Articles

Allosteric Modulation of Monomeric Proteins*

Received for publication, March 3, 2005

Paolo Ascenzi‡§, Alessio Bocedi‡§, Alessandro Bolli‡, Mauro Fasano�, Stefania Notari§,and Fabio Polticelli‡

From the ‡Dipartimento di Biologia and Laboratorio Interdipartimentale di Microscopia Elettronica, Universita“Roma Tre,” Viale Guglielmo Marconi 446, I-00146 Roma, Italy; §Istituto Nazionale per le Malattie InfettiveI.R.C.C.S. “Lazzaro Spallanzani,” Via Portuense 292, I-00149 Roma, Italy; and �Dipartimento di BiologiaStrutturale e Funzionale and Centro di Neuroscienze, Universita dell’Insubria, Via Alberto da Giussano 12,I-21052 Busto Arsizio (VA), Italy

Multimeric proteins (e.g. hemoglobin) are considered to be the prototypes of allosteric enzymes, whereasmonomeric proteins (e.g. myoglobin) usually are assumed to be nonallosteric. However, the modulation ofthe functional properties of monomeric proteins by heterotropic allosteric effectors casts doubts on thisassumption. Here, the allosteric properties of sperm whale myoglobin, human serum albumin, and human�-thrombin, generally considered as molecular models of monomeric proteins, are summarized.

Keywords: Allostery, monomeric proteins, myoglobin, albumin, thrombin.

When allosteric effectors bind to specific site(s) on agiven protein, they influence the tendency to bind otherligand(s) at other site(s), e.g. they modulate the enzymeactivity. Therefore, an allosteric protein exists in two ormore different conformations with different (re)activityproperties. The relative populations of the different confor-mations (e.g. active and inactive) are determined by bind-ing of allosteric effectors. They fall into two categories,homotropic and heterotropic allosteric effectors. Hetero-tropic allosteric effectors modulate the functional proper-ties of both monomeric and multimeric allosteric proteinsby binding to different sites within the protein structure,whereas homotropic effectors modulate the functionalproperties of proteins containing at least two equivalentbinding clefts. This condition is fulfilled only in the case ofmultimeric allosteric proteins with hemoglobin represent-ing the prototype of this class. In multimeric allostericproteins, cooperativity can be achieved by conformationalchanges caused by ligand binding to one of the equivalentsites that propagate to the other cleft(s) and facilitate orimpair binding of other ligand molecule(s). However, itmust be kept in mind that proteins containing multiple(equivalent) binding sites are not necessarily allostericmacromolecules [1–5].

Multimeric proteins (e.g. hemoglobin) are considered tobe the prototypes of allosteric enzymes, whereas mono-meric proteins (e.g. myoglobin) usually are assumed to be

nonallosteric. However, the modulation of the functionalproperties of monomeric proteins (e.g. sperm whale myo-globin, human serum albumin, and human �-thrombin) byheterotropic allosteric effectors casts doubts on this as-sumption [6–18]. Here, the allosteric properties of spermwhale myoglobin, human serum albumin, and human�-thrombin, generally considered as molecular models ofmonomeric proteins, are summarized.

LACTATE FACILITATES O2 DELIVERY FROMSPERM WHALE MYOGLOBIN

Myoglobin is a monomeric globular hemoprotein de-voted to the storage and facilitated diffusion of O2 incardiac and striated muscles [19–21]. Very recently, myo-globin has been reported to catalyze pseudo-enzymati-cally NO scavenging, protecting cellular respiration, and toinfluence redox pathways in cardiac muscle, protectingthe heart from oxidative damage [22–26]. Pseudo-enzy-matic reactions can be controlled because myoglobin ex-ists in distinct conformations with different ligand bindingand catalytic properties. Moreover, cavities within the pro-tein matrix of myoglobin facilitate ligand channeling to andfrom the heme, multiple ligand copies storage, multiligandreactions, and conformational transitions supporting li-gand binding [13, 16, 22, 27, 28]. Myoglobin undergoesfunctionally relevant ligand-linked tertiary conformationalchanges, and the allosteric equilibrium between the differ-ent structural arrangements and the heme reactivity isaffected by binding of nonheme ligands [8, 13, 29–32].

