A semi-empirical method for the estimation of the hydration number of Mn(II)-complexes

: A semi-empirical equation to estimate the hydration number of Mn(II) complexes was derived from a database of 49 previously published 1 H longitudinal Nuclear Magnetic Relaxation Dispersion proﬁles. This equation has the longitudinal 1 H relaxivity and the molecular weight of the Mn(II) complex under consideration as parameters.


Introduction
During the last decades, Gd 3+ -chelates, such as Gd-DTPA (DTPA = diethylenetriamine-N,N,N',N",N",N"'-pentaacetate) and Gd-DOTA (DOTA = 1,4,7,10-tetraazacyclododecane-N,N',N",N"'tetraacetate), have become indispensable tools for the contrast enhancement of magnetic resonance images [1][2][3]. These contrast agents (CAs) have proven to be generally extremely safe; only 0.03% of all administrations (about 100 million worldwide) gave rise to serious adverse effects. However, during recent years, some concerns have arisen because of (i) incidents of nephrogenic systemic fibrosis (NSF) associated with Gd 3+ -complexes of linear DTPA derivates and (ii) observation of Gd-accumulation in the brains of patients with normal renal function, after repeated administrations of DTPA-type of CAs [4][5][6][7][8]. Both effects may be ascribed to the relative kinetic instability of Gd 3+ complexes of linear polyaminocarboxylates. The concerns about Gd-based CAs have led to an increased interest in application of Gd 3+ -free CAs. High spin d 5 Mn 2+ is an attractive alternative for f 7 Gd 3+ , because it has only two less unpaired electrons, whereas a high electronic symmetry is favorable for high relaxivities. Moreover, Mn 2+ is less toxic than Gd 3+ , as reflected in its important role as a cofactor in many enzymatic reactions, including the anti-oxidant enzyme superoxide dismutase, as well as in enzymes involved in neurotransmitter synthesis and metabolism in the brain. However, the concentration of free Mn 2+ in organisms is very low, for instance 0.3-1 µg·L −1 in human blood [9]. High concentrations are neurotoxic, and therefore, Mn 2+ preferably needs to be sequestered for safe application as CA in humans. Due to its lower charge, complexes are usually less stable than their Gd 3+ -counterparts. It is important to assure that the dissociation of Mn 2+ -based CAs is minimized in order to avoid any neurotoxic side effects. On the other hand, the relaxation rate enhancing efficacy should be as high as possible in view of the inherently low sensitivity of magnetic resonance imaging (MRI) CAs. Finding a balance between kinetic stability and optimal sensitivity is a challenge during the design of novel Mn 2+ -based CAs.
The efficacy of a CA is usually expressed by its relaxivity, the longitudinal, or transverse relaxation rate enhancement normalized for a solution containing 1 mM of paramagnetic metal ions (r 1 and r 2 , respectively). One of the most important parameters governing the longitudinal relaxivity (r 1 ) of paramagnetic CAs is the number of water molecules in the first coordination sphere of the metal ion (q), since the inner sphere contribution to r 1 is linearly proportional to it. Several good methods are available to evaluate q for Gd 3+ -based CAs, including measurements of lanthanide induced 17 O NMR shifts (NMR = Nuclear Magnetic Resonance) of water [1,10,11], determination of the 17 [12]. The determination of q for Mn 2+ -complexes is more challenging. The luminescence decay method is impossible, because the complexes are not luminescent. 17 O NMR methods are more ambiguous since they rely on the assumption that A O /h is almost independent on the coordination environment of Mn 2+ , which appears not to be case: values in the range −(26-73) × 10 6 rad·s −1 have been evaluated from 17 [13]. By contrast, the range of A O /h values observed for Gd 3+ -complexes observed is narrower (−(3.6-4.2) × 10 6 rad·s −1 ) [11], allowing reasonable accurate estimations of q using a 17 O NMR shift and/or relaxation rates. Moreover, accurate Mn 2+ -induced shift measurements are difficult because of the very large line broadenings and relatively small induced shifts.
