Proton MRD Profile Analysis in Intracellular Hemoglobin Solutions: A Three Sites Exchange Model Approach

Abstract

The Three Sites Exchange Model (3SEM) was properly used to describe Proton (¹H) Magnetic Relaxation Dispersion (¹HMRD) in intracellular samples of hemoglobin A (HbA) and hemoglobin S (HbS) at 310 K. The HbA and HbS samples were obtained from whole blood of voluntary donors and patients, respectively, and processed by classical methods (centrifugation, decanting and freezing–thawing cycles). The ¹HMRD profiles (20 kHz–60 MHz) were obtained using a Fast Field Cycling NMR Relaxometer (Stelar FFC 2000 Spinmaster) and Minispec relaxometers (Mq20, Mq60). The 3SEM used includes the contribution of labile protons from the structure of the protein; and the contribution of the cross relaxation to dispersion was estimated as: at least one order of magnitude lower than the total dispersion. Two dispersions were found: one of them properly describing the hemoglobin rotational correlation time and another one probably related to internal and/or hydrated water molecules with effective correlation times higher than 1 ns and main residence times less than the rotational correlation time of the protein. The use of the 3SEM creates the conditions to properly explain proton magnetic relaxation during the HbS polymerization process.

Appendices

Appendix 1

The two sites water exchange model (2SWEM) for ¹H magnetic relaxation in protein solutions.

This approach [11,12,13,14,15] considers a fast exchange of water molecules between the solvent and the n_ws reduced amount of sites at the protein surface available for water binding. The water is considered irrotationally bound to these sites and the ¹H-¹H intramolecular dipolar interaction is defined as the main contribution to ¹H water relaxation. Moreover, a mono-exponential autocorrelation function is used to describe the dipolar couplings and the interacting spins are considered as included in one spherical molecule rotating isotropically in a continuous media.Thus, the ¹H water magnetic relaxation can be described as:

$$ \begin{gathered} R_{1} \left( {\omega_{0} } \right) = R_{1w}^{bulk} + \delta_{1} \tau_{C} \left[ {\frac{0.2}{{1 + \omega_{0}^{2} \tau_{C}^{2} }} + \frac{0.8}{{1 + 4\omega_{0}^{2} \tau_{C}^{2} }}} \right] \hfill \\ \delta_{1} = P_{b} \frac{3}{2}\left( {\frac{{\mu_{0} }}{4\pi }} \right)^{2} \frac{{\gamma^{4} \hbar^{2} }}{{r^{6} }} \hfill \\ R_{2} \left( {\omega_{0} } \right) = R_{2w}^{bulk} + \delta_{2} \tau_{C} \left[ {\frac{3}{5} + \frac{1}{{1 + \omega_{0}^{2} \tau_{C}^{2} }} + \frac{0.4}{{1 + 4\omega_{0}^{2} \tau_{C}^{2} }}} \right] \hfill \\ \delta_{2} = P_{b} \frac{3}{4}\left( {\frac{{\mu_{0} }}{4\pi }} \right)^{2} \frac{{\gamma^{4} \hbar^{2} }}{{r^{6} }} \hfill \\ \end{gathered} $$

(6)

In the Eq. (6), R₂ = 1/T₂ and $R_{2w}^{bulk}$ is the transverse proton magnetic relaxation rate in the solvent.τ_C is the effective correlation time of the bound water, which includes the contributions of the bound water residence time (τ_res) and τ_R according to [16]:

$$ \frac{1}{{\tau_{C} }} = \frac{1}{{\tau_{res} }} + \frac{1}{{\tau_{R} }} $$

(7)

As the bound water has been considered irrotationally bound to the protein (τ_res >  > τ_R), then τ_C = τ_R. P_b is a function of n_ws, the molar concentration of Hb (N_Hb), the molarity of water (N_w) and the volume fraction occupied by the macromolecules (V) [11, 13, 17]:

$$ P_{b} = \frac{{n_{ws} N_{Hb} }}{{N_{w} \left( {1 - V} \right)}} $$

(8)

Assuming n_ws ≤ 10, as suggested by several authors [16], P_b is in the order of 10^–4 for intracellular Hb (N_Hb = 5 mM/l) and the fraction of free water (solvent), which appears multiplying to $R_{1w}^{bulk}$ and $R_{2w}^{bulk}$ in the Eq. (6), can be considered equal to one.

