**2. Residual anisotropic spin interactions**

Conventional NMR structure analysis based on the short-range spin interactions does not give any global structural information required in protein morphology study. The residual anisotropic spin interactions, which become apparent for a '*weakly*' aligned protein, give molecular orientation information relative to the magnetic field. And the information gives the clues to analyze protein morphology.

Two different types of residual anisotropic spin interactions are observed in the amide 1H-15N spin pair on a peptide plane; one is the nuclear spin dipolar-dipolar interaction and the other is anisotropic shielding effect against the external magnetic field, which is called as chemical shift anisotropy (CSA) (Fig. 2).

These anisotropic spin interactions are not observed on a NMR spectrum for protein in an isotropic solution, where protein rapidly tumbles. Because of the rapid rotation that allows for protein to direct the entire angles against a magnetic field, anisotropic spin interactions are completely canceled for isotropic sample. In contrast, protein dissolved in a magnetically ordering liquid crystalline medium, for example, experiences some rotational restrictions by steric bump with the liquid crystalline molecules. Thus, it leads to incomplete cancellation of anisotropic spin interactions, which should give the residual anisotropic spin interactions observed on a spectrum. In a properly tuned liquid crystalline concentration, there is enough space to allow protein tumbling to some extent. The protein in an aligned liquid crystalline medium, accordingly, still can give high resolution NMR signals. In the higher liquid crystalline concentration, the protein tumbling becomes substantially limited and its NMR signals become severely broadened to prohibit their spectral observation. The condition that protein tumbling is slightly restricted to achieve incomplete cancellation of the anisotropic spin interactions, but it still gives narrow lines enough for the observation on a NMR spectrum is called as a '*weak*' alignment (Bax 2003).

Fig. 2. Two anisotropic spin interactions observed for amide 1H-15N spin pair. (a) dipolar interaction. (b) 15N chemical shift anisotropy (CSA) effect.

Complementary Use of NMR to X-Ray Crystallography

(b). Split with for each paired signals corresponds to <sup>1</sup>

averaged expectation value for the dipolar Hamiltonian, Eq. (1).

.

aligned sample. The change in <sup>1</sup>

below.

is defined by polar angles (

,

described by the time dependent polar angles (

direction cosines to three molecular axes.

for the Analysis of Protein Morphological Change in Solution 413

Fig. 3. RDC measurement with 1H coupled HSQC. Isotropic sample (a) and aligned sample

The experimentally measured residual dipolar couplings, *res Dij* , is represented as the time-

2 3

The angle brackets denote the time-averaged resultant. The molecule in an isotropic tumbling gives no RDC, <sup>2</sup> *P t* (cos ( )) 0 , due to the isotropic distribution of molecular orientation. On the other hand, the molecule in an anisotropic tumbling incompletely vanishes the term, <sup>2</sup> *P t* (cos ( )) 0 . The magnitudes of the residual values are related to the molecular orientation angle and the alignment order against a magnetic field. The theoretical description to relate the RDCs to the orientation angles and orders will be given

Fig. 4. Three different representations for NH bond vector against a magnetic field. (a) Angle dependency of the 1H-15N RDC is described by , the angle between the NH bond vector and a magnetic field. (b) NH bond vector on a molecular coordinate system, whose direction

> 

). Molecular reorientation relative to a magnetic field is

( ), ( ) *t t* ). (c) NH bond vector is recast by

<sup>4</sup> <sup>2</sup> <sup>2</sup> ( ) (cos ( ))

<sup>0</sup>

 *i j ij*

 

*h res*

*NH J* from the isotropic value is RDC.

*NH J* . The apparent <sup>1</sup>

*ij <sup>r</sup> D t P t* (2)

*NH J* is changed for

The spin interactions apparent only under a weakly aligning condition are the residuals by incomplete cancellation of the spatially anisotropic spin interactions. The effects are, therefore, called as '*residual*' anisotropic spin interactions. In the following text, we will focus on two types of the residual anisotropic spin interactions on a peptide plane prior to describe the DIORITE; they are the residual dipolar coupling (RDC) and the residual CSA (RCSA). DIORITE relies on these two anisotropic spin interactions, and their theoretical basics will be helpful in understanding DIORITE approach.

#### **2.1 Residual dipolar coupling, RDC**

Residual nuclear spin dipolar interaction (RDC) gives the direction of the internuclear vector against the magnetic field. In the peptide bond of a protein labeled with 15N, the RDC for amide 15N nucleus defines the angle between the NH bond vector and a magnetic field (Fig. 2a).

