Supporting Information. Differential Binding Models for Direct and Reverse Isothermal Titration Calorimetry

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1 Supportng Informaton Dfferental Models for Drect and everse Isothermal Ttraton alormetry Isaac Herrera, Mtchell A. Wnnk* hemstry Department, Unversty of Toronto, 8 St. George Street Toronto ON, anada, M5S 3H6 1. Dfferental Models for Drect and everse alormetrc Ttratons of Homotropc Systems. Dfferental bndng models (DMs) are dfferental equatons wth general form d[s]/dξ where [S] represents the concentraton of the bndng speces n the ttraton cell and ξ the progress of the ttraton experment. DMs can be appled to calormetry ttratons when the sgnal s evaluated as the rato between the dfferental heat and a varable representng the amount of ttrant added dq/dξ. ecently, we presented a theoretcal DM for drect calormetry ttratons of receptors wth multple stes for one class of lgand speces, or homotropc bndng, where we assumed a constant receptor concentraton to smplfy the dervaton. 1 Ths strategy allowed us to evaluate the propertes of a calormetry ttraton curve, and more mportantly, to compare our theoretcal model to other dfferental models also derved wth the assumpton of a constant receptor concentraton, such as the Wseman sotherm and others. 5 The prevous models can be appled to drect calormetry ttratons after the concentratons of the bndng speces are corrected for any dluton effects. However, we could not fnd a satsfactory answer to the followng queston: How can a model derved wth the assumpton of a constant receptor concentraton be later appled to a calormetrc ttraton where both receptor and lgand concentratons vary? To answer ths queston, we retraced our dervaton steps and obtaned dfferental equatons as a functon of the total volume of ttrant njected to represent the dluton of both ttrant and analyte speces. S1

2 We follow four general steps to derve dfferental bndng models for calormetrc ttratons. Frst, we use equlbrum reactons, bndng constants, mass-balance equatons, and bndng polynomals to descrbe a bndng system. Second, we derve exact dfferental equatons from the mass-balance expressons to represent the changes n concentraton of both free and bound speces n the ttraton mxture n terms of the amount of ttrant added. We refer these expressons as dfferental bndng models (DMs). Thrd, we constran the DMs for the drecton of the ttraton and the type of ttraton cell. Fnally, we defne expressons that relate the heat sgnal of IT experments to the DM by consderng the heat and mass balances of the bndng speces n the calormeter. These steps are descrbed below n more detal Polynomal and Mass-alance Expressons for a Homotropc System One of the most general models for a receptor-lgand ttraton s a homotropc bndng model, whch descrbes an equlbrum reacton of a receptor () wth multple bndng stes for one type of lgand () speces, as gven by [ ] +, β = (S1) [ ][ ] where β s the cumulatve bndng constant for a receptor bound to lgand speces. The bndng polynomal P descrbes the dstrbuton of receptor speces from the lowest saturaton state to the hghest saturaton state N, usng the cumulatve bndng constants and the concentraton of free lgand [] n soluton as gven by P N = β [ ] (S) = The dervatves of the bndng polynomal wth respect the free lgand descrbe mportant statstcal and bndng propertes of the system such as the average number of lgand speces bound to the receptor ( N ) and the bndng capacty ( ). 6 The expressons for gven by N and are S

3 N β[ ] ln P [ ] P = = = P P = ln[ ] [ ] N (S3a) ln P N N (ln[ ]) ln[ ] [ ] = = = [ ] N N β[ ] β[ ] = = = P P (S3b) In a prevous publcaton, we evaluated the theoretcal propertes of eqs S3a and S3b n a ttraton experment for receptors wth one and two bndng stes (.e., N = 1, ). 1 A detaled explanaton of the dervatves of the bndng polynomal can also be found n ref. 6. The macroscopc propertes of the ttraton experment can be expressed n terms of the bndng polynomal P and ts dervatves. For nstance, the mass-balance equatons for the total receptor concentraton ( ) and total lgand concentraton ( ) are gven by = N = [ ] = [ ] P (S4a) = N = [ ] + [ ] = [ ] + N (S4b) = whle the mole fracton α, whch represents the normalzed probablty of fndng a receptor bound to lgand speces at a gven free lgand concentraton, s gven by [ ] [ ] β α = = (S5) P 1.. Dfferental Models for Homotropc Systems. To derve dfferental bndng models, we apply the mplct functon theorem to the massbalance expressons n eqs S4 and obtan a system of dfferental equatons wth the general form d[s]/dξ to represent the change n the concentraton of the bndng speces wth respect to the change n the ndependent varable(s) that represents the progress of the ttraton experment. To begn, we defne a total explct functon for the lgand speces (F ) n terms of the dependent ([]) and the ndependent varables (, ) for the ttraton usng the mass balance expresson n eq S4b S3

4 F ([ ],, ) = [ ] + N = (S6) where both and are nested functons of the volume of ttrant njected V (e.g., = f(v)). To contnue, we derve exact dfferentals for the dependent varables n the ttraton. For nstance, the exact dfferental for the free lgand s gven by [ ] [ ] d[ ] = d + d (S7) and the partal dfferental terms n eq S7 are obtaned by applyng the mplct functon theorem to eq S6, as gven by [ ] ( F ) 1 ( F [ ] [ ] [], = = ), 1 + [ ] ( F ) N F [ ] [], = = ( []), 1 + (S8a) (S8b) The term /[ ] n eqs S8a,b s obtaned from eq S3b and corresponds to the partal dervatve N / [ ]. Evaluatng the term /[ ] s mportant when calculatng the concentraton of free lgand by numercal ntegraton (Secton ). y combnng eqs S7 and S8, we obtan a general soluton for the exact dfferental d[] d N d d [ ] = (S9) 1 + / [ ] Smlarly, we derve the total dfferentals for the free and bound receptor speces (.e., d[ ] wth ) by applyng the mplct functon theorem to the total functon F ([ ],, ) derved from eq S5, where = f([], ). As prevously shown durng the dervaton of our theoretcal model, 1 the exact dfferentals can be obtaned by applyng the chan rule to eq S5, whch gves dα = α ( N ) [ ] d [ ] (S1a) d[ ] = dα + α d (S1b) to represent the dfferental change n the mole fracton (dα ) and the concentraton of a receptor bound to lgands (d[ ]), respectvely. Then, by combnng eqs S9, S1a, and S1b, we obtan an exact dfferental expresson to evaluate d[ ] n terms of d and d, as gven by S4

