A Computational Investigation of a Biologically Revelant Rearrangement: Significance of Molecular Structure and Orbital Interactions as Related to the Transformation of a Model Endoperoxide into a Hydroperoxide Product


Density functional calculations have been performed on a cyclohexadienol endoperoxide model system derived from the tyrosine model compound p-cresol and singlet molecular oxygen. Results indicate that the endoperoxide undergoes a 1,3 hydrogen migration to form a cyclohexadienone hydroperoxide. The rearrangement is an exothermic (-18.83 kcal/mol) process proceeding via a concerted mechanism. The transition state is more polar (4.29 D) than either the reactant endoperoxide (2.79 D) or product hydroperoxide (3.61 D). A constrained conformation in both the reactant and transition state favor n → σ* orbital interactions between the nσ and np lone pairs on the exocyclic oxygen in the reactant and the scissile C-O endoperoxide bond to which it is bonded, as well as, other orbital-orbital interactions characteristic of endoperoxide cleavage and hydrogen migration.


Significance of Endoperoxide Formation

The reaction between singlet oxygen in its 1Δgstate (1O2) and proteins results in the formation of long-lived peroxidic species derived from amino acids (Davies 2004, Wright 2002). Singlet O2 is generated in biological systems via a range of processes (e.g., enzymatic, chemical and photochemical) (Davies 2004). Singlet O2-mediated peroxide formation has been shown to occur on a variety of proteins having diverse amino acid composition and function (Wright 2000). Modification of biomolecules, particularly proteins by 1O2 can result in the induction of primary damage that changes both the properties and function of affected proteins (Davies 2004). Several examples include, but are not limited to cataract formation, altered mechanical properties of connective tissue proteins (e.g. collagen), and destabilization leading to unfolding. The interaction of peroxide-containing peptides and proteins with other biomolecules results in secondary damage which could potentially lead to disruption of metabolic and regulatory processes and contribute to disease progression.

Interaction of 1O2 with amino acids, peptides, and proteins can occur via two different mechanisms: that of chemical reaction and intermolecular energy transfer (e.g., physical quenching). The latter process produces ground-state triplet O2 and an electronically excited acceptor molecule. The second-order rate constants for the chemical reaction between 1O2 and amino acid residues in proteins are highly variable (Wilkinson 1995), with tryptophan, tyrosine, histidine, methionine, and cysteine reacting most rapidly at physiological pH with rate constants on the order of 107-108 M-1sec-1 (Wilkinson 1995). Of these reactive amino acids, tryptophan is thought to yield a dioxetane across the C2-C3 double bond of the indole, although a C3 hydroperoxide may form as well (Nakagawa 1976, Saito 1977). Tyrosine and histidine have been shown to undergo reaction to yield hydroperoxides (Davies 2004, Kang 2000, Tomita 1969, Wright 2002). Peroxide formation is believed to proceed via an unstable endoperoxide intermediate. Such an intermediate has only been observed experimentally in organic solvents at low temperature using highly substituted histidine derivatives (Kang 2000).

In an effort to better understand the reactivity of the proposed endoperoxide, we have performed computations that examine the energetic and stereoelectronic features of the transition state separating the reactant endoperoxide from the product hydroperoxide using a tyrosine model system based on p-cresol. Results of this study will provide insights into the possible mechanism of this rearrangement by providing quantitative data of: 1) electronic descriptors related to the topology of the electron density (i.e., the laplacian of the electron density), 2) the path followed by bonded electrons connecting two atomic centers (i.e., ellipticity), 3) atomic charges, 4) molecular orbital hybridizations and electron occupancy, and 5) stabilization energies resulting from donor-acceptor orbital interactions.


Procedural Overview

The reaction between 1O2 and tyrosine was studied instead of that involving tryptophan or histidine for several reasons. First, the tyrosine endoperoxide model system (C7H8O2) contains fewer atoms than the tryptophan model (C9H9NO2), which decreases computational time significantly. Second, the tryptophan dioxetane is believed to undergo reactions which can produce other products in addition to the hydroperoxide (Nakagawa 1976, Saito 1977). Third, the histidine endoperoxide gives rise to a number of peroxide containing products, the structures of which have not been determined (Kang 2000).