The RDCs amide 1H-15N spin pairs are the measurement of common choice, primarily due to the good spectral dispersion of the 1H-15N correlation signals on a 2D spectrum. The RDCs are obtained from a pair of 1H-15N HSQC spectra measured without 1H decoupling during t1 evolution period of 15N spin; one spectrum is for the isotropic sample, and the other is for the aligned (Fig. 3). On a 1H-coupled 1H-15N HSQC spectrum, each peak gives a pair of signals split along the 15N axis (Fig. 3). The split width of signal for the isotropic sample corresponds to single-bond spin coupling ( <sup>1</sup> *NH J* ) between the covalently bonded 1H and 15N nuclei, around 93 Hz (Fig. 3a). Split width for the aligned sample gives the value deviated from <sup>1</sup> *NH J* (Fig. 3b). The difference in the apparent <sup>1</sup> *NH J* values between the isotropic and aligned sample is the RDC. In this section, we will describe how the RDCs can define the alignment tensor of protein, by describing its theoretical background.

#### **2.2 Theoretical description on RDC**

We focus on the amide 1H-15N spin pair here; i and j represent 1H and 15N nucleus, respectively. The dipolar Hamiltonian to describe the magnitude of dipolar split is written in the following equation in a laboratory frame under the high-field limit condition:

$$H\_{\ddot{y}}^{\;\;D}(t) = -\left(\frac{\mu\_0}{4\pi}\right) \frac{\gamma\_i \gamma\_\gamma h}{2\pi^2 r\_\eta^3} I\_{i\bar{z}} I\_{j\bar{z}} P\_2(\cos\Theta(t))\tag{1}$$

where *r*ij is the distance between 1H and 15N atoms, i and j are the gyromagnetic ratios for 1H and 15N nuclear spins, respectively. As physical constants, *h* is Planck constant and *<sup>0</sup>* is vacuum permeability. Iiz and Sjz represent the spin angular-momentum operators for 1H and 15N spins, respectively. The angular part of the dipolar Hamiltonian is described by the second rank Legendre function, <sup>2</sup> *P t* (cos ( )) , here ( )*t* is a time-dependent angle between the magnetic field and the internuclear vector (NH bond vector in the present context) (Fig. 4a). In determining molecular orientation by RDC, bond libration effect is negligible due to its faster motion relative to molecular tumbling; bond libration occurs in a few hundred psec time regimes, whilst molecular tumbling happens in nsec range. In this sense, *r*ij is defined as time-averaged effective internuclear distance, thus it is estimated to be slightly longer (1.04 Å) than the static bond length (1.02 Å).

The spin interactions apparent only under a weakly aligning condition are the residuals by incomplete cancellation of the spatially anisotropic spin interactions. The effects are, therefore, called as '*residual*' anisotropic spin interactions. In the following text, we will focus on two types of the residual anisotropic spin interactions on a peptide plane prior to describe the DIORITE; they are the residual dipolar coupling (RDC) and the residual CSA (RCSA). DIORITE relies on these two anisotropic spin interactions, and their theoretical

Residual nuclear spin dipolar interaction (RDC) gives the direction of the internuclear vector against the magnetic field. In the peptide bond of a protein labeled with 15N, the RDC for amide 15N nucleus defines the angle between the NH bond vector and a magnetic field (Fig.

The RDCs amide 1H-15N spin pairs are the measurement of common choice, primarily due to the good spectral dispersion of the 1H-15N correlation signals on a 2D spectrum. The RDCs are obtained from a pair of 1H-15N HSQC spectra measured without 1H decoupling during t1 evolution period of 15N spin; one spectrum is for the isotropic sample, and the other is for the aligned (Fig. 3). On a 1H-coupled 1H-15N HSQC spectrum, each peak gives a pair of signals split along the 15N axis (Fig. 3). The split width of signal for the isotropic

1H and 15N nuclei, around 93 Hz (Fig. 3a). Split width for the aligned sample gives the

the isotropic and aligned sample is the RDC. In this section, we will describe how the RDCs can define the alignment tensor of protein, by describing its theoretical

We focus on the amide 1H-15N spin pair here; i and j represent 1H and 15N nucleus, respectively. The dipolar Hamiltonian to describe the magnitude of dipolar split is written

