Thermodynamic Modeling for Cancer Cells

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Thermodynamic Modeling for Cancer Cells

February 23, 2016 by danutdaagmailcom

Thermodynamic Modeling for Cancer Cells

Author: Danut Dragoi, PhD

As scientific facts about cancers become more available, a physical modeling become more transparent as a basis to quantify various forms of energies involved in cancer cells interacting collectively with healthy cells. At this point in time the parametrization of a Hamiltonian equation is an important theoretical step of modeling cancer cells. Since some input parameters for a Hamiltonian or free energy thermodynamic functions are well known, recent SEM images of cancer cells can define, see link here and in here. the use of a phenomenological approach that takes into consideration the thermodynamic Gibbs free energy function. The two pictures below taken from here and from here show that the cancer cell is an evolving spheroid covered with nano-tubes shaped tentacles that attache to the healthy cells.

Image SOURCE

http://www.24matins.fr/cancer-pancreas-decouverte-a-toulouse-281754

Image SOURCE

http://news.mit.edu/2015/cancer-cells-escape-blood-vessels-121

At the molecular level, the attaching tentacles have an opposite electric charge than the healthy cells, link in here, and perhaps different handedness to accommodate a strong bond. In literature attempts to describe a living cell in resting state were done, see link in here . Because the cancer cell is an unstable system, growing cell(s), the process can be imagined as a non-equilibrium thermodynamic process. The authors of the paper, link in here, discuss the properties of cancer cells from a new perspective based on an analogy with phase transitions in physical systems. Similarities in terms of instabilities and attractor states are outlined and differences discussed. While physical phase transitions typically occur at or near thermodynamic equilibrium, a normal-to cancer (NTC) transition is a dynamical non-equilibrium phenomenon, which depends on both metabolic energy supply and local physiological conditions.

Assuming a dynamic equilibrium of a cancer cell with normal cell at a given moment in time t in its evolution we can adapt a Gibbs free thermodynamic function with addition of a misfit potential of different DNA strands and amino acids with different chiralities developed inside the cancer cell

G(U, T, S, P, V, R(+),L(-) ;t)=U-TS +PV+RT[ln(Xn)-ln(R(+)/L(-))] (1)

In expression (1) the variables have the usual known significance, link in here. The term RTln(Xn) is related with the chemical potential of component n. The R(+)=Right and L(-)=Left handedness variables are the percentage of DNA strands and/or chiral object / molecules right handed and left handed inside the cancer cell. The expression (1) is a time implicit form of Gibbs free energy for a cancer cell in which each term has some variation with time specific to cancer cells. For meaningful physical states not far from equilibrium, we select time to be sufficient small in order to have low contributions from the diverging terms. The last term in expression (1) was preferred in that way because in real situations, as the reference link here, shows that the DNA of every organism on Earth is a right-handed double helix. In same paper the authors explain what is the physical reason for organisms on Earth to have right handedness. Notice the components amino acids of a human DNA are all natural amino acids are left handed except aspartame. However, the DNA strands do not keep the left handed character of each amino acid in composition of the DNA. As we know, for healthy living cells the handedness of DNA is right.

Therefore, in relation (1) we differentiate two complex systems of enantiomers in a racemic mixture. As found in literature, link here, two enantiomers react with different reaction rates in a chemical reaction with a chiral catalyst or reagent, resulting in an enantioenriched sample of the less reactive enantiomer. This enantiomeric excess (ee) of the unreacted starting material continually rises as more product is formed, reaching 100% just before full completion of the reaction. In equation (1) the form of the last term is coming from enantiomeric excess (ee), R(+)/(R(+)+L(-))-L(-)/(R(+)+L(-)), in which each term is a probability to have an R(+) DNA and L(-) DNA within the cancer cell. From thermodynamic point of view, when we consider the energies in the Gibbs free energy expression, we need to take the logarithm of the probabilities as a difference, in which we get: ln(R(+)/(R(+)+L(-)))-ln( L(-)/(R(+)+L(-))=ln(R(+))-ln(R(+)+L(-))-ln(L(-))+ln(R(+)+L(-))=ln(R(+)/L(-)).

Knowing that for healthy living cells, like in a healthy pancreas cells, we have a shift in energy when a phase transition Normal to Cancer (NTC) state occurs. The link in here describes the NTC concept. A simple explanation of difference in energy of two chiral different objects is given here .

Moving all terms that do not include the R/L ratio on the left side of the equation (1) we get equation (2):

ΔG=-RTln(R(+)/L(+)) (2)

in which we distinguish two fundamental cases. First when G<0, metabolic reactions are going well as in a healthy organism, and second when G>0, metabolic reactions produce cancer. In the first case R(+)>L(-) as should be. For cancer living cell R(+)<L(-) we have G>0 where the cells are on unstable conditions. To cure them, we need to take energy from them in order to get them back to their original lower energy state. Surgery comes naturally here. We have to not forget that on each side of the transition NTC the physical and chemical coefficients have shifts that can be exploited in curing the cancer. As practice shows the cancer cells produce specific proteins that are used as identifiers of the cancers. Below the NTC transition those specific proteins are missing, indicating that the organism is healthy.

In general the lower the G is for a healthy human body the better.

From practical point of view, it is difficult to measure the ratio within the logarithm function. For this reason we introduce an approximation in which we use the measurements on the aspartame amino acid, that has both chiral forms among all natural amino-acids, see link in here , used by human living cells. Therefore the ratio R(+)/L(-) can be replaced in this approximation with D/L ratio for aspartame amino acid. Using appropriate samples and HPLC analytical method, which is a very sensitive method using micro-mole amounts of sample in solution, the D/L ratio can be easily determined and the G Gibbs free energy evaluated using equation (3):

ΔG=-RTln(D/L) (3)

Other experimental adjustments may be required for real situations, measurements and validations.