5 [ ] [ ] d[ ] = d + d (S11) where the partal dfferental terms are gven respectvely by [ ] α( N) = [ ] + (S1a) [ ] N ( N) = α 1 [ ] + (S1b) Usng eqs S7 and S11, we obtan a general expresson to calculate the change n concentraton of both free and bound speces ([S]) n the ttraton mxture of a homotropc bndng system as a functon of any ndependent varable ξ representng the progress of the ttraton experment. d[ S] [ S] d [ S] d = + dξ dξ dξ (S13) For example, the total lgand concentraton could be used n place of ξ for a drect or lgandto-receptor ttraton (T), whle could be used for a reverse or receptor-to-lgand ttraton (T). We prefer to descrbe a more generc ttraton experment by usng the total volume of ttrant njected V n place of ξ (eq 9, man text) d[ S] [ S] d [ S] d = + dv dv dv (S14) whch can be appled to drect, reverse, or any other ttraton setup after constranng the dluton expressons for the lgand (d /dv) and receptor (d /dv) speces Dfferental Dluton Expressons for Ttrant and Analyte Speces The dfferental dluton expressons for the lgand and receptor speces n a calormetry ttraton depend on the type of ttraton cell, the drecton of the ttraton, and the njecton protocol. The two most common desgns for cells n a ttraton calormeter are an overfll, or perfuson cell, 7 9 and a partal-fll cells (Fgure S1). In an overfll cell, materal flows both n and out of the ttraton cell, whle n a partal fll cell materal flows only nto the cell. S5

6 Fgure S1. Ttraton cells used n sothermal ttraton calormetry: (a) Overfll cell: the njecton of ttrant (ΔV n ) dsplaces an equal volume of the ttraton mxture (ΔV out ) from the cell. In thermodynamc terms, the system s open to the flow of materal both n and out of the reacton cell. (b) Partal-fll cell: the njecton of ttrant (ΔV n ) ncreases the ntal volume of the ttraton mxture (V ) by V = V + ΔV n. To derve the dluton expressons for d and d, we use a generc notaton for a substance S wth an ntal concentraton n the cell of S, a concentraton n the syrnge of S, and a total concentraton n the ttraton cell S. After dervng a general dluton expresson, we set substance S as ether the analyte A, the ttrant, or any other background speces. Fnally, we apply boundares to the dluton dfferentals by usng the ntal concentratons used for the bndng speces Dluton Expressons for an Overfll Ttraton ell In an overfll type calormeter (Fgure S1a), the soluton of S flls the ttraton wth a constant volume V (hatched area) and the connectng port. The syrnge serves for two purposes: to delver the soluton of ttrant and to str and homogenze the ttraton mxture. The Teflon rod decreases the nose generated by the strrng mechansm but also prevents the homogenzaton between the solutons n the cell and the connecton tube durng the ttraton. As a result, the njecton Vn wth concentraton area) from the ttraton mxture at concentraton S causes an equvalent volume dsplacement out V (crossed S. We use a sngle varable ΔV to represent S6

7 both Vn and Vout. Under these assumptons, the mass-balance equaton for the moles of S durng the stepwse njecton of ttrant n a constant-volume ttraton cell are gven by where S n = n + n n (S15) n out S S S S n corresponds to the ntal moles, n n to the moles njected, and n out to the moles S ejected. Smlarly, the total concentraton when substance S s njected sequentally nto an overfll ttraton cell s gven by S S n V = = + V (S16) V V V S S S S n In eq S16, the term S V V s the effectve concentraton njected ( S ) whle the term S V V s the effectve concentraton ejected from the ttraton cell ( out S ). We obtan a general expresson for the change n concentraton of substance S when a small volume of ttrant s njected n relaton to the cell volume V by evaluatng eq S16 n the lmt ΔV, as gven by d = d d = dv (S17) S n out S S S S V whch s equvalent to the dluton expresson of substance S n a contnuously strred tank reactor (ST) operatng wth a sem-batch addton of reactants. 1 From eq S17, we obtan the dluton expressons for any ttraton experment by settng the ntal concentratons for analyte A and ttrant n both the syrnge and the ttraton cell. For a typcal bndng experment, ttrant s njected nto analyte A. In ths setup, the ntal concentratons n the ttraton cell corresponds to A for the analyte and = for the ttrant. Smlarly, the ttrant concentraton n the syrnge corresponds to whle the analyte concentraton n the syrnge s =. After applyng these constrants to eq S17, we obtan an A mplct soluton to the dfferental dluton expressons for the analyte and ttrant speces (eq 1 of man text), as gven by d dv A V A = (S18a) d = (S18b) dv V S7

8 Eqs S18a and S18b can be ntegrated algebracally from V = to V usng the ntal concentratons A and = for the analyte and ttrant speces, respectvely, to gve exp V V A = A ( / ) = (1 exp( V / V )) where the rato between eqs S19b and S19a corresponds to the extent of ttraton (Φ A ) 11 A A (S19a) (S19b) Φ = ( exp( V / V ) 1) / (S) Fnally, after substtutng eqs S19a,b nto eqs S18a,b, we obtan the explct soluton to the dfferental dluton of analyte and ttrant speces da A e VV = dv V (S1a) d VV = e dv V (S1b) Addtonal explct solutons can be obtaned by applyng the respectve boundary condtons to eq S17. For example, a substance S that s ntally present n both the syrnge and the ttraton cell, such as the buffer or a background on, can be evaluated wth the expresson d e dv V S S S VV = (S) whle a dluton ttraton, 1 where both lgand and receptor speces are ntally present n the syrnge, but not n the ttraton cell, s represented wth the expresson d dv V S S VV = e (S3) Dluton Expressons for a Partal-Fll Ttraton ell Usng a smlar procedure, we can derve complementary dluton expressons for the speces n a partally-flled ttraton cell (Fgure S1b). Frst, we assume that a known volume (V, hatched area) of substance S wth concentraton S s placed n the ttraton cell. Then, the volume njected (ΔV) wth concentraton S ncreases the total volume of ttraton mxture (.e. V = V + ΔV) and the total concentraton S of substance S n the cell, as gven by S8

9 S V S + S V = (S4) V + V We obtan a general expresson for the dluton of S n a partal-fll ttraton cell by dfferentatng eq S4 wth respect to the total volume njected, as gven by d S ( S S) V ( V + V ) = dv (S5) and after applyng the ntal boundary condtons to eq S5, we obtan the dluton expressons for a partal-fll ttraton cell where the ttrant s njected nto analyte A, as gven by da AV = dv ( V + V ) (S6a) d dv V = (S6b) ( V +V ) Expressons equvalent to eqs S6a and S6b have been derved by others 13 and should only be appled to ttraton setups where the total volume of the ttraton mxture ncreases wth each addton of ttrant General Features of the Dfferental Dluton Expressons The relatve change n the total concentraton of analyte and ttrant speces s gven by the dfferental d A /d. Interestngly, ths dfferental expresson gves a constant value n both partal-fll and overfll type calormeters, as gven by d d A = (S7) whch means that the relatve change n concentraton between the analyte and ttrant speces s lnearly dependent. Eqs S and S5 show that the dluton expressons d S /dv are equal to zero when the ttrant and analyte solutons have the same concentraton; and hence, the concentraton remans constant. Although ths observaton mght seem trval, t hghlghts the mportance of preparng ttrant and analyte solutons wth matchng concentratons of background speces such as protons, buffer, counter ons, and compettve lgands. 14 As a result, fewer partal dfferental A S9

10 terms are evaluated n the general form of eq S14 when the dluton terms for the background speces are canceled out Heat alance Expressons for a alormeter wth an Overfll Ttraton ell The total heat content (q tot ) n a perfuson type calormeter 15 can be expressed as the sum of the heat content of the speces njected (q n ), the heat of reacton for the speces n the ttraton cell (q cell ), the heat of dluton (q d ), and the heat content of the speces that flow out of the cell (q out ). The expresson to evaluate q tot n a closed thermodynamc system that encompasses both the syrnge and the ttraton cell s gven by qtot = qn + qcell + qd qout (S8) Each term n eq S8 can be wrtten as a weghted sum of the concentraton, or moles, of each substance and ts cumulatve enthalpy of bndng ΔH. For example, the heat of reacton for the ttraton of a homotropc bndng system (eq 5, man text) s gven by N q = V H [ ] (S9) cell = 1 The sgnal measured n IT s proportonal to the change n the heat content of the ttraton cell or dfferental heat dq. Most calormeters used n IT nclude an overfll or perfuson cell where the heat compensaton sensors are only n contact wth the ttraton cell. Ths means that the heat for the materal flowng out of the cell cannot be measured (q out = ). 16 Thus, the dfferental heat n a calormeter wth an overfll ttraton can be evaluated term-by-term wth the expresson. dqobs = dqn + dqcell + dqd (S3) In eq S3, there are two alternatves to evaluate the term dq n. Frst, we could assume that the ttrant speces are n a free or unbound state as shown for the reactants n the equlbrum reacton of a homotropc bndng system (eq S1). y defnton, free or unbound speces exhbt enthalpes equal to zero. As a result, we set dq n = when evaluatng calormetry ttratons of receptor-lgand speces wth homotropc nteractons. As an alternatve, we could evaluate dq n wth the followng expresson when bound speces n the syrnge dssocate upon njecton. N b dqn = V H j ( d [ j ] d [ j ] ) (S31) j= 1 S1

11 In eq S31, d[ j ] represents the ntal concentraton of the j-th ttrant speces and d[ j ] s ther correspondng equlbrum concentraton n the ttraton cell. The term d[ j ] s proportonal to the mole fracton of the j-th bound speces n the syrnge ( α The terms α j and j ) and ts total concentraton. reman constant durng the ttraton experment. The term d[ j ], however, s lkely dependent on the equlbrum concentraton of the bound speces n the ttraton cell (.e., d[ j ] ~ g(ξ) d[ ], where g(ξ) s a proportonalty functon dependent on the progress of the ttraton). Eq S31 can be used to evaluate the heat sgnal n a dluton ttraton 1 or to evaluate the heat contrbuton for speces that release protons after njecton. 17,18 The major contrbuton to the heat measured wth eq S3 s dq cell. The term dq cell accounts for changes n the heat content of the ttraton cell due to the equlbrum reacton between the receptor and lgand speces, as gven by N dq = V H d[ ] (S3) cell = 1 After substtutng eqs S11 and S17 nto eq S3, we observe that the term d[ ] ncludes contrbutons from the ttrant njected and the overflown ttraton mxture, as gven by N [ ] n out [ ] n out dqcell = V H ( d d ) + ( d d ) (S33) = 1 We apply two constrants to eq S33. Frst, we elmnate the contrbuton from the materal flowng out of the ttraton cell by cancelng out the terms out d and out d. Second, we assume n that the concentraton of analyte n the syrnge s zero ( = and d = ). After applyng these constrants, we obtan the general soluton to evaluate the heat sgnal for drect and reverse ttratons of homotropc bndng systems n overfll ttraton cells d q N A cell = H = 1 A [ ] dv (S34) Eq S34 s equvalent to the dfferental heat per volume of ttrant (eq 1, man text) after evaluatng the rato between δ q cell A and the volume of ttrant njected dv A A S11

12 d q [ ] dq H (S35) A N A cell V = = dv = 1 In IT, most ttraton curves are plotted usng the mole-normalzed values of the dfferental heat (δh = δq/δn) aganst the mole rato between ttrant and analyte speces (Φ A, eq S). The term δn corresponds to the product of the ttrant concentraton n the syrnge and the volume njected (.e., d n = dv ). Thus, the expresson for the dfferental heat per mole δh n terms of δq V (eq 15, man text) s equvalent to δ H N A = H = 1 A [ ] (S36) We use the nexact dfferental notaton n eqs S34-S36 to represent the dependence between the heat measured and the path or ttraton drecton used to evaluate the bndng system. More mportantly, we would lke to emphasze that the heat sgnal n a calormeter wth an overfll ttraton cell, a thermodynamcally open system, should not be evaluated wth an exact dfferental d[ ]. Groler and del ío 19 presented a detaled dervaton of the dluton and the heat balance expressons for a calormeter based on the nexact dfferental formalsm. Addtonal heat contrbutons that result from the njecton of ttrant, mechancal strrng, or heat leakage are all taken nto account n eq S3 wth the dfferental heat of dluton term dq d. Ideally, the dfferental heat of dluton s determned n a background ttraton experment where the ttrant s njected nto a buffer soluton. There are three alternatves to evaluate the heat of dluton n a ttraton calormetry experment. The frst alternatve s to assume that the dfferental heat of dluton δq d depends on ether the volume added at a gven njecton pont (dv) or the total concentraton of ttrant, as gven by d qd dv d A = h (S37) We recommend usng eq S37 for ttraton experments where the background ttraton of ttrant nto buffer shows constant values for δq, whch s the case for ttratons wth matchng concentratons of background ons (for an example, see ref. 1). The second alternatve could be appled to experments where the background ttraton has a lnear response n relaton to the total volume njected, as gven by S1

13 d qd dv ( ) = a+ bv (S38) where a s equvalent to Δh d and corresponds to the lnear ntercept of the fttng curve at V =, and b s the slope of dfferental heat n the background ttraton (for an example, see ref. 16). A thrd opton s to evaluate the dfferental heat of dluton usng the dfferental term for the change n concentraton of free ttrant speces and the molar heat of dluton, as gven by A d qd [ ] [ ] = hd dv A (S39) In eq S39, the term ( []/ ) A corresponds to eq S8a for a drect or lgand-to-receptor ttraton, and to eq S1b wth = for the reverse ttraton (.e., ( []/ ) ). In Fgures S5E,D and S7E,D as well as Fgures A, of the man text, we observe that the curves for the partal dfferental terms ( []/ ) and ( []/ ) exhbt sgmodal behavour. Hence, we recommend applyng eq S39 for ttraton experments n whch the njecton of ttrant nto buffer has a nonlnear response.. Numercal Integraton Approach to Evaluate the Dfferental Model for a Homotropc System We calculate the concentraton at equlbrum of the free lgand [], the mole fractons (α ), and the total concentraton for the receptor and lgand speces (, ) n the ttraton of a homotropc bndng system by numercally ntegratng the DMs wth respect to the total volume njected V. For ths purpose, we represent the DM as a system of coupled ordnary dfferental equatons (ODEs) and apply the unge-kutta-fehlberg algorthm ncluded n the software Igor Pro (Wavemetrcs Inc., ake Oswego, O, USA) to ntegrate the ODE from an ntal value at V =. We nclude the dfferental terms d[], dα, and d[ ] for = {,, N }, (eqs S9, S1a and S1b, respectvely) n the defnton of the coupled ODEs, as gven by S13

