Page 255 - Thermodynamics of Biochemical Reactions

Apparent Equilibrium Constants 255

10
h (2.95121 lo6 + 3.99052 10 h)
6 10 h2
1 + 2.95121 10 h + 1.99526 10
(b) This equation can be integrated to obtain lnp:

Integratel(nH/h),hl

-7
1. Log[3.39624 10 + hl + 1. Log[0.000147571 + hl
This should be lnp, which is

6 10 2
Log[l + 2.95121 10 h + 1.99526 10 h ]
except for an integration constant.
We can compare the two expressions for In p by calculating numerical values at pH 2,2.5, 3, 3.5, ... 10:

ph=Table[n,{n,2,10,.5}]
I2, 2.5, 3., 3.5, 4., 4.5, 5., 5.5, 6., 6.5, 7., 7.5, 8., 8.5, 9., 9.5, 10.1

-6
I-, 1 0.00316228, 0.001, 0.000316228, 0.0001, 0.0000316228, 0.00001, 3.16228 10 , 1. 1(
100
-10
3.16228 1. 3.16228 1. 3.16228 lo-’, 1. lo-’, 3.16228 10 , 1.
LOg[p]/.h->hh

I14.521, 12.2494, 10.0391, 7.98259, 6.20586, 4.73863, 3.48147, 2.35442, 1.37906, 0.6602:
0.0892422, 0.0290869, 0.00928946, 0.00294688, 0.000932821, 0.000295078}

These are the values of In p calculated from pK1 and pK2.

Integrate[(nH/h),hl/.h->hh

{-9.19566, -11.4672, -13.6775, -15.734, -17.5108, -18.978, -20.2352, -21.3622, -22.3376,
-23.4579, -23.6274, -23.6875, -23.7073, -23.7137, -23.7157, -23.7163)

These are the values of In p calculated by integration of the binding curve (that is, nH versus h)

(Integrate[ (nH/h) ,hI/.h->hh)-(Log[pl /.h->hh)
{-23.7166, -23.7166, -23.7166, -23.7166, -23.7166, -23.7166, -23.7166, -23.7166, -23.711
-23.7166, -23.7166, -23.7166, -23.7166, -23.7166, -23.7166, -23.7166)
This is the integration constant. Now calculate values of p at 0.5 pHs.

6
{2.02472 10 , 208854., 22904.2, 2929.44, 495.634, 114.275, 32.5065, 10.5318, 3.97106, 1.
1.29529, 1.09332, 1.02949, 1.00931, 1.00292, 1.00091, 1.00027)