The role of the cell volume-area ratio in 2 thermodynamic analysis of the cancer growth control 3 for in vitro experiments 4

From a thermodynamic point of view, living cell life is no more than a cyclic process. It 12 starts with the newly separated daughter cells and restarts when the next generations grow as free 13 entities. In this cycle the cell changes its entropy. In cancer the growth control is damaged. In this 14 paper we analyze the role of the volume-area ratio in cell in relation to the heat exchange between 15 cell and its environment in order to point out the effect on the cancer growth. The result holds to a 16 possible control of the cancer growth based on the heat exchanged by the cancer towards its 17 environment, and the membrane potential variation, with the consequence of controlling the ions 18 fluxes and the related biochemical reactions. This second law approach could represent a starting 19 point for a possible future support for the anticancer therapies, in order to improve their 20 effectiveness for the untreatable cancers. 21


Introduction
Living cells live in an environment which slowly changes its chemical and physical properties.
In mammalian animals, organs and tissues are generally endowed with a homeostatic capability, which is characterized by internal thermal regulation.
A living system can be defined as an open system, so non-equilibrium thermodynamics, can be applied just to living systems [1].
Cancer is no more than a living cell with a different growth behaviour in relation to the cell we consider normal.At a definite phase of its life any cancer cell divides into two cells.The size of the single cell at the beginning of division can vary [2]; consequently, also the relative size of the two daughter cells can vary, but lower and upper bounds on cell volume have been observed [3].
Living cell life, cancer included, is a cyclic process, which starts with the newly separated daughter cells and restarts when the next generations grow as free entities.Consequently, thermodynamics of cycle processes can represent a good tool for the analysis of cancer.In thermodynamics is fundamental the definition of the control volume: in this context it could be useful to consider the single cell as the observed control volume, but the usual experimental setup doesn't allow us to introduce such approach, so we are forced to observe a multitude of single cells and consider this multitude as a complex cooperative system of single cells.This system, considered as a single cell or as the cooperative system of all cells in a culture, is no more than an open system from a thermodynamic point of view, so, energy and matter flow through the border of the system, while biochemical and biophysical transformations occur within the system, with a related net production of entropy.The environment of the system considered is composed by the suspending aqueous solution of cell nutrients, the substances discarded by the cells, the gaseous atmosphere above the suspending solution.Consequently, the bio-system results composed of:

•
The cell membrane, which delimits the volume of the cell, controls inflows and outflows of molecules; • The cytoplasm which is an aqueous solution of molecules that fills the cell interior; • The organelles, suspended in the cytoplasm.
The system may contain many substances not initially present in its environment.Many enzymes found in fragments of cytoplasm membranes are often not directly accessible to system environment, while, in the environment, the concentrations of some molecular species decrease in time, and nutrients must flow into the system in order to allow the biochemical reactions to occur and produce macromolecular cell components, with a related increase of the living cell mass and volume [4].These biochemical reactions require energy.This energy is obtained from the same reactions, which involve the nutrients, with a related waste of heat towards the living cell environment.The net effect of all these biochemical reactions is to reduce the entropy of the system, with an increase of the entropy generation in the environment [5].
The analysis of the chemical species and of their reactions in the living cells have led to some sequences which start with nutrient molecules and end with the formation of living cells substance and waste molecules and waste heat.Too numerous reaction sequences are considered to exist.
Moreover, many molecules can be part of more than one sequence, providing coupling of sequences.
The characteristics of the living systems have been determined by using batch cultures, grown at the optimum temperature for the species employed by the experimenter, even if these experimental method presents disadvantages because the living system is forced to live in a continually changing environment, due to supply and decrease of the environmental concentration of nutrients related to the cells growth, while both the environmental concentration of the waste molecules and the waste heat rise proportionally.
Cells exchange energy and matter through their membrane, in order to maintain their living conditions.All the biophysical and biochemical processes require fluxes of energy, ions and molecules which are controlled by the endogenous electric fields and accumulated in the nm-thin layer of water.Cells live in an environment which slowly changes its chemical and physical properties.From a thermodynamic point of view, a living system is defined as an open system, in non-equilibrium thermodynamic states with its environment.The 1931 cancer cells were proved to be fermentative, as consequence of a metabolic injury [6] and genetic effects.In normal cells, mitosis is synchronized with cell growth in order to maintain their size during replication.This biomedical result represents the bioenergetic bases to link the production of lactic acid and extracellular-intratumoral acidification to cancer growth and metastasis.Moreover, both the pH of the cell cytoplasm and the extracellular environment are controlled by the living cell membrane potential.
