Strange kinetics of single molecules in living cellsEli Barkai, Yuval Garini, and Ralf Metzler Citation: Phys. Today 65(8), 29 (2012); doi: 10.1063/PT.3.1677 View online: http://dx.doi.org/10.1063/PT.3.1677 View Table of Contents: http://www.physicstoday.org/resource/1/PHTOAD/v65/i8 Published by the American Institute of Physics. Additional resources for Physics TodayHomepage: http://www.physicstoday.org/ Information: http://www.physicstoday.org/about_us Daily Edition: http://www.physicstoday.org/daily_edition

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For centuries, physical imaging tools havebeen opening new frontiers in biology. Thediscovery of the cell nucleus by Scottishbotanist Robert Brown was made possibleby early-19th-century light microscopes,

and DNA was unveiled by mid-20th-century x-raydiffraction imaging.

During his observations in the 1820s, Brownmade another discovery, which has come to bear hisname. He was startled to see the jittering, lifelikemotion of small particles enclosed in pollen grains.He used control experiments with dust particles torule out the notion that the movers had to be living“animalcules.” In the early 20th century, Brownianmotion became the subject of theoretical investiga-tions by Albert Einstein, Paul Langevin, MarianSmoluchowski, and others.

Following single moleculesNow once again, another connection between biol-ogy and physics is being forged, this time by a newimaging technique called single-molecule spec-troscopy.1 Tracking individual molecules or smalltracer particles in living cells yields insight into themolecular pathways that underlie cellular regula-tion, signaling, and gene expression. Researchers

may soon be able to follow the trajectory of an indi-vidual messenger RNA molecule from its produc-tion—by the transcription of a sequence encoded ina specific gene on the cell’s DNA—to its conversioninto a protein by a ribosome. Although some indi-vidual proteins are too small to follow by single-molecule tracking, certain proteins that occur in ex-tremely low concentrations could be followed bymolecular buoys that emit light when the proteinstemporarily dock at them.

The light emitted from a single molecule mov-ing through a living cell is just one example of dy-namics in complex animate or inanimate systems inwhich one encounters complicated time variation ofobservables. Usually there’s little hope of determin-ing those variations in detail, except for some aver-aged features. Such averages are usually taken oversuitable ensembles: One observes many moleculesand averages the results. But in single-molecule ex-periments, one observes the same particle for a long

www.physicstoday.org August 2012 Physics Today 29

The irreproducibility of time-averaged observables in living cellsposes fundamental questions for statistical mechanics and

reshapes our views on cell biology.

Eli Barkai and Yuval Garini are professors of physics at Bar-Ilan University in Ramat Gan, Israel. Ralf Metzler is a professor of physicsat the University of Potsdam in Germany and Finland Distinguished Professor at Tampere University of Technology in Finland.

of single moleculesin living cells

Eli Barkai, Yuval Garini, and Ralf Metzler

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time, and the reported quantities are then time av-erages rather than ensemble averages.

Statistical physics usually deals with so-calledergodic systems, for which time and ensemble av-erages are the same. That equality is codified in theergodic theorem at the heart of statistical mechan-ics. But when tracking chemically identical mole-cules diffusing in living cells, one routinely findsthat the time averages vary from one molecule tothe next. Such apparent randomness of time aver-ages is in complete contrast to our experience of theBrownian motion of molecules in the dilute condi-tions of a test tube.

In that sense, single-molecule tracking is shift-ing our point of view away from the usual ergodicline of thought. One can no longer safely assumethat measurement of one molecule’s motion yieldsthe dynamical behavior of another identical mole-cule under the same physical conditions. This arti-cle seeks to provide an overview of the current ex-perimental state of single-molecule tracking inliving cells and of how statistical physicists are de-veloping new tools to interpret those measure-ments. In particular, we focus on the observation ofdistinctly nonergodic behavior and large devia-tions from Brownian motion. We will also discusssome potential implications of that “strange kinet-ics” for cell biology.

