Information transfer by leaky, heterogeneous, proteinkinase signaling systemsMargaritis Voliotisa, Rebecca M. Perrettb, Chris McWilliamsa, Craig A. McArdleb, and Clive G. Bowshera,1

aSchool of Mathematics and bLaboratories for Integrative Neuroscience and Endocrinology, School of Clinical Sciences, University of Bristol, Bristol BS8 1TW,United Kingdom

Edited* by Ken A. Dill, Stony Brook University, Stony Brook, NY, and approved November 20, 2013 (received for review July 31, 2013)

Cells must sense extracellular signals and transfer the informationcontained about their environment reliably to make appropriatedecisions. To perform these tasks, cells use signal transductionnetworks that are subject to various sources of noise. Here, westudy the effects on information transfer of two particular typesof noise: basal (leaky) network activity and cell-to-cell variability inthe componentry of the network. Basal activity is the propensityfor activation of the network output in the absence of the signalof interest. We show, using theoretical models of protein kinasesignaling, that the combined effect of the two types of noisemakes information transfer by such networks highly vulnerable tothe loss of negative feedback. In an experimental study of ERKsignaling by single cells with heterogeneous ERK expression levels,we verify our theoretical prediction: In the presence of basalnetwork activity, negative feedback substantially increases in-formation transfer to the nucleus by both preventing a near-flataverage response curve and reducing sensitivity to variation insubstrate expression levels. The interplay between basal net-work activity, heterogeneity in network componentry, and feed-back is thus critical for the effectiveness of protein kinase sig-naling. Basal activity is widespread in signaling systems underphysiological conditions, has phenotypic consequences, and isoften raised in disease. Our results reveal an important role fornegative feedback mechanisms in protecting the informationtransfer function of saturable, heterogeneous cell signaling sys-tems from basal activity.

cell sensing | MAPK signaling | mutual information | ultrasensitivity |biomolecular networks

Cells must sense extracellular concentrations and transfer theinformation contained about their environment reliably to

make appropriate decisions. Understanding the process of in-formation transfer from the biological environment to the nu-cleus (1) and studying quantitatively how information about thesignal is lost along the way are essential in understanding cellulardecision-making (2, 3). The signal transduction networks used bycells are subject to various sources of noise, and we are onlybeginning to explore how these affect the process of informationtransfer (4, 5). Here we focus, in the context of protein kinase(PK) signaling, on the little-studied effects of two importanttypes of biological noise: cell-to-cell variability in the compo-nentry of the network and basal network activity. The effect oninformation transfer of cell-to-cell variation (heterogeneity) inthe protein componentry of signaling networks remains largelyunexplored. Such variation is expected under physiological con-ditions (6) and underlies the variable responses observed forgenetically identical cells exposed to the same stimulus or drugtreatment (7–9). By basal activity, we mean the propensity foractivation of the signaling system in the absence of the stimulusor signal of interest. Basal activity is widespread in signalingsystems under physiological conditions (with observable pheno-typic consequences) and is often raised in disease. Although itssource is not always well-characterized, such activity arises innumerous ways—for example, the constitutive activity of recep-tors, basal enzymatic activity caused by the conformational

heterogeneity of proteins over time (10), and the promiscuity ofsignaling intermediaries (11).Many extracellular stimuli control cell fate decisions using PK

signaling, often through MAPK cascades such as the well-knownERK pathway (12, 13). Basal MAPK activity is known to befunctionally relevant. Basal ERK activity enhances cell substrateinteractions (14) and is essential for survival of quiescent,unstimulated cells (15). In yeast, basal activity in one branch ofthe Hog1 MAPK pathway, combined with negative feedbackfrom Hog1, results in faster response to osmotic stress andhigher sensitivity to external signals (16).Most mammalian cells express receptor tyrosine kinases

(RTKs) and G protein-coupled receptors (GPCRs) that act tostimulate the Raf-MEK-ERK cascade (12, 13, 17). Basal activityis often observed in signaling from RTKs (18) and is evident inthe measured levels of activated ERK in unstimulated cells (14,19, 20). Most antagonists of GPCRs behave as inverse agonists,suppressing the response caused by the basal (or constitutive)activity of the receptor, and most drugs acting on GPCRs thatare in clinical use are now recognized as inverse agonists (21).Raised basal activity is also known to be important in variousdiseases (21), including the roles of the NGF receptor in breastcancer (22) and the V600EB-Raf protein in melanoma. In fact, theERK pathway is dysregulated in about one-third of all humancancers, frequently through gain-of-activity mutations in the RASand RAF genes (23).Here, we use mutual information to quantify the effects on

information transfer of basal activity and heterogeneity in net-work componentry, comparing our systems with and without

Significance

Extracellular concentrations convey information to cells abouttheir environment. To sense these signals, cells use biomolec-ular networks that exhibit inevitable cell-to-cell variability andbasal activity. Basal activity is widespread under physiologicalconditions (with phenotypic consequences), is often raised indisease, and can eradicate the transfer of information. In anexperimental study of ERK signaling by single cells exhibitingheterogeneous ERK expression and basal activity, we verifyour central theoretical prediction: Negative feedback sub-stantially increases information transfer to the nucleus bypreventing a near-flat average response curve and reducingsensitivity to variation in the ERK expression level. Our resultsreveal an important role for negative feedback mechanismsin protecting information transfer by saturable cell signalingsystems from basal activity.

