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# 物理代写|电磁学代写Electromagnetism代考|PHYS404 The Poisson-Boltzmann Equation

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## 物理代写|电磁学代写Electromagnetism代考|The Poisson-Boltzmann Equation

Here, we introduce the Poisson-Boltzmann equation as described elsewhere. ${ }^4$ In the Poisson-Boltzmann approach, all the macromolecular atoms are considered explicitly as particles with partial point charges at the atomic positions, and the dielectric constant of the macromolecule is $\varepsilon_p$ (it is often considered to be low, typically, $\varepsilon_p$ is in the range of 2-4). The solvent environment surrounding the macromolecule is taken implicitly into account as a dielectric medium with the dielectric constant of $\varepsilon_w$ (typically, about 80). The macromolecular dielectric value does not take into account the rearrangement of polar and charged amino acids with external electric fields, which could result into a larger dielectric constants. For example, it is suggested that the increase of the dielectric can compensate for the need for group re-orientations.
In non-homogeneous interacting particles system, density of a particle at any point $\mathbf{r}$ can be written as
$$\sigma_{I, i}(\mathbf{r})=g_i(\mathbf{r}) \sigma_{I, i}^0(\mathbf{r})$$
where $\sigma_{I, i}^0(\mathbf{r})$ is the particle density of the same system considered as ideal gas (i.e., non-interacting particles system), and $g_i(\mathbf{r})$ is the $i$-th particle distribution, which is taken to follow the Boltzmann distribution
$$g_i(\mathbf{r})=\exp \left(-\beta W_i(\mathbf{r})\right) .$$

## 物理代写|电磁学代写Electromagnetism代考|Calculation of pKa of Amino Acids in Macromolecules

Using Eq. (11.18), and again assuming that the probabilities are proportional to concentrations, we write
$$\frac{c_{\mathrm{M}^{-}}}{c_{\mathrm{HM}}}=\exp \left(\frac{-\Delta G}{k_B T}\right) \text {. }$$
In Eq. (11.71), $\Delta G$ is the free energy of the ionized state relative to neutral state of a macromolecule. Therefore, Eq. (11.18) can also be written as
$$\mathrm{pKa}=\mathrm{pH}-\log _{10}\left(\exp \left(\frac{-\Delta G}{k_B T}\right)\right) .$$
Or,
$$\Delta G=2.303 k_B T \gamma(\mathrm{pH}-\mathrm{pKa}) .$$

## 物理代写|电磁学代写Electromagnetism代考|The PoissonBoltzmann Equation

$$\sigma_{I, i}(\mathbf{r})=g_i(\mathbf{r}) \sigma_{I, i}^0(\mathbf{r})$$

$$g_i(\mathbf{r})=\exp \left(-\beta W_i(\mathbf{r})\right) .$$

## 物理代写|电磁学代写Electromagnetism代考|Calculation of pKa of Amino Acids in Macromolecules

$$\frac{c_{\mathrm{M}^{-}}}{c_{\mathrm{HM}}}=\exp \left(\frac{-\Delta G}{k_B T}\right) .$$

$$\mathrm{pKa}=\mathrm{pH}-\log _{10}\left(\exp \left(\frac{-\Delta G}{k_B T}\right)\right) .$$

$$\Delta G=2.303 k_B T \gamma(\mathrm{pH}-\mathrm{pKa}) .$$

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