REVIEW

Using Gene Expression Noise toUnderstand Gene RegulationBrian Munsky,1* Gregor Neuert,2* Alexander van Oudenaarden2,3

Phenotypic variation is ubiquitous in biology and is often traceable to underlying genetic andenvironmental variation. However, even genetically identical cells in identical environmentsdisplay variable phenotypes. Stochastic gene expression, or gene expression “noise,” has beensuggested as a major source of this variability, and its physiological consequences have beentopics of intense research for the last decade. Several recent studies have measured variability inprotein and messenger RNA levels, and they have discovered strong connections between noise andgene regulation mechanisms. When integrated with discrete stochastic models, measurements ofcell-to-cell variability provide a sensitive “fingerprint” with which to explore fundamental questionsof gene regulation. In this review, we highlight several studies that used gene expression variabilityto develop a quantitative understanding of the mechanisms and dynamics of gene regulation.

Identical genotype and environmental expo-sure are not sufficient to guarantee a uniquephenotype. Consider a single mother cell di-

viding into two daughter cells of equal volume.During the division process, all the molecules inthe mother cell are in Brownian motion accord-ing to the laws of statistical mechanics. The prob-ability that each daughter cell inherits the samenumber of molecules is infinitesimally small.Even in the event that the two daughter cells re-ceive exactly one copy of a particular transcrip-tion factor, each transcription factor will performa Brownian random walk through its cellularvolume before finding its target promoter andactivating gene expression. Because Brownianmotion is uncorrelated in the two daughter cells,it is statistically impossible for both genes to be-come activated at the exact same time, furtheramplifying the phenotypic difference betweenthe two daughter cells. These are just two exam-ples of the many sources of gene expression var-iability that arise in isogenic cells exposed to thesame environment.

The origins and consequences of stochasticgene expression, or gene expression “noise,”have been studied extensively during the last dec-ade and have recently been reviewed in detail(1–6). Here we focus on recent works that in-tegrate experimental and computational analy-ses of gene expression noise to systematically testand refine our understanding of regulation in dif-

ferent genes, regulatory pathways, and organisms.We discuss how combining single-cell measure-ments and stochastic analyses can reveal qualita-tive and quantitative features of gene regulationthat are hidden by bulk assays or deterministicanalyses. In the first part of this review, wediscuss how the cell-to-cell variability in gene

expression of a particular transcript or protein hasbeen used to develop a quantitative understand-ing of the underlying gene regulation. In the secondpart, we cover how statistical correlations in thefluctuations of different transcripts and/or proteinscan be used to infer gene regulatory interactions.

Inferring Models of Gene Regulation fromVariability in Gene Expression

Poisson expression statistics. In the simplestpossible model of constitutive gene expression(Fig. 1A), a transcript is produced at a constant ratekR and destroyed in a first-order reaction with rateconstant gR. If the total number of a particular tran-script m is large, the kinetics can be approximatedby the following deterministic differential equation:

dm

dt¼ kR − gRm ð1Þ

This approximation breaks down in cells whenthe copy numbers of transcripts are small. For ex-ample, the average transcript copy numbers of theconstitutive housekeeping genesMDN1,KAP104,and DOA1 in budding yeast are 6.1, 4.9, and 2.6,respectively (7). These low copy numbers suggesta probabilistic reformulation of Eq. 1. In the con-stitutive expression model, transcript births anddeaths occur as uncorrelated events, such that inany short time interval, dt, the probability of onetranscript production is kR dt, and the probabil-ity of one transcript degradation is gR m dt. Forequilibrium to be possible, the probability ofhaving m transcripts, Prob[m], and producing an-other must be equal to the probability of hav-ing (m + 1) transcripts, Prob[m + 1] and havingone degrade. That is, kR Prob[m] = gR (m + 1)Prob[m + 1] for anym, which is only possible ifthe copy-number distribution follows a Poissondistribution (8).

It is possible to quantify the variability intranscriptional regulation at themRNA level usingsingle-molecule fluorescent in situ hybridization(smFISH) (9, 10). In this technique, endogenousmRNA transcripts are labeled with a large num-ber of fluorescently modified DNA oligonucleo-tides. As a result, a fluorescence microscopecan detect the precise location of each individualmRNA molecule as a diffraction-limited spot.

Zenklusen et al. (7) used smFISH to countspecific mRNA molecules in intact fixed yeastcells and found that the constitutive gene expres-sion model offers surprisingly good quantitativematches to transcriptional behaviors for the house-keeping genes MDN1, KAP104, and DOA1 inbudding yeast. The measured numbers of mRNAtranscripts per cell were well described by Poissondistributions for all three genes. By measuringthe number of partially formed nascent mRNA ineach nucleus, Zenklusen et al. also determinedthat subsequent transcript production events wereuncorrelated (7), again consistent with the con-stitutive expression model.

Two-state model of gene regulation.Althoughthe constitutive gene expression model capturesthe fluctuations of several housekeeping genes inbudding yeast (7), it does not perform as well whengene expression is regulated. Because deviations

1Center for Nonlinear Studies, the Information Sciences Group,and the National Flow Cytometry Resource, Los Alamos Nation-al Laboratory, Los Alamos, NM 87545, USA. 2Departments ofPhysics and Biology, Massachusetts Institute of Technology, Cam-bridge, MA 02139, USA. 3Hubrecht Institute, Royal NetherlandsAcademy of Arts and Sciences and University Medical CenterUtrecht, Uppsalalaan 8, 3584 CT, Utrecht, Netherlands.

*These authors contributed equally to this work. To whomcorrespondence should be addressed. E-mail: munsky@lanl.gov (B.M); gneuert@mit.edu (G.N.)

A BConstitutivegene expression

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Fig. 1. Constitutive versus regulated gene expression. (A) Schematic of a constitutive gene expressionmodel with transcription rate kR and mRNA degradation rate constant gR. (B) Schematic of a two-state(On, Off) model with transition rates kOn and kOff.

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from Poisson behavior indicate regulation, canquantifying these deviations reveal the mech-anism of regulation? A key parameter to quantifythe deviation from Poisson statistics is the Fanofactor, which is the ratio between the variance,s2, and the mean,m, of the mRNA copy-numberdistribution,s2=m (8). For a Poisson distribution,the Fano factor equals 1. Golding et al. (11)determined a Fano factor of 4.1 for a synthetictranscript driven by the PLAC/ARA promoter inEscherichia coli, indicating that the transcriptdistribution was significantly wider than a Poissondistribution. A two-state model of gene expres-sion (12–16) can fit these data much better. Thismodel considers two promoter states: an Off state,in which no transcription occurs, and an On state,which has transcription rate kR. The constants kOnand kOff define the transition rates between thetwo states, and gR is a first-order rate constant fortranscript degradation (Fig. 1B). The Off state isusually associated with a closed chromatin statein which the binding sites for transcription factorsare inaccessible, whereas the On state is asso-ciated with the open active chromatin state (14).

According to the two-state model, the aver-age fraction of cells in the On state is fOn =kOn/(kOn + kOff), and the average number ofmRNA molecules in each cell is m ¼ fOnkR=gR.The expression for the Fano factor in steady statecan be written as (12):

s2m ¼ 1þ ð1 − fOnÞkRðkOn þ kOff þ gRÞ

ð2Þ

Figure 2A uses a heatmap to illustrate the Fanofactor’s dependence upon kOff and kOn for a fixedtranscription rate kR. To compare the variabilityat equal expression levels, the three dashed linesdenote parameter combinations that produce anaverage of 2, 25, and 75mRNAs per cell. Althoughthe average expression level is constant along theselines, the Fano factor varies significantly, as doesthe qualitative shape of the mRNA distribution.For example, Fig. 2B shows the distributions cor-responding to the filled squares on them ¼ 25 linein Fig. 2A. Although each parameter set yields anaverage of 25 molecules per cell, they exhibitthree distinctly different behaviors for the varia-bility of m between cells. On the basis of differ-ences in the mean, Fano factor, and qualitativeshapes of distributions, we can dissect the param-eter space into three different “phenotype” classes(14, 17). In class I, both kOn and kOff are slow,and cells separate into distinct On and Off pop-ulations, yielding a bimodal mRNA distribution(Fig. 2B, left) and resulting in a large Fano fac-tor. In class II, kOn is slow and kOff is fast, andtherefore most cells are Off. In this case, the lowvalue of fOn contributes to low means and Fanofactors, but occasional mRNA bursts give rise tolong exponential tails in the mRNA distribution(Fig. 2B,middle). Finally, in class III, kOn is fast incomparison to either gR or kOff, and the systemspends very short periods in the Off state. The

dynamics of this special case collapses down tothat of the constitutive expression model, with aneffective transcription rate keffR ¼ kR fOn and aPoisson-like mRNA distribution (Fig. 2B, right).

