Modelling Permittivity in Proteins: A User-Friendly Approach

1. Introduction

Proteins are increasingly exploited in bioelectronics, most notably in biosensors [1,2,3,4,5] and optoelectronic devices [6,7]. Indeed, in vivo, proteins respond to environmental changes—pH, temperature, solute concentration, specific ligands—through conformational rearrangements [2,8,9,10] that trigger a cascade of biochemical events culminating in genuine biological “sensing,” from antigen capture to vision. In vitro biosensors harness part of this same cascade to detect the binding of a specific ligand and convert it into a chemical or physical signal [11,12]. Finally, in silico, increasingly sophisticated computational tools now support the design of biosensors and the prediction and quantification of their output signal [13,14,15,16,17].

Among the many transduction strategies available—mass, electrical, piezoelectric, chemical, optical, and so on—electrical biosensors occupy a prominent place, owing to how readily their output integrates with standard electronics. Yet several basic questions about the electrical and dielectric behavior of proteins remain open. Electrical transport is known to proceed via sequential tunneling [18,19], but the absolute resistivity of individual amino acids remains unknown [8,20], and it is not known whether—or how—this resistivity depends on the surrounding environment, unlike resistance, which is known to depend on the protein’s three-dimensional structure and to vary with conformation [8,20].

The permittivity of proteins has been calculated and measured under dry conditions [21,22,23,24], consistently yielding a narrow range, roughly 2–4. Its value in the hydrated state, presumed to be considerably larger, is still the object of active computational [25,26,27] and experimental [28] investigation. The picture is further complicated by the fact that permittivity depends strongly on temperature, degree of hydration, and protein conformation [23]. There is, however, growing consensus that permittivity should be regarded as a combination of an inner, dry-like contribution and an outer, solvent-like one, continuously distributed across the protein—that is, permittivity is intrinsically non-uniform inside a protein [23]. How large the corresponding inner and outer regions are, and how they depend on the environment, remains an open computational question [23,29]. In conclusion, calculating or measuring the permittivity of hydrated proteins remains a genuine challenge in fundamental biophysics.

Addressing permittivity is, however, also a key step in interpreting—and ultimately designing—biosensors. When measurement data are analyzed, the capacitance change associated with ligand binding is typically attributed only to macroscopic effects, i.e., changes in protein geometry [30], while the possibility of a genuine change in permittivity itself is generally overlooked. This is difficult to justify: the permittivity of a hydrated protein is known to depend substantially on the local degree of solvent exposure [23], and ligand binding necessarily changes that exposure—particularly around the binding pocket, through desolvation. It is therefore natural to reconsider the role of permittivity itself in binding processes, and hence in biosensing devices [31].

In this paper, we carry out in silico studies of the permittivity of hydrated proteins, building on a strategy introduced in [31] that complements existing analytical and computational approaches with a computationally inexpensive framework suited to the rapid analysis of large protein datasets. Our approach is grounded in well-established continuum models [23,32,33]—coarse-grained procedures that distribute permittivity continuously across the atoms or amino acids of a protein. Typically, this distribution is organized around the radius of gyration, effectively describing the protein as a shell-like structure; while this is adequate for near-spherical proteins, it becomes ineffective for markedly elongated ones [32].

Alternatively, we implement the calculation of static permittivity within the proteotronics framework [8,10], using as a structural proxy the coordination number of each amino acid, i.e., its number of nearest neighbors. This procedure has already been benchmarked against an independent approach based on the protein’s macroscopic dipole, yielding results in qualitative agreement with the sparse experimental data available [31]. In this paper, we extend the procedure to a structurally and functionally heterogeneous sample of proteins, diverse in structure and function. We find that the normalized permittivity depends only weakly on the specific interpolation function used to distribute it among amino acids—indicating that the topology of the folded protein, rather than the particular mathematical form adopted, is the dominant factor shaping the effective dielectric landscape. At the same time, it is worth focusing on functions whose parameters, once tuned, span a wide range of values, so as to capture the substantial dependence of permittivity on environmental conditions; we introduce functions whose parameters can plausibly be associated with the degree of hydration. We further show that the calculated permittivity correlates with the protein’s compactness. A non-trivial consequence of this heterogeneous dielectric model is that impedance measurements should depend on the position of the electrical contacts—a dependence that disappears under the assumption of a uniform dielectric constant, and that therefore constitutes a possible experimental signature of spatially varying protein permittivity. We test this prediction by comparing the calculated electrical impedance of apo-azurin [34] and of the novel protein NP_888769.1 [35], and find that the results indeed depend on the position of the electrical contacts, differing substantially from those obtained assuming a uniform permittivity distribution.