Author contributions: C.A.M. and C.G.B. designed research; M.V., R.M.P., and C.M. per-formed research; M.V., R.M.P., and C.G.B. contributed new reagents/analytic tools; M.V.,C.M., and C.G.B. analyzed data; and C.A.M. and C.G.B. wrote the paper.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.1To whom correspondence should be addressed. E-mail: C.Bowsher@bristol.ac.uk.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1314446111/-/DCSupplemental.

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negative feedback. Mutual information and information capacityare becoming increasingly important for quantifying informationtransfer by biochemical systems (4, 24–26). Mutual informationallows the comparison of systems in terms of how well a givensignal can be inferred using the Bayesian posterior distribution(27) based on each system’s intracellular output. We use statis-tical decision theory to explain in what sense higher mutual in-formation (SI Appendix, Eq. S6) implies better inferences of thebiological environment in question (SI Appendix). This approachmotivates our experimental measurement of the mutual in-formation between stimulus concentration and nuclear activatedERK levels for single cells. We propose to quantify the effect ofheterogeneity in network componentry by computing the in-crease in mutual information under the assumption that thelevels of the variable components in each cell are monitored inaddition to the outputs of the signaling network. Recent methodsfor decomposing variation in the output of biochemical systems(28, 29) are used to provide additional insight into the effects of

negative feedback and of different sources of noise that harmsignaling. In our experimental study of ERK signaling, we ex-amine the effect of cell-to-cell variation in expression levels ofthe ERK protein itself and show that the interplay between suchvariation, basal activity, and negative feedback is critical for theeffectiveness of PK signaling.

ResultsLoss of Negative Feedback Can Eradicate Information Transfer byLeaky Heterogeneous PK Signaling Systems. Negative feedback isa common feature of PK signaling systems. For example, acti-vated ERK inhibits Raf by phosphorylation (30), and ligand-induced phosphatases (e.g., MAPK phosphatases and Sproutyproteins) negatively regulate MAPK activation (31, 32). In thispaper, we argue that negative feedback protects informationtransfer by these systems from the combined effects of basalactivity and cell-to-cell variation in network componentry. Webegin with the simple building block in which a PK phosphorylates

A B

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D

H I

E F

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Fig. 1. Output–signal relationships for phosphorylation–dephosphorylation cycles: the effects of basal activity and negative feedback. We consider enzy-matic cycles (A) without and (B) with basal kinase activity [i.e., catalytic activity of the kinase in the absence of ligand (L)] and in each case, without and withnegative feedback (no FB and FB, respectively; indicated by dotted lines). p-ase, phosphatase; Sub, substrate. Basal activity eradicates the switch-like response,but the output dynamic range is mostly restored by negative feedback. (C–F) For each value of the signal (total level of L) and five different fixed levels oftotal substrate (color-coded), we show the average level of the output (solid lines) together with the 0.1 and 0.9 percentiles of the output distribution (outputis the log of the level of pSub). Rate parameters in C and E, and in D and F, are identical except for the presence of basal kinase activity. (G–J) The same as inC–F but with stochasticity in the level of total substrate (the log10 of total substrate is approximately normally distributed as shown, with a mean of 3.9 andconstant over time); output distributions are depicted by the probabilities attaching to each bin. Results are based on stochastic simulation of the reactionnetworks (SI Appendix, Table S1 shows networks and parameters).

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its substrate and a phosphatase dephosphorylates it. This phos-phorylation–dephosphorylation cycle is a widespread motif inbiochemical signaling networks, including MAPK cascades. Be-cause cellular signaling networks are complex and partiallycharacterized, this cycle also provides us with a parsimonioustheoretical model for generating hypotheses, which we then in-vestigate experimentally.

