Microhydration of cytosine and its radical anion: cytosine.(H2O)n (n=1-5)

Microhydration effects on cytosine and its radical anion have been investigated theoretically, by explicitly considering various structures of cytosine complexes with up to five water molecules. Each successive water molecule (through n=5) is bound by 7-10 kcal mol(-1) to the relevant cytosine complex. The hydration energies are uniformly higher for the analogous anion systems. While the predicted vertical detachment energy (VDE) of the isolated cytosine is only 0.48 eV, it is predicted to increase to 1.27 eV for the lowest-lying pentahydrate of cytosine. The adiabatic electron affinity (AEA) of cytosine was also found to increase from 0.03 to 0.61 eV for the pentahydrate, implying that the cytosine anion, while questionable in the gas phase, is bound in aqueous solution. Both the VDE and AEA values for cytosine are smaller than those of uracil and thymine for a given hydration number. These results are in qualitative agreement with available experimental results from photodetachment-photoelectron spectroscopy studies of Schiedt et al. [Chem. Phys. 239, 511 (1998)].     

Photoelectron spectrum of valence anions of uracil and first-principles calculations of excess electron binding energies

The photoelectron spectrum (PES) of the uracil anion is reported and discussed from the perspective of quantum chemical calculations of the vertical detachment energies (VDEs) of the anions of various tautomers of uracil. The PES peak maximum is found at an electron binding energy of 2.4 eV, and the width of the main feature suggests that the parent anions are in a valence rather than a dipole-bound state. The canonical tautomer as well as four tautomers that result from proton transfer from an NH group to a C atom were investigated computationally. At the Hartree-Fock and second-order Moller-Plesset perturbation theory levels, the adiabatic electron affinity (AEA) and the VDE have been converged to the limit of a complete basis set to within +/-1 meV. Post-MP2 electron-correlation effects have been determined at the coupled-cluster level of theory including single, double, and noniterative triple excitations. The quantum chemical calculations suggest that the most stable valence anion of uracil is the anion of a tautomer that results from a proton transfer from N1H to C5. It is characterized by an AEA of 135 meV and a VDE of 1.38 eV. The peak maximum is as much as 1 eV larger, however, and the photoelectron intensity is only very weak at 1.38 eV. The PES does not lend support either to the valence anion of the canonical tautomer, which is the second most stable anion, and whose VDE is computed at about 0.60 eV. Agreement between the peak maximum and the computed VDE is only found for the third most stable tautomer, which shows an AEA of approximately -0.1 eV and a VDE of 2.58 eV. This tautomer results from a proton transfer from N3H to C5. The results illustrate that the characteristics of biomolecular anions are highly dependent on their tautomeric form. If indeed the third most stable anion is observed in the experiment, then it remains an open question why and how this species is formed under the given conditions.     

Microsolvation pattern of the hydrated radical anion of uracil: U-(H2O)n (n = 3-5)

The microsolvation patterns of the uracil radical anion in water clusters U-(H2O)n with n ranging from 3 to 5 were investigated by the density functional theory approach. The electron detachment energies (VDE) of the stable anionic complexes with different numbers of hydration water are predicted. The linear dependence of the VDE value of the most stable anionic complexes with respect to the hydration number suggests the importance of the clustered waters in the microsolvation of the radical anion of the nucleobases. The formation of the water clusters is found to be necessary in the most stable conformers of the tri-, tetra-, and pentahydrated radical anion of uracil. The microsolvation pattern with three or more well-separated hydration water molecules in the first hydration layer is less stable than the arrangement with the waters in tight clusters. The charge transfer between the anionic uracil and the hydration water is high. Good agreement between the experimental and the theoretical vertical detachment energy yield in this study further demonstrates the practicability of the B3LYP/DZP++ approach in the study of radical anions of the DNA subunits.     