Brownian motionThree years after Einstein’s historic 1905 paper onBrownian motion, Jean Perrin in Paris introducedsystematic single-particle tracking. Because theBrownian trajectories were relatively short, heused ensemble averages over many particle tracesto obtain meaningful statistics. A few years later,Ivar Nordlund in Uppsala, Sweden, conceived amethod for recording much longer time series.

That let him determine time averages over indi-vidual trajectories and thus avoid averages overensembles of particles that were probably notidentical (see figure 1).

To understand how the approaches of Perrinand Nordlund are connected to each other, imaginedripping a drop of ink into water. The initially lo-calized blob will spread according to the laws of dif-fusion such that its mean squared displacement(MSD),

(1)

grows linearly in time. The proportionality factor D1is called the diffusion constant. The MSD representsan ensemble average in the sense that it measuresthe spreading of many particles, characterized bythe spatial average of r 2 over the probability densityfunction P(r, t) of finding a particle at position r attime t. (Angle brackets denote ensemble averages.)

In single-particle analyses such as Nordlund’s,by contrast, one measures the trajectory of a singleparticle in terms of the time series r(t′) over a totalmeasurement time t. Typically one measures a time-averaged MSD

(2)

which integrates the squared displacement betweentrajectory points separated by the lag time Δ muchshorter than t. (Overbars denote time averages.) ForBrownian motion of a particle in water at room tem-perature over long measurement times,

(3)

That long-time convergence is essentially identicalwith the ensemble average in equation 1. The equiv-alence of time and ensemble averaging is the hallmark

⟨ ( )⟩ = ( , ) = 6 ,r r r r2 2 31t P t d D t∫

‾‾ ‾‾‾‾δ2 ( ) = ( ′ + ) − ( ′) ′ ,Δ Δ( )r rt t dt21

t − Δ ∫0

t − Δ

‾‾δ2 → Δ6 .D1

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Strange kinetics

Figure 1. Analyzing Brownian motion by differentapproaches. (a) In 1908 Jean Perrin recorded individ-ual trajectories of small putty particles in water at30-second intervals (red dots). (b) He then plottedall the 30-second displacements, shifted to a com-mon origin, and obtained an ensemble diffusionconstant by fitting a Gaussian to the distribution ofpoints. (Adapted from ref. 15.) (c) Six years later, IvarNordlund traced, on moving film strips, individualtrajectories of mercury particles in water as theyslowly settled to the bottom. The waviness of thecurves is due to Brownian motion. He analyzed thetrajectories to obtain time-averaged mean squareddisplacements. (Adapted from ref. 16.)

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of ergodicity. In that sense, the experiments of Perrinand Nordlund are indeed equivalent.

Anomalous diffusion in living cellsSingle-molecule tracking to evaluate time-averagedMSD in cells is usually based on video microscopyof fluorescently labeled molecules (see the boxbelow). Alternatively, one can use indirect trackingwith optical tweezers.

What can be seen in such experiments? IdoGolding and Edward Cox at Princeton Universityhave tracked the motion of single messenger RNAmolecules in bacteria cells.2 They found that diffu-sion of those molecules is anomalous—relative toBrownian diffusion—in two important regards. Parameterizing the time-averaged MSD by

(4)

they found, first of all, that the anomalous diffusionexponent α is about 0.7, which means that the mes-senger RNA diffusion in vivo has a weaker time de-pendence than the Brownian diffusion described inequation 3 with α = 1. Furthermore, the anomalousdiffusion constant Dα deduced from a single trajec-tory exhibits a pronounced scatter from one trajec-tory to another (see figure 2a). It looks random.

The randomness and the anomalous time dependence persisted when the Princeton experi-

menters changed physiological conditions or evendisrupted the bacterium’s cytoskeletal internalstructure. But figure 2b suggests that the confiningcell walls play some role in the anomalous results.