Loss of Negative Feedback Can Eradicate Output Dynamic Range inthe Phosphorylation–Dephosphorylation Cycle with Basal Activity.The standard model of the phosphorylation–dephosphorylationcycle has no basal activity or negative feedback. It is well-knownthat this model generates a sharp, switch-like (or ultrasensitive)response when the modifying enzymes are well-saturated withsubstrate (33). We performed simulations of the stochastic,steady state dynamics of the cycle in this regime, explicitlyallowing for the binding of an activating ligand to the PK. Thesignal value, S, is the total number of ligand molecules in thesystem. Initially, only the liganded kinase can bind the substrate(Fig. 1A). Fig. 1C depicts the signal dependence of the mean andvariability of the phosphorylated substrate (pSub) level for dif-ferent total numbers of substrate proteins in the system. Theresponse is sharp, as expected, with high response uncertaintynear to the threshold (34). The effect of negative feedback onthe properties of the standard cycle does not seem to have beenstudied. To introduce negative feedback, we assume that bindingof pSub to either form of the kinase causes its reversible in-activation. We observe a more graded response as a function ofthe signal in the presence of negative feedback, as has beenreported for some other biochemical networks (35–37), while thethreshold value is unchanged (Fig. 1D).Next, we introduce basal activity into the system (Fig. 1B, red

arrow). To do so, we hold other rate parameters constant andallow substrate phosphorylation by the unliganded kinase toproceed at a reduced rate compared with phosphorylation bythe liganded kinase. With basal activity, the response becomesa shallow, linear function of the signal (Fig. 1E). Due to itsswitch-like response, the signaling system is particularly vulner-able to basal activity because the switch can be activated in theabsence of the appropriate signal—if the basal activity is suffi-ciently high compared with the phosphatase activity (SI Appen-dix), there is no signal value for which low output is a steady stateof the switch mechanism. As expected, the effect emerges ina sudden transition as the basal catalytic activity is increased (SIAppendix, Fig. S1) and, depending on the phosphatase activity,can occur despite the basal catalytic activity being small relativeto that of the liganded kinase (10% in Fig. 1). The range of thecycle’s output collapses to near zero, and with some variation(uncertainty) present in the total amount of substrate in thesystem, different signal values become essentially impossible totell apart (Fig. 1I). Variability in the output is now mostly be-cause of the sensitivity of the average response to the value oftotal substrate. This is a surprising effect: Loss of negativefeedback results in loss of all information transfer. The zeroinformation transfer arises because the output distributions areessentially the same for different values of the signal in the nofeedback cycle (Fig. 1I). Importantly, negative feedback providesa solution to the signaling problem posed here by basal activity,restoring the (dynamic) range of the output of the leaky system(Fig. 1 F and J). In the presence of variation in the total substratelevel, negative feedback substantially increases the mutual in-formation between the phosphorylated substrate level and thesignal, I(pSub; S), from 0.00 in Fig. 1I to 0.84 bits (for the log-normal signal distribution yielding the highest mutual information)in Fig. 1J.To understand further how different sources of noise in the

output affect information transfer, we make use of recently in-troduced methods for decomposing variation in the output of

biochemical systems (28). The fidelity of signaling (29) is basedon the output-signal relationships that play a prominent role inmost cell signaling research, is computed by decomposing theoutput variance, and can be seen as a generalized signal-to-noiseratio. The components of the output variance used correspondto widespread measures for understanding biochemical systems(dynamic range, intrinsic variation, etc.). The fidelity is related tomutual information, because it sets a lower bound on mutualinformation equal to log(1 + fidelity)1/2 when evaluated underthe appropriate signal distribution (28).Because we are interested in the effects of heterogeneity in the

componentry of signaling networks, we focus here on the kinase–phosphatase systems where the total substrate level has a distri-bution (Fig. 1 G–J), modeling cell-to-cell variability in the time-invariant total substrate level. We can explain almost all of thelarge increase in mutual information caused by negative feed-back using the fidelity measure. It is given here by the dimen-sionless expression (28, 29),

fidelity ¼ VarfE½ZjS�gEnðE½ZjS; tSub�−E½ZjS�Þ2

oþ EfVar½ZjS; tSub�g

; [1]

with Z as the output (log of the level of pSub), S as the signal,tSub as the total level of substrate, and E denoting the expecta-tion or average. The numerator is a precisely defined measure ofthe output dynamic range, while the denominator gives the noisearising from two sources: first, the global sensitivity of the averageresponse to total substrate levels and second, the stochasticityof the biochemical reactions comprising the transduction mech-anism. Because there is no upper limit mathematically onhow large fidelity can be, it is sometimes convenient to usethe measure 1 − ( fidelity + 1)−1, the fraction of the variancein Z explained by variation in the signal. This measure is anincreasing function of fidelity that takes values between zeroand one.Negative feedback causes the fidelity of the leaky phosphor-