Lactate, an obligatory product of glycolysis under an-aerobic conditions, allosterically affects myoglobin [8] asmuch as organic phosphates and/or protons (not effectivein myoglobin under physiological conditions) influence thehemoglobin function [33–37]. Lactate, behaving as a het-erotropic allosteric effector, brings about the decrease of

* This work was partially supported by grants from Ministry ofEducation, University, and Research of Italy (MIUR, FIRB-2001and PRIN-COFIN 2003 to P. A.) and University “Roma Tre”(CLAR-2004 to F. P. and P. A.).

‡ To whom correspondence should be addressed: Diparti-mento di Biologia, Universita “Roma Tre,” Viale GuglielmoMarconi 446, I-00146 Roma, Italy. Tel.: 39-06-55173200(2); Fax:39-06-55176321; E-mail: ascenzi@uniroma3.it.

© 2005 by The International Union of Biochemistry and Molecular Biology BIOCHEMISTRY AND MOLECULAR BIOLOGY EDUCATIONPrinted in U.S.A. Vol. 33, No. 3, pp. 169–176, 2005

This paper is available on line at http://www.bambed.org 169

the O2 affinity in myoglobin (i.e. the increase of P50 from0.27 to 2.8 mm Hg) at moderately acid pH (i.e. 6.5) [8], acondition that may be achieved in vivo under a physicaleffort [38]. The decrease of the O2 affinity in myoglobin bylactate reflects lowering of the second order rate constantfor the hemoprotein oxygenation by one order of magni-tude. According to linked functions [1–5], the affinity oflactate for deoxygenated myoglobin is higher than that forthe oxygenated hemoprotein by one order of magnitude(i.e. Kd for lactate binding to myoglobin increases from2.5 � 10�3 to 2.6 � 10�2 M upon oxygenation) [8].

The effect of lactate on O2 binding to myoglobin under-lines the existence of a ligand-linked structural change(s)(Fig. 1). Although the lactate binding site of myoglobin hasnot been yet identified, the inspection of the crystal struc-ture of different myoglobin derivatives has shown an anion(e.g. sulfate) binding site at hydrogen bonding distancesfrom ArgCD3, HisE7, and ThrE10 residues [39]. Therefore,lactate might impair the access of O2 to the heme distalpocket by stabilizing the “closed” HisE7 conformation.Note that the HisE7 residue represents the “gate” for ligandaccess to and escape from the hemoprotein active center[40]. Moreover, lactate binding might impair the movementof the iron atom to the heme plane appearing as a crucialstep for ligand binding [30]. Interestingly, ArgCD3, HisE7,and ThrE10 mutations profoundly alter ligand binding prop-erties of myoglobin [41, 42].

The observation that lactate at the tens of mM levelaffects the O2 affinity of myoglobin in vitro [8] suggests thatan allosteric mechanism is at work in vivo as lactate con-centration can change from 2 to 20 mM between restedand exhausted muscle [43]. The effect of lactate on the

functional properties of myoglobin has important physio-logical implications. During a prolonged physical effort,such as diving for sperm whale, muscle cells are continu-ously consuming ATP, which cannot be fully reconstitutedbecause of the decreased intracellular O2 concentrationand the subsequent impairment of the oxidative phospho-rylation. As a consequence, the glycolytic pathway pro-gressively shifts from pyruvate to lactate production withATP formation being further reduced. Hence, lactate in-crease may have useful consequences because it maytrigger a compensatory mechanism whereby the O2 affinityof myoglobin is reduced and the O2 release helps in keep-ing constant the ATP formation [8].

ENDOGENOUS LIGANDS MODULATE DRUG BINDINGTO HUMAN SERUM ALBUMIN

Albumin is the most prominent protein in plasma, itsconcentration being 45 mg/ml in the serum of humanadults, but it is also found in tissues and secretionsthroughout the body. Albumin is constituted by a singlechain containing three homologous domains (labeled I, II,and III), each composed of two (A and B) subdomains.Owing to its flexible three-domain architecture, albuminhas an exceptional ligand binding capacity and displaysallosteric properties. In particular, the conformationaladaptability of albumin involves more than the immediatevicinity of the binding site(s), affecting both the structureand the ligand binding properties of the whole macromol-ecule [6, 12, 14, 15, 44, 50].

The three domains of albumin have different bindingcapacity for a broad variety of ligands such as aminoacids, fatty acids, hormones, metal ions, heme, bilirubin,and drugs. The interaction of ligands with albumin occursat several functionally linked regions as shown in Fig. 2 [12,14, 44–54].

