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\DOI{10.5802/crchim.318}
\datereceived{2023-10-19}
\daterevised{2024-04-17}
\dateaccepted{2024-05-16}
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\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Protein-based Josephson junction}

\alttitle{Jonction Josephson \`{a} base de prot\'{e}ines}

\author{\firstname{Gleb} \lastname{Zilberstein}\CDRorcid{0009-0000-4200-6599}\IsCorresp}
\address{SpringStyle Tech Design Ltd, str, Oppenheimer 7, 7670107, Rehovot, Israel}
\email[G. Zilberstein]{post4gleb@gmail.com}

\author{\firstname{Emmanuil} \lastname{Baskin}}
\addressSameAs{1}{SpringStyle Tech Design Ltd, str, Oppenheimer 7, 7670107, Rehovot, Israel}

\author{\firstname{Svetlana} \lastname{Zilberstein}}
\addressSameAs{1}{SpringStyle Tech Design Ltd, str, Oppenheimer 7, 7670107, Rehovot, Israel}

\shortrunauthors

\keywords{\kwd{Biological molecules}\kwd{Josephson
junctions}\kwd{Superconductivity}\kwd{Structure analysis of biological
particles}\kwd{Nanoparticles}}

\altkeywords{\kwd{Mol\'{e}cules biologiques}\kwd{Jonctions
Josephson}\kwd{Supraconductivit\'e}\kwd{Analyse de la structure des
particules biologiques}\kwd{Nanoparticules}}

\begin{abstract}
For the first time, a Josephson junction based on a biological molecule
has been created.
A device consisting of a single macromolecule or biological particle
(MM or BNP) and a Josephson junction (JJ) is proposed. The MM or BNP is
placed in the gap between the superconducting electrodes of the JJ. The
choice of the Josephson contact as a device for studying the properties
of nanometer macromolecules or bionanoparticles makes sense because the
scales of the coherence length of Cooper pairs in superconductors and
the sizes of MMs or BNPs coincide. 
\end{abstract}

\begin{altabstract}
Pour la premi\`{e}re fois, une jonction Josephson bas\'{e}e sur une
mol\'{e}cule biologique a \'{e}t\'{e} cr\'{e}\'{e}e.
Un dispositif constitu\'{e} d'une seule macromol\'{e}cule ou particule
biologique (MM ou BNP) et d'une jonction Josephson (JJ) est
propos\'{e}. La MM ou BNP est plac\'{e}e dans l'espace entre les
\'{e}lectrodes supraconductrices de la JJ. Le choix du contact
Josephson comme dispositif pour \'{e}tudier les propri\'{e}t\'{e}s de
macromol\'{e}cules nanom\'{e}triques ou de bionanoparticules a du sens
car les \'{e}chelles de la longueur de coh\'{e}rence des paires de
Cooper dans les supraconducteurs et les tailles des MM ou des BNP
co\"{i}ncident.
\end{altabstract}

\maketitle

\vspace*{1pc}

\twocolumngrid

\end{noXML}

\section{Introduction}

Nanotechnology makes it possible to create nanoscale structures that
can be used to study the electronic properties and electronic structure
of a macromolecule or biological nanoparticle (MM or BNP) \cite{1,2,3}.
In single-molecule electronics~\cite{7}, it was proposed to use the
Josephson junction (JJ)~\cite{4,5,6} to study the electronic properties
of small organic molecules and for different versions of Andreev
spectroscopy and molecular electronics methods and devices.

The purpose of this work is to show the possibility of the Josephson
effect in a JJ based on an insulator barrier of MM or BNP. To do this,
we propose to use special superconducting nanoscale devices including
the studied MM or BNP. In this case, the barrier size is determined by
that of the MM, 2--2000~nm. Nevertheless, if the coherence length of
Cooper pairs in superconductors and the sizes of MMs or BNPs are of the
same order of magnitude, the Josephson effect may take place.
Realization of the Josephson effect let us to measure electrophysical
parameters (\mbox{tensors} conductivity, electronic
\mbox{polarizability} and magnetic susceptibility) of the MM and its
resonance frequency. The use of a JJ to determine the electronic
properties of single MMs or BNPs can be an additional approach to
conventional structural analysis. 