Differentiated cells are hyperpolarized as compared to quiescent or cycling cells, and the hyperpolarization increases the efflux of some ions (Ca 2+ , K + , Zn 2+ etc.).The fundamental role of inhibition of proliferation in normal as well as neoplastic cells has recently been pointed out as well as the interactions between the ion channels and the other elements of the signalling network.Direct cell migration is fundamental in tissue formation, but, if proliferation and invasion is out of control then a new behaviour occurs and cancer emerges as a disease of abnormal growth.At a definite phase of its life any cancer cell divides into two cells.The size of the single cell at the beginning of division can vary; consequently, also the relative size of the two daughter cells can vary, but lower and upper bounds on cell volume have been observed.All these processes are driven by fluxes of energy and mass, and the cell shape results fundamental in their analysis.From a thermodynamic point of view, living systems are no more than complex systems, open with a control of fluxes.
Considering the chemical reaction at constant pressure and temperature the Gibbs Free Energy could seem the selected function for the study of the steady states of the living systems, but its decrease as a criterion for occurrence of a spontaneous evolution is limited to the complex phenomena which occur at constant temperature and pressure inside the living system.But, a general objective function for the analysis of the living systems is required.Moreover, considering that the system wastes energy, and mass, it generates irreversibility, so the general criterion for study its spontaneous evolution is the entropy generation related to the changes of the system.
Entropy generation always increases in any spontaneous and irreversible evolution.
The aim of this paper is to analyze deeply the use of the heat role in the study of the cancer systems.Indeed, recently we have used the entropy generation to introduce a thermodynamic approach to the analysis of the cancer system in order to design a possible support to the present anticancer therapies.But, the thermodynamic approach introduces a new viewpoint for the analysis of the living systems.This paper wishes to explain and improve the thermodynamic formulation of this approach, with the second aim to highlight how to support the biomedical sciences in the comprehension of the possible thermodynamic support to the present anticancer therapies.

Materials and Methods
Life involves organizational and thermodynamic processes, which tend towards the maximum conversion of available energy [1][2][3][4][5][6][7][8].The biochemical reactions produce or consume external metabolites, and connect internal metabolites, at constant concentrations in the cells at their steady states.In order to do so, cells must exchange energy and matter through their membrane.Indeed, many processes such as replication, transcription and translation, require fluxes of ions and molecules which are driven by the endogenous electric fields and accumulate in the nm-thin layer of water [11,14,15].These fluxes of ions induce biochemical reactions within cells and tissues.
Moreover, the 1931 Nobel laureate Otto Warburg proved that cancer cells are fermentative, pointing out that this was the consequence of a metabolic injury [16].In normal cells, mitosis is synchronized with cell growth for cells to maintain their size during replication.Even if the tumor behavior is more complex and it is probably based also on genetic structures of the cells, the results of Warburg highlights the important role of energy conversion in cells.From a thermodynamic point of view, this result can represent the starting point of the analysis; indeed, the genetic processes have the consequence of regulation for the cell behavior, but, if we consider the cell as a black box, as usually done in thermodynamics, then the genetic regulation is no more than the "mind" of the cell, without any direct consequence on the thermodynamic balances.Indeed, the thermodynamic approach evaluates the life cycle of the cell by considering only the energy and mass fluxes balances during the whole cycle of cell life, and not considering the gene activities, but evaluating only their consequences expressed by the energy conversion in cell.
Living cells metabolism implies flows of matter and heat into and out of the cells.What are the consequences of these fluxes and metabolic reactions?To answer to this question, we consider the Gibbs free energy variation in time, due to the mass fluxes.It results [12,[17][18][19]: where G is the Gibbs free energy, W  is the useful power done by the cell, n  is the molar flow, Ze is the molar electrochemical potential, with μ chemical potential, Ze charge of the ion considered, and φ cell membrane electric potential, 0 T is the temperature of the cell environment, t is the time, in means inflow, out means outflow, and g S  is the entropy generation rate, defined as [12,[17][18][19]: where Q  is the heat power exchanged, s is the molar specific entropy, and f means formed.This equation highlights as these flows cause entropy variation; moreover, these fluxes imply also a great number of chemical reactions within the cells, accompanied by entropy generation.Now, we introduce the equation ( 2) into the equation (1) obtaining: From a thermodynamic point of view, the cell life is no more than a thermodynamic stationary state, at constant environmental temperature and pressure, and this can be obtained by introducing the two conditions [12,[17][18][19]: which hold to the equation: which expresses the fundamental role of heat fluxes between the cell and its environment.
