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1.
Pathogens ; 12(6)2023 Jun 18.
Artigo em Inglês | MEDLINE | ID: mdl-37375532

RESUMO

Hepatitis C virus (HCV) is a dangerous virus that is responsible for a large number of infections and deaths worldwide. In the treatment of HCV, it is important that the drugs are effective and do not have additional hepatotoxic effects. The aim of this study was to test the in silico activity of 1893 terpenes against the HCV NS5B polymerase (PDB-ID: 3FQK). Two drugs, sofosbuvir and dasabuvir, were used as controls. The GOLD software (CCDC) and InstaDock were used for docking. By using the results obtained from PLP.Fitness (GOLD), pKi, and binding free energy (InstaDock), nine terpenes were finally selected based on their scores. The drug-likeness properties were calculated using Lipinski's rule of five. The ADMET values were studied using SwissADME and pkCSM servers. Ultimately, it was shown that nine terpenes have better docking results than sofosbuvir and dasabuvir. These were gniditrin, mulberrofuran G, cochlearine A, ingenol dibenzoate, mulberrofuran G, isogemichalcone C, pawhuskin B, 3-cinnamyl-4-oxoretinoic acid, DTXSID501019279, and mezerein. Each docked complex was submitted to 150 ns-long molecular dynamics simulations to ascertain the binding stability. The results show that mulberrofuran G, cochlearine A, and both stereoisomers of pawhuskin B form very stable interactions with the active site region where the reaction product should form and are, therefore, good candidates for use as effective competitive inhibitors. The other compounds identified in the docking screen either afford extremely weak (or even hardly any) binding (such as ingenol dibenzoate, gniditrin, and mezerein) or must first undergo preliminary movements in the active site before attaining their stable binding conformations, in a process which may take from 60 to 80 ns (for DTXSID501019279, 3-cinnamyl-4-oxoretinoic acid or isogemichalcone C).

2.
ACS Catal ; 12(4): 2589-2605, 2022 Feb 18.
Artigo em Inglês | MEDLINE | ID: mdl-36568346

RESUMO

Light-dependent protochlorophyllide oxidoreductase is one of the few known enzymes that require a quantum of light to start their catalytic cycle. Upon excitation, it uses NADPH to reduce the C17-C18 in its substrate (protochlorophyllide) through a complex mechanism that has heretofore eluded precise determination. Isotopic labeling experiments have shown that the hydride-transfer step is very fast, with a small barrier close to 9 kcal mol-1, and is followed by a proton-transfer step, which has been postulated to be the protonation of the product by the strictly conserved Tyr189 residue. Since the structure of the enzyme-substrate complex has not yet been experimentally determined, we first used modeling techniques to discover the actual substrate binding mode. Two possible binding modes were found, both yielding stable binding (as ascertained through molecular dynamics simulations) but only one of which placed the critical C17=C18 bond consistently close to the NADPH pro-S hydrogen and to Tyr189. This binding pose was then used as a starting point for the testing of previous mechanistic proposals using time-dependent density functional theory. The quantum-chemical computations clearly showed that such mechanisms have prohibitively high activation energies. Instead, these computations showed the feasibility of an alternative mechanism initiated by excited-state electron transfer from the key Tyr189 to the substrate. This mechanism appears to agree with the extant experimental data and reinterprets the final protonation step as a proton transfer to the active site itself rather than to the product, aiming at regenerating it for another round of catalysis.

3.
Antibiotics (Basel) ; 11(6)2022 Jun 07.
Artigo em Inglês | MEDLINE | ID: mdl-35740185

RESUMO

Resistance to trimethoprim and other antibiotics targeting dihydrofolate reductase may arise in bacteria harboring an atypical, plasmid-encoded, homotetrameric dihydrofolate reductase, called R67 DHFR. Although developing inhibitors to this enzyme may be expected to be promising drugs to fight trimethoprim-resistant strains, there is a paucity of reports describing the development of such molecules. In this manuscript, we describe the design of promising lead compounds to target R67 DHFR. Density-functional calculations were first used to identify the modifications of the pterin core that yielded derivatives likely to bind the enzyme and not susceptible to being acted upon by it. These unreactive molecules were then docked to the active site, and the stability of the docking poses of the best candidates was analyzed through triplicate molecular dynamics simulations, and compared to the binding stability of the enzyme-substrate complex. Molecule 32 ([6-(methoxymethyl)-4-oxo-3,7-dihydro-4H-pyrano[2,3-d]pyrimidin-2-yl]methyl-guanidinium) was shown by this methodology to afford extremely stable binding towards R67 DHFR and to prevent simultaneous binding to the substrate. Additional docking and molecular dynamics simulations further showed that this candidate also binds strongly to the canonical prokaryotic dihydrofolate reductase and to human DHFR, and is therefore likely to be useful to the development of chemotherapeutic agents and of dual-acting antibiotics that target the two types of bacterial dihydrofolate reductase.

