Probing the limits to microRNA-mediated control of gene expression

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by  Araks Martirosyan

ProbingtheLimitstoMicroRNA-MediatedControlofGeneExpression
Araks Martirosyan,1,2 Matteo Figliuzzi,3,4,5 Enzo Marinari,1,6,∗ and Andrea De Martino1,2,7,∗
1Dipartimento di Fisica, Sapienza Universita` di Roma, Rome, Italy
2SoftandLivingMatterLab,InstituteofNanotechnology(CNR-NANOTEC),ConsiglioNazionaledelleRicerche,Rome,Italy
3Sorbonne Universite´s, UPMC, Institut de Calcul et de la Simulation, Paris, France
4SorbonneUniversite´s, UPMC,UMR7238, ComputationalandQuantitativeBiology, Paris, France
5CNRS, UMR 7238, Computational and Quantitative Biology, Paris, France
6INFN, Sezione di Roma 1, Rome, Italy
7Center for Life Nano Science@Sapienza, Istituto Italiano di Tecnologia, Rome, Italy
Abstract – According to the ‘ceRNA hypothesis’, microRNAs (miRNAs) may act as mediators of an
effectivepositiveinteractionbetweenlongcodingornon-codingRNAmolecules,carryingsignificantpotential
implications for a variety of biological processes. Here, inspired by recent work providing a quantitative
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descriptionofsmallregulatoryelementsasinformation-conveyingchannels,wecharacterizetheeffectiveness
1
of miRNA-mediated regulation in terms of the optimal information flow achievable between modulator
0
(transcription factors) and target nodes (long RNAs). Our findings show that, while a sufficiently large
2
degree of target derepression is needed to activate miRNA-mediated transmission, (a) in case of differential

n mechanisms of complex processing and/or transcriptional capabilities, regulation by a post-transcriptional
a miRNA-channel can outperform that achieved through direct transcriptional control; moreover, (b) in the
J
presence of large populations of weakly interacting miRNA molecules the extra noise coming from titration

6 disappears, allowing the miRNA-channel to process information as effectively as the direct channel. These
2 observationsestablishthelimitsofmiRNA-mediatedpost-transcriptionalcross-talkandsuggestthat, besides
providingadegreeofnoisebuffering,thistypeofcontrolmaybeeffectivelyemployedincellsbothasafailsafe

] mechanismandasapreferentialfinetunerofgeneexpression,pointingtothespecificsituationsinwhicheach
N
ofthesefunctionalitiesismaximized.
M
Author Summary – The discovery of RNA interference has revolutionized the decades’ old view of
.
o RNAs as mere intermediaries between DNA and proteins in the gene expression workflow. MicroRNAs (or
i miRNAs),inparticular,havebeenshowntobeabletobothstabilizetheproteinoutputbybufferingtranscrip-
b
tional noise and to create an effective positive interaction between the levels of their target RNAs through a
-
q simplecompetitionmechanismknownas‘ceRNAeffect’. WithmiRNAscommonlytargetingmultiplespecies
[ ofRNAs,thepotentialimplicationisthatRNAscouldregulateeachotherthroughextendedmiRNA-mediated

interaction networks. Such cross-talk is certainly active in many specific cases (like cell differentiation),
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but it’s unclear whether the degree of regulation of gene expression achievable through post-transcriptional
v
miRNA-mediated coupling can effectively overcome the one obtained through other mechanisms, e.g. by
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directtranscriptionalcontrolviaDNA-bindingfactors. Thisworkquantifiesthemaximalpost-transcriptional
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regulatorypowerachievablebymiRNA-mediatedcross-talk,characterizingthecircumstancesinwhichindirect
1
control outperforms direct one. The emerging scenario suggests that, in addition to its widely recognized
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noise-bufferingrole,miRNA-mediatedcontrolmayindeedactasamasterregulatorofgeneexpression.
0
.
1
0 INTRODUCTION ers.
6
Asthemajordirectregulatorsofgeneexpression,transcrip-
1
: The problem of tuning protein expression levels is cen- tionfactors(TFs)aremostimmediatelyidentifiedasthekey
v tral for eukaryotic cell functionality. A variety of molecular potentialmodulatorsofproteinlevels[4]. Inasomewhatsim-
i
X mechanismsareimplementedtoguarantee,ononehand,that plified picture, one may imagine that a change in amount of
protein copy numbers stay within a range that is optimal in a TF can induce a change in the expression level of the cor-
r
a the given conditions and, on the other, that shifts in expres- respondinggene,andthattheabilitytoregulatethelatter(the
sionlevelscanbeachievedefficientlywhenevernecessary[1– outputnode)viatheformer(theinputnode)canbeassessed
3](whereby‘efficiency’hereencompassesbothadynamical byhowstronglythetwolevelscorrelate. Theeffectivenessof
characterization,intermsofthetimesrequiredtoshift,anda a regulatory element is however limited by the stochasticity
static one, in terms of moving as precisely as possible from ofintracellularprocesses,fromtheTF-DNAbindingdynam-
onefunctionalrangetoanother). Quantifyingandcomparing icstotranslation[5]. Aconvenientframeworktoanalyzehow
theireffectivenessindifferentconditionsisanimportantstep noiseconstrainsregulationisprovidedbyinformationtheory
to both deepen our fundamental understanding of regulatory [6, 7]. In particular, the simplest situation in which a sin-
circuitsandtogetcase-by-casefunctionalinsightaboutwhy gle TF modulates the expression of a single protein can be
aspecificbiochemicalnetworkhasbeenselectedovertheoth- characterizedanalyticallyundertheassumptionthatthenoise
affecting the input-output channel is sufficiently small. The
mutual information between modulator and target –a conve-
nient quantity through which regulatory effectiveness can be
∗Authorscontributedequally characterized– depends on the distribution of modulator lev-
2
els and can be maximized over it. Remarkably, in at least canthereforegenericallybeobtainedbysolvinganoptimiza-
onecasethismaximumhasbeenfoundtobealmostsaturated tionproblemoverthedistributionofinputs. Thistypeofap-
by the actual information flow measured in a living system proachprovidesanupperboundtotheeffectivenessofareg-
(for more details see [8, 9]). In other terms, for sufficiently ulatory mechanism as well as indications concerning which
smallnoiselevelsinthechannelthatlinksTFstotheirfunc- parameters,noisesourcesand/orinteractionsmosthamperits
tionalproducts,onemayquantifytheoptimalregulatoryper- performance.