At present, systematic measurements of the dielectric properties of hydrated proteins remain too sparse and heterogeneous to allow quantitative calibration of any coarse-grained dielectric model. The goal of this work is therefore not quantitative—we do not aim to fix a single model or a definitive set of parameters—but rather to construct a physically motivated theoretical framework capable of generating experimentally testable predictions.

2. Materials and Methods

2.1. Materials

To test our procedure, we selected a sample of 60 proteins with significant differences in structure and function, including the same proteins in different activation states or taken from different organisms. The goal was to verify the ability of the procedure to group proteins that share similar characteristics or belong to the same family and to distinguish different configurations of the same protein. Additionally, for the same reason, proteins in the free-state or bound to other related proteins were also included. The structures were selected among the models obtained with X-ray diffraction or NMR, at high or very high resolution [36], which, at present, constitute the best representation of a hydrated protein [37]. Selection covers proteins with amino acid number,$N_{aa}$, between 55 and 828.

Among these proteins, there are 3 different models of the azurin from different species (1E65, 1AIZ, 1NZR) [34,38,39] and 4 models of lysozyme from different species (3LZT,1LZ1,162L,168L) [40,41,42,43], the bacteriorhodopsin light-driven proton pump in the light and dark-adapted configuration (2NTW,2NTU) [44], three complexes made of 2 and 3 units (2GY7, 1P9M, 2J8C) [45,46,47]; the first is angiopoietin-2 complexed with its natural receptor tyrosine kinase-2, Tie-2; the second is the cytokine IL-6 complexed with its two natural receptors, RecA and RecB; and the third is the photosynthetic reaction center with its three subunits, L, M, and H. Finally, the models of: a. IL-6 in its native state (1ALU) [48]; b. Angiopoietin-2 (Ang-2) in its native state (1Z3S) [49]; c. Receptor alpha of IL-6 in its native state (1N26) [50]; d. Ang-2 receptor, Tie-2, in its native state (2GY5) [45]. More details are reported in Table A1.

2.2. Theory & Methods

Our research is developed within the field of proteotronics, a coarse-grained description of the topological and electrical properties of proteins. This approach is time-efficient and still allows us to estimate the sensing signal provided by proteins used as active materials in biosensors [8,10,16,19,20].

We produce an equivalent of protein static relative permittivity, say k, starting from the intrinsic polarizability of each amino acid and its position in the 3D structure. In particular, each amino acid is assigned a relative permittivity$\epsilon \left(n\right),$which depends on its specific (dry) polarizability,${\epsilon }_i$, [31] and the coordination number, $L$ : The greater the number of neighbors, the fewer solvent molecules can reach it, or, in other words, the greater the probability that its effective permittivity remains close to its dehydrated value, and this holds regardless of the residue’s position relative to the protein’s center of mass . The coordination number is obtained by representing the protein with a network of nodes corresponding to the amino acid centroids and edges connecting close nodes. How close two nodes are considered, and the resulting network complexity, depend on the choice of a distance parameter, $R_C$ [8,10]. For appropriate $R_C$ values, the protein backbone is faithfully reproduced, and the coordination number can be used to describe the node position.

2.2.1. Models

We analyze the outputs of a pool of parametric continuous functions (models) suitable for distributing $\epsilon \left(n\right)$ values between the intrinsic values ${\epsilon }_i$, and the value of the solvent, $\mu$, here chosen to be the typical bulk value of 80. The values of ${\epsilon }_i$ were calculated as described in [31]; by using polarizability data from [21]. Different choices [22] slightly change the value of $k$ [51]. In principle, there are no constraints on the choice of a specific function or, in other words, here we do not attempt to propose a microscopic constitutive law. Rather, we compare different smooth distributions with the aim of assessing the robustness of the framework.

We therefore consider three families of functions: polynomial, exponential, and sigmoidal. In all the following models we use at most 3 different topological variables, i.e., the maximum and minimum number of nearest neighbors, $L_{MAX},L_{min}$calculated for the whole protein, and the average value $<L>.$ Finally, the independent variable is the coordination number, $L$, normalized to its maximal value: $x\left(n\right)={\displaystyle \frac{L\left(n\right)}{L_{MAX}}}$ and the average value of $\epsilon \left(n\right),<\epsilon \left(n\right)> =k$ is taken to be the effective protein permittivity .

• Linear

$$\epsilon \left(n\right)={\displaystyle \frac{\left(\mu -{\epsilon }_i\left(n\right)\right)}{1-\frac{L_{min}}{L_{MAX}}}}\left(1-x(n)\right)+{\epsilon }_i(n),$$

(1a)

• B. Polynomial

$$\epsilon \left(n\right)={\epsilon }_i\left(n\right){x\left(n\right)}^m+\mu \left(1-{x\left(n\right)}^m\right),$$

(1b)

• C. Sigmoidal 1

$$\epsilon \left(n\right)=\mu \left(1-w\right)\left(1-{x\left(n\right)}^m\right)+{\epsilon }_i\left(n\right){x\left(n\right)}^{m}w, w={\displaystyle \frac{{\left(x\left(n\right)\right)}^q}{{{\left(x\left(n\right)\right)}^q+\left(<x\left(n\right)>\right)}^q}}$$

(1c)

• D. Sigmoidal 2

$$\epsilon \left(n\right)=\mu -\left(\mu -{\epsilon }_i\left(n\right)\right){\displaystyle \frac{{\left(x\left(n\right)\right)}^q}{{{\left(x\left(n\right)\right)}^q+\left(<x\left(n\right)>\right)}^q}},$$

(1d)

• E. Exponential 1

$$\epsilon \left(n\right)=\left(\mu -{\epsilon }_i\left(n\right)\right)e^{-{\left({\displaystyle \frac{L\left(n\right)}{<L>}}\right)}^q}+{\epsilon }_i\left(n\right),$$

(1e)

• F. Exponential

$$\epsilon (n)=\mu \left(1-{x\left(n\right)}^m\right)e^{-{\left({\displaystyle \frac{L(n)}{<L>}}\right)}^q}+{\epsilon }_i\left(n\right){x\left(n\right)}^m$$

(1f)

Figure 1 shows the different shapes of the listed functions. In particular, the most probable value of the protein permittivity, obtained for $L=<L>$ , decreases with the value of the free parameter ($m$) (B,C,F). On the other hand, another parameter,$q,$ (C,D,E,F) tunes the sharpness of the function, not affecting the most probable value of the protein permittivity. Therefore, we propose to interpret $m$ as the hydration parameter, because for very high values of $m$ (>6) most of the residue permittivity, $\epsilon \left(n\right)$, is close to $\mu$ as in a completely hydrated protein, while, for $m$ smaller than 0.1, most of residues have values of $\epsilon$ small as in a dry condition [23]. Finally, the choice of $q$ regulates the sharpness of the function, thus setting the relative extent of the protein’s inner and outer regions [23]. Only comparison with new experimental data will be able to guarantee the validity of this interpretation and refine the parameters of the function.

