PHYS3410
BIOPHYSICS
II
Lecture Notes
Section
I:
Thermodynamics of Biological Systems
Lecture
8 : Active Transport
The membranes
found in living cells are asymmetric and the coupling of scalar
chemical reactions driven by affinities to vectorial flows
of material (neutral molecules and ions) is basic to life
processes.
Chemical
reactions "pump" material (e.g. Na+ in
animal and plant cells, H+ in plant cells) in direction
opposite to electrochemical gradient.
Jk = LkkFk + LkrAr
(8.1)
where
Fk = -grad mk
is the
conjugate force for Jk
Chemical
reaction coupled to Jk may involve several
components:
u1m1
+ u2m2 +.....uimi 
ui+1mi+1 + ui+2 mi+2... (8.2)
mi
components participating in the reaction
ui
stoichiometric coefficients
Affinity
of this reaction:
 (8.3)
From
Onsager relations (valid at least for steady state, close
to equilibrium), the reverse process is also possible. Flow
Jk of ions, produced by the electrochemical
gradient of these ions, can drive a chemical reaction in a
direction opposite to that dictated by the affinity Ar
for that reaction.
Jr
= LrkFk + LrrAr (8.4)
The scalar
quantity Lrr is related to the rate constant for
the reaction in the absence of vectorial coupling (i.e. when
Lrk = 0). The relative magnitudes of the
coefficients are constrained:
(LrrLkk
- LrkLkr) ³ (8.5)
Chemiosmotic
Hypothesis
Charge
separation and ion movement are responsible for phosphorylation
of ADP in mitochondrial membranes and during photosynthesis
in chloroplast membranes (Robertson, 1960). Peter Mitchell
(1961, 1966) proposed that synthesis of ATP
from ADP in mitochondria promoted by the flow of protons
driven by concentration gradient and trans-membrane potential
difference (p.d.) Dy.
The mitochondria
are found in both plant and animal cells. They oxidize organic
molecules, releasing energy, part of which is used to form
ATP. It is thought that mitochondria (and chloroplasts) are
bacteria, which formed symbiosis with other organisms.
Synthesis
of ATP from ADP and phosphate in mitochondrial membranes.
During oxidation-reduction protons are extruded from the mitochondrion
and this establishes a difference in the electrochemical potential
(proton motive force) for H+ across the membrane.
The flow of protons back into the mitochondrion down the electrochemical
gradient drives the synthesis of ATP. The ATPase in mitochondrial
membrane is called ATP synthase.
Jr
is the flow of reaction ATP 
ADP:
 
 (8.6)
Where
i and o refer to inside and outside of mitochondrial membrane
and K is equilibrium constant.
At constant
temperature and pressure the Gibbs free energy :
 (8.9)
At equilibrium
the change in Gibbs free energy (at constant temperature and
pressure) = 0
(8.10)
Substitute
dx = dNi/ui
(8.11)
Neglecting
the  ip term:
 (8.12)
where
mio are standard chemical
potentials dependent on temperature only, DGo
is the change in standard free energy and K is the equilibrium
constant.
Can express
eqn. (8.12) as:
 (8.13)
For reaction
in eqn. (8.6):
 (8.14)
Can express
(8.14)
(8.15)
where c stands
for concentration, which is more often used than activity
a.
Rearrange:
(8.16)
and
 (8.17)
Mitchell
called eqn. (8.17) the proton-motive-force (difference in
electrochemical potentials on each side of the membrane).
The magnitude
of the right hand side of (8.16) can be obtained from the
equilibrium constant for the hydrolysis of ATP in free solution
(uncoupled to any transmembrane transport). This corresponds
to value of DGo
= -29kJ/mole.
Using these values (cH2O = 55 kmol/m3),
concentrations of ATP and ADP can be calculated for any value
of DmH+.
For instance, for cATP/cADP = 1, with
a value of cP =10 mM, requires DmH+
= -19.97 kJ/mole
Can consider
two special situations:
(a) no
pH gradient: the reaction requires a membrane potential difference
yi - yo
= -207 mV.
(b) no
membrane potential present: pHi - pHo
~ 3.5 (pH = -log10cH+ ).
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