ylation–dephosphorylation cycle to increase from 0.01 to 2.0 asa result of the large increase in the output dynamic range in thenumerator of Eq. 1 (Fig. 1 I and J). This corresponds to an in-crease in the implied lower bound on the mutual information, I(pSub; S) = I(pSub; logS), from 0.00 to 0.80 bits (using that logSis normally distributed and E[ZjS] is approximately linear here).The sensitivity to total substrate decreases slightly, whereas thecontribution of biochemical stochasticity increases (but is anorder of magnitude smaller than the other terms for the systemwith feedback). We conclude that it is the effect of negativefeedback on the output dynamic range which explains the sub-stantial increase in mutual information (SI Appendix, Table S4).It is worth noting that the response function for the leaky cyclewith negative feedback (Fig. 1J) is not obtained simply bytranslating to the left the one for the same cycle without basalactivity (Fig. 1H).What is unusual about the effect of negative feedback in this

leaky signaling system? Previous work has found that negativefeedback tends to decrease both the output dynamic range andthe variance caused by biochemical stochasticity, because itsuppresses means as well as variances (29, 38). Informationtransfer can increase or decrease, depending on which of the twoeffects dominates (4). Here, as a result of the basal activity, theoutput dynamic range dramatically increases with negativefeedback (and variance caused by biochemical stochasticity alsoincreases) (Fig. 1E cf. Fig. 1F), the opposite effects to thosereported previously.

Negative Feedback Protects Information Transfer from the CombinedEffects of Basal Activity and Heterogeneity in Substrate Levels. Fig. 2explores the effects in the leaky cycle of varying both the signal

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distribution and the extent of variation in the total substratelevel. We use a lognormal distribution for total substrate tocapture the nonnegativity and skewness of the distribution ofnumbers of proteins and consider, for the lognormal signals,a broad range of locations and scale (or range) parameters. Forthe vast majority of cases considered, negative feedback increa-ses the mutual information between the signal and the phos-phorylated substrate level (Fig. 2 A–D, Upper). An exception isthe case with no variation in total substrate level, where themutual information, I( pSub; S), for the feedback and no feed-back cycles is broadly comparable in magnitude. The resultshighlight the benefit of careful quantification of informationtransfer using mutual information—it is the combined effects ofbasal activity and heterogeneity that make information transfervulnerable to loss of negative feedback. In the complete absenceof variability in total substrate, the cycle without feedback cangive surprisingly good information transfer if the level of sub-strate is not too high (Figs. 1E and 2A). Although its outputdynamic range is small, variation around the mean response isalso small. Nevertheless, in the absence of feedback, the infor-mation in the phosphorylated substrate level is nonrobust andfalls rapidly to low levels when variation in total substrate is in-troduced. Variability in the output caused by the sensitivity ofthe average response to the total substrate level then overwhelmsthe small output dynamic range. Negative feedback, however, isable to substantially increase the information transfer, I(pSub; S),in these circumstances, for the majority of the signal distributionsconsidered (Fig. 2 B–D).The information loss caused by variation in total substrate can

be evaluated (SI Appendix) as the difference between I(pSub,tSub; S)—the joint information when the level of total substrateis also monitored in interpreting the output—and I( pSub; S).This difference is equal to the amount that the information in

phosphorylated substrate would increase, on average, if totalsubstrate were fixed (SI Appendix, Eq. S5). As the variation intotal substrate increases, the joint information I(pSub, tSub; S)remains fairly constant for the cycle with feedback (Fig. 2 B–D,Lower). However, the information loss due to total substratevariation tends to increase, as expected, and the information inthe phosphorylated substrate level, I( pSub; S), is graduallyeroded (Fig. 2 A–D, Upper compared with Fig. 2 B–D, Lower).Analogous results for the cycle without basal activity are given inSI Appendix. The entropies of the signal distributions used in Fig.2 vary between 2.3 and 10.4 bits approximately (as a result ofdiscretizing the distribution of S so that its value is an integernumber of molecules). We thus observe substantial loss of in-formation relative to the maximum possible information transferfor these distributions (the signal entropy), despite the kinase,phosphatase, and substrate not being present in low numbersof molecules.Finally, in the context of Fig. 1, we explored the relationship

between the strength of the negative feedback, the magnitudeof the basal kinase activity, and information transfer, I(pSub; S), bythe leaky cycle. We parameterize negative feedback strength bythe binding rate of pSub to the kinase. Information transfer ishump-shaped as a function of feedback strength, peaking at anoptimal strength that is quite insensitive to the magnitude of thebasal activity (SI Appendix, Fig. S3). Intuitively, informationtransfer can decrease for higher feedback strengths because ofthe detrimental effects of biochemical stochasticity encounteredwith lower numbers of molecules. The optimal mutual infor-mation decreases as the basal activity is increased, as expected.We then varied the negative feedback mechanism, instead havingthe phosphorylated substrate activate gene expression of phos-phatase (or an allosteric activator of the phosphatase). Negativefeedback can again restore information transfer to 1 bit or

