The MWC model considers an oligomeric macromolecule consisting of n identical subunits containing a single ligand-binding site each. Those subunits may adopt two conformations, R (relaxed) and T (tense), with different ligand-binding affinities (KR > KT). Within a given oligomer, all subunits exhibit the same conformation and ligand binding occurs to any subunit independently. However, although T state predominates initially, as a consequence of the higher ligand-binding affinity for R conformation, ligand binding will shift the conformational equilibrium by indirectly eliciting a concerted conformational change of all subunits toward R conformation within the same oligomer at once, resulting in a cooperative binding behavior. The MWC model can only reproduce positive cooperativity, while the Koshland-Nemethy-Filmer (KNF) model can reproduce both negative and positive cooperativities. Both models, MWC and KNF, represent reduced and limited versions of a general allosteric model (44, 59) that takes into account all possible conformational states (e.g., mixed conformations within a protein oligomer) and all possible liganded states, but, in practice, they are considerably more useful and manageable than the general allosteric model.

The binding polynomial for a macromolecule with n subunits that can exist, all at once within a given oligomer, in two different conformations, R and T, and each subunit with a single ligand-binding site, is given byZ=i=0n[PLi][P]=i=0n([RLi][R]+[TLi][R])(1)where PLi represents the protein complex with i ligand molecules bound, and RLi and TLi refer to the complexes of each oligomeric conformational state with i binding sites occupied by ligand molecules. In terms of site-specific binding parameters, the MWC binding polynomial is written as followsZ=(1+KR[L])n+γ(1+KT[L])n=i=0n(ni)(KRi+γKTi)[L]i(2)where KR and KT are the site-specific microscopic intrinsic association constants for a binding site in a protein subunit in the R and T states, respectively, with KR > KT, and γ is the equilibrium constant for the conformational equilibrium between the T and R oligomers (γ = [Tn]/[Rn]). Because initially the oligomer in T conformation predominates, the equilibrium constant γ is larger than 1. Each term represents the binding subpolynomial considering n independent ligand-binding sites in each conformational state, R and T.

Valuable information can be extracted from the binding polynomial. In particular, the molar fraction of each liganded speciesFRLi=(ni)KRi[L]iZFTLi=(ni)γKTi[L]iZ(3)

The two first derivatives of the binding polynomial relative to ligand concentration and temperature provide two fundamental quantities, the average number of ligand molecules bound per oligomeric macromoleculenLB=i=lnZln[L]=nZ(KR[L](1+KR[L])n1+γKT[L](1+KT[L])n1)(4)that can be expressed more conveniently asnLB=i=lnZln[L]=1Z(i=0n(ni)i(KRi+γKTi)[L]i)=i=0niFRLi+i=0niFTLi(5)and the average molar excess binding enthalpyH=RT2lnZT=1Z(nKR[L](1+KR[L])n1ΔHR+γ(1+KT[L])nHγ+nγKT[L](1+KT[L])n1HT)(6)that can also be expressed more conveniently asH=RT2lnZT=1Z(i=0n(ni)(KRi[L]iiHR+γKTi[L]i(Hγ+iHT)))=i=0nFRLiiHR+i=0nFTLi(Hγ+iHT)(7)where ∆HR and ∆HT are the site-specific microscopic intrinsic ligand-binding enthalpies for a binding site in a protein subunit in the R and T states, respectively, and ∆Hγ is the enthalpy associated to the concerted conformational change between the R and T states.

The binding equations corresponding to the binding equilibrium derive from the mass conservation and the chemical equilibrium[P]T=[P]Z[L]T=[L]+[P]TnLB=[L]+[P]TlnZln[L](8)

