An equilibrium DNA stretching curve, F(x), is extremely sensitive to the presence of DNA binding proteins, such as NC, that bind stoichiometrically along the whole length of the DNA molecule. It, therefore, can be used to monitor this binding. To do this, we obtain the stretching curves of indi- vidual DNA molecules undergoing the force-induced melting transition. At large enough extensions the DNA under tension is a combination of melted and double helical regions, with the relative proportion of both forms varying with the molecular extension. Because the HIV-1 NC proteins are known to bind dsDNA and single-stranded DNA (ssDNA) with comparable afﬁnities (Table 2), we will not try to distinguish between these modes of binding, but instead use our F(x) curves in order to obtain an estimate of the average composite binding constant to both forms of DNA. While this method cannot separately determine dsDNA and ssDNA binding, it allows us to quantify the interaction of NC with polymeric natural DNA. Such DNA more strongly resembles the actual substrate for NC binding in vivo relative to other artiﬁcial constructs such as oligonucleotides and homopolymers. This method was used successfully in a previous work to obtain binding constants for LINE-1 ORF1p and several mutants, and the results obtained were in agreement with bulk binding experiments (38). In order to estimate the equilibrium binding constant of the protein to DNA, we can follow the variation of the force-extension curve F(x) with the protein concentration c. The most distinct feature of the changes induced by the pres- ence of the NC proteins in DNA stretching and relaxation behavior is the signiﬁcant increase in the slope of the force- induced melting transition upon addition of the protein. This increase in transition width is likely due to NC’s sequence- speciﬁc binding and subsequent reduction in DNA melting cooperativity, as well as intercalation of NC into dsDNA upon stretching. All of these effects contribute to NC’s nucleic acid chaperone capabilities. Here, we use this growing trans- ition width as a quantitative parameter to monitor the protein binding to ssDNA. We can deﬁne the transition width DF as a force interval over which the DNA molecule switches from an entirely ds state to an entirely ss state. To estimate this trans- ition width, we ﬁt a tangent line to the transition midpoint (determined by linear regression analysis of several stretching curves), as shown in Figure 3. This midpoint corresponds to half the extension change between dsDNA only (solid line, closed circle), and ssDNA only (solid line, closed triangle) in the presence of protein. The slope of this line is the change in force from the beginning of the transition to the end of the transition, divided by the change in DNA length during the transition. Thus, the force transition width DF is the difference in forces at which this tangent line intersects tangent lines to the dsDNA and ssDNA extension curves (Figure 3). The val- ues obtained in this way are plotted as a function of protein concentration c for all four proteins studied (Figure 4). In all cases the transition width ﬁrst increases linearly with protein concentration, and saturates at DFsat. We see that the plots for all three forms of NC protein used in this study are similar within the experimental errors and very different from the plot for Gag. The slope of the linear region of this plot can be used to estimate the binding constant. Indeed, at sub-saturating protein concentrations, the fraction Q of DNA binding sites bound by protein is very small, Q<<1, and increases linearly 2 3 4 Nucleic Acids Research, 2006, Vol. 34, No. 2 597 Figure 3. Stretching curves for l-DNA in the absence of protein (open circle), in the presence of 20 nM NCp7 (wild type) protein (closed circle), and for ssDNA with NC bound on it (closed triangle), all in FIM buffer (see Materials and Methods) and room temperature. For ssDNA with NC bound, data are from ref. (35), and was taken with 10 nM NCp7 in 25 mM NaCl, 10 mM HEPES, pH 7.5, room temperature. Stretching curves for dsDNA (open square) and ssDNA (open triangle), fit to standard polymer models, in the absence of protein (90) are also shown. The method used to determine DF, the force transition width, is shown schematically. 1 K ¼ dFsat · dFðcÞ c 1 · dðdFðcÞÞ ¼ dFsat dc jc!0 The fact that these stretching curves are not completely reversible does not preclude the measurement of equilibrium binding isotherms. The stretching curves used for these meas- urements did not depend on pulling rate (at the low pulling rates used here) and the measurements therefore are indistin- guishable, within the reported experimental error, from equilibrium measurements (40). Note that irreversible thermal melting curves (which do not depend on heating rate) are routinely used to obtain equilibrium thermodynamic measurements (41). with the concentration of the protein c: Q ¼ K · c‚ where K is the apparent binding constant of protein to DNA. Note that the free protein concentration does not depend on the extent of binding because the protein is in great excess over the DNA in this experiment. In addition, NCp7 is known to exist as a monomer in solution even at high concentrations (39), and while Gag can form dimers in solution, at the con- centrations used here the number of dimers should be negli- gible (S. A. Datta and A. Rein, unpublished data). Assuming that the change in the transition width, dF ¼ DF DF0, is proportional to the fraction of sites bound by protein, Q, and with the saturated value of this parameter, dFsat, we can also write: dFðcÞ ¼ Q · dFsat Here, DF0 is the transition width in the absence of protein. Combining Equations 2 and 3 we can estimate the protein– DNA equilibrium binding constant as: