Paramagnetism in Experimental Biomolecular NMR / Edition 1

by Claudio Luchinat, Giacomo Parigi, Enrico Ravera

Overview

Paramagnetic NMR is a growing technique that represents an increasingly important tool for the investigation of biomolecules. This book presents an update and overview of the paramagnetic NMR phenomena and effects as well as guidelines for practical implementation of state-of-the-art experiments. All experiments are supported by a solid theoretical foundation. Areas mentioned are the development of solid state NMR, the use of paramagnetic tags providing information on the structure and mobility of the investigated systems, and dynamic nuclear polarization to increase sensitivity.

Compiled by experts in the field, this book has international appeal for researchers as well as students interested in magnetic resonance and structural biology who require experimental support and accessible information.

Product Details

ISBN-13:	9781788010863
Publisher:	RSC
Publication date:	08/07/2018
Series:	ISSN , #16
Pages:	316
Product dimensions:	6.14(w) x 9.21(h) x (d)

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CHAPTER 1

NMR Consequences of the Nucleus–Electron Spin Interactions

GIACOMO PARIGI AND CLAUDIO LUCHINAT

1.1 The Effect of Paramagnetism on NMR Spectra

The presence of unpaired electrons in molecules makes them paramagnetic and largely affects their NMR spectra. The main effects related to the presence of a paramagnetic center, i.e., of atoms or ions with unpaired electrons, are (see Figure 1.1):

(1) the NMR shifts are perturbed, so that the shifts measured for the paramagnetic molecule (δ) and for a diamagnetic analogue (δdia) (i.e., for the same molecule without the paramagnetic center or with the paramagnetic metal ion substituted by a diamagnetic one) differ. These differences are called hyperfine shifts (δhs):

δ = δdia + δhs (1.1)

(2) the nuclear relaxation rates of the paramagnetic molecule (R1 longitudinal; R2 transverse) are increased with respect to those of a diamagnetic analogue (Rdia1 and Rdia2). This difference is called paramagnetic relaxation enhancement (R1M and R2M):

[MATHEMATICAL EXPRESSION OMITTED] (1.2)

(3) the probabilities for the different orientations of the paramagnetic molecule in a magnetic field are not the same, so a partial self-orientation occurs. This partial self-orientation is responsible for the occurrence of residual dipolar couplings (Δvrdc), analogously to what happens when partial molecular orientation is driven by external devices. The presence of paramagnetic residual dipolar couplings affects the J-coupling between nuclei:

[MATHEMATICAL EXPRESSION OMITTED] (1.3)

This chapter deals with the description of these effects, and of their physical origin, which is related to the so-called hyperfine coupling. In eqn (1.1)–(1.3) other minor effects, namely residual anisotropic chemical shifts, cross relaxation terms, and dynamic frequency shifts, respectively, have been neglected (see later).

1.1.1 The Hyperfine Coupling

The interaction between the magnetic moment associated to a nuclear spin I and the magnetic moment associated to an unpaired electron spin S is called hyperfine coupling and is traditionally described by the Hamiltonian

Hhc = S x A x I (1.4)

The energy corresponding to this interaction fluctuates with time (and thus this interaction represents a source of relaxation for the nuclear spin, see Section 1.4) as a result of the molecular motions and of changes in the electron spin state. The other terms describing the contributions to the energy of the system are:

(i)the Zeeman interaction between the applied magnetic field B0 and the nuclear magnetic moment µI=hγII, where h is Planck's constant divided by 2π and γI is the nuclear magnetogyric ratio,

(ii)the Zeeman interaction between B0 and an 'effective' electron magnetic moment taking into account both the electron spin and the orbital magnetic moments. This is parameterised using the g tensor, with the effective electron magnetic moment given by µS=-µBLD[g x S, where µB is the electron Bohr magneton. Therefore, the g tensor couples the 'effective' electron spin to B0, thus accounting for the presence of the anisotropic orbital contributions to the electron magnetic moment. As a result, the latter depends on the orientation of the molecule with respect to the magnetic field direction,

(iii) the zero-field splitting (ZFS) interaction, due to the spin–orbit coupling, for systems with more than one unpaired electron (S>1/2), and

(iv)all other diamagnetic interactions present in the molecule.

The whole Hamiltonian describing a paramagnetic system is thus composed of the terms

[MATHEMATICAL EXPRESSION OMITTED] (1.5)

where D is the ZFS tensor.

The A tensor in Hhc is composed of two terms: the isotropic Fermi-contact term and the anisotropic electron–nucleus dipole–dipole interaction term

A = AFC1 + ADIP (1.6)

where AFC is a scalar and ADIP is a traceless symmetric tensor. The scalar term, called the contact coupling constant, is given by

[MATHEMATICAL EXPRESSION OMITTED] (1.7)

where µ0 is the permeability of the vacuum, ge is the free-electron g factor and p is the electron spin density at the nucleus (normalized to one electron). Anisotropic orbital contributions are excluded from AFC because the coupling responsible for the contact interaction is only with the unpaired electron spin density residing on the nucleus (and thus in s orbitals).