formanceachievableintermsofthemaximumnumberofbits It would be especially important to understand in which
ofmutualinformationthatcanbeexchangedbetweenmodu- conditions the degree of control of the output variable (i.e.
latorandtarget. the ceRNA/protein level) that can be accomplished through
Several control mechanisms however act at the post- post-transcriptional miRNA-mediated cross-talk may exceed
transcriptional level [10–12]. Among these, regulation by that obtainable by different regulatory mechanisms. In this
small regulatory RNAs like eukaryotic microRNAs (miR- work we characterize the maximal regulatory power achiev-
NAs) has attracted considerable attention over the past few able by miRNA-mediated control and compare it with that
years[13–15]. Inshort,miRNAsaresmallnon-codingRNA of a direct, TF-based transcriptional unit [45]. In principle,
moleculesencodedbynuclearDNA,thatcaninhibittransla- since fluctuations can be reduced by increasing the number
tionorcatalyzedegradationofmRNAswhenboundtothem ofmolecules,an(almost)arbitraryamountofinformationcan
viaprotein-mediatedbase-pairing. miRNAsappeartobecru- betransmittedthroughabiochemicalnetwork.However,cells
cial in an increasing number of situations ranging from de- havetofacetheburdenofmacromolecularsynthesis[46–48].
velopment to disease [16–18]. Their function however can Optimality is therefore the result of a trade-off between the
differsignificantlyfromcasetocase. Forinstance,theyhave benefitsofreducedfluctuationsandthedrawbacksoftheas-
been well characterized as noise buffering agents in protein sociated metabolic costs. For this reason, we start by fixing
expression[19,20]oraskeysignalingmoleculesinstressre- a maximal rate of transcription (or, alternatively, the maxi-
sponse[21]. Recently,though,investigationsofthemiRNA- mal number of output molecules) so as to have a simple but
mediatedpost-transcriptionalregulatory(PTR)networkhave reasonableframeworktocharacterizeandcomparethecapac-
hintedatapossiblymoresubtleandcomplexrole. Itisindeed ities of the different regulatory channels. Next, we quantify
nowclear[22–27]thatthemiRNA-RNAnetworkdescribing how an input signal is processed by the transcriptional (TF-
thepotentialcouplingsstretchesacrossamajorfractionofthe based)andpost-transcriptional(miRNA-mediated)regulatory
transcriptome, withalargeheterogeneitybothinthenumber elementsbycharacterizingtheresponseintheoutputceRNA’s
of miRNA targets and in the number of miRNA regulators expressionlevels. Insuchasetting,informationflowisham-
foragivenmRNA.Thecompetitioneffectsthatmayemerge pered by intrinsic noise if the target gene is weakly dere-
insuchconditionssuggestthatmiRNAsmayactaschannels pressedbytheactivationofitscompetitor. Otherwise, target
through which perturbations in the levels of one RNA could derepressionappearstohaveastrongimpactonaregulatory
betransmittedtootherRNAspeciessharingthesamemiRNA element’scapacity. Uponvaryingthemagnitudeofderepres-
regulator(s). Such a scenario has been termed the ‘ceRNA sion by tuning the kinetic parameters, we then show that in
effect’, whereby ceRNA stands for ‘competing endogenous certain regimes miRNA-mediated regulation can indeed out-
RNA’ [28]. In view of its considerable regulatory and ther- perform direct control of gene expression. Finally, we argue
apeutic implications, the ceRNA effect has been extensively that the presence of miRNA molecules in large copy num-
analyzedboththeoreticallyandexperimentally[29–44]. bersnotablyreducesthelevelofintrinsicnoiseonweaklytar-
getedtranscripts.Inthiscase,themutualregulationofceRNA
Theapparentubiquityofpotentiallycross-talkingceRNAs
molecules by miRNA-mediated channels may become a pri-
howeverraisesanumberoffundamentalquestionsaboutthe
marymechanismtofinelytunegeneexpression.
effectivenessof“regulationviacompetition”perse.Although
Besides providing a quantitative characterization of the
hundredsoftargetsarepredictedforasinglemiRNA,obser-
maximalregulatorypowerachievablethroughmiRNA-based
vationsshowthatonlyfewofthemaresensitivetochangesin
post-transcriptional control, these results provide important
miRNA expression levels. Most targets are likely to provide
hintsonthecircumstancesinwhichregulationbysmallRNAs
a global buffering mechanism through which miRNA levels
mayfunctionasthemaintunerofgeneexpressionincells.
areoverallstabilized[28,29]. Effectivecompetitionbetween
miRNA targets requires that the ratio of miRNA molecules
to the number of target sites lies in a specific range, so that
RESULTS
the relative abundance of miRNA and RNA species must be
tightly regulated for the ceRNA mechanism to operate [29–
34]. On the other hand, the magnitude of the ceRNA ef- MathematicalmodelofceRNAcompetition
fect is tunable by the miRNA binding and mRNA loss rates
[33,34,42]. Theperformanceofaregulatoryelement,how- We consider (see Fig. 1) a system formed by two ceRNA
ever,doesnotonlydependonkineticparameters,butalsoon species (whose levels are labeled m and m , respectively)
1 2
therangeofvariability(andpossiblyonthedistributionitself) and one miRNA species (with level labeled µ), whose tran-
ofmodulatorlevels(e.g. TFs)[8,9]. Themaximalregulatory scriptions are activated by a single TF each (with levels la-
effectivenessofagivengeneticcircuit–quantifyinghowpre- beled, respectively, f , f andf ). BothceRNAsareinturn
1 2 µ
cisely the output level can be determined by the input level– targetedbythemiRNA.miRNA-ceRNAcomplexes(levelsla-
3
catalytically(i.e. withmiRNArecycling)atratesκ [49]. In
i
addition, ceRNA and miRNA molecules degrade (resp. syn-
thesize) at rates d (resp. b ) and δ (resp. β), respectively.
i i
StepsleadingtotheformationoftheRNA-inducedsilencing
complex (RISC), allowing for the miRNA-ceRNA binding,
areneglectedforsimplicity.TFlevelsaretreatedasexternally
controlledparameters.
