2.4: Coordinated Water and NMR

It is quite relevant to know whether a water molecule is coordinated to the metal ion in a metalloenzyme, and whether it is still coordinated in the presence of substrates and inhibitors. The presence or absence of H₂O coordinated to a paramagnetic center can in principle be monitored by solvent water ¹H NMR,⁶⁹ by exploiting the occurrence of a magnetic interaction between the magnetic moments of the unpaired electrons and the nuclear magnetic moments of the water protons. When this interaction fluctuates with time, it causes a shortening of the water-proton relaxation times.*

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* The nuclear longitudinal relaxation time, T₁ can be defined as the rate constant by which the populations of the M_I = \(\frac{1}{2}\) and M_I = \(-\frac{1}{2}\) (for protons) levels reach their equilibrium value after an external perturbation (e.g., a radiofrequency pulse in an NMR experiment). The transverse relaxation time, T₂, can be defined as the average lifetime of a hydrogen nucleus in a given spin state. The NMR linewidth is inversely proportional to T₂. The relation T₂ ≤ T₁, always holds.

The longitudinal relaxation rate values, T₁^-1, of all the solvent water protons increase when even a single water molecule interacts with a paramagnetic center, provided that this bound water exchanges rapidly with free water molecules. To obtain the necessary experimental data, a methodology has been developed based on the measurement of water ¹H T₁^-1 values at various magnetic fields (Nuclear Magnetic Relaxation Dispersion, NMRD).^69-71 The experimental data contain information on the correlation time, i.e., the time constant for the dynamic process that causes the proton-unpaired electron interaction to fluctuate with time; furthermore, under certain conditions, they may provide quantitative information on the number of interacting protons and their distance to the metal. The enhancement of T₁^-1, called T_1p^-1, is caused by the paramagnetic effect on bound water molecules and by the exchange time \(\tau\)_m, according to the relationship

\[(T_{1p})^{-1} = f_{M}(T_{1M} + \tau_{M})^{-1} \tag{2.11}\]

where f_M is the molar fraction of bound water and T_1M is the relaxation time of a bound water proton. Therefore we measure the water ¹H T₁^-1, subtract the diamagnetic effect (i.e., the water-proton relaxation rate measured in a solution of a diamagnetic analogue), obtain T_1p^-1, then check that \(\tau\)_m is negligible with respect to T_1M For high-spin cobalt(II), T_1M is of the order of 10^-3 s, whereas \(\tau\)_m is about 10^-5 s. Then the experimental T_1p can be safely related to T_1M. It is now important, in order to proceed with the analysis, to define the correlation time for the interaction between proton nuclei and unpaired electrons, \(\tau\)_c. Its definition is important in order to obtain a physical picture of the system, and to quantitatively analyze the obtained T_1M values.⁶⁹ \(\tau\)_c is defined by

\[\tau_{c}^{-1} = \tau_{r}^{-1} + \tau_{s}^{-1} + \tau_{m}^{-1} \tag{2.12}\]

where (\tau\)_r is the rotational correlation time, (\tau\)_s is the electronic relaxation time, and (\tau\)_mhas been previously defined. (\tau\)_r depends on the size of the molecule, which can be calculated rigorously if the molecule is spherical, or approximately if it is not. The appropriate expression is

\[\tau_{r} = \frac{4\pi \eta a^{3}}{3k_{B}T} \tag{2.13}\]

where \(\eta\) is the microviscosity of the solution, a is the radius (or approximate radius) of the molecule, k_B is the Boltzmann constant, and T is the absolute temperature. For CA, \(\tau\)_r can be safely calculated to be \(\simeq\)10^-8 s at room temperature. Since the correlation time \(\tau\)_c in high-spin cobalt proteins varies between 10^-11 and 10^-12 s, it must therefore be determined by the electronic relaxation time.

Water ¹H NMRD profiles are often analyzed by using the classical dipolar interaction approach, as first described by Solomon:⁷²

\[T_{1M}^{-1} = \frac{2}{15} \left(\frac{\mu_{0}}{4 \pi}\right)^{2} \frac{\gamma_{I}^{2} g_{e}^{2} \mu_{B}^{2} S(S+1)}{r^{6}} \left(\frac{7 \tau_{c}}{1 + \omega_{S}^{2} \tau_{c}^{2}} + \frac{3 \tau_{c}}{1 + \omega_{I}^{2} \tau_{c}^{2}} \right) \tag{2.14}\]

where \(\mu\)₀ is the permeability of vacuum, \(\gamma\)_I is the nuclear magnetogyric ratio, g_e is the electron g-factor, S is the electron spin quantum number, r is the electron-nucleus distance, and \(\omega\)_S and \(\omega\)_I are the electron and nuclear Larmor frequencies, respectively. This equation describes the dipolar interaction between the magnetic moment of nucleus \(I(\hbar \gamma_{I} \sqrt{I(I+1)})\) and the magnetic moment of the electrons \(S(g_{e} \mu_{B} \sqrt{S(S+1)})\)as a function of the correlation time (\(\tau\)_c) and of the magnetic field (expressed as (\omega\)_I and \(\omega\)_S). Neglect of the zero-field splitting of the S = \(\frac{3}{2}\) manifold may introduce an error in the quantitative estimates within a factor of two. ⁷³

Fitting of the data for pseudotetrahedral complexes shows that they have \(\tau\)_s of 10^-11 s, whereas five-coordinate complexes have a shorter \(\tau\)_s, on the order of 10^-12 s. The latter derivatives also have exchangeable protons that could correspond to a water molecule in the coordination sphere, whereas the former do not.²⁵ The \(\tau\)_s values are thus proposed as indicators of the coordination number in low-symmetry, four- and five-coordinate cobalt complexes. The shorter electronic relaxation times are related to low-lying excited states, which, independently of the particular mechanism, favor electron relaxation.⁷⁴

Short electronic relaxation times in paramagnetic compounds cause only minor broadening of ¹H NMR lines, whereas the isotropic shifts (i.e., the shifts due to the presence of unpaired electron(s), usually very large) are independent of the value of the electronic relaxation times. For cobalt-substituted carbonic anhydrase, the ¹H NMR spectra have been recorded for several derivatives, and the proton signals of histidines coordinated to the metal were found to be shifted well outside the diamagnetic region (Figure 2.14).⁷⁵ Five-coordinate species give sharper signals than four-coordinate ones. The spectra in D₂O for both kinds of derivatives show three fewer isotropically shifted signals than in H₂O. These signals are assigned to histidine NH protons, which are replaced by deuterons in D₂O. Five-coordinate species provide ¹H NMR spectra with many signals slightly shifted from the diamagnetic position. It is believed that such complexes have relatively large magnetic anisotropy, which, summed up to the external magnetic field, provides further differentiation in shifts of the protons. Such shift contributions are called pseudocontact shifts. These shifts depend on the third power of the distance from the metal and on the position of the proton with respect to the molecular axes. These signals belong to protons of noncoordinated residues from 5 to 10 Å from the metal. Their assignment in principle provides further information on the structure in the vicinity of the metal ion. The ¹H NMR spectra of cobalt(II) enzymes thus afford a powerful method for monitoring structure and reactivity of the metal-bound residues. This is one task for future investigations of the enzyme.

Figure 2.14 - ¹H NMR spectra of cobalt(II)-substituted bovine carbonic anhydrase II and some inhibitor derivatives. The three sharp downfield signals in each spectrum disappear in D₂O and are assigned to the exchangeable ring NH protons of the three coordinated histidines. The sharp signal labeled with an arrow is assigned to the H\(\delta\)2 proton of His-119, which is the only non-exchangeable ring proton in a meta-like rather than in an ortho-like position with respect to the coordinating nitrogen. The T₁ values (ms) of the signals for the I^- and C₂O₄^2- derivatives are also shown.^25,75