The heme binding cleft and Sudlow’s site I are function-ally and spectroscopically linked. In fact, the affinity of theheme for albumin decreases by about one order of mag-nitude upon warfarin binding (i.e. Kd increases from 1.3 �10�8 to 1.5 � 10�7 M). According to linked functions [1–5],heme binding to albumin decreases the warfarin affinity bythe same extent (i.e. Kd increases from 3.0 � 10�6 to 2.0 �10�5 M). This indicates that drugs binding to Sudlow’s siteI (e.g. warfarin) act as allosteric effectors for heme asso-ciation and vice versa. By contrast, the heme binding cleftand Sudlow’s site II are functionally uncoupled, thus drugsbinding to Sudlow’s site II (e.g. ibuprofen) do not affectheme association and vice versa [12, 14].

Benzodiazepines bind to several functionally and allos-terically linked albumin clefts depending on their opticalconformation and substitution. Allosteric interactions havebeen reported to affect stereoselective binding equilibriabetween albumin and ibuprofen, warfarin, and lorazepam.In particular, binding of either ibuprofen enantiomers toalbumin affects the interaction mode of lorazepam. More-over, lorazepam binding to albumin affects the bindingmode of warfarin enantiomers [15, 55, 56]. Furthermore,allosteric effects have been reported for S-oxazepamhemisuccinate/R-oxazepam hemisuccinate, ibuprofen/S-lorazepam acetate, and L-tryptophan/phenytoin binding toalbumin [17, 57]. Carbamazipine has direct competition

FIG. 1. Comparison of the structures of deoxygenated (yel-low, PDB entry 1MBD) [98] and carbonylated (red, PDB entry1MBC) [99] sperm whale myoglobin. Carbon monoxide isshown as a red stick. A sulfate ion mimicking lactate binding tosperm whale myoglobin is shown in blue. Lactate stabilizes thedeoxygenated form of myoglobin lowering the O2 affinity [8]. Thisand the following figures have been made with Grasp [100].

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with L-tryptophan by binding to Sudlow’s site II, but allos-teric interactions occur with ligands that bind to Sudlow’ssite I, the tamoxifen cleft, and the digitoxin site [58].

Remarkably, NO, NNO, halothane, and chloroform bindto albumin cleft(s), such as Sudlow’s site I, which appearsspectroscopically and allosterically linked to Cys34. It isnoteworthy that NO can modify albumin structure withoutnitrosylation of the Cys34 residue. These findings suggestthat NO may modulate anesthetic binding to albumin andsupport the hypothesis that some physiological effects ofNO result from anesthetic-like noncovalent bonding toproteins [59].

Fatty acids (FAs)1 are effective in the allosteric regulationof the albumin binding properties, especially of Sudlow’ssite I. Myristate regulates the albumin binding properties ina complex manner, involving both competitive and allos-teric mechanisms. The structural changes associated withFA binding can be regarded essentially as relative domainrearrangements to the I-II and II-III interfaces. This allos-teric regulation is not observed for short FAs (e.g. octano-ate) that preferably bind to Sudlow’s site II, displacing thespecific ligands (e.g. ibuprofen) without inducing albuminallosteric rearrangement(s). This indicates that the hydro-phobic interactions between the long FA polymethylenictail and albumin drives allosteric rearrangements. In turn,Sudlow’s site I ligands (e.g. warfarin) displace FA2, andSudlow’s site II ligands (e.g. ibuprofen) displace FA5 andFA6. Moreover, heme binding to albumin displaces FA1[14, 50, 52, 60].

An overview of conformational changes following myris-tate binding to albumin and following heme and warfarinassociation to the myristate-albumin binary complex areshown in Fig. 2. The conformation of the subdomain IA isremarkably affected by binding of myristate and seems tobe mostly unaffected by the presence of additional ligandsbound to the myristate-albumin complex. It should benoticed that the IA-IB interface hosts the heme bindingsite, thus providing an explanation for the allosteric inter-action between heme and FAs. On the other hand, thesubdomain IIB conformation in the myristate-albumin-heme ternary complex is more similar to the ligand-freealbumin conformation than that of the myristate-albumincomplex in the absence and presence of warfarin. More-over, the long �-helix that connects subdomains IIB to IIIAis remarkably tilted, thus affecting both geometry and ste-reoselectivity of Sudlow’s site II [49, 50, 52–54, 60, 61].