Al, Sn, Nb have stable oxide films on the surface due to interaction
with atmospheric oxygen.\ Oxides of these metals are very often used in
chemistry and biochemistry as adsorption, separation and bioconjugate
media for chromatography of biological molecules and sorbents for a
wide range of biochemical and chemical compounds. This makes it
possible to use ``Metal--Oxide--(MM or BNP)'' structures to create a
JJ. Types of chemical bonding between a BNP (protein, biological
membrane, biological complex, DNA, RNA) at room temperature are
electrostatic and hydrophobic interactions~\cite{12}. At room
temperature the types of interactions between the metal oxide and the
MM or BNP depend on the type and number of ionic, hydrophobic, and
hydrophilic groups on the BNP surface.

For example, an aluminum oxide film on the surface of an aluminum
electrode can be a good material to create a JJ based on a conjugate
with a MM or BNP. For example, the coherence length is 1600~nm for
aluminum in the superconducting state~\cite{14}.

\section{Materials and methods} \label{sec2}
\vspace*{-.15pc}
\subsection{JJ with protein-based insulation barrier} \label{sec2.1}
\vspace*{-.15pc}
\subsubsection{Preparation of JJ with protein on metal{\hfill\break}
electrode surface} \label{sec2.2.1}
\vspace*{-.1pc}

To prepare a protein--metal structure on the metal (Sn or Nb) electrode
surface, we used a polyacrylamide gel (PAAG) bead with a single green
fluorescent protein (GFP) molecule (28~kDa; length 4.2~nm; diameter
${\sim}$2.4~nm; Merck Catalog N~14-392) in an individual gel bead. We
prepared an individual bead (diameter ${\sim}$1~$\upmu$m) from PAAG with
a single GFP molecule inside the bead using 10\% PAAG and immobilines
(pH ${=}$ 9)~\cite{8}. Beads of 10\% PAAG with immobilines (pH ${=}$ 9)
(PAAG with functional acidic and basic groups) were placed in contact
with flat PAAG. Initially GFP molecules were located in the pores of
the flat PAAG. The concentration of the GFP in the flat gel was
0.01~$\upmu$g/mL. GFP was introduced into PAAG using electrophoresis.
The distribution of GFP molecules in the gel pores and transfer of the
GFP from the gel to the gel bead was controlled using a fluorescent
microscope~\cite{8,9,10,11}. The pore size of the 10\% PAAG was
${\sim}$70~nm. A bead made from immobilines with a PAAG adsorbed
(extracted) single GFP molecules from a flat PAAG. At pH ${=}$ 9 in
immobiline, the GFP protein is well transferred into the immobiline
bead from a flat PAAG with pH ${=}$ 7. We then transferred the immobiline
beads from the individual GFP molecules to the surface of the Nb or Sn
electrode. The gel bead transfer to the metal surface was carried out
using a nanomanipulator.

Then we applied a 1~V electrical signal to transfer the GFP molecule
from the immobiline gel bead to the surface of the Nb or Sn electrode
(5~s period of electrophoresis). The counter electrode was another Nb
(or Sn) electrode. After the transfer of the GFP to the surface of the
Nb (or Sn) electrode, fluorescence was used to control the presence of
the GFP on the surface of the Nb (or Sn) electrode. The presence of GFP
fluorescence on the surface of the electrode indicated the transfer of
a single GFP molecule to the electrode. The voltage for electrophoresis
is less than the decomposition potential of water and therefore no
chemical reactions should have occurred with the protein molecule when
it was transferred from the gel bead to the metal (Nb or Sn) surface.

At room temperature the binding mechanism of the GFP to the Nb or Sn
electrode is associated with electrostatic and hydrophobic interactions
between the metal, metal oxide layer, and the protein
\mbox{molecule}~\cite{12}. 

