N ODE’s:

dfidt

= fi−1k+i−1 + fi+1k

−i − fi(k

+i + k−i−1), (6)

with k+i = Nki and k−i = Nkie

−A/N . These equationsalways have a fixed point at the uniform state where fi =1N for all i. The linearized dynamics around the uniformstate can be written as:

dδfidt

=1

N

∑j

[∂k+

i−1

∂fj+∂k−i∂fj−

(∂k+

i

∂fj+∂k−i−1

∂fj

)]δfj

+ (δfi−1 − δfi)N + (δfi+1 − δfi)Ne−A/N (7)

=∑j

Kijδfj (8)

where the last line defines the matrix Kij . Oscillating so-lutions are possible when Kij acquires an eigenvalue witha positive real part, making this fixed point unstable. Infigure 6 (b), we plot the critical affinity Ac where thesepositive real parts first appear, as a function of the num-ber of internal states N . We confirmed that the dominantpair of eigenvalues contains nonzero imaginary parts at

11

A = Ac for all points plotted, so that the transition is atrue Hopf bifurcation to an oscillatory phase.

Note that in the limit A → ∞, N → ∞, this modelbecomes identical to the irreversible limit of a fully-connected driven XY model.

V. SIMULATIONS AND ANALYSIS

To measure N and ∆S in the KaiC model and ournew toy model, we generated an ensemble of trajectoriesfor each set of parameters. Each ensemble of the KaiCmodel contained 1,200 trajectories, while each toy modelensemble contained 1,120 trajectories. Before collectingdata, we initialized each trajectory by running the dy-namics for longer than the empirically determined relax-ation time of the system, in order to obtain a steady-stateensemble.

After projecting each trajectory onto the first two prin-cipal components and computing the n-cycle first passage

times as described above, we obtained the variance inτn as a function of n for each ensemble. We computedbootstrapped 64% confidence intervals for the estimate ofthe variance using the Python module “bootstrapped,”available at https://github.com/facebookincubator/bootstrapped. This data is plotted for all the KaiC en-sembles in figure 7. The data is well fit by a straightline even for low n in each of the plots. We obtained theslope D of these lines using a weighted least-squares fit,also shown in the figure, with the weights provided bythe inverse of the bootstrapped confidence intervals.

We used these confidence intervals to obtain the boot-strap estimate for the uncertainty in D. The size of theconfidence interval was proportional to n, as expectedfrom a simple multiplicative noise model where the slopeD is a random variable. The constant of proportional-ity yields an estimate for the standard deviation of thedistribution from which D is sampled. We obtained thisvalue for each data point with another least-squares lin-ear fit, and used it to set the size of the error bars infigures 2 and 4.

12

0

100

ATPfrac = 0.64 ATPfrac = 0.67 ATPfrac = 0.71 ATPfrac = 0.74

0

100

ATPfrac = 0.76 ATPfrac = 0.79 ATPfrac = 0.81 ATPfrac = 0.83

0

100

Varia

nce

of C

ompl

etio

n Ti

me

ATPfrac = 0.85 ATPfrac = 0.86 ATPfrac = 0.88 ATPfrac = 0.89

0

100

ATPfrac = 0.90 ATPfrac = 0.91 ATPfrac = 0.92 ATPfrac = 0.93

0 20Number of Cycles

0

100

ATPfrac = 0.93

0 20Number of Cycles

ATPfrac = 0.94

0 20Number of Cycles

ATPfrac = 0.95

0 20Number of Cycles

ATPfrac = 0.99

FIG. 7. Estimating N from KaiC simulations. Each panel shows the estimated variance and bootstrapped 64% confidenceintervals in the n-cycle first passage time τn. Straight black lines are linear fits, whose slopes provide the values of D used inthe computation of N for the main text figures.