Appendix 2

The three sites exchange model (3SEM) for ¹H magnetic relaxation in protein solutions.

The longitudinal magnetic relaxation of the ¹H in aqueous solutions of proteins is dominated by the R₁ of the water protons (R_1w) and the labile protons at the protein structure (R_1p) [1, 11, 13, 15, 16, 22]. In aqueous solutions of proteins there are three types of water: internal (in), hydrated (hy) and free (bulk) water [16, 22]. The internal water is extensively bounded to the protein, through hydrogen bonds, in small cavities and deep crevices localized at the macromolecular structure [23, 24], having main residence times ($\tau_{res}^{in}$) from10⁻¹⁰ s to 10⁻³ s [22,23,24,25], and rotational correlation times ($\tau_{c}^{in}$) greater than 10⁻⁹ s [26]. The hydrated water is bounded to the external protein surface through hydrogen bonds, having main residence times ($\tau_{res}^{hy}$) and rotational correlation times ($\tau_{c}^{hy}$) in the range from 10⁻¹¹ s to 10⁻¹⁰ s [23, 25]. The free water is characterized by rotational correlation times ($\tau_{c}^{bulk}$) in the order of 10⁻¹² s [16]. The labile protons are short lived protons (low values of residence times:$\tau_{res}^{p}$) located in specific residues at the protein structure. The internal and hydrated water, as well as the labile protons, exchange fast with the solvent ($\tau_{res}^{in} ,\tau_{res}^{hy} ,\tau_{res}^{p} < < R_{1}^{ - 1}$). The ¹HMRD profiles in aqueous solutions of proteins can be described using the following equation system [22]:

$$ \begin{gathered} R_{1} \left( {\omega_{0} } \right) = R_{1w}^{bulk} + \alpha + \beta \tau_{C}^{{}} \left[ {\frac{0.2}{{1 + \left( {\omega_{0} \tau_{C}^{{}} } \right)^{2} }} + \frac{0.8}{{1 + 4\left( {\omega_{0} \tau_{C}^{{}} } \right)^{2} }}} \right] \hfill \\ \alpha = \left( {\frac{{N_{hy} }}{{N_{T} }}} \right)\left( { < R_{1w}^{hy} >_{AV} - R_{1w}^{bulk} } \right) + 0.1\beta_{{{\text{int}} er}} \tau_{C}^{{}} \hfill \\ \beta = \beta_{{{\text{int}} ra}} + 0.9\beta_{{{\text{int}} er}} + \beta_{p} \hfill \\ \beta_{{{\text{int}} ra}} = \frac{1}{{N_{T} }}\frac{3}{2}\sum\limits_{\mu } {\left( {D_{\mu }^{{{\text{int}} ra}} A_{\mu }^{{{\text{int}} ra}} } \right)}^{2} \hfill \\ \beta_{{{\text{int}} er}} = \frac{1}{{N_{T} }}\sum\limits_{\mu } {\sum\limits_{i} {k_{i} } } \frac{1}{2}\left[ {\left( {D_{\mu 1i}^{{{\text{int}} er}} A_{\mu 1i}^{{{\text{int}} er}} } \right)^{2} + \left( {D_{\mu 2i}^{{{\text{int}} er}} A_{\mu 2i}^{{{\text{int}} er}} } \right)^{2} } \right] \hfill \\ D = \left( {\frac{{\mu_{0} }}{4\pi }} \right)^{{}} \frac{{\gamma^{2} \hbar }}{{\left( {r_{H - H}^{{}} } \right)^{3} }} \hfill \\ \beta_{p} = \frac{{\left[ {R_{1p} \left( 0 \right) - R_{1p} \left( {\omega_{\alpha } } \right)} \right]}}{{\tau_{C}^{{}} }} \hfill \\ R_{1p} \left( {\omega_{{}} } \right) = \frac{1}{{2\left( {N_{T} + N_{p} } \right)}}\sum\limits_{k} {\frac{{N_{pk} }}{{T_{1pk} \left( \omega \right) + \left( {\tau_{res}^{p} } \right)_{k} }}} \hfill \\ \hfill \\ \end{gathered} $$

(9)

where $R_{1w}^{hy}$ is the longitudinal magnetic relaxation rate of the ¹H at the hydrated water molecules and the symbol <  > _AV is used to indicate average value. N_hy and N_in are the numbers of hydrated and internal water molecules per protein molecule. A is a generalized orientational order parameter and D is the dipole coupling constant [22]. $A_{\mu }^{{{\text{int}} ra}}$ and $D_{\mu }^{{{\text{int}} ra}}$ correspond to the intramolecular dipolar interaction between the protons belonging to the internal water molecules. $A_{\mu 1i}^{{{\text{int}} er}}$,$A_{\mu 2i}^{{{\text{int}} er}}$, $D_{\mu 1i}^{{{\text{int}} er}}$ and $D_{\mu 2i}^{{{\text{int}} er}}$ correspond to the intermolecular dipolar interactions between the protons belonging to the internal water molecules and the ith proton at the macromolecular structure. Here has been considered that the R₁ of the protons at the internal water molecules and the R₁ of the labile protons disperse with the same effective correlation time: τ_C.

In the Eq. (9), the μ sum is over the N_in internal water molecules, the i sum is over all protons partners in long-lived intermolecular dipole couplings, the k sum is over all labile protons groups and subscripts 1 and 2 refer to the two protons at the internal water molecule [22]. N_T is the total number of bound water molecules per protein molecule (N_T = N_in + N_hy), N_p is the total number of labile protons per protein molecule and N_pk is the number of labile protons in each group. For intermolecular dipole couplings within the cluster of the internal water molecules k_i is a function of ω₀ and τ_R, and for intermolecular dipole couplings with protein protons k_i = 1 [22]. ω_α is a resonance frequency at the high-frequency relaxation rate plateau. The Eq. (9) is strictly valid for an isolated pair of dipole-coupled equivalent I = 1/2 nuclei.

To take into account the contribution from cross-relaxation, a negative term is added to the first equation of the Eq. (9) to obtain:

$$ \begin{gathered} R_{1} \left( {\omega_{0} } \right) = R_{1w}^{bulk} + \alpha + \beta \tau_{C}^{{}} \left[ {\frac{0.2}{{1 + \left( {\omega_{0} \tau_{C}^{{}} } \right)^{2} }} + \frac{0.8}{{1 + 4\left( {\omega_{0} \tau_{C}^{{}} } \right)^{2} }}} \right] - \beta_{cross} \tau_{C}^{{}} F_{cross} \left( {\omega_{0} \tau_{C} } \right) \hfill \\ F_{cross} \left( {\omega_{0} \tau_{C} } \right) = \frac{{\left[ {\frac{1.2}{{1 + 4\left( {\omega_{0} \tau_{C} } \right)^{2} }} - 0.2} \right]^{2} }}{{\left[ {0.1 + \frac{0.3}{{1 + \left( {\omega_{0} \tau_{C}^{{}} } \right)^{2} }} + \frac{0.6}{{1 + 4\left( {\omega_{0} \tau_{C}^{{}} } \right)^{2} }}} \right]}} \hfill \\ \end{gathered} $$

(10)

Here F_cross(ω₀τ_C) is obtained for the particular case in which the protons of the bound water interact with only one proton belonging to the protein structure [22]. More general and inclusive cases give place to much more complex terms describing cross-relaxation contribution with non-analytical solutions [22].

If $\tau_{res}^{in} ,\tau_{res}^{p}$ >  > τ_R, then τ_C = τ_R in the equations for R₁, β_p and F_cross at the Eqs. (9) and (10) considering that:

$$ \frac{1}{{\tau_{c}^{i} }} = \frac{1}{{\tau_{res}^{i} }} + \frac{1}{{\tau_{R}^{{}} }} $$

(11)

where the superscript i can be referred to the internal water molecules, or to labile protons at the protein structure. In the specific case of α (Eq. (9)), the second term corresponds to intermolecular dipole couplings of the internal water molecules, in which τ_C=$\tau_{c}^{in}$ is lower than 1 ns, thus contributing to the non-dispersive relaxation.