# Non-Equilibrium Thermodynamic Approach to Ca\({}^{2+}\)-Fluxes in Cancer
## 2 Materials and Methods  
The living cell membrane is characterized by a different permeability in relation to the distinct ions (Na\({}^{+}\), K\({}^{+}\), Cl\({}^{-}\), Ca\({}^{2+}\), etc.) which cause an electric potential difference, \(\Delta\phi\), between the cytoplasm and the extracellular environment, measured in reference to the environment [35; 36]. Since 1956, it has been clear that cancer cells are electrically different from normal ones [37]. [PERSON] pointed out that hyperpolarization:  
* Characterises the start of the cell M phase [38];
* Can reversibly block the synthesis of DNA and the mitosis [39];
* Was found to be a characteristic of the normal cells: the lowered membrane potential was identified as a cause of an increase in proliferation of the cancer cells [40]. Consequently, in 1971, [PERSON] conjectured a relation between the cell cycle progression and the membrane electric potential changes [40]: this hypothesis has always been experimentally confirmed [41; 42; 43; 44; 45]. Moreover, the fundamental role of the membrane electric potential has recently been highlighted in relation to the control of the critical cell functions (proliferation, migration, and differentiation) [46; 47; 48]. In this context, the role of the ion fluxes has also been highlighted; indeed, an increase in the Na\({}^{+}\) intracellular concentration in tumour causes a depolarisation, during malignant transformation of normal cells [41; 49; 50]. On the other hand, the K\({}^{+}\) intracellular concentration remains approximately constant [51]. The membrane electric potential can be theoretically described by the Goldman-Hodgkin-Katz equation [52; 53; 54]  
\[\Delta\phi=\frac{RT}{F}\,\ln\left(\frac{P_{\text{Na}^{+}}[\text{Na}^{+}]_{ \text{outside}}+P_{\text{K}^{+}}[\text{K}^{+}]_{\text{outside}}+P_{\text{ Cl}^{-}}[\text{Cl}^{-}]_{\text{outside}}}{P_{\text{Na}^{+}}[\text{Na}^{+}]_{ \text{inside}}+P_{\text{K}^{+}}[\text{K}^{+}]_{\text{inside}}+P_{\text{Cl}^{- }}[\text{Cl}^{-}]_{\text{inside}}}\right) \tag{1}\]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, \(F=96,485\) C mol\({}^{-1}\) is the Faraday constant, and \(P\) is the relative permeability [55; 56; 57], such that \(P_{\text{Na}^{+}}=0.04\), \(P_{\text{K}^{+}}=1\) and \(P_{\text{Cl}^{-}}=0.45\)[55; 56; 57]. In order to develop a non-equilibrium thermodynamic analysis of the cell membrane, we must consider the interrelationship between the fluxes through the cell membrane (heat and ion fluxes) and the potentials at the borders of the membrane itself (temperature and electric potential). To do so, we follow the Onsager approach, by introducing the phenomenological equations [34; 58; 59; 60; 61]  
\[\left\{ \begin{array}{l}\mathbf{J}_{e}=-L_{11}\,\frac{\
abla\phi}{T}-L_{12}\, \frac{\
abla T}{T^{2}}\\ \mathbf{J}_{Q}=-L_{21}\,\frac{\
abla\phi}{T}-L_{22}\,\frac{\
abla T}{T^{2}} \end{array} \right. \tag{2}\]  
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 \(L_{12}=L_{21}\) in the absence of magnetic fields, and \(L_{11}\geq 0\) and \(L_{22}\geq 0\), and \(L_{11}L_{22}-L_{12}^{2}>0\)[34; 58; 59; 60; 61; 62; 63]. The phenomenological coefficients in the Equations (2) are constant over the range where the linear laws hold, and they must be determined experimentally [60; 64]: \(L_{11}\) is named the heat conductivity, \(L_{22}\) is commonly called the electrical conductivity, while \(L_{12}\) and \(L_{21}\) are named the cross coefficients. Moreover, the cross coefficients are independent of both \(L_{11}\) and \(L_{22}\)[62; 63]. When ion fluxes occur \(\mathbf{J}_{e}\
eq\mathbf{0}\), it follows that [58; 59]  
\[\frac{dc_{i}}{dt}=-\
abla\cdot\mathbf{J}_{i} \tag{3}\]  
where \(c_{i}\) is the concentration of the \(i\)-th ion (Na\({}^{+}\), K\({}^{+}\), Ca\({}^{2+}\), Cl\({}^{-}\), etc.