PAAG and other gels have been successfully used for fluorescence
spectroscopy of individual (single) macromolecules, such as proteins.
Gel pores with cell sizes commensurate with the size of macromolecules
(proteins) fix the macromolecule well and facilitate the task of
manipulation and transfer of the macromolecule during the manufacture
of the experimental setup~\cite{8,9,10,11}.
The properties of GFP are well defined and it is a well observed single
molecule, which facilitates the JJ assembly process and quality
control.

We used the fluorescence microscopy method from~\cite{30,31} to achieve
the experimental conditions to work with a single GFP molecule.
Isolation of single molecules in PAAGs is routinely and reliably used
for the isolation and detection of single GFP molecules in single
molecule fluorescence spectroscopy~\cite{32}. To monitor the formation
of ``single GFP molecule''--metal structure on the electrode surface,
we measured the fluorescence of the GFP under a fluorescent
microscope~\cite{8,9,10,11,31}.

The GFP on the Nb or Sn electrode was dried at room temperature in a
helium atmosphere.\ After drying the protein layer, the protein-coated
Nb or Sn electrode was cooled. Then we assembled the JJ.

A top ``needle'' electrode covered with a GFP on a metal surface
touched the lower metal electrode. The electrode with GFP was moved
using a vertical nanomanipulator. The contact area was
$10^{-4}~\upmu\text{m}^2$. The electrical circuit (JJ made from
Sn--GFP--``needle'' Sn electrode; or JJ made from Nb--GFP--``needle'' Nb
electrode) was closed and the voltmeter connected to the JJ for
measurements (Figure~\ref{fig1}).

\begin{figure*}
\includegraphics{fig01}
{\vspace*{-.3pc}}
\caption{\label{fig1}(a)~Scheme of Sn--GFP--Sn Josephson junction.
(b)~Nb or Sn metal surface bounding with GFP. Metal oxide layer
electrostatic bounding with protein's ionogenic groups.}
{\vspace*{-.4pc}}
\end{figure*}

For control experiments, we prepared a JJ without GFP molecules. For
this purpose, we used beads made from immobilines without GFP molecule.
The immobiline beads were applied to the surface of the electrodes.
Then the immobiline gel beads were removed and the final assembly of
the control JJ was carried out.

\subsection{JJ with GFP-based insulation barrier} \label{sec2.2}

We made measurements of low-temperature phenomena in the Me--MM--Me
structure.\ The voltage (V) on the structures was measured by the
\mbox{4-probe} method using digital voltmeters of the HP 34401A type at
a constant current set by a large load resistance.\ Temperatures below
4.2~K were obtained by evacuating helium vapor and fixed with a
monostat with an accuracy of 0.01~K. The temperature above 4.2~K was
obtained by heating helium vapor with a microheater, and controlled
using an exemplary germanium thermometer of the TSG-1 type.
Sn and Nb with a purity of at least 99.999\% were used as electrodes
(purity control made by X-ray \mbox{photoelectron}
spectroscopy---XPS---analysis).

Figures~\ref{fig2} and~\ref{fig3} show the low-temperature dependences
of the resistances (R) for the Nb--GFP--Nb (Figure~\ref{fig2}) and
Sn--GFP--Sn (Figure~\ref{fig3}) structures.\ It is known that Sn passes
into the superconducting state at $T_c = 3.72$~K, and Nb at $T_{c} =
9.3$~K. In our \mbox{M--MM--M} structures, upon cooling starting from
$T_{c}$, a decrease in resistance is observed up to technical zero,
determined by the resolution limit of the equipment used. For the
Nb--GFP--Nb (Figure~\ref{fig2}) and Sn--GFP--Sn
(\mbox{Figure}~\ref{fig3}) structures, the transition to technical zero
is observed around 4.5~K and 3.65~K, respectively. 

\begin{figure}
\includegraphics{fig02}
{\vspace*{-.2pc}}
\caption{\label{fig2}Low-temperature dependences of the resistances $R$
for the Nb--GFP--Nb structure from temperature.}
\end{figure}

\begin{figure}
\includegraphics{fig03}
{\vspace*{-.2pc}}
\caption{\label{fig3}Low-temperature dependences of the resistances $R$
from temperature for the \mbox{Sn--GFP--Sn} structure.}
\end{figure}