3. Results

3.1. Compactness Index

The compactness index of a protein is introduced as the ratio between the maximum value of the coordination number, say $L_{MAX}$, and its average value$, <L>, C_{index}={\displaystyle \frac{L_{MAX}}{<L> }}$. This quantity captures the degree of symmetry of the nearest neighbor distribution. Indeed, while $C_{index}~2$ represents a fairly symmetric distribution, $C_{index}~1$represents a distribution dominated by high values of $L$, corresponding to internal amino acids and therefore a very compact structure (Fig. A1). The extreme case in which the mean and maximum values of $L$ are very close, for example in coiled-coil structures, is not analyzed here. Consequently, higher values of $C_{index}$ correspond to higher values of the protein permittivity. Thus, while not an independent confirmation of the model itself, we can identify $C_{index}$ as a tool for easily interpreting the permittivity as defined here. In Figure 2 we report the permittivity values vs $C_{index}$ . Data are calculated with the 6 models, using different values of the $m,q$, parameters and different values of $R_C$ . All the reported curves can be closely fitted by the linear relation $k\propto ln\left(C_{index}\right)$. This relationship is very strong (correlation larger than 98%) for Models B,C with values of the $m$parameter from small (0.25) to intermediate (2) — i.e., in our interpretation, under moderate hydration conditions. A weaker relationship is observed for Model F (correlation larger than 95%), while both in Models D and E the effective permittivity depends only weakly on $C_{index}$. Some more details in Appendix B.

In general, it can be observed that, compared to this relationship, the interaction radius, $R_C$ , plays a marginal role, independently on the value of parameters; specifically, in Figure 2 data for different values of $R_C$ are reported for $m$ =1 in model B, for $m$ =1, $q$ =1 in models C, F, and for $q$ =1 for models D,E. Only model A shows a moderate dependence on $R_C$ . In this model, $k$ has also a bimodal behaviour which is presumably due to presence of the third topological parameter, $L_{min}$.When the test function has two different parameters, $m$ and $q$, we report data with $q$ =1 using filled symbols and data with $q$ =6 using empty symbols. These figures provide, for the sample set, a preview of the range of permittivity values, $k,$ as implemented by the specific model, furthermore it easy to observe that as $m$ and/or $q$ increase, so does $k$ . Finally, in models D and E, $k$ remains limited to a very narrow range across the whole sample and model A, which has no parameters, provides quite large values of $k$.

3.2. Role of $R_C$

Although the value of $R_C$ does not alter the $k$($C_{index}$) dependence, the distribution of $k$ among the sample set depends on this parameter . Because of the scarcity of experimental data, we selected its value based on a plausibility test: In our sample, we identified several reference proteins useful to test the method. Specifically, these included three different azurins and four different lysozymes, from different biological species, a bacteriorhodopsin in different activation states, some different complexes and their constituents ( see also section 2.1).

We expect: a. very similar values for similar proteins; b. a slight difference between bacteriorhodopsin in the dark/light -adapted structure [20,44]; c. substantial differences between the compact angiopoietin-2 and the highly elongated Tie2 receptor, both composing the heterodimer 2GY7; e. Substantial differences among the constituents of the 1P9M trimer, particularly between the dense and compact interleukin IL-6 and the two receptors, which are more elongated and less compact; f. Substantial differences in the 2J8C trimer, where the two L and M chains are much more compact than the H chain.

We observed that different values of $R_C$ satisfy only some of these conditions and that only $R_C=6Å$ satisfies all of them, independently of the chosen model. This result agrees with previous investigations on the ability of a topological network to capture the main structural properties of proteins [52].

3.3. Model Selection

The comparison of the six models (A-F) is performed through the normalized permittivity, i.e., ($k-<k>$) . All models show a common trend over the set of selected proteins, while both the average$<k>$ , and the variance, $\sigma$, i.e., the relative difference among protein permittivities, are rather different (Figure 3a). Furthermore, not all proposed models allow one to adjust $<k>$ and$\sigma$ freely. In particular, Model A has the maximum variance, $\sigma$, and quite a high value of <k>, but neither of them can be adjusted because the model has no free parameters (Figure 3a). Models D and E, which use a single parameter, q, have very small variance and a rather high value of <k>. On the other hand, by adjusting the parameters of Models B (m), and C, F (m,q), it is possible to precisely modulate the mean value of <k>.

In the 4 panels of Figure 3 we report the main results:

The normalized permittivity for the six models is reproduced in Figure 3a. Notice that while Model A exhibits the maximal variance, model D shows the minimal one.

