Tranfer interligand NOEs between two ligands binding competitively to the same binding pocket of a common receptor have first been observed in our group.10 1 These spectra, called INPHARMA spectra, can be used to determine the binding modes of two ligand-receptor complexes from a single measurement.11, 13 They are based on the trNOE effect (transfer nuclear Overhauser effect). In INPHARMA spectra, in contrast to trNOE spectra, not only one but two ligands will be in solution together with a common receptor. Normally, the receptor is big enough not to be seen in the NMR spectra, due to line broadening. The measurement of a normal NOESY spectrum on this sample will produce cross peaks between atoms of the two different ligands, indicating magnetization transfer. This transfer between the ligands is purely mediated by the receptor. The ligands don’t bind to the same binding pocket at the same time, nor do they transfer magnetization when they are in the bulk solvent. Possibly occurring minor contributions to interligand magnetization transfer, like from unspecific binding of the ligands to the receptor, are neglected for now.
The application of INPHARMA spectra for structure determination, the determination of the binding epitopes, has first been done during my work on my diploma thesis.11, 12 There a model of epothilone A, an anti cancer agent, binging to tubulin, was determined. This model explained virtually all available SAR (structure activity relationship) data and mutagenesis data. In 2008 the method was successfully validated on a molecular reference system of complexes with structures known from X-ray crystallography.13 During the INPHARMA experiments the concentrations of both ligands, the receptor and the complexes are in chemical equilibrium. All species exist in a free and bound form. The principle of the method shall be illustrated in figure (fig. 1.1).
If the concentration of the ligands is large enough the free, uncomplexed state of the receptor can be neglected.14 Experimentally, for the tubulin-epothilone A-baccatin III system, the concentration ratio receptor:ligandA:ligandB was approximately 1:50:50.10 Thus the following (1.2) simplified model was used for the simulation the the INPHARMA spectra. It will be used during the remaining part of this document.
For two given complex structures TA and TB of the 2-step model the re-sulting INPHARMA spectrum can be simulated. As the free ligands, due to their small size, don’t contribute significantly to the spectra their con-formation is mostly irrelevant, though considered. In the ideal case only
1Similar spectra involving two ligands binding close to each other in the same binding pocket at the same time had first been reported by Dawei Li et al.8
TA T TB
Figure 1.1: Sketch of the 3-step model of INPHARMA magnetization transfer.
Thez-magnetization (red) of the protons H1 and H2 of ligand A can be partially transferred to ligand B. Adapted from a figure by Dr. Víctor Sánchez-Pedregal.
TA TB
Figure 1.2: Sketch of the 2-step model of INPHARMA magnetization trans-fer. This model can be used if the ligands are in large excess in respect to the receptor.14 Adapted from a figure by Dr. Víctor Sánchez-Pedregal.
a single pair of complexes can be found to explain the experimental data.
This would mean that the structure of both complexes would have been de-termined. The INPHARMA spectra will depend on the orientations of the ligands in the binding pocket, their conformation, and their precise relative position to neighboring atoms of the receptors, and so on the structure of the binding pocket. This dependence is illustrated in fig. 1.3.
H5
Figure 1.3: Model of three different complex pairs and the resulting INPHARMA spectra. Purely intra-molecular peaks are highlighted in the color of the respective ligand. Inter-molecular INPHARMA peaks are highlighted in red. It is clearly visible how the different orientations of ligand B change the patter of the IN-PHARMA peaks. At the same time the intra-molcular trNOE peaks stay hardly unchanged. In this figure only the orientation of ligand B is modified. The ori-entation of the other ligand shall be assumed to be known. If also ligand A is allowed to be moved, having an unknown orientation in respect to the receptor, there might be some different orientations leading to the same INPHARMA spec-tