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The influence of radiofrequency electromagnetic exposure on ligand binding to hydrophobic receptor proteins is a plausible early event of the interaction mechanism. A comprehensive quantum Zeeman-Stark model has been developed which takes into account the energy losses of the ligand ion due to its collisions inside the receptor crevice, the attracting nonlinear endogenous force due to the potential energy of the ion in the binding site, the out of equilibrium state of the ligand-receptor system due to the basal cell metabolism, and the thermal noise. The biophysical "output" is the change of the ligand binding probability that, in some instances, may be affected by a suitable low intensity exogenous electromagnetic "input" exposure, e.g., if the depth of the potential energy well of a putative receptor protein matches the energy of the radiofrequency photon. These results point toward both the possibility of the electromagnetic control of biochemical processes and the need for a new database of safety standards.

A quantum–mechanical model is proposed to evaluate the bioelectrochemical transduction of low-intensity acoustic exposure, with reference to the binding process of a ligand (messenger) ion to a receptor protein. The interaction of the acoustic exposure with the ligand–receptor chemical reaction is assumed to be mediated by the endogenous electric field of the receptor binding crevice. The displacement of the solvent (water) molecules surrounding the protein, when exposed to the acoustic field, induces the displacement of the receptor molecules from their initial thermal equilibrium position, so that the attractive endogenous force exerted by the receptor on the ligand becomes time-varying at the frequency of the acoustic exposure. A suitable perturbing Hamiltonian operator is evaluated, which transduces the exogenous acoustic exposure into a kind of Stark effect. The density operator method is utilized to statistically evaluate the binding probability in presence of thermal noise. The typical reference intensity (power density) of the incident acoustic wave is 0.1 mW cm −2 in the frequency range 0.5–5 MHz. Usual first-order approximation techniques are employed to evaluate in closed form the change in the binding probability, as compared as usual with its value at thermal equilibrium. By doing so, it is possible to identify a range of parameter values where resonant effects due to the acoustic exposure occur. Thus, a direct numerical integration of the equations is performed in order to validate these first-order findings. The resulting effects are very small, as compared with the binding probability under unperturbed conditions, i.e., at thermal equilibrium, which appears to be the limiting factor of the effectiveness of low-intensity acoustic exposure. These conclusions suggest future research directions, accounting for the out-of-equilibrium living state of a cell and its related basal metabolism, in order to explain those published experimental results which suggest that the binding probability may be significantly affected by a low-intensity exogenous acoustic field.

This paper addresses the problem of evaluating theoretically the effects of electromagnetic (EM) exposure on biological systems. EM fields can affect the chemical activity of ions while they interact with the membrane receptors of a cell during a binding process. A coulombic three-state Zeeman model for the binding site is proposed, and the ion adsorption under low-frequency magnetic exposure is described by means of the density operator. The system interactions with the thermal bath are accounted for by a suitable set of lifetime parameters. The limits of a first-order perturbation approach, which allows us to find a closed-form expression for the ion-binding probability, are discussed. The biological effectiveness of a Helmholtz coil exposure system is rather small if computed according to the simplifying assumptions of the paper.

This paper attempts to elucidate the mechanism of action of electromagnetic exposure on ligand-receptor binding. The classical Langevin-Lorentz model, which can be used to describe the adsorption process of a messenger ion, is reviewed and discussed. We conclude that low intensity exposure does not affect appreciably the ion dynamics in the presence of background thermal white noise. A more realistic evaluation of the endogenous field present in a binding site leads to a quantum model based on weak Zeeman-Stark effects. The case of the Zeeman effect is studied in detail assuming a three-state binding site. The density operator method is used, introducing suitable lifetimes which model the thermal bath interaction. The closed form expression for the binding probability is found, as a function of the ligand-receptor parameters and of the electromagnetic sinusoidal exposure.

The influence of electromagnetic exposure on ligand binding to receptor proteins is a putative early event of the interaction mechanism leading to biological effects. The most recent development of the quantum Zeeman-Stark model is reviewed, addressing the following points: losses due to the collisions of the ligand ion inside the hydrophobic binding crevice and thermal noise; evaluation of the attracting endogenous force of the binding site from the protein data base; out of equilibrium state of the ligand-receptor system due to the basal cell metabolism.

The binding site of most metalloproteins is hydrophobic, so that the ligand metal ion (mass M and charge Q) is attracted by the receptor protein in a dehydrated environment1. Therefore the ion collision frequency can be several order of magnitude lower than in bulk water because the residual water molecules are few2. Furthermore, the protein atoms are slightly displaced by the docking/sailing ion, with respect to their equilibrium positions obtained from the protein data bank. The actual net attracting force is lower than the value calculated on the basis of these fixed atomic positions. Both features are necessary conditions for a ligand-protein system to be susceptible to low-intensity e.m. exposure3. Once the resulting ion potential energy Uend(r) [J] is available versus the ion distance \(\vec r\) [m] from the centre of the binding site, a quantum Zeeman-Stark model of the binding process allows the prediction of its susceptibility to low intensity, e.m. exposure4, i.e., below the current safety standards, at r.f.. A simple and isotropic relationship for Uend(r) [J] has been developed, which can be fitted to the characteristics of the ion-protein system obtained from the protein data bank (fig.1). The binding potential energy depends on four parameters only, i.e. U0 [J], ωend [Hz], ξ [Jm] and R0 [m]. The minimum value of the potential energy at \(\vec r\) = 0 is Uend (0)= -U0. For small values of \(\vec r\) it is Uend ≅-U0 +(1/2)Mω end 2 r2 and, for large values of \(\vec r\), Uend≅-ξ/r, where the order of magnitude of the characteristic distance R0 corresponds to the maximal attracting force (-∇Uend, where ∇ is the nabla operator): $${U_{end}}\left( r \right) = - \xi /r + \left\{ {\xi /r - {U_0} + \xi /{R_0} + \left( {\xi /2R_0^2 - {U_0}/{R_0}} \right)r + \left( {M\omega _{end}^2/2 + \xi /6R_0^3 - {U_0}/2R_0^2} \right){r^2}} \right\}\exp \left( { - r/{R_0}} \right)$$ (1)

The dynamics of the Zn-protoporphyrin IX has been assessed by means of molecular dynamics methods. The conformational rearrangements of the protoporphyrin atoms during the Zn2+ docking (sailing) into (from) the center of its binding site occurs in the picosecond scale, so that any microwave electromagnetic exposure can be considered as almost constant, with respect to the dynamics of the constituent atoms induced by the ion displacement. The computer simulation of the complexation process with and without a low-intensity electromagnetic field has demonstrated the inhibitory effect of the exposure on the complex formation.1 This result has been confirmed experimentally at 2.45 GHz.1,2 We have tested the predictive ability of the quantum Zeeman-Stark (Z-S) model3 of ligand binding against the Zn-protoporphyrin system.

The published experimental results suggest that e.m. exposure (from sub-ELF to RF) at fields or S.A.R. values below the current safety standards, can affect biological processes. Two elementary processes are good candidates to be the first interaction steps: the binding of messenger ions to their receptor proteins and the transport of messenger ions inside protein channels. In both cases the ion dynamics can be often described in terms of the classical Langevin-Lorentz (L-L) model1–4. The closed form integration of the L-L equations under ad hoc simplifying conditions was performed in1–3, for the first time. This rather comprehensive approach includes, as particular cases, all the classical models so far published. In order to analyse or predict the experimental results, the values of the various parameters which enter the L-L equation must be physically plausible, keeping the number of fitting parameters as low as possible. Moreover, their value must be consistent with the effectiveness of the low value of the exposure density. Toward such goals, the following features must be considered. The binding or channel crevice must be hydrophobic, so that it repels the solvent water molecules and strips away the ion hydration shells. If the gradient of the ion endogenous potential energy inside the crevice is an highly non-linear function of the spatial coordinates, the water dipoles can be drifted away from the protein crevice by the resulting endogenous dielectrophoretic force. Therefore, the ion moves in the ballistic Knudsen regime and its Langevin collision frequency can be several order of magnitude lower than in bulk water5. The resulting energy losses are small, so that the ion dynamics become more sensitive to low-intensity e.m. exposure. However, the exogenous signal is typically unable to overcome the Langevin random force (thermal noise) if the ion-protein system, in absence of the e.m. exposure, is at thermal equilibrium. On the other hand, the basal state of any ion-protein systems in a living cell is maintained out of thermal equilibrium by the cell metabolism.

One-dimensional polyelectrolytes are proposed as molecular devices based on soliton propagation. The use of such “molecular wires” as delay lines or shift registers raises the problem of their electrical screening, because, when a large number of these wires are densely packed, cross-talking between them might occur. In order to overcome such a drawback, a detailed study of the polyelectrolyte behavior in an ionic solution has been developed, under equilibrium conditions. The results bring into evidence a condensation process, around each molecular wire, of the mobile ions in the solution. The consequent self-screening effect could be relevant in reducing cross-talking.

It has been shown that low-power ultrasound (US) has a significant effect on the rate of fresh fracture healing in animals and humans. The action of US can involve a direct mechanical effect (microstrain) or an indirect electrokinetic effect. To test the electrokinetic hypothesis, a four-electrode measurement technique is used to evaluate the effects of externally applied US on cell membrane impedance. Results indicate that US significantly affects the membrane impedance in the frequency range predicted by electrochemical modeling to be relevant to specific absorption, 5-100 kHz. The action of US via an electrokinetic pathway implies the displacement of ions in the diffuse double layer via the local pressure wave. Results obtained with a chloride-ion-depleted solution showed a significant US modification of membrane impedance in a higher frequency range (50-300 kHz), which appears to confirm that the effect of US at the membrane surface is ion-dependent. >