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Page 1: One Is Not Enough

doi:10.1016/j.jmb.2009.07.050 J. Mol. Biol. (2009) 392, 1133–1144

Available online at www.sciencedirect.com

One Is Not Enough

Robert Daber, Kim Sharp and Mitchell Lewis⁎

Department of Biochemistry andBiophysics, University ofPennsylvania School ofMedicine, 37th and HamiltonWalk, Philadelphia, PA19104-6059, USA

Received 3 March 2009;received in revised form15 July 2009;accepted 16 July 2009Available online22 July 2009

*Corresponding author. E-mail [email protected] used: MWC model

Changeux model; GFP, green fluore

0022-2836/$ - see front matter © 2009 E

In both prokaryotic and eukaryotic organisms, repressors and activators areresponsible for regulating gene expression. The lac operon is a paradigm forunderstanding how metabolites function as signaling molecules andmodulate transcription. These metabolites or allosteric effector moleculesbind to the repressor and alter the conformational equilibrium between theinduced and the repressed states. Here, we describe a set of experimentswhere we modified a single inducer binding site in a dimeric repressor andexamined its effect on induction. Based upon these observations, we havebeen able to calculate the thermodynamic parameters that are responsiblefor the allosteric properties that govern repressor function. Understandinghow effector molecules alter the thermodynamic properties of the repressoris essential for establishing a detailed understanding of gene regulation.

© 2009 Elsevier Ltd. All rights reserved.

Edited by J. Karn

Keywords: lac repressor; allostery; induction; repression; inducer binding

Introduction

Cellular proteins that perform specific metabolictasks are often regulated to meet the needs of theorganism. In many instances, regulating the fluxthrough a pathway is achieved by adjusting theconcentration of an enzyme that controls the rate-determining step in a pathway. In both prokaryoticand eukaryotic organisms, enzyme concentration isregulated by transcription, which is frequentlycontrolled by repressors and activators. Thesemolecules either directly or indirectly monitor theaccumulation or diminution of a metabolite andrespond like a molecular switch—increasing ordecreasing the rate of gene expression. Effectormolecules are the chemical signals that convey themetabolic state of the cell to the genetic machinery.They can function as inducers, anti-inducers, or co-repressors, and in each instance, they alter theequilibrium between two unique conformationalstates of the molecular switch.The switch that regulates the lac operon is arguably

the most well characterized and therefore serves asthe paradigm for understanding gene regulation(recently reviewed by Wilson et al.1). The switch is atwo-component system and consists of a repressormolecule and an operator. The repressor is a 360-amino-acid protein that has a modular structure

ress:

, Monod–Wyman–scent protein.

lsevier Ltd. All rights reserve

composed of an NH2-terminal or “headpiece”domain (∼60 residues) and a COOH-terminal“core” domain. The headpiece contains the classichelix–turn–helix motif that recognizes and binds toan operator sequence, while the core domain isresponsible for inducer binding and contains thedimerization interface. The second component ofthe switch is the operator, a short stretch of DNAthat is pseudo-palindromic. In the lac operon, theprimary operator is positioned just upstream of thegene for β-galacotosidase.2 As a negative regulator,the repressor associates with the operator sequenceand physically blocks transcription of the genes thatare necessary for lactose metabolism. The inducermolecule relieves repression by altering the repres-sor–operator equilibrium, stabilizing a conforma-tion that is incompatible with operator binding.3

The natural inducer of the lac operon is an analogueof lactose, allolactose, but a gratuitous inducer, suchas 1-isopropyl-β,D-thiogalactopyranoside (IPTG),can also effectively decrease the binding affinitybetween the repressor and its operator. Since theinducer binds to a site that is distal to the DNAbinding domain, the signal is transmitted throughthe molecule by altering the conformation of therepressor; this structural rearrangement of therepressor results in an allosteric transition.Monod, Wyman, and Changeux first described

how structural changes could alter a protein's abilityto perform a given function and coined the termallostery.4 The Monod–Wyman–Changeux MWCmodel assumes that (a) allosteric proteins areoligomeric with at least one axis of symmetry, (b)these proteins adopt two distinct conformations

d.

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1134 One Is Not Enough

(designated as “R” and “T” by Monod et al. anddescribed here as “R” and “R⁎”), and (c) theconformational states have different affinities forthe substrates. In this instance, the R and R⁎conformations have different affinities for theinducer and the operator; (d) each substrate (i.e.,operator or inducer) binds to independent sites oneach monomer and, finally, (e) the molecularsymmetry is conserved in the transition from onestate to another. The lac repressor is an allostericprotein; it is oligomeric, and each monomer has twowell-separated substrate binding sites, one for theoperator and another for the inducer. Biochemicaland structural studies have also demonstrated thatthe repressor adopts two distinct quaternary struc-tures that correspond to the induced and repressedstates (see Wilson et al.1).Here, we describe a set of experiments that

allowed us to elucidate the in vivo thermodynamicparameters that characterize the binding of inducerto the two conformational states of the repressor, aswell as the equilibrium constant between these twostates. By measuring the effect of zero, one, and twoinducers binding to the repressor, we were able toextract equilibrium constants and develop a thermo-dynamic model to describe the allosteric process.

Fig. 1. The surface of the dimeric repressor depictingthe locations of mutants with the Is phenotype. Themutants shown in yellow surround or are within theinducer binding pocket. The surface shown in redrepresents Is mutants believed to interfere with transmis-sion of the allosteric signal.

Results

In vivo induction analysis

Several mutant repressor molecules that bind tothe operator DNA with wild-type affinity but areincapable of induction have been identified.5 Thesesubstitutions, classified by an Is phenotype, eitherare defective in inducer binding or cannot transmitthe allosteric signal to the DNA binding domain.The position of the Is point mutations appear in fivegeneral locations with respect to the primarysequence and includes residues 70–80, 90–100,190–200, 245–250, and 272–277. When the positionsof these mutations are mapped onto the proteinstructure, as is illustrated in Fig. 1, they cluster eitherat the dimer interface or in close proximity to theinducer binding site. Presumably, mutants withinthe ligand binding pocket interfere with the abilityof the repressor to bind the inducer, while mutantsat the dimer interface disrupt the signaling process.6

The crystal structure of the repressor bound to theinducer illustrates that IPTG forms hydrogen bondsto the amino acid side chains of Asp149, Arg197,Asn246, and Asp274, as well as van der Waalsinteractions with a hydrophobic surface created byIle79, Leu148, Phe161, Asn291, Phe293, andLeu296.7 In fact, mutating any of these residuesfrequently results in the Is phenotype.5 However, asone would anticipate, not all mutations in theinducer binding pocket affect induction equally.One particularly potent mutation in the ligandpocket, R197G, completely disrupts inducer bindingbut does not alter the repressor's ability to fold or

bind to the operator.8 The side chain of R197 anchorsthe galactose ring of the inducer by forming twohydrogen bonds with the C2 and C3 hydroxyls. Toestablish how inducer binding affects the allostericproperties of the repressor, we used the R197Gmutation and created heterodimeric repressors withzero, one, or two functional inducer binding sites.As described previously,9 heterodimeric repres-

sors were created by altering the C-terminal dimerinterface of the wild-type dimeric lac repressor.These heterodimeric repressors have a uniquemonomer–monomer interface that allows us toprobe the effect of asymmetric changes in the ligandpocket. The R197G mutation was incorporated intoone or both monomers, producing heterodimericrepressors with zero, one, or two nonfunctionalinducer binding pockets (Table 1). Subsequently, theheterodimeric repressors were introduced into cellscontaining the GFPmut3.1 reporter, whose tran-script is controlled by the binding of the repressor toa chimeric operator.9 Cells transformed with plas-mids containing the heterodimers were then grownin the presence and in the absence of 2.5 mM IPTG,and the levels of transcription were determined bymeasuring green fluorescent protein (GFP) fluores-cence. As shown in Fig. 2a, the heterodimericrepressors exhibit tight repression, but disruption

Page 3: One Is Not Enough

Table 1. Sequence of repressors used in in vivo assays

Monomer 1 Monomer 2Functionalinducer sitesRepressor 197 282 17 18 22 251 255 278 281 282

Wild typea R Y Y Q R L R D C Y 2R197A A Y Y Q R L R D C Y 0R197G G Y Y Q R L R D C Y 0Y282S and Het2S R S T A N K L T W T 2Y282S/R197A and Het2S A S T A N K L T W T 1Y282S/R197G and Het2S G S T A N K L T W T 1

This table represents the various repressor constructs used in experimentation. The top sequence corresponds to the wild-type sequencefound in the lac repressor.

a Both a full-length tetrameric repressor and a dimeric construct truncated after residue 332 were analyzed. All constructs containingY282S, R197G/A or Het2S were also truncated after residue 332.

1135One Is Not Enough

of the ligand binding sites alters the relativeinduction. The heterodimeric repressor with bothsites intact has near-full induction (76% maximalexpression), similar to both the dimeric form and thetetrameric form of the wild-type lac repressors (Fig.S1). When either of the inducer binding sites iseliminated, induction is far from optimal and

repression is only slightly relieved, leading to lessthan 20% maximal induction. As expected, whenboth inducer binding pockets are destroyed, repres-sion can only be slightly relieved at the highestconcentration of the inducer tested. Since thisheterodimeric repressor binds to an asymmetricoperator, we also explored if the orientation of the

Fig. 2. Repression and induc-tion data. (a) The histogram illus-trates the levels of GFP productionin the induced and the repressedstates. For comparison, GFP pro-duction is shown for the wild-type(tetrameric) repressor and for thenatural operator. The next fourpairs of bar graphs illustrate thelevel of GFP in the induced and therepressed states when an R197Gmutation is placed in (a) bothpockets, (b) neither pocket, (c) thepocket proximal to the promoterand (d) the pocket distal to thepromoter in the heterodimericrepressor. Repression by the hetero-dimers is with respect to the chi-meric operator. (b) Response ofeach repressor mutant to a rangeof IPTG concentrations. Fractionalexpression is defined as the signalof the sample divided by the signalwhen no repressor is presentHigher concentrations of IPTGtested produced mixed resultswith no increase in signal for thetwo-site case. Each construct withan R197G mutation began inducingagain, consistent with reducedIPTG affinity for that mutation.

,

.

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1136 One Is Not Enough

repressor containing a single functional ligandbinding site with respect to the promoter affectedinduction. The asymmetric operator was inserted inboth orientations into the reporter plasmid. Thelevels of induction were identical irrespective ofwhether the inducible monomer was proximal ordistal to the promoter (Fig. 2a).The induction measurements were repeated over

a wide range of inducer concentrations (0.1 nM to50 mM) to ensure complete induction (Fig. 2b).Again, when both inducer sites were functional, weobserved 80% of the maximal expression of GFPeven at the highest concentration of inducer; thefluorescent signal never reaches the same level asthat obtained when no repressor is present. Thissuggests that even when both inducer sites are fullyoccupied, there is still residual binding to theoperator resulting in incomplete induction. Consis-tent results were also obtained with both thetetrameric and the dimeric wild-type repressors,and the lack of full induction may represent therepressor–operator–inducer complex recently iden-tified with small-angle scattering.10 These wild-typerepressor constructs behaved similarly since theauxiliary operator sites required for increasedrepression by the tetrameric repressor were absent.Induction was further limited by inactivating one

of the binding sites. Repressors with one functionalligand binding site never exceed 20% of the maximalinduction even at very high concentrations of theinducer. The titration curves suggest that althoughrepression can be partially relieved upon binding asingle inducer, two molecules of IPTG bound to therepressor are needed for complete induction, suchthat one is not enough. To ensure that the observedphenotype was not an artifact of the R197Gmutation, we also tested additional constructscontaining the R197A mutation. The mutationsdemonstrated identical phenotypes (Fig. S2), sup-porting the working hypothesis that mutatingresidue R197 affects inducer binding alone. Byfitting the induction data to a “two-state model,”we can establish the conformational equilibriumconstant in the MWC model, as well as the bindingaffinities of the inducer to the repressed and inducedconformations.