All calculations were performed using Gaussian 03W suite of programs (Frisch 1998), and all visualization was performed using GaussView. Geometry optimizations were performed using the unrestricted hybrid density functional of Perdue, Berke, and Ezerhof (UPBE1PBE) (Perdew 1997) and the 6-31+G(d,p) basis set (Francl 1982). A basis set is the mathematical description of the orbitals within an atomic or molecular system. The basis set chosen for this study allow orbitals to change size and shape. Location of the transition state was done using the Synchronous Transit-Guided Quasi-Newton (STQN) Method (Peng 1993). Optimized structures of the reactant and product were used to generate an initial guess of the transition state structure (QST2 search), which was then refined using a QST3 search. This latter process requires reactant, product, and initial guess transition state structure files as input. The structures of the reactant and product were based on the findings of Wright et al. (2002). Frequency analysis was performed on all optimized structures to determine zero-point corrected energies and vibrational frequencies; the latter was analyzed to confirm that only the transition state structure exhibited a single imaginary frequency. Reported corrected energies are unscaled. The connectivity of atoms is shown in Figure 1.



Electronic descriptors characteristic of the bond critical point, namely the electron density, (ρ(r)), the laplacian of the electron density, (∇2ρ(r)), and ellipticity, (ε), were determined based on the Atoms in Molecules Theory of Bader (Bader 1985) using the algorithm of Cioslowski et al. (Stefanov 1995). These single-point calculations were performed using the UPBE1PBE functional and the 6-31+G(d,p) basis set. A very nice tutorial describing this theory and its applications is available at the following URL: http://www.chemistry.mcmaster.ca/faculty/bader/aim/aim_0.html.

In an effort to understand the role of stereo-electronic effects, natural bond orbital (NBO) analysis was used to calculate atomic charges, identify significant n → σ*, π → σ* and σ → σ* interactions, and determine hybridization exponents for select bonded atoms (i.e., sp2.24, sp3.35, etc.) (Reed 1985). These single-point calculations were performed using the restricted UPBE1PBE functional and the 6-31+G(d,p) basis set. More information about NBO analysis may be found at http://www.chem.wisc.edu/~nbo5/.

Results and Discussion

Transition State Energy, Structure, and Polarity

Optimized structures of the reactant endoperoxide (I), the transition state (TS), and product hydroperoxide (II) are shown in Figure 1. Table 1 lists the associated zero-point energies and dipole moments for each structure. The data in Table 1 indicates that conversion of I to II is an exothermic process, with a Gibbs energy (ΔG) of -18.8 kcal/mol. This exothermicity is consistent with the relief of strain upon scission of the C5-O17 bond and conversion of the 6-member ring containing atoms C1 thru C6 from a boat to planar conformation. Increased entropy of II relative to I also contributes to the driving force for this reaction. Specifically, the entropy change for transformation of I into to II results in a 6 cal/mol-K increase in entropy. Another contribution to the exothermicity comes from the formation of a more stable bond carbonyl bond in II relative to the C-O bond in I. Significantly, the reaction must overcome a 43.9 kcal/mol gas phase activation barrier. Given this large barrier, this result would appear to be inconsistent with the findings that the tyrosine derived endoperoxide has never been isolated or observed, and observation of the histidine endoperoxide occurred only at low temperature in a nonreactive organic solvent composed of acetone and trichlorofluoromethane (Kang 2000). This apparent contradiction may be reconciled by considering the structure of the transition state, and its polarity.

Transformation of the endoperoxide into a hydroperoxide requires the migration of H9 from O8 to O17 with concomitant cleavage of the C5-O17 bond, the ultimate formation of a carbonyl bond between atoms C5 and O8, and flattening of the cyclohexadienone ring (see Figure 1c). Examination of the transition state structure in Figure 1b reveals a constrained molecule containing three distorted 6-member rings, and one 4-member ring composed of atoms C5, O8, H9, and O17. The presence of a 4-member ring is consistent with a concerted 1,3 H migration. Specifically, the migrating atom H9 resides approximately halfway between O8 and O17 at 1.278Å and 1.247Å, respectively, while the distance between C5 and O17 is more indicative of a broken bond (1.952 Å), the 1.296 Å distance between C5 and O8 is intermediate between the a 1.373 Å single bond in I, and the 1.224 Å double bond in II. Given the high degree of strain in the TS structure, it is reasonable to observe a high TS energy.