A B C D

Fig. 2. Information transfer by leaky, heterogeneous phosphorylation–dephosphorylation cycles: the effect of negative feedback. We consider enzymaticcycles with basal kinase activity and heterogeneity in total substrate level, with and without negative feedback (FB and no FB, respectively). Negativefeedback protects the information transmitted in the level of pSub. The variance of total substrate increases from A to D as shown; the distribution of totalsubstrate is lognormal, and mean[log10tSub] is 3.9. A–D plot mutual information for the network with negative FB against mutual information for thenetwork without feedback (no FB), with lines having a gradient of one shown in black: Upper shows I(pSub; S), and Lower shows the joint information I(pSub,tSub; S) (SI Appendix, Eq. S5). The changing scales of the axes are highlighted by red boxes. Each dot corresponds to a particular discretized lognormal signaldistribution (with color indicating the entropy of the signal, S). The means of the distributions of log10(S) vary uniformly between 1.0 and 2.8, and the SDs varybetween 0.025 and 0.25. Results are based on stochastic simulation of the reaction networks (SI Appendix, Table S1 shows networks and parameters).

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more. Interestingly, the response functions with negative feed-back are switch-like, and the feedback, therefore, causes the op-posite effect to response linearization (because it converts a linearresponse to a sigmoidal one) (SI Appendix, Fig. S5). Expressingadditional phosphatase molecules when signal detection is needed(in the presence of ligand) is an attractively simple, although en-ergetically costly, solution to the basal activity problem.A main conclusion, then, of our theoretical modeling (extended

below) is the prediction that information transfer by leaky heter-ogenous PK signaling systems is highly vulnerable to mutationscausing loss of negative feedback. We now turn to experimentaltesting of this prediction in the context of Raf/MEK/ERK sig-naling in HeLa cells, where there is cell-to-cell variation in theexpression level of ERK, the final substrate in the cascade.

Information Transfer by ERK Signaling Systems: The Interplay of BasalActivity, Heterogeneity, and Negative Feedback. Raf kinases phos-phorylate and activate MEK which, in turn, phosphorylates ERK1

and ERK2 on Thr and Tyr residues of a TEY activation loop (12,13). ERK (used here to denote ERK1 and ERK2) is mostly cy-toplasmic in resting cells, but activation causes nuclear accumu-lation of dual phosphorylated ERK (ppERK) (39), enablingphosphorylation of nuclear substrates, including transcriptionfactors that control cell fate. ERK-mediated negative feedback isknown to influence response kinetics rapidly by inhibitory phos-phorylation of Raf or SOS (30), and also more slowly by in-creasing expression of MAPK phosphatases (31). Furthermore,recent work emphasizes that the ERK cascade exhibits theproperties of a negative feedback amplifier, with ERK-medi-ated negative feedback limiting the responsiveness of tumor cellsto MEK inhibitors (37).Here, we use automated cell imaging to monitor both the level

of ppERK in the nucleus and the total level of ERK expressionin HeLa cells (SI Appendix). We stimulate the cells (after 24 h inlow-serum conditions) with either EGF (to activate ErbB1receptors) or phorbol 12,13 dibutyrate (PDBu; to activate PKC),

A B C

D E

Fig. 3. ERK-mediated negative feedback substantially increases information transfer to the nucleus. (A) The Raf-MEK-ERK signaling network is shown ac-tivated by both RTK and PKC, with negative feedback from activated ERK. (B and C) Time courses of mutual information for stimulation by a constant level of(B) PDBu or (C) EGF. Comparison of the signaling systems with feedback (FB; solid lines) and without feedback (no FB; dashed lines) from ppERK. Mutualinformation between nuclear levels of ppERK and the signal, I(nppERKt; S) are shown in green, and joint mutual information (assuming that the total ERKlevel is also taken into account), I(nppERKt, TotERKt; S), is shown in black. Results based on uniform signal distributions with signal concentrations as in D and E(yielding signal entropies of 2.8 and 3.0 bits, respectively). Time is time from stimulus. (D) For the responses to PDBu at an early and a late time, the dis-tributions of the output (nppERKt) corresponding to each signal concentration, with the probability in each output bin color-coded. Means (dark blue lines)and SDs (light blue lines; right axes) of output are also shown. AFU; arbitrary fluorescence units. Data are the same as data used in B. (E) The same as in D butfor the EGF stimulus and using the data from C.

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studying populations of cells at a range of times after initialstimulation. We use siRNA to knockdown endogenous ERK andrecombinant adenovirus to add back previously characterizedimaging reporters (20, 40) consisting of either a WT ERK2-GFPfusion or a catalytically inactive mutant (K52R ERK2-GFP).Expression of the inactive mutant allows us to examine thenetwork in the absence of ERK-mediated feedback. Our aim isto study the transfer of information by the signaling system to thenucleus, which is reflected in the nuclear levels of ppERK thatoccur in response to the different stimulus concentrations.