14 d [ ] dv = d d N dv dv 1 + / [ ] (S4a) dα α( N) d [ ] = (S4b) dv [] dv d [ ] d d dv = α α dv + dv (S4c) where the dluton expressons d /dv and d /dv can be adjusted to represent any type of ttraton cell used or ttraton drecton (e.g., eqs S18 or S6). For example, a drect ttraton n an overfll cell (Fgure S1a) where receptor s njected nto a lgand soluton (T) s represented wth the dluton expressons d d e dv V dv V VV VV =, = (S41a) whle the reverse ttraton n the same type of ttraton cell s represented by e d d dv V dv V VV VV = e, = e (S41b) We nclude the dfferental terms dα /dv and d[ ]/dv n the defnton of the ODE (eq S7) for completeness. However, the ntegrated values for α and [ ] were calculated wth eq S5 usng the ntegrated values of [] and. The numercal ntegraton approach replaces the steps shown n our prevous manuscrpt to calculate the dluton-adjusted concentratons of and usng eqs S19a-b, and to calculate the equlbrum concentratons of free lgand at every njecton step usng a root-fndng algorthm. The system of coupled ODEs s ntegrated numercally usng ntal values for each dfferental term and a set of guess values for the cumulatve bndng constants β = {β,, β N }. The ntal values correspond to the concentratons and mole fractons for the bndng speces at V =. For example, the ntal values for a drect or T ttraton are defned as = [ ] =, = [ ] =, [ ] =, > a = 1, and a = > (S4) In other words, the ntal concentratons and mole fractons of the lgand-contanng speces are zero, whle the free receptor concentraton s equvalent to the ntal concentraton of analyte S14

15 placed n the ttraton cell. Ths also means that the mole fracton of free receptor s one. Smlarly, the ntal values for a reverse or T ttraton are defned as [ ], [ ], [ ] = = = = > = (S43) where the ntal values for mole fractons n the lmt of are evaluated wth the expresson α = β ( ) N β ( ) = (S44) For the T ttraton, n partcular, we requre numercally stable expressons for the terms /[ ] and α ( N )/[ ] n eqs S4a and S4b, respectvely, that can be evaluated n the lmt of []. The numercally stable expresson for /[ ] s gven by N N 1 1 [ ] [ ] β β = = = [ ] [ ] P P (S45) In the same way, the numercally stable expresson for the term dα /d[] wth > n eq S4b s gven by 1 ( ) dα ( ) [ ] α N β N = = (S46) d [ ] [ ] P whle the numercally stable expresson for the term dα /d[] s gven by N 1 β[ ] dα = = (S47) d [ ] ( P ) To demonstrate the use of eqs S4a-c n the analyss of bndng assays between receptor and lgand speces, we smulated cumulatve and dfferental ttratons for receptors wth one bndng ste as shown n Fgure S. Frst, we selected the followng expermental parameters to provde a c-value of 1: =.5 mm, = 5. mm, V = 1.4 m, and a bndng constant β 1 = x 1 6 M 1. Addtonally, we assumed an overfll ttraton cell; hence, the concentraton of lgand n the syrnge ( ) could be calculated wth an expresson derved from eq S to ensure that S15

16 approxmately.4 mole equvalents of lgand (Φ T.4) are present n the ttraton cell after njectng the volume of ttrant avalable n the syrnge (.e., V = 3 µ), as gven by Φ exp 1 T = ( VV) (S48) Fgure S. Smulated ttraton curves of the cumulatve and dfferental ttratons for a receptor wth one bndng ste n an overfll type cell. The bndng polynomal for the receptor s P = 1+ β 1 [] wth β 1 = x 1 6 M 1, and the ttraton parameters are =.5 mm (c = 1), = 5. mm, V = 1.4 m. (A) umulatve ttraton curves for a receptor wth one bndng stes. The dstrbuton of bndng speces as a functon of the volume njected are calculated numercally by ntegratng the system of ODEs n eq S4a-c. () Dfferental bndng curves for the total d[s]/d and the partal dfferentals ( [S]/ ). The total dfferental curves were obtaned by S16

17 evaluatng eq S13 wth ξ = and the ntegrated values shown n Fgure SA for [] and. The partal dfferental curves ( [S]/ ) were obtaned by evaluatng eqs S8a and S1a wth the ntegrated values shown n Fgure SA for [] and Fgure SA shows the cumulatve ttraton of a receptor wth one bndng ste wth ntegrated values for the bndng speces [] and [], and total concentratons and as a functon of the total volume of ttrant njected V. In ths smulaton, the receptor s nearly saturated at the equvalence pont of the ttraton (Φ T = 1.) and the hyperbolc curve of [] and has an upper lmt defned by at Φ T > 1.. The total lgand concentraton can be evaluated as a functon of Φ T wth an expresson obtaned from eqs S48 and eq S19 ΦT = (S49) 1 +Φ ( ) T where the term s equvalent to the dluton rate obtaned n eq S7. Usng eq S49, we Sr y calculate the total concentratons of lgand speces at the equvalence pont to be =.455 mm, whch represents nearly 1% dluton from the ntal receptor concentraton. The dfferental ttraton of a receptor wth one bndng ste s shown n Fgure S. The total dfferentals for d[s]/d corresponds to eq S13, where the varable representng the progress of the ttraton s set as ξ = and the dfferental d /d corresponds to eq S7. The partal dfferentals ( [S]/ ) corresponds to eq S8a for the free lgand and to eq S1a for the bound receptor. The expressons for the dfferentals d[s]/d and ( [S]/ ) were evaluated usng the ntegrated values for [] and. The curves for the total dfferentals d[s]/d account for the combned effect of bndng speces that flow n and out of the ttraton cell. Ths means that n a drect ttraton the partal dfferentals ( [S]/ ) account for the change n the concentraton for the bndng speces nsde the ttraton cell, whle the partal dfferentals wth the general form ( [S]/ ) account for the materal flowng out of the ttraton cell. The curves for the partal dfferentals ( [S]/ ) are not shown n Fgure S, but they correspond to the dfference between the total and partal dfferentals (.e., ( [S]/ ) = d[s]/d ( [S]/ ) ). As a result, the curve for d[]/d has values lower than zero once the receptor approaches full saturaton at approxmately Φ T = 1.5. At ths pont n the ttraton, the change n the concentraton of s lower than the absolute value of the dluton rate (eq S7). On the other hand, the dfferental S17