Extracting the thermodynamic parameters

A fundamental tenet of the MWC model is thatallosteric proteins exit in two conformations withdifferent relative activities. In the case of the lacrepressor, the two conformations are differentiatedby their preferential binding to either the operator,O, or the inducer ligand, I. Here, we designate theconformation that prefers to bind DNA as the activeform R and the conformation that prefers to bindinducer as the inactive form R⁎. The two conforma-tions are in equilibrium (Eq. (1)), such that KR⁎R=[R⁎]/[R]:

R fKR⁎R

R⁎ ð1Þ

Both of these repressor conformations can bind tothe operator in an equilibrium manner, but equili-brium values are not identical:

R +O ZKRO

RO ð1aÞ

R⁎ +O ZKR⁎O

R⁎O ð1bÞ

where KRO=[RO]/[R][O] and KR⁎O=[R⁎O]/[R⁎][O]are the respective association constants. Althoughboth conformations can bind to the operator, theratio of equilibrium constants, s=KR⁎O/KRO, must beless than 1 for induction to occur.In the classic MWC model, the inducer also binds

to both conformations of the repressor with differentaffinities; it binds to the active (R) conformation withan affinity given by the binding constant KIR= [RI]/[R][I] and to the inactive (R⁎) conformation with anaffinity given by the binding constant KIR⁎=[R⁎I]/[R⁎][I]. Since the functional unit of the repressor isdimeric, the repressor can bind two inducer mole-cules. The two binding sites are distant from oneanother, and there is no direct interaction betweeninducers such that the inducer molecules bindindependently to each of the binding pockets withidentical affinities (i.e., there is no contribution tocooperativity from direct ligand–ligand interaction).This assumption is borne out by the observation thatthe dependence of the apparent rate of dissociationof the repressor–operator complex with respect toinducer concentration is noncooperative.11 In accor-dance with the MWC model, inducer binding to therepressor drives the repressor equilibrium towardthe inactive R⁎ conformation and, therefore, induc-tion only requires that the inducer has a greateraffinity for the inactive conformation R⁎ than for theactive conformation R (i.e., KIR⁎NKIR).The level of transcript produced in our GFP assay

can be modeled using a standard ligand bindingisotherm. The signal (E) is a function of theconcentration of unbound repressor with respect toits operator dissociation constants (the repressorconcentrations at which there would be 50% siteoccupancy).

EEmax

=1

1 +Rtot½ �R50½ �

� � ð2Þ

In the absence of repressor, the fluorescent signal(Emax) is constitutively produced in this system.Since the repressor can adopt two distinct con-formations, R and R⁎, that have different operatorbinding dissociation constants, then the bindingisotherm takes the following form:

EEmax

=1

1 +Ra½ �R50½ � +

R⁎a½ �R⁎50½ �

� � ð3Þ

The relative binding affinity is a function of theconcentration of unbound repressor species in the two

Page 5: One Is Not Enough

Fig. 3. A diagram illustrating the linked equilibria thatresult from inducer binding and operator binding. KRR⁎ isthe equilibrium constant that describes the induced andthe repressed states. KIR and KIR⁎ refer to the binding of theinducer to these two conformational states. KRO is theequilibrium constant that describes the operator binding.

1137One Is Not Enough

forms [Ra] and [R⁎a] and their respective dissociationconstants for operator binding ([R50]=1/KRO and[R⁎50]=1/KR⁎O). As the concentration of the repres-sor in the active form increases, the fractionalexpression decreases. Similarly, increasing the affinityof the repressor for its operator also decreases thefractional expression.The fractional expression described in Eq. (3) can

be expressed as the product of two dimensionlessratios: the product of the ratio of the total concen-tration of the repressor to its operator bindingdissociation constant in the active form, [Rtot]/[R50], and the fractional amount of unboundrepressor in the active form relative to the totalconcentration of repressor, [Ra]/[Rtot]. Similarly,[R⁎a]/[Rtot] is the fractional amount of unboundrepressor in the induced form relative to the totalconcentration of repressor, such that Eq. (3) takes thefollowing form:

EEmax

=1

1 +Rtot½ �R50½ �

Ra½ �Rtot½ � +

Rtot½ �R⁎50½ �

R⁎a½ �Rtot½ �

� � ð4Þ

We defined a variable f as [Ra]/[Rtot], whichdepends upon both the inducer concentration andthe conformational equilibrium constant, andanother variable r as the [Rtot]/[R50] ratio. Sincethe amount of repressor species bound to operator isa small fraction of the unbound species, we canassume that f⁎=1− f and that the ratio of theoperator affinities for the two repressor forms foroperator is simply KR⁎O/KRO= s. This allows us toexpress the relative binding affinity in terms of threevariables, r, f, and s, where e=E/Emax.

e =1

1 + r f + s 1� fð Þð Þð Þ ð5Þ

The linked equilibria that describe the binding ofthe repressor to its operator and to the inducer areillustrated in Fig. 3. There are 12 repressor speciesthat contribute to the total repressor concentration.

Rtot½ � = R½ � + 2 RI½ � + R⁎½ � + 2 R⁎I½ � + RI2½ � + R⁎I2½ �f g+ RO½ � + 2 RIO½ � + RI2O½ �f g + R⁎O½ �+ 2 R⁎IO½ � + R⁎I2O½ �f g ð6aÞ

When there are many more copies of the repressorthan operators, the species bound to the operator(RO, RIO, RI2O, R⁎O, R⁎IO, and R⁎I2O in Eq. (6))represent a small fraction of the total repressorconcentration (an approximation of repressor andoperator copy numbers was derived from plasmidconstruction; see Materials andMethods) and can beneglected. This simplifies the subsequent equationsand permits a closed-form solution. The concentra-tion of unbound repressor that exists in the R and R⁎

conformations is the total of all unbound repressorspecies [Ra] and [R⁎a], respectively:

Ra½ � = R½ � + 2f g RI½ � + RI2½ �f g ð6bÞ

R⁎a½ � = R⁎½ � + 2f g R⁎I½ � + R⁎I2½ �f g ð6cÞFor the heterodimeric repressor with two inducer

binding sites, there are six unbound species inequilibrium, namely R, RI, RI2, R⁎, R⁎I, and R⁎I2 (thetwo-site case). When one inducer binding site isremoved, the terms contained in curly brackets inEq. (6a) to Eq. (6c) disappear, and there are only fourunbound species: R, RI, R⁎, and R⁎I (the one-sitecase). In both instances, the amount of repressor [R]that can bind to the operator depends upon thevalue of the conformational equilibrium constantKRR⁎, the inducer concentration [I], and relativeinducer binding affinities to the R and R⁎ conforma-tions. These three parameters are responsible forestablishing the relative amounts of active andinactive repressors, R and R⁎, as described below.When the heterodimeric repressor has a singleinducer binding pocket, the amount of the variousrepressor/inducer species is governed by the linkedequilibria on the left half of Fig. 3. The concentra-tions of unbound repressor in the active and inactiveforms are the sum of the first two terms in Eq. (6a)and Eq. (6b), respectively, while the total repressorconcentration is the sum of the first four terms in Eq.(6c). When both ligand pockets are functional, thelinked equilibria include the right half of Fig. 3 andthe total repressor concentrations include the curlybracket terms in Eq. (6a) to Eq. (6c).The concentrations of all repressor species can be

evaluated in terms of the concentration of onespecies, namely the free active repressor [Ra], usingthe appropriate equilibrium constants and thefraction of repressor in the unbound R form, f=[Ra]/[Rtot], which is given by the ratio of two bindingpolynomials. For the heterodimeric repressor that

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1138 One Is Not Enough

has two functional inducer binding pockets, the ratioof two binding polynomials is given by:

f2 =Ra½ �Rtot½ � =

1 + I½ �KIRð Þ21 + I½ �KIRð Þ2 + KRR⁎ 1 + I½ �KIR⁎

� �2 ð7aÞ

We note that, as in the original MWC model, thepresence of factors having the form (1+ [L]K)n

results from independent ligand binding at nidentical sites, and it properly accounts for thestatistical factor arising from the macroscopic indis-tinguishability of distinct microscopic species IR, RI,etc. Similarly, the fraction of repressor in the Rconformation when there is a single inducer bindingsite can be described by the ratio of correspondingpolynomials:

f1 =Ra½ �Rtot½ � =

1 + I½ �KIRð Þ1 + I½ �KIRð Þ + KRR⁎ 1 + I½ �KIR⁎

� � ð7bÞ

When both pockets are eliminated or in the absenceof inducer, the fraction of the repressor in the Rconformation reduces to:

f0 =1

1 +KRR⁎ð7cÞ

At saturating concentrations of inducer, thefraction of free repressor depends only upon theconformational equilibrium, KRR⁎, and the relativebinding of the inducer to the repressor in the R andR⁎ conformations, as described by Eq. (8a) and Eq.(8b):

f V1 = limI½ �Yl

f1 Ið Þ = 1

1 +KRR⁎KIR⁎

KIR

ð8aÞ

f V2 = limI½ �Yl

f2 Ið Þ = 1

1 +KRR⁎KIR⁎

KIR

� �2 ð8bÞ

By substituting Eq. (7) and Eq. (8) for each of the 0,1 and 2 site cases into Eq. (5), we can relate therelative amount of transcript produced at limitingconditions to four parameters: (a) the ratio of totalrepressor concentration to the active form operatorbinding dissociation constant, r; (b) the ratio ofinducer binding to the inactive and active repressorconformations, x = KIR⁎

KIR; (c) the equilibrium between

these two different conformations, KRR⁎; and (d) theratio of operator affinities of the two repressorforms, s:

en = 1= 1 + r 1=1 +KRR⁎xn� �

+ rs 1� 1= 1 + KRR⁎xn� �� �� �ð9Þ

The three values of ei, for n=0, 1, and 2, are theexperimentally observed levels of GFP expressionwith zero inducer (or with zero sites) at a highinducer concentration for the one-site repressor anda high inducer concentration for the two-site

repressor, respectively. Rearranging Eq. (9), wedefine the variable yi as:

yn = 1=en � 1 = rð1= 1 +KRR⁎xn� �

+ s 1� 1= 1 +KRR⁎xn� �� �Þ ð10Þ

By evaluating the ratio of the one-site and two-sitecases to the zero inducer case, we can eliminate thevariable r and obtain two equations in the threevariables s, x, and KRR⁎.

yn=yo = 1= 1 + KRR⁎xn� �

+ s 1� 1= 1 + KRR⁎xn� �� �� �

= 1= 1 + KRR⁎� �

+ s 1� 1= 1 + KRR⁎� �� �� � ð11Þ

where n=1 or 2.The two parameters x and KRR⁎ were first fit to the

experimentally observed values y1 and y2 on the left-hand side of Eq. (11) by a straightforward gridsearch assuming there is no binding of the inactiverepressor (R⁎) to the operator (O) (i.e., s=0). We thensolved for r using Eq. (10). Subsequently, theabsolute value of the inducer binding constant,KIR⁎, was determined by fitting the expression levelcurves versus inducer concentration for both theone-site and two-site dimers using the parametervalues x, r, and KRR⁎. The value of KIR⁎ [1/Kd(IR⁎)]was refined by bisection to minimize the meanabsolute error in fractional expression. The otherinducer binding constant was then determined asKIR=xKIR⁎.