Notably, albumin undergoes pH- and allosteric effector-dependent reversible conformational isomerization(s). Be-

1 The abbreviations used are: FA, fatty acid; PAR1, protease-activated receptor 1; PDB, Protein Data Bank.

[61]. B, comparison of the domain I structures, corresponding toresidues 1–204. C, comparison of the subdomain IIB structures,corresponding to residues 306–399. The structures of domain Iand domain IIB refer to ligand-free albumin (blue, PDB entry 1E78)[60], myristate-albumin (red, PDB entry 1BJ5) [52], myristate-albumin-heme (green, PDB entry 1O9X), and warfarin-myristate-albumin (yellow, PDB entry 1H9Z) [61]. The four albumin struc-tures are superimposed by a least-squares procedure using onlythe backbone atoms from Ala175 to Glu204.

FIG. 2. Ligand binding to human serum albumin. A, albuminstructure. The four globular domains are colored as follows: sub-domain IA, cyan; subdomains IB�IIA, green; subdomainsIIB�IIIA, yellow; subdomain IIIB, red. Ligand binding sites areoccupied by warfarin (Sudlow’s site I in subdomain IIA), ferricheme (heme site in subdomain IB), and myristate anion (Sudlow’ssite II in subdomain IIIA). Other ligand binding sites are occupiedby myristate anions. All ligands are rendered with blue sticks.Atomic coordinates are taken from PDB entries 1O9X and 1H9Z

171

tween pH 2.7 and 4.3, albumin shows a fast form, char-acterized by a dramatic increase in viscosity, low solubility,and a significant loss in the �-helical content. Between pH4.3 and 8.0 and in the absence of allosteric effectors,albumin displays the normal form, which is characterizedby the heart-shaped structure. Between pH 4.3 and 8.0in the presence of allosteric effectors (e.g. drugs andlong-chain FAs) and at pH greater than 8.0 in the ab-sence of ligands, albumin changes conformation to thebasic (B) form with the loss of �-helix and an increasedaffinity for some ligands, like warfarin (i.e. Kd decreasesfrom 2.1 � 10�6 M at pH 6.1 to 6.7 � 10�7 M at pH 9.3)[12, 48, 62–67].

As a whole, albumin binds different classes of ligands atmultiple allosterically and spectroscopically linked bindingsites, and the three-domain design of monomeric albuminis at the root of its functional properties which are remi-niscent of those of multimeric proteins.

Na� TRIGGERS ANTICOAGULANT AND PROCOAGULANTACTIVITIES OF HUMAN �-THROMBIN

Thrombin, the key (chymo)trypsin-like serine proteaseresponsible for blood coagulation, vascular development,and signaling [10, 11, 18, 68–74], is a Na�-activated mo-nomeric allosteric enzyme [75]. Thrombin exists in vivo intwo tertiary conformations (i.e. slow and fast forms), show-ing opposite roles. The slow form is anticoagulant becauseit activates more specifically protein C, a potent inhibitor ofblood coagulation. By contrast, the fast form cleaves pref-erentially fibrinogen and the protease-activated receptor 1(PAR1), promoting procoagulant, prothrombotic, and sig-naling functions [9–11, 18, 76–79].

The anticoagulant and procoagulant activities of throm-bin are modulated by Na� binding [75]. Under physiolog-ical conditions, the procoagulant activity of thrombin isintermediate between those of the slow and fast formsbecause the value of the equilibrium constant for Na�

binding (110 mM) is comparable to the plasma Na� con-centration (140 mM); indeed thrombin is 60% bound toNa� [80]. Remarkably, Na� binding to thrombin is requiredfor the optimal cleavage of fibrinogen and PAR1 but not ofprotein C. The Na�-bound thrombin (i.e. the fast form)cleaves fibrinogen and PAR1 with kcat/Km values that are,respectively, 170-fold and 260-fold higher than that ofprotein C. However, the Na�-free thrombin (i.e. the slowform) cleaves fibrinogen and PAR1 with kcat/Km values thatare, respectively, 4.7-fold and 4.4-fold higher than that ofprotein C. Because the enzymatic hydrolysis of protein C isessentially Na�-independent, the anticoagulant activity ofthe Na�-free thrombin reflects an exclusive drop of therates of fibrinogen and PAR1 cleavage. Furthermore, Na�

binding to thrombin enhances the kcat/Km value for thehydrolysis of chromogenic substrates. In turn, the Na�

affinity for the free enzyme (i.e. Kd � 22 mM) is lower thanthat reported for the enzyme:chromogenic substrate re-versible complex (i.e. Kd � 10 mM), whereas the acyl-enzyme formation decreases the Na� affinity (i.e. Kd � 17mM). Support to the procoagulant role of Na�-bound formof thrombin comes from the observation that changes inNa� concentration of plasma resulting in hypernatriemia([Na�] � 145 mM) or hyponatriemia ([Na�] � 135 mM) are

often associated with thrombosis or bleeding. Moreover,genetic defects of prothrombin leading to impaired Na�

binding are associated with bleeding phenotypes [10, 11,18, 75–83].