X-ray crystallography may provide structures of Mn 2+ complexes in the solid state, but these structures are not necessarily the same in solution. Frequently dimeric or oligomeric assemblies occur in crystal structures, which will dissociate upon dissolution in water. Often, a value of q is estimated based on the denticity of the organic ligand and assuming a total Mn 2+ -coordination number of 6. However, high spin Mn 2+ -complexes have almost no ligand stabilization energy, and consequently, do not show a clear preference for a specific coordination number. Others use the magnitudes of the longitudinal relaxivities to estimate q, which is ambiguous as well, as will be demonstrated below. Recently, Póta et al. suggested that DFT calculations (at the M062X/TZVP level) may provide reliable q-values [14]; however, these calculations require excessive amounts of computing time with supercomputers.
Here, we will demonstrate that reliable q values for Mn 2+ -complexes can be obtained from experimental r 1 values at low Larmor frequencies (LF) and the formula weight (FW) of the complexes.

Results and Discussion
A database of 49 previously published 1 H longitudinal Nuclear Magnetic Relaxation Dispersion profiles (plots of r 1 versus LF) was constructed (see Table 1 and Figure 1). The relaxivity has an inner-sphere contribution by water molecules exchanging between the first coordination sphere of the metal ion and the bulk (r i,IS ) and an outer-sphere contribution due to water molecules in the bulk that diffuse in the surroundings of the metal ion without being bound to it (r 1,OS ) (see Equation (1)). The amount of a Mn 2+ -based CA needed is always very small (in the mM range) and the Mn 2+ -induced chemical shifts of the water protons are negligible. Under those conditions, r 1,IS was related to the longitudinal relaxation time of a Mn 2+ -bound water molecule (T 1M ), according to Equation (2) [15,16]: Here, τ M is the residence time of a water molecule in the first coordination sphere of Mn 2+ . Since the electronic relaxation time (T 1e ) for Mn 2+ -complexes was generally larger than the rotational correlation time (τ R ), the contribution of the Curie relaxation mechanism to the longitudinal relaxation was negligible; only the dipolar and the scalar mechanisms needed to be taken into consideration. These contributions (T 1M,DD −1 and T 1M,SC −1 , respectively) may be estimated by using the Solomon-Bloembergen-Morgan (SBM) equations (Equations (3)- (7)) [17,18]: Here, (µ 0 /4π) is the magnetic permeability in vacuum, r MnH is the distance between Mn 2+ and the H-atom of a bound water molecule, S is the electron spin (S = 5/2 for Mn 2+ ), γ I is the 1 H nuclear gyromagnetic ratio, γ S is the electron gyromagnetic ratio, and ω I and ω S are the Larmor frequencies of the proton and electron spin, respectively, A H /h is the hyperfine coupling constant between Mn 2+ and the bound water proton, The electronic relaxation times are often interpreted in terms of the zero-field splitting (ZFS) interaction using Equations (6) and (7) [19]: In these equations, ∆ 2 represents the mean-squared fluctuation of the ZFS, and τ v is the correlation time for the instantaneous distortion of the coordination polyhedron of Mn 2+ . From these equations, it can be derived that the low field limiting value of the longitudinal and transverse electronic relaxation rates (τ S0 −1 ) were equal, given by Equation (8): The outer-sphere contribution to the relaxivity (r 1,OS ) is described by Equations (9) and (10) [20]: Here, N A is Avogadro's number, a MnH is the distance of closest approach of a diffusing water molecule to Mn 2+ , D MnH stands for the diffusion coefficient, and J os (ω, T je ) (j = 1, 2) are spin density functions. The diffusion correlation time (τ MnH ) is given by a MnH 2 /D MnH .
