Systems Biology (Methods in Molecular Biology)

2 Thermodynamics Framework

From the formalism of the classical thermodynamics [12] entropy

production can be evaluated through the variation of Gibbs’s free

energydGTpwhen the system evolves subjected to the constraints

the temperatureTand the pressurepconstants as

δSi¼

1

T

dGTp ð 1 Þ

The temporal variation of the expression of Eq. (1) represents

the entropy production rate as

δSi

dt

¼

1

T

dGTp

dt

ð 2 Þ

whereδdtSiS_irepresents the entropy production rate. The termdGdtTp

can be developed by means of the chain rule as a function of the

degree of advance of the reactionξas

dGTp

dt

¼

∂G

∂ξ



Tp

dξ

dt

ð 3 Þ

where ∂∂Gξ



Tp

, according to De Donder and Van Rysselberghe

[13],represents the affinityA¼∂∂Gξ



Tp

, and the termddtξis the

reaction rateξ_.

The rate of entropy production (Eq.3) can be written as

δSi

∂t

¼S_i¼

1

T

Aξ_¼

1

T

ΔG_ξ ð 4 Þ

whereA¼ΔG. The affinityAcan be evaluated from the iso-

therm of the reaction [14] by the equation

A¼RTlnKCRT

Xk

i¼ 1

νklnCk¼RTln

KC

∏Cνkk



ð 5 Þ

where KC¼kkbf is the Guldberg-Waage constant;kf,kbare the

specific rate constants of the direct and inverse reaction stepsf,b,

respectively;Ckis the concentration of thekth specie; and theνkare

the stoichiometric coefficients that are taken, by agreement, as

positive for the products and negative for the reactants. Therefore,

Eq. (5) can be written as

A¼RTln

kf∏C

νkfðÞ

kfðÞ

kb∏C

νkbðÞ

kbðÞ

!

ð 6 Þ

The rate of reactionξ_can be written as

ξ_¼ ξ_fξ_b



¼kf∏C

νkfðÞ

kfðÞkb∏C

νkbðÞ

kbðÞ ð^7 Þ

130 Sheyla Montero et al.