It turns out that anomalous diffusion and theirreproducibility of time averages are common inliving cells. Similar results have been found for lipidgranules in yeast cells,3 for channel proteins (pore-forming molecules in cell membranes),4 and fortelomeres (chromosomal end parts) in human cellnuclei.5 Control experiments in artificially dilute en-vironments exhibit anomalous diffusion in which αdecreases with increasing concentration of crowd-ing agents and reaches a saturation value at typicalphysiological conditions.

Those results, in vivo and vitro, challenge ourpreconceptions. We would anticipate that an un-bounded molecule not actively driven by cellularmotors exhibits ordinary Brownian motion. More-over, trained in the spirit of the ergodic theorem, oneexpects sufficiently long measurements of ‾δ2 to bereproducible.

There’s another difference between Brownianmotion and diffusion in dense biological environ-ments. For a Brownian process, a measurement of‾δ2 and therefore D1 in the time interval (0, t) will beidentical to a measurement in the interval (t, 2t). Abiological cell, however, is constantly changing and

‾‾δ2 ~ ,.DααΔ

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5 µm

1 µm 1 µm

a b c

Even in simple cells such as bacteria, the interior is a super-dense mix of proteins, nucleic acids, semiflexible polymerssuch as actin, lipid membranes, and more. To follow individualmolecules in such an environment, one has to label them withsmall fluorescent marker molecules. The panels show suchlabeled molecules in different living cells: (a) chromosomeends (telomeres) in a human cell nucleus,5 (b) trajectory of achannel protein molecule in the plasma membrane of ahuman kidney cell,4 and (c) a fluorescent messenger RNA tag(bright spot) in an Escherichia coli cell (gray oval).3

Such markers function as molecular navigation lights.Interacting with an exciting laser field, they fluoresce. Foradequate resolution, labeled molecules must be sufficientlyfar apart and distinguishable from other objects by emissionwavelength. A green fluorescent protein (GFP) is ideal in thatregard. But an unbound GFP would move too fast to beobserved. Beyond the signal-to-noise problem, many fluo-rescent probes blink and eventually go dark (see the article

by Fernando Stefani, Jacob Hoogenboom, and Eli Barkai inPHYSICS TODAY, February 2009, page 34). To overcome thosedifficulties, experimenters at first tagged only relatively large,slow moving objects.

For a robust signal, one can add many markers to a largesingle molecule. Multiple marking can, however, change themolecule’s behavior.3 But attaching markers doesn’t alwayscompromise the biological system. For telomeres and viruses,fluorescent tagging doesn’t interfere with biological activity ordynamics.4,17 Sufficiently large objects like lipid granules orplastic beads can even be tracked with light microscopes.2

Sunney Xie and colleagues at Harvard University havedeveloped a method they call detection by localization, whichlets them observe molecules much smaller than messengerRNA. The team’s emphasis is on genetic kinetics rather thanrecording the paths of individual molecules.14 Advances inboth optical technology and the biochemistry of fluorescentmarkers should usher in a new era in cell biology.

Tracking in vivo

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aging; some divide and some die. Therefore onemight imagine that diffusion properties are not al-ways invariant under time translation.

Models of anomalous diffusionLet us consider further the origin of anomalous dif-fusion and its deep connection to ergodic principles.6

Physicists have been studying anomalous diffusionprocesses in disordered materials (see the article byHarvey Scher, Michael Shlesinger, and John Bendlerin PHYSICS TODAY, January 1991, page 26) and turbu-lent systems (see the article by Joseph Klafter,Shlesinger, and Gert Zumofen in PHYSICS TODAY,February 1996, page 33). Most of that work involvedlarge ensembles of particles—for example, chargecarriers in amorphous semiconductors. Promptedby the new technologies of single-molecule tracking,we now need to deal with single trajectories and con-sider time averages instead of ensemble averages.