The equation ( 5) is very difficult to be numerically evaluated, because it implies the knowledge of all the balance at the second member of the equation for each cell.So, we must find and alternative way to evaluate the cell heat flux.
The heat flux is no more than the heat power exchanged by the cell and its environment.We can consider that, inside the experimental setup usually used in the biophysical and biochemical analysis of cells, the heat flux is exchanged by convection with the suspending aqueous solution around any cell, so we can write: where α is the coefficient of convection, ( ) with the result that greater is the volume-area ratio lower is the thermal exchange, being α , cell ρ and cell c approximately constant.It occurs because the cell adapts its volume/area rate in order to optimize the cell membrane fluxes to obtain the work it needs, and, conversely, this geometric rate controls also the heat exchange.The results obtained highlight the role of the volume-area ratio of the cells in relation to their heath exchange in in vitro experiments.This effect is fundamental when the heat exchange plays a fundamental role in the analysis of the experimental data.

Preprints
In particular, we wish to highlight that this effect is particularly interesting in the study of the cancer growth compared with normal cell growth, because cancer presents a different metabolic cycle.This is important when we study the cancer growth control.Indeed, we can point out that the equation ( 9) is also the equation which links a frequency, i.e. the inverse of the time ( ) , to a structural and geometrical properties of the cell and its environment, in relation to the heat exchange.But, what is this frequency?
It is difficult to find an answer without considering the thermodynamic approach.Indeed, each system presents a proper time of answer to the external thermal perturbation.We suggest that this frequency is no more than the inverse of the cell proper time of answering to the external thermal perturbation, or the heat exchange rate.Indeed, the heat flow can be written as: where Q is the heat wasted during the cell life.Moreover, we can consider also that the ions fluxes are controlled by the cell membrane potential, which is related to the Gibbs free energy by the relation [17][18][19]: where R is the universal constant of gasses and F is the Faraday constant.At the stationary states, remembering the relations (4), it follows: which links the variation of the cell membrane electric potential to the variation of the pH, related to the ions fluxes.Conversely, we can try to force a variation of the pH by a variation of the cell membrane electric potential.
How can we try to do so?We can induce a variation in the cell membrane electric potential by using an electromagnetic wave, with a frequency just equal to the proper frequency of our system.
The results are summarized in the Table 1.It is possible to highlight that the electromagnetic waves induce a different behavior in the cancer cells considered; indeed, they decrease their growth if compared with the cancer cells outside of the electromagnetic field, proving that there exists a forcing phenomena of heat flux control which controls the related ions fluxes and, consequently, induce a different behavior to the biosystem.

Table 1. Cell parameters and calculated ELF-EMF frequencies.
For each cell line, thirty cells in different fields were evaluated in their size.Cell volumes were estimated and used to calculate the forcing frequencies of ELF-EMF, using data in our previous experimental paper [21] The experimental results confirm the results here obtained, in particular that the cancer cells modify their behavior only if irradiated by an electromagnetic field at the proper frequency for the cell line considered in our previous experimental paper [21].

Discussion
The results, here obtained, point out the fundamental role of the cell volume-are ratio in relation to the fluxes control.
Indeed, there is a temperature difference between the interior of a living cell and its environment.This is a thermodynamic necessity for life.Sensible heat is exchanged between inside and outside of the cell due to this temperature difference.This heat flow contribute to entropy generation.Part of the entropy generated appears outside the cell as sensible heat.The fraction of all entropy generation that appears in this form depends on the nature and number of processes occurring in the cell.Consideration of the temperature difference between environment and cell interior allows the introduction of non-equilibrium thermodynamics for the analysis of cells behavior.Brock suggested that the stability of thermophilic organisms can be attributed to membrane structure properties of these organisms [22].
The temperature gradient contribution to the flow of substances through the cell membranes of the cell with a consequent influence on metabolic processes [23][24][25][26].The approach here suggested allows us to evaluate the homeostatic cellular response to external perturbations.This answer is no more than a thermo-chemical output of the cell in the environment.So, we can suggest that the thermodynamic approach holds to a model of analysis of the action and reaction in terms of membrane flux variation.This approach could represent a new approach to design possible support to the present anticancer therapies, by introducing external fields variation, at the proper answer time, in the therapeutic protocols.From the experimental results it is clear a reduction of the growth of the cancer with the consequence of improving the effects of the present therapies.
The growth rate of cells, cancer included, at a fixed temperature is a function of both composition of the medium and chemical potentials of the component substances.This represents a sort of control to the growth, because there is a maximum rate at which each bio-chemical reaction can occur under the existing constraints.This rate is conditioned by the volume-are ratio, because it controls the ions fluxes, i.e. the fluxes of the chemical reactants.