Barrier-free intermolecular proton transfer induced by excess electron attachment to the complex of alanine with uracil

The photoelectron spectrum of the uracil-alanine anionic complex (UA)(-) has been recorded with 2.540 eV photons. This spectrum reveals a broad feature with a maximum between 1.6 and 2.1 eV. The vertical electron detachment energy is too large to be attributed to an (UA)(-) anionic complex in which an intact uracil anion is solvated by alanine, or vice versa. The neutral and anionic complexes of uracil and alanine were studied at the B3LYP and second-order M?ller-Plesset level of theory with 6-31++G(*) (*) basis sets. The neutral complexes form cyclic hydrogen bonds and the three most stable neutral complexes are bound by 0.72, 0.61, and 0.57 eV. The electron hole in complexes of uracil with alanine is localized on uracil, but the formation of a complex with alanine strongly modulates the vertical ionization energy of uracil. The theoretical results indicate that the excess electron in (UA)(-) occupies a pi(*) orbital localized on uracil. The excess electron attachment to the complex can induce a barrier-free proton transfer (BFPT) from the carboxylic group of alanine to the O8 atom of uracil. As a result, the four most stable structures of the uracil-alanine anionic complex can be characterized as a neutral radical of hydrogenated uracil solvated by a deprotonated alanine. Our current results for the anionic complex of uracil with alanine are similar to our previous results for the anion of uracil with glycine, and together they indicate that the BFPT process is not very sensitive to the nature of the amino acid's hydrophobic residual group. The BFPT to the O8 atom of uracil may be relevant to the damage suffered by nucleic acid bases due to exposure to low energy electrons.     

Electron attachment to a hydrated DNA duplex: the dinucleoside phosphate deoxyguanylyl-3',5'-deoxycytidine

The minimal essential section of DNA helices, the dinucleoside phosphate deoxyguanylyl-3',5'-deoxycytidine dimer octahydrate, [dGpdC](2), has been constructed, fully optimized, and analyzed by using quantum chemical methods at the B3LYP/6-31+G(d,p) level of theory. Study of the electrons attached to [dGpdC](2) reveals that DNA double strands are capable of capturing low-energy electrons and forming electronically stable radical anions. The relatively large vertical electron affinity (VEA) predicted for [dGpdC](2) (0.38 eV) indicates that the cytosine bases are good electron captors in DNA double strands. The structure, charge distribution, and molecular orbital analysis for the fully optimized radical anion [dGpdC](2)(·-) suggest that the extra electron tends to be redistributed to one of the cytosine base moieties, in an electronically stable structure (with adiabatic electron affinity (AEA) 1.14 eV and vertical detachment energy (VDE) 2.20 eV). The structural features of the optimized radical anion [dGpdC](2)(·-) also suggest the probability of interstrand proton transfer. The interstrand proton transfer leads to a distonic radical anion [d(G-H)pdC:d(C+H)pdG](·-), which contains one deprotonated guanine anion and one protonated cytosine radical. This distonic radical anion is predicted to be more stable than [dGpdC](2)(·-). Therefore, experimental evidence for electron attachment to the DNA double helices should be related to [d(G-H)pdC:d(C+H)pdG](·-) complexes, for which the VDE might be as high as 2.7 eV (in dry conditions) to 3.3 eV (in fully hydrated conditions). Effects of the polarizable medium have been found to be important for increasing the electron capture ability of the dGpdC dimer. The ultimate AEA value for cytosine in DNA duplexes is predicted to be 2.03 eV in aqueous solution.     

Electron affinities of small uracil-water complexes: a comparison of benchmark CCSD(T) calculations with DFT

We present benchmark CCSD(T) calculations of the adiabatic electron affinities (AEA) and the vertical detachment energies (VDE) of the uracil molecule interacting with one to three water molecules. Calculations with rather large aug-cc-pVTZ basis set were only tractable when the space of virtual orbitals was reduced to about 60% of the full space employing the OVOS (Optimized Virtual Orbital Space) technique. Because of the microhydration, the valence-bound uracil anion is stabilized leading to gradually more positive values of both AEA and VDE with increasing number of participating water molecules. This agrees with experimental findings. Upon hydration by three water molecules, the electron affinity of uracil increased in comparison with AEA of the isolated molecule by about 250 up to 570 meV, depending on the geometry of the complex. CCSD(T) results confirm trends determined by DFT calculations of the microhydrated uracil and its anion, even if electron affinities of the free and hydrated uracil molecule are overestimated by DFT by up to 300 meV.     

Electron attachment to gas-phase uracil

We present results about dissociative electron attachment (DEA) to gas-phase uracil (U) for incident electron energies between 0 and 14 eV using a crossed electron/molecule beam apparatus. The most abundant negative ion formed via DEA is (U-H)-, where the resonance with the highest intensity appears at 1.01 eV. The anion yield of (U-H)- shows a number of peaks, which can be explained in part as being due to the formation of different (U-H)- isomers. Our results are compared with high level ab initio calculations using the G2MP2 method. There was no measurable amount of a parent ion U-. We also report the occurrence of 12 other fragments produced by dissociative electron attachment to uracil but with lower cross sections than (U-H)-. In addition we observed a parasitic contaminating process for conditions where uracil was introduced simultaneously with calibrant gases SF6 and CCl4 that leads to a sharp peak in the (U-H)- cross section close to 0 eV. For (U-H)- and all other fragments we determined rough measures for the absolute partial cross section yielding in the case of (U-H)- a peak value of sigma (at 1.01 eV)=3 x 10(-20) m2.     

Effects of microsolvation on the adenine-uracil base pair and its radical anion: adenine-uracil mono- and dihydrates

Microhydration effects upon the adenine-uracil (AU) base pair and its radical anion have been investigated by explicitly considering various structures of their mono- and dihydrates at the B3LYP/DZP++ level of theory. For the neutral AU base pair, 5 structures were found for the monohydrate and 14 structures for the dihydrate. In the lowest-energy structures of the neutral mono- and dihydrates, one and two water molecules bind to the AU base pair through a cyclic hydrogen bond via the N(9)-H and N(3) atoms of the adenine moiety, while the lowest-lying anionic mono- and dihydrates have a water molecule which is involved in noncyclic hydrogen bonding via the O4 atom of the uracil unit. Both the vertical detachment energy (VDE) and adiabatic electron affinity (AEA) of the AU base pair are predicted to increase upon hydration. While the VDE and AEA of the unhydrated AU pair are 0.96 and 0.40 eV, respectively, the corresponding predictions for the lowest-lying anionic dihydrates are 1.36 and 0.75 eV, respectively. Because uracil has a greater electron affinity than adenine, an excess electron attached to the AU base pair occupies the pi* orbital of the uracil moiety. When the uracil moiety participates in hydrogen bonding as a hydrogen bond acceptor (e.g., the N(6)-H(6a)...O(4) hydrogen bond between the adenine and uracil bases and the O(w)-H(w)...N and O(w)-H(w)...O hydrogen bonds between the AU pair and the water molecules), the transfer of the negative charge density from the uracil moiety to either the adenine or water molecules efficiently stabilizes the system. In addition, anionic structures which have C-H...O(w) contacts are energetically more favorable than those with N-H...O(w) hydrogen bonds, because the C-H...O(w) contacts do not allow the unfavorable electron density donation from the water to the uracil moiety. This delocalization effect makes the energetic ordering for the anionic hydrates very different from that for the corresponding neutrals.     

Microsolvation effect, hydrogen-bonding pattern, and electron affinity of the uracil-water complexes U-(H2O)n (n = 1, 2, 3)

To achieve a systematic understanding of the influence of microsolvation on the electron accepting behaviors of nucleobases, the reliable theoretical method (B3LYP/DZP++) has been applied to a comprehensive conformational investigation on the uracil-water complexes U-(H(2)O)(n) (n = 1, 2, 3) in both neutral and anionic forms. For the neutral complexes, the conformers of hydration on the O2 of uracil are energetically favored. However, hydration on the O4 atom of uracil is more stable for the radical anions. The electron structure analysis for the H-bonding patterns reveal that the CH...OH(2) type H-bond exists only for di- and trihydrated uracil complexes in which a water dimer or trimer is involved. The electron density structure analysis and the atoms-in-molecules (AIM) analysis for U-(H(2)O)(n) suggest a threshold value of the bond critical point (BCP) density to justify the CH...OH(2) type H-bond; that is, CH...OH(2) could be considered to be a H-bond only when its BCP density value is equal to or larger than 0.010 au. The positive adiabatic electron affinity (AEA) and vertical detachment energy (VDE) values for the uracil-water complexes suggest that these hydrated uracil anions are stable. Moreover, the average AEA and VDE of U-(H(2)O)(n) increase as the number of the hydration waters increases.     

Successive attachment of electrons to protonated Guanine: (G+H)* radicals and (G+H)- anions

The structures, energetics, and vibrational frequencies of nine hydrogenated 9H-keto-guanine radicals (G+H)(*) and closed-shell anions (G+H)(-) are predicted using the carefully calibrated (Chem. Rev. 2002, 102, 231) B3LYP density functional method in conjunction with a DZP++ basis set. These radical and anionic species come from consecutive electron attachment to the corresponding protonated (G+H)(+) cations in low pH environments. The (G+H)(+) cations are studied using the same level of theory. The proton affinity (PA) of guanine computed in this research (228.1 kcal/mol) is within 0.7 kcal/mol of the latest experiment value. The radicals range over 41 kcal/mol in relative energy, with radical r1, in which H is attached at the C8 site of guanine, having the lowest energy. The lowest energy anion is a2, derived by hydride ion attachment at the C2 site of guanine. No stable N2-site hydride should exist in the gas phase. Structure a9 was predicted to be dissociative in this research. The theoretical adiabatic electron affinities (AEA), vertical electron affinities, and vertical detachment energies were computed, with AEAs ranging from 0.07 to 3.12 eV for the nine radicals.     

The ability of silylenes to bind excess electrons: electron affinities of SiX(2), and SiXY species (X,Y=H,CH(3),SiH(3),F,Cl,Br)

Recently, Ishida and co-workers have isolated silylene radical anions via the one-electron reduction of isolable cyclic dialkylsilylenes, discovering these corresponding radical anions to be relatively stable at low temperatures. Herein we report theoretical predictions of the adiabatic electron affinities (AEA), vertical electron affinities, and vertical detachment energies of a series of methyl, silyl, and halosubstituted silylene compounds. This research utilizes the carefully calibrated DZP++ basis with the combination of the popular nonhybrid and hybrid DFT functionals, BLYP, B3LYP, and BHHLYP. The level of theory employed and the ensemble of species under study confirm the ability of silylenes to bind excess electrons with Si(SiH(3))(2) being the most effective, having a predicted AEA of 1.95 eV. While it is known that methyl substituents have a diminishing effect on the computed electron affinities (EAs), it is shown that fluorine shows an analogous negative effect. Similarly, previous suggestions that Si(CH(3))(2) will not bind an electron appear incorrect, with EA[Si(CH(3))(2)] predicted here to be 0.46 eV.     

Vibrational Feshbach resonances in uracil and thymine

Sharp peaks in the dissociative electron attachment (DEA) cross sections of uracil and thymine at energies below 3 eV are assigned to vibrational Feshbach resonances (VFRs) arising from coupling between the dipole bound state and the temporary anion state associated with occupation of the lowest sigma* orbital. Three distinct vibrational modes are identified, and their presence as VFRs is consistent with the amplitudes and bonding characteristics of the sigma* orbital wave function. A deconvolution method is also employed to yield higher effective energy resolution in the DEA spectra. The site dependence of DEA cross sections is evaluated using methyl substituted uracil and thymine to block H atom loss selectively. Implications for the broader issue of DNA damage are briefly discussed.     

Vibrational Spectroscopy of the Dehydrogenated Uracil Radical by Autodetachment of Dipole‐Bound Excited States of Cold Anions

Molecules with large enough dipole moments can bind an electron by the dipole field, which has little effect on the molecular core. A molecular anion can be excited to a dipole‐bound state, which can autodetach by vibronic coupling. Autodetachment spectroscopy of a complex anion cooled in a cryogenic ion trap is reported. Vibrational spectroscopy of the dehydrogenated uracil radical is obtained by a dipole‐bound state with partial rotational resolution. Fundamental frequencies for 21 vibrational modes of the uracil radical are reported. The electron affinity of the uracil radical is measured accurately to be 3.4810±0.0006 eV and the binding energy of the dipole‐bound state is measured to be 146±5 cm^?1. The rotational temperature of the trapped uracil anion is evaluated to be 35 K.     