Analysis of the extracted parameters

The value of KRR⁎ was determined to be 2±0.5,which is in good agreement with previous in vitromeasurements12 and consistent with our observa-tion that the repressor prefers to crystallize in theinduced R⁎ conformation in the absence of theligand, regardless of the crystallization conditions.7

The ratio of the inducer binding constants, x=KIR⁎/KIR=15±3, illustrates that the inducer preferentiallybinds to the R⁎ conformation. The in vivo ratio ofequilibrium constants is however somewhat lessthan that observed by O'Gorman et al. in vitro.12 Thecalculated dissociation constant for active repressor–inducer binding, Kd(IR), is 4±2 μM. Since the ratio ofthe two binding constants, KIR⁎/KIR, is ∼15, we caninfer that the corresponding dissociation constantfor the inducer to the inactive repressor, Kd(IR⁎), is15×4 μM or 60 μM. Overall, the values of KRR⁎

and x are consistent with our structural observa-tion that the conformations of the apo repressorand the repressor–IPTG complex are essentiallyisomorphous.7

The ratio of the total repressor concentration to itsoperator dissociation constant, [Rtot]/[R50], is acomplex quantity that reflects the binding affinityof the repressor for the operator and the concentra-tions of the repressor and operator in the cell. Wefound that this ratio is 150±50. This implies thatthere is a vast excess of repressor compared withthat required for 50% operator binding, which is

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1139One Is Not Enough

consistent with our in vivo assay system (seeMaterials and Methods). The model also predictsthat the switch is leaky. In the absence of inducer,only a fraction, f0=1/(1+KRR⁎), of the repressormolecules adopt the R conformation, leading to a netactive repressor concentration of f0=1/(1+2), whichis “only” ∼50-fold above the repressor–operatordissociation constant. Although the repressor isoperating under saturating conditions and the vastmajority of the operators are bound by the repressor,a small amount of transcript, roughly 2%–3%,would be produced since E/Emax=1/(1+50). Inthe absence of inducer, a detectable level of back-ground expression is observed, consistent with thevalues calculated from the model.According to the MWC model, the contribution

the inactive conformation makes to the binding ofthe repressor to the operator is negligible. Thepreviously discussed parameters were determinedunder this assumption that s=0; however, to test itsvalidity, we repeated the above fitting procedure,allowing the induced conformation R⁎ to bind Owith a range of affinities (0≤ s≤1). The resultingparameters are tabulated in Table 2. When the R⁎conformation binds the operator with an affinity of1/10,000 or less than the active form R, the modelproduces quality fits very similar to the limitingexpression and IPTG titration curves, with almostidentical output parameters. Assuming a somewhathigher relative affinity of 1/1000 produces a slightlypoorer fit to the limiting expression data but anequally good fit to the whole induction curve. At ahigher relative affinity of s=1/100, the fit is poor.Essentially, when the relative affinity of R⁎ for Obinding is large, the inducer cannot pull therepressor off the operator and, as a consequence,the system is not inducible. Overall, the best-fitparameters are slightly different from those whenwe assume s=0; notably, the conformational equili-brium is somewhat closer to unity, but the differ-ences fall within the estimated uncertainty rangesand the same qualitative picture emerges: there is asmall but significant preference for the inactive formof repressor in the absence of inducer, a large excessof repressor relative to its Kd for operator binding,and binding of inducer to the inactive repressorform that is tighter by 1 order of magnitude. We

Table 2. Fitted model parameters

s x KRR⁎ r

Kd(IR⁎

(μM)a

0 15±3 2.3±0.5 150±50 4±21/100,000 15±3 2.3±0.5 170±50 4±21/10,000 15±3 2.4±0.5 170±50 4±21/1000 19±5 1.3±0.5 106±30 4±21/100 1160±600 0.01±0.005 41±25 0.3–1.

Uncertainty limits were determined from parameter ranges that douba Defined as Kd(IR

⁎)=1/KIR⁎.

b Percentage of error fitting limiting expression values e0, e1, and e2entire induction curves with the same three parameters plus KR

⁎I.c Dependent parameter fixed by Kd(IR

⁎)/x.

therefore conclude that the s value must be on theorder of 1/1000 or less to explain the in vivo data.This allows us to effectively ignore R⁎ binding tooperator.The quality of the model and the accuracy of the

parameters can be assessed by comparing thecalculated and observed induction curves (Fig. 4).The experimental data fit the calculated model verywell, yielding a mean unsigned error of 2%. Thebest-fit model slightly overestimates the slope ofexpression versus I at the midpoint. This could resultfrom systematic overestimation of the effective IPTGactivity in vivo—that is, the actual activity of IPTG issomewhat lower than the nominal concentrationvalues used in plotting the data. Another measure ofthe validity of the derived constants was assessed bydetermining the ratio of the [I]50 values for the one-site to two-site cases. A calculated value of 2.5 agreeswell with the experimentally observed value of ≈3.The agreement between the observed and calculatedratios is a goodmeasure of self-consistency since thisratio is not an adjustable parameter. In the fitting tothe model (although not in the model itself), weassumed that the concentration of repressor wasmuch greater than that of the operator such that theamount of repressor bound to operator is smallcompared with the total and free amounts ofrepressor. Estimates of the repressor copy numberare imprecise, but the ratio of repressor to operatorin a similar system was estimated to be 100:1.13 Thisratio is sufficiently large such that the fractionalerror that results from neglecting the concentrationof the bound repressor species in Eq. (6a) is 1/100,less than a few percentages. This permits us toobtain closed-form expressions for limiting expres-sion levels and a straightforward physical inter-pretation in terms of the basic model parameterswith little numerical error. A more exact model maybe applied for low repressor copy number bykeeping all the terms in Eq. (6a) and fittingiteratively, assuming that the total repressor remainsconstant at each step.The expression of GFP in the reporter system

depends upon the inducer concentration and itsrelative affinity for the R⁎ and R conformations ofthe repressor, which we estimate to be 4 and 60 μM,respectively. This difference in binding affinities

)

Fit error for(limiting expression

values (%)b

Fit errorfor entire induction

curves (%)bKd(RI)(μM)c

4 1.9 604 1.9 604 1.8 606 2.1 79

1 13 2.7 812

le the best-fit error.

with the parameters x, KRR⁎, and r and percentage of error fitting

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Fig. 4. The graph is a plot of thefractional induction as a function ofthe log of the inducer concentrationfor the heterodimeric repressorswith a single binding pocket andtwo intact pockets. The curves arethe expected values calculated fromthe model.