<sup>4</sup> 2 3 <sup>2</sup> <sup>2</sup> ( ) (cos ( ))

where *r*ij is the distance between 1H and 15N atoms, i and j are the gyromagnetic ratios for 1H and 15N nuclear spins, respectively. As physical constants, *h* is Planck constant and

vacuum permeability. Iiz and Sjz represent the spin angular-momentum operators for 1H and 15N spins, respectively. The angular part of the dipolar Hamiltonian is described by the second rank Legendre function, <sup>2</sup> *P t* (cos ( )) , here ( )*t* is a time-dependent angle between the magnetic field and the internuclear vector (NH bond vector in the present context) (Fig. 4a). In determining molecular orientation by RDC, bond libration effect is negligible due to its faster motion relative to molecular tumbling; bond libration occurs in a few hundred psec time regimes, whilst molecular tumbling happens in nsec range. In this sense, *r*ij is defined as time-averaged effective internuclear distance, thus it is estimated to be slightly longer

*ij iz jz <sup>r</sup> H t II P t* (1)

in the following equation in a laboratory frame under the high-field limit condition:

<sup>0</sup>

 *i j ij*

 

*D h*

*NH J* (Fig. 3b). The difference in the apparent <sup>1</sup>

*NH J* ) between the covalently bonded

*NH J* values between

*<sup>0</sup>* is

basics will be helpful in understanding DIORITE approach.

sample corresponds to single-bond spin coupling ( <sup>1</sup>

**2.1 Residual dipolar coupling, RDC** 

2a).

value deviated from <sup>1</sup>

**2.2 Theoretical description on RDC** 

(1.04 Å) than the static bond length (1.02 Å).

background.

Fig. 3. RDC measurement with 1H coupled HSQC. Isotropic sample (a) and aligned sample (b). Split with for each paired signals corresponds to <sup>1</sup> *NH J* . The apparent <sup>1</sup> *NH J* is changed for aligned sample. The change in <sup>1</sup> *NH J* from the isotropic value is RDC.

The experimentally measured residual dipolar couplings, *res Dij* , is represented as the timeaveraged expectation value for the dipolar Hamiltonian, Eq. (1).

$$D\_{\vec{y}}^{\prime \text{res}}(t) = -\left(\frac{\mu\_0}{4\pi}\right) \frac{\gamma\_i r\_j h}{2\pi^2 r\_{\vec{y}}^3} < P\_2(\cos \Theta(t)) > \tag{2}$$

The angle brackets denote the time-averaged resultant. The molecule in an isotropic tumbling gives no RDC, <sup>2</sup> *P t* (cos ( )) 0 , due to the isotropic distribution of molecular orientation. On the other hand, the molecule in an anisotropic tumbling incompletely vanishes the term, <sup>2</sup> *P t* (cos ( )) 0 . The magnitudes of the residual values are related to the molecular orientation angle and the alignment order against a magnetic field. The theoretical description to relate the RDCs to the orientation angles and orders will be given below.

Fig. 4. Three different representations for NH bond vector against a magnetic field. (a) Angle dependency of the 1H-15N RDC is described by , the angle between the NH bond vector and a magnetic field. (b) NH bond vector on a molecular coordinate system, whose direction is defined by polar angles (, ). Molecular reorientation relative to a magnetic field is described by the time dependent polar angles ( ( ), ( ) *t t* ). (c) NH bond vector is recast by direction cosines to three molecular axes.

Complementary Use of NMR to X-Ray Crystallography

as the most ordered axis. An asymmetry,

non-zero average spherical harmonics:

, , 

where the polar angles (

molecular coordinate system.

the Euler angles (

described on the molecular coordinate system by Euler angles.

( ( ), ( )) ( ( ), ( ))

Using the parameters, the *res Dij* is re-represented in a more intuitive form:

2 3

*i j*

 

<sup>0</sup>

 

orientation independent *'isotropic chemical shifts'* on a spectrum.

RCSA under a weakly aligning state (Lundstrom, Hansen and Kay 2008).

 

*h res*

,

**2.3 Residual chemical shift anisotropy, RCSA** 

as residual CSA, RCSA (Kurita et al. 2003).

 

for the Analysis of Protein Morphological Change in Solution 415

The Saupe order matrix describes the angle and the order for aligned protein relative to the magnetic field. Diagonalization of the Saupe order matrix gives the alignment tensor; its principal values represent the alignment order along each principal axis and the principal axes define the direction of the aligned molecule. In the alignment tensor, z-axis is defined

axially symmetric ordering. The orientation of the alignment tensor axes (principal axes) is

In the alignment tensor axes system (the alignment tensor frame), the residual dipolar coupling *res Dij* is represented by the following two parameters, *A*a and *A*r, that include two

2 2 24 32

*Y tt SS A* 

tensor frame. *A*a and *A*r represent the axial and rhombic components of the order magnitudes, respectively. The orientation of protein relative to a magnetic field is defined by