The Na� binding site of thrombin is located in closeproximity to the primary specificity subsite S1 nestledbetween the 186 and 220 loops that contribute to sub-strate specificity in serine proteinases. The bound Na� iscoordinated octahedrally by the backbone atoms ofArg221A and Lys224 and four buried water molecules,which are anchored to the side chains of Asp189 and

FIG. 3. Comparison of the structures of the slow (red, PDBentry 1SGI) [87] and fast (yellow, PDB entry 1SG8) [87] form ofhuman �-thrombin. The Na�-induced slow to fast transition ismediated by: (i) the formation of the ion pair between Arg187 andAsp222, (ii) the orientation of Asp189 in the primary specificitysubsite S1, (iii) the conformational shift of Glu192 at the entrance ofthe active site, (iv) the orientation of the catalytic Ser195 residue,and (v) the architecture of the water network spanning from theNa� binding site to the catalytic site. The Na� cation is shown asa blue sphere.

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Asp221, and to the backbone atoms of Tyr184A and Gly223

(Fig. 3) [84–87].Amino acid residues Asp189, Glu217, Asp222, and Tyr225,

all in close proximity to the bound Na�, form the allostericcore of thrombin and are responsible for the enhancementof the catalytic activity following monovalent cation bind-ing. These residues, together with Thr172, Arg187, Tyr184A,Ser214, and Gly223, link the Na� binding site to the primaryspecificity subsite S1 and the S3-S4 specificity pocket. TheNa�-induced slow to fast transition is mediated by: (i) theformation of the ion pair between Arg187 and Asp222 con-necting the 186 and 220 loops that define the Na� bindingsite, (ii) the orientation of Asp189 in the primary specificitysubsite S1 for electrostatic coupling with the P1 Arg resi-due of the substrate, (iii) the conformational shift of Glu192

at the entrance of the active site, (iv) the orientation of thecatalytic Ser195, and (v) the architecture of the network ofwater molecules spanning �20 Å from the Na� bindingsite to the catalytic site. In the thrombin slow form (i.e.Na�-free), the orientation of Glu192 away from the activesite region compensates changes around Asp189 andSer195, reducing the electrostatic clash with the P3 and P3�acidic residues of protein C. This explains the intrinsicanticoagulant nature of Na�-free thrombin. The changes inthe allosteric core and in the water network explain thethermodynamic and kinetic signatures linked to Na� bind-ing and mechanism of thrombin activation by Na� [18, 87].

Na� activation is also observed in other enzymes in-volved in blood coagulation and in the immune responsebut not in digestive and degradative (chymo)trypsin-likeserine proteases. Notably, thrombin-like procoagulant ef-fects of Na� have been reported for Factor Xa. The Tyr225

residue plays a crucial role in determining the Na�-de-pendent allosteric nature of serine proteases by allowingthe correct orientation of the backbone O atom of theresidue 224, which contributes directly to the coordinationof Na�. Moreover, the side chain of Tyr225 secures theintegrity of the water channel embedding the primaryspecificity subsite S1, which is required for the correctsubstrate recognition [83, 86, 88, 89].

Ligand binding to the thrombin exosites I and II, flankingthe catalytic groove, also induces conformational changesin the active site and modulate allosterically the enzymeactivity [10, 11, 18, 76–79]. Binding of thrombomodulin, anintegral membrane protein found on endothelial cells es-pecially in the microcirculation, to thrombin exosite I inhib-its fibrinogen cleavage, promotes rapid activation of pro-tein C, increases the rate of thrombin inhibition by thebovine basic pancreatic trypsin inhibitor or the protein Cinhibitor, and increases the rate of activation of the throm-bin-activable fibrinolysis inhibitor [78, 90–93]. Moreover,binding of heparin and glycosaminoglycans to exosite IIgreatly increases the rate of enzyme inactivation by anti-thrombin and heparin cofactor II [94–96]. Interestingly,ligand binding to exosite I and exosite II does not play asignificant role in the allosteric modulation mediated byNa� [87].