Coupled-cluster and explicitly correlated perturbation-theory calculations of the uracil anion

A valence-type anion of the canonical tautomer of uracil has been characterized using explicitly correlated second-order Moller-Plesset perturbation theory (RI-MP2-R12) in conjunction with conventional coupled-cluster theory with single, double, and perturbative triple excitations. At this level of electron-correlation treatment and after inclusion of a zero-point vibrational energy correction, determined in the harmonic approximation at the RI-MP2 level of theory, the valence anion is adiabatically stable with respect to the neutral molecule by 40 meV. The anion is characterized by a vertical detachment energy of 0.60 eV. To obtain accurate estimates of the vertical and adiabatic electron binding energies, a scheme was applied in which electronic energy contributions from various levels of theory were added, each of them extrapolated to the corresponding basis-set limit. The MP2 basis-set limits were also evaluated using an explicitly correlated approach, and the results of these calculations are in agreement with the extrapolated values. A remarkable feature of the valence anionic state is that the adiabatic electron binding energy is positive but smaller than the adiabatic electron binding energy of the dipole-bound state.     

Electron affinities of uracil: microsolvation effects and polarizable continuum model

We present adiabatic electron affinities (AEAs) and the vertical detachment energies (VDEs) of the uracil molecule interacting with one to five water molecules. Credibility of MP2 and DFT/B3LYP calculations is supported by comparison with available benchmark CCSD(T) data. AEAs and VDEs obtained by MP2 and DFT/B3LYP methods copy trends of benchmark CCSD(T) results for the free uracil and uracil-water complexes in the gas phase being by 0.20 - 0.28 eV higher than CCSD(T) values depending on the particular structure of the complex. AEAs and VDEs from MP2 are underestimated by 0.09-0.15 eV. For the free uracil and uracil-(H(2)O)(n) (n = 1,2,3,5) complexes, we also consider the polarizable continuum model (PCM) and discuss the importance of the microsolvation when combined with PCM. AEAs and VDEs of uracil and uracil-water complexes enhance rapidly with increasing relative dielectric constant (ε) of the solvent. Highest AEAs and VDEs of the U(H(2)O)(5) complexes from B3LYP with ε = 78.4 are 2.03 and 2.81 eV, respectively, utilizing the correction from CCSD(T). Specific structural features of the microsolvated uracil-(H(2)O)(n) complexes and their anions are preserved also upon considering PCM in calculations of AEAs and VDEs.     

Electrochemical reduction of uracil in dimethyl sulfoxide: Autoprotonation of the anion radical

The electrochemical reduction of uracil in dimethyl sulfoxide was investigated, using d.c.and a.c. polarography, cyclic voltammetry, and controlled potential electrolysis. Uracil is reduced in a one-electron step (E_1/2=?2.3 V); the apparent number of electrons transferred (n) decreases from one at infinite dilution to one-half at concentrations above 1mM. The concentration dependent n-value is due to proton transfer by the parent compound to the radical anion formed on reduction. Such a proton transfer, which has been observed for 2-hydroxypyrimidine, deactivates part of the uracil, which would otherwise be available for reduction, by formation of the more difficultly reducible conjugate base. The uracil anion forms insoluble mercury salts, producing two oxidation waves (E_1/2 of ?0.1 and ?0.3 V); the latter wave is due to formation of a passivating film on the electrode. Digital simulations indicate that the protonation rate exceeds 10⁵M^?1 s^?1 and that, at low uracil concentration, some of the free radical formed on protonation is further reduced. At concentrations exceeding 1 mM, all of the free radical dimerizes. The effect of added acids and base on the electrochemical behavior is described.     

Solvent-induced shifts in electronic spectra of uracil

Highly accurate excitation spectra are predicted for the low-lying n-π* and π-π* states of uracil for both the gas phase and in water employing the complete active space self-consistent field (CASSCF) and multiconfigurational quasidegenerate perturbation theory (MCQDPT) methods. Implementation of the effective fragment potential (EFP) solvent method with CASSCF and MCQDPT enables the prediction of highly accurate solvated spectra, along with a direct interpretation of solvent shifts in terms of intermolecular interactions between solvent and solute. Solvent shifts of the n-π* and π-π* excited states arise mainly from a change in the electrostatic interaction between solvent and solute upon photoexcitation. Polarization (induction) interactions contribute about 0.1 eV to the solvent-shifted excitation. The blue shift of the n-π* state is found to be 0.43 eV and the red shift of the π-π* state is found to be -0.26 eV. Furthermore, the spectra show that in solution the π-π* state is 0.4 eV lower in energy than the n-π* state.     

