Calculation of site enthalpy for binding of ligands and protons to macromolecules

Abstract

The analysis of equilibria in solution by the partition function method has shown how the total chemical amounts [T[ M], [T A], and [T H] of the components M, A, and H respectively can be expressed and determined as functions of different powers of site affinity constants k j and cooperativity functions γ( i j,b j ), whereas the cumulative formation constants β PQR cannot be used as statistically independent parameters to be refined in least-squares processes. The same drawback holds for molar enthalpies, Δ H PQR . A special algorithm has been developed by which the heat evolved can be expressed as a function of specific site enthalpies Δ j and specific cooperativity enthalpies Δ h γj for each class j of sites. The algorithm represents the mathematical analog of the interconnections between components in the types of complex macrospecies M PA QH R, M PA Q, A QH R, M P H R , M P (A Q H R ), etc., of the chemical model assumed and their deconvolution into microspecies M P A Q H R . Each class j of sites with site constant k j and cooperativity coefficient b j is described by a polynomial ▪ where Y j is any ligand M, A, or H and i t is the number of sites in the class j. The concentrations of the microspecies are calculated as single terms of the polynomials J j or by products of more terms, each of which belongs to a different J j . Each term of the polynomial is labeled by its own index ( p j , or q j , or r j , or in general i j ), which is the exponent of the term and contains the statistical factor m ij . calculated as the coefficient of the term of the polynomial. The product of more terms is labeled by the indices of the component terms. The relations are therefore represented by combinations of indices ▪. In order to perform the calculation of concentrations of the microspecies and macrospecies by a procedure suitable for computer programming, each polynomial J j is associated with a vector J j whose elements [ i j ] are the terms of J j . The cooperativity factors are set in a diagonal matrix Γ j , whose elements are ▪ and then introduced into the non-cooperative polynomials J j by vector products ▪. The product of terms giving for each microspecies the contribution to the total concentrations [T M], [T A], and [t H] is calculated as the element of a matrix ▪ obtained as tensor product: (1,2= j 1J 2 or ▪ etc. Depending on the chemical model, there are additional different matrices L l. The combination of indices of each element of L l is ▪. The indices are said to define an index space {i.s.}, parallel to the affinity cooperativity space. The elements of the matrices L l are also used to set a matrix ΔC, whose elements Δ c pqr are the changes of concentration of the microspecies during the thermochemical reaction. The i.s. is parallel to the concentration space also. The enthalpy change at the nth experimental point for each microspecies M p,A qH r is calculated from the quantities ▪ ▪ which are then summed for all the indices p,q,r. This relation can also be put in a matrix form: ▪. All these matrices define spaces which are parallel to {i.s.}. The observed heat at the nth experimental point is the scalar product ▪ where the elements of the vector X j, with 1 ⩽ j' ⩽2j max are the whole set of j couples of variables Δ h j , Δ h yj , and the elements of the vector a j' are weighted experimental thermochemical values. By repeating the calculations and summing successively the values for all the n experimental points, the system of normal equations Ax j' = ΔQ is set up, where the elements of ΔQ are sums of n weighted experimental heats. By solution of the system, the values of Δ h j and Δ h yj for all the j classes are obtained.