In the control M--M structure where there was no MM between the metal
electrodes and the electrodes directly touched each other, the
transition to technical zero resistance during cooling occurred very
abruptly. 
XPS elemental analysis and electric resistance measurements as a
function of temperature indicated a sufficiently high purity of the
metals used as electrodes. The smearing of the transition may indicate
fluctuation phenomena in the vicinity of the critical temperature
$T_{c}$. In the vicinity of $T_{c}$, fluctuations of the order
parameter cause the appearance of resistance in the superconducting
state at $T < T_{c}$ and superconducting electrons at $T > T_{c}$.\ In
two-dimensional and one-dimensional superconductors, allowance for such
fluctuations should lead to smearing of the temperature transition. In
pure bulk superconductors, allowance for fluctuations does not
noticeably affect the nature of the superconducting transition. In thin
superconducting filaments and particles with diameter $d \ll \xi_0$ the
coherence length (2300~{\AA}~\cite{14} or 2100~{\AA}~\cite{15} for Sn,
430~{\AA} for Nb~\cite{13}), the influence of fluctuations on the
temperature transition smearing is observed. In films these values are
typically smaller.

Twenty GFP-based JJ devices were tested to assess device-to-device
reproducibility.\ Current--voltage characteristics are given in the
Supplementary Material. There is an oxidated film on the surface of Sn
or Nb. The orientation and attachment (interaction) of the GFP molecule
on oxide surfaces during the production of the JJ is carried out
through certain ionogenic groups. GFP's electrostatic charge
distribution at the time of device production is not random~\cite{28}.
There are references in the literature about the orientation of the GFP
molecule on crystalline surfaces. The orientation of the MM or BNP
inside the JJ insulator barrier can be controlled by fluorescence
spectroscopy methods~\cite{26} or running electrostatic interactions
control~\cite{27} at the time of GFP transfer to the electrode surface.
It is also possible to check the degree of isotropy (or anisotropy) of
the MM or BNP in this way~\cite{26}.

\section{Results} \label{sec3}

The electronic structure of MMs and BNPs is mostly often anisotropic in
three-dimensional space.\ This anisotropy gives different profiles of
the phase transition dependence on the normal to superconducting state
for superconductor--MM contacts.\ Dependence of the ``electrical
resistance--temperature''{\break} (R--T) curve on direction or position
of the MM or BNP inside the JJ may be a new characteristic for the MM
or BNP. It is also possible to check the degree of isotropy (or
anisotropy) of nanoparticles and MMs in this way.

For the Josephson effect, the protein molecule's dielectric barrier
thickness must be much smaller than the Cooper pair coherence length. 
Figures~\ref{fig4}(a) and~(b) show an example of the
current--voltage characteristics (CVCs) of the \mbox{Nb--GFP--Nb} and
Sn--GFP--Sn structures prepared by the method described above in the
text.

{\def\ffigcaption{\label{fig4}}
\xsubfig[t!]{04a}{}
\figcnt
\begin{figure*}
\includegraphics{fig04}
\xcaption{\label{fig4}\ffigcaption}{Current--voltage characteristics of
the following Josephson junctions: (a)~Nb--GFP--Nb and Nb--Nb (without
GFP), (b)~Sn--GFP--Sn and Sn--Sn (without GFP), (c)~Nb--mGFP--Nb with
modified GFP, and (d)~Sn--mGFP--Sn with modified GFP.}
\end{figure*}}
\setcounter{figure}{4}

It is possible that the low critical current can be associated with
induced superconductivity in the MM channel, as is the case in
superconductor--normal metal--superconductor structures. 
In the presence of a ``coupling'', the critical current should
oscillate in weak magnetic fields~\cite{29}. The dependence $I_{c}(H)$
was measured in weak magnetic fields at a current transverse to the
magnetic field, flowing through the Sn--GFP--Sn structure, at which a
voltage drop across the M--MM--M structure is recorded.\ 
Current--voltage characteristic for the Nb--GFP--Nb structure are
presented in Figure~\ref{fig4}(a).
Measurement set-up was shielded from the Earth's magnetic field.

Figures~\ref{fig4}(c) and~(d) show the CVCs for Nb--mGFP--Nb
and Sn--mGFP--Sn with a modified GFP (mGFP) (another GFP structure).
This modified GFP is eGFP His-tag from Cube Biotech (catalogue number
29901)~\cite{36}. The CVCs for GFP and mGFP are \mbox{different}.
Figure~\ref{fig5} shows the observed dependence $I_{c}(H)$, which
clearly indicates an oscillation of the critical superconducting
current.
Thus, in the case of the transition of electrodes to the
superconducting state in Me--MM--Me structures, the Josephson effect is
observed.

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig5}(a)~Sn--GFP--Sn and (b)~Nb--GFP--Nb JJ critical
current versus magnetic field applied in the plane of the
substrate---perpendicular to the current direction.}
\end{figure}

When analyzing the CVCs of a JJ, it is necessary to divide it into
sections~\cite{19,35}. At a voltage of 0~V,{\break} a superconducting
current is observed in both JJs (Sn--GFP or Nb--GFP).
The magnitude of the total step of the JJ's superconducting current is
equal to the sum of the steps from the elementary JJ contacts of the
superconductor and the GFP subunits (amino acid or beta helix). These
macromolecule's subunits influence the formation of a path for
superconducting current tunneling through the JJ. Thus, based on the
height of the general stage of the JJ, one can find the number of
subunits of the macromolecule. As the electrical voltage increases, the
normal current component begins to appear. This normal current
component occurs in addition to the superconducting current. The normal
component of the current is carried out by electrons and not by Cooper
pairs. Hysteresis on the CVC of the Nb--GFP JJ means that the JJ has a
certain electrical capacity. In the CVC of the Sn--GFP JJ, the
electrical capacitance is quite small. When an external electrical
voltage is applied to a JJ, the current is carried by Cooper pairs and
electrons. When the JJ's internal energy ($E_{c}$) is exceeded,
electromagnetic radiation is generated, determined by the formula
{\begin{eqnarray*}
&\displaystyle E_c = \frac{\hbar I_c}{2e}\raisebox{-.8pc}{}\\
&\displaystyle \omega = \frac{2e}{\hbar}R\sqrt{I^2-I_c^2}\raisebox{-.7pc}{}\\
&\displaystyle 2e\overline{V} = \hbar\omega
\end{eqnarray*}}\unskip
$I_{c}$---critical current through JJ, $R$---resistance of MM- or
BNP-based JJ, $V$---voltage on JJ's electrodes, $e$---electron charge,
$I$---full current through JJ, $\hbar$---Plank constant,
$\overline{V}$---average value of voltage on JJ's electrodes.

The electromagnetic radiation mode manifests itself in the form of a
flatter CVC section, where \mbox{excess} JJ energy is emitted in the
form of electromagnetic radiation. The radiation frequency is
proportional to the voltage (potential difference) across the JJ. The
greater the voltage, the greater the frequency of the emitted
radiation. Measuring the radiation frequency in this section of the CVC
provides information about the ohmic resistance of the JJ and the
contributions of normal and superconducting currents. The magnitudes of
the current steps are the sum of the contributions of steps from
elementary contacts from the superconductor to the subunits of the
macromolecules (amino acids). After increasing the electrical voltage
to more than 450~${\upmu}$V for \mbox{Sn--GFP} and 280~${\upmu}$V for
Nb--GFP, the main contribution is made by the normal component of the
current (electrons). Differences in the CVCs of Nb--GFP and Sn--GFP
appear due to different values of the elements in the circuit diagram
(ohmic resistance, electrical capacitance and inductance). The critical
level of the superconducting current and the normal electron current
level when an electrical voltage is applied determine the type of
superconductive electrodes CVC for each type of JJ. The CVC is
determined by the electronic characteristics of the dielectric placed
in the JJ gap.