Anomalous diffusion, irreproducibility of timeaverages, and violation of time-translational invari-ance are prominent features of a widely applicablestochastic process known as the continuous-timerandom-walk (CTRW) model. In traditional ran-dom-walk models, a particle jumps around a latticein discrete time steps. In CTRW, by contrast, the par-ticle remains immobile after each jump for a randomwaiting time τ. One assumes that the distribution ofwaiting times follows the power-law form

(5)

Unlike Einstein’s approach to Brownian motion,which corresponds to a finite-average sojourn timebetween jump events, here the average waiting timediverges. That is, ⟨τ⟩ = ∫0

∞ τψ(τ)dτ = ∞.We will see that such scale-free dynamics repre-

sents a possible scenario that leads to the strange ki-netics under discussion. The CTRW picture can be jus-

tified by microscopic models, with α in equation 5 de-pending on specific system properties. For example,the distribution of waiting times might correspond toa random walker continually caught in potential wellswhose depths are distributed exponentially.

In Einstein’s theory of Brownian motion, the en-semble-averaged MSD ⟨r2(t)⟩ grows linearly in time.It’s proportional to t/⟨τ⟩, the number of steps formean duration ⟨τ⟩. For anomalous diffusion, we usescaling arguments to set ⟨τ⟩ = ∫0

t τψ(τ)dτ ~ t1 − α. Thatassignment yields the anomalous-diffusion result

(6)

Thus scale-free waiting times do indeed yield diffu-sion processes that are slower than Brownian motion.

The CTRW model has a more drastic effect on‾δ2, the time-averaged MSD. For Brownian motion,time and ensemble averages become identical whenthe measurement time is long compared to the timescale ⟨τ⟩. But CTRW yields an infinite ⟨τ⟩. No matterhow long one measures ‾δ2, it doesn’t converge to⟨r2(t)⟩. Ergodicity is broken, and ‾δ2 remains random.Averaging ‾δ2 over many individual trajectories, onefinds an ensemble average7,8

(7)

Here, unlike in equation 4, the dependence on the lagtime Δ is linear, despite the underlying anomalous dif-fusion. Therefore some care is needed when interpret-ing experiments; what seems to be normal diffusionmay well be anomalous. In equation 7, the anomaly isa kind of aging process. That is, the ensemble average⟨‾δ2⟩ decreases with increasing experimental time t.

There’s a scaling argument for that aging behavior: For Brownian motion, one has⟨‾δ2⟩ → 6D1Δ = (⟨r 2(t)⟩/t) Δ. One then gets equation 7by replacing ⟨r 2(t)⟩/t with Dαt α − 1.

ψ τ τ α( ) ~ with 0 < < 1 .−1 − α

⟨ ( )⟩ ~ .r2 t tα

‾‾⟨ ⟩δ2 ~ .Dα

Δ

t1 − α

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Strange kinetics

1 2 3

1

0.1

0.01

0.00110 1.5

1.5

2.0

2.0

3.0

3.0

2.5

2.5

3.5LAG TIME (s)Δ

TIM

Eδ2

-AV

ER

AG

ED

(μm

)2‾‾

x (μm)

y(μ

m)

a b

α = 1 (in vitro)

α = 0.7 (in vivo)

Figure 2. Motion of labeled molecules of messenger RNA in a living Escherichia coli bacterium. (a) Time-averaged meansquared displacements ‾δ2 of individual trajectories, plotted as functions of lag time Δ in equation 2, display pronounced trajectory-to- trajectory scatter. But all have roughly the same logarithmic slope, corresponding to an anomalous diffusion exponent α ≈ 0.7 in equation 4. By contrast, the same molecules in water (starred data points) exhibit the α = 1 slope of normal Brownian diffusion. (b) A single messenger RNA molecule exploring a large fraction of the bacterium’s interior collides repeatedly with its confining cell walls. (Adapted from ref. 3.)