In the point-dipole approximation (see Section 1.2.2), the dipolar term S · ADIP · I should describe the energy related to the dipolar magnetic field generated by the magnetic moments, which, in the classical approach, is

[MATHEMATICAL EXPRESSION OMITTED] (1.8)

where the effective electron magnetic moment is considered to take into account the presence of the orbital contributions. Therefore,

[MATHEMATICAL EXPRESSION OMITTED] (1.9)

where the superscript T indicates the matrix transpose, and r is the vector connecting the positions of the nuclear and electron spins. Of note, the components of the g tensor are not the experimental EPR values obtained for the ground state, but the molecular g values along the main directions of the tensor.

In quantum chemistry computations, a different formulation for the ADIP tensor is used (see Section 1.6).

1.1.2 The Curie Spin

The tiny difference in population between the nuclear spin states in the presence of a magnetic field, at room temperature, is responsible for the NMR signal originated by that nucleus. In fact, this difference causes an induced average magnetic moment, which is proportional to the observed NMR signal (see Section 8.1.1). The larger the magnetic field (and the smaller the temperature), the larger the difference in spin population, and thus the induced average magnetic moment.

The difference in population between the different spin states is much larger for electrons than for protons, due to their larger magnetogyric ratio (γS/γH=658.2). This difference is also of paramount importance for the understanding of the NMR spectra of paramagnetic systems. The electron spin S is thus conveniently separated into two terms, corresponding to the null average part s (<s>=0) and to the thermal average of S, SC:

S = s + Sc = s +κ (1.10)

where κ is the versor of SC, which in general differs from the magnetic field direction (see later). Therefore, the electron (thermal average) induced magnetic moment is given by

[MATHEMATICAL EXPRESSION OMITTED] (1.11)

In quantum mechanics formalism the magnetic moment is defined as the partial derivative of the energy with respect to the magnetic field

[MATHEMATICAL EXPRESSION OMITTED] (1.12)

so that

[MATHEMATICAL EXPRESSION OMITTED] (1.13)

where Ei is the energy of the electron i-th state, k is the Boltzmann constant and T is the temperature. Following van Vleck, let us suppose that Ei can be written to a good approximation through perturbation theory limited to second-order correction to the ground state energy

[MATHEMATICAL EXPRESSION OMITTED] (1.14)

In our case H(1) is the Zeeman energy (H(1)=-B0 · µS), so that E(1) and E(2) for relatively small B0 are much smaller than the eigenvalues of H(0), E(0), independent of the magnetic field. Therefore,

[MATHEMATICAL EXPRESSION OMITTED] (1.15)

Then, to the first order in Ei/(kT)

[MATHEMATICAL EXPRESSION OMITTED] (1.16)

Since [MATHEMATICAL EXPRESSION OMITTED], and retaining only the linear terms in B0

[MATHEMATICAL EXPRESSION OMITTED] (1.17)

In the second term of the numerator, the average of the values calculated over the i and k states is

[MATHEMATICAL EXPRESSION OMITTED]

so that

[MATHEMATICAL EXPRESSION OMITTED] (1.18)

This equation is derived on the assumption that all states at zero field are not degenerate. In the presence of states (indicated by the indexes λ and η) with degeneracy (indicated by n and m),

[MATHEMATICAL EXPRESSION OMITTED] (1.19)

Eqn (1.18) and (1.19) can be written as

[MATHEMATICAL EXPRESSION OMITTED] (1.20)

where <SS> is the effective electron spin dyadic equal to

[MATHEMATICAL EXPRESSION OMITTED] (1.21)

where E0n is the energy of the state |n> at zero magnetic field.

Using eqn (1.11) and (1.20),

[MATHEMATICAL EXPRESSION OMITTED] (1.22)

Therefore, <µS> and SC are anisotropic quantities, i.e., they have different values for the different directions of the magnetic field with respect to the molecular frame.

As an example, we can consider the case of an S=1 system with the states |+1> and |-1> higher in energy than the state |0> of a quantity Δ (axial ZFS). Let us suppose that excited states have energy much larger than kT. Recalling that [MATHEMATICAL EXPRESSION OMITTED], the non-null elements of <SS> are:

[MATHEMATICAL EXPRESSION OMITTED]

and thus

[MATHEMATICAL EXPRESSION OMITTED] (1.23)

1.1.3 The Magnetic Susceptibility

As we have seen, a magnetic field induces magnetic moments with non-null thermal average (eqn (1.20)). Usually, the larger the field, the larger the magnetization (until saturation conditions are met). The magnetic susceptibility is defined as the derivative of the magnetization of a substance with respect to the magnetic field strength; if the magnetic field is weak enough not to reach the saturation conditions and thus the magnetization increases linearly with B0, the magnetic susceptibility is independent of the magnetic field strength. In paramagnetic systems, the magnetization is determined by the induced electron magnetic moments; therefore,

[MATHEMATICAL EXPRESSION OMITTED] (1.24)

and thus (using eqn (1.20))

[MATHEMATICAL EXPRESSION OMITTED] (1.25)

In the linear regime, the proportionality constant relating the induced electron magnetic moment and the magnetic field is thus dictated by the magnetic susceptibility per molecule χ according to the following equation:

[MATHEMATICAL EXPRESSION OMITTED] (1.26)

The tensor χ takes into account both the anisotropy of g and the anisotropy of SC due to ZFS or coupling of the ground state with excited states, and it is thus an anisotropic quantity. The principal values of χ (χxx, χyy, χzz) are labelled in order to have the χzz component as the most separated of the three components, so that the anisotropy parameters

[MATHEMATICAL EXPRESSION OMITTED] (1.27)

obey the relationship |Δχrh| ≤ 2|Δax|/3.

The energy related to the (thermal average) induced electron magnetic moment (see eqn (1.12)) is thus

[MATHEMATICAL EXPRESSION OMITTED] (1.28)

and therefore it depends on the orientation of B0 with respect to the molecule.

In the case of S=1 and axial ZFS (see eqn (1.23)),

[MATHEMATICAL EXPRESSION OMITTED] (1.29)

Analogous equations can be calculated for other spin multiplicities.

1.2 The Hyperfine Shift

The hyperfine shift is the difference in the resonance frequency of a nuclear transition between states differing by ΔmI=[+ or -] 1 in the absence of the paramagnetic center and in its presence, scaled by the nuclear Larmor frequency (γIB0/2π). It is thus related to the energy of the hyperfine coupling (eqn (1.4)) between nuclear and electron magnetic moments.

Electrons change their spin state much faster than protons, due to their more efficient relaxation pathways. As a consequence, the nucleus sees the unpaired electron(s) changing its state rapidly among the possible mS levels before being able to change its own mI energy level. This implies that the nuclear magnetic moment actually interacts with the electron induced magnetic moment, calculated from the average of their population distribution at the temperature of the system (see previous section). Therefore, using eqn (1.4) and (1.22), the hyperfine shift is

[MATHEMATICAL EXPRESSION OMITTED] (1.30)

where ι indicates the magnetic field direction. It is convenient to define a hyperfine shielding tensor σhs:

[MATHEMATICAL EXPRESSION OMITTED] (1.31)

from which the shifts can be easily calculated for the different directions of the magnetic field. In the case of isotropic molecular reorientation, i.e. of isotropically oriented magnetic fields, the hyperfine shift (called in this case the isotropic hyperfine shift) becomes

[MATHEMATICAL EXPRESSION OMITTED] (1.32)

where Tr indicates the trace, i.e. the sum of the diagonal terms, of the σhs matrix. In writing eqn (1.32) it was also assumed that the magnetic field direction changes on a time scale slower than the electron relaxation time, i.e. that the molecular reorientation time is longer than the electron relaxation time.

In the quantum chemistry formalism, the shielding tensor is defined as the second partial derivative of the thermally averaged energy with respect to the magnetic field and to the nuclear magnetic moment, calculated at zero magnetic field and zero magnetic moment:

[MATHEMATICAL EXPRESSION OMITTED] (1.33)

From eqn (1.5), it is found that

[MATHEMATICAL EXPRESSION OMITTED] (1.34)

As already anticipated in Section 1.1, the operator A is split in the Fermi-contact (AFC) and dipolar (ADIP) components, which in solution provide the contact (δcs) and pseudocontact (δpcs) shifts, respectively.

1.2.1 The Fermi-contact Shift

The contact shift arises from the delocalization of the unpaired electron spin(s), in particular at the site of the detected nucleus. It is thus proportional to the unpaired electron spin density, ρ, present at the nucleus (i) through direct overlap between the molecular orbital(s) containing the unpaired electron(s) and the s orbitals (allowing for a non-zero electron density at the nucleus) of the observed nucleus, and (ii) due to spin polarization of paired electrons. When an unpaired electron in a d orbital of a metal is partially delocalized to an overlapping p or d orbital of a covalently bound atom, in fact, the spin polarization mechanism induces unpaired electron spin density at the nucleus of this atom because, among the paired electrons in an s orbital, the electron with spin parallel to the unpaired electron spin has a slight preference to occupy the region of space closer to that of the unpaired electron (Hund's rule). In this way the electron with spin antiparallel to the unpaired electron has a larger spin density on the nucleus, and the difference in the spin densities corresponding to the two states provides the unpaired spin density. In the presence of direct molecular overlap with an s orbital, the spin density at the nucleus is positive, whereas in the case of spin polarization, it is negative.

(Continues…)

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Table of Contents

NMR Consequences of the Nucleus-Electron Spin Interactions; Intrinsic and Extrinsic Paramagnetic Probes; Structural and Dynamic Characterization of Protein Domains Using Paramagnetic Data; Treating Biomacromolecular Conformational Variability; Protein-Protein Interactions; Solid-State NMR of Paramagnetic Proteins; Relaxometry and Contrast Agents; Dynamic Nuclear Polarization; Paramagnetic NMR in Drug Discovery; Small Paramagnetic Co-solute Molecules; Subject Index