ERK-Mediated Negative Feedback Substantially Increases InformationTransfer to the Nucleus. We measure information transfer to thenucleus using I(nppERKt; S), the mutual information betweennuclear levels of ppERK at time t and the concentration of signalmolecules (either PDBu or EGF). Comparing time courses ofthis mutual information for cells expressing WT and catalyticallyinactive ERK2-GFP (Fig. 3 B and C, solid and dashed greenlines), we observe that negative feedback is needed to maintaineffective signaling to the nucleus. We assume uniform dis-tributions of the signal throughout Fig. 3 across the concen-trations shown, giving a signal entropy of 2.8 bits for PDBu and3.0 bits for EGF (we generalize our conclusions to a broad rangeof signal distributions below). The results from a duplicate ex-periment are shown in SI Appendix, Fig. S6.For the system with feedback from ERK, measured values of

I(nppERKt; S) range for PDBu stimulation from 0.7 to 1.1 bits(Fig. 3B) and for EGF stimulation from 0.4 to 0.7 bits (Fig. 3C).Information transfer peaks more slowly for PDBu than EGF.The majority of the information transferred to the nucleus by thissystem is lost by the no feedback system (>60% for all times inFig. 3 B and C, except 120 min for EGF). Recall that the jointmutual information, I(nppERKt, TotERKt; S), measures the in-formation transfer when the cell-wide level of ERK expression(TotERKt) in each cell is taken into account. Much of this jointinformation transferred by the feedback system is again lost bythe no feedback system under stimulation with PDBu (except at5 min). The difference between the black and green lines in Fig.3 B and C quantifies the information loss when the cell monitorsnuclear ppERK levels alone. The minimum, average, and max-imum such losses measured for the system with feedback were0.14, 0.27, and 0.41 bits for PDBu and 0.10, 0.17, and 0.35 bits forEGF. Notice that the maximum mutual information that wouldbe obtained on average if total ERK expression was fixed (SIAppendix, Eq. S5) is no higher than 1.3 bits for PDBu and 1.1 bitsfor EGF.To understand the effect of the ERK-mediated feedback, we

examine the dependence on signal concentration of the distri-bution of nuclear ppERK levels across cells, focusing on an earlyand a late time point (Fig. 3 D and E). Here, the feedbackconverts an approximately linear mean response function, withsmall slope, to a sigmoidal one with substantially reduced basallevel (reduced nuclear ppERK at zero ligand concentration).The no feedback system has diffuse and similar output dis-tributions at all signal levels, including the zero one. By contrast,the output distributions for the system with feedback overlap toa lesser extent. Evaluating the contributions of the three terms inEq. 1 to our fidelity measure provides additional insight. ForPDBu at 30 min, ERK-mediated feedback increases fidelity from0.16 to 2.7 [I(nppERKt; S) increases from 0.11 to 1.0 bits], and forEGF at 5 min, the increase is from 0.10 to 1.3 [I(nppERKt; S)increases from 0.08 to 0.70 bits]. The feedback increases theoutput dynamic range and decreases the sensitivity of the meanresponse to the total ERK level (Fig. 4), and usually also resultsin a small decrease in other sources of output variation (using thedata in Fig. 3 D and E and a uniform signal distribution) (SIAppendix, Table S4). Overall, the fidelity therefore increases,driven by the first two effects. The increased sensitivity of the

mean response to the total ERK level results in the diffuseoutput distributions seen for the no feedback system in Fig. 3 Dand E. Under chronic rather than acute stimulation as here, ref.30 also finds that negative feedback from ERK (by phosphory-lation of Raf-1) confers increased robustness of the mean re-sponse to total ERK level.We propose that, in our cell system, negative feedback me-

diated by ERK is needed to preserve information transfer to thenucleus as a result of the combined effect of heterogeneity inERK expression and the level of basal activity in the ERKpathway. The importance of negative feedback from ERK iswell-established, and has already been discussed. Althoughpositive feedback in other MAPK pathways and other cell types,such as oocytes, is known (41), there seem to be no previousreports of positive feedback from ERK in HeLa cells. Basalactivity of ppERK has been shown previously (19, 20), and isclearly manifested on breaking feedback from ERK here (out-puts for zero stimulus are shown in Fig. 3 D and E) and as shownin ref. 9.Our interpretation of the experimental results is strongly

supported by the effects of breaking negative feedback that arepredicted by our theoretical work. In the phosphorylation–de-phosphorylation cycle with basal activity and heterogeneoussubstrate levels, we find near-zero information transfer and near-zero output dynamic range in the absence of negative feedback(Fig. 1 I and J, and Fig. 2 B–D, Upper). Both information transferand output dynamic range can be restored by negative feedback.We also performed stochastic simulation of a MAPK cascademodel in which Raf is activated by PKC, and the latter directly

A B

C D

Fig. 4. Variation in network componentry: the effects of the expressionlevel of the ERK protein in the systems without (no FB) and with feedback(FB). (A and B) For each concentration of PDBu, we plot the average output,E[nppERKtjS, TotERKt], at t = 30 min for six values of total ERK correspondingto the

�17,27, . . . ,

67

�quantiles of the observed distribution of total ERK [mea-

sured in log arbitrary fluorescence units (AFU)]. For the same values of totalERK, we show the band given by ±1 SD, V[nppERKtjS, TotERKt]

1/2. Alsoplotted is the population average output, E[nppERKtjS], which is shown asa black line. (C and D) The same as in A and B but for EGF stimulus and theresponses at 5 min. Results are computed using the same data as in Fig. 3 Dand E and smoothing spline regression.