So, our results show a method for the design of therapies and experiments for their analysis.
Indeed, the specific effect of the single frequencies has two important consequences: • In every cell type different parameters of electromagnetic waves impact differently the energy utilization and proliferation, with different inhibition effects on the cell growth; • The same electromagnetic wave has distinct effects on different cells, with a selectivity behavior.Recently, the Bandyopadhyay research team has demonstrated that the spontaneous oscillations of neurons microtubules of the frequency of around 1 MHz oscillation of electrical dipole moments of free electrons and conformational switching, cause wave interference which are the origin of the characteristic shape of the electrical oscillations of the brain at the electroencephalographic signal of 4-40 Hz nested gestalts, named beat frequencies [27,28].The result is no more than the link between the brain synchronization of consciousness with the quantum mechanical behaviours of microtubules.This result proves that quantum vibrations microtubules are entangled across neuronal networks via the gap junction, interconnecting channels which physically link neurons together.This result highlights the fundamental role of microtubules, and their quantum effect, with particular interest for resonance, in cell behaviour.But the proven theory doesn't suggest any link between microscopic and macroscopic behaviours of cells.This result is no more than a thermodynamic resonance for the whole cell.Microtubules are electrical polar structures with power supply from hydrolysis (around 10-14 W cm -1 per unit length of the microtubule) of guanosine triphosphate to guanosine diphosphate: the related energy can excite vibrations.Microtubules lose part of the energy by viscous damping of the surrounding cytosol [29].
Microtubules plays a fundamental role both in the organisation activities of the living cells cytoskeleton and in the intracellular transport [30].As Bandyopadhyay showed the microtubules of the frequency of around 1 MHz, but the global effect of the neurons microtubules synchronisation is the brain 4-40 Hz signal, lower than five orders of magnitude.In particular, Poznanski et al. pointed out how intracellular capacitive effects of bound electrical charges within mitochondrial membranes can influence electrotonic signals expressed as solitary waves [21] and that the outer mitochondrial membrane acts as an amplifier of the ingoing waves.Moreover, they showed the fundamental role of the changes of the mitochondrial membrane equilibrium potential in sustaining solitons with self-regulation in their amplitude [21].In physics and chemistry, resonance is the phenomenon in which a vibrating system or external force drives another system to oscillate with greater amplitude at specific frequencies.At resonant frequencies, small periodic driving forces have the ability to produce large amplitude oscillations, due to the storage of vibration energy.So, the effect shown by Bandyopadhyay is no more that the amplification of a resonance electromagnetic interaction between external electromagnetic waves and cells microtubules, which generates also the microtubules synchronisation.This is a biological resonant effect!But, microtubules are in all cells and they play the same role in all human cells.So, what Bandyopadhyay has pointed out, must occur in all cells, with different global effects in relation to their specialized functions.In particular, the mitochondrial respiratory chain and oxidative phosphorylation cause a dispersion of energy as heat, related to the energy from nutrients converted in proton-motive force driving ATP synthesis, necessary to transport molecules against gradient [31].In this context, it is possible to introduce a thermodynamic approach which realize this biophysical model is to consider the cells as adaptive thermal engines, able to convert energy from one form to another by coupling metabolic and chemical reactions with transport processes [24][25][26]; cells irreversibly consume free energy to maintain thermal and chemical processes and to sustain the transport of matter, energy and ions [24][25][26].Human cells must exchange their wasted heat with a constant temperature environment (the human body around them), so if a difference in the metabolism and in the efficiency of the cell system occurs, as in cancer, the cell encounters a difficult in maintain its optimal inside temperature for life.But, in any heat exchange there is a characteristic temperature function of physical (density, specific heat, convective coefficient) and geometrical (cell volume and membrane area) quantities.This time represents a specific time for any cells lines.But, this time is also the resonant time, and its inverse is no more than the biological tuning fork frequency.So, by inflowing an electromagnetic wave with a frequency evaluated by the biological tuning fork frequency for any cell line, we can obtain a forced behaviour of the considered cell line, a modification in the inside organisational process due to the synchronisation of the microtubules involved in the resonant interaction, just as in the Bandyopadhyay effect.So, external short frequency can produce high microtubules resonant effect, with the control of the cell behaviour.
