This paper briefly walks through the effects of entropy in the molecular dynamics of cellular structures. The main forces opposing entropy are the Coulomb interactions. The Poisson–Boltzmann equation is then obtained to avoid using the constrained partition function which is ultimately dependent on the electrostatic energies. The Poisson–Boltzmann describes the solvent using continuum mechanics, applications of which are beyond the scope of this paper.
Physical forces cause molecules in living systems to become performing structures which are sequence-dependent. The specific forces that form these molecules come from electromagnetic interactions between nuclei and electrons. From strongest to weakest, these Coulomb interactions are: the strong covalent bonds, the hydrogen bonds, and the Van der Waals interactions. Furthermore, in order to have the molecules correctly operate in the cell, their energies should be \( \sim k_{\mathrm{B}} T \). Energies that are lower than this order can be ignored as they pale in comparison to the widespread thermal fluctuations; energies higher than this order modify the degrees of freedom in an undesirable way. The contributions of entropy to the free energy are in order of \(k_{\mathrm{B}} T\), as well; ergo, entropy indeed effects physics at the scale of interest (e.g., molecular scale).
Numerous electrolytes dissociate and ionize in solution due to entropy. Charges opposing this decrease the produced energy; the minimization of this energy is stabilized by the configurational entropy gain. This can be mathematically written into the equation for change of Helmholtz free energy: \[ \Delta F \,=\, \Delta E - T\Delta S \,=\, -\varepsilon_b + k_{\mathrm{B}} T\log\left(\frac{V}{NV_0}\right)\,, \tag{1} \] where \(\varepsilon_b\) is the binding energy and \(V_0\) is some characteristic volume [1]. Letting \(\Delta F=0\) and solving for the equilibrium concentration \(c= N/V\); \[ \begin{equation} c \,=\, \frac{1}{\lambda^3}e^{-\beta\epsilon_b}\,, \tag{2} \end{equation} \] where \(\beta\) is the thermodynamic beta and \(\lambda\) is the thermal wavelength \((\lambda^3 = V_0)\).
Moreover, the electrostatic contribution \(\epsilon_c\) to the binding energy of opposite charges is estimated using Coulomb's law, \[ \epsilon_c \,=\, -\frac{q^2 z^2}{\kappa r} \,, \tag{3} \] where \(q\) is the electron charge, \(z\) is the valence, and \(\kappa\) is the dielectric constant of medium. (Gaussian units have been used here.) It is important to note that the ratio of \(\epsilon_c\) to \(k_{\mathrm{B}} T\) is the quantity of physical interest; \[ \left\vert \frac{\epsilon_c}{k_{\mathrm{B}} T} \right\vert \,=\, \frac{z^2 l_{\mathrm{B}}}{r}\,, \tag{4} \] where \(l_{\mathrm{B}}\) is the Bjerrum length defined as [2]: \[ l_{\mathrm{B}} \,\equiv\, \frac{\beta q^2}{\kappa}\,. \tag{5} \] (From Mehran Kardar, for the instance of water, \(\kappa\approx 81\) and at room temperature \( l_{\mathrm{B}} \approx 7.1\) Å.)
In addition to their self binding, some protein macroions bind to DNA. By holding these macroion at a fixed distance from each other an effective Coulomb interaction can be found. The restricted partition function is given by integrating over all the other degrees of freedom; \[ Z \,=\, \int \prod_{\jmath} d^3r_\jmath \, e^{-\beta \mathcal{H}_c} \,, \label{6}\tag{6} \] where \(\mathcal{H}_c\) is the Hamiltonian including Coulomb interaction related energies.
Due to the complexity in evaluating the constrained partition function \(\eqref{6}\) an approximation should be utilized. Assume each counterion to its corresponding macroion experiences an effective potential \(\phi(\boldsymbol{\mathrm{r}})\). In this "mean-field" approximation, the density of counterions \(\gamma\) adjusts to the potential through the Boltzmann weight, \[ n_\gamma(\boldsymbol{\mathrm{r}}) \,=\, \overline{n}_\gamma\, e^{-\beta\phi(\boldsymbol{\mathrm{r}}) qz_\gamma } \,, \tag{7} \] where \(\overline{n}_\gamma\) is the adjustable parameter which must be set so that when \(n_\gamma(\boldsymbol{\mathrm{r}})\) is integrated the correct number of counterions is acquired. Nevertheless, the potential should satisfy the Poisson equation; \[ \nabla^2 \phi\left(\boldsymbol{\mathrm{r}}\right) \,=\, -\frac{4\pi}{\kappa}\rho\left( \boldsymbol{\mathrm{r}} \right)\,, \label{8}\tag{8} \] where the charge density \(\rho\) is given by, \[ \rho\left( \boldsymbol{\mathrm{r}} \right) \,=\, \rho_{\mathrm{macroions}}\left( \boldsymbol{\mathrm{r}} \right) + \sum_\gamma qz_\gamma \, \overline{n}_\gamma \, e^{\beta \phi(\boldsymbol{\mathrm{r}}) qz_\gamma } \,, \label{9}\tag{9} \] note that the second term is the fluctuation averaged counterion density. Equation \(\eqref{8}\) with \(\rho(\boldsymbol{\mathrm{r}})\) given in Equation \(\eqref{9}\) is the Poisson–Boltzmann equation: \[ \nabla^2 \phi \,=\, -\frac{4\pi}{\kappa}\left( \rho_{\mathrm{macroions}} + \sum_\gamma qz_\gamma \, \overline{n}_\gamma \, e^{\beta \phi qz_\gamma } \right) \,. \label{10}\tag{10} \]
The Poisson–Boltzmann equation in line \(\eqref{10}\) is used in ionic solutions research. As a non-linear partial differential equation, its exact solutions are generally difficult to obtain; be that as it may, it can be solved using numerical methods as it is at least less complicated than the restricted partition function given by Equation \(\eqref{6}\).
The partition function \(Z\) is a function of \(\mathcal{H}_c\), itself incorporating \(\epsilon_c\). The electrostatic energy is one of the two main contributions to the free binding energy, the other being all non-electrostatic energies [3]. The binding energy is the energy which opposes entropy minimizing the energy produced by the dissociation of charge.