The dependence of the permittivity average, $<k>,$ and variance $\sigma$ on the model parameters is shown in Figure 3b, for model B. Notice that <k> grows with $m$ while $\sigma$ has its maximum at intermediate values of $m$ . This is in agreement with the interpretation of $m$ as a hydration indicator: for small and high values it tends to reproduce the dry and fully hydrated conditions, respectively, and both are limiting conditions in which distinguishability is lost. Similar results may be obtained also with models C and F.

An overview of absolute permittivity values is given by Figure 3c,d. Specifically, it is plotted for models A, D, E in Figure 3c and for models C,F in Figure 3d, in both cases using a wide range of parameters. Models D,E show very limited distinguishability, in particular for q=1, and cover a limited range of values of <k> . On the other hand, models C,F, as previously seen for model B, can cover a wider range of values of <k>, and have also a larger distinguishability.

In conclusion, the normalized permittivity is substantially independent of the choice of a specific model, suggesting that the proposed procedure is, in general, insensitive to the choice of the specific continuous function [31]. To simulate possible experimental data, however, some models work better than others — in particular the B, C, and F models, thanks to the wide range of permittivity values they can cover.

Therefore, we performed the following investigations using only one of them, specifically model B, which is a single-parameter (m) function. In particular, using $m$ = 0.25, we obtain, for the four lysozyme representations in our dataset, the permittivity value observed for a solvent content of 20% [23], i.e., $k$ close to 10.

3.4. Impedance Calculation

As a final step of the present investigation we numerically explore the response of a small (128 amino acids) blue-copper protein, azurin from Pseudomonas aeruginosa. This protein is taken as a model for studying electron transfer in metalloproteins [53,54,55,56,57,58] owing to its unusual copper oxidation–reduction mechanism, which is accompanied by distinctive optical properties [59]. Furthermore, its conductive response has been studied both experimentally [56,60,61] and theoretically [53,57,58], using monolayer junctions [60] as well as in single-molecule apparatus [62,63]. Here, for the sake of simplicity, we calculate the impedance of this protein in the apo form, in an ideal single-molecule apparatus [63], i.e., in a solvent such as PBS [62]. The procedure is numerically implemented in the framework of proteotronics. Specifically, after the protein equivalent network is built up, each link between a couple of nodes $a,b$ is dressed with the elementary impedance [8]:

$$Z_{ab}=g_{ab}\left({\displaystyle \frac{\rho }{1+i{\epsilon }_{ab}{\epsilon }_0\rho \omega }}\right)$$

(2)

where$g_{ab}$ is a geometrical factor accounting for the distance and relative orientation of the corresponding amino acids, ${\epsilon }_0$ is the vacuum dielectric constant, ${\epsilon }_{ab}$=${(\epsilon }_a+{\epsilon }_b)/2$ is the intermediate value of permittivities, ${\epsilon }_a, {\epsilon }_b$ are here calculated according to model B $m$=0.25.

We perform calculations in a wide range of frequencies (0-10⁵ rad/s), by attributing, for simplicity, a single value of resistivity to all amino acids, specifically, =610⁶ m, which is a typical resistivity value for non-metal bulk materials at room temperature.

The analysis is performed by solving the protein-network equivalent which is ideally contacted with an AC bias, $V\left(\omega \right)$. The flow of current inside the network, $I\left(\omega \right)$ , follows, as usual, the path of minimum resistance. Finally, we calculate the protein complex impedance $Z\left(\omega \right)={\displaystyle \raisebox{1ex}{$V\left(\omega \right)$}\!\left/ \!\raisebox{-1ex}{$I\left(\omega \right)$}\right.}$.

Figure 4a shows the imaginary part of impedance of apo-azurin [34] normalized to the protein resistance, $ReZ\left(0\right)$. Imaginary impedance, $ImZ\left(\omega \right)$ is a very sensitive tool to test the dielectric properties of a material, since it allows fine determination of the characteristic time(s) of the process and, consequently, the protein permittivity value. It is commonly analyzed in terms of the equivalent Randles circuit [17,30,64] which mainly consists of a constant-phase element$, C^{\alpha }{C}^1 represents the ideal capacitor$ , in parallel with a resistive element, $R$. Additional elements may be added to represent the role of the solution, the roughness of interfaces, and so on [17]. Calculations were performed using two different contact positions, in particular, up-down, as detailed in [63], using CYS3 as input and ASN42 as output, and left-right (ASN16 as input, SER66 as output). Since apo-azurin has quite a spherical shape, the global impedance is only weakly affected by the position of contacts. In both contacting configurations, the protein permittivity, say${\epsilon }_r$ , is calculated from the maximal impedance value, using the relation $\rho {\epsilon }_r{\epsilon }_0=\tau ={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\omega }_{MAX}$}\right.}$ , with$\tau$the protein characteristic time and ${\omega }_{MAX}$the frequency producing the maximal value of the imaginary impedance,$-ImZ\left(\omega \right)$ [8]. We found for ${\epsilon }_r$ values similar to, or very close to, $k$, the average value calculated in Section 3. By contrast, we have also explored the possible effects of an elongated configuration on impedance measurements. In particular, we analyzed the features of protein NP_888769.1 from the Gram-negative bacterium Bordetella bronchoseptica (PDB code 2L25) [35]: it is similar to azurin in size (141 amino acids) and in the calculated average permittivity, $k$ , although structurally quite different [36]. In this case, changing the position of the electrical contacts produces quite different results: the left-right (PRO76-GLU141) configuration produces an impedance very close to that of an ideal resistance-capacitive (RC) parallel circuit, i.e., describes a system with a single characteristic time, while the up-down (TYR47-ALA89) configuration shows a flattened shape typical of a system of multiple characteristic times, in this case, instead of an ideal capacitor, the equivalent circuit is better described by a constant-phase element ($C^{\alpha },\alpha <1$) . These results strongly suggest that the broad distribution of amino acid permittivities may produce both coherent (single-characteristic-time) and incoherent (multiple-characteristic-time) responses depending on the probe (electrical current) pathways. Finally, the up-down configuration shows a considerably higher value of permittivity. This is presumably due to the different pathways followed by the ideal current in the two configurations: the first one crosses a smaller number of boundary amino acids than the latter, thus producing a permittivity value closer to the average, while the latter highlights the boundary amino acids thus producing a higher permittivity value.

To test this hypothesis, we calculated the impedance of the two proteins in both the contacting configurations by attributing each of the amino acids the same value. The result is plotted in Figure 4c, where the 4 different impedances are calculated assuming $k$ =13 for all amino acids. In this case, all of them produce the ideal result (a single characteristic time) with the maximum equal to $ReZ\left(0\right)$/2, which fully agrees with the expected response of an ideal RC parallel circuit. This difference should be used as an ultimate test to understand if the distribution of permittivity is uniform or inhomogeneous. A pictorial representation of the proteins with the two different contact positions is reported in Figure 5.

4. Discussion and Conclusions

This paper introduces a computationally efficient, topology-based framework for estimating the effective static permittivity, k, of hydrated proteins—a quantity that, despite being well defined in classical electrodynamics for bulk dielectrics, has so far resisted any unified computational treatment for proteins: existing methods are either too complex for routine use or insufficiently precise [23], and direct measurement remains experimentally challenging. The advance we propose is not a new microscopic constitutive law, but a coarse-grained, structure-only proxy—the coordination number of each amino acid—that is inexpensive to compute, requires no fitting to atomic-level chemistry, and can therefore be applied uniformly across large, structurally diverse protein datasets.