Denoting the TF-DNA binding/unbinding rates by k and
in
k , respectively (for simplicity, these parameters are taken
out
to be the same for all involved TFs), the TF binding sites’
fractional occupancies n (0 ≤ n ≤ 1, (cid:96) ∈ {1,2,µ}) obey
(cid:96) (cid:96)
thedynamics
dn
(cid:96) =k (1−n )fh−k n , (1)
dt in (cid:96) (cid:96) out (cid:96)
accordingtowhichtranscriptionalactivationrequirestheco-
FIG.1. SchematicrepresentationofthebasicceRNAnetwork
operative binding of h TF molecules for each RNA species
model. Themodelcomprisesthreetranscriptionfactors(TF , TF
1 2 involved. In general, the occupancies by different TFs equi-
andTF )controllingthesynthesisoftwoceRNAspecies(ceRNA
µ 1 librate on different timescales [50]. However it is often as-
andceRNA )andonemiRNAspecies,respectively. Thefractional
2
sumedthatthetranscriptionalon/offdynamicsismuchfaster
occupanciesofTFbindingsitesaredenotedbyn ,n andn ,re-
1 2 µ
spectively.BothceRNAsaretargetsforthemiRNA,withwhomthey than transcription itself [51, 52]. As a consequence, each n(cid:96)
formcomplexesdenotedrespectivelyasC andC . Theamountof canbefixedatits‘equilibrium’value.
1 2
moleculesforeachspeciesisdenotedbythevariablenexttothecor-
respondingnode.Reactionratesarereportednexttothecorrespond-
ingarrow. n = kinf(cid:96)h . (2)
(cid:96) k fh+k
in (cid:96) out
Everyprocessintheaboveschemecontributestotheover-
beledc withi=1,2)assembleanddisassembleatratesk±, all level of noise. We represent the mass-action kinetics of
i i
respectively, whereas complexes can be degraded both stoi- the system through the set of coupled Langevin processes
chiometrically(i.e. withoutmiRNArecycling)atratesσ and (i=1,2)[41,53]
i
dm
i =−d m +b n −k+µm +k−c +ξ −ξ++ξ− ,
dt i i i i i i i i i i
dc
i =k+µm −(k−+κ +σ )c +ξσ+ξ+−ξ−−ξκ , (3)
dt i i i i i i i i i i
dµ (cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
=−δµ+βn − k+µm + (k−+κ )c +ξ − ξ++ ξ−+ ξκ ,
dt µ i i i i i µ i i i
i i i i i
where the mutually independent random (Poisson) ‘forces’
ξ , ξ±, ξ , ξκ and ξσ denote, respectively, the intrinsic
i i µ i i
noiseinceRNAlevels(duetorandomsynthesisanddegrada-
tionevents),intheassociation/dissociationprocessesofcom- (cid:104)ξ (t)ξ (t(cid:48))(cid:105)=(d m +b n )δ(t−t(cid:48)) , (4)
plexes, in the miRNA level, in the catalytic complex decay i i i i i i
and in the stoichiometric complex decay. Each of the above (cid:10)ξi+(t)ξi+(t(cid:48))(cid:11)=ki+miµδ(t−t(cid:48)) , (5)
noisetermshaszeromean,whilecorrelationsaregivenby (cid:10)ξ−(t)ξ−(t(cid:48))(cid:11)=k−c δ(t−t(cid:48)) , (6)
i i i i
(cid:104)ξ (t)ξ (t(cid:48))(cid:105)=(δµ+βn )δ(t−t(cid:48)) , (7)
µ µ µ
(cid:104)ξκ(t)ξκ(t(cid:48))(cid:105)=κ c δ(t−t(cid:48)) , (8)
i i i i
(cid:104)ξσ(t)ξσ(t(cid:48))(cid:105)=σ c δ(t−t(cid:48)) , (9)
i i i i
whereweintroducedthesteadystatemoleculenumbers
4
ceRNA degradation but, again, the ceRNA level is
weaklysensitivetosmallchangesinµasmostceRNAs
areboundincomplexeswiththemiRNA;
• if µ (cid:39) µ , ceRNA is susceptible to µ: spontaneous
0,i i
andmiRNA-mediateddecaychannelshavecomparable
weights and the ceRNA level is very sentitive to small
changesinµ.
The behaviour of the FFs (see Fig. 2B) emphasizes how
noise patterns change in the different regimes. In the dis-
FIG. 2. Cross-talk scenario in the ceRNA network. Steady
played example, ceRNA and ceRNA become susceptible
statevaluesof(A)theceRNAandmiRNAlevels, and(B)thecor- 1 2
respondingFanofactors. MarkersdenoteGAcomputations,curves forln(f1) (cid:39) 2.5andln(f1) (cid:39) 2.1,respectively. Indeed,one
correspondtoapproximateanalyticalsolutionsobtainedbythelinear observes that the corresponding FFs peak close to these val-
noiseapproximation. Valuesofthekineticparametersarereported ues, in accordance with the observation that stochastic fluc-
inTableI. tuations are enhanced when the rates of substrate supply are
adequately balanced in a stoichiometrically coupled system
[34,54,55]. TheFFforceRNA appearstoapproachonefor
2
largevaluesoff ,asexpectedforthepurePoissonbirth/death
1
b n +k−c process that characterizes the free regime [56]. On the other
mi = id i+k+iµi , (10) hand, for very small but nonzero mean fractional occupancy
i i n (corresponding to small values of f ), m will with high
1 1 1
µ= βnµ+(cid:80)i(ki−+κi)ci , (11) probability only take on the values 0 or 1, as a transcribed
δ+(cid:80) k+m moleculewillquicklyundergodegradationorsequestrationin
i i i
acomplex. Insuchasituation,themeanandvarianceofm
k+µ m 1
c = i i . (12) will be calculated by summing up zeros and ones over time,
i σi+ki−+κi leadingtoaFFequaltoone.
Weshallbeinterestedinthefluctuationsofmolecularlevels
aroundthesteadystatethatariseduetointrinsicnoisesources.
Measuringtheperformanceofregulatoryelements
The stochastic dynamics of the system can be simulated via
the (GA, see ). Fig. 2A shows typical GA results for m ,
1 TheamplitudeofceRNAderepression
m and µ, with the corresponding Fano factors (FFs) shown
2
inFig. 2B,asfunctionsoff . Numericalresultsarematched
1
against analytical estimates obtained by the linear noise ap- We shall focus on quantifying the regulatory capacity of
proximation(see). thepost-transcriptionalandtranscriptionalchannelsthatlink,
Inparticular,inFig.2oneseesthatanincreaseofm1isac- respectively,TF1toceRNA2andTF2toceRNA2,asdepicted
companiedbyaconcomitantincreaseofm andbyadecrease inFig. 3.