Another anisotropic spin interaction associated with the amide 15N nucleus comes from chemical shift anisotropy, CSA (Fig. 2b). This effect is observed as 15N chemical shift change induced by a weak alignment (Fig. 5). The alignment induced chemical shift change is called

The chemical shift for each nucleus is related to the shielding against the magnetic field by the electrons surrounding the nucleus, which is called as chemical shielding. In general, the spatial distribution of electrons around the nucleus is not uniform but rather spatially anisotropic. Therefore, the chemical shielding effect to the nucleus should change according to the orientation molecule against a magnetic field; the chemical shift for the nucleus can be orientation dependent. However, the molecule in an isotropic solution averages the orientation dependent shielding effects by its rapid isotropic tumbling, thus giving the

As in the case of the dipolar interaction, a weak alignment makes the orientation dependency of CSA observable, due to the incomplete averaging of anisotropic shielding effects. The amide 15N has significantly large CSA, because of the substantial anisotropic distribution of the electron on a peptide plane. Protein in a weakly aligned state, thus, shows significant chemical shift changes along 15N axis relative to its isotropic positions (Fig. 5). Although it is not described in detail in this manuscript, carbonyl 13C has also large CSA and it shows significant

 

*zz a*

5 15

2 2 2 4 3 <sup>2</sup> (3cos 1) sin cos2

*ij a r <sup>r</sup> DAA* (8)

) that describes the position of the alignment tensor frame on the

*i j*

*xx yy r*

 

define the orientation of NH bond vector in the alignment

(7)

 

5 5

( )/ *SS S xx yy zz* , represents the deviation from

20 4 4

*Y tt S A*

Protein structure is represented by the atom positions in a Cartesian coordinate system. This coordinate system is referred as the molecular coordinate system here. The orientation of each NH bond vector in a protein is described by polar angles () in the molecular coordinate system (Fig. 4b). Now, we neglect the local NH bond libration, thus, the angles and are time-independent. Likewise, at any instant in time, the orientation of the magnetic field is described by the time dependent polar angles, ( ( ), ( ) *t t* ) (Fig. 4b). The molecule reorients against a magnetic field in a time dependent manner; this molecular motion is represented by the directional change of the magnetic field on the molecular coordinate system (Fig. 4b). The polar angles defining a magnetic field direction, therefore, are described as time dependent parameters.

The Legendre function <sup>2</sup> *P t* (cos ( )) is expanded by the spherical harmonics according to the spherical harmonic addition theorem:

$$P\_2(\cos \theta(t)) = \frac{4\pi}{s} \sum\_{q=-2}^{2} Y\_{\, \, 2q}^\*(\theta, \varphi) Y\_{2q}(\zeta'(t), \zeta'(t)) \tag{3}$$

where the *Y*2q's are the normalized spherical harmonics. Using Eq. (3), the residual dipolar coupling (Eq. (2)) is expressed with spherical harmonics:

$$D\,\mu^{\prime \text{as}} = -\left(\frac{\mu\_0}{4\pi}\right)\frac{\gamma\_{,\gamma}\gamma\_{,\delta}}{2x^2r\_{i\_{\parallel}}^{\gamma\_{\perp}}}\frac{4x}{3}\sum\_{q=-2}^{2}Y\_{\,\,2q}^{\ast}(\theta,\varphi)\left\langle Y\_{2q}(\zeta^{\prime}(t),\zeta^{\prime}(t))\right\rangle\tag{4}$$

In Eq. (4), the five time-averaged spherical harmonics <sup>2</sup> ( ( ), ( )) *Y tt <sup>q</sup>* with *q* = -2, -1, 0, 1, 2 defines the molecular alignment angles and its magnitudes. Recasting these five terms makes more intuitively acceptable representation to describe the molecular alignment state, which is Saupe's order matrix.

The use of Saupe's order matrix reforms Eq. (4).

$$D\_{\vec{y}}{}^{\rm res} = -\left(\frac{\mu\_0}{4\pi}\right) \frac{\gamma\_{\uparrow}\gamma\_{\uparrow}\hbar}{2\pi^2 r\_{i\_{\uparrow}}^{\rm s}} \frac{4\pi}{^{5}} \sum\_{k,l=x,y,z} S\_{kl} \cos(\alpha\_k) \cos(\alpha\_l) \tag{5}$$

where, *Skl* is an element in Saupe's order matrix. The Saupe order matrix is a traceless, symmetric, 3 3 matrix, thus it comprises of five independent elements. In the representation using Saupe's order matrix, each bond vector orientation is defined by direction cosines against the molecular coordinate axes (Fig. 4c). The Saupe's order matrix elements that are made of the time-averaged spherical harmonics <sup>2</sup> ( ( ), ( )) *Y tt <sup>q</sup>* are shown below:

$$\begin{aligned} S\_{zz} &= \sqrt{\frac{5}{8}} \sqrt{\frac{4\pi}{\zeta}} \left( \left\{ Y\_{21}(\tilde{\xi}(t), \zeta'(t)) \right\} - \left\{ Y\_{2\cdot 1}(\tilde{\xi}(t), \zeta'(t)) \right\} \right) \\ S\_{yz} &= i\sqrt{\frac{5}{8}} \sqrt{\frac{4\pi}{\zeta}} \left( \left\{ Y\_{21}(\tilde{\xi}(t), \zeta'(t)) \right\} + \left\{ Y\_{2\cdot 1}(\tilde{\xi}(t), \zeta'(t)) \right\} \right) \\ S\_{xy} &= i\sqrt{\frac{5}{8}} \sqrt{\frac{4\pi}{\zeta}} \left( \left\{ Y\_{21}(\tilde{\xi}(t), \zeta'(t)) \right\} - \left\{ Y\_{2\cdot 2}(\tilde{\xi}(t), \zeta'(t)) \right\} \right) \\ S\_{zx} &- S\_{yy} = \sqrt{\frac{3}{2}} \sqrt{\frac{4\pi}{\zeta}} \left( \left\{ Y\_{22}(\tilde{\xi}(t), \zeta'(t)) \right\} + \left\{ Y\_{2\cdot 2}(\tilde{\xi}(t), \zeta'(t)) \right\} \right) \\ S\_{zz} &= \sqrt{\frac{4\pi}{\zeta}} \left\{ Y\_{20}(\tilde{\xi}(t), \zeta'(t)) \right\} \end{aligned} \tag{6}$$

Protein structure is represented by the atom positions in a Cartesian coordinate system. This coordinate system is referred as the molecular coordinate system here. The orientation of

coordinate system (Fig. 4b). Now, we neglect the local NH bond libration, thus, the angles

reorients against a magnetic field in a time dependent manner; this molecular motion is represented by the directional change of the magnetic field on the molecular coordinate system (Fig. 4b). The polar angles defining a magnetic field direction, therefore, are

The Legendre function <sup>2</sup> *P t* (cos ( )) is expanded by the spherical harmonics according to the

where the *Y*2q's are the normalized spherical harmonics. Using Eq. (3), the residual dipolar

 

2 4 \* 2 5 2 2 2 (cos ( )) ( , ) ( ( ), ( ))

 *q q*

2 4 \* 4 5 2 2 <sup>2</sup> <sup>2</sup>

defines the molecular alignment angles and its magnitudes. Recasting these five terms makes more intuitively acceptable representation to describe the molecular alignment state,

where, *Skl* is an element in Saupe's order matrix. The Saupe order matrix is a traceless, symmetric, 3 3 matrix, thus it comprises of five independent elements. In the representation using Saupe's order matrix, each bond vector orientation is defined by direction cosines against the molecular coordinate axes (Fig. 4c). The Saupe's order matrix

> 

( ( ), ( )) ( ( ), ( )) ( ( ), ( )) ( ( ), ( )) ( ( ), ( )) ( ( ), ( ))

 

 

 

( ( ), ( )) ( ( ), ( ))

 

 

*ij kl k l <sup>r</sup> kl xyz*

*ij q q <sup>r</sup> <sup>q</sup>*

*q*

<sup>0</sup>

 

In Eq. (4), the five time-averaged spherical harmonics <sup>2</sup> ( ( ), ( ))

<sup>0</sup>

 

 

*h res*

3 4

3 4

3 4

4 5 20

*zz*

*xx yy*

*xz yz xy*

*S Y tt*

3 4

 

 

( ( ), ( ))

2 3 4 4 5 <sup>2</sup> , ,,

elements that are made of the time-averaged spherical harmonics <sup>2</sup> ( ( ), ( ))

8 5 21 2 1

*S Ytt Y tt Si Y tt Y tt Si Y tt Y tt*

8 5 21 2 1

8 5 22 2 2

2 5 22 2 2

*SS Y tt Y tt*

 *i j i j*

 

*h res*

2 3

 *i j i j*

coupling (Eq. (2)) is expressed with spherical harmonics:

are time-independent. Likewise, at any instant in time, the orientation of the magnetic

 

  *P t Y Y tt* (3)

( , ) ( ( ), ( ))

*D Y Y t t* (4)

 

> 

cos( )cos( )

*D S* (5)

 ( ), ( ) *t t* ) (Fig. 4b). The molecule

*Y tt <sup>q</sup>* with *q* = -2, -1, 0, 1, 2

 *Y tt <sup>q</sup>* are shown

(6)