18 bndng curve for ( []/ ) tends towards zero when no addtonal bndng speces are formed nsde the ttraton cell. In IT, the ttraton curves of the dfferental heat per volume (δq V ) or the dfferental heat per mole (δh) n a calormetry ttraton of a receptor wth one ste reach values close to zero for experments wth matchng concentratons of the buffer or any addtonal background speces. Hence, IT experments for receptors wth one ste should be evaluated usng the dfferental bndng curve for ( []/ ), as derved wth eq S35 and S36. We prepared an EXE fle that uses AGI, 1 an open-source lbrary, to numercally ntegrate the ODE defned n eq S4a and to smulate drect and reverse ttraton experments of receptors wth homotropc bndng nteractons and N bndng stes. Ths EXE fle s avalable upon request. 3. omparng Fnte Dfference and Dfferental Models In ths secton, we present our ratonale for usng dfferental bndng models to evaluate sothermal ttraton calormetry experments. Ths explanaton s necessary because the fnte dfference approach has been consdered by some as the only sutable opton to evaluate ttraton calormetry experments.,3 We beleve both fnte and dfferental models are equally vald alternatves to evaluate ttraton experments, where each numercal approach dffers n the assumptons to account for the dluton of the bndng speces and to evaluate the change n the heat content of the ttraton cell. To evaluate the dluton of the bndng speces, most fnte dfference models assume an nstantaneous njecton wth the term 1 ΔV p /V. 3 In comparson, dfferental models assume nstantaneous mxng wth the term exp( V/V )(eqs S19a and S19b). In our experence, both expressons provde accurate values when the volume njected s small n relaton to the cell volume. For example, the dluton coeffcent calculated for the 3 th njecton n a ttraton experment, where 1 µ alquots are njected nto a 14 µ ttraton cell, corresponds to (1 1 µ/14 µ)^ Smlarly, the dluton coeffcent calculated wth a dfferental expresson s exp( (3 x 1 µ)/14 µ) The dfference between the dluton coeffcents s approxmately.61, or.77%, and ths dfference decreases when smaller volumes are njected at every ttraton step. Hence, both fnte and dfferental approaches provde accurate results for the dluton of the bndng speces when usng a typcal ttraton setup. S18

19 To evaluate the change n the heat content of the ttraton cell Δq p, fnte dfference models use the expresson shown n eq 6 of the man text N q = V H [ ] (S5) p p = 1 where the term Δ[ ] p s typcally calculated usng ether the backward dfference formula or the mdpont formula 4 gven by [ ] = [ ] [ ] + ( V V )[ ] (S51) p p p 1 p p 1 V [ ] + [ ] p p p 1 [ ] p = [ ] p [ ] p 1 + V (S5) In both representatons of the fnte dfference model, the term [ ] p [ ] p 1 mples that the concentraton changes lnearly between p and p 1, whle the term wth the coeffcent ( V V ) accounts for the materal flowng out of the ttraton cell. p The term Δ[ ] p n eq S5 can also approxmated wth a dfferental expresson. For ths purpose, we frst represent the concentraton [ ] p as functon of the total volume njected or f(v). The concentraton [ ] p-1 for the precedng njecton step s equvalent to f(v ΔV P ) and can be evaluated wth the Taylor expanson 3 Vp Vp f( V Vp) f( V) Vp f ( V) = + f ( V) f ( V) + (S53) 3! After rearrangng eq S53 and cancelng the hgher order dfferentals, we obtan the frst order approxmaton d[ ] [ ] p = f( V) f( V Vp) Vp (S54) dv As shown n the dervaton of eqs S33, the soluton for the exact dfferental d[ ] n eq S54 can be obtaned by combnng eqs S11 and S17 to gve [ ] n out [ ] n out d[ ] = ( d A d A ) + ( d d ) (S55) A where the exact dfferental reduces to the nexact dfferental A S19

20 d [ ] [ ] = dv V A (S56) after cancellng the contrbuton of the materal flowng out of an overfll ttraton cell. Fnally, we obtan an expresson equvalent to S34 to evaluate the change n the heat content Δq p wth a dfferental expresson after combnng eqs S5, S54, and S56 N [ ] qp = H Vp = 1 (S57) where the dfferental ( [ ]/ ) A was evaluated usng the concentratons calculated wth a total volume of ttrant V. Hence, eqs S34 and S5 are equvalent approaches to evaluate a ttraton experment after usng a frst order Taylor approxmaton to evaluate the term Δ[ ] p. A 4. Supportng Tables In the Tables presented below that show correlaton matrces, we fnd hgh correlatons (.e., (.e.,.9 < r j < 1.) among the cumulatve bndng constants β and bndng enthalpes ΔH. A hgh correlaton among the bndng parameters could suggest that the model was overparameterzed. 5 As shown wth our theoretcal model, the hgh correlaton also ndcates that one can factorze the cumulatve bndng parameters usng ether sequental or ndependent bndng constants. 1 umulatve parameters descrbe the overall equlbrum reacton of a free receptor bound to lgand speces (eq S1). Ths means that the ftted value of β +1 ncreases when β ncreases. As a result, β and β +1 wll lkely show a hgh correlaton coeffcent. For nstance, the correlaton coeffcents among the cumulatve bndng constants β 1, β, and β 3 were all hgher than.9 for the ndvdual fts of the drect and reverse ttratons between DGA and Gd 3+ (Table S). However, the correlaton coeffcents decrease f one constrans the model usng ether sequental or ndependent bndng constants. For example, only the correlaton coeffcent between K 1 and K was hgher or equal to.9 after applyng a sequental bndng model to the drect and reverse ttratons between DGA and Gd 3+, whle those between K 1 and K 3 or K and K 3 were lower than.9 (see Table S3). For both representatons of the DMs wth N = 3 and ether cumulatve or sequental bndng stes, we obtan the same statstcal ch-square and reduced-ch square values. Ths means that both models are mathematcally equvalent. For ths reason, we rely on F-tests to S

21 dscrmnate among models wth dfferent bndng stochometres rather than on the correlaton coeffcents for the ndvdual fts. S1

22 Table S1. Indvdual Fts and orrelaton Matrces for the Drect and everse Ttratons between Dglycolc Acd and Gd n MES buffer (1 mm, ph 5.5) usng a DM wth Two Stes. orrelaton Matrx from Drect Ttraton wth DM (N = ) Value T T f log β 1 log β ΔH 1 ΔH Δh d T f 1.58 ± log β 1 (-log M) 5.76 ± log β (- log M) 9.94 ± ΔH 1 (kj mol 1 ).63 ± ΔH (kj mol 1 ) ± T Δh d (kj mol 1 ) ±.1 1 ν 51 χ χ ν.88 orrelaton Matrx from everse Ttraton wth DM (N = ) Value T T f log β 1 log β ΔH 1 ΔH Δh d T f 1.84 ± log β 1 (-log M) 5.44 ± log β (- log M) 9.31 ± ΔH 1 (kj mol 1 ).8 ± ΔH (kj mol 1 ) ± T Δh d (kj mol 1 ) -.35 ±. 1 ν 5 χ χ ν 33.6 orrelaton Matrx from Global Ft of Drect and everse Ttratons wth DM (N = ) Value T Tl T f f log β 1 log β ΔH 1 ΔH Δh d T f 1.64 ± T f 1.73 ± log β 1 (-log M) 5.56 ± log β (- log M) 9.59 ± ΔH 1 (kj mol 1 ).87 ± ΔH (kj mol 1 ) -9.5 ± T Δh d (kj mol 1 ) -.68 ±.1 1 T Δh d (kj mol 1 ) -.36 ±. ν 14 χ χ ν 5.1 S