1140 One Is Not Enough

results in curves that are slightly sigmoidal whenexpression/repression values are plotted againstinducer concentration. The midpoint, somewherebetween the two individual dissociation constantvalues, is ∼25 μM, and the plot has a Hill coefficientof ∼1.25 (Fig. S3), consistent with numerousstudies.14–16 This mild cooperativity arises fromthe inducer-promoted shift of repressor conforma-tional equilibrium toward the higher-affinity form.However, since these curves are all characterized bya single inflection point, without further indepen-dent data, only a single “apparent” inducer affinityconstant is extractable.In the above model of heterodimeric repressor

action, we have assumed that the binding pocketmutation R197G/A in one of the monomers acts byblocking binding of inducer to one site and that thisis responsible for the drop in maximal expressionfrom 75% to 18%. Inducibility of the repressorwould also be reduced if the mutation caused theconformational equilibrium to shift toward the Rform (i.e., KRR⁎ decreases). While it is unlikely that abinding pocket mutation would exert its effectthrough KRR⁎ rather than KIR and KIR⁎, we canconsider this possibility in our model. Using thetwo-site Eq. (5a) rather than Eq. (5b) to fit the “one-site mutation” induction curve and adjusting onlyKRR⁎, we can model a reduced maximal expressionlevel of 18% by setting KRR⁎=0.15. In this model, theheterodimeric repressor with one-site R197G/Amutation still binds two inducers with unchangedaffinity, but the mutation has pushed the repressorfurther into the active conformation. However, thismodel predicts that at zero inducer, there is nodetectable expression (i.e., the switch is not leaky),which is inconsistent with the experimental data.

Discussion

A thermodynamic perspective

The lac repressor is a model system for exploringgene regulation and the mechanism by whicheffector molecules regulate transcription. Measuringinduction using the heterodimeric repressors with

zero, one, and two inducer pockets allowed us tounambiguously extract three important thermody-namic quantities: the equilibrium constant betweenthe induced and the repressed conformations of therepressor KRR⁎, the relative affinities of each con-formation for inducer (KIR⁎/KIR), and the amount ofrepressor in the cell relative to its operator bindingdissociation constant. In addition, determining theseparameters allowed us to begin exploring how eachaspect of the repressor contributes to its function-ality by modeling theoretical changes in eachparameter.From modeling the in vivo data, it appears that

effective repression in the absence of inducer (b4%expression) requires a large excess of active repres-sor with respect to its operator binding constant (50-fold). While a background expression of 4% may beless than ideal, given the other thermodynamicparameters of the repressor, an r value of 150appears to be optimal. To better understand therelationship between the ratio of active repressorconcentration and operator affinity, we generated aseries of simulated induction curves by varying the[Rtot]/[R50] ratio (Fig. 5a). By increasing the ratio to500 or 1000, which can be achieved by increasingeither the active repressor concentration or therepressor–operator affinity, constitutive expressioncan be minimized to less than 1%. However,increases in the ratio also affect the dynamic rangeof the switch. Increased repression comes at theexpense of induction, and the inducer cannot relieverepression to more than 50% maximal expression.Conversely, if the ratio is too small, then the switchcan be induced but background expression is veryhigh. When the ratio of r is 150, the backgroundexpression appears to be minimized while balancingmaximal induction near 75%. Experimentally, boththe one-site and the two-site inducer binding casesshow a plateau in expression level at transcriptionallevels less than 100%, suggesting that there is alwaysa small fraction of the repressor that can adopt theactive (R) conformation and block transcription.The equilibrium distribution of repressors in both

the active conformation and the inactive conformationis an additional parameter that controls the repressionand induction profile of the repressor. Frommodeling

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Fig. 5. Simulated plots of fractional induction with respect to inducer concentration (plotted on a log axis). With theuse of the determined thermodynamic parameters for the repressor, theoretical plots were simulated for stepwise changesin (a) r, the ratio of [RTOT]/[R50]; (b) KRR⁎, the conformational equilibrium (L); (c) x, the ratio of inducer binding affinitiesfor both the active and the inactive repressor conformations; and (d) s, the ratio of operator binding affinities for each ofthe repressor conformations. The plots shown in black correspond to the value derived from the experimental data.

1141One Is Not Enough

the experimental data, there appears to be a rathersmall energy difference between the two conforma-tions of the repressor (R and R⁎), ΔG=kTln(KRR⁎)=0.4 kcal/mol. This modest difference in energyallows the repressor to easily switch between activeand inactive states. Again, we simulated inductioncurves by altering the equilibrium constant (desig-nated as “L” according to the MWC nomenclature) toexplore the impact of this parameter on repressoractivity (Fig. 5b). Since the active conformation of therepressor has preferential binding to the operator,when the apparent equilibrium constant is less than 1and the active conformation of the repressor isdominant, repression is strong and there is minimalbackground expression. However, altering KRR⁎ toincrease the concentration of active repressor alsoincreases the ratio r discussed previously. This limitsthe dynamic range of expression and permits only afraction of the possible induction. In contrast, whenthe equilibrium constant is large, the majority of therepressor is in the R⁎ conformation, resulting in aswitch that is leaky but fully inducible. At the fixedvalues for the other thermodynamic properties, thedynamic range of the switch is optimal when L (KRR⁎)

is between 1 and 10. There is a low level of basalexpression in the absence of inducer and a nearly fullinduction at high concentrations of inducer.Since repressor activity is a product of linked

equilibria, the distribution of active and inactiverepressors can be altered by adding either operatoror inducer ligands. The system is repressed whenthere is an excess of active repressor with respect toits operator dissociation constant. Induction, on theother hand, requires an excess of inactive repressor.The inducer relieves repression by shifting theapparent conformational equilibrium, decreasingthe concentration of active repressor. A singleinducer binds 15 times more tightly to the R⁎conformation than the R conformation, which inenergetic terms is about 1.6 kcal/mol. In the presenceof a single bound inducer, the apparent equilibriumconstant increases, K′RR⁎=exp(0.4 + 1.6)/kY≈30, stabi-lizing the R⁎ conformation; the [Ra]/[R50] ratiodrops by ∼5-fold (150/30). The additional energyshifts the apparent equilibrium further toward theR⁎ conformation, thereby reducing the effectiveconcentration of the repressor in the R conformation.Since there is less active repressor, polymerase has a