2
of the average number of free miRNA molecules. This is an Inwhatfollows, weshalltakethenumberoftranscription
instanceofmiRNA-mediatedceRNAcross-talk.Indeed,upon factorsasthekeyinputvariables(f1forthemiRNA-mediated
up-regulating m1 by injecting the corresponding TF (i.e. by channelandf2 forthedirectchannel),andthenumberoftar-
increasingf1),theleveloffreemiRNAswilldecreaseasmore get RNAs m2 as the key output variable. It is reasonable to
andmoremoleculeswillbeactivelyrepressingceRNA ,caus- think that an effective regulator should have a sensible im-
1
inginturnanup-regulationofceRNA .f willthuspositively pactonthetargetmolecule. Asafirstmeasureofregulatory
2 1
correlate with m . One easily sees that steady-state ceRNA performancewewillthereforeconsidertheamplitudeofvari-
2
levelsdependonµthroughasigmoidalfunction,namely[42] ation(AOV)ofmiviafj,definedas
m = biniF [µ] , F [µ]= µ0,i . (13) ∆ij =mi(fjmax)−mi(fjmin) , (14)
i d i i µ +µ
i 0,i corresponding to the difference between the maximum and
theminimumsteadystatelevelofceRNA obtainedbyvary-
Theconstantµ0,i = kdi+i (1+ σik+i−κi)actsasa‘soft’threshold ingfj fromfjmintofjmax. Clearly,largervialuesof∆ij imply
forthemiRNAlevel,allowingtodistinguishthreesituations: stronger degrees of derepression of ceRNA induced by the
i
correspondingTF.Forsakesofsimplicity,weshalladoptthe
• if µ (cid:28) µ , ceRNA is free or unrepressed: spon-
0,i i following notation to distinguish the AOV induced by tran-
taneous ceRNA degradation dominates over miRNA-
sciptionalandmiRNA-mediatedcontrolchannels:
mediated decay channels, so that, effectively, the
ceRNAlevelisweaklysensitivetosmallchangesinµ;
∆ ≡∆ , (15)
• if µ (cid:29) µ , ceRNA is bound or repressed: miRNA- 22 TF
0,i i
∆ ≡∆ . (16)
mediated ceRNA decay dominates over spontaneous 21 miRNA
5
FIG.3. Schematicrepresentationofthechannelsunderconsideration.(A)miRNA-channel,(B)TF-channel.
Note that the AOV only involves mean levels. As such, it is where
uthneabsylestteoma.ccount for the intrinsic degree of stochasticity of Z =(cid:90) fjmaxdf (cid:32) 1 (cid:20)dm2(fj)(cid:21)2(cid:33)1/2 (20)
j 2πeσ2 (f ) df
fjmin m2 j j
isanormalizingfactor. Forp = p oneobtainsthechannel
Maximalmutualinformation opt
capacity
Inordertoevaluatetheimpactofnoise,asafurtherquanti- Iopt(m2,fj)=log2Z . (21)
fierofregulatorypowerweshallemploythemutualinforma-
S1Figureprovidesaqualitativeyetintuitiverepresentationof
tion (MI), a standard information-theoretic measure of chan-
howtheshapeoftheinput/outputrelationandthesizeoffluc-
nelperformancegivenby(i,j =1,2)[52]
tuations affect information flow. Intuitively, the normalizing
(cid:90) p(m ,f ) factorZ’counts’thenumberofdistinguishableoutputlevels
I(mi,fj)= p(mi,fj)log2 p(m )ip(fj )dmidfj , (17) comparing the local slope of the input/output curve with the
i j
noise strength: if the slope is too low or the fluctuations are
wherep(m ,f )standsforthejointprobabilitydistributionof toohighnoinformationcanbetransmittedthroughthechan-
i j
m andf , whilep(m )andp(f )denoteitsmarginals. The nel. We shall use the following notation to distinguish the
i j i j
MI is a non-negative quantity that vanishes when m and f capacitiesofthedirectandindirectchannels:
i j
are independent. Conversely, when I(m ,f ) = I one can
i j
reliably distinguish (roughly) 2I different values of the out-
I (m ,f )≡I , (22)
putvariableupontuningtheinput(see[9,45,52]forfurther opt 2 2 TF
detailsabouttheusefulnessofthisschemefortheanalysisof Iopt(m2,f1)≡ImiRNA . (23)
transcriptionalregulatoryorbiochemicalmodules).Themax- Weshallfurthermoredenotetheirdifferenceby∆I = I −
TF
imalachievableinformationflowforafixedchannel(i.e. for I .
miRNA
given kinetic parameters) is called the channel capacity and We are henceforth going to adopt the following protocol.
can be computed by maximising the MI over the input vari- Aftergeneratingtheoutputdata(m ,σ )bysimulatingthe
able (in this case, over the distribution of the TF level). In circuit’s dynamics via the GA, the o2ptimm2al input distribution
the simplest case, one can assume that the channel is Gaus- will be constructed according to Eq. (19). We shall then let
sian, meaning that for every given input fj the output m2 is thecircuitprocessaninputsignalsampledfromthatdistribu-
describedbytheconditionalprobabilitydensity tionandcalculatetheMIforboththetranscriptionalandpost-
1 (cid:20) (m −m (f ))2(cid:21) transcriptionalchannels(seetheflowchartinFig. 4). Channel
p(m |f )= exp − 2 2 j , (18) capacities will then be compared, with the goal of clarifying
2 j (cid:112)2πσ2 2σ2 (f )
m2 m2 j the conditions under which the former can be more effective
where both the average m (f ) and variance σ2 (f ) = thanthelatter.
(cid:10)(m −m (f ))2(cid:11) depend 2onjthe input variable mf2. Ijn the
2 2 j j
limit of small variance, the distribution p (f ) of TF levels
opt j ComparingindirectmiRNA-mediatedregulationwithdirect
thatmaximisestheMIbetweenm andf reads[45]
2 j transcriptionalcontrol
1 (cid:32) 1 (cid:20)dm (f )(cid:21)2(cid:33)1/2
p (f )= 2 j , (19) Target prediction algorithms suggest that miRNA binding
opt j Z 2πeσm22(fj) dfj affinitiesvarysignificantlyamongtheirtargets[57].Suchhet-
6
As displayed in Fig. 5, the maximal amount of information
transmittedthroughthepost-transcriptionalchannel(Fig.5A)
increasesmonotonicallywithk+ (forfixedk+)anddisplays
1 2
amaximumversusk+ (forfixedk+). Indeed,thehigherk+,
2 1 1
the higher c , and the more information about changes in f
1 1
canbetransmittedthroughthemiRNA-channel. Ontheother
hand, ceRNA is unrepressed by the miRNA for small val-
2
uesofk+whileitisstronglyrepressedforhighvaluesofk+,
2 2
for any value of the input. In both regimes, ceRNA is only
2
weakly sensitive to changes in the level of its competitor. In
such conditions, no information can be transmitted through
themiRNA-channel. Theinformationflowisthereforemaxi-
malbetweenthesetwoextremes.
Forthetranscriptionalchannel,weexpecttoseemaximum
information transmission when the target, ceRNA , is unre-
2
pressed by the miRNA, in which case the channel is effec-
tivelydecoupledfromtherestofthenetwork,whilenoinfor-
mation transmission is expected when the target is fully re-
pressedbythemiRNA.Indeed, inFig. 5Boneseesthat, for
smallvaluesofk+(unrepressedtarget),I isthehighestand
2 TF
itdecreasestozeroask+increasestogetherwiththedegreeof
2
target repression. Moreover, since a higher k+ moves m to
FIG.4. Flowchartofthemethod.AC++implementationisavail- 1 2
largervalues(i.e. towardsthefreeregime),I increaseswith
ableathttps://github.com/araksm/ceRNA. TF
k+ and decreases with k+ in the susceptible regime. Notice
1 2
that, when post-transcriptional capacity is highest, miRNA-
erogeneitiesindeedaremappedtothemiRNAbindingkinet- mediated regulation is as effective as transcriptional control
ics and are shown to influence the susceptibility of the tar- (seeFig. 5C).
gets to the miRNA molecules [36]. Moreover, the level of Theintuitionthatthechannelcapacityisstronglycoupled
complementarity between the regulator and the target seems tothetarget’ssizeofderepression(itsAOV)isconfirmedby
to be decisive for the selection of a decay channel (catalytic thefactthatthelatterindeeddisplaysaverysimilarbehaviour
or stoichiometric) for the miRNA-ceRNA complex [15, 27]. (seeS2Figure).
One may therefore expect that the effectiveness of miRNA-
mediatedpost-transcriptionalcontroldependsstronglyonthe
kineticparameterscharacterizingthenetwork. miRNA-mediatedregulationmayrepresentthesolecontrol
Hereinparticular, wearegoingtoinvestigatehowtheca- mechanismincaseofdifferentialcomplexprocessing
pacitiesoftheseregulatoryelementsareaffectedbychanges
in(i)miRNA-ceRNAbindingkinetics,(ii)miRNArecycling Direct experimental measurements of miRNA-recycling
rates,and(iii)effectivetranscriptionratesofallRNAspecies ratesarechallenging[59]. Howeveritiswellknownthatthe
involved. degreee of complementarity of a miRNA to its target lead to
In order to contrast the performances of miRNA- and TF- different repression pathways [60] and that differential com-
channelsweshallstartbyanalyzingtheirrespectiveresponses plex processing strongly affects reciprocity of competition
tothesameinputsignal. Moreprecisely,theinputvariablefj [33]. Inordertobetterquantifythedegreeofregulatorycon-
willbevariedfrom0toavaluefmaxdefinedbythecondition trolachievableincaseofdifferentprocessingmechanismswe
nj(fmax)=0.99(i.e. fromasituationinwhichthepromoter changedthevaluesofmiRNA-recyclingratesκ1andκ2keep-
is always free to one in which it is essentially always occu- ing fixed all the other parameters, obtaining the results dis-
pied). played in Fig. 6. Both I and I are higher for small
TF miRNA
values of the recycling rates, and decrease as miRNA recy-
cling increases. The observed dependence of I on κ
miRNA 1
ThecapacityofthemiRNA-channelismaximalinaspecificrange agreeswiththeresultsof[42],wherethesensitivityofmiRNA
ofmiRNA-ceRNAbindingrates moleculestochangesinthelevelsoftheirtargetsdecreasesas
themiRNArecyclingrateincreases,eventuallyvanishingfor
AvailableestimatesofRNAbindingenergeticsindicatethat κ (cid:29)σ . Indeed,forlargeκ miRNAsbecomeinsensitiveto
1 1 1
the target sites of an individual small non coding RNA can changesinf ,andhencenoinformationcanbetransmitted.
1
spanawiderangeofaffinitybothinmammaliancells[57]and Remarkably (see Fig. 6C), for large κ and small κ
2 1
in bacteria [58]. To explore the functional consequences of miRNA-mediatedregulationmaybethesolemechanismable
such heterogeneities we analyse the behaviour of the regula- toeffectivelycontrolthetargetlevel. Inthiscase,infact,I
TF
torychannelschangingthevaluesofthecomplexassociation becomesespeciallysmallasthetargetiscompletelyrepressed
rates k+ and k+ while keeping fixed all others parameters. bythemiRNAwhileI staysfinitesincetheactivationof
1 2 miRNA
7
FIG.5. Dependenceofthetranscriptionalandpost-transcriptionalchannelcapacitiesonmiRNA-ceRNAassociationrates.(A)I .
miRNA
(B)I (C)∆I =I −I .ValuesofthekineticparametersarereportedinTableI.
TF TF miRNA
FIG.6. Dependenceofthetranscriptionalandpost-transcriptionalchannelcapacitiesonmiRNArecyclingrates. (A)I . (B)I
miRNA TF
(C)∆I =I −I .ValuesofthekineticparametersarereportedinTableI.
TF miRNA
ceRNA cantitrateawaythemiRNAandthereforederepress lation outperforms TF-based control can occur. Such a pos-
1
m . sibility might indeed be realized for relatively large values
2
Asforthepreviouscase, thebehaviouroftheAOVcorre- ofk+,wherethemiRNA-mediatedchannelworksoptimally.
