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