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1142 One Is Not Enough

greater opportunity to transcribe its message. There-fore, in the presence of high concentrations ofinducer, the single-site repressor can achieve afractional induction level of 1/(1+5)=16%, whichis consistent with our experimental observation. Asecond inducer molecule shifts the apparent equili-briummore dramatically. An additional 1.6 kcal/molof binding energy increases KRR⁎ to∼450, driving therepressor further toward the R⁎ conformation andlowering the [Ra]/[R50] ratio to less than unity, ∼0.4.As a consequence, in the presence of high levels ofinducer, we expect 1/(1+0.33)=75% induction,which is very close to the experimental observation.Since the calculated energetic effects appear to bepurely additive, we would suspect cooperativity tobe of little importance when it comes to induction.In fact, the Hill coefficient (at the inducer midpoint)is a rather modest 1.25 and concludes thatcooperativity is not an essential requirement ofthe switch. Cooperativity is important for shiftingthe window in which an effector can elicit aresponse to some relatively narrow range of non-zero concentrations. For example, the cooperativityseen with hemoglobin is probably due to the factthat the partial pressures of O2 are never zero andmay only range from 40 to 80 mmHg. In contrast,in noncooperative (Langmuir isotherm type) bin-ding, the rate of increase is greatest starting from azero effector, which works better as a switch in thelac system given the actual inducer concentrationrange.In the previous examples, the experimentally

determined value of x=15 was used to modelinduction as a consequence of altering the otherparameters. Since the repressor's response to induceris dependent upon the ratio of relative affinities ofthe inducer for each conformation of the repressor,simulated induction curves were also producedwithvariations in x (Fig. 5c). For x=1, the inducer binds toboth the active and the inactive conformations withthe same affinity and expression levels becomeinsensitive to inducer. When the ratio is less thanunity, the inducer binds more tightly to the activeconformation and actually decreases induction. Theinducer ligand would then function more like a co-repressor. Onlywhen the ratio is significantly greaterthan 1 does the system work as a true switch—having full repression at low inducer concentrationsand complete induction at high concentrations ofinducer. Unlike the other parameters that alter bothrepression and induction, no such balance exists forthe value of x and the switch functions even better asx increases to greater values. The measurements andanalysis described were made using the gratuitousinducer IPTG; it would be interesting to see if thevalue for x, and thus the maximal expression level, ishigher when the natural inducer, allolactose, isutilized. Regardless, this analysis suggests thatinduction of the lac repressor with IPTG can beimproved by finding mutations that increase theratio of binding affinities beyond 15.As the MWCmodel suggests, the repressor adopts

two distinct conformations, but only one of the

conformations is functional. Therefore, the switchalso depends on the relative operator binding of thetwo repressor conformations. To explore the impor-tance of this parameter on repressor activity, weexamined how variations in s alter the inductioncurves (Fig. 5d). When the value of s is greater thanunity, the inactive conformation of the repressor bindsto the operator and the inducer again behaves as a co-repressor by increasing repression. At unity, bothconformations have the same affinity for the operatorand inducer has no effect. The switch will functiononlywhen the value of s is≪1.As sdecreases, there isgreater induction that quickly approaches the limit ass=0. Based upon these curves, when s is less than0.0005, the system functions as if s=0.

A kinetic perspective

The thermodynamic description of the switch isextremely useful for understanding the molecularmechanism of gene regulation. However, to fullyappreciate how this switch functions, we mustconsider its activities in the context of its kineticproperties. The binding of the lac repressor to itsoperator has been well studied, and all of the data areconsistent with a bimolecular reaction (Eq. (12))having an association constant, KRO, of 1.5×10

10 M−1

under standard conditions.11,15 The equilibrium con-stant, of course, is the ratio of the association anddisassociation rate constants:

KRO = RO½ �= R½ � O½ � =Ka=Kd ð12ÞThe dissociation constant has been measured experi-mentally for the wild-type repressor and its naturaloperator, as Kd=6×10

−4 s−1; using the above valueof KRO, this gives Ka=7.5×10

6 M−1 s−1.15,17,18 Aconsiderably greater association rate has beenmeasured directly by Riggs et al., who found avalue of 7×109 M−1 s−1.15 In general, the decrease by3 orders of magnitude in affinity that results frominducer binding can arise from a decrease in the on-rate, an increase in the off-rate, or any combination ofthe two. Changes in the off-rate have been estimatedto range from 6- to 18-fold at inducer concentrationsof 10 μM, well above the inducer midpoint.15,16 Thisincrease, while significant, implies that a largeportion of the affinity loss is due to a decrease inthe on-rate. We therefore deduce that the inducerstabilizes the R⁎ conformation of the repressor bydecreasing the on-rate conservatively by 10-fold and,arguably, by 100-fold or more. It is instructive tospeculate how altering these kinetic parameters canaffect gene regulation and whether modulation ofaffinity through the association rate rather than thedissociation rate has any functional significance.On average, the fraction of the operator that is free

of repressor is f0=1/(1+[R]KRO) for a given repres-sor concentration [R]. Since operator binding isactually a stochastic process, at any instance, theoperator is either occupied or not and there will be adistribution of occupation times. From the theory ofstochastic processes,19 the time that the operator is