1
latesstronglywiththisscenario,i.e. largerAOVcorresponds ThisregimeisexploredinFig. 7.
to higher achievable information flow in the regulatory ele- Remarkably,onefindsthatpost-transcriptionalcontrolout-
ment(seeS2Figure). Thissuggeststhat,althoughtargetsthat performstranscriptionalonewhen
aredegradedpurelycatalyticallycannotcompeteformiRNA
atsteadystate(asalsoshownforbacterialsmallRNAs[61]), (i) the effective trascription rate of ceRNA exceeds that
2
they can be notably derepressed if one of their competitors ofceRNA (b n >b n ,seeinTableI)inpresenceof
1 2 2 1 1
decaysstoichiometrically. weakcatalyticdegradation(Fig. 7A);
(ii) the effective transcription rate of ceRNA is smaller
1
Effectivederepressionofarepressedtargetmaybeconditionalon thanthatofthemiRNA(b n <βn ,seeinTableI)in
1 1 µ
theactivationofitscompetitorandofmiRNA presenceofstronglycatalyticallydegradedtarget. (see
Fig. 7B).
Transcriptional and post-transcriptional channel capacities
are expected to depend on the effective transcription rates of A more detailed analysis shows that, in these scenarios,
the ceRNAs and the miRNA, described respectively by the wheneverI >I thetarget’sderepressionsizeislarger
miRNA TF
quantities b n and βn , since these bear a direct impact on whenthesignalisprocessedpost-transcriptionallyratherthan
i i µ
the regime (repressed, susceptible or unrepressed) to which throughthedirecttranscriptionalchannel,asshowninS3Fig-
each ceRNA belongs. In particular, upon changing the frac- ure. Again,theAOVappearsthereforetobeagoodproxyfor
tional occupancies, situations where miRNA-mediated regu- thechannelcapacity.
8
FIG.7. Dependenceof∆IonthefractionaloccupancyoftheTF
bindingsiteintheoptimalityrangeforthepost-transcriptional
miRNA-mediatedchannel.(A)Caseoftargetswithweakcatalytic
degradation(smallκ andκ ). (B)Caseofastronglycatalytically
1 2
degradedtarget(largeκ )withaweaklycatalyticallydegradedcom-
2 FIG. 8. Channel capacities as a function of the target’s de-
petitor(smallκ ).Notethatn (cid:39)1for(A)whilen (cid:39)0.5for(B).
1 µ µ greeofderepression(AOV).Theblackcurvecorrespondstodirect
ValuesofthekineticparametersarereportedinTableI.
transcriptionalcontrolinabsenceofmiRNAs,whilethebluecurve
describesthebehaviorofthemiRNA-mediatedpost-transcriptional
channel.Forbothchannelslnmmin =4.62.Thepredictedmaximal
2
Previousworkhadpointedtonear-equimolarityofceRNAs MIinthePoissonianlimitgiveninS1Textisshownasadashedline.
andmiRNAswithinanetworkasoneofthecentralprecondi- ValuesofthekineticparametersarereportedinTableI.
tions for significant ceRNA regulation [34, 42, 62]. On the
other hand, experiments on liver cells have shown that such
conditionsmaynotbemetinphysiologicalconditions,where vanishinglysmall.
the number of miRNA target sites can vastly exceed that of Similarly in Fig. 8 one can see that I increases
miRNA
miRNA molecules [29, 30]. Our results suggest that target (roughly) logarithmically with ∆ if the latter exceeds
miRNA
derepression may be significant even if the competitor is in a certain threshold, whereas it is essentially zero below this
lowcopynumbers,providedacertainheterogeneityinkinetic value.Inotherwords,themiRNA-mediatedchannelisunable
parameters(e.g. foracatalyticallydegradedtargetandasto- to convey any information unless the degree of target dere-
ichiometrically degraded competitor) is present, in line with pression is sufficiently high. Hence I can be close to
miRNA
recentexperimentspointingtoakeyroleofparameterdiverse- zero even though ∆ is finite. Notice that the derepres-
miRNA
ness[36]. sionthresholdformiRNA-mediatedinformationtransmission
islargerthanthecorrespondingthresholdfordirecttranscrip-
tionalinformationflow.
Asufficientlyhighdegreeoftargetderepressionisrequiredfor Acloseanalysisrevealsthatsuchafeatureisduetotheran-
informationtransmission domfluctuationsonthelowestexpressionlevelofthetarget,
mmin. Belowthreshold,infact,thedegreeofderepressionof
2
thetargetissmallerthanthefluctuationsonmmin causedby
We have seen that AOV displays a dependence on kinetic 2
intrinsic noise (reported in Fig. 8). Therefore the response
parameters that very closely matches the one found for the
observed at the level of the output is indistinguishable from
channel capacities. In particular, whenever I > I ,
miRNA TF
thenoise. InthemiRNA-mediatedpost-transcriptionalchan-
we found ∆ > ∆ . One may be lead to think that
miRNA TF nelthethresholdAOVislargersincenoiseonmminishigher.
theAOVbyitselfmayfullydescribethecapacityofmiRNA- 2
AssoonastheAOVovercomesrandomfluctuationsonmmin,
mediatedpost-transcriptionalinformationprocessingandthat 2
thechannelcapacitiesstartstoincreasewith∆ .