\section{Discussion} \label{sec4}
\subsection{JJ with MMs or BNPs as device for detection of their
electronic properties} \label{sec4.1}

The implementation of the Josephson effect on MMs or BNPs makes it
possible to develop a method and device for determining the electronic
properties of these particles. In the proposed device and method, only
one single MM or BNP is required to determine its electronic properties
and structure.

By placing a particle in a JJ, it is possible to measure the tensor
components (tensor of the second rank) of the electric polarizability,
magnetic susceptibility of the MM or BNP for different directions and
magnitudes of the external electric or magnetic field's strength
vector. By measuring the ``plasma frequency'' of a superconducting
resonator based on a JJ with a MM or BNP, in the dielectric layer (or
``superconducting current---MM or BNP''), one can obtain projections of
the electron density distribution of the MM or BNP. When measuring
various (all) projections in space, one can obtain the volume
distribution of the electron density of the MM or BNP placed in the JJ.
A JJ-based superconducting resonator is preferred over a capacitor due
to the absence of thermal fluctuations and the ability to exploit the
unique properties of superconductors near the superconducting-to-normal
transition.

To study the electrical properties of MMs or BNPs, the JJ has
advantages over a conventional RLC circuit based on normal metals with
normal electronic conductivity. These advantages are: (1)~a higher
quality factor of the JJ oscillatory circuit than that of an
oscillatory circuit with normal conductivity; (2)~the ability to record
Josephson radiation when voltage is applied to the JJ's electrodes (use
this radiation for ``Josephson'' spectroscopy of MMs or BNPs); (3)~the
tunneling length of Cooper pairs in a JJ exceeds the length of electron
tunneling from the surface of a normal metal electrode. Accordingly,
superconducting current as a probe has advantages over the tunnel
current of normal electrons. Low temperatures when measuring the
electrical characteristics (electronic parameters---structure) of MMs
or BNPs rigidly fixed in the superconducting resonator make it possible
to avoid the influence of movement and reorientation of MMs or BNPs in
the electromagnetic field of the superconducting resonator. To describe
the operation of a JJ with a MM or BNP insulator layer, we can use a
RCSJ model (Resistively \& Capacitively Shunted Junction model) with an
electric capacitance~C.

When using a ``thin'' JJ, the measured characteristic is the magnitude
of the superconducting current, which depends on the phase difference
of the superconducting current on the JJ's superconducting electrodes.
When placing a dielectric MM or BNP in the gap between the JJ's
superconducting electrodes, the system becomes similar to a parallel
connection of a superconducting shunt, capacitance and electrical
resistance.

A JJ with a MM or BNP in the gap between the superconducting electrodes
acts as a superconducting resonator.\ In contrast to the ``thin'' JJ
implementations, we use a dielectric barrier (gap) between the
superconducting electrodes to measure the main components of the
electronic characteristics and electronic structure of the MM or BNP
that is part of this barrier. The difference between such a
device and a ``thin'' JJ is the presence of electrical resistance,
capacitance, inductance when an external field (electric and/or
magnetic) is applied. MMs or BNPs usually have certain chemical groups
and types of chemical bonds (C--C, C--N, C--O, \mbox{C--H}, O--H,
C${=}$O, N--H, C${=}$C, aromatic groups, indole, etc.). These chemical
groups and bonds have certain dipole and magnetic moments, and
dielectric constants. Dielectric constants, constants of magnetic
susceptibility are tensors of the second rank.

A JJ with a single MM or BNP placed in a dielectric gap between the
superconducting electrodes behaves like a very sensitive RLC
oscillating circuit with effective capacitance, magnetic inductance and
superconducting shunt. In such an oscillatory circuit, with the help of
an external potential difference or a magnetic field, resonance
frequencies are excited. The resonance frequencies of the circuit
depend on the electronic structure of the MM or BNP placed in the
dielectric gap of the superconducting resonator. Also, the resonance
frequencies depend on the orientation of the MM or BNP relative to the
JJ's superconducting electrodes. A single MM or BNP must be fixed
between the superconducting electrodes of the Josephson superconducting
resonator (Figure~\ref{fig6}). Superconducting electrodes are placed
radially with the MM or BNP at the center and this device looks like a
``rotunda''.