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The CTRW theory describes processes inwhich the random walker becomes localized forwaiting-time periods governed by ψ(τ). BenoîtMandelbrot proposed a different model of anom-alous diffusion, which he called fractional Brown-ian motion (FBM). Here a stochastic differentialequation with random noise ξ(t),

(8)

describes a component x(t) of r(t). Unlike CTRW,Mandelbrot’s model requires that the dynamics bestationary, which means that the noise correlationfunction ⟨ξ(t2)ξ(t1)⟩ depends only on the time dif-ference ∣t2 − t1∣. In that statistical sense, then, thenoise is time-translation invariant. But unlike con-ventional Brownian noise, the FBM noise is corre-lated in time. The correlation function goes like(α − 1)∣t2 − t1∣α − 2. Its power-law decay with increas-ing time difference eventually yields anomalousdiffusion.

As with CTRW, the FBM ensemble-averagedmean squared displacement increases with time liket α. But FBM’s stationary noise restores the equivalenceof ensemble and time averages. Indeed ergodicity andstationary dynamics are in many cases related.

Being ergodic and exhibiting no aging, FBM isfundamentally different from CTRW processes. TheFBM model can be derived from microscopic sce-narios. It might describe, for example, a coordinateof a single particle in an interacting many-body sys-tem—a monomer in a polymer chain or some probeparticle in a membrane.

Interpreting experiments in living cellsWhat is the origin of the randomness of time-averaged observables? Is it the nonergodicity of theCTRW model? Or is it a result of spatial inhomo-geneities? The latter would imply that the environ-ment sampled by the molecule during its motionthrough the cell differs from one trajectory to an-other. Generally, it’s hard to determine whether theobserved randomness of ‾δ2 is due to ergodicitybreaking or random environments.

The specialist community is developing diag-nostic tools to answer such questions.7,9 To distin-guish between different stochastic models, onemight try to measure the waiting-time distributionψ(τ) directly or probe for the aging effects pre-dicted by the CTRW approach. David Weitz’sgroup at Harvard University has measured a long-tailed ψ(τ) like that of equation 5 for micron-sizedbeads diffusing in a cross-linked actin network. Re-cently, Diego Krapf and coworkers at ColoradoState University observed power-law waitingtimes in the motion of channel proteins in mem-branes of living cells (see figure 3a).4 They alsodemonstrated the occurrence of ergodicity break-ing and aging by showing that ‾δ2 decreases withincreasing measurement time according to equa-tion 7 (figure 3b). All those observed behaviors arepredicted by CTRW theory.

But what is the influence of the cell walls thatconfine molecular motion? While the CTRW model

predicts that ‾δ2 is proportional to Δ, the data in fig-ure 3b show power-law scaling proportional to Δα.As demonstrated theoretically7,10 and experimen-tally,2 confinement induces an apparent scaling ofthe form ‾δ2 ~ Δ β in the CTRW model, provided thatthe molecule under observation interacts with thecell boundaries during the experimental time.

A different kind of experiment was performedby one of us (Garini) and coworkers at Bar-Ilan Uni-versity in Israel.5 The group recorded the trajectoriesof individual telomeres within cell nuclei (see the boxon page 31) and found pronounced scatter of ‾δ2.Other experiments had also seen such scatter. But theBar-Ilan team saw something new: The labeledtelomeres do not explore the volume of the nucleus.Attached to the large chromosomes, they remainfairly localized.

The observed telomere motion yielded an α ofroughly 0.3, close to the ‾δ2 ~ Δ1/4 scaling predictedfor motion in a polymer melt in Pierre-Gillesde Gennes’s reptation model (see the article by TomMcLeish in PHYSICS TODAY, August 2008, page 40).In that model, a polymer moves like a snake to cir-cumnavigate the topological obstacles created bysurrounding polymers in a polymer melt or densesolution. Because of the telomere’s connection to thelong polymeric chromosome, we expect its diffu-sion to be governed by FBM. And that’s what de-tailed analysis of the data seems to show. In partic-ular, there’s no evidence of aging.

Relevance of anomalous diffusionAnomalous diffusion of molecules in living cells isslower than normal Brownian processes. Thereforeit’s sometimes called subdiffusion. What is its bio-logical significance? Might subdiffusion be benefi-cial for the cell’s function? Naively, one might expectBrownian motion to be more efficient because theparticles move faster and therefore speed up chem-ical reactions and the search for physiological tar-gets. Why, then, is anomalous diffusion so commonin living systems?

Those questions are difficult to answer with ourlimited current knowledge of the exact dynamicsunderlying the various biochemical processes in liv-ing cells. Anomalous diffusion of large molecules isrelated to the high density of the cell environment,which creates many obstacles for the molecule alongits path.11 One can speculate that such crowding issimply a tradeoff between the need to assemble alarge number of different molecular and structuralcomponents for complex tasks and the requirementthat the cell be compact. From that point of view,anomalous diffusion is a consequence of evolution-ary optimization.

There are, in fact, several good arguments forwhy anomalous diffusion might be advantageous.It might, for example, lead to higher reaction effi-ciency. Biochemical reactions often involve initia-tion barriers. A reactant that diffuses normallycould swiftly escape its target before it’s had timeto interact.3 In certain models, the chance of findinga nearby target is explicitly increased by anom-alous diffusion.12

Recent simulation studies further underline

dx t( )dt

= ( ),ξ t

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the biological significance of anomalous diffu-sion.13 Enzymatic reaction cascades have been re-ported in which subdiffusion optimizes the finalproduct by keeping intermediate products fromwandering off. Also, it’s been shown that some cel-lular defense mechanisms with very low bindingrates to their targets are rendered surprisingly ef-ficient by subdiffusion.

Another important idea relates subdiffusion tothe organization of the cell nucleus.5 Most of theDNA in human cell nuclei is tightly wound in 46chromosomes. The chromosomes are spatially sep-arated into territories. In the box on page 31, each ofthe fluorescent molecules in panel a is presumablyensconced in such a territory.

That separation of chromosomes is essentialfor the cell’s genomic function. It may be that suchordering into territories is achieved by physicalbarriers. Alternatively, the territorial separationmight be connected to the extremely slow Δ1/4 dif-fusion measured for the telomeres, which maysimply be due to the crowded and viscous environ-ment. In that case, the chromosomes remain com-partmentalized without the need for physicalboundaries; they are like tightly packed com-muters in a subway car at rush hour, where jam-ming maintains the ordered state.

Thus far, the tracking and simulation resultsare just single pieces of the puzzle. But they alreadyshow that subdiffusion and efficient cellular dy-namics are not mutually exclusive. Recent bioinfor-matics findings suggest that critically interactingparts of the genome are often arrayed in close prox-imity on the DNA. That arrangement provides an-other argument for the benefits of anomalous diffu-sion. Efficient cell function requires reactants to beproduced near their intended reaction centers.Anomalous diffusion can ensure efficiency by keep-ing reactants from escaping.

Such a local picture of cellular regulation and sig-naling would not only be compatible with anomalousdiffusion, it would also be energetically economicaland make possible high physiological accuracy withlow copy numbers of individual reactants. Location-specific single-molecule targeting could thus becomethe new paradigm for cell biology, replacing the con-ventional conception of the cell as a small, well-mixedreaction flask. It would seem that cells have learnedways to use subdiffusion to their advantage.

Nonetheless, some processes involving transferof chemical information or cargo have to be fast. Insuch cases, anomalous diffusion poses problems.When necessary, cells might overcome such prob-lems by active motion along cytoskeletal motorways,along which motor proteins move cargo. Inside somelong human neurons, for instance, small vesicles aretransported along tubular structures for up to ameter. Such motion is “ super- diffusive” in the sensethat the exponent α in equation 4 exceeds 1.