A second factor concerns the polarity of the TS structure. The dipole moments given in Table 1 reveal that the TS structure is more polar (4.29 D) than both structures I or II, exhibiting a 54% and 19% increase in dipole moment, respectively. The increased polarity of the transition state would be expected to destabilize, and hence increase the activation energy if the reaction occurred in nonpolar medium. Thus, it is reasonable to conclude that at least part of the calculated TS energy results from performance of the calculation en vacuo.

Based on these results, it is plausible to hypothesize that a tyrosine endoperoxide formed within a protein would be highly susceptible to rearrangement, due to the thermodynamic driving force to form the more stable hydroperoxide, and the net stabilization of the polar transition state when this structure is embedded in a polarizable medium consisting of amino acids.

Electronic Descriptors Characteristic of the Bond Critical Point

Additional support for the concerted nature of the transformation of this rearrangement was obtained by calculating the bond critical point (BCP) properties ρ(r), ∇2ρ(r), and ε (Bader 1985). Changes in the electron density ρ(r) at the bond critical point parallel changes in bond length, which also correlate well with bond strength. Moreover, the laplacian of the electron density, ∇2ρ(r), which is the sum of the three curvature eigenvalues λ1, λ2, and λ3 characteristic of the BCP, is an indicator of the amount of charge located in the internuclear region between bonded atoms. When ∇2ρ(r)< 0, charge is concentrated in the internuclear region between atoms, and is indicative of a shared interaction. Conversely, a closed-shell interaction is characterized by ∇2ρ(r)> 0, and a depletion of charge in the internuclear region.

The third descriptor, ε, the bond ellipticity, is described by the relationship (&#9551/&#9552)-1. It indicates the extent of bond elongation in the plane perpendicular to the bond path, as well as, the general shape of the bond, and the degree of π-character. In general when &#9551/&#9552 > 1, a bond is considered to have π-character, and when &#9551/&#9552 approaches 1, the bond is considered to have increased π-character. The value of ε is also sensitive to delocalization resulting from conjugation and hyperconjugation (Bader 1985). In this instance, formal double bonds involved in conjugation tend to experience decreased ε values, while formal single bonds tend to exhibit increased ε values. When double-bond character is induced in a formal single bond, this hyperconjugative interaction tends to increase ε. Determination of these descriptors for reactant and transition state structures disclose a great deal about this reaction and reinforce the premise that conversion of I to II proceeds via a concerted process that passes through a polar transition state. Table 2 lists these descriptors for structures I and TS.



Overall, the data provide a quantitative measure of the changes I undergoes upon reaching the transition state. Most importantly, these changes are consistent with the structures shown in Figure 1a and b. For instance, contraction of bond C5-O8 in the transition state corresponds to a significant accumulation of charge density as indicated by an increase in ρ(r) and decrease in ∇2ρ(r). These changes being indicative of enhanced shared interactions. Likewise, the lengthening and partial cleavage of bond C5-O17 corresponds to a significant decrease in ρ(r) and an increase in ∇2ρ(r), changes indicative of strong closed-shell interactions (i.e., depletion of charge density in the region between atomic basins).

The results also bring to light the concerted nature of this rearrangement. The approximately equidistant location of H9 between O8 and O17 (see Figure 1b), results in nearly equal values of ρ(r) and similar values of ∇2ρ(r) for the two interactions in the transition state. Even more significantly, changes in ρ(r) and ∇2ρ(r) characteristic of the interaction between H9 and atoms O8 and O17 coincide with decreased bonding between H9 and O8 (i.e., decreased ρ(r) and increased ∇2ρ(r), and increased bonding between H9 and O17 as evidenced by increased ρ(r) and larger negative∇2ρ(r) at the bond critical point compared to bond H9-O8. When compared to the intact bond in I (i.e., bond H9-O8) the transient nature of these interactions, it is apparent that the trio of interactions between atoms O8, H9, and O17 are interaction is considerably weaker than those characteristic of bond H9-O8 where ρ(r) is 0.365 eÅ3 and ∇2ρ(r) is -2.13 eÅ5.

Bond critical point descriptors reveal two other interesting characteristics of the transition state. First, the results indicate that conversion of I to TS strengthens the O17-O18 bond by increasing ρ(r) and decreasing ∇2ρ(r). This change parallels the decreased O17-O18 bond length, which changes from 1.435 Å in I to 1.387 Å in TS. Second, there is evidence for conjugative effects, as revealed by changes in ρ(r) and ∇2ρ(r) for bonds C1-C6, C3-C4, and C2-O18. Specifically, the formal double bonds C1-C6 and C3-C4 exhibit a decrease in ρ(r), with the latter being least affected. Compared to these doubles bonds, bond C2-O18 undergoes a 12.5% reduction in ρ(r). It is highly probable that the observed change is due to activation of O17. Taken together, changes in these two descriptors are consistent with an increased delocalization of electron charge in the transition state. Such delocalization would serve to stabilize atoms directly involved in the rearrangement (i.e., C5, O8, H9, and O17).

In this study, changes in ε are perhaps best understood in terms of charge density expansion/contraction and conjugation. If one considers the ratio λ12 a reflection of the direction of charge density in the plane perpendicular to the bond path, ε values support a concerted transition state model in which bond C5-O8 is becoming more π-like exhibiting a λ12 ratio of 1.124 compared to this ratio in I (1.117). Similarly, bond C5-O17 while essentially broken, exhibits a λ12 ratio of 1.402, which indicates significant thru-space conjugative interactions between atoms C5 and O17. The nearly two-fold increase in ε for bond C2-O18 is clearly indicative of induced conjugation. This increase is undoubtedly due to the altered electronic character of O17 in the transition state. Conjugative interactions are also observed for bonds C1-C6 and C3-C4, with these formal double bonds exhibiting a decrease in ε. Lastly, ellipticity values for bonds H9-O8 and H9-O17 again support the concerted nature of this reaction with their values yielding λ12 ratios of 1.081 and 1.085, respectively. These values are indicative of nearly cylindrical contraction of charge density along the bond path between atoms O8 and O17 connected by H9.

Natural Bond Orbital (NBO) Analysis

NBO analysis has been used to identify important conformation specific orbital interactions, determine atomic charges, and obtain descriptions of orbital hybridizations (Reed 1985). The NBO analysis transforms canonical delocalized Hartree-Fock molecular orbitals (MOs) into localized orbitals via sequential transformation of nonorthogonal atomic orbitals into sets of natural atomic orbitals, natural hybrid orbitials, and natural bond orbitals (NBOs). Filled NBOs describe the hypothetical ideal localized Lewis bonding structure. Interactions between filled and antibonding (or Rydberg) orbitals are indicative of a deviation from ideal Lewis structure and can be viewed as a measure of delocalization. Delocalizing interactions are determined by a second-order perturbation approach. Using this perturbation approach it is possible to calculate stabilization energies for orbital interactions between filled donor NBOs and empty or partially filled acceptor NBOs. The stabilization energy, ΔEij is the difference in energy between the donor orbital and the new lower energy orbital formed from the mixing of the donor and acceptor orbitals. The stabilization energy is obtained from Equation 1, where qi, F(ij), and εji represent the donor orbital occupancy, the off-diagonal NBO Fock matrix element, and the energies of the donor (εi) and acceptor (εj) orbitals (diagonal elements of the NBO Fock matrix), respectively.



Donor-acceptor interactions are favored when an occupied donor orbital of appropriate energy transfers electron charge to an empty or partially occupied acceptor orbital of higher energy that can overlap with the donor orbital. Favorable overlap is indicated by the magnitude of F(ij). It is important to note that stabilization energies are not absolute, but instead reveal trends that provide an improved understanding of a chemical system. The results shown in Tables 3-5 compliment the bond critical point data shown in Table 2, and bring to light the significance of orbital interactions.



Specifically, the data in Table 3 indicate that migration of H9 from O8 to O17, occurs via the movement of a charge depleted H9 to a charge enriched O17. The charge depletion observed on atom C5 in [b[I is the result of it being bonded to two atoms of greater electronegativity (ie., O8 and O17). Transformation of I to TS further depletes charge on C5 as a result of C5-O17 bond scission and contraction of bond C5-O8. It is interesting to note that compared to O8, O17 possesses less charge, yet this latter atomic center is more nucleophilic than O8. This character may be understood by considering two points. First, the role of O18 must be considered, and the data reveal that this atom donates more charge in the transition state than in structure I. If the sum of the charges on O17 and O18 in I are compared to that in structure TS, the charge is still less than that of O8. However once that transition state is reached, the additive charge of O17 and O18, although still less than O8, is significantly greater. This conjugative contribution of O18 is consistent with the increased ρ(r) and decreased ∇