4.
Biophys Chem ; 269: 106512, 2021 02.
Artigo em Inglês | MEDLINE | ID: mdl-33307371

RESUMO

The most recent contribution by Sunil Nath in these pages is, mostly, a repetition of his previous claims regarding failures of the chemiosmotic hypotheses, supplemented with some fresh misunderstandings of the points I had sought to clarify in my previous critique⁠. Considerable portions rehash 50-60 years-old controversies, with no apparent understanding that the current chemiosmotic hypothesis, while birthed by Mitchell, differs from Mitchell's details in many respects. As such, Nath has devoted much time dealing with a few errors (or wrong hypotheses) by Mitchell (in a few places I would almost venture to say "typographical mistakes in typesetting") and presents the ensuing conclusions as "refutations" of the chemiosmotic paradigm, completely neglecting that such details (such as the precise H+/ATP or H+:O ratios) are completely irrelevant to the reality (or not) of an electron-transport chain that uses the free energy liberated by electron-transfer to remove H+ from a compartment, to which it returns through and ATP synthase which uses the energy in that spontaneous return to drive ATP synthesis. The thermodynamical mistakes and misunderstandings of the relevant literature present in Nath's new contribution are so numerous, though, that I feel forced to call the attention of the readers of "Biophysical Chemistry" to them.


Assuntos
Trifosfato de Adenosina , Prótons , Trifosfato de Adenosina/metabolismo , Transporte de Elétrons , Oxirredução , Termodinâmica
5.
J Biomol Struct Dyn ; 39(15): 5579-5587, 2021 09.
Artigo em Inglês | MEDLINE | ID: mdl-32662753

RESUMO

There has recently been interest in the development of small-molecule inhibitors of the oligomerization of Bacillus anthracis protective antigen for therapeutic use. Some of the proposed lead compounds have, however, unfavorable solubility in aqueous medium, which prevents their clinical use. In this computational work, we have designed several hundreds of derivatives with progressively higher hydro-solubility and tested their ability to dock the relevant binding cavity. The highest-ranking docking hits were then subjected to 125 ns-long simulations to ascertain the stability of the binding modes. Several of the potential candidates performed quite disappointingly, but two molecules showed very stable binding modes throughout the complete simulations. Besides the identification of these two promising leads, these molecular dynamics simulations allowed the discovery of several insights that shall prove useful in the further improvement of these candidates toward higher potency and stability.Communicated by Ramaswamy H. Sarma.


Assuntos
Bacillus anthracis , Antígenos de Bactérias , Simulação de Acoplamento Molecular , Simulação de Dinâmica Molecular
6.
Biophys Chem ; 264: 106424, 2020 09.
Artigo em Inglês | MEDLINE | ID: mdl-32717593

RESUMO

Recent publications have questioned the appropriateness of the chemiosmotic theory, a key tenet of modern bioenergetics originally described by Mitchell and since widely improved upon and applied. In one of them, application of Gauss' law to a model charge distribution in mitochondria was argued to refute the possibility of ATP generation through H+ movement in the absence of a counterion, whereas a different author advocated, for other reasons, the impossibility of chemiosmosis and proposed that a novel energy-generation scheme (referred to as "murburn") relying on superoxide-catalyzed (or superoxide-promoted) ADP phosphorylation would operate instead. In this letter, those proposals are critically examined and found to be inconsistent with established experimental data and new theoretical calculations.