Michael Elbaum and coworkers at Israel’sWeizmann Institute of Science have investigatedsuch behavior by tracking microspheres in livingcells. Like the groups that track molecules, they alsofind that the time-averaged MSD is random fromone trajectory to another.

TrendsWhile the experiments we have surveyed here focusmainly on the diffusion of single molecules in livingcells, the single-molecule approach is far more gen-eral. Recent experiments show how a cell’s fate canbe determined by a stochastic single-moleculeswitch. It’s known that genetically identical cells cancome in different phenotypes. For example, Es-cherichia coli bacteria with the same genotype canhave different resistivities to antibiotics. Sunney Xie’sgroup at Harvard University has used single-molecule techniques to reveal the mechanism leading

34 August 2012 Physics Today www.physicstoday.org

Strange kinetics

MEASUREMENT TIME (s)tWAITING TIME (s)τ110.1 1010 100

5

10

20

30

104

103

102

101WA

ITIN

G-T

IME

DIS

TR

IBU

TIO

N(

)ψ

τa b111 ms222 ms333 ms444 ms{Lag time Δ

δ2 ‾‾(μ

m)2

TIM

E-A

VE

RA

GE

D

τ−1.9

Figure 3. Tracking individual channel protein molecules in human cell walls. (a) Observed distribution ψ(τ) of waiting times τ between observed steps approximates a power-law decay with exponent α of about 0.9 (see equation 5).That’s taken as evidence for the continuous-time random-walk (CTRW) model of anomalous diffusion. (b) The time-averaged mean squared displacements ‾δ2 for different data-taking lag times Δ (see the color key) all decrease with increasing measurement time t. That’s indicative of an aging effect predicted by the CTRW model (see equation 7).(Adapted from ref. 4.)

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to the creation of such a phenotype.14 Interestingly,they find that whether the cell develops into one phenotype or another depends on a single bindingevent of a repressor molecule to the DNA.

Since the cell’s fate in such a scenario is deter-mined by a single molecular event, one is again farfrom the realm of conventional thermodynamics,where the phase of a macroscopic system neverturns on a single microscopic event—paceSchrödinger’s cat. Finding a case where the flippingof one molecular coin actually does determine thefate of a living organism has been made possible bysingle-molecule detection techniques.

As investigators in this young field accumulatemore and better data, they will have the opportunityto categorize the motions and reactions of a wide va-riety of molecules in living cells and relate them tocellular functions. For example, we would like to seethe correlation between the exponent α and the sizeof diffusing molecules. When do smaller molecules,usually unhampered by dense cellular environ-ments, exhibit normal ergodic diffusion?

Future optical challenges include improvingtemporal resolution and finding smaller andbrighter light emitters that don’t disturb biologicalfunction. Finally, the fundamental difference be-tween ensemble and time averages is certainly notlimited to a single observable like the mean squareddisplacement of a particle diffusing in living cells.Such departures from ergodicity have broad conse-quences for the dynamics of disordered inanimatesystems, in which single-particle behavior can bevery different from that of the ensemble.

This work was supported by the Israel Science Foundationand the Academy of Finland. We thank Ido Golding andDiego Krapf for providing experimental data and for usefuldiscussions.

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Natl. Acad. Sci. USA 108, 6438 (2011).5. I. Bronstein et al., Phys. Rev. Lett. 103, 018102 (2009).6. J. P. Bouchaud, J. Phys. I France 2, 1705 (1992); G. Bel,

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11. M. J. Saxton, Biophys. J. 72, 1744 (1997).12. G. Guigas, M. Weiss, Biophys. J. 94, 90 (2008).13. M. Hellmann, D. W. Heermann, M. Weiss, Europhys.

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(1908).16. I. Nordlund, Z. Phys. Chem. 87, 40 (1914).17. G. Seisenberger et al., Science 294, 1929 (2001). ■

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