## 2 Materials and Methods  
The membrane potential of a living cell can be evaluated by using the modified Goldman-Hodgkin-Katz equation [2; 3]:  
\[\Delta\phi=\frac{RT}{F}\,\ln\,\left(\frac{P_{\text{Na}^{+}}[\text{Na}^{+}]_{ \text{outside}}+P_{\text{K}^{+}}[\text{K}^{+}]_{\text{outside}}+P_{\text{Cl} ^{-}}[\text{Cl}^{-}]_{\text{outside}}}{P_{\text{Na}^{+}}[\text{Na}^{+}]_{ \text{inside}}+P_{\text{K}^{+}}[\text{K}]_{\text{inside}}+P_{\text{Cl}^{-}}[ \text{Cl}^{-}]_{\text{inside}}}\right) \tag{1}\]  
where [A] is the concentration of the ion A, \(R=8.314\) J mol\({}^{-1}\)K\({}^{-1}\) is the universal constant of ideal gasses, \(T\) is the absolute temperature, \(F\) is the Faraday constant and \(P\) is the relative permeability such that \(P_{\text{Na}^{+}}=0.04\), \(P_{\text{K}^{+}}=1\) and \(P_{\text{Cl}^{-}}=0.45\). Our aim was to introduce a non-equilibrium thermodynamic approach; therefore, we must use the general phenomenological relations [40; 41; 42]:  
\[\left\{ \begin{array}{l}\mathbf{J}_{e}=-L_{11}\,\frac{\
abla\phi}{T}-L_{12} \,\frac{\
abla T}{T^{2}}\\ \mathbf{J}_{Q}=-L_{21}\,\frac{\
abla\phi}{T}-L_{22}\,\frac{\
abla T}{T^{2}} \end{array} \right. \tag{2}\]  
where \(\mathbf{J}_{e}\) is the current density [A m\({}^{-2}\)], \(\mathbf{J}_{Q}\) is the heat flux [W m\({}^{-2}\)], \(T\) is the living cell temperature and \(L_{ij}\) are the phenomenological coefficients, such that [42]\(L_{12}=L_{21}\) in absence of magnetic fields, and \(L_{11}\geq 0\) and \(L_{22}\geq 0\), and [42]\(L_{11}L_{22}-L_{12}^{2}>0\). At the stationary state, the net ion fluxes is null, \(\mathbf{J}_{e}=\mathbf{0}\), so the previous equations hold to:  
\[\left\{ \begin{array}{l}\frac{\
abla T}{T}=-\frac{L_{11}}{L_{12}}\,\
abla \phi\\ \mathbf{J}_{Q}=\left(L_{22}\frac{L_{11}}{L_{12}}-L_{12}\right)\frac{\
abla \phi}{T}\end{array} \right. \tag{3}\]In relation to the first equation, it is possible to highlight that a Seebeck-like effect occurs in the cell membrane [39], while the other equation expresses an analytical, but also biophysical, relation between the heat flux towards the environment and the membrane electric potential gradient. Considering that:  
\[\dot{Q}=\int_{A}\mathbf{J}_{Q}\cdot\mathbf{\hat{n}}\,dA \tag{4}\]  
where \(A\) is the area of the membrane external surface, we can write:  
\[\delta\dot{Q}=\left(L_{22}\frac{L_{11}}{L_{12}}-L_{12}\right)\frac{\
abla\phi} {T}\cdot\mathbf{\hat{n}}\,dA=\frac{k}{T}\,\
abla\phi\cdot\mathbf{\hat{n}}\,dA \tag{5}\]  
where \(k=(L_{22}L_{11}/L_{12})-L_{12}\) is a thermoelectric property of the cell, which links the heat flux to the membrane electric gradient. Equation (5) relates the heat power, exchanged with the cell environment, to the cell membrane potential gradient. Cells exchange heat power with their environment by convection [43]:  
\[\delta\dot{Q}=\rho\,c\,\frac{dT}{dt}\,dV=-\alpha\,(T-T_{0})\,dA \tag{6}\]  
where \(\rho\approx 10^{3}\) kg m\({}^{-3}\) is the cell density, \(c\approx 4186\) J kg\({}^{-1}\) K\({}^{-1}\) is the specific heat of the cell, \(\alpha\approx 0.023\,Re^{0.8}Pr^{0.35}\lambda/\langle R\rangle\) is the coefficient of convection, with \(\lambda\approx 0.6\) W m\({}^{-1}\)K\({}^{-1}\) conductivity, \(Re\approx 0.2\) the Reynolds number and \(Pr\approx 0.7\) the Prandtl number [43], \(A\) area of the cell membrane, \(V\) is the cell volume and \(\langle R\rangle=dV/dA\approx V/A\) is the mean radius of the cell. So, it follows that:  
\[\frac{d\phi}{d\ell}=-\frac{\alpha}{\left(L_{22}\frac{L_{11}}{L_{12}}-L_{12} \right)}\,T\,(T-T_{0})=-\frac{\alpha}{k}\,T\,(T-T_{0}) \tag{7}\]  
which highlights the relation between the membrane gradient of the membrane electric potential and the temperature of the cell, being \(\ell\) the length of the membrane.