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1143One Is Not Enough

occupied by repressor is exponentially distributedwith a mean of τon=1/Kd. Similarly, the mean timethe operator is free of repressor also varies exponen-tially, with a mean of τoff=1/Ka[R]. Based upon thedata of Riggs et al.,15 we can estimate that in theabsence of inducer, the mean occupancy time is1600 s. With an estimated repressor concentration of20 nM (O'Gorman et al.12), the mean time theoperator is free of repressor in the absence ofinducer is 6 s using the lower estimate of Ka and is7 ms using the upper estimate.Transcription initiation takes a finite amount of

time. Consequently, the fraction of time the operatoris free and the duration of time it is free are bothrelevant for effective transcription. We hypothesizethat the promoter must be free of repressor longenough that the transcriptional machinery canprogress to the point where it is not affected byrepressor binding. The kinetics of transcriptionalinitiation at the lac promoter has been wellstudied.20 RNA polymerase binds to the promoterregion, forming a “closed” complex, which thenundergoes an “isomerization” to form the open(strand separated) initiating complex. Assuming theisomerization step, which has a first-order rateconstant of 1.6×10−2 s−1, approximates the time toinitiate transcription, and then the promoter must befree from the repressor for roughly 60 s. This isconsiderably longer than the mean time it is freeunder high repression conditions. Now if the effectof the inducer were solely on the dissociation rate ofthe repressor, the mean occupancy time would besignificantly reduced, but the mean time theoperator is free of repressor would be unchanged;therefore, on average, the time frame is still too shortfor transcription to initiate. If the inducer affectedrepressor affinity by depressing the on-rate, then theaverage time the promoter is free of repressor wouldincrease, allowing transcription to initiate.

Conclusions

Elucidating the thermodynamic properties of themolecular switch is essential for developing acomplete understanding of gene regulation. TheMWC model for the allosteric transition appliesbeautifully to the lac molecular switch and accountsfor the conformations of the repressor observed inthe crystal structures. Here, we have experimentallydetermined the equilibrium constants that accountfor the observed properties of the molecular switch.Our analysis suggests that the transition from theinduced conformation to the repressed conforma-tion requires a relatively small amount of energy(0.4 kcal/mol) and only subtly favors the inducedconformation. This soft equilibrium allows theactivity of the repressor to be modified by adjustingthe conformational equilibrium through themechanisms of linked equilibria. Since both theinducer and the operator preferentially bind todifferent conformations, binding of inducer pro-vides energy to compete with operator binding. By

increasing the effective concentration of the inactiverepressor, inducer binding reduces the number ofrepressors in the active conformation and thereforereduces repression. Our analysis demonstrates thata single inducer provides enough energy to increaseexpression levels to 20% and that two inducers arerequired for maximal induction. In addition, fromaltering various parameters of the repressor andmonitoring the effect on repressor activity, we havegained valuable insight into designing improvedswitches. Tuning the allosteric properties of therepressor will allow us to create better and novelmolecular switches.

Materials and Methods

Repressor plasmid construction

Constructions of the dimeric wild-type repressorplasmid (pBD21004), the heterodimeric plasmid(pBD22010), and the plasmids containing each of theconstituent monomers (pBD21008 and pBD21903) havebeen described previously.9 To introduce the R197A andR197G mutations into the pBD21004 and pBD21008plasmids, we used full-circle PCR mutagenesis withprimers BD030 (5′ GCT CTG GCT GGC TGG CAT AAATAT C 3′) and BD031 (5′ CAG ACG CGC CGA GAC AGAAC 3′) and primers BD032 (5′ GGT CTG GCT GGC TGGCAT AAA TAT C 3′) and BD031, respectively. Afterintroducing the point mutations into each of theserepressor genes, we reintroduced the two monomericrepressor genes into the heterodimeric plasmid. In short,the pBD21903 plasmid was digested with DrdI and therepressor gene was isolated via gel purification. Subse-quently, the vector containing the recently mutated R197Aor R197G gene (pBD21012, pBD21016) was digested withDrdI and purified. These vectors (also containing theY282S monomeric mutation) were then ligated with theinsert containing the complementary repressor gene,creating plasmids pBD22011 and pBD22012. The repres-sor plasmid is derived from the pACYC vector with acopy number around 15. Expression of lacI is under thecontrol of the constitutive lac promoter. Estimates ofrepressor copy number are comparable with those ofOehler et al.13

Reporter plasmid construction

Construction of the reporter plasmid with GFPmut3.1under the control of the lac promoter/operator has beendescribed previously.9 In short, GFPmut3.1 was intro-duced into the pBR322 vector and placed under control ofthe lac promoter/operator. To introduce operator changesproducing chimeric operators, we used full-circle PCRmutagenesis (operator 212:411, primers BD024 andBD021; operator 411:212, primers BD020 and BD025). Toexplore the role of directionality on induction withrepressor mutants containing asymmetric ligand pocketknockouts, we created and utilized two reporters. Each ofthese reporters contained a chimeric operator sequencewith the orientation of the half sites inverted with respectto each other. In assays using the wild-type (tetrameric ordimeric) repressor, the reporter containing the naturaloperator was used.

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1144 One Is Not Enough

In vivo repression/induction assay

To analyze the phenotypes of various repressor–operator combinations, we used an in vivo fluorescentassay. To quantify the level of fluorescence and thereforeindirectly measure the degree of transcription, we grewthen analyzed cells in a Perkin Elmer Victor3 plate reader.In short, combinations of repressors and operators weretransformed and colonies were selected in triplicate forovernight culture growth. In addition, cells controlling thereporter only were also chosen to establish the level ofmaximal expression under nonrepressing conditions. Forinduction analysis, samples were grown in the absence ofIPTG, as well as in the presence of various amounts ofinducer. Once samples reached saturation, 200-μl aliquotswere taken and introduced into flat-bottom 96-well plates.A dilution plate was also prepared so that the opticaldensity of the cultures could more accurately be deter-mined. Each of these plates was then measured forGFPmut3.1 fluorescence (495-nm excitation wavelengthand 510-nm emission wavelength) and optical density(A590) on a Perkin Elmer Victor3 Plate reader. Fluorescencedata were then processed to remove background fluores-cence and normalize the GFP signal to the cell count. Thenormalized signals for biological replicates were thenaveraged to provide a relative fluorescent signal for eachsample. Errors for each sample were determined from thestandard deviation of the biological replicates. Fractionalexpression levels were calculated by dividing the averageGFP signal for the sample with a given repressor by thesignal of the sample containing no functional repressor.

Supplementary Data

Supplementary data associated with this articlecan be found, in the online version, at doi:10.1016/j.jmb.2009.07.050

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