As a whole, the Na� binding site, exosite I, exosite II, andthe active site of thrombin are allosterically linked. Ligandbinding modulates thrombin activity, which has targeteddistinct tertiary conformational states toward its two fun-

damental and competing roles in hemostasis, i.e. proco-agulant and anticoagulant [9–11, 18, 76–79, 87].

CONCLUSIONS

Although tetrameric hemoglobin is considered to be theprototype of allosteric proteins, monomeric sperm whalemyoglobin, human serum albumin, and human �-thrombinrepresent invaluable model systems for teaching allostery.Indeed, as stated by Di Cera et al. [76], the allosteric mech-anism modulating thrombin function(s) represents one of thesimplest and important structure-function correlations everreported for monomeric enzymes in general.

Last but not least, the creation of monomeric allostericenzymes has been reported by covalent linkage of nonin-teracting and nonallosteric monomeric proteins with theprerequisite effector-binding and catalytic functionalities,respectively. With the combinatorial process called “ran-dom domain insertion,” the fragment of the TEM-1 �-lac-tamase gene coding for the mature protein lacking itssignal sequence was randomly inserted into the Esche-richia coli maltose-binding protein gene to create a domaininsertion library. From a library of �2 � 104 in-frame fu-sions, two allosteric enzymes coupling maltose bindingand nitrocefin hydrolysis were identified. Remarkably,maltose binding to the regulatory domain of these bifunc-tional fusions determines a conformational change(s) thatis transmitted to the catalytic domain modulating the rateof nitrocefin hydrolysis [97].

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APPENDIX

The reaction mechanism, in terms of association equi-librium constants, which describes linked functions

(Scheme 1) [1–5], considers a protein, P, that can bind twodifferent ligands L1 and L2, each at a distinct binding site.Because each equilibrium is assumed to be rapid and notindependent of the others, according to Equation 1, itfollows that [1–5]:

K� � KB � [L2PL1]/([P] � [L1] � [L2]) � KA � K� (Eq. 1)

if K� � K�, then KA � KB; and if K� � K�, then KA � KB. Inother words, thermodynamics requires that if the bindingof component L1 is affected by the addition of componentL2, then there must be a reciprocal effect of added L1 onthe interaction of L2 with P. Moreover, if the two bindingsites are functionally unlinked, then K� � K� and KA � KB.

Taking into account all the species involved in Scheme1, it is possible to describe the variation of the L2-depend-ent apparent equilibrium constant K� (between the proteinP and the ligand L1) as well as its reciprocal effect, repre-sented by the change of the L1-dependent apparent equi-librium constant K (between the protein P and the ligandL2) by Equations 2 and 3 [1–5]:

K� � ([PL1] � L2PL1�)/� P� � L2P�� � L1��

� K� � � 1 � KB � L2��/ 1 � KA � L2��� (Eq. 2)

K � L2P� � L2PL1��/� P� � PL1�� � L2��

� KA � � 1 � K� � [L1��/ 1 � K� � L1])} (Eq. 3)

Values of K�, K�, KA, and KB relevant to Scheme 1 may beeasily obtained from plots of K� versus [L2] (according toEquation 2) and of K versus [L1] (according to Equation 3).When P is ligand-free, then K� � K� (according to Equation2) and K � KA (according to Equation 3), and when P is L2

and L1 saturated, then K� � K� � K� � KB/KA (according toEquation 2) and K � KB � KA � K�/K� (according toEquation 3), respectively [1–5].

The reaction mechanism, in terms of association equi-

SCHEME 1

SCHEME 2

175

librium constants, which describes linked functions forenzyme action (Scheme 2) [1–5], considers the enzyme P,the substrate L1, the allosteric effector L2, the reversibleenzyme:substrate complex PL1, the reversible allostericeffector:enzyme:substrate complex L2PL1, and the reac-tion product(s) Pr, k� and k� as the rate-limiting step incatalysis for the conversion of PL1 and L2PL1 to P � Pr andL2P � Pr, respectively. Because each equilibrium is as-sumed to be rapid and not independent of the others, it is

possible to describe the variation of the L1- and L2-de-pendent apparent initial velocity (vi) by Equation 4 [1–5]:

vi � k � �L1� � KB–1 � k� � L2�/k���/�L1� � L2�

� K�/K� � KA–1 � L1�

� K�–1 � L2� � K�

–1 � KA–1�� (Eq. 4)

where k (�Vmax/[P]) is the apparent catalytic constant.

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