0README hemoglobin-oxygenation |-- 0-Old directory |-- 0README obvious |-- exam1.tyuma.data hemoglobin oxygention data |-- commands.s R commands to analyze data |-- weight_functions.s source these before above analysis |-- tyuma.ps plots generated by analysis `-- tyuma_results_summary text generated by analysis BINDING OF OXYGEN TO HEMOGLOBIN -- DATA The data in the file "exam1.tyuma.data" (from Figure 3 of Tyuma et al., Biochemistry 12, 1491-1498 (1973)) are for the oxygenation of stripped human adult hemoglobin, in 0.1 M NaCl and 0.05 M Tris buffer (pH 7.4) at 25 C. The fractional saturation with oxygen is given as a function of the oxygen partial pressure (mm Hg). The data are exceptionally accurate, particularly at very low and very high saturations. MWC AND ADAIR MODELS The Monod-Wyman-Changeux model for cooperative binding of oxygen to hemoglobin gives for the fractional saturation the expression L*(p/KT)*(1+(p/KT))^(n-1) + (p/KR)*(1+(p/KR))^(n-1) y = --------------------------------------------------- L*(1+(p/KT))^n + (1+(p/KR))^n where L is the allosteric constant, equal to the ratio [T]/[R] for the unliganded states; KR and KT are, respectively, the oxygen dissociation constants for the high affinity R state and the low affinity T state; and p is the oxygen partial pressure. For human hemoglobin, the value of L is about 105, and the value of n is 4. For the Adair model of coopoerative binding, the fractional saturation is given by K1*p + 3*K1*K2*p^2 + 3*K1*K2*K3*p^3 + K1*K2*K3*K4*p^4 y = ----------------------------------------------------------- 1 + 4*K1*p + 6*K1*K2*p^2 + 4*K1*K2*K3*p^3 + K1*K2*K3*K4*p^4 where K1, K2, K3 and K4 are the association contants for binding of the first, second, third and fourth molecules of oxygen, respectively; P is the oxygen partial pressure. FIT TO THE DATA Nonlinear least-squares fits to the data are done with the R-language statements in the file "commands.s", by use of the function "nls". Starting values are from the literature. An unweighted fit (uniform weights for the oxygen saturation data) was carried out for the MWC model. Weighted fits were done for the MWC and Adair models; for these the oxygen saturation data were weighted uniformly for the Hill function (log(y/1-y)), which weights data for low and high oxygen partial pressure most heavily. For the weighted fits, the file "weight_functions.s" must have been sourced. The file "tyuma.ps" has four plots of the fits: Plot 1: Fractional satn, y, vs. Oxygen pressure, p Fits to MWC equation: black "+"'s, data red dashed line & squares, uniformly weighted green solid line & triangles, weighted by 1/(y*(1 - y)) Plot 2: log(y)) vs. log(p) Fits to MWC equation: black "+"'s, data red dashed line & squares, uniformly weighted green solid line & triangles, weighted by 1/(y*(1 - y)) Plot 3: log(y/(1 - y)) vs. log(p) Fits to MWC equation: black "+"'s, data red dashed line & squares, uniformly weighted green solid line & triangles, weighted by 1/(y*(1 - y) Plot 4: log(y/(1 - y)) vs. log(p) Fits weighted by 1/(y*(1 - y)): black "+"'s, data red dashed line & squares, Adair eqn green solid line & triangles, MWC eqn The text file "tyuma_results_summary" has: 1. trace output for each of the three fits: unweighted fit of Tyuma data to MWC model; weighted fit of Tyuma data to MWC model; weighted fit of Tyuma data to Adair model 2. Summaries of the fits: parameter estimates, std. dev. of estimates, and significance; Residual standard error of fit; Correlation of Parameter Estimates. 3. Residuals: Unweighted variance, sum((y - ycalc)^2)/df, for unweighted and weighted fits; Weighted variance = sum(((y - ycalc)/(y*(1-y)))^2)/df, for unweighted and weighted fits. COMMENTS The "standard" (unweighted) nonlinear least-squares fit of the MWC model to the Tyuma oxygenation data is shown in Plot 1 (red calculated points and red dashed curve (conincident with the green curve)). The quality of the fit is good: residual standard error ~ 5 percent of fractional oxygenation (see Summary for unweighted fit to MWC model, in file "tyuma_results_summary"). The parameter estimates are plausible: Estimate Std. Error t value Pr(>|t|) L 1.593e+06 2.340e+06 0.681 0.5052 Kt 2.927e+01 1.659e+00 17.649 2.29e-12 *** Kr 1.469e-01 5.479e-02 2.682 0.0158 * However, the standard error of the estimate for "L" shows that it does not differ significantly from zero. Low saturation data, which are accurately determined in Tyuma's measurements, contribute little to the fit. Plot 3, a Hill plot (log(y/(1-y)) vs. p, spreads the low and high saturation data. The unweighted MWC fit (red calculated points and red dashed curve) deviates from the data (black "+") at low saturation. A fit weighting residuals appropriately for the Hill plot (green calculated points and green dashed curve) gives a better fit for low and high saturation. This is seen in Plot 3 and in quantile plots: qqnorm(residuals(ty.mwc)); qqline(residuals(ty.mwc)) qqnorm(residuals(ty.mwc.wt)); qqline(residuals(ty.mwc.wt)) qqnorm(residuals(ty.ad.wt)); qqline(residuals(ty.ad.wt)) For the weighted fit, all parameters are at greater than 99% significance. A weighted fit with the Adair model is similar to and even slightly better than the weighted fit with the MWC model (Plot 4). This is unsurprising, since the Adair model has one more adjustable parameter. John Rupley rupley@u.arizona.edu -or- jar@rupley.com 30 Calle Belleza, Tucson AZ 85716 - (520) 325-4533; fax - (520) 325-4991 Dept. Biochemistry & Molecular Biophysics, Univ. Arizona, Tucson AZ 85721