Uric acid is one of the most interesting molecules in the human body susceptible to theoretical studies. Large amount of information can be obtained online nowadays, although some websites may contain serious errors. While there is no disagreement in uric acid being a diprotic acid, there is no uniformity in the name given to its –1 anion. Let us call the –1 anion by the name biurate (analogous to bicarbonate and bisulfate) and the –2 anion by the name of urate (in contrast to the name given to the –1 anion in many websites). Uric acid is a product of purine in food intake and is excreted in urine. A proper balance is maintained in the human body. Out of balance, hypouricemia is believed to be associated with multiple sclerosis, while hyperuricemia is linked to gout and some form of kidney stones. Gout is caused by precipitation of uric acid in blood, usually in the joints and some kidney stones are caused by precipitation of sodium biurate, according to several websites.

There is also some confusion to the values of the acid dissociation constants. Early studies have been summarized by Wilcox et al. [1], whose determination gave pK_a1=5.62 at 26°C and 5.57 at 37°C, as well as pK_a2=9.05 at 20°C. Also, in the 1970s, Finlayson and Smith [2] obtained a pK_a1 of 5.61 at 25°C and 5.47 at 38°C. In the 1980s, Simic snd Jovanovic [3] reported the acid dissociation constants of 5.40 and 9.80 at 20°C, while Wang and Königsberger [4] found pK_a1=5.26 at 25°C and 5.19 at 37°C. Surprisingly, one finds 3.89 for pK_a1 in the online Handbook of Chemistry and Physics. To get a semi-quantitative picture, we take 5.4 and 9.8 as the two pKa values and construct the distribution diagram [5], shown in Figure 1, assuming unity activity coefficients. Although there may be a slight decrease in pKa at 37°C, one expects over 98% of total uric acid in humans exists in the form of biurate and less than 1% as undissociated uric acid at the normal blood pH of 7.4. However, at a urine pH of 6.5, for example, the distribution is approximately 12% uric acid and 88% of biurate. Since uric acid is about 20 times less soluble than sodium biurate, the likely effect of hyperuricemia is that uric acid would precipitate in the blood vessels in one’s foot and that sodium biurate would precipitate as kidney stones (barring supersaturation). More recently, Ngo and Assimos [6] gave a thorough review of the epidemiology and pathophysiology of uric acid stones.

Figure 1: Distribution diagram showing the fractions of (a) uric acid H₂U, (b) biurate (HU^–)and (c) urate (U^2–), as functions of pH for pK_a1=5.4 and pK_a2=9.8, assuming unit activity coefficients.

In the present work, we make a theoretical study of uric acid and its ions because of their interesting properties, mainly using density functional theory (DFT). Throughout this paper, we use the Pople shorthand notation of method1/basis1//method2/basis2 to denote that method 1, with basis set 1 is used to compute the property of interest at the geometry optimized by method 2 with basis set 2. All of our predictions of properties were only calculated at the equilibrium geometry optimized by B3LYP/6-311+(d,p). A convenient shorthand notation for uric acid is H₂U and that the biurate anion, HU−. The salt NaHU is sometime known as monosodium monohydrogen urate (MSMU).

During the past few decades, many chemical and physical properties have been studied with density functional theory (DFT). Some workers try to design better and better functionals for all properties, while we among others look for best procedures for each property of interest. In both cases, experimental results are needed to validate the methods used. In this work, we wish to calculate several properties of uric acid and its ions using the best DFT method for each property, gained from experience with other molecules.

In this study, we used two programs: Gaussian09 [7] and ADF2012 [8]. Within the Gaussian09 package, many basis sets of contracted Gaussian functions are available. Workers tend to use Dunning’s correlation–consistent basis sets for small molecules and Pople’s split-valence sets, such as 6-31G for very large systems. For uric acid and its ions studied in this work, as well as other medium-size organic molecules, we opt for the polarized split-valence triple-zeta set 6-311+G(d,p). On the other hand, the ADF Program [8] provides a wide selection of basis sets of Slater-type orbitals. As in our recent studies, we favor the two sets of our own design [9,10]. For general properties of small to medium-size molecules, our favourite choice is the even-tempered polarized quadruple-zeta set (et-pVQZ) [9]. However, for polarizability and excitation calculations, a better basis set is the augmented polarized triple-zeta (aug-TZP) set [10]. From past experience, the smaller set of polarized triple-zeta (TZP) functions seems to be quite adequate for vertical ionization energies (VIEs) of both valence and core orbitals [11,12]. Hence, for larger molecules, one can save computing times by using the smaller TZP set. However, uric acid and its ions are small enough so that we can perform the calculations with the larger et-pVQZ basis set.

As stated above, all of our predictions of properties (with the exception of the UV absorption spectra in aqueous solutions) were calculated at the equilibrium geometry optimized by B3LYP/6- 311+(d,p). The B3LYP method has been thoroughly tested by Riley et al. [13] and Tirado-Rives and Jorgensen [14], on closed-shell organic molecules. For the geometry of uric acid and its ions in aqueous solution, we used the Polarizable Continuum Model (PCM) in the Gaussian09 program [7], by including

SCRF=(Solvent=Water) to the input

For the time-dependent DFT calculation of the excitations, we used the Conductor-like Screening Model (COSMO) in the ADF2012 program [8], by adding the following input lines

Solvation

SOLV Name=Water

End

The methods used in this work have been published many times in the literature. Because different methods were used for different properties, we need to give a brief description of each method used. For the geometry optimization of the structures of uric acid and its ions, we used the DFT method of B3LYP with an adequate basis set known as 6-311+G(d,p), available in the Gaussian09 [7] program. The results of the most stable tautomers shown in Figure 2 and Table 1, agree with previous studies. The vibrational frequencies and IR intensities of H₂U vapor were calculated at the same B3LYP/6-311+(d,p) level of theory. However, as an approximate correction to the harmonic approximation used, the predicted wave numbers have been scaled by a factor of 0.958, following the work of Asami et al. [15]. For the dipole moment and polarizabilty, as well as UV excitation spectrum, we preferred the exchange-correlation potential V_xc known as statistical averaging of orbital potentials (SAOP) [16-18]. For VIEs of valence electrons, the method abbreviated as ΔPBE0 (SAOP) was used [19]. It means the energy difference between parent and cation calculated with the parameter-free exchange-correlation functional E_xc known as PBE0 [20-22], for the electron density computed with V_xc=SAOP [16-18]. Finally, for core-electron binding energies (CEBEs), we use the method developed in 1999 [11,23,24], namely ΔPW86-PW91+C_rel, which stands for the energy difference between parent and cation calculated with the exchange functional PW86 [25] and the correlation functional PW91 [26]. A small relativistic correction C_rel, derived empirically in 1995 [27], was added. These methods have been tested on many organic molecules. See our recent paper [28] and references therein.

Figure 2: Structures of the most stable tautomers of (a) uric acid H₂U, (b) biurate (HU^–) and (c) urate (U^2–).

Table 1: Optimized bond lengths (in Ångstroms) of uric acid and its anions.

The optimized bond lengths of uric acid H₂U we obtained from B3LYP/6-311+G(d,p) are compared with previous studies [29-31] in Table 1, together with those of H₂U, HU^-, and U²− in aqueous solution. Many of the changes in bond lengths of the anions can be rationalized in terms of resonance, such as the lengthening of C4-C5 and the three C-O bonds, as well as the shortening of C2-N3, N3-C4, C5-C6, and C8-N9 in urate(aq). However, simple valence-bond bond order cannot explain the changes in the other bonds.

In Table 2, we present the vibrational wave numbers of uric acid vapor after scaling with a constant factor of 0.958, following Asami et al. [15] and the IR intensities in km mol^-1, together with the experimental values in gas phase [15], and in KBr pellets [32-37]. The agreement seems quite reasonable, especially when we notice the discrepancies between the results from different workers.

Table 2: Vibrational wave numbers (in cm^-1) and IR intensities (in km mol^-1) of uric acid.

We believe that use of V_xc=SAOP leads to reliable electron densities. The dipole moments from SAOP are 3.349, 3.444, and 3.384 D from et-pVQZ, aug-TZP, and aug-et-pVQZ basis sets, respectively, to be compared with 3.92 D from the B3LYP/6-31G** calculation of Altarsha et al. [38]. On the other hand, the average static dipole polarizabilities we obtained are 99.40, 100.55, and 100.28 au from et-pVQZ, aug- TZP, and aug-et-pVQZ basis sets, respectively; while the polarizability anisotropies are 132.29, 133.26, and 130.97 au from et-pVQZ, aug- TZP, and aug-et-pVQZ, respectively. We believe that the results from aug-et-pVQZ basis to be more reliable.

The study of UV absorption of uric acid and its anions is more difficult. Each of the three species, H₂U, HU^- and U²− absorbs UV light to be excited to two or more excited states. First, we summarize the wavelengths (in nm) of the experimental absorption maxima in Appendix A and include the most likely (in our opinion) vertical excitation energies (in eV) for the observed transitions. We do not have any experience in modeling UV-visible absorptions of organic molecules in aqueous solutions by continuum dielectric approximations and decide to test over 30 functionals in this study. In Table 3, we report the results of time-dependent DFT calculations of the three species in aqueous solutions. Only transitions with oscillator strengths greater than 0.1 are included. Calculated vertical excitation energies within 0.1 eV of the best estimates of the excitation energies are considered excellent and are indicated in red, while those within 0.2 eV of the best estimates fairly good and are shown in green. We see that many of the functionals tested do not give reliable estimates of the excitation energies. The best functionals appear to be PBE0 and mPW1PW, while the functional, X3LYP and B1LYP can also provide reasonable predictions. In Table 4, we present some of our results, now with f-values, to compare with the recent calculation of Altarsha et al. [39]. Note that our dielectric continuum models for the effect of solvation are undoubtedly a crude approximation for aqueous solutions, in which hydrogen bonds with the solutes are present. To include the effects of hydrogen bonds, would require inclusion of discreet water molecules with procedures, such as the ONIOM method [G09], which is beyond the scope of the present study. Of all the solvents for UVvisible excitation calculations, water is probably the most difficult to model. It would be interesting to see whether or not such a method applies to UV-visible absorption of other organic molecules in aqueous solution.

Table 3: Calculated observable (f >0.1) excitations of uric acid and its ions in aqueous solution.

Table 4: Excitation energies in eV with f-values in parentheses calculated with continuum dielectric models compared to observed absorption maxima^a of uric acid and its ions in aqueous solution.

In Table 5, we compare the vertical ionization energies of valence electrons computed by various methods, all calculated at the geometry optimized by B3LYP/6-311+(d,p). From our experience with other molecules, the reliability of the prediction is ranked as follows: Koopmans’ theorem (KT)41] somehow missed one of the low-lying cations). On the other hand, our DFT approach of ΔPBE0(SAOP) has been tested with many organic molecules and the basis set of et-pVQZ is large enough for almost all applications. Therefore, it is our belief that the column labeled DFT in Table 5 to be most reliable. Although the DFT calculations of some of the cations have not converged, we based the assignment of the observed values mainly on the DFT values. It would be interesting to have a new measurement of the photoelectron spectrum of uric acid, hopefully with better resolution extending to higher VIEs.

Table 5: Vertical ionization energies (in eV) of uric acid vapor.

Finally, the core-electron binding energies from our DFT calculations are presented in Table 6, in which calculated charges and the electrostatic potentials from the SAOP/et-pVQZ method are also listed for comparison. None of the calculated quantities offers a good correlation with the CEBEs. Since it is possible to measure the valence VIEs of uric acid vapor even back in 1978, we hope that an ESCA experiment with X-ray or synchrotron radiation can be performed on the core-electron binding energies also.

Table 6: Calculated core electron binding energies of uric acid vapor.

In this study, we have relied on our experience with DFT to predict the various properties of uric acid with what we believe to be the best method for each property. The properties include (a) the equilibrium geometry, (b) the vibrational spectrum, (c) the dipole moment, (d) the static dipole polarizability, (e) the UV absorption spectrum, (f) the vertical ionization energies of the valence electrons, and (g) the core-electron binding energies. In addition, we determined the best method (namely time-dependent DFT with the PBE0 functional) for the calculation of vertical excitation energies of uric acid and its anions in aqueous solution, even using the continuum dielectric models. It would be interesting to see whether such a method applies to UV-visible absorption other organic molecules in aqueous solution. Moreover, we encourage experimentalists to make new measurements of the ionization energies of both valence and core electrons.

Appendix A- Experimental UV spectra

There have been many measurements of the UV absorption spectra of uric acid and its anions in aqueous solution, some with specific pH controlled by buffers and some without specified pH values [37,42-52]. In Supplementary table, we summarize the measurements in order to assess the reliability of calculated vertical excitation energies and present the midrange value for each transition, after rejecting some likely outliers, indicated in italics. Those without specific pH values are included in the likely range based on the observed wavelengths of maximum absorption. It should be noted that absorption maxima do not necessarily correspond to vertical excitations because of vibrational effects.

Appendix B-Personal example

It may be of interest to some readers to see an actual example of the effect of pH on the possibility of gout and kidney stone. The following reasoning is based on several approximations, including unit activity coefficients. The acid dissociation constants of uric acid determined by Wilcox et al. [1] correspond to pK_a1=5.57 and pK_a2>10.70 at 37°C. Using these values, the fractions of H₂U, HU^- and U²− are 0.0145, 0.982 and 0.004, respectively, at pH=7.4 and 0.105, 0.8945, and 0.000 at pH=6.5. Wilcox et al. [1] also determined that the solubility of H₂U is 0.35 mM and the K_sp of NaHU to be 4.15x10^-5 M² or 7.92x10^-5 M², depending on which method one uses. Now, my recent urine analysis gave pH=6.5. If we assume that the total uric acid concentration=0.458 mM, and [Na⁺]=0.139 M are the same in my blood as in my urine, we can reach the following conclusions: The total uric acid concentration times the fractions give [H₂U]=0.0075 mM in my blood and [HU^-]=0.410 mM in my urine. In my urine, the concentration product [Na⁺][HU^-] can be estimated to be 5.7x10^-5 M², and therefore,there is a possibility of NaHU precipitating in my kidney in principle, although it is known that supersaturation is common with NaHU. On the other hand, the estimated concentration of H₂U in my blood is well below the solubility of 0.35 mM, thanks to the drug allopurinol. (The interested reader can see Ref. [53] for a DFT study of allopurinol). Since both gout and sodium biurate stones are caused by precipitations, drinking more water is certainly advisable. On the other hand, some internet websites advise drinking aqueous solution of baking soda, increasing the blood pH, to reduce the risk of uric acid precipitation. However, from the estimates given above, we see that such a procedure would not be advisable in my case.

The financial support of the Natural Sciences and Engineering Research Council (NSERC) of Canada is gratefully acknowledged.