The last equation can be transformed into a (n + 1)th-degree polynomial equation in [L] (in this case, a 15th degree polynomial equation) with coefficients that are functions of KR, KT, γ, [P]T, and [L]T and that can be solved numerically (e.g., Newton-Raphson method) for the unknown [L]. Equation 8 must be solved for each experimental point in the calorimetric titration (i.e., each ligand injection j), for which the total concentrations of protein and ligand after each injection j are calculated as follows[P]T,j=[P]0(1vV0)j[L]T,j=[L]0(1(1vV0)j)(9)where [P]0 is the initial macromolecule concentration in the cell, [L]0 is the ligand concentration in the syringe, V0 is the calorimetric cell volume, and v is the injection volume. Once the free ligand concentration is known, the concentration of each complex after each injection can be calculated (subscript j omitted for the sake of clarity)[RLi]=[P]TFRLi=[P]T(ni)KRi[L]iZ[TLi]=[P]TFTLi=[P]T(ni)γKTi[L]iZ(10)The heat effect, qj, associated with each injection j is calculated, considering that it reflects the change in the average excess molar binding enthalpy or in the concentration of all complexes in the calorimetric cell between injection j and j − 1qj=V0([P]T,jHj[P]T,j1Hj1(1vV0))=V0([P]T,j1(1+KR[L]j)n+γ(1+KT[L]j)n(nKR[L]j(1+KR[L]j)n1HR+γ(1+KT[L]j)nHγ+nγKT[L]j(1+KT[L]j)n1HT)[P]T,j11(1+KR[L]j1)n+γ(1+KT[L]j1)n(nKR[L]j1(1+KR[L]j1)n1HR+γ(1+KT[L]j1)nHγ+nγKT[L]j1(1+KT[L]j1)n1HT)(1vV0))=V0([P]T,ji=0n(FRLi,jiHR+FTLi,j(Hγ+iHT))[P]T,j1i=0n(FRLi,j1iHR+FTLi,j1(Hγ+iHT))(1vV0))=V0(i=0n(iHR([RLi]j[RLi]j1(1vV0))+(Hγ+iHT)([TLi]j[TLi]j1(1vV0))))(11)Last, qj is normalized by the amount of the ligand injected during each injection, and an adjustable parameter qd accounting for the background injection heat (due to solution mismatch, turbulence, etc.) is also includedQj=qjv[L]0+qd(12)In addition, a normalizing parameter N is included in Eq. 9 by multiplying [P]0 to account for the active or binding-competent fraction of macromolecule (percentage of protein able to bind ligand).

Nonlinear least squares regression allows the determination of the binding parameters KR, KT, γ, ∆HR, ∆HT, ∆Hγ, N, and qd. A fitting routine was implemented in Origin 7.0 (OriginLab).

For the sake of simplicity, a non-normalized binding polynomial has been used in Eq. 2, where only the ligand-free R state has been taken as the reference state. A renormalized binding polynomial can be constructed taking the subensemble of ligand-free states, R and T, as the reference (ensemble) state, resulting in a standard binding polynomial with a leading term equal to 1 when grouping terms according to powers in [L]. The new binding polynomial is given byZ*=Z1+γ=11+γ(1+KR[L])n+γ1+γ(1+KT[L])n(13)and the only difference in the subsequent development is that a different reference value for the excess average ligand-binding enthalpy is usedH*=Hγ1+γHγ(14)

The second term in the right-hand side of Eq. 14 after the minus sign is a constant value equal to the average excess enthalpy of the non-liganded fraction of protein (i.e., ligand-free R and T states)H*=HHR+T(15)

Thus, that term introduces a shift in the enthalpy scale, and an additional constant term in the average excess molar enthalpy makes no difference when calculating the heat effect for each injection, as can be easily provenqj=V0([P]T,jHj[P]T,j1Hj1(1vV0))=V0[P]T,j(HjHj1)=V0[P]T,j(H*jH*j1)(16)

A nested MWC cooperativity model for the behavior of ClpP in which each heptamer might undergo a specific concerted conformational change could have been applied to ClpP but that would have added three additional equilibrium constants and three additional enthalpy changes, which would result in overparameterization in the fitting function and correlation/dependency between fitting parameters. In addition, the KNF cooperativity model in which sequential conformational changes are occurring as the oligomer is being occupied by the ligand could have been applied to ClpP, but, again, more parameters should have been considered in the fitting function, resulting in overparameterization. Besides, the KNF model cannot reproduce the activation effect induced by bortezomib because, in the KNF model, the conformational change induced by the ligand is specifically restricted to those subunits binding the ligand, with no possibility of ligand-free subunits undergoing the activating conformational change. Thus, the MWC cooperativity model is the minimal model able to reproduce the behavior of ClpP.

The concentration (and the evolution along a titration) of the different protein species (R and T conformations) can be readily calculated from the previous equations. The total fractions of protein subunits in the conformation R or T, FR,T and FT,T, respectively, are given byFR,T=(1+KR[L])nZ=1Zi=0n(ni)KRi[L]i=i=0nFRLiFT,T=γ(1+KT[L])nZ=1Zi=0n(ni)γKTi[L]i=i=0nFTLi(17)and the fractions of protein subunits in the conformation R or T that are bound to the ligand, FR,B and FT,B, respectively, are given byFR,B=KR[L](1+KR[L])n1Z=1nZi=0n(ni)iKRi[L]i=1ni=0niFRLiFT,B=γKT[L](1+KT[L])n1Z=1nZi=0n(ni)γKTi[L]i=1ni=0niFTLi(18)

The fractions of ligand-free protein subunits in the conformation R or T, FR,F and FT,F, respectively, can be calculated as the following differencesFR,F=FR,TFR,B=(1+KR[L])n1Z=i=0nFRLi1ni=0niFRLiFT,F=FT,TFT,B=γ(1+KT[L])n1Z=i=0nFTLi1ni=0niFTLi(19)

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