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig6}``JJ-ROTUNDA''---radial geometry of JJ's array
for 2 or 3D measurements of single molecule (or nanoparticle) electric
polarization and magnetic susceptibility.}
\end{figure}

The presence of electric capacitance leads to the possibility of
creating a circuit to measure the parameters of the dielectric layer.
The JJ can be described as a nonlinear inductive element which stores
energy when the current through it changes~\cite{18,19,20,21,22,23,24}.
In the general case, within the framework of the resistive model, the
equivalent circuit is supplemented by a capacitance C connected in
parallel (Figure~\ref{fig7}).

\begin{figure}
\includegraphics{fig07}
\caption{\label{fig7}Schematic diagram of a resistive model with
electric capacitance.}
\end{figure}

Accordingly, we can consider the natural resonance frequency of the
circuit with capacitance $C$ and ``self-inductance'' JJ:
{\begin{eqnarray*}
&\displaystyle \omega_p = \sqrt{\frac{2eI_c}{\hbar C}}\raisebox{-1pc}{}\\
&\displaystyle \omega_p = \frac{1}{\sqrt{L_{\mathrm{J}}C}}\raisebox{-1.3pc}{}\\
&\displaystyle L_{\mathrm{J}} = \frac{\hbar c^2}{2eI_c} = \frac{c\Phi_0}{2\rmpi I_c}\raisebox{-1pc}{}\\
&\displaystyle \Phi_0 = \rmpi\hbar c/e
\end{eqnarray*}}\unskip
$I_{c}$---critical current through JJ, $C$---capacitance of JJ contact,
$\Phi_{0}$---magnetic flux quantum, $I_{s}$---superconductive current,
$I_{n}$---critical current (normal current), $I_{d}$---displacement
current through capacitor, $c$---light speed.

From measurements of the JJ's electric capacitance, it is possible to
determine the MM or NP's polarizability tensor components. The
components of the magnetic susceptibility tensor are determined from
measurements of the inductance. The measurements from each pair of JJ
electrodes correspond to certain tensor components. To determine the
three parameters of electronic polarizability,{\break} conductivity,
and magnetic susceptibility, it is sufficient to measure the normal
current, superconducting current, and frequency.

Based on the experimental values of the current through the JJ on the
CVC, the parameters of the circuit diagram of the modeling JJ
(inductance, capacitance, conductivity) are determined. These
parameters are the parameters of the molecule between the
superconducting electrodes. Based on conductivity, inductance and
capacitance, one can determine magnetic susceptibility, polarizability,
dielectric constant, and electrical conductivity. These quantities are
tensors of the second rank with 9 components. These components provide
information about the spatial distribution of the electron density of
the protein molecule.

To find the parameters of circuit diagram elements, it is necessary
measure the response to an external magnetic or electric (applied
voltage) field. By measuring the resonance frequency of current
oscillations in the circuit, we can find the electric capacitance and
magnetic inductance using the formula:
{\[
\omega_p = \frac{1}{\sqrt{L_{\mathrm{J}}C}}.
\]}\unskip
To find the structural elements of a protein molecule, it is necessary
to perform the following steps: (1)~acquire the CVC and measure the
JJ's resonance frequency without the protein molecule; (2)~measure the
parameters of the JJ's electrical circuit with a protein molecule;
(3)~acquire the CVC and measure the resonance frequencies of all amino
acids separately in a JJ of the same design; measure the circuit
diagram parameters in the JJ with amino acids; (4)~it is necessary to
compare the CVC and resonance frequencies of amino acids and protein
molecules; (5)~determine the relative values of the steps of the
general CVC of the protein molecule and the intensity of the of
electromagnetic radiation at the resonance frequencies of each amino
acid. The magnitude of the steps on the CVC will be proportional to the
number of amino acids in that protein molecule. The frequency of the
emitted electromagnetic radiation and the voltage of each stage on the
CVC are determined by Josephson's formula ($V$---voltage;{\break}
$e$---electron charge).
{\[
\omega_{\mathrm{J}} = \frac{2eV}{\hbar}.
\]}\unskip