23 Table S. Indvdual Fts, Global Ft, and orrelaton Matrces for the Drect and everse Ttratons between Dglycolc Acd and Gd n MES buffer (1 mm, ph 5.5) usng a DM wth Three Stes and umulatve. orrelaton Matrx from Drect Ttraton wth DM (N = 3) Value T f log β 1 log β log β 3 ΔH 1 ΔH ΔH 3 T f 1.3 ± log β 1 (-log M) 6.48 ± log β (- log M) ± log β 3 (-3 log M) ± ΔH 1 (kj mol 1 ).55 ± ΔH (kj mol 1 ) ± ΔH 3 (kj mol 1 ) ±.1 1 ν 5 χ χ ν.3 orrelaton Matrx from everse Ttraton wth DM (N = 3) Value T T f log β 1 log β log β 3 ΔH 1 ΔH ΔH 3 Δh d T f.989 ± log β 1 (-log M) 6.51 ± log β (- log M) 11.4 ± log β 3 (-3 log M) ± ΔH 1 (kj mol 1 ).81 ± ΔH (kj mol 1 ) ± ΔH 3 (kj mol 1 ) ± T Δh d (kj mol 1 ) -.3 ±.1 1 ν 48 χ χ ν.78 T f T f Value f T orrelaton Matrx from Global Ft of Drect and everse Ttratons wth DM (N = 3) T log β 1 log β log β 3 ΔH 1 ΔH ΔH 3 Δh d f T 1. ± ± log β 1 (-log M) 6.46 ± log β (- log M) 11.3 ± log β 3 (-3 log M) ± ΔH 1 (kj mol 1 ).54 ± ΔH (kj mol 1 ) -6.3 ± ΔH 3 (kj mol 1 ) ± (kj mol 1 ) -.3 ±.1 1 ν 13 χ χ ν.91 Δh d T S3

24 Table S3. Indvdual Fts, Global Ft, and orrelaton Matrces for the Drect and everse Ttratons between Dglycolc Acd and Gd n MES buffer (1 mm, ph 5.5) usng a DM wth Three Stes and Sequental. orrelaton Matrx from Drect Ttraton wth DM (N = 3) Value T 1 f log K 1 log K log K 3 ΔH 1 ΔH ΔH 3 T f 1.3 ± log K 1 (-log M) 6.48 ± log K (- log M) 4.91 ± log K 3 (-3 log M) 3.7 ± ΔH 1 (kj mol 1 ).55 ± ΔH (kj mol 1 ) ± ΔH 3 (kj mol 1 ) -8.5 ±.7 1 ν 5 χ χ ν.3 orrelaton Matrx from everse Ttraton wth DM (N = 3) Value T 1 T f log K 1 log K log K 3 ΔH 1 ΔH ΔH 3 Δh d T f.989 ± log K 1 (-log M) 6.51 ± log K (- log M) 4.89 ± log K 3 (-3 log M).99 ± ΔH 1 (kj mol 1 ).51 ± ΔH (kj mol 1 ) ± ΔH 3 (kj mol 1 ) -9.7 ± T Δh d (kj mol 1 ) -.3 ±.1 1 ν 48 χ χ ν.78 T f T f Value f T orrelaton Matrx from Global Ft of Drect and everse Ttratons wth DM (N = 3) 1 log K 1 log K log K 3 ΔH 1 ΔH ΔH 3 f T 1. ± ± log K 1 (-log M) 6.46 ± log K (- log M) 4.85 ± log K 3 (-3 log M) 3.3 ± (kj mol 1 ).54 ± (kj mol 1 ) ± (kj mol 1 ) ± (kj mol 1 ) -.3 ±.1 1 ν 13 χ χ ν.91 ΔH 1 1 ΔH ΔH 3 T Δh d Δh d T S4

25 Table S4. Indvdual Fts and orrelaton Matrces for the Drect and everse Ttratons between Dglycolc Acd and Gd n MES buffer (1 mm, ph 5.5) usng a DM wth Four Stes. orrelaton Matrx from Drect Ttraton wth DM (N = 4) Value T f log β 1 log β log β 3 log β 4 ΔH 1 ΔH ΔH 3 ΔH 4 T f 1.8 ± log β 1 (-log M) 7.43 ± log β (- log M) ± log β 3 (-3 log M) ± log β 4 (-4 log M).74 ± ΔH 1 (kj mol 1 ).55 ± ΔH (kj mol 1 ) ± ΔH 3 (kj mol 1 ) ± ΔH 4 (kj mol 1 ) ±.5 1 ν 48 χ 7.6 χ ν.15 f T Value f T orrelaton Matrx from everse Ttraton wth DM (N = 4) T log β 1 log β log β 3 log β 4 ΔH 1 ΔH ΔH 3 ΔH 4 Δh d.99 ± log β 1 (-log M) 7. ± log β (- log M) 1.8 ± log β 3 (-3 log M) 16.9 ± log β 4 (-4 log M) 19.7 ± ΔH 1 (kj mol 1 ).5 ± ΔH (kj mol 1 ) -6. ± ΔH 3 (kj mol 1 ) -8.5 ± ΔH 4 (kj mol 1 ) ± T Δh d (kj mol 1 ) -.3 ±.1 1 ν 46 χ 8.75 χ ν.63 Table S5. F-Tests for the Drect and everse Ttraton between Dglycolc Acd and Gd n MES buffer (1 mm, ph 5.5) usng DMs wth 4 Stes. Fttng Drect (T) everse (T) (N = ) (N = 3) (N = 4) (N = ) (N = 3) (N = 4) ν χ χ ν F rt F χ S5

26 Table S6. Indvdual Fts and orrelaton Matrces for the Drect and everse Ttratons between Dglycolc Acd and Gd n MES buffer (1 mm, ph 5.5) usng a DM wth Two Stes and f = 1.. Value orrelaton Matrx from Drect Ttraton wth DM (N = ) and f = 1. log β 1 log β ΔH 1 ΔH Δh d T log β 1 (-log M) 5.6 ± log β (- log M) 9.71 ± ΔH 1 (kj mol 1).97 ± ΔH (kj mol 1) ± T Δh d (kj mol 1) ±.1 1 ν 5 χ χ ν 59. Value orrelaton Matrx from everse Ttraton wth DM (N = ) and f = 1. log β 1 log β ΔH 1 ΔH Δh d T log β 1 (-log M) 5.79 ± log β (- log M) 9.96 ± ΔH 1 (kj mol 1 ).89 ± ΔH (kj mol 1 ) -8.4 ± T Δh d (kj mol 1 ) ±. 1. ν 51 χ χ ν 81.6 Table S7. Indvdual Fts and orrelaton Matrces for the Drect and everse Ttratons between Dglycolc Acd and Gd n MES buffer (1 mm, ph 5.5) usng a DM wth Three Stes and f = 1.. Values orrelaton Matrx from Drect Ttraton wth DM (N = 3) and f = 1. log β 1 log β log β 3 ΔH 1 ΔH ΔH 3 log β 1 (-log M) 6.64 ± log β (- log M) ± log β 3 (-3 log M) ± ΔH 1 (kj mol 1 ).6 ± ΔH (kj mol 1 ) ± ΔH 3 (kj mol 1 ) ±.7 1 ν 51 χ 14.3 χ ν.79 Values orrelaton Matrx from everse Ttraton wth DM (N = 3) and f = 1. log β 1 log β log β 3 ΔH 1 ΔH ΔH 3 Δh d T log β 1 (-log M) 6.33 ± log β (- log M) 11.6 ± log β 3 (-3 log M) ± ΔH 1 (kj mol 1 ).83 ± ΔH (kj mol 1 ) -6.6 ± ΔH 3 (kj mol 1 ) ± T Δh d (kj mol 1 ) -.31 ±.1 1 ν 49 χ χ ν 1.4 S6