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binds the signaling ligand (thus mimicking our system withPDBu stimulation). Basal activity is introduced through a con-stant level of a second, weakly activating ligand. Otherwise, themodel is taken from ref. 37. Again, for the leaky heterogeneoussystem, negative feedback (from ERK to Raf) converts an ap-proximately linear mean response function with small slope toa sigmoidal one with reduced basal level, substantially increasingthe output dynamic range (SI Appendix, Fig. S7). These meanresponse functions are also less sensitive to variation in the totallevel of ERK than the response functions of the feedback intactsystem (SI Appendix, Table S4). The information transfer fromthe stimulus to activated ERK is, therefore, substantially in-creased by the negative feedback from 0.08 to 0.91 bits (undera uniform signal distribution). The inclusion of additional neg-ative feedback by expression of phosphatase molecules (actingon phosphorylated ERK) increases the output dynamic rangestill further (SI Appendix, Fig. S8).For the experiments described above, we knocked down en-

dogenous ERK and infected cells with adenovirus-expressingERK2-GFP at 10 pfu/nL. This titer was used to obtain a broadrange of ERK expression levels, but we also performed experi-ments with a lower adenoviral titer equal to 1 pfu/nL. With thelower titer, ERK-mediated feedback again improves informationtransfer to the nucleus by ppERK (PDBu stimulus) (SI Appendix,Fig. S9) and, for the uniform signal distribution, >44% of theinformation transferred is lost by preventing feedback. Breakingthe feedback again reduces the output dynamic range, increasesthe sensitivity to total ERK variation, and reduces the fidelity (SIAppendix, Table S4). We also performed experiments with cellsexpressing endogenous ERK (i.e., without ERK knockdownor adenovirus treatment), and information transfer was mostlyintermediate between the higher and lower titer systems. In-formation losses caused by variation in total ERK by all threesystems with feedback intact were broadly similar. To assess theimpact of the signal distribution (SI Appendix, Fig. S10), weevaluated the information transfer, I(nppERKt; S), for all signaldistributions having the same support as the uniform one alreadyused and with probabilities either zero or a multiple of 1/7 forPDBu (1/8 for EGF). We find, using either adenoviral titer, that ourconclusion holds for this wide range of signal distributions: ERK-mediated feedback enhances information transfer to the nucleus inthe presence of basal activity and heterogeneity in ERK expression.

DiscussionCells must sense concentrations in their environment and transferthe information contained reliably to make appropriate deci-sions (29, 42). The effects of different kinds of noise on the signaltransduction networks that perform this task are still poorlyunderstood. A recent study of TNF signaling by single cells (4)concluded that noise can substantially restrict information transferabout stimulus intensity, particularly when just one or two bio-chemical outputs are examined, but the principle sources of thisnoise are not known. In general, different types of noise arelikely to play a role in reducing information transfer: basal net-work activity, cell-to-cell variation (or heterogeneity) in networkcomponentry, promiscuity of signaling intermediaries (5), andfluctuations from the stochastic nature of the transduction mecha-nism. Here we focus, in the context of protein kinase (PK) sig-naling, on the effects of the first two of these types of noise. Wequantify the effect of variation in network componentry bycomputing the increase in mutual information under the as-sumption that the levels of the variable components (e.g., levelsof total substrate) are monitored in addition to the outputs of thesignaling network. Our estimation algorithm for mutual in-formation avoids the need to discretize or bin the output.Along with refs. 4 and 26, we provide measurements of mutual

information for cell signaling experiments. We find, in commonwith refs. 4 and 26, values of ∼1 bit or less for the single cell