A cell can increase the average mRNA copynumber m from 2 (indicated by the blue star inFig. 2A) to 25 by either decreasing kOff (purplearrow in Fig. 2A) or increasing kOn (red arrow inFig. 2A). Increasing kOn converts a class II phe-notype into a class III phenotype, resulting inmorePoisson-like expression. Conversely, decreasingkOff shifts the system to class I, corresponding tobimodal expression. Thus, although both modu-lation mechanisms yield the same change in av-erage mRNA levels, their single-cell statistics arequantitatively and qualitatively different. Belowwe highlight several studies that exploit thesedifferences to learn more about the gene regu-latory control mechanisms.

As in studies of constitutive expression, single-cell responses of regulated genes have been ex-amined at the mRNA level. Raj et al. (15) usedsmFISH to study gene expression variability inmammalian cells. They integrated an inducibletetO promoter into the genome and quantifiedmRNA numbers and locations. The measuredmRNA distributions had long exponential tailsthat closely matched those of the class II pheno-type, corresponding to bursts of mRNA that wereshort, infrequent, and intense. Furthermore,Raj et al. observed that On cells exhibited extra-bright clusters of nascent transcripts and elevatedlevels of nuclear mRNA,whereas Off cells lackedthese transcription site spots and had far fewernuclear mRNAs. In this context, spatial variabil-ity provided quantitative insight into transcrip-tional dynamics. More recent smFISH studieshave also discovered mRNA distributions from

B

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bab

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00 50 100 150 200 50 100 150 200

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64

Fig. 2. Effects of transcriptional control onmRNA distributions. (A) Heat map of the cell-to-cell variability(Fano factor, s2/m), versus normalized gene activation rate kOn /gR and normalized deactivation ratekOff/gR with fixed production and degradation rates (kR = 100, gR = 1). Lines of equal average mRNAexpression are shown for 2, 25, and 75 molecules. The parameter space is separated into three classes(I, II, III) that exhibit different types of cell-to-cell variability. (B) Representative distributions from eachclass: Class I corresponds to systems with long Off and On periods, giving rise to bimodal distributionswith clearly delineated On/Off populations. Class II corresponds to populations with short On and longOff periods, giving rise to occasional mRNA bursts and long distribution tails. Class III includes systemswith short Off periods, giving rise to continuous production and more graded unimodal distributions.All three distributions have the same average of 25 mRNAs and correspond to the squares in (A).Distributions were computed with the finite state projection approach (34).

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all three classes for a myriad of other genes inseveral model organisms, including bimodal dis-tributions of class I in cell-cycle and induciblegenes in yeast (18); long exponential distributiontails corresponding to class II in E. coli (19),ribosomal RNA (20), and coding and long non-coding RNA (21) transcription in yeast; and uni-modal Poisson-like distributions of class III inyeast (7).

Single-molecule FISH has also been used toexplore how transcriptional regulation changesbetween conditions (15, 19). Raj et al. (15) showedthat an increase in the transcriptional activatortTA or in the number of activator binding sitesincreased the transcriptional activity of the tetOpromoter in mammalian cells. Quantitative com-parisons of measured mRNA distributions withthe two-state model revealed that activation wasconsistent with either kOff modulation (purple ar-row in Fig. 2A) or kR modulation. In a recentstudy, So et al. also integrated experimental andcomputational analyses to find that kOff modula-tion is a common motif, which regulates mRNAexpression in 20 independent E. coli genes, whosemRNA expressions span four orders of magnitude(19). Using smFISH (10), they measured mRNAdistributions in 150 different combinations ofgenes and growth conditions that modulate thosegenes. After correcting for different gene copynumbers and mRNA lifetimes, the mRNA meanand Fano factor were computed and plotted forevery gene and experimental condition, and theresulting scatter plot was closely fit by Eq. 2,where kOn and kR were constant, and kOff wasselected to match the mean expression.