27 Table S8. Indvdual Fts and orrelaton Matrces for the Drect and everse Ttratons between Dglycolc Acd and Gd n MES buffer (1 mm, ph 5.5) usng a DM wth Four Stes and f = 1.. Value orrelaton Matrx from Drect Ttraton wth DM (N = 4) log β 1 log β log β 3 log β 4 ΔH 1 ΔH ΔH 3 ΔH 4 log β 1 (-log M) 6.89 ± log β (- log M) 1. ± log β 3 (-3 log M) ± log β 4 (-4 log M) 16.9 ± ΔH 1 (kj mol 1 ).57 ± ΔH (kj mol 1 ) ± ΔH 3 (kj mol 1 ) ± ΔH 4 (kj mol 1 ) ± ν 49 χ χ ν 1.79 Value orrelaton Matrx from everse Ttraton wth DM (N = 4) log β 1 log β log β 3 log β 4 ΔH 1 ΔH ΔH 3 ΔH 4 Δh d T log β 1 (-log M) 7.65 ± log β (- log M) ± log β 3 (-3 log M) ± log β 4 (-4 log M) 1.14 ± ΔH 1 (kj mol 1 ).54 ± ΔH (kj mol 1 ) ± ΔH 3 (kj mol 1 ) -7.9 ± ΔH 4 (kj mol 1 ) ± T Δh d (kj mol 1 ) -.3 ±.1 1 ν 47 χ 6.47 χ ν 1.33 Table S9. F-Tests for the Drect and everse Ttratons between Dglycolc acd and Gd n MES buffer (1 mm, ph 5.5) Usng DMs wth 4 Stes and f = 1.. Fttng Drect (T) everse (T) (N = ) (N = 3) (N = 4) (N = ) (N = 3) (N = 4) ν χ χ ν F rt F χ S7

28 Table S1. Indvdual Analyses for the Drect and everse Ttratons Ntrlotracetc Acd (NTA) and Gd n MES buffer (1 mm, ph 5.5) usng a DM wth Two Stes and f = 1.. Value orrelaton Matrx from Drect Ttraton wth DM (N = ) and f = 1. log β 1 log β ΔH 1 ΔH Δh d T log β 1 (-log M) 7.13 ± log β (- log M) 11.7 ± ΔH 1 (kj mol 1) 7.44 ± ΔH (kj mol 1) ± T Δh d (kj mol 1).75 ±.1 1 ν 53 χ χ ν Value orrelaton Matrx from everse Ttraton wth DM (N = ) and f = 1. log β 1 log β ΔH 1 ΔH Δh d T log β 1 (-log M) 7.54 ± log β (- log M) 1.5 ± ΔH 1 (kj mol 1 ) 6.55 ± ΔH (kj mol 1 ) ± T Δh d (kj mol 1 ) -.37 ±.1 1 ν 5 χ χ ν 6.66 Table S11. Indvdual Fts and orrelaton Matrces for the Drect and everse Ttratons Ntrlotracetc Acd (NTA) and Gd n MES buffer (1 mm, ph 5.5) usng a DM wth Three Stes and f = 1.. Value orrelaton Matrx from Drect Ttraton wth DM (N = 3) and f = 1. log β 1 log β log β 3 ΔH 1 ΔH ΔH 3 Δh d T log β 1 (-log M) 7.13 ± log β (- log M) 11.7 ± log β 3 (-3 log M) 7.7 ± 1.7x ΔH 1 (kj mol 1 ) 6.69 ± ΔH (kj mol 1 ) ± ΔH 3 (kj mol 1 ) ± 5.x T Δh d (kj mol 1 ).75 ±.1 1 ν 51 χ χ ν Value orrelaton Matrx from everse Ttraton wth DM (N = 3) and f = 1. log β 1 log β log β 3 ΔH 1 ΔH ΔH 3 Δh d T log β 1 (-log M) 7.89 ± log β (- log M) 1.89 ± log β 3 (-3 log M) ± ΔH 1 (kj mol 1 ) 6.8 ± ΔH (kj mol 1 ) ± ΔH 3 (kj mol 1 ) -17. ± T Δh d (kj mol 1 ) -.35 ±.1 1 ν 5 χ 13.5 χ ν.7 S8

29 Table S1. F-Tests for the Drect and everse Ttratons between Ntrlotracetc Acd (NTA) and Gd n MES buffer (1 mm, ph 5.5) Usng DMs wth and 3 Stes and f = 1.. Fttng Drect (T) everse (T) (N = ) (N = 3) (N = ) (N = 3) ν χ χ ν F rt F χ Table S13. Indvdual Fts and orrelaton Matrces for the Drect and everse Ttratons between trc Acd (IT) and Gd and Gd n MES buffer (1 mm, ph 5.5) usng a DM wth Two Stes and f = 1.. Values orrelaton Matrx from Drect Ttraton wth DM (N = ) and f = 1. log β 1 log β ΔH 1 ΔH Δh d T log β 1 (-log M) 7. ± log β (- log M) ± ΔH 1 (kj mol 1 ) ± ΔH (kj mol 1 ) ± T Δh d (kj mol 1 ). ±.1 1 ν 5 χ χ ν 3.83 Values orrelaton Matrx from everse Ttraton wth DM (N = ) and f = 1. log β 1 log β ΔH 1 ΔH Δh d T log β 1 (-log M) 6.83 ± log β (- log M) 1.86 ± ΔH 1 (kj mol 1 ) ± ΔH (kj mol 1 ) ± T Δh d (kj mol 1 ) -.9 ±.1 1 ν 5 χ χ ν.43 S9

30 Table S14. Indvdual Fts for the Drect and everse Ttratons between trc Acd (IT) and Gd n MES buffer (1 mm, ph 5.5) usng a DM wth Three Stes and f = 1.. Values orrelaton Matrx from Drect Ttraton wth DM (N = 3) and f = 1. log β 1 log β log β 3 ΔH 1 ΔH ΔH 3 Δh d T log β 1 (-log M) 7.34 ± log β (- log M) ± log β 3 (-3 log M) ± ΔH 1 (kj mol 1 ) ± ΔH (kj mol 1 ) ± ΔH 3 (kj mol 1 ) ± T Δh d (kj mol 1 ). ±.3 1 ν 5 χ χ ν 1.9 Values orrelaton Matrx from everse Ttraton wth DM (N = 3) and f = 1. log β 1 log β log β 3 ΔH 1 ΔH ΔH 3 Δh d T log β 1 (-log M) 6.87 ± log β (- log M) 1.93 ± log β 3 (-3 log M) ± ΔH 1 (kj mol 1 ) ± ΔH (kj mol 1 ) ± ΔH 3 (kj mol 1 ) ± T Δh d (kj mol 1 ) -.9 ±.1 1 ν 5 χ χ ν.6 Table S15. F-Tests for the Drect and everse Ttratons between trc Acd (IT) and Gd n MES buffer (1 mm, ph 5.5) Usng DMs wth and 3 Stes and f = 1.. Fttng Drect (T) everse (T) (N = ) (N = 3) (N = ) (N = 3) ν χ χ ν F rt F χ 5..9 S3

