Proton motive force

Calculations on the energy contained in the proton gradient across the mitochondrial or bacterial inner membrane

G. Woehlke, 07. Nov. 2013

Units and constants:

General conditions (one charge per transported ion, ambient temperature is body temperature):

z=1;

T=Quantity[(273.15+37),"K"]

As pH is defined as ‘the negative base-10 logarithm of the H3O* concentration”, but the Nernst equation is based on the natural logarithm, the following conversion constant is useful:

convertLn2Log10=Log[2.]/Log10[2.]

Assumptions on the pH and electrical gradients over the membrane:

pHout=6.5;pHin=7.;

deltapH=pHin-pHout

deltaPsi=Quantity[-0.180,"Volt"]

Molar units:

R=Quantity[8.31451,"J/(K mole)"]

F=Quantity[96485.309,"C/mole"]

Single-molecule units:

nA=Quantity[6.0221367 10^23,"mole^-1"]

kB=R/nA

eQ=Quantity[1.6021773 10^-19,"Coulomb"]

kB T

Biophysicists tend to prefer thinking in single-molecular dimesions, biochemicists in large-scale, molar dimensions.

Calculation of Gibb’s free enthalpy:

Energy from electrical potential:

z F deltaPsi

Energy from pH difference:

convertLn2Log10 R T deltapH

Sum of both:

z F deltaPsi +convertLn2Log10 R T deltapH

Plot3D[convertLn2Log10 R T x+z F y,{x,0,1},{y,Quantity[-0.,"Volt"],Quantity[-0.25,"Volt"]}]

Some more advanced versions of the matter:

dGmolar[{deltapH_,deltaPsi_}]:=z F deltaPsi+convertLn2Log10 R T deltapH

dGmolar[{1,Quantity[-.0,"Volt"]}]

dGmolar[{1,Quantity[-.140,"Volt"]}]/nA

dG[{deltapH_,deltaPsi_}]:=z eQ deltaPsi+convertLn2Log10 kB T deltapH

dG[{1,Quantity[-.140,"Volt"]}]

protonKBT=dG[{0.5,Quantity[-.140,"Volt"]}]/(kB T)

Comparison to the free enthalpy of ATP hydrolysis:

deltaG0primeATP=Quantity[-30500,"Joule/mol"]

deltaG0primeATP/nA

atpKBT=deltaG0primeATP/(nA kB T)

With the energy that one proton contains at physiological pH and electric gradients over a biological membrane, the following number of ATP molecules can be synthesized:

atpKBT/protonKBT