Although the former studies explored generegulation at the mRNA level (7, 15, 18–21),similar conclusions have been reached throughsingle-cell analyses at the protein level. Raser andO’Shea (14) used single-cell measurements offluorescent protein concentrations to show thatinduction of the PHO5 promoter in buddingyeast increases the expression level while reduc-ing the cell-to-cell variability. This trend wasexplained as the system starting in class II andincreasing kOn to switch toward class III (redarrow in Fig. 2A). Similarly, Octavio et al. (22)explored the regulation of the FLO11 gene inyeast. Using inducible promoters to control theregulatory proteins Flo8, Sfl1, Tec1, Ste12, Phd1,Msn1, and Mss1, they pushed the system intoeach of the three phenotypes. Then, by elucidat-ing how each transcription factor altered thevariability in gene expression, they determinedthe mechanisms by which each factor modu-lated transitions between an Off state, an inter-mediate “competent” state, and the fully activeOn state.

Inferring Gene Regulatory Interaction fromCorrelations Between Fluctuating GenesThe examples above illustrate how the expres-sion distribution of a particular transcript or flu-

orescent protein reporter can be used to quantifythe transitions between active and inactive tran-scription states and to determine the mechanismby which regulators modulate this process. Inmany of these studies, the analysis of regulatorybehavior required the application of an externalinput or a change in environmental conditions. Itis not always easy to introduce such a perturba-tion, but what if they already existed in nature?As discussed above, most cellular proteins under-go stochastic fluctuations, which can activate orrepress downstream processes and thereby intro-duce valuable perturbations. As a result, whenmultiple transcript or protein species are moni-tored in the same cell, important additional infor-mation can be extracted by analyzing howdifferentspecies correlate with one another. This correla-tion analysis was used in experiments focused onsynthetic gene networks in E. coli, where expres-sion levels of several genes weremonitored in thesame cell with fluorescent reporters. By analyz-ing the pairwise correlation between the differ-ent fluorescent reporters, the major fluctuationsources could be determined (23, 24).

In a recent study, Stewart-Ornstein et al. (25)used fluorescent proteins to examine the pairwisecorrelations of hundreds of different yeast genes,whose expression levels varied over three ordersof magnitude. Even without using exogenous per-turbations, single-cell steady-state measurementscould reveal clear groups of genes whose stochas-tic fluctuations were strongly coordinated. Thesecollections of genes, which they labeled “noiseregulons,” corresponded to functional groups re-lated to stress response, mitochondrial regula-tion, and amino acid biosynthesis. Furthermore,Stewart-Ornstein et al. showed that steady-statecorrelations were strongly predictive of the pro-teins’ dynamic response to heat shock.

Using a two-color RNA fluorescent in situhybridization assay, Gandhi et al. (18) measuredpairwise correlations between RNA species regu-lated by the same promoter or by two differentpromoters. TheGal4-regulated genesGAL1,GAL7,andGAL10were induced with 2% galactose, andtheir distributions were measured at steady state.As expected, single-cell correlation analyses showedstrong correlations between GAL1 and GAL7, aswell as between GAL1 and GAL10. mRNA cor-relations were also found in other regulatory genes.Transcripts of the genes SWI5 and CLB2, whichare expressed in the G2/M stages of the cell cycle,were strongly correlated with each other, butweakly anticorrelated with NDD1, which domi-nates during the S phase. By contrast, constitutivegenes such asMDN1 (ribosome biogenesis),PRP8(pre-mRNA splicing), and KAP104 (nucleocyto-plasmic transport) exhibitedmuch less coordination.