2ρ(r) of the peroxide bond in the transition state (see Table 2). A second point worth considering takes into account the conformational change that occurs upon reaching the transition state. In the reactant the dihedral angle C5-O8-H9-O17 is gauche (-60.3o), hence H9 is farther from O17 and engaged in a strong bond with O8. Upon reaching the transition state, this dihedral changes to -19o which weakens the interaction between H9 and O8 and strengthens the interaction between it and O17. The decreased distance separating H9 and O17 in the transition state resulting from the conformational change (ie., gauche to eclipsed), together with the increased nucleophilicity of O17 is consistent with a process that proceeds via a concerted mechanism, and clearly reveal the shift in electron charge accompanying this reaction.



The role of oxygen lone pairs is revealed upon examination of the data presented in Table 4. The data reveal a number of stabilizing n → σ* and π → σ* donor-acceptor interactions in both the endoperoxide form (I), and the transition state (TS). One significant σ → σ* interaction characteristic of the transition state is also observed. Results of this NBO analysis reveal three types of the lone pairs. Lone pair one (n1) is an s-type, the second lone pair (n2) a p-type, and the third (n3), which is specific to the transition state, is a p-type. Hybridizations and occupancies of the lone pair orbitals on atoms O8, O17, and O18 are given in Table 5. The data in Table 5 show the change in hybridization that occurs upon reaching the transition state. Specifically, the data reveal that decreased occupancy is associated with orbital contraction (see O8 lone pairs n1 and n2; and O18 lone pair n2), while increased occupancy is associated with orbital expansion (see O17 lone pair n1). It is worth noting that changes in occupancy upon reaching the transition state coincide with the role of the lone pair in the reaction. That is to say, the decreased occupancy on all lone pairs except lone pair n1 of O17 is consistent with the concerted nature of this rearrangement. The data also reveal what appears to be a lone pair exchange between lone pair n3 on atom O8 and n2 on atom O17. Specifically, n3 is absent in the reactant and present in the transition state, while the opposite holds for n2 on atom O17. It is intriguing to speculate that this exchange is characteristic of this type of hydrogen migration, and reveals the importance of orbital involvement.



Conformational effects are significant and play an important role in this rearrangement. First, the gauche conformation characteristic of I favors stereoelectronic interactions between n2 and the σ* orbital of bond C5-O17, and both n1 and n2 with the σ* orbital of bond C4-C5. Of these interactions, the n → σ* between n2 and C5-O17 provides approximately 4-fold greater stabilization (see Table 4). Second, decreasing the C5-O8-H9-O17 dihedral angle to that observed in the transition state shifts the interactions of n1 and n2 to the bond forming between atoms H9 and O17 (See Table 4) and generates the third "lone pair". Figure 2 contains views of

these lone pair donor and acceptor orbitals. The strongest interactions (106.1 and 90.2 kcal/mole) involve the lone pair n3. This lone pair is oriented along the z-axis of the molecule, and is characteristic of the transition state. The interaction between n3 and bonds H9-O17 and C5-O17 is maximized by an energy gap of less than 1 kcal/mol between the donor and acceptor orbital (εji), and the large Fock matrix element which is the result of the coincidence of the donor and acceptor orbitals on atom O8 (see Figure 2).



Two other TS-specific stabilizing interactions also occur between lone pair n1 and n2 of O8 and bond H9-O17. As was the case for the interaction involving n3, these interactions result from the small energy difference between the donor and acceptor orbitals, and the significant Fock matrix element. The decreased stabilization energy (ΔEij) for the interactions involving n1 and n2 of O8 is most likely a result of the less than optimal orientation of the donor and acceptor orbitals (see Figure 2). This interpretation is consistent with the effect that eclipsed and gauche conformations have on orbital alignment and their ability to act as donors of electron charge (Juaristi 1995). Compared to the contributions of n3, these other stabilizing lone pair interactions, albeit weaker, are also important.