) in the molecular

each NH bond vector in a protein is described by polar angles (

field is described by the time dependent polar angles, (

described as time dependent parameters.

spherical harmonic addition theorem:

which is Saupe's order matrix.

below:

The use of Saupe's order matrix reforms Eq. (4).

and  The Saupe order matrix describes the angle and the order for aligned protein relative to the magnetic field. Diagonalization of the Saupe order matrix gives the alignment tensor; its principal values represent the alignment order along each principal axis and the principal axes define the direction of the aligned molecule. In the alignment tensor, z-axis is defined as the most ordered axis. An asymmetry, ( )/ *SS S xx yy zz* , represents the deviation from axially symmetric ordering. The orientation of the alignment tensor axes (principal axes) is described on the molecular coordinate system by Euler angles.

In the alignment tensor axes system (the alignment tensor frame), the residual dipolar coupling *res Dij* is represented by the following two parameters, *A*a and *A*r, that include two non-zero average spherical harmonics:

$$\begin{aligned} \left\{ Y\_{20} \left( \xi'(t), \zeta'(t) \right) \right\} &= \sqrt{\frac{5}{4\pi}} S\_{\frac{\pi}{4\pi}} = \sqrt{\frac{5}{4\pi}} A\_a \\ \left\{ Y\_{2\pm 2} \left( \xi'(t), \zeta'(t) \right) \right\} &= \sqrt{\frac{5}{24\pi}} \left( S\_{\infty} - S\_{yy} \right) = \sqrt{\frac{15}{32\pi}} A\_r \end{aligned} \tag{7}$$

Using the parameters, the *res Dij* is re-represented in a more intuitive form:

$$D\_{\vec{y}}{}^{\prime \kappa} = -\left(\frac{\mu\_0}{4\pi}\right)\frac{\gamma\_\gamma r\_\gamma h}{2x^2 r\_{i\_\parallel}^{\prime}} \left[A\_a(\Im \cos^2 \theta^{\prime} - \mathbf{l}) + \frac{2}{3} A\_r \sin^2 \theta^{\prime} \cos 2\varphi^{\prime}\right] \tag{8}$$

where the polar angles (, define the orientation of NH bond vector in the alignment tensor frame. *A*a and *A*r represent the axial and rhombic components of the order magnitudes, respectively. The orientation of protein relative to a magnetic field is defined by the Euler angles (, , ) that describes the position of the alignment tensor frame on the molecular coordinate system.

#### **2.3 Residual chemical shift anisotropy, RCSA**

Another anisotropic spin interaction associated with the amide 15N nucleus comes from chemical shift anisotropy, CSA (Fig. 2b). This effect is observed as 15N chemical shift change induced by a weak alignment (Fig. 5). The alignment induced chemical shift change is called as residual CSA, RCSA (Kurita et al. 2003).

The chemical shift for each nucleus is related to the shielding against the magnetic field by the electrons surrounding the nucleus, which is called as chemical shielding. In general, the spatial distribution of electrons around the nucleus is not uniform but rather spatially anisotropic. Therefore, the chemical shielding effect to the nucleus should change according to the orientation molecule against a magnetic field; the chemical shift for the nucleus can be orientation dependent. However, the molecule in an isotropic solution averages the orientation dependent shielding effects by its rapid isotropic tumbling, thus giving the orientation independent *'isotropic chemical shifts'* on a spectrum.

As in the case of the dipolar interaction, a weak alignment makes the orientation dependency of CSA observable, due to the incomplete averaging of anisotropic shielding effects. The amide 15N has significantly large CSA, because of the substantial anisotropic distribution of the electron on a peptide plane. Protein in a weakly aligned state, thus, shows significant chemical shift changes along 15N axis relative to its isotropic positions (Fig. 5). Although it is not described in detail in this manuscript, carbonyl 13C has also large CSA and it shows significant RCSA under a weakly aligning state (Lundstrom, Hansen and Kay 2008).

Complementary Use of NMR to X-Ray Crystallography

structure specific manner.

**3. Achieving weak alignment** 

section, we will review some media for weak alignment.