Assuntos
Metabolismo Energético , Mitocôndrias/metabolismo , Difosfato de Adenosina/metabolismo , Concentração de Íons de Hidrogênio , Osmose , Fosforilação , Prótons
7.
J Phys Chem A ; 122(37): 7497-7507, 2018 Sep 20.
Artigo em Inglês | MEDLINE | ID: mdl-30141932

RESUMO

The cycloaddition of azides to iodoalkynes is strongly enhanced by some Cu+-complexes. We have studied computationally six reaction pathways for the cycloaddition of 24 combinations of azide and iodoalkyne to identify the dominant pathways and the influence of reactant structure on the evolution of the reaction. Two pathways were found to be operating for distinct sets of reactants. In the first pathway, initial complexation of iodoalkyne by Cu+ is followed by the binding of the azide to the metal through its substituted nitrogen atom, followed by attack of the nonhalogenated alkyne carbon by the terminal nitrogen atom. This pathway is generally followed by aromatic or electron-deficient azides, unless the iodoalkyne bears an electron-withdrawing group. The second pathway is a single-step mechanism similar (apart from the alkyne bond weakening caused by complexation) to that observed in the absence of catalyst. Electron-deficient iodoalkynes and methyl azides strongly prefer this mechanism, regardless of the identity of the reaction partners. The catalytic gain obtained through the use of Cu+ depends only partially on its direct effect on the energy of the transition state (relative to that of the infinitely separated reactants) and may be lost if the iodoalkyne itself strongly interacts with the catalyst through the formation of too strong a π-complex.

8.
PeerJ ; 4: e2805, 2016.
Artigo em Inglês | MEDLINE | ID: mdl-28028471

RESUMO

Cofactor-less oxygenases perform challenging catalytic reactions between singlet co-substrates and triplet oxygen, in spite of apparently violating the spin-conservation rule. In 1-H-3-hydroxy-4-oxoquinaldine-2,4-dioxygenase, the active site has been suggested by quantum chemical computations to fine tune triplet oxygen reactivity, allowing it to interact rapidly with its singlet substrate without the need for spin inversion, and in urate oxidase the reaction is thought to proceed through electron transfer from the deprotonated substrate to an aminoacid sidechain, which then feeds the electron to the oxygen molecule. In this work, we perform additional quantum chemical computations on these two systems to elucidate several intriguing features unaddressed by previous workers. These computations establish that in both enzymes the reaction proceeds through direct electron transfer from co-substrate to O2 followed by radical recombination, instead of minimum-energy crossing points between singlet and triplet potential energy surfaces without formal electron transfer. The active site does not affect the reactivity of oxygen directly but is crucial for the generation of the deprotonated form of the co-substrates, which have redox potentials far below those of their protonated forms and therefore may transfer electrons to oxygen without sizeable thermodynamic barriers. This mechanism seems to be shared by most cofactor-less oxidases studied so far.

9.
PeerJ ; 4: e2299, 2016.
Artigo em Inglês | MEDLINE | ID: mdl-27547588

RESUMO

1,2-dihydro-1,2-azaborine is a structural and electronic analogue of benzene which is able to occupy benzene-binding pockets in T4 lysozyme and has been proposed as suitable arene-mimicking group for biological and pharmaceutical applications. Its applicability in a biological context requires it to be able to resist modification by xenobiotic-degrading enzymes like the P450 cytochromes. Quantum chemical computations described in this work show that 1,2-dihydro-1,2-azaborine is much more prone to modification by these enzymes than benzene, unless steric crowding of the ring prevents it from reaching the active site, or otherwise only allows reaction at the less reactive C4-position. This novel heterocyclic compound is therefore expected to be of limited usefulness as an aryl bioisostere.

10.
R Soc Open Sci ; 3(2): 150582, 2016 Feb.
Artigo em Inglês | MEDLINE | ID: mdl-26998328

RESUMO

We have studied the role of Cu(+)-phenantroline as a catalyst in the cyclization of N-aryl-enaminones using density-functional theory computations. The catalyst was found to bind the substrate upon deprotonation of its eneaminone, and to dramatically increase the acidity of the carbon adjacent to the ketone functionality. The deprotonation of this carbon atom yields a carbanion which attacks the aryl moiety, thereby closing the heterocycle in the rate-determining step. This C-C bond forming reaction was found to proceed much more rapidly when preceded by re-protonation of the substrate N-atom (which had lost H(+) in the initial step). Hydride transfer to the catalyst then completes the indole synthesis, in a very fast step. The influence of Li(+) and K(+) on the regio-selectivity of the cyclization of bromo-substituted analogues could not, however, be reproduced by our model. Alternative pathways involving either single-electron transfer from the catalyst to the substrate or ring cyclization without previous carbon α-deprotonation were found to be kinetically or thermodynamically inaccessible.

