2.2 Calculation Methods and Models for Simulating Dopachrome Conversion
39
also complementarily participate in the selective conversion to DHICA by inhibiting
decarboxylation as a minor factor.
For the possible coordination geometry of Cu(II)-dopachrome complex, a fourcoordinate model was used. In this structure, Cu(II) is planarly bound with the
quinonoid group and two H 2 O molecules, as shown in Fig. 2.3. Although Cu(II)
aqua complex may be present in a five-fold-coordinated structure [31, 32], one of the
coordination with H 2 O is very weak (weaker than the hydrogen bond formed in the
H 2 O molecular cluster). Since this weak coordination shows fluctuating behavior of
the coordination position, we did not include this in our model.
To calculate the Gibbs free energy, we first computed the Hessian matrix, and
then diagonalized it to find the normal mode frequencies of the molecule. A partition
function was obtained by considering the degrees of freedom for the molecular
vibration, rotation, and translation. The temperature was set to 309.5 K considering
the human body temperature, and the pressure was set to 1.0 atm. This thermodynamic
model assumes a non-interacting dopachrome ideal gas (with PCM correction), and
also that the pressure–volume product is uniquely determined by the temperature.
Theoretical methods for calculating the solvation free energy are still in the developing stages and the complete description cannot be expected in the short term.
Although we used PCM for the qualitative description of the solvent–solute interaction, this can be extended into the following model; the solute molecules (dopachrome
or dopachrome-Cu(II) complex) are surrounded by H 2 O molecules, and form several
hydration layers. This hydration layer is considered to have a strong interaction with
the solute molecules so that the position of the O atoms in H 2 O do not significantly
change by thermal fluctuation.
Keeping in mind the presence of the above-mentioned hydration layers, we also
conducted calculations with several H 2 O molecules when considering deprotonations, decarboxylation, and complex formation between Cu(II) and dopachrome.
The entire solution is divided into two liquid phase regions: a liquid phase region
including at least solute molecules and the corresponding hydration layer, and a
liquid phase region surrounding the hydrated region. (As described above, the solute
concentration is sufficiently low so that the distances between solute molecules are
large enough to ignore the interactions.) It is assumed that the heat exchange, expansion and compression work, and proton exchange are possible between the two liquid
phase regions, and have the same temperature, pressure, and pH.
In this model, the protonations and deprotonations are affected by pH. Since
pH determines the chemical potential of protons, the deprotonated states become
stable at high pH, and the protonated states are favorable at low pH. The major
protonated/deprotonated states in the equilibrium state under a given pH are determined by the acid dissociation constant K a of the corresponding functional group.
The concentrations of the protonated and deprotonated states become almost identical at the condition pH = pK a as described by Henderson-Hasselbalch equation.
For example, to calculate the acid dissociation constant of the carboxyl group in
dopachrome, it is necessary to compute the Gibbs free energy change for the proton
dissociation. For this purpose, we considered a thermodynamic cycle as shown in
Fig. 2.4.
39
also complementarily participate in the selective conversion to DHICA by inhibiting
decarboxylation as a minor factor.
For the possible coordination geometry of Cu(II)-dopachrome complex, a fourcoordinate model was used. In this structure, Cu(II) is planarly bound with the
quinonoid group and two H 2 O molecules, as shown in Fig. 2.3. Although Cu(II)
aqua complex may be present in a five-fold-coordinated structure [31, 32], one of the
coordination with H 2 O is very weak (weaker than the hydrogen bond formed in the
H 2 O molecular cluster). Since this weak coordination shows fluctuating behavior of
the coordination position, we did not include this in our model.
To calculate the Gibbs free energy, we first computed the Hessian matrix, and
then diagonalized it to find the normal mode frequencies of the molecule. A partition
function was obtained by considering the degrees of freedom for the molecular
vibration, rotation, and translation. The temperature was set to 309.5 K considering
the human body temperature, and the pressure was set to 1.0 atm. This thermodynamic
model assumes a non-interacting dopachrome ideal gas (with PCM correction), and
also that the pressure–volume product is uniquely determined by the temperature.
Theoretical methods for calculating the solvation free energy are still in the developing stages and the complete description cannot be expected in the short term.
Although we used PCM for the qualitative description of the solvent–solute interaction, this can be extended into the following model; the solute molecules (dopachrome
or dopachrome-Cu(II) complex) are surrounded by H 2 O molecules, and form several
hydration layers. This hydration layer is considered to have a strong interaction with
the solute molecules so that the position of the O atoms in H 2 O do not significantly
change by thermal fluctuation.
Keeping in mind the presence of the above-mentioned hydration layers, we also
conducted calculations with several H 2 O molecules when considering deprotonations, decarboxylation, and complex formation between Cu(II) and dopachrome.
The entire solution is divided into two liquid phase regions: a liquid phase region
including at least solute molecules and the corresponding hydration layer, and a
liquid phase region surrounding the hydrated region. (As described above, the solute
concentration is sufficiently low so that the distances between solute molecules are
large enough to ignore the interactions.) It is assumed that the heat exchange, expansion and compression work, and proton exchange are possible between the two liquid
phase regions, and have the same temperature, pressure, and pH.
In this model, the protonations and deprotonations are affected by pH. Since
pH determines the chemical potential of protons, the deprotonated states become
stable at high pH, and the protonated states are favorable at low pH. The major
protonated/deprotonated states in the equilibrium state under a given pH are determined by the acid dissociation constant K a of the corresponding functional group.
The concentrations of the protonated and deprotonated states become almost identical at the condition pH = pK a as described by Henderson-Hasselbalch equation.
For example, to calculate the acid dissociation constant of the carboxyl group in
dopachrome, it is necessary to compute the Gibbs free energy change for the proton
dissociation. For this purpose, we considered a thermodynamic cycle as shown in
Fig. 2.4.