## Theoretical model of the electrical properties of a single olfactory receptor sensing action

ORs belong to the largest and important class of GPCR receptors (G protein-coupled receptors), sharing with this kind of receptors the seven-helices transmembrane structure. Unfortunately, the information about the structure at atomic level is missing for ORs while it is available for rhodopsin (GPCR photonic receptor). Therefore, we have developed a model, which takes advantage of the common coarse grain structure of ORs and bovine rhodopsin.

Our device under test (DUT in the following) is a single molecule of bovine rhodopsin, inserted between two metallic contacts in some environment (membrane, biological solution, antibodies, etc.).

The DUT is modelled as an equivalent circuit consisting of an impedance network (IN). The equivalent circuit can be presented as a simple non-directed graph. The nodes (vertices) of this graph correspond to the aminoacids (aminoacid residues) of the protein (348 for the bovine rhodopsin) and the links (edges) between any couple of nodes characterise some kind of interaction between aminoacids, which are neighbouring in space within a given radius. Two more nodes acting as contacts are then introduced.

The contacts are linked to a given set of aminoacids (each contact linked to at least one aminoacid). The environment is not taken into account at this stage. The elementary impedance (impedance of the link) is taken as the most usual passive equivalent AC circuit made of a resistor in parallel with a capacitor.

To construct the Impedance Network, the aminoacids are considered as uniform spheres with the center in the alpha carbon atom and a typical van der Waals radius, Ra=5Å. When two spheres overlap they are assumed to be neighbouring and the impedance edge should be inserted between the corresponding nodes of the circuit. Otherwise there is no interaction between aminoacids and thus no link between the corresponding nodes.

The determination of the value of each impedance can be fixed with increasing degree of complexity. Here we consider three different approaches.

(i) As first approach, the impedances of the edges are taken to be all the same.

(ii) In a second approach, Zi,j is taken to be proportional to the li,j as for a simple ohmic resistor and planar homogeneous capacitor.

(iii) In a third approach, by assuming that the cross-sectional area of resistor and capacitor is equal to the cross-section area of overlapping spheres.

Further approaches can be introduced to allow including properties of real aminoacids.

In going from model (i) to (iii) we exploited a better sensitivity of the network to the change of its structure. For the three models presented it can be observed a systematic decrease of |Z| at increasing Ra (radius of electrical interaction) reflecting the increasing importance of parallel with respect to series connections. The continuous increase of Ra is equivalent to a continuous three-dimensional contraction of the structure and represents an example of a spatial conformation dependence of |Z|. Everywhere below, by default we use the model (iii).

Such an equivalent circuit can be used for modelling the network with the value of all the elementary impedances fixed (perfect impedance network) or with each elementary impedance allowed to take two or more values stochastically thus becoming a Random Impedance Network.

For the comparison of two spatial configurations of the same molecule one should build equivalent circuits for both of them. Even if the primary structure of the molecule remains the same, distances between aminoacids change and therefore the topology of the network will in general change. In the second configuration some new links should arise, some old should disappear. In particular, when the elementary impedances depend on distances, the circuit should become even more sensitive to the change of configuration.

The main conclusion that can be drawn is that the maximum variation of impedance due to a conformational change is expected up to about 30% and corresponds to a an interacting radius in the range 5 < R < 25Å

This prediction concerns a frozen network (static network) in which the atomic positions are fixed at the equilibrium values. On the other hand, fluctuations of the atomic positions, due to thermal motion imply an impedance noise, whose level, in comparison with the impedance variation due to the conformational transition and with the electrode/amplifier noise, is crucial to the actual detection of the ligand capture by the GPCR. Therefore, here we extended the previous model to include the effect of the thermal motion on the electrical response to an AC field. To this purpose, we applied a classical harmonic oscillator and then consider the case of a quantum harmonic oscillator.

We have shown that the thermal random fluctuations will not mask the expected electrical response upon the conformational change.

Our device under test (DUT in the following) is a single molecule of bovine rhodopsin, inserted between two metallic contacts in some environment (membrane, biological solution, antibodies, etc.).

The DUT is modelled as an equivalent circuit consisting of an impedance network (IN). The equivalent circuit can be presented as a simple non-directed graph. The nodes (vertices) of this graph correspond to the aminoacids (aminoacid residues) of the protein (348 for the bovine rhodopsin) and the links (edges) between any couple of nodes characterise some kind of interaction between aminoacids, which are neighbouring in space within a given radius. Two more nodes acting as contacts are then introduced.

The contacts are linked to a given set of aminoacids (each contact linked to at least one aminoacid). The environment is not taken into account at this stage. The elementary impedance (impedance of the link) is taken as the most usual passive equivalent AC circuit made of a resistor in parallel with a capacitor.

To construct the Impedance Network, the aminoacids are considered as uniform spheres with the center in the alpha carbon atom and a typical van der Waals radius, Ra=5Å. When two spheres overlap they are assumed to be neighbouring and the impedance edge should be inserted between the corresponding nodes of the circuit. Otherwise there is no interaction between aminoacids and thus no link between the corresponding nodes.

The determination of the value of each impedance can be fixed with increasing degree of complexity. Here we consider three different approaches.

(i) As first approach, the impedances of the edges are taken to be all the same.

(ii) In a second approach, Zi,j is taken to be proportional to the li,j as for a simple ohmic resistor and planar homogeneous capacitor.

(iii) In a third approach, by assuming that the cross-sectional area of resistor and capacitor is equal to the cross-section area of overlapping spheres.

Further approaches can be introduced to allow including properties of real aminoacids.

In going from model (i) to (iii) we exploited a better sensitivity of the network to the change of its structure. For the three models presented it can be observed a systematic decrease of |Z| at increasing Ra (radius of electrical interaction) reflecting the increasing importance of parallel with respect to series connections. The continuous increase of Ra is equivalent to a continuous three-dimensional contraction of the structure and represents an example of a spatial conformation dependence of |Z|. Everywhere below, by default we use the model (iii).

Such an equivalent circuit can be used for modelling the network with the value of all the elementary impedances fixed (perfect impedance network) or with each elementary impedance allowed to take two or more values stochastically thus becoming a Random Impedance Network.

For the comparison of two spatial configurations of the same molecule one should build equivalent circuits for both of them. Even if the primary structure of the molecule remains the same, distances between aminoacids change and therefore the topology of the network will in general change. In the second configuration some new links should arise, some old should disappear. In particular, when the elementary impedances depend on distances, the circuit should become even more sensitive to the change of configuration.

The main conclusion that can be drawn is that the maximum variation of impedance due to a conformational change is expected up to about 30% and corresponds to a an interacting radius in the range 5 < R < 25Å

This prediction concerns a frozen network (static network) in which the atomic positions are fixed at the equilibrium values. On the other hand, fluctuations of the atomic positions, due to thermal motion imply an impedance noise, whose level, in comparison with the impedance variation due to the conformational transition and with the electrode/amplifier noise, is crucial to the actual detection of the ligand capture by the GPCR. Therefore, here we extended the previous model to include the effect of the thermal motion on the electrical response to an AC field. To this purpose, we applied a classical harmonic oscillator and then consider the case of a quantum harmonic oscillator.

We have shown that the thermal random fluctuations will not mask the expected electrical response upon the conformational change.