\subsection{Some aspects of preparing a MM- or{\hfill\break} BNP-based
JJ} \label{sec4.2}

For easy handling and preparation of the ``rotunda'' device, one can
use MMs or BNPs inside gel beads. It is possible to use different types
of gels (polyacrylamide, agarose, silica gel, etc.) with a pore size of
tens or hundreds of nanometers to isolate a single protein molecule or
other MMs or BNPs. The pore size of the gels is commensurate with the
coherence length of the Cooper pairs. After evaporation of water from
the gel, the volume of gel pores and the volume of gel pieces decrease
tenfold. Therefore, for a simpler assembly of a MM-based JJ between
superconducting electrodes, one can use submicron gel pieces (beads,
other shapes) with single MMs (proteins, DNA, or BNP) inside the gel's
pore cells.

\section{Conclusion} \label{sec5}

For the first time, a JJ was created on the basis of a biological
molecule. A device is proposed that consists of a macromolecule or
biological particle (MM or BNP) and a JJ. The MM or BNP is placed in
the gap between the superconducting electrodes of the JJ. The choice of
the JJ as a device for studying the properties of MMs or BNPs makes
sense because the scales of the coherence length of Cooper pairs in
superconductors and the sizes of MMs or BNPs \mbox{coincide}.

The Josephson effect manifests itself not only in tunnel junctions, but
also when connecting two (or more) superconductors through a weak link
of any kind. In this work study case the weak link is a protein
molecule placed between superconducting \mbox{electrodes}.

The presence of oxide films on the surface of superconductors provides
a good opportunity to create covalently bonded stable metal--oxide--(MM
or BNP) structures (produced at room temperature). By varying the
conditions for the synthesis of metal--oxide--(MM or BNP) structures,
it is possible to selectively use ionic or hydrophobic groups in MMs or
BNPs. The choice of superconducting electrode material allows one to
vary different coherence lengths and for different types of MMs or BNPs
(with different sizes). For each size of BNP it is proposed to select a
specific superconductor material.

The proposed method could be used to study the mechanisms of
interaction of drugs with membrane target proteins, or in virology, for
example, to study the process of how a virus interacts with a cell.
Note that endogenous viruses are very large formations. Their diameter
reaches 25~nm and their mass can exceed 1~MDa. Viruses, even of the
same species, are very heterogeneous. This may enable the detection of
the fold and fill of DNA under the envelope of the virus.

It is discussed that conventional electrophysical measurements of the
resulting JJ structure with MM or BNP (CVC, dependence of the critical
current on the external magnetic field, and others) make it possible to
obtain information about the distribution of electron density in the MM
or BNP.

Prior to this work, no one had used a JJ to determine the structural
and electronic properties of dielectrics and MMs or BNPs.\ Much remains
to be understood in the Josephson effect itself in molecular compounds
based on biological molecules and \mbox{particles}~\cite{25,33,33a,34}.
{\vspace*{-.2pc}}

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.
{\vspace*{-.2pc}}

\section*{Acknowledgements}

We thank to Professor Matvey V.~Entin from the Institute of
Semiconductor Physics of the Russian Academy of Sciences (Novosibirsk)
and Professor Pier Giorgio Righetti from Politecnico di Milano for
interest in our work and very fruitful discussions.
{\vspace*{-.2pc}}

\back{}

\section*{Supplementary data}

Supporting information for this article is available on the journal's
website under \printDOI\ or from the author. 

\CDRsupplementaryTwotypes{supplementary-material}{\cdrattach{crchim-318-suppl1.pdf}}
\CDRsupplementaryTwotypes{supplementary-material}{\cdrattach{crchim-318-suppl2.pdf}}

{\vspace*{-.2pc}}

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