under conditions of constant stimulation using a single stimulusand output (the different studies, however, use different stimulusentropies). When the mutual information measured is 1 bit, anoften-stated conclusion is that the cell can therefore distinguish(implicitly, without error) two states of the biological environ-ment or signal but not more than two states, suggesting theprevalence of binary decisions by cells. We believe that thisconclusion can be misleading. It seems to stem from the Shannontheory of communications engineering (43), particularly theasymptotic equipartition property, which considers infinitely(asymptotically) long sequences of inputs and outputs arisingfrom the repeated use of the signaling mechanism. We prefera different interpretation of the mutual information between thebiological environment and the biochemical network output(perhaps at steady state or some fixed time, t): Mutual infor-mation quantifies the extent to which the Bayesian posteriorprediction of the environment based on the network outputimproves on the (prior) prediction using the distribution ofthe signal in the environment alone. This interpretation recog-nizes that cellular inference about the environment is usuallyimperfect and makes clear that an output with mutual infor-mation of 1 bit can provide information about the state of non-binary environments. We provide additional discussion and math-ematical derivations in SI Appendix.Basal network activity describes the propensity for activation

of the network output in the absence of the signal of interest. Basalactivity is widespread in signaling systems under physiologicalconditions (with observable phenotypic consequences) and isoften raised in disease. Its presence has been widely reported incell signaling, including the fields of MAPK and GPCR signal-ing. Basal activity comes as no surprise given the modernunderstanding of the conformational heterogeneity of proteins(10, 44): Receptors and enzymes constantly transition among anensemble of states, some of which are catalytically active (even inthe absence of activating ligands and allosteric modulators).Due to their switch-like, nearly off–on responses, PK signaling

systems without negative feedback are particularly vulnerable tobasal activity, because the switch can be activated in the absenceof the appropriate signal. With even small amounts of variationin protein substrate levels, different signal values are difficult totell apart, and information transfer falls dramatically to nearzero. Using theoretical models of leaky heterogeneous PK sig-naling systems, we predict that negative feedback substantiallyincreases information transfer by both preventing the near-flataverage response curve and reducing sensitivity to variation inexpression levels of the substrate protein. This prediction isverified in an experimental study of ERK signaling by single cellswith heterogeneous ERK expression.A conventional view of negative feedback is that it reduces

both output dynamic range and variation from other sources, sothat information transfer can increase or decrease dependingon which effect dominates (4, 29). By contrast, negative feedbackin our leaky systems substantially increases output dynamicrange, reduces variation from other sources, and therefore sub-stantially increases information transfer. Negative feedback byexpression of additional phosphatase molecules, such as MAPKphosphatases, when signal detection is needed (in the presenceof ligand) is an attractively simple negative feedback solution tothe basal activity problem. However, the mechanism and speedof negative feedback turn out to not be crucial for the protectionof information transfer at steady state.The Ras/Raf/MEK/ERK pathway is often dysregulated in

human cancer, frequently through gain-of-function mutationsin the RAS and RAF genes (45). The B-RAF mutation occursin the majority of melanomas and results in increased activationof ERK, with the V600EB-RAF single substitution mutationoccurring commonly (46). The V600EB-Raf protein has raised ki-nase activity, activating the pathway independently of Ras, and is

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insensitive to negative feedback from ERK (47). Mutated B-Rafproteins are transforming in NIH 3T3 cells and disrupt the balancebetween signal-independent (basal) kinase activity and negativefeedback. Our work shows in a quantitative, experimental frame-work how sufficient basal activity combines with loss of negativefeedback to cause elevated levels of activated ERK that are largelyinsensitive to the levels of proliferative stimuli (as observed intransformed cells). Basal activity makes the effectiveness of PKsignaling particularly vulnerable to mutations causing the loss ofnegative feedback. Such mutations can result in almost completeabrogation of information transfer. Our findings should proveuseful in comparing transformed with untransformed cells anddesigning drug interventions.Negative feedback protects information transfer in leaky

heterogeneous PK signaling networks: In the presence of basalactivity, it can convert a near-flat average response curve to asigmoidal one, with output that has a much higher dynamic range.This effect combines with improved robustness to cell–cell varia-tion in substrate expression levels to maintain the effectivenessof signaling. Our results reveal an important role for negative

feedback mechanisms in protecting the information transferfunction of saturable, heterogeneous cell signaling systems frombasal activity. They are an important step toward deciphering thedesign of biochemical signaling networks, their role in cellulardecision-making, and their malfunction in disease.

MethodsTo estimate mutual information for a continuous, possibly multivariateoutput, Z, we write I(Z; S) = −Es[h(ZjS = s)] + h(Z) and use nearest neighbormethods to estimate each term (SI Appendix). For the first term, we estimatethe conditional entropy, h(ZjS = s), for each signal value s and then take theaverage according to the signal distribution. For the second term, we use aniterative scheme to sample from the joint distribution corresponding to thatsignal distribution. Cell culture, transfection, and imaging are described inResults and SI Appendix.

ACKNOWLEDGMENTS. The authors thank Ramon Grima, Peter Swain, andPhilipp Thomas for insightful feedback. M.V. acknowledges a MedicalResearch Council Fellowship in Biomedical Informatics, and C.A.M. acknowl-edges Biotechnology and Biological Sciences Research Council GrantJO14699/1 and Medical Research Council Grant G0901763.

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