11.
PeerJ ; 3: e1127, 2015.
Artigo em Inglês | MEDLINE | ID: mdl-26246970

RESUMO

Bacterial populations present in Hg-rich environments have evolved biological mechanisms to detoxify methylmercury and other organometallic mercury compounds. The most common resistance mechanism relies on the H(+)-assisted cleavage of the Hg-C bond of methylmercury by the organomercurial lyase MerB. Although the initial reaction steps which lead to the loss of methane from methylmercury have already been studied experimentally and computationally, the reaction steps leading to the removal of Hg(2+) from MerB and regeneration of the active site for a new round of catalysis have not yet been elucidated. In this paper, we have studied the final steps of the reaction catalyzed by MerB through quantum chemical computations at the combined MP2/CBS//B3PW91/6-31G(d) level of theory. While conceptually simple, these reaction steps occur in a complex potential energy surface where several distinct pathways are accessible and may operate concurrently. The only pathway which clearly emerges as forbidden in our analysis is the one arising from the sequential addition of two thiolates to the metal atom, due to the accumulation of negative charges in the active site. The addition of two thiols, in contrast, leads to two feasible mechanistic possibilities. The most straightforward pathway proceeds through proton transfer from the attacking thiol to Cys159 , leading to its removal from the mercury coordination sphere, followed by a slower attack of a second thiol, which removes Cys96. The other pathway involves Asp99 in an accessory role similar to the one observed earlier for the initial stages of the reaction and affords a lower activation enthalpy, around 14 kcal mol(-1), determined solely by the cysteine removal step rather than by the thiol ligation step. Addition of one thiolate to the intermediates arising from either thiol attack occurs without a barrier and produces an intermediate bound to one active site cysteine and from which Hg(SCH3)2 may be removed only after protonation by solvent-provided H3O(+). Thiolate addition to the active site (prior to any attack by thiols) leads to pathways where the removal of the first cysteine becomes the rate-determining step, irrespective of whether Cys159 or Cys96 leaves first. Comparisons with the recently computed mechanism of the related enzyme MerA further underline the important role of Asp99 in the energetics of the MerB reaction. Kinetic simulation of the mechanism derived from our computations strongly suggests that in vivo the thiolate-only pathway is operative, and the Asp-assisted pathway (as well as the conversion of intermediates of the thiolate pathway into intermediates of the Cys-assisted pathway) is prevented by steric factors absent from our model and related to the precise geometry of the organomercurial binding-pocket.

12.
PeerJ ; 2: e551, 2014.
Artigo em Inglês | MEDLINE | ID: mdl-25237602

RESUMO

The addition of two electrons and two protons to the C17=C18 bond in protochlorophyllide is catalyzed by a light-dependent enzyme relying on NADPH as electron donor, and by a light-independent enzyme bearing a (Cys)3Asp-ligated [4Fe-4S] cluster which is reduced by cytoplasmic electron donors in an ATP-dependent manner and then functions as electron donor to protochlorophyllide. The precise sequence of events occurring at the C17=C18 bond has not, however, been determined experimentally in the dark-operating enzyme. In this paper, we present the computational investigation of the reaction mechanism of this enzyme at the B3LYP/6-311+G(d,p)//B3LYP/6-31G(d) level of theory. The reaction mechanism begins with single-electron reduction of the substrate by the (Cys)3Asp-ligated [4Fe-4S], yielding a negatively-charged intermediate. Depending on the rate of Fe-S cluster re-reduction, the reaction either proceeds through double protonation of the single-electron-reduced substrate, or by alternating proton/electron transfer. The computed reaction barriers suggest that Fe-S cluster re-reduction should be the rate-limiting stage of the process. Poisson-Boltzmann computations on the full enzyme-substrate complex, followed by Monte Carlo simulations of redox and protonation titrations revealed a hitherto unsuspected pH-dependence of the reaction potential of the Fe-S cluster. Furthermore, the computed distributions of protonation states of the His, Asp and Glu residues were used in conjuntion with single-point ONIOM computations to obtain, for the first time, the influence of all protonation states of an enzyme on the reaction it catalyzes. Despite exaggerating the ease of reduction of the substrate, these computations confirmed the broad features of the reaction mechanism obtained with the medium-sized models, and afforded valuable insights on the influence of the titratable amino acids on each reaction step. Additional comparisons of the energetic features of the reaction intermediates with those of common biochemical redox intermediates suggest a surprisingly simple explanation for the mechanistic differences between the dark-catalyzed and light-dependent enzyme reaction mechanisms.