31 Table S16. omparson between the Fttng and Statstcal Obtaned by Applyng Dfferental Models (DM) and Fnte-Dfference Models (FDM) for eceptors wth Two Stes to the Drect and everse Ttratons between Gd and NTA n MES uffer (1 mm, ph 5.5) DM (N = ) FDM (N = ) orrelaton Matrx from Global Analyss wth DM (N =) T T Global T T Global T f f T log β 1 log β ΔH 1 ΔH Δh T d T Δh d T f 1.36 ± ± ± ± T f ± ± ± ± log β 1 (-log M) 7.43 ±. 7.5 ± ± ± ± ± log β (- log M) 1.14 ± ± ± ±. 1.1 ± ± ΔH 1 (kj mol 1 ) 6.5 ± ± ± ± ± ± ΔH (kj mol 1 ) ± ± ± ± ± ± T Δh d (kj mol 1 ).7 ± ±.1.69 ± ± T Δh d (kj mol 1 ) ± ± ± ±.1 1 ν χ χ ν Table S17 omparson between the Fttng and Statstcal Obtaned by Applyng Dfferental Models (DM) and Fnte-Dfference Models (FDM) for eceptors wth Two Stes to the Drect and everse Ttratons between Gd and IT n MES uffer (1 mm, ph 5.5) DM (N = ) FDM (N = ) orrelaton Matrx from Global Analyss wth DM (N =) T T Global T T Global f T f T log β. 1 log β ΔH 1 ΔH Δh T d T Δh d T f 1.11 ± ±.1.99 ± ± T f ± ± ± ± log β 1 (-log M) 7.7 ± ± ± ± ± ± log β (- log M) 11.1 ± ± ± ± ± ± ΔH 1 (kj mol 1 ) ± ± ±.1 -. ± ± ± ΔH (kj mol 1 ) ± ± ± ± ± ± T Δh d (kj mol 1 ).19 ± ±.1.17 ± ± T Δh d (kj mol 1 ) ± ± ±.1 -. ±.1 1 ν χ χ ν S31

32 5. Supportng Fgures Fgure S3. aselne-corrected thermograms for one of the replcates n the drect and reverse ttratons between Gd and dglycolc acd. The bottom panels show the smoothed baselnes and thermograms before baselne subtracton. The ntegrated values for each of the njecton peaks were used to calculate the averaged dfferental heat per volume dq V shown n Fgure 1 of the man text. S3

33 Fgure S4. Drect and reverse calormetrc ttratons between ntrlotracetc acd (NTA) and Gd n MES uffer (1 mm, ph 5.5). (A, ) aselne-corrected thermograms for one of the replcates n the drect and reverse ttratons between Gd and NTA. The bottom panels show the smoothed baselnes and thermograms before baselne subtracton. (, D) Averaged values of the dfferental heat per volume of ttrant njected for the drect and reverse ttratons. The error bars show the standard devaton (σ p ) obtaned from three replcates for each ttraton drecton. The dashed lnes (black) ndcate the fttng curves obtaned from ndvdual analyses n each ttraton drecton whle the sold lnes (red) ndcate the fttng curve from the global analyss of both ttraton drectons. The bottom panels show the resduals obtaned from the ndvdual (sold symbols) and global (open symbols) analyses. The resduals are randomly dstrbuted around the zero lne wth values that oscllate between +1 and 1 J 1. The standard devatons are hgher for the njectons near the nflecton ponts on the ttraton curve. As a result, these njectons receve lower statstcal weghts when fttng the ttraton curves and the correspondng resduals also have larger values. S33

34 Fgure S5. Mole fracton dstrbutons α and dfferental bndng curves for bndng speces S n the drect and reverse ttratons, where represents Gd, and represents NTA lgand. The dash-n-dotted lnes represent the Gd-(NTA) n complexes and the dotted lnes represent free NTA speces. (A, ) The curves for the mole fractons show that n the T ttraton the receptor saturaton ncreases sequentally from free Gd speces (α ) to the ternary complexes Gd-(NTA) (α ), whle the receptor saturaton decreases n the reverse order n a T ttraton. (,D) The weghted sums of the dfferental bndng curves ( [ ]/ ) A and the cumulatve enthalpes of bndng ΔH n Table S16 were used to calculate the fttng curves n Fgure S4 for the global analyss (sold lnes, red) wth at DM as descrbed wth eq S34 (eq 7, man text). The lmtng values for the dfferental ( [ ]/ ) and the mole fracton α are nearly equvalent at the begnnng of the ttraton (see eq 7, man text). The term ( β / β1) NTA has a value hgher than 1 when calculated wth the global fttng parameters n Table S16 (.e., 1 ( ) M 1 *.5 x 1 3 M = 5.7). Hence, the ntal contrbuton of the fully saturated complex Gd-(NTA) to the heat sgnal n reverse calormetry ttraton s hgher than 9%. S34

35 Fgure S6. Drect and reverse calormetrc ttratons between ctrc acd (IT) and Gd n MES uffer (1 mm, ph 5.5). (A) aselne-corrected thermogram for one of the replcates n the drect ttraton taken from ref 1. () aselne-corrected thermogram for one of the replcates n the reverse ttratons between Gd and IT. ottom panels n A, show the smoothed baselnes and the thermograms before baselne subtracton. (, D) Averaged values of the dfferental heat per volume of ttrant njected for the drect and reverse ttratons where the error bars represent the standard devaton (σ p ) obtaned from three replcates for each ttraton drecton. The dq V values n panel were reprocessed from ref 1. The dashed lnes (black) n, D ndcate the fttng curves from the ndvdual analyses n each ttraton drecton whle the sold lnes (red) ndcate the fttng curve from the global analyss of both ttraton drectons. The resduals obtaned from the ndvdual (sold symbols) and global (open symbols) analyses are shown n the bottom panels of, D. Smlar to the NTA-Gd ttratons, the resduals are randomly dstrbuted around the zero lne wth values that oscllate between +1 and 1 J 1 for the drect ttraton and between +5 and 5 J 1 for the reverse ttraton. S35

36 Fgure S7. Mole fracton dstrbutons α and dfferental bndng curves for bndng speces S n the drect and reverse ttratons, where represents Gd, and represents IT lgand. The dash-n-dotted lnes represent the Gd-(IT) n complexes and the dotted lnes represent free IT speces. Smlar to the Gd-NTA ttratons n Fgure S5, the mole fractons n the T ttraton between Gd and IT ncrease sequentally from α to α, but decrease n the reverse order n a T ttraton. In addton, the lmtng values for the mole fracton α and the dfferental ( [ ]/ ) and are nearly equvalent at the begnnng of the ttraton. The term ( β / β1) IT has a value lower than 1 when calculated wth the global fttng parameters n Table S17 (.e., 1 ( ) M 1 *.5 x 1 3 M = 6.55). As a result, the ntal contrbuton of the fully saturated complex Gd-(IT) to the heat sgnal n a reverse calormetry ttraton s slghtly lower than 9% (see eq, man text). S36