Conjugative stabilization is provided by carbon-carbon double bonds C1-C6 and C3-C4. In the case of these two bonds, donation of charge via a π orbital to the C5-O17 antibonding orbital provides 6.18 and 5.93 kcal/mol, respectively of stabilization in the reactant. In the transition state, the stabilization contributed by these bonds increases significantly (see Table 4), with bond C3-C4 contributing ~ 20 kcal/mol of stabilization compared to bond C1-C6 which contributes ~ 14 kcal/mol of stabilization energy. The difference in stabilization for these bonds can be understood in terms of symmetry. In the reactant (I), the symmetrical nature of the molecule allows for essentially equal contribution. The small difference in the calculated stabilization values is not significant. Upon reaching the transition state, however, the symmetry is destroyed which results in nonequivalent contributions. A second transition state specific interaction, a σ → σ* between bonds C5-O8 and H9-O17 is observed. This hyperconjugative interaction elicits 9.22 kcal/mol of stabilization energy. Hyperconjugative interactions can stabilize transition states via donation of electron charge into a vacant σ* orbital of the incipient bond (Cieplak 1989), which in this case corresponds to H9-O17. Via this interaction, the donor C5-O8, may be viewed as providing electron charge to push (steer) H9 towards the acceptor O17.

Earlier in this paper results relating to changes in ellipticity, ε were linked to charge density contraction and conjugation. The relationship between charge contraction and conjugation to ε is enhanced upon considering the orbital hybridizations of bonds C5-O8, C5-O17, and H9-O17. Table 6 presents the hybridization and ellipticity values for these bonds. Recalling that ε = λ12 -1), with the ratio λ12 reflecting how much a bond is elongated in the plane perpendicular to the bond path, the orbital hybridizations listed in Table 6 reveal a significant trend. Specifically, the value of ε for bonds C5-O8, C5-O17, and H9-O17 is linked to orbital hybridization, with bond C5-O17, which has undergone scission yet still possesses two interacting atoms, exhibiting the largest hydridization exponents (sp22.30) for C5 and (sp26.16d0.01) for O17, as well as the largest ε (0.402). Hybridization exponents for bonds C5-O8 and H9-O17 indicate that these bonding orbitals have significant sp2 character, and hence are predicted to be less expansive than the bonding orbital of C5-O17. The smaller ε values for C5-O8 (0.124) and H9-O17 (0.0805) support this prediction.

In all, results of the NBO analysis reinforce the importance of conformation with respect to stereo-electronic effects as evidenced by the contributions of O8 lone pairs, and the role of π bonds in conjugative interactions (Isaacs 1995 (chapter 2), Juaristi 1995). The results of these calculations also support a model for the rearrangement of endoperoxide (I) to hydroperoxide (II) that proceeds via a concerted process involving the stereo-electronically assisted migration of H9 from O8 to O17. It should be noted that this investigation revealed one possible pathway for the transformation of I to II, and one cannot rule out other mechanisms such as a step-wise process or even one that proceeds via a radical intermediate.


The goal of this study was to acquire data that could be used to form a hypothesis regarding the reactivity of the endoperoxide formed as a result of the reaction of 1O2 with a tyrosine model compound. Results indicate that rearrangement to the product hydroperoxide occurs via a concerted 1,3 hydrogen migration. Energetically, the reaction is exothermic, passing over an estimated 43.9 kcal/mol gas phase barrier. Net stabilization of the polar transition state when this structure is embedded in a polarizable medium such as a protein or polar solvent is predicted to decrease the height of this barrier, and hence increase the probability of reaction. Changes in bond critical point descriptors when the reactant reaches the transition state correlate well with the concerted nature of this rearrangement, and reveal the importance of conjugative interactions. Conformation dependent stereoelectronic interactions allow for a number of favorable n → σ* orbital interactions between the nσ and np lone pairs on the exocyclic oxygen in the reactant and the scissile C-O endoperoxide bond to which it is bonded. Additional orbital-orbital interactions were identified which serve to stabilize the transition state via n → π* and σ → σ* interactions. In all, these orbital interactions provide significant stabilization making rearrangement to the hydroperoxide a facile process. Lastly, the data obtained in this study will be used as a reference point for future calculations that investigate the reactivity of I when H9 is substituted with deuterium. These calculations will allow for the determination of a theoretical kinetic isotope effect (KIE). Determination of a KIE will provide additional information about the nature of the bond breaking and making steps in the transition state, and make it possible to further refine the proposed model.


The author and advisor thank Stetson University for providing support and an ideal environment for pursuing intellectual development.


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