First, changes in sample volume during a titration should only be ignored if the error involved is less than about 2%. Healy’s titration in Figure 2 shows results from eighteen successive 10 µL aliquot additions of DNA stock solution to a 3.00 mL DAPI sample. Ignoring the volume change leads to a final DNA concentration that is 6% (i.e., 3.183.00) too high. Using spreadsheets, it is easy enough for students to correct for the volume change due to each aliquot addition.¹

Second, of the eighteen titration points in Figure 2, Healy omits the first two because in a “reciprocal plot, the largest errors are found with small values of ƒ_bound”,² and he omits the last seven points because “eq 5 is only valid for conditions under which the macromolecule is not saturated,” that is, ƒ_bound is less than 0.9. Regarding the latter assertion, I could find no mention of this proviso in any of the experimental biochemistry texts that I checked (e.g., 2–5). Because the Scatchard analysis is derived from simple thermodynamic binding equilibria, it should hold true for any equilibrium mixture, regardless of the level of saturation of the macromolecule’s ligand binding sites.

Healy’s point about reciprocal plots exaggerating the impact of small errors in small values is an important one (6, 7, and references therein). This is a particular problem in the type of Scatchard analysis that Healy uses in his Figure 3, plotting [DNA]_total/ ƒ_bound versus 1(1 – ƒ_bound). When ƒ_bound is low, small errors are magnified on the y axis,³ and when ƒ_bound is high (i.e., close to one), small errors are magnified on the x axis. For this reason, it is expedient and instructive to use nonlinear regression to fit the simple hyperbolic saturation exemplified in these binding equilibria (6, 7). The equation for hyperbolic saturation, familiar to all biochemistry students from Michaelis–Menten kinetics, is

where r_max is the total number of identical independent ligand binding sites on a single macromolecule; K_d is the equilibrium dissociation constant for these binding sites; and [ligand]_free, the dependent variable, is the equilibrium concentration of unbound ligand free in solution. Equation 3 in Healy’s article is a version of this equation.

Healy varies [ligand]_free, the dependent variable, by adding aliquots of the macromolecule, DNA, and this in turn causes [ligand]_free to decline. The dependent variable measured in the experiment is the number of ligands bound per macromolecule, often called r (or N_occ in Healy’s article). Healy explains that ƒ_bound is calculated from experimental fluorescence intensities, and r and [ [ligand]_free can then be calculated from fbound as follows:

Fitting Healy’s data ([ [ligand]_free, r) to the equation for hyperbolic saturation returns two key fitted values: rmax = Nmax, the y axis asymptote; and K_d = C_1/2, the [ [ligand]_free necessary to reach half-saturation. Using sixteen of Healy’s eighteen titration points in his Figure 2, Kaleidagraph returns values of 1/r_max = 30.1 ± 0.6 nucleotides per DAPI (compared to Healy’s reported value of 31 and literature value of 36), and K_d = 167 ± 18 nM (compared to Healy’s reported 100 nM and the literature range 100–200 nM). Even more important, the hyperbolic fit to Healy’s data shows that there is a nonzero y intercept, 1/r₀ = 117 ± 9 nucleotides per DAPI. Thus the main reason why Healy had to eliminate the seven highest titration points from his linearized Scatchard plot in Figure 3 was not because of a breakdown in the validity of eq 5, nor because of exaggerated errors in the reciprocals. The problem stems from a fluorescence that increases with added DNA, even in the absence of free DAPI (i.e., at very high [DNA]). This fluorescence is equivalent to one bound DAPI per 117 nucleotides and could be due to intrinsic DNA fluorescence or to a fluorescent contaminant in the DNA.

It is interesting to note that nonlinear regression serves four beneficial purposes when applied to Healy’s DNA–DAPI binding fluorescence data: (i) it avoids the error exaggeration due to reciprocals; (ii) it yields a familiar hyperbolic saturation plot with easily interpretable fitted parameters (y asymptote and C_1/2); (iii) it allows the inclusion of seven data points that Healy had to omit from his linearized plot, Figure 3; and (iv) it yields a hypothesis other than reciprocal error exaggeration as to why these seven points did not fall on the line in Figure 3. Clearly, having students use nonlinear regression can yield substantial benefits.

Notes

1. Note also the typo on p 1305 in the middle of the right hand column: the concentration of the DNA stock solution is actually 400 µg/mL, not 400 mg/mL.
2. ƒ_bound is defined as the fraction of total ligand that is bound to the macromolecule:
   
   
   

   
   
3. It is interesting to note that the discrepancy in Healy’s first two titration points (i.e., lowest ƒ_bound) is quite substantial. In order for these two points to lie on the same line as the nine linear points in Figure 3, their fluorescence intensities would have to be 20–50% higher. It seems probable that this large discrepancy may be endemic to the DNA–DAPI system, rather than due to random experimental error.

Literature Cited

1. Healy, E. F. J. Chem. Educ. 2007, 84, 1304–1307.
2. Boyer, R. Modern Experimental Biochemistry, 3rd ed.; Benjamin-Cummings: San Francisco, 2000; pp 244–249.
3. Wilson, K.; Walker, J. Principles and Techniques of Practical Biochemistry, 5th ed.; Cambridge University Press: Cambridge, 2000; pp 407–410.
4. van Holde, K. E.; Johnson, W. C.; Ho, P. S. Principles of Physical Biochemistry, Prentice Hall: Upper Saddle River, NJ, 1998; pp 605–616.
5. Freifelder, D. Physical Biochemistry: Applications to Biochemistry and Molecular Biology, 2nd ed.; W. H. Freeman: New York, 1982; pp 655–666.
6. Silverstein, T. P. J. Chem. Educ. 2004, 81, 485.
7. Martin, R. B. J. Chem. Educ. 1997, 74, 1238–1240.