Thermodynamics of Bioreactions
113
^
- =
Ap =
A ¥ - 2.303— A
pH
= A ¥ -0 .0 6 0
6pH
(4)
F
n
A p
has the unit volts per mole. A
y
/is the membrane potential (more negative on the inside, hence Ay/-is
positive and has been found to be approximately 0.14 V). A
pH
is the difference in pH between the outside
and the inside of the membrane (about 0.05 pH units and negative). The total value of A
p
is about 0.143 V
per mole of FT, equivalent to a release of free energy of 96.5 ■
0.143 = 13.8 kJ per mole of protons
transported. The required inward flow of protons is provided for by the redox reaction in Eq (1), which
expends some of its free energy in pumping protons in the opposite direction, against the steep
electrochemical gradient The two flows are vectorially arranged, and they are mediated by different and
mutually isolated transmembrane proteins. Thus the total oxidative phosphorylation process is conceived as
a cyclic process with four steps, two scalar chemical reactions and two vectorial transport processes. The
affinity of one transport process and of one chemical process is negative, whereas the other two processes
are energetically coupled by the requirement that the total rate of free energy dissipation
D
must be non-
negative.
D
is given as the sum of four terms
vAi,
where V
| is the “flow” (moles per second converted or
transported) and
A,
is the corresponding drive force, i.e. the affinity of the ith process.
D = ^ ,v lA ,kO
(5)
It is convenient to combine the four reactions into two pairs. One is the redox process and the associated
outward proton transport; the other is the phosphorylation and its associated downhill flow of protons. Let
A0
and
Ap
be the total affinity of each of these two process pairs:
A = A ° - nA
h*
(
6
)
A =Al+n,
A „+
P
P
P
H
(7)
A°gis
large and positive, and
A ^is
negative.
na
and
rip
denote the moles of protons excluded or taken up by
the cell (or mitochondria). From Eq. (5) it follows that
A0v0 + Apvp
is nonnegative, v0 and vp being the overall
rates of the oxidation and phosphorylation reactions, respectively.
Now, by the theory of non-equilibrium thermodynamics (see
e.g
Katchalsky and Curran (1965)) - the
driving forces and the flows of two energetically coupled processes are related by
c A + cnnA n
oo
o
op
p
(
8
)
v„
= c A + cnnA„
p
po
o
pp
p
(9)
The linear equations hold very close to equilibrium where microscopic reversibility leads to Onsager’s
reciprocal relations, which state that
c0p - Cpo
(or more generally that the
cn
matrix is symmetric). Whether
enzymatic processes that may work far from equilibrium really conform to the assumptions that permit
linearization of much more complicated relations between v, and
A}
is a moot question, which Rottenberg
(1979) purports to answer by reference to the near constancy of overall enzymatic conversion rates when
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