**3.1 Magnetically aligning liquid crystalline media** 

perpendicular to the magnetic field (Fig. 6a).

for the Analysis of Protein Morphological Change in Solution 417

The 15N CSA tensor is known to be significantly dependent on the local structure, particularly backbone torsion angle (Yao et al. 2010). Experimentally determined 15N CSA tensors are reported with a various method (Fushman, Tjandra and Cowburn 1999, Cornilescu and Bax 2000, Boyd and Redfield 1999, Hiyama et al. 1988, Kurita et al. 2003). These data give consensus 15N CSA tensor values for the residue in each type of secondary structure, -helix, -sheet and others. Practically, the use of the secondary structure specific 15N CSA tensor values can determine the alignment tensor within experimental errors. Experimental determination of 15N CSA tensor for protein in solution is possible using the weak alignment technique. We previously proposed the method using magic-angle sample spinning to determine the accurate secondary structure specific 15N CSA tensor, in which the bicellar media was used for a weak alignment (Kurita et al. 2003). In this experiment, we used only one aligned state, thus, only determined the 15N CSA tensors in a secondary

Recently, Bax and co-workers have applied this approach to determine the residue specific 15N CSA tensors for a protein, where they used five more different aligning states to solve the Saupe order matrix for each residue (Yao et al. 2010). The residue specific 15N CSA tensor determination that requires multiple aligned states of protein is rather demanding experiments, which require a various loop mutant to change aligning angle (Yao et al. 2010). However, the continuous effort to collect the residue specific 15N CSA tensors in the similar way by Bax and co-workers will establish a clearer correlation between the 15N CSA tensor and backbone torsion angles and also local interactions like hydrogen bonding, which may allow the prediction of the appropriate 15N CSA tensor values from the structure. The refined 15N CSA tensors will further improve the quality of alignment tensor analysis with

In applying the residual anisotropic spin interactions described above, it is required to make a protein in a weakly aligned state. The aligning protein has to be carefully tuned to make the anisotropic interactions observable in a spectrum with keeping the spectral resolution and intensity. Alignment order is practically tuned to approximately 10-3, giving about 20 Hz in maximum absolute magnitude for amide 1H-15N RDC. To achieve a weak alignment, some artificial medium has to be used, because the inherent magnetic susceptibility of a globular protein is too small to align to the desired extent, except for some heme-containing proteins having substantial magnetic susceptibility associated with a heme group. In this

Magnetically ordering liquid crystalline media are commonly used. Discoidal phospholipid assembly, bicelle, is one of the prevailingly used materials for a weak alignment of protein (Ottiger and Bax 1999, Ottiger, Delaglio and Bax 1998, Tjandra and Bax 1997).. The bicelle is composed of a mixer of dimyristoylphosphatidylcholin (DMPC) and dihexynoylphosphatidylcholine (DHPC) in a ratio of 3:1. This phospholipid binary mixture forms lipid bilayers disks 30 nm – 40 nm in diameter. Bicelle has substantial magnetic susceptibility, and it spontaneously aligns under magnetic field with the normal of the bicelle surface staying

In the experiments to measure the anisotropic spin interactions, an appropriate amount of bicelle is put into protein solution. In a high magnetic field, bicelles align and the aligned

the RCSA, although the present RCSA based approach gives an acceptable result.

Fig. 5. 15N chemical shift change induced by a weak alignment. 1H-15N HSQC spectra are compared for 15N labeled ubiquitin dissolved in 7.5% (w/v) DMPC/DHPC/CTAB ternary bicelle solution between isotropic and aligned states: (a) under the magic angle sample spinning condition, giving an isotropic spectrum, (b) the aligned condition. Chemical shift changes along the 15N axis caused by a weak alignment are clearly observed, for examples the signals in dotted square. These spectral changes come from the incomplete cancellation of 15N chemical shift anisotropy (CSA) effect under a weak aligning condition. The change along the 15N axis is the value for the residual CSA, RCSA.

The alignment induced chemical shift change, or RCSA, , is expressed with Sauce's order matrix, *ij S* , as for the RDC, Eq. (5). Based on this relation, we can also determine the alignment tensor of the molecule as done with the RDC.

$$
\Delta \mathcal{S} = \frac{2}{3} \sum\_{i=\mathbf{x}, \mathbf{y}, z} \sum\_{j=\mathbf{x}, \mathbf{y}, z} \sum\_{k=\mathbf{x}, \mathbf{y}, z} S\_{ij} \cos(\theta\_{ik}) \cos(\theta\_{jk}) \delta\_{ik} \tag{8}
$$

Here, cos*ij* denotes the direction cosine between *i*-axis in the molecular coordinate system and *j-*axis in the CSA tensor principal axis. The principal value for the CSA tensor axis *k* is represented by *kk* . The spatially anisotropic distribution of the electrons around a nucleus is represented by a tensor (CSA tensor), where each principal axis represents the shielding magnitude. The 15N CSA tensor axes are drawn on a peptide plane (Fig. 2b).

The alignment tensor is determined based on the Saupe order matrix obtained through the relation in Eq. (8) for a set of data. It is noted that in calculating the Saupe order matrix, we require the knowledge of the CSA tensor that is defined by the relative orientation of the CSA tensor frame on a peptide plane and the magnitudes of the principal values. In the case of the RDC, we assumed the effective interatomic distance, *rij*, to a fixed value. In contrast to the RDC based analysis that requires one pre-defined value for *r*ij, the RCSA analysis requires five predefined parameters; two principal values , *zz xx* , with 0 *xx yy zz* , and Euler angles (, , ) that define the CSA tensor orientation on a peptide plane. The RCSA based alignment tensor determination is therefore, much more knowledge demanding.

Fig. 5. 15N chemical shift change induced by a weak alignment. 1H-15N HSQC spectra are compared for 15N labeled ubiquitin dissolved in 7.5% (w/v) DMPC/DHPC/CTAB ternary bicelle solution between isotropic and aligned states: (a) under the magic angle sample spinning condition, giving an isotropic spectrum, (b) the aligned condition. Chemical shift changes along the 15N axis caused by a weak alignment are clearly observed, for examples the signals in dotted square. These spectral changes come from the incomplete cancellation of 15N chemical shift anisotropy (CSA) effect under a weak aligning condition. The change

matrix, *ij S* , as for the RDC, Eq. (5). Based on this relation, we can also determine the

and *j-*axis in the CSA tensor principal axis. The principal value for the CSA tensor axis *k* is

is represented by a tensor (CSA tensor), where each principal axis represents the shielding

The alignment tensor is determined based on the Saupe order matrix obtained through the

require the knowledge of the CSA tensor that is defined by the relative orientation of the CSA tensor frame on a peptide plane and the magnitudes of the principal values. In the case of the RDC, we assumed the effective interatomic distance, *rij*, to a fixed value. In contrast to the RDC based analysis that requires one pre-defined value for *r*ij, the RCSA analysis requires five pre-

) that define the CSA tensor orientation on a peptide plane. The RCSA based alignment

 ,*zz xx* , with

,, ,, ,,

*i x y z j x y zk x y z*

*S*

magnitude. The 15N CSA tensor axes are drawn on a peptide plane (Fig. 2b).

cos( )cos( ) *ij ik jk kk*

 

data. It is noted that in calculating the Saupe order matrix, we

 

(8)

*kk* . The spatially anisotropic distribution of the electrons around a nucleus

*ij* denotes the direction cosine between *i*-axis in the molecular coordinate system

, is expressed with Sauce's order

0 *xx yy zz* , and Euler angles

along the 15N axis is the value for the residual CSA, RCSA. The alignment induced chemical shift change, or RCSA,

alignment tensor of the molecule as done with the RDC.

Here, cos

(, ,  relation in Eq. (8) for a set of

defined parameters; two principal values

represented by

2 3

tensor determination is therefore, much more knowledge demanding.

The 15N CSA tensor is known to be significantly dependent on the local structure, particularly backbone torsion angle (Yao et al. 2010). Experimentally determined 15N CSA tensors are reported with a various method (Fushman, Tjandra and Cowburn 1999, Cornilescu and Bax 2000, Boyd and Redfield 1999, Hiyama et al. 1988, Kurita et al. 2003). These data give consensus 15N CSA tensor values for the residue in each type of secondary structure, -helix, -sheet and others. Practically, the use of the secondary structure specific 15N CSA tensor values can determine the alignment tensor within experimental errors.

Experimental determination of 15N CSA tensor for protein in solution is possible using the weak alignment technique. We previously proposed the method using magic-angle sample spinning to determine the accurate secondary structure specific 15N CSA tensor, in which the bicellar media was used for a weak alignment (Kurita et al. 2003). In this experiment, we used only one aligned state, thus, only determined the 15N CSA tensors in a secondary structure specific manner.

Recently, Bax and co-workers have applied this approach to determine the residue specific 15N CSA tensors for a protein, where they used five more different aligning states to solve the Saupe order matrix for each residue (Yao et al. 2010). The residue specific 15N CSA tensor determination that requires multiple aligned states of protein is rather demanding experiments, which require a various loop mutant to change aligning angle (Yao et al. 2010). However, the continuous effort to collect the residue specific 15N CSA tensors in the similar way by Bax and co-workers will establish a clearer correlation between the 15N CSA tensor and backbone torsion angles and also local interactions like hydrogen bonding, which may allow the prediction of the appropriate 15N CSA tensor values from the structure. The refined 15N CSA tensors will further improve the quality of alignment tensor analysis with the RCSA, although the present RCSA based approach gives an acceptable result.