The method is implemented within proteotronics [8,10], a bottom-up procedure that derives global protein properties—such as the impedance response to an AC bias—directly from amino-acid-level features. The effective permittivity is built by continuously distributing permittivity values across the amino acids of a folded protein, using the three-dimensional structure deposited in the Protein Data Bank [36] as the reference standard. Compared with earlier implementations of the continuous-distribution approach [23,32,33], our strategy is coarse-grained at the amino-acid level rather than the atomic level, and distributes permittivity by ordering amino acids according to coordination number; at this stage it does not yet capture residue-specific chemical features (each amino acid is treated as electrically neutral [21], with no explicit propensity to interact with water). Among the possible structural descriptors of the local dielectric environment, the coordination number offers an attractive compromise between physical significance and computational efficiency: residues with many neighbors are generally less solvent-exposed and therefore experience an environment closer to the protein interior, while poorly coordinated residues are more strongly hydrated. Other descriptors, such as solvent-accessible surface area (SASA) [65,66], provide a more direct estimate of solvent exposure at the atomic level, but the coordination number is purely topological, computationally inexpensive, and robust to small structural fluctuations.

Using several smooth interpolation functions to distribute permittivity among amino acids, we obtain the effective protein permittivity, k, as the average of the individual amino-acid permittivities—the quantity we expect to be measurable at the macroscopic scale, for example via optical spectroscopy [67], though not necessarily identical to values obtained from smaller-scale experiments. A central finding of this work is that the normalized permittivity, (k − ⟨k⟩), follows a closely similar trend across the entire protein sample regardless of the specific interpolation function adopted. This indicates that the effective dielectric landscape of a protein is governed primarily by its topology, rather than by the particular mathematical form used to distribute permittivity—a degree of robustness that was not guaranteed a priori. This microscopic, topology-based estimate also compares well with an entirely independent, macroscopic approach based on the protein’s global dipole moment [31].

By contrast, the absolute value of the permittivity does depend on the specific interpolation function, which should therefore be chosen to match what is experimentally known. We selected a small number of functions whose parameters, once calibrated, can reproduce values ranging from those typical of a dehydrated protein to those of the bulk solvent. We further show that the calculated permittivity scales logarithmically with a purely topological quantity that captures the compactness of the protein, C_index—providing a simple, structure-only estimator of a protein’s expected permittivity range ahead of any dielectric calculation.

As a concrete application, we compute the electrical imaginary impedance, ImZ, of two proteins—azurin and the novel protein NP_888769.1—a typical output of electrochemical impedance spectroscopy (EIS). These results substantiate Warshel’s point that the very definition of permittivity is subtler than it appears [68,69]: in both proteins, the permittivity extracted from the impedance calculation is close to, but distinct from, the plain average of the amino-acid permittivities, k. The discrepancy arises because solving the equivalent impedance network does not weight every amino acid’s permittivity equally, unlike a simple average: the network is built from elementary impedances, each tied to a specific amino acid’s permittivity, and the total impedance—and hence the effective permittivity it implies—depends on the current pathways selected by the network, which in turn depend on where the electrical contacts are placed. This dependence is most pronounced for a non-spherical protein such as NP_888769.1, and it constitutes a previously unrecognized experimental signature of internal dielectric heterogeneity: a measurable, contact-position-dependent response that simply disappears if permittivity is assumed uniform. This is, we believe, the paper’s most practically relevant advance: it gives experimentalists a concrete, testable way to probe whether a protein’s permittivity is uniformly or heterogeneously distributed, using nothing more than a standard EIS setup with two different contact geometries.

The present framework is intentionally coarse-grained. It neglects explicit solvent molecules, residue-specific hydrogen-bond networks, conformational dynamics, and the frequency dependence of the permittivity itself. These simplifications are what make rapid, large-scale calculations possible while still capturing the dominant topological features that shape the heterogeneous dielectric landscape; the trade-off is that the framework requires experimental input to be properly calibrated.

Ultimately, this work raises more questions than it answers: how much genuine variability in permittivity exists among proteins—modest, as suggested in [70], or substantial —and how large should the corresponding variance be in our model? What changes should be expected when structure, temperature, pressure, frequency, or hydration are varied? How should singular configurations, such as coiled coils—where the mean and maximum coordination number, $⟨L⟩,L_{MAX}$, nearly coincide—be described within this framework? And how can permittivity values obtained from different types of measurement, such as EIS versus refractive index [67], be reconciled with one another? We regard these open questions not as weaknesses of the approach, but as the natural next steps that a physically grounded, computationally accessible framework now makes possible to formulate precisely.