13.
PeerJ ; 2: e470, 2014.
Artigo em Inglês | MEDLINE | ID: mdl-25071993

RESUMO

The binding of several rubromycin-based ligands to HIV1-reverse transcriptase was analyzed using molecular docking and molecular dynamics simulations. MM-PBSA analysis and examination of the trajectories allowed the identification of several promising compounds with predicted high affinity towards reverse transcriptase mutants which have proven resistant to current drugs. Important insights on the complex interplay of factors determining the ability of ligands to selectively target each mutant have been obtained.

14.
J Mol Model ; 19(12): 5457-67, 2013 Dec.
Artigo em Inglês | MEDLINE | ID: mdl-24220926

RESUMO

We have evaluated the performance of 15 density functionals of diverse complexity on the geometry optimization and energetic evaluation of model reaction steps present in the proposed reaction mechanisms of Cu(I)-catalyzed indole synthesis and click chemistry of iodoalkynes and azides. The relative effect of the Cu(+) ligand on the relative strength of Cu(+)-alkyne interactions, and the strong preference for a π-bonding mode is captured by all functionals. The best energetic correlations with MP2 are obtained with PBE0, M06-L, and PBE1PW91, which also provide good quality geometries. Furthermore, PBE0 and PBE1PW91 afford the best agreement with the high-level CCSD(T) computations of the deprotonation energies of Cu(+)-coordinated eneamines, where MP2 strongly disagrees with CCSD(T) and the examined DFT functionals. PBE0 also emerged as the most suitable functional for the study of the energetics and geometries of Cu(+) hydrides, while at the same time correctly capturing the influence of the Cu(+) ligands on the metal reactivity.


Assuntos
Catálise , Química Click , Cobre/química , Ligação de Hidrogênio , Ligantes , Metais/química , Modelos Moleculares , Teoria Quântica , Termodinâmica
15.
J Org Chem ; 77(10): 4653-9, 2012 May 18.
Artigo em Inglês | MEDLINE | ID: mdl-22530929

RESUMO

Substituted pyrroles may be synthesized from selected 1,2-pyridazines through a reductive ring contraction involving the addition of four electrons and four protons. Our density functional theory computations of this reaction mechanism show that the first reduction event must be preceded by the uptake of one proton by 1,2-pyridazine and that the reaction proceeds through a 2e(-)/3H(+)-bearing intermediate. In the absence of electron-withdrawing groups able to resonate charge away from the ring, this intermediate lies too high in energy, making the reaction sequence thermodynamically inaccessible. After another two-electron reduction and the addition of two more protons, the original 1,2-pyridazine ring opens. Ring contraction and ammonia elimination then proceed with very small barriers, irrespective of the substituents present in the original 1,2-pyridazine. By establishing the need for electron-withdrawing resonant groups in the 3- and 6-positions to stabilize the critical intermediate in the initial stages of the reaction, this work suggests that the scope of the reductive ring contraction of 1,2-pyridazines may be expanded to pyridazines bearing COCH(3) groups, amides or aryls in these positions. We also explain the lack of reactivity of unsubstituted 1,2-pyridazine and analyze the feasibility of bypassing the high energy 2e(-)/3H(+)-intermediate through disproportionation of earlier 2e(-)/2H(+)-bearing intermediates.

16.
Phys Rev Lett ; 107(18): 181601, 2011 Oct 28.
Artigo em Inglês | MEDLINE | ID: mdl-22107621

RESUMO

Superconductors in a cylindrical geometry respond periodically to a cylinder-threading magnetic flux, with the period changing from hc/2e to hc/e depending on whether the Aharonov-Bohm effects are suppressed. We show that holographic superconductors present a similar phenomenon, and that the different periodicities follow from classical no-hair theorems. We also give the Ginzburg-Landau description of the period-doubling phenomenon.

