Execution

B. Multifit

2. Execution

The first step toward creating a procedure to minimize SSR* was to develop the means to evaluate SSR* itself. This required identifying the fundamental random variables Xj, their uncertainties cr~i' and the derivatives 8(Dcalepi-Dobspi)/8xj.

The identities of the fundamental random variables of an experiment depend on the design of the experiment itself. Strictly, every measurement performed is a random variable. Binding studies are typically performed in the Dougherty group by combining stock solutions of host and guest together with additional buffer in an NMR sample tube, and recording the spectrum. The values of [H]o and [G]o

(5) Bevington, P.R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill;

New York, 1969; p 60.

are then altered by adding more host solution, guest solution, or buffer, and the NMR spectrum is again recorded. The steps of adding solution and recording the spectrum are repeated several times, and the spectra of uncomplexed host and guest are measured independently.

The fundamental random variables contributing to a single observation are:

• The host and guest concentrations, [H]s and [G]5 , of every stock solution used to make up the sample,

• the volume Va of each solution aliquot added to the sample tube,

• The calibration I of the delivery devices (pipets or syringes) employed to add the aliquots, and

• the NMR peak position measurements Oobspi and Ofreep·

The stock solution concentrations are perhaps the most significantly mis-meas- ured quantities in a binding study. Ordinarily, these concentrations are determined by NMR integration against a known standard, such as 3,3-dimethylglutaric acid or potassium hydrogen phthalate. NMR integrations are notoriously imprecise, and are not considered valid to within less than about five percent. Aliquot volumes are determined by two related but independent random variables: delivery device precision and delivery device accuracy. Precision is the reproducibility of volumes added by a device. The precision errors of aliquots added by the same device are independent and identically distributed with a mean of zero. Accuracy is a measure of the likely calibration error of the delivery device. The volumes of all aliquots delivered by a single device will be mis-measured by the same proportional amount;

for instance, they all may be two percent too low. Thus, the difference between an aliquot's true and measured volumes is (measured value) x (calibration error)

+

(reproducibility error). The final fundamental random variables considered are the

NMR measurements. Because NMR signals are well-resolved and reproducible, the errors in these variables are very small. The principal source of such error is the digitization of the spectrum: the peak position cannot be known more specifically than the distance between two points. Another possible but not always present contributor to peak position measurement error is peak width. If a peak is very broad, it is difficult to tell exactly where its center lies.

Once the fundamental random variables have been identified, it is necessary to determine their impacts upon the observations according to equation 18. This task is tedious but straightforward: it requires only differentiation of ( Ocalc pi - Oobs pi)

with respect to each of the fundamental variables. Substitution of equations 12 and 6 into equation 18 put it in terms of the the fundamental random variables Oobspi

and Ofreep' and the not-so-fundamental random variables

[H]o

and

[G]o.

These concentrations can in turn be expressed in terms of fundamental random variables.

In any sample solution created by adding aliquots of other solutions together, the total host and guest concentrations are given by equations 20 and 21.

V

=Vt

+IVa

[H]o

=

([Ht]o Vt +

[Ha]oiVa) /V

[G]o

=

([Gt1o Vt +

[Ga]oiVa) /V

(19)

(20) (21)

V is the total sample volume; it is the sum of

Vt,

the volume of solution in the sample tube before addition of the most recent aliquot, and IVa, the volume of the most recent aliquot. The calibration of the delivery device adding the most recent aliquot is I; its "measured" value is unity. The nominal volume of the aliquot is Va.

[Ht]o

and

[Gt]o

are the total host and guest concentrations of the sample before the addition of the most recent aliquot, and

[Ha]o

and

[Ga]o

are the total

host and guest concentrations of the added solution. The only random variables in equations 19-21 that are necessarily fundamental are Va and I. All of the other variables, however, can eventually be decomposed into fundamental variables if they are not fundamental themselves.

Vt,

[Ht]o, and [Gt]o are the V, [H]o, and [G]o of the sample previously in the tube; consequently, these values are all zero for the first solution in a tube. The added solutions may be but are not required to be stock solutions. If they are stock solutions, then [Ha]o and [Ga]o are the fundamental random variables [H]s and [G]s; if they are not, then they are ultimately composed of stock solutions added together. The details of determining these derivatives are given in the Appendix to this chapter.

D. Method of Creswell and Allred.

Another popular method for finding ]( from NMR titration experiments was developed independently by Creswell and Allred6 and by Horman and Dreux,7 and is currently championed by Wilcox. 8 This method involves using t5free as an ad- justable parameter instead of an independently-measured variable. It is claimed to be superior to methods in which t5rree is directly measured, because the parameter estimates are unaffected by errors in the determination of t5free· In other methods (such as Ernul), if t5free is measured erroneously, the model is systematically com- promised. The method of Creswell and Allred determines t5rree from the entire data set, instead of relying on a single measurement.

(6) Creswell, Clifford J .; Allred, A. L. "Thermodynamic constants for hydrogen bond formation in the chloroform-benzene-cyclohexane system," J. Pl1ys. Chern. 1962, 66, 1469-1472.

(7) Horman, Ian; Dreux, Bernard "Estimation of Association constants of bimolecular organic complexes," Anal. Chern. 1983, 55, 1219-1221.

(8) Wilcox, Craig S. "Design, synthesis, and evaluation of an efficacious functional group dyad.

Methods and limitations in the use of NMR for measuring host-guest interactions," In Frontiers in Supramolecular Organic Cl1emistry and Plwtochemistry, Schneider, H.-J .; Diirr, H., Ed.; VCR: Weinheim, 1990.