21.2: Statistical Thermodynamics of Biomolecular Reactions

Statistical mechanics can be used to calculate K_a on the basis of the partition function. The canonical partition function Q is related to the Helmholtz free energy through

\[ F= -k_bT\ln Q \]

\[ Q = \sum_{\alpha}e^{-E_{\alpha}/k_BT} \]

where the sum is over all microstates (a particular configuration of the molecular constituents to a macroscopic system), Boltzmann weighted by the energy of that microstate E_α. The chemical potential of molecular species i is given by

\[ \mu_i = -k_BT \left( \dfrac{\partial \ln Q}{\partial N_i} \right)_{V,T, \{ N_{j \neq i} \} } \]

We will assume that we can partition Q into contributions from different molecular components of a reacting system such that

\[ Q =\prod_iQ_i \]

The ability to separate the partition function stems from the assumption that certain degrees of freedom are separable from each other. When two sub-systems are independent of one another, their free energies should add (F_TOT = F₁ + F₂) and therefore their partition functions are separable into products: Q_TOT = Q₁Q₂. Generally this separability is a result of being able to write the Hamiltonian as H_TOT = H₁ + H₂, which results in the microstate energy being expressed as a sum of two independent parts: E_α= E_α,1+E_α,2. In addition to separating the different molecular species, it is also very helpful to separate the translational and internal degrees of freedom for each species, Q_i = Q_i,transQ_i,int. The entropy of mixing originates from the translational partition function, and therefore will be used to describe concentration dependence.

For Ni non-interacting, indistinguishable molecules, we can relate the canonical and molecular partition function q_i for component i as

\[ Q_i= \dfrac{q_i^{N_i}}{N_i!} \]

and using Sterling’s approximation we obtain the chemical potential,

\[ \mu_i = -RT\ln \dfrac{q_i}{N_i} \]

Following the reasoning in eqs. (2)–(5), we can write the equilibrium constant as

\[ K_a = \dfrac{N_C}{N_AN_B}=\dfrac{q_C}{q_Aq_B}V \]

This expression reflects that the equilibrium constant is related to the stoichiometrically scaled ratio of molecular partition functions per unit volume \(K_a = \prod_i(q_i/V)^{v_i}\). Then the standards binding free energy is determined by eq. (4).

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