Although correlations at a single time pointcan reveal static relationships among differentmRNA and protein species, this view lacks in-formation about the system’s history and causalrelationships. If two proteins X and Y are cor-

related, the questions remain: Does X activate Y;does Y activate X; or does a third protein Wcontrol them both? To illustrate this situation,Fig. 3A shows simple motifs by which proteinsW, X, and Y could relate to one another, and Fig.3B shows typical scatter plots of the single-cellexpression for proteins X and Y for these motifs.When static correlations cannot discriminate be-tween thesemotifs, dynamic correlations in single-cell fluctuations may help (26). Such analysesmake use of the cross-correlation function (26),RXYðtÞ ¼ ⟨X ðt þ tÞY ðtÞ⟩=sXsY, which mea-sures how fluctuations in Yat time t relate to thoseinX at time t + t. Here, ⟨:::⟩ denotes the covarianceof two variables, and sX and sY are the standarddeviations of X and Y, respectively. The magni-tude of RXYðtÞ reveals positive or negative regu-lation, and the timing of peaks in RXYðtÞ revealscausality in this regulation. As examples, Fig. 3Cplots the cross-correlation functions between pro-teins X and Y for each of the motifs in Fig. 3A.For the first motif, where X activates Y, the blueline in Fig. 3C (left) shows that RXYðtÞ has amaximum, and because X is upstream of Y, thispeak occurs at a negative delay time. Conversely,when protein Y is a repressor of X, RXYðtÞ has aminimum at a positive t (Fig. 3C, second col-umn, red line). If both X and Y were controlledby W, the maximum or minimum would occurat t = 0, and its sign would be positive or neg-ative depending upon whether W has the sameor different effects on X and Y (Fig. 3C, righttwo columns).

Dunlop et al. (26) tested this dynamic cor-relation approach in live cells by inserting threefluorescent protein reporters of different colorsinto the E. coli genome. Yellow fluorescent pro-tein (YFP) was fused to the l CI repressor, whichcontrolled expression of red fluorescent protein(RFP). Cyan fluorescent protein (CFP) was placedon a separate constitutive promoter. With the useof fluorescence time-lapse microscopy, all threecolors could be monitored simultaneously overseveral hours. Dynamics of the YFP-RFP pairwere anticorrelatedwith a delay of about 120min,clearly revealing that CI-YFP repressed RFP (sim-ilar to Fig. 3C, second column, blue line). Con-versely, the unregulated YFP-CFP pair exhibiteda delay-free correlation characteristic of commonupstream regulators (extrinsic noise) that affectboth YFP and CFP in a similar fashion (similar toFig. 3C, third column). Thus, the causal relation-ships of all three reporters were uniquely deter-mined. Extending and applying this approach totheCRP-GalS-GalE feed-forward loop in E. coli,they analyzed how the relationship betweenGalSand GalE varies under different fucose concen-trations and under the influence of GalR (26).

Although correlations at either mRNA or pro-tein levels can reveal gene regulatory relation-ships, the two do not always perform equallywell. To illustrate this scenario, Fig. 3, D and E,show scatter plots and cross-correlations between

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Fig. 3. Different regulatory motifs yield different steady-state correla-tions. (A) Schematics of four possible regulator motifs: X activates Y; Xrepresses Y; W activates both X and Y; and W activates X but represses Y.For each motif, mRNA is produced according to the constitutive model;protein is translated from mRNA as a first-order reaction; and both mRNAand protein degrade as a first-order reaction. Regulation changes to thetranscription rate are defined as kR(X) = aX4/(M4 + X4) for activationand kR(X) = aM4/(M4 + X4) for repression. (B) Scatter plots of the pop-ulations of protein X and protein Y at steady state. (C) Dynamic cross-

correlation functions of protein X and protein Y, versus the correlation timedelay. The magnitude of RXY(t) indicates how strongly X(t + t) is correlated(positive) or anticorrelated (negative) with Y(t). For causal events, where Xactivates (or represses) Y, peaks (or dips) appear in RXY(t) at negative valuesof t. Blue lines correspond to the motif in (A), and red lines correspond tothe same motif in which X and Y have been interchanged. (D) Scatter plotsfor mRNA X and mRNA Y populations. (E) Dynamic cross-correlation formRNA X and mRNA Y. Simulations were conducted with the stochasticsimulation algorithm (35).