Structures and energetics of electrosprayed uracil(n)Ca2+ clusters (n = 14-4) in the gas phase

Clusters of uracil (U) about a calcium dication, U(n)Ca(2+) (n = 14-4), have been studied in the gas phase by both experimental and theoretical methods. Temperature dependent blackbody infrared radiative dissociation (BIRD) experiments were performed on U(n)Ca(2+) clusters with n = 14-5 and the observed Arrhenius parameters are reported here. Master equation modeling of the BIRD kinetics data was carried out to determine threshold dissociation energies. Initial geometry calculations were performed using the B3LYP density functional and 3-21G(d) basis set. A sample of ten conformations per cluster was obtained through a simulated annealing study. These structures were optimized using B3LYP/6-31G(d) level of theory. Fragment-based hybrid many body interaction (HMBI) MP2/6-311++G(2df,2p)/Amoeba calculations were performed on representative conformations to determine theoretical binding energies. Results were examined in relation to cluster size (n). A significant increase in the energy required to remove uracil from U(6)Ca(2+) when compared to larger clusters supports previous reports that the calcium ion is coordinated by six uracil molecules in the formation of an inner shell. For clusters larger than n = 6, an odd-even alternation in threshold dissociation energies was observed, suggesting that the outer shell uracil molecules bind as dimers to the inner core. Proposed binding schemes are presented. Multiple structures of U(5)Ca(2+) are suggested as being present in the gas phase where the fifth uracil may be either part of the first or second solvation shell.     

The influence of microhydration on the ionization energy thresholds of uracil and thymine

In the present study the ionization energy thresholds (IET's) of uracil and thymine have been calculated (with the B3LYP, PMP2, and P3 levels of theory using the standard 6-31++G(d,p) basis set) with one to three water molecules placed in the first hydration shell. Then (B3LYP) polarizable continuum model (PCM) calculations were performed with one to three waters of the hydration shell included. Calculations show there is a distinct effect of microhydration on uracil and thymine. For uracil, one added water results in a decrease in the IET of about 0.15 eV. The second and third water molecules cause a further decrease by about 0.07 eV each. For thymine, the first water molecule is seen to decrease the IET by about 0.1 eV, while the second and third water molecules cause a further decrease of less than 0.1 eV each. The changes in IET calculated here for thymine with one to three waters of hydration are smaller than the experimental values determined by Kim et al. (Kim, S. K.; Lee, W.; Herschbach, D. R. J. Phys. Chem. 1996, 100, 7933). Preliminary results presented here indicate that the experimental results may involve keto-enol tautomers of thymine. The results of placing the microhydrated structures of uracil and thymine in a PCM cavity was seen to make very little difference in the IET when compared to the IET of ordinary uracil or thymine in a PCM cavity. The implications are that accurate calculations of the IET's of uracil and thymine can be obtained by simply considering long-range solvation effects.     

Do Photolyases Need To Provide Considerable Activation Energy for the Splitting of Cyclobutane Pyrimidine Dimer Radical Anions?

cis-syn Cyclobutane pyrimidine dimers, major UV-induced DNA lesions, are efficiently repaired by DNA photolyases. The key step of the repair reaction is a light-driven electron transfer from the FADH(-) cofactor to the dimer; the resulting radical anion splits spontaneously. Whether the splitting reaction requires considerable activation energy is still under dispute. Recent reports show that the splitting reaction of a dimer radical anion has a significant activation barrier (0.45 eV), and so photolyases have to provide considerable energy. However, these results contradict observations that cis-syn dimer radical anions split into monomers at -196 degrees C, and that the full process of DNA photoreactivation was fast (1.5-2 ns). To investigate the activation energies of dimer radical anions, three model compounds 1-3 were prepared. These include a covalently linked cyclobutane thymine dimer and a tryptophan residue (1) or a flavin unit (3), and the covalently linked uracil dimer and tryptophan (2). Their properties of photosensitised splitting of the dimer units by tryptophan or flavin unit were investigated over a large temperature range, -196 to 70 degrees C. The activation energies were obtained from the temperature dependency of splitting reactions for 1 and 2, 1.9 kJ mol(-1) and 0.9 kJ mol(-1) for the thymine and uracil dimer radical anions, respectively. These values are much lower than that obtained for E. coli photolyase (0.45 eV), and are surmountable at -196 degrees C. The activation energies provide support for previous observations that repair efficiencies for uracil dimers are higher than thymine dimers, both in enzymatic and model systems. The mechanisms of highly efficient enzymatic DNA repair are discussed.