Indeed, the fluxes across the cell membrane represent a fundamental quantity for cell behaviour control [24][25][26]: the cells reach their optimal asset by a selective process of interactions with their environment, with the consequent effect of the redistribution of energy and mass flows in their metabolic network, enabled by regulatory proteins [24][25][26].Indeed, the cell mitotic cycle is composed of a sequence of processes such as DNA replication, chromosome condensation and segregation, duplication and migration of the spindle pole, breakdown of the nuclear envelope, and cytokinesis [20].Moreover, the cell cycle has been highlighted to be controlled by a control system which monitors DNA integrity before any transition to the next phase.In eukaryotic cells, the main control process occurs at the G1/S transition, in late S (DNA synthesis) phase, at mitosis (M) entry and at the metaphase to anaphase transition.Any process is controlled by the cyclin-dependent kinases, regulated by the oscillatory expression of G1 and G1/S-cyclins, S-cyclins, and M-cyclins.The transition between metaphase to anaphase is triggered by the anaphase-promoting complex/cyclosome (APC/C).Mitogens stimulate the entry into the cell cycle from a quiescent (G0) phase.Exit from mitosis can lead to differentiation, apoptosis, or return to quiescence [20].These mechanisms are altered in neoplastic cells .Cell metabolism is constrained by the maximum amount of macromolecules that can be contained in the intracellular volume: any biochemical process requires energy, and any energy conversion generates outflows of energy, due to the second law of thermodynamics.So, the fundamental thermodynamic approach to the behaviour of the cell systems is to consider inflows and outflows of energy and masses (ions included).Indeed, it was experimentally pointed out that: • Any increase in K + -channel expression and activity of K + at the G1/S boundary is often necessary for cell cycles; • Ca2 + fluxes, which can bind with tubulin, control the membrane potential, regulate the mitotic spindle and cytokinesis, regulate DNA transcription, modulate the expression and activity of the transcription factors that control expression of the G1 cyclins, producing direct effects on cyclins, cyclin kinases, and the associated proteins; • Ions-fluxes vary the membrane potential which determines pH variation inside and outside the cell, with a consequent variation of the metabolic cycle [24][25][26].

Conclusions
In this paper, we have developed the analysis of a thermodynamic approach to cancer cells, with particular regards to the role of the volume-area ratio in the heat exchange and the consequences to the cancer cells behavior.
We have pointed out the existence of a proper time of answer of any cell line to the heat exchange.This time results related to the cells volume-area ratio, a geometrical parameter fundamental for the considerations on the fluxes and cells membrane electric potential variation.
Then, starting from some previous experimental results [23][24][25][26], we have obtained also an experimental proof of the present results.
The results highlight how the irreversibility plays a fundamental role also in biophysical systems; indeed, the geometrical rate is completely related to the entropy generation as it is clear by introducing the relation ( 6) into the relation (2).This holds to a new approach to biological physics, based on the first and second law of thermodynamics.
The results here obtained confirm the biochemical results obtained by other scientists who used an entropic approach; indeed, Luo has demonstrated that a low-frequency and low-intensity electromagnetic field or ultrasound irradiation may increase the entropy production rate of a cell in normal tissue than that in cancer; consequently, it reverse the direction of entropy current between two kinds of cells.The modification of PH value of cells may also cause the reversal of the direction of entropy flow between healthy and cancerous cells [31,32].
is the surface area of the cell, which changes with the phases of the development of the cell, V is the volume of the cell, R is the volume/area ratio, a parameter which influences the chemical reaction time and the fluxes through the cell membrane, cell ρ is the cell mass density, and cell c is the specific heat of the cell, and () temperatures between the cell temperature and the environment temperature.The term/ A V R =is the geometric shape of the cell in relation to convection.We must introduce this quantity because at a stage of his life a cell has a definite volume, but it can change its shape in relation to its duplication phase at that time.Indeed, in eukaryotic cells, the main control process occurs at the G1/S transition, in late S (DNA synthesis) phase, at mitosis (M) entry and at the metaphase to anaphase transition.Any process is controlled by the cyclin-dependent kinases, regulated by the oscillatory expression of G1 and G1/S-cyclins, S-cyclins, and M-cyclins.The transition between metaphase to anaphase is triggered by the anaphase-promoting complex/cyclosome (APC/C).Mitogens stimulate the entry into the cell cycle from a quiescent (G0)phase.Exit from mitosis can lead to differentiation, apoptosis, or return to quiescence[20].These mechanisms are altered in neoplastic cells.Now, from equation (6), we can obtain: (www.preprints.org)| NOT PEER-REVIEWED | Posted: 11 December 2018 Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 11 December 2018 doi:10.20944/preprints201811.0134.v2 .preprints.org) | NOT PEER-REVIEWED | Posted: 11 December 2018 Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 11 December 2018 doi:10.20944/preprints201811.0134.v2