17.
J Phys Chem B ; 115(8): 1903-10, 2011 Mar 03.
Artigo em Inglês | MEDLINE | ID: mdl-21291195

RESUMO

Oxygen-dependent coproporphyrinogen III oxidase catalyzes the sequential decarboxylation of the propionate substituents present on the A and B rings of coproporphyrinogen III in the heme biosynthetic pathway. Although extensive experimental investigation of this enzyme has already afforded many insights into its reaction mechanism, several key features (such as the substrate binding mode, the characterization of the active site, and the initial substrate protonation state) remain poorly described. The molecular dynamics simulations described in this paper enabled the determination of a very promising substrate binding mode and the extensive characterization of the enzyme active site. The proposed binding mode is fully consistent with the known selectivity of the active site toward substituted tetrapyrroles and explains the lack of activity of the H131A, R135A, D274A, and R275A mutants and the reasons behind the nonoccurrence of catalysis on the C and D rings of the tetrapyrrole. An important role in this binding mode is fulfilled by G276, as its carbonyl oxygen intervenes in the substrate anchoring by hydrogen bonding its ring D pyrrole NH group. The presence of this interaction (which is only possible with the protonated NH pyrrole group) and the absence of positively charged side chains close to the pyrrole nitrogen (which might stabilize the N-deprotonated pyrrole postulated in some mechanistic proposals) show that the pyrrole ring is very unlikely to undergo deprotonation during the catalytic cycle and allow the discrimination between the previously postulated mechanistic proposals.


Assuntos
Coproporfirinogênio Oxidase/química , Substituição de Aminoácidos , Sítios de Ligação , Domínio Catalítico , Biologia Computacional , Coproporfirinogênio Oxidase/genética , Coproporfirinogênio Oxidase/metabolismo , Ligação de Hidrogênio , Simulação de Dinâmica Molecular , Mutação , Ligação Proteica , Especificidade por Substrato , Tetrapirróis/química
19.
J Phys Chem B ; 114(27): 8994-9001, 2010 Jul 15.
Artigo em Inglês | MEDLINE | ID: mdl-20553007

RESUMO

Uroporphyrinogen III decarboxylase catalyzes the fifth step in heme biosynthesis, the elimination of carboxyl groups from the four acetate side chains of uroporphyrinogen III to yield coproporphyrinogen III. We have previously found that the rate-limiting step of uroporphyrinogen III decarboxylase is substrate protonation rather than the decarboxylation reaction. This protonation can be effected by an arginine residue (Arg37) in close proximity to the substrate. In this report, we present evidence for the function of this arginine residue as a general acid catalyst. Although substrate protonation by H(3)O(+) is both exergonic and very fast, our density functional calculations show that in the presence of a protonated Arg37 substrate, decarboxylation becomes rate-limiting, and the substrate spontaneously breaks upon protonation. These results suggest that the active site must be shielded from solvent protons. Consequently, H(3)O(+) can be excluded from a role in both protonations proposed for the enzyme mechanism. In agreement with these conclusions, a second arginine residue (Arg41) is uniquely positioned to act as donor of the second proton, with an activation barrier below 2 kcal mol(-1). Generated mutant uroporphyrinogen III decarboxylase variants carrying amino acid exchanges in the position of both arginine residues (R41A, R41K, R37A, and R37K) failed to produce coproporphyrinogen III. The proposed unusual use of two basic residues as general acids in two different proton donation steps by uroporphyrinogen III decarboxylase provides an elegant solution to the problem of simultaneously binding the very negative uroporphyrinogen (which requires a positively charged active site), and selectively protonating it while preventing excessive carboxylate stabilization by positive charges.


Assuntos
Arginina , Oniocompostos , Uroporfirinogênio Descarboxilase/química , Uroporfirinogênio Descarboxilase/metabolismo , Domínio Catalítico , Humanos , Simulação de Dinâmica Molecular , Mutagênese Sítio-Dirigida , Prótons , Uroporfirinogênio Descarboxilase/genética , Uroporfirinogênios/metabolismo
20.
Phys Rev Lett ; 103(9): 091601, 2009 Aug 28.
Artigo em Inglês | MEDLINE | ID: mdl-19792782

RESUMO

A gravity dual of a superconductor at finite temperature has been recently proposed. We present the vortex configuration of this model and study its properties. In particular, we calculate the free energy as a function of an external magnetic field, the magnetization, and the superconducting density. We also find the two critical magnetic fields that define the region in which the vortex configurations are energetically favorable.

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