stochasticity could, to a large degree, be neglected. In some miRNA
situations,however,thesizeoftheoutputrangeispoorlyin-
formativeaboutthechannelperformance. Thisbecomesclear
byexploringtherelationshipbetweentheAOVandthechan- Noisereductionmakesthecapacityofthepost-transcriptional
nel capacity at a quantitative level. In Fig. 8 the depen- channelcomparabletothatofthetranscriptionalchannelinthe
dence of the optimal MI on the AOV is discussed for two limitoflargepopulationsofweaklyinteractingmiRNAs
cases: (i)directtranscriptionalregulationbyTFinabsenceof
miRNAmolecules(blackcurve), (ii)miRNA-mediatedpost- We will now focus specifically on the influence of the
transcriptionalregulation(bluecurve). miRNA-ceRNA binding component of the intrinsic noise,
InFig. 8weindeedseethatforlargeenoughAOV,where which will be studied upon varying miRNA-ceRNA bind-
fluctuationsoneachlevelaresmallcomparedto∆ ,theTF- ing rates and miRNA population size. We shall consider
TF
channel capacity approaches the theoretical limit of a Pois- twodifferentscenarios,namely(a)direct,TF-basedtranscrip-
sonian channel worked out in S1 Text. On the other hand, tional regulation in absence of miRNAs (corresponding to
forsmallvaluesof∆ ,thecapacityofTF-channelbecomes k+ = 0 for i = 1,2), and (b) indirect post-transcriptional
TF i
9
themiRNA-ceRNAbindingaffinitiesareexperimentallychal-
lenging. However,computationalmethodspredictaconsider-
abledegreeofheterogeneityacrossdifferentmiRNA-ceRNA
pairs[57]. Remarkably, forthemajorityofcasesreportedin
the literature, the predicted miRNA-binding energies of the
RNAsonthe‘inputside’ofthechannelarelowerthanthose
oftheRNAsonthe‘outputside’,inlinewiththeoptimalcon-
ditionshighlightedbyourmodel.Forexample,bindingaffini-
tiesbetweenthelongnon-codingRNAlinc-MD1anditsreg-
ulatorymiRNAs(playingacentralroleinskeletalmusclecell
FIG.9. Comparisonoftranscriptionalandpost-transcriptional differentiation)aresignificantlylowerthanthosecharacteriz-
regulationforafixedoutputvariationrange. (A)Dependenceof ingmiRNAinteractionswithlinc-MD1’scompetitors,namely
ITFandImiRNAonω.(B)FFofm2atthecorrespondingsteadystate. the MAML1 and MEF2C mRNAs, as predicted by miRanda
AOVandmm2in arethesameforbothchannels,namely∆miRNA = algorithm [39, 64]. Likewise, the circular RNA CDR1 has
∆ =210±1andmmin =11±1.Valuesofthekineticparameters
TF 2 beenfoundtocontainaround70bindingsiteswithhighcom-
arereportedinTableI.
plementarity for miR-7, corresponding to a strong effective
couplingthroughwhichitcanregulatetheexpressionofmiR-
7’s target genes [37, 65, 66]. Finally, the high sequence ho-
miRNA-mediatedregulation(correspondingtonon-zerocom-
mology of pseudogenes (long non-coding RNA genes devel-
plex binding rates). Kinetic parameters will however be re-
oped from protein-coding genes but unable to produce pro-
scaledas
teins) with their parental gene allows them to compete for a
largenumberofsharedmiRNAs[28,38,67].
OurstudyalsopointstothepotentialrelevanceofmiRNA-
δ → δ∗ =ωδ ,
ceRNAcomplexdecaychannels. Itisknownthat, incaseof
k+ → (k+)∗ =ωk+ , (24) sufficientcomplementarity,miRNAmoleculescanfunctionas
i i i
siRNAs and cleave their targets after binding them [15, 27].
where ω > 0 is the re-scaling factor. Note that the above Such targets decay catalytically and are therefore effectively
choicedoesnotchangem aslongask−c (cid:28)b n and(k−+ degraded by the miRNAs. Our model predicts that miRNA-
2 i i i i i
κ )c (cid:28)βn (bothofwhicharetrueinthecaseweconsider). mediatedcontrolmaybethepreferredregulatorymechanism
i i µ
Bychangingωonemaythereforecharacterizethedependence in presence of kinetic heterogeneities at the level of miRNA
of the channel capacities onthe kinetic parameters δ and k+ recycling rates. In particular, targets that undergo catalytic
i
whilekeepingtheoutputrangeapproximatelyfixed,suchthat degradationmaybeefficientlyderepressedbytheircompeti-
∆ (cid:39)∆ foreveryω. tors. Thistypeofscenariohasbeenobservedinexperiments
miRNA TF
Fig. 9Ashowshowthethechannelcapacitieschangewith concerning the bacterial small RNA Qrr [61]. Qrr represses
ω. One sees that in general I < I , so that at fixed its targets by distinct mechanisms. For instance, luxR is re-
miRNA TF
AOVthemiRNA-mediatedchanneltypicallyhassmallerca- pressed catalytically, luxM stoichiometrically, while luxO is
pacitythanthedirectone.However,inthelimitofsmallω,i.e. silencedthroughtranslationalrepression. luxMandluxOare
whenmiRNApopulationsaresufficientlylargeandmiRNA- howeverabletoderepressLuxRinthepresenceofQrr. Inthe
ceRNAcouplingsareweak,themiRNAchannelcanperform lightofourresults,theseobservationsmaythereforepointto
aseffectivelyasapuretranscriptionalchannel.Thisresultcan ahigherthanexpectedrolefortheceRNAeffectinvivo,espe-
be understood by examining the noise levels (see Fig. 9B). ciallyincasesinwhichheterogeneitiesinkineticparameters
For small ω (or high µ ), in particular, the FF approaches arethoughttobestrong[34,59,62,68,69].
0,2
thePoissonianlimitsince(i)forµ(cid:38)µ relativefluctuations
02
inmiRNAlevelsaresmall,extranoisecomingfrommolecu-
lar titration disappears and the target degradation constant is DISCUSSION
rescaledasd →d +k+µ(f );(ii)µ(cid:28)µ targetceRNA
2 2 2 1 0,2
is seen in free regime, where it is not sensitive to miRNA
Non-codingRNAmolecules,andmiRNAsspecifically,are
molecules.Insuchconditions,miRNA-mediatednoisebuffer-
increasinglyassociatedtoregulatoryfunctions. Besidesmak-
ing[20]cannotbeobserved, neverthelesstranscriptionaland
ing it mandatory to characterize the specific role of ncRNAs
miRNA-mediatedregulationareeffectivelyequivalent.
onacase-by-casebasis, especiallyforsituationslikedisease
ordifferentiation,thisfactalsoraisesthequestionofwhatin-
gredients can make miRNAs a preferred tool to regulate the
ObservedheterogeneitiesinmiRNA-targetinteractionstrengthmay level of a target RNA over, for instance, the target’s TF. A
enableregulationbymiRNA-mediatedcross-talk possibleanswerliesinthenoise-bufferingrolethatmiRNAs
canplay,whichisespeciallyevidentingeneticcircuitrieslike
The quality of miRNA-target interaction influences the incoherentfeed-forwardloops[19,20]. Byreducingrelative
binding kinetics and may be decisive for the activation of fluctuationsintheoutputlevel,miRNAscanconferrobustness
the target decay channel [15, 27, 63, 64]. Estimations of to gene expression profiles. However the so-called ‘ceRNA
10
hypothesis’opensthewaytothepossibilitythattheirregula- tiveness of microRNA-mediated post-transcriptional control
toryfunctionsarecarriedoutatabroader, thoughmoresub- of gene expression by computing the capacity of the corre-
tle,level. Inshort,accordingtotheceRNAscenariomiRNAs sponding regulatory channel and comparing it to that of di-
canmediateaneffectivepositiveinteractionbetweentheirtar- rect,TF-driventranscriptionalregulation. Evidently,multiple
get RNAs driven by the targets’ competition to bind them. factorscaninfluencetheflowofinformationacrossnodesin
In this sense, miRNAs can be seen as a sort of ‘channel of abiochemicalnetwork,startingfromtheintrinsicnoisinessof
communication’ between RNAs through which RNA levels eachreactionstep.