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the mRNA X and mRNA Y corresponding toprotein X and protein Y, respectively. Althoughprotein X and protein Y are coordinated for allfour motifs in Fig. 3, this is not the case for theirmRNA levels. This can be explained by the dis-parate time scales of mRNA and protein. Fast-degrading mRNA may exhibit fluctuations witha broad frequency bandwidth. Conversely, slowdegradation of proteins filters out fast fluctua-tions but keeps slow fluctuations. Constitutivelyexpressed mRNA X has both fast and slow fluc-tuations, but protein X only transmits the slowfluctuations downstream. The result is that thedynamics of mRNA X and mRNA Y are dom-inated by uncorrelated fast fluctuations, whichovershadow their correlated slow fluctuations. Onthe other hand, protein X and protein Y onlycontain the better-correlated slow fluctuations.That is, two mRNA species can be mostly un-correlated with one another, yet produce proteinin a coordinated fashion. Gandhi et al. (18) ob-served such a circumstance in budding yeast,when they found very little correlation betweenpairs of transcripts that encode coordinated pro-teins of the same protein complex, including pro-teasome and RNA polymerase II subunits. Theyeven found correlation lacking in two alleles ofthe same gene. In a related study, Taniguchi et al.(27) analyzed more than 1000 genes in E. coliand measured both mRNA and protein copynumbers in single cells. They found that for mostgenes, even the numbers of mRNA and proteinmolecules were uncorrelated. These studies sug-gest that understanding of regulatory phenomenarequires one to consider regulation at both themRNA and the protein level.

From these studies, it is now clear that var-iability in single-cell measurements contains awealth of information that can reveal new in-sights into the regulatory phenomena of specificgenes and the dynamic interplay of entire genenetworks. As modern imaging techniques beginto beat the diffraction limitations of light (28) andflow cytometers become affordable for nearlyany laboratory bench (29), we find ourselves inthe midst of an explosion in single-cell research.With the advent of single-cell sequencing (30, 31),it might be possible to determine the full tran-scriptome of many single cells in the near futureand to determine the full expression distributionsand correlations for all genes in the genome. Weexpect that the approaches described in this re-view, which have been pioneered with the modelmicrobial systems, will be readily applied tomam-malian cells and tissues (32, 33).

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910 (2011).4. A. Eldar, M. B. Elowitz, Nature 467, 167 (2010).5. M. E. Lidstrom, M. C. Konopka, Nat. Chem. Biol. 6,

705 (2010).6. B. Snijder, L. Pelkmans, Nat. Rev. Mol. Cell Biol. 12,

119 (2011).7. D. Zenklusen, D. R. Larson, R. H. Singer, Nat. Struct.

Mol. Biol. 15, 1263 (2008).8. M. Thattai, A. van Oudenaarden, Proc. Natl. Acad.

Sci. U.S.A. 98, 8614 (2001).9. A. M. Femino, F. S. Fay, K. Fogarty, R. H. Singer, Science

280, 585 (1998).10. A. Raj, P. van den Bogaard, S. A. Rifkin, A. van Oudenaarden,

S. Tyagi, Nat. Methods 5, 877 (2008).11. I. Golding, J. Paulsson, S. M. Zawilski, E. C. Cox, Cell 123,

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13. T. B. Kepler, T. C. Elston, Biophys. J. 81, 3116 (2001).14. J. M. Raser, E. K. O’Shea, Science 304, 1811 (2004).15. A. Raj, C. S. Peskin, D. Tranchina, D. Y. Vargas, S. Tyagi,

PLoS Biol. 4, e309 (2006).16. V. Shahrezaei, P. S. Swain, Proc. Natl. Acad. Sci. U.S.A.

105, 17256 (2008).17. S. Iyer-Biswas, F. Hayot, C. Jayaprakash, Phys. Rev. E Stat.

Nonlin. Soft Matter Phys. 79, 031911 (2009).18. S. J. Gandhi, D. Zenklusen, T. Lionnet, R. H. Singer,

Nat. Struct. Mol. Biol. 18, 27 (2011).19. L.-H. So et al., Nat. Genet. 43, 554 (2011).20. R. Z. Tan, A. van Oudenaarden, Mol. Syst. Biol. 6,

358 (2010).21. S. L. Bumgarner et al., Mol. Cell 45, 470 (2012).22. L. M. Octavio, K. Gedeon, N. Maheshri, PLoS Genet. 5,

e1000673 (2009).23. J. M. Pedraza, A. van Oudenaarden, Science 307,

1965 (2005).24. N. Rosenfeld, J. W. Young, U. Alon, P. S. Swain,

M. B. Elowitz, Science 307, 1962 (2005).25. J. Stewart-Ornstein, J. S. Weissman, H. El-Samad,

Mol. Cell 45, 483 (2012).26. M. J. Dunlop, R. S. Cox III, J. H. Levine, R. M. Murray,

M. B. Elowitz, Nat. Genet. 40, 1493 (2008).27. Y. Taniguchi et al., Science 329, 533 (2010).28. B. Huang, M. Bates, X. Zhuang, Annu. Rev. Biochem. 78,

993 (2009).29. L. Bonetta, Nat. Methods 2, 785 (2005).30. T. Kalisky, P. Blainey, S. R. Quake, Annu. Rev. Genet. 45,

431 (2011).31. F. Tang et al., Nat. Methods 6, 377 (2009).32. S. Itzkovitz et al., Nat. Cell Biol. 14, 106 (2012).33. P. Dalerba et al., Nat. Biotechnol. 29, 1120 (2011).34. B. Munsky, M. Khammash, J. Chem. Phys. 124,

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Acknowledgments: This work was funded by the NationalScience Foundation (ECCS-0835623) and a NIH Pioneer award(1DP1OD003936).

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REVIEW

Computational Approachesto Developmental PatterningLuis G. Morelli,1,2,3 Koichiro Uriu,1,4 Saúl Ares,2,5,6 Andrew C. Oates1*

Computational approaches are breaking new ground in understanding how embryos form. Here,we discuss recent studies that couple precise measurements in the embryo with appropriatelymatched modeling and computational methods to investigate classic embryonic patterningstrategies. We include signaling gradients, activator-inhibitor systems, and coupled oscillators,as well as emerging paradigms such as tissue deformation. Parallel progress in theory andexperiment will play an increasingly central role in deciphering developmental patterning.

Animal and plant patterns amaze and per-plex scientists and lay people alike. Buthow are the dynamic and beautiful pat-

terns of developing embryos generated? Usedappropriately, theoretical techniques can assistin the understanding of developmental processes(1–5). There is considerable art in this, and thekey to success is an open dialogue between exper-

imentalist and theorist. The first step in this dia-logue is to formulate a theoretical description ofthe process of interest that captures the propertiesand interactions of the most relevant variablesof the system at a level of detail that is both use-ful and tractable. Once formulated, the secondstep is to analyze the theoretical model. If themodel is sufficiently tractable, it may be possible

to understand its behavior with “pencil-and-paper” analysis and compare this analytical solu-tion directly with experimental data. Very often,however, the number of variables and the com-plexity of their interactions preclude this ap-proach, and the behavior ofmodelsmust be solvedor simulated by using computers in order to beunderstood and compared with data. This com-bined approach, which we refer to as computa-tional biology, has become popular recently withthe availability of powerful computers and in-creasingly sophisticated numerical algorithms.

1Max Planck Institute of Molecular Cell Biology and Genetics,Pfotenhauerstrasse 108, 01307 Dresden, Germany. 2MaxPlanck Institute for the Physics of Complex Systems, NöthnitzerStrasse 38, 01187 Dresden, Germany. 3Consejo Nacional deInvestigaciones Científicas y Técnicas, Departamento de Física,Universidad de Buenos Aires, Ciudad Universitaria, 1428Buenos Aires, Argentina. 4Theoretical Biology Laboratory, RIKENAdvanced Science Institute, Saitama 351-0198, Japan. 5Logic ofGenomic Systems Laboratory, Centro Nacional de Biotecnología–Consejo Superior de Investigaciones Científicas (CSIC), CalleDarwin 3, 28049 Madrid, Spain. 6Grupo Interdisciplinar deSistemas Complejos (GISC), Spain.

*To whom correspondence should be addressed. E-mail:oates@mpi-cbg.de

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