An inspection of Equations (4) and (5) indicates that in an NMRD profile, two inflection points may be expected: one originating from 1/T 1,DD when ωτ d2 = 1 (at LF ≈ 10 MHz), and another one due to 1/T 1,SC when ωτ S2 = 1 (at LF ≈ 0.06 MHz) [21,22]. Until now, only two Mn II -based CAs have been observed that have NMRD profiles showing these two inflection points: [Mn II (H 2 O) 6 ] 2+ [21,22] and [Mn II It should be noted that the SBM equations, particularly Equations (6) and (7), only hold under certain conditions, such as (i) the Redfield condition of extreme narrowing (∆ s τ RH << 1, ∆ s is the static ZFS), (ii) the ZFS energy should be much larger than the Zeeman energy (E ZFS >> E Zeeman ), (iii) the electronic decay should be mono-exponential [24,25]. Outside these limits, a proper description of the electronic relaxation requires more complicated calculations that also take the static ZFS into account. Although the conditions for applying the SBM theory for the modeling of NMRD profiles are not always valid for Mn 2+ complexes, this theory has been applied in almost all studies on the complexes mentioned in Table 1. It may be justified if the Redfield condition applies and if only data for LF > 10 MHz were considered [26], because then the influence of the electronic relaxation can be neglected. Otherwise, the SBM model can be applied for a qualitative description of structure relaxivity relationships when the best fit parameters obtained, particularly those concerning the electronic relaxation, are considered as effective rather than physically relevant [27]. The most relevant parameters governing the relaxivity as determined by fitting of NMRD profiles with the SBM equations are included in Table 1.  Table 1.
For the evaluation of q, we focused our attention to the low field parts of the NMRD profiles. Below the dispersion at LF ≈ 10 MHz, the profiles always had a plateau, where r1 was rather large and almost independent of LF, which allows an accurate measurement of r1. At low field, r1 was dominated by q and the electronic relaxation, whereas at higher field strengths the nuclear relaxation dominated, which in turn was dominated by q, τR, and τM. This might lead to local maxima in the profile for high molecular weight compounds and relatively low relaxivities for low molecular compounds [2].  Table 1. For the evaluation of q, we focused our attention to the low field parts of the NMRD profiles. Below the dispersion at LF ≈ 10 MHz, the profiles always had a plateau, where r 1 was rather large and almost independent of LF, which allows an accurate measurement of r 1 . At low field, r 1 was dominated by q and the electronic relaxation, whereas at higher field strengths the nuclear relaxation dominated, which in turn was dominated by q, τ R , and τ M . This might lead to local maxima in the profile for high molecular weight compounds and relatively low relaxivities for low molecular compounds [2].  2 ], the r 1 was measured at about 1 MHz (at the plateau behind the first dispersion), where the scalar contribution may be neglected. Figure 2 displays a plot of the collected low field r 1 -values versus the reported q values for the Mn 2+ complexes concerned (q lit ). Many of the q lit values were obtained by applying a combination of several of the methods described above. For q = 0, the datapoints are in a narrow range of r 1 -values around an average of 2.4 ± 0.2 s −1 ·mM −1 . This value can therefore be applied as a good estimate of the outer sphere contribution of all Mn 2+ complexes. For q > 0, Figure 2 clearly shows that the ranges of r 1 values for various q-values overlap. Hence r 1 ranges alone cannot be applied for the evaluation of q.  A closer inspection of the structures corresponding with datapoints at q = 1 and q = 2 indicates that r1 increased with the molecular volume of the Mn 2+ -complexes at each q, which suggests that the r1 at low field also depended on the rotational correlation time (τR). Therefore, we next considered the datapoints for q ≠ 0. For these data, r1/q appeared not to be linearly proportional to the molecular weight (FW). Regression of r1/q as a function of FW with the computer program CurveExpert 1.4 using all in-built models afforded a good fit with Equation (11): The hydration number can be calculated with Equation (12): Figure 3 shows the good agreement between the calculated hydration numbers (qcalc) and the values mentioned in the publications from which the r1 values were taken (qlit). Fractional q numbers are also correctly predicted. From Figure 3, the accuracy of qcalc is estimated to be ±0.4. The good agreement between all calculated and reported q values indicates that the applied previously reported q values are correct. Moreover, it indicates that q and FW are the main parameters that determined the relaxivity; apparently, the influence of other parameters was rather small. This is surprising, since r1 at low LF was expected to be dominated by the electronic relaxation.  