canbealteredandnoisecanbeprocessed(bothbufferedand Our basic challenge was therefore understanding in which
amplified). Previous work [42] has shown that the ceRNA circumstances miRNA-mediated control can outperform the
effect may generate both highly plastic and highly selective TF-based one, thereby obtaining insight on why the ceRNA
ceRNA-ceRNA couplings, thereby representing a potentially effect appears to be so often employed by cells in situations
powerfulmechanismtoimplementgeneregulationatthepost- where accurate tuning and/or shifts of expression levels are
transcriptionallevel. required.Wehavethereforeconsidered,alongthelinesof[33,
Although predicted theoretically, the extent and relevance 34, 41–44], a mathematical model of the ceRNA effect and
of ceRNA effect in vivo is poorly understood. On one hand, characterizeditssteadystateintermsofbothmeanmolecular
considerable evidence points to the ceRNA effect playing a levels and regulatory capacities of the miRNA-mediated and
majorroleincertaindis-regulatedortransientcellularstates. TF-basedchannelsviastochasticsimulations.
For instance, it has been shown that the expression of the Wehavefirstconsideredhowthetwochannelsprocessin-
tumor-suppressorgenePTENcanberegulatedbyitsmiRNA- puts (the TF levels f and f ) that vary in the same range.
1 2
mediatedcompetitorsVAPA,CNOT6L,SERINC1orZNF460 We have shown that, while the capacity of the TF-channel
[70]. Furthermore, many pseudogenes have been found to dependsmonotonouslyoneachmiRNA-ceRNAbindingrate
compete with their parental genes for a shared pool of com- and is largest when the target is unrepressed by miRNAs
monmicroRNAs,thusregulatingtheirexpressionascompet- (as might have been expected), the capacity of the post-
itive endogenous RNA [28, 38, 67, 71] . Such mechanisms transcriptionalchannelismaximalinaspecificrangeofval-
seem to be of particular relevance in cancer. For instance, ues of the miRNA-ceRNA binding rates. In agreement with
murinemodelsengineeredtooverexpressthepseudogenesof [33],wefoundthatmiRNA-channel’sefficiencyistunableto
theproto-oncogeneBRAFdevelopanaggressivemalignancy optimality by the binding kinetics. Furthermore, our model
resemblinghumanBcelllymphomasince,byfunctioningas suggests that both capacities decrease as the miRNA recy-
ceRNAs, they elevate BRAF expression both in vitro and in clingratesincrease,confirmingpreviousindicationsobtained
vivo[38]. Likewise, thelongnoncodingRNAlinc-MD1has bydifferentanalyticaltechniques[42]. Consistentlywiththe
been shown to regulate the skeletal muscle cell differentia- scenarioobservedexperimentallyforthebacterialsmallRNA
tionclockbyspongingmiRNAsfromitscompetitors,thereby Qrr [61], our model finally suggests that catalytically regu-
enacting a ceRNA mechanism. In particular, MAML1 and lated targets are weakly capable of competing for miRNAs
MEF2C(codingfortranscriptionfactorsthatactivatemuscle- butmightbesignificantlyderepressedbytheircompetitors.
specific gene expression) compete with linc-MD1 for miR- Inaddition,post-transcriptionalmiRNA-mediatedinforma-
133andmiR-135respectively[39]. Takentogether,theavail- tionprocessingwasshowntobecharacterizedbyathreshold
able evidence indicates that miRNA activity depends on the behaviourasafunctionoftheAOV.Inotherterms,noinfor-
miRNA:targetratio, onmiRNAtargetsiteabundanceandon mation can be transmitted across the channel unless the tar-
miRNA binding affinities. Further analyses of high through- get’sdegreeofderepressionissufficientlylarge. Thisimplies
put datasets confirm this observation [30, 36]. One may thattheregulatoryeffectivenessofthechanneliswellencoded
therefore question how said factors may influence miRNA- by the degree of target derepression when the latter is suffi-
mediatedpost-transcriptionalcontrol. ciently high, in which case it is possible to identify regimes
The problem however arises of quantifying the degree of inwhichpost-transcriptionalregulationismoreaccuratethan
control that can be exerted through miRNAs. Taking the transcriptionalcontrol.
‘channel’ analogy more strictly (as done before for simpler Togetadeeperinsightontheoriginoftheobservedthresh-
regulatory elements [9, 45, 52]), one may resort to informa- oldbehaviouronemusthowevergobeyondtheAOVandcon-
tiontheoreticconceptsandtoolstocharacterizepreciselyhow sider more carefully how the miRNA-ceRNA binding noise
wellmiRNAscanprocessfluctuationscomingfromthemod- affectstheoverallpicture.AftershowingthatmiRNA-ceRNA
ulatornodesandtransferthemtothetargetnodes. Thisissue bindingnoiseisindeedattheoriginofthethresholdbehaviour
goes beyond noise buffering, specifically including the abil- that limits the miRNA-channel capacity, we have uncovered
ity to respond to large changes in the mean levels as well as the rather remarkable property that in presence of large but
to changes in the structure of fluctuations. As the properties weaklyinteractingmiRNApopulationstheceRNAeffectcan
of a channel are conveniently encoded in the mutual infor- regulategeneexpressionaseffectivelyasthetarget’smodula-
mationbetweentheinputandoutputnodes,askinghowwell tornodeitself.
a channel can function amounts to asking what is the chan- The present work has focused on a small genetic circuit
nel’scapacity,i.e. themaximumvalueoftheinput-output(or made up of a single miRNA species and two target RNA
modulator-target)mutualinformationachievablethroughthat species at steady state. Previous work has however shown
channel. Thisworkaimedpreciselyatquantifyingtheeffec- thatcross-talkispossibleevenduringtransients[44]. Going