Detailed study of the dielectric function of a lysozyme solution studied with molecular dynamics simulations

Abstract

The spread of microwave technology and new microwave applications in medicine have revitalized interest in the dielectric behavior of biological systems. In this work, the Fröhlich–Kirkwood approach and the linear response theory have been applied in conjunction with molecular dynamics simulations to study the dielectric response of a lysozyme solution as a model. The overall experimental dielectric behavior of a 9.88 mM lysozyme solution has been reproduced in a quantitative manner by employing a method based on the decomposition of the hydration shells close to the solute. Detailed analysis of the calculated spectra identified two δ-processes located at 200 MHz (δ1) and about 1 GHz (δ2), respectively. δ1 is associated mainly with the first hydration shell, while δ2 mainly with bulk water and the second hydration shell. Moreover, indications for the existence of an even faster relaxation in the 10¹¹-Hz frequency range were found for the first time. Finally, the static dielectric constants of lysozyme and its first and second hydration shells were calculated based on the Fröhlich–Kirkwood and the linear response theory approaches.

Appendix

As mentioned in section “Theoretical Background” (Eq. 3), the susceptibility χ can be expressed in terms of a Fourier–Laplace transform of the dipole moment time auto- (or cross-) correlation function. In the case of ions it takes the form:

$$\begin{aligned} \chi_{ij} - \frac{i\sigma ( 0)}{\omega } = \frac{ 1}{{ 3Vk_{B} T}}L \left[- \frac{\text{d}}{{{\text{d}}t}}\left\langle {{\mathbf{M}}_{\text{I}} ( 0 )\cdot {\mathbf{M}}_{\text{I}} (t )} \right\rangle \right]- \frac{i\sigma ( 0)}{\omega } \hfill \\ \begin{array}{*{20}c} {} & {} & {} \\ \end{array} = \frac{i}{{3Vk_{B} T}}L [\left\langle {{\mathbf{J}}_{\text{I}} ( 0 )\cdot {\mathbf{J}}_{\text{I}} (t )} \right\rangle ]- \frac{i\sigma ( 0)}{\omega } \hfill \\ \end{aligned}$$

(12)

where the term of the static conductivity iσ(0)/ω is simply subtracted from both sides, because it is typically eliminated from the experimental spectrum. $\varPhi_{\text{II}} (t )= \left\langle {{\mathbf{J}}_{\text{I}} ( 0 )\cdot {\mathbf{J}}_{\text{I}} (t )} \right\rangle$ is the time autocorrelation function of the ionic current, which in our case includes also the protein current due to its charge (+8e). That is:

$${\mathbf{J}}_{\text{I}} (t) = {\mathbf{J}}_{\text{II}} (t) + {\mathbf{J}}_{\text{PI}} (t) = \sum\limits_{i = 1}^{N} {q_{i} {\mathbf{v}}_{i} + Q{\mathbf{v}}_{\text{MC}} }$$

(13)

Here v _i is the velocity of ion i, Q is the total charge of protein and v _MC the velocity of its center of mass.

The expression for the conductivity of the ionic current can be written as:

$$\sigma (\omega )= \frac{ 1}{{ 3\varepsilon_{ 0} Vk_{B} T}}L [\left\langle {{\mathbf{J}}_{\text{I}} ( 0 )\cdot {\mathbf{J}}_{\text{I}} (t )} \right\rangle ]$$

(14)

As in the case of dipole moment contribution, $\left\langle {{\mathbf{J}}_{\text{I}} ( 0 )\cdot {\mathbf{J}}_{\text{I}} (t )} \right\rangle$ in the above equation is replaced by a fit function

$$f (t )= \sum\limits_{n = 1}^{ 3} {A_{n} {\text{sin(}}\omega_{n} t + \varphi_{\text{n}} ) {\text{exp(}} - t /\tau_{n} )}$$

(15)

with fit parameters reported in Table 6. Figure 9 represents the current autocorrelation function of our system as well as an appropriate fit.

Table 6 Fit parameters for the current autocorrelation function f(t)

Fig. 9

The autocorrelation function of the current of our system (protein + ions) at 300 K. The blue diamonds represent the simulation data, while the red line is the corresponding fit

The static value of σ(0) is given from:

$$\sigma ( 0 )= \frac{ 1}{{ 3\varepsilon_{ 0} Vk_{B} T}}\mathop { \lim }\limits_{{{\text{t}} \to \infty }} \int_{ 0}^{t} {{\mathbf{J}}_{\text{I}} ( 0 )\cdot {\mathbf{J}}_{\text{I}} ( {\text{t}}^{'} ) {\text{d}}t^{'} }$$

(16)

Using Eqs. (12–16), the real and the imaginary parts of the current contribution to the frequency-dependent dielectric constant can be found to be:

$$\begin{aligned} \text{Re} [\varepsilon_{I} (\omega )] = \frac{1}{{3\varepsilon_{0} Vk_{B} T}}\sum\limits_{n = 1}^{3} {\frac{{A_{n} t_{n} }}{2\omega }} \left[\frac{{t_{n} (\omega_{n} + \omega )\sin \phi_{n} - \cos \phi_{n} }}{{1 + t_{n}^{2} (\omega + \omega_{n} )^{2} }} - \hfill \right.\\ \left. \begin{array}{*{20}c} {} & {} & {} \\ \end{array} \frac{{t_{n} (\omega_{n} - \omega )\sin \phi_{n} - \cos \varphi_{n} }}{{1 + t_{n}^{2} (\omega_{n} - \omega )^{2} }}\right] \hfill \\ \end{aligned}$$

(17)

$$\begin{aligned} \text{Im} [\varepsilon_{I} (\omega )] = \frac{1}{{3\varepsilon_{0} Vk_{B} T}}\sum\limits_{n = 1}^{3} {\frac{{A_{n} t_{n} }}{2\omega }} \left[\frac{{\sin \phi_{n} - t_{n} (\omega - \omega_{n} )\cos \phi_{n} }}{{1 + t_{K}^{2} (\omega - \omega_{K} )^{2} }} + \hfill \right. \\ \left. \begin{array}{*{20}c} {} & {} & {} \\ \end{array} \frac{{\sin \phi_{n} + t_{n} (\omega + \omega_{n} )\cos \phi_{n} }}{{1 + t_{n}^{2} (\omega + \omega_{n} )^{2} }} - \frac{{2\sin \phi_{n} + t_{n} \omega_{n} \cos \phi_{n} }}{{1 + t_{n}^{2} \omega_{n}^{2} }}\right] \hfill \\ \end{aligned}$$

(18)

It should be noted that the above expressions have the form of 0/0 for ω → 0 and this is the result of subtracting the term iσ(0)/ω (see Eq. 1). Using this, we can write:

$$\mathop {\lim }\limits_{\omega \to 0} \text{Re} [\varepsilon_{\rm I} (\omega )] = \frac{1}{{3\varepsilon_{0} Vk_{B} T}}\sum\limits_{n = 1}^{3} {\frac{{A_{n} t_{n} }}{2}\left[\frac{{2t_{n} \sin \phi_{n} + 4t_{n}^{2} \omega_{n} \cos \phi_{n} - 2t_{n}^{3} \omega_{n}^{2} \sin \phi_{n} }}{{(1 + t_{n}^{2} \omega_{n}^{2} )^{2} }}\right]}$$

(19)

Using fit parameters from Table 6 we simply find $\mathop {\lim }\nolimits_{\varpi \to 0} \text{Re} \left[ {\varepsilon_{I} (\omega )} \right]$ = −0.007. This is a negligible contribution compared to that of the dipole moment (Fig. 3) and consequently is ignored in our calculations.

In Fig. 10, the real and the imaginary part of the current contribution to the frequency-dependent dielectric constant is shown.

Fig. 10

The ionic contribution (protein + ions) to the real and imaginary part of the frequency-dependent dielectric constant of our system at 300 K