Table 1) seem to be predicted correctly. To explain this, simulations of r1 at LF = 0.01 MHz as a function of τM, τR, and τS0 were made using the SBM equations. Figure 4 shows, for example, the simulations for typical q = 1 Mn 2+ complexes. Figure 4A shows that r1 is unsensitive to variations in τM; however, it is strongly dependent on τR, and Figure 4B shows that r1 was only very sensitive to variations of τS0 for complexes for which τR > 150 ps. The complexes in the database used to deduce Equations (11) and (12) are exclusively of low and medium molecular weight compounds (FW < 1250). A closer inspection of the structures corresponding with datapoints at q = 1 and q = 2 indicates that r 1 increased with the molecular volume of the Mn 2+ -complexes at each q, which suggests that the r 1 at low field also depended on the rotational correlation time (τ R ). Therefore, we next considered the datapoints for q = 0. For these data, r 1 /q appeared not to be linearly proportional to the molecular weight (FW). Regression of r 1 /q as a function of FW with the computer program CurveExpert 1.4 using all in-built models afforded a good fit with Equation (11): The hydration number can be calculated with Equation (12): Figure 3 shows the good agreement between the calculated hydration numbers (q calc ) and the values mentioned in the publications from which the r 1 values were taken (q lit ). Fractional q numbers are also correctly predicted. From Figure 3, the accuracy of q calc is estimated to be ±0.4. The good agreement between all calculated and reported q values indicates that the applied previously reported q values are correct. Moreover, it indicates that q and FW are the main parameters that determined the relaxivity; apparently, the influence of other parameters was rather small. This is surprising, since r 1 at low LF was expected to be dominated by the electronic relaxation. Even the q-values of complexes with relatively large τ S0 6 ] 2+ , see Table 1) seem to be predicted correctly. To explain this, simulations of r 1 at LF = 0.01 MHz as a function of τ M , τ R , and τ S0 were made using the SBM equations. Figure 4 shows, for example, the simulations for typical q = 1 Mn 2+ complexes. Figure 4A shows that r 1 is unsensitive to variations in τ M ; however, it is strongly dependent on τ R , and Figure 4B shows that r 1 was only very sensitive to variations of τ S0 for complexes for which τ R > 150 ps. The complexes in the database used to deduce Equations (11) and (12) are exclusively of low and medium molecular weight compounds (FW < 1250). The simulations in Figure 4 confirmed that for those complexes, r 1 can be expected to be insensitive to variations in electronic relaxation rates and τ M . However, it may be expected that these Equations are no longer valid for high molecular weight Mn 2+ complexes. However, in those cases, q can generally be estimated from low molecular analogs.
Inorganics 2018, 6, x 7 of 10 An attempt to apply a similar procedure for the evaluation of q from r1 values at LF = 60 MHz resulted in unreasonable qcalc values that deviated substantially from qlit.  Table 1

Conclusions
An inspection of published 1 H NMRD profiles of Mn 2+ -complexes of low and medium molecular  Table 1  An attempt to apply a similar procedure for the evaluation of q from r1 values at LF = 60 MHz resulted in unreasonable qcalc values that deviated substantially from qlit.  Table 1

Conclusions
An inspection of published 1 H NMRD profiles of Mn 2+ -complexes of low and medium molecular weights (FW < 1250) shows that the magnitudes of r1 at LF ≤ 0.1 MHz is dominated by the parameters An attempt to apply a similar procedure for the evaluation of q from r 1 values at LF = 60 MHz resulted in unreasonable q calc values that deviated substantially from q lit .

Conclusions
An inspection of published 1 H NMRD profiles of Mn 2+ -complexes of low and medium molecular weights (FW < 1250) shows that the magnitudes of r 1 at LF ≤ 0.1 MHz is dominated by the parameters q and τ R . Using a dataset of 49 published NMRD profiles as learning set, Equations (11) and (12) were evaluated for a quick and reliable estimation of q from r 1 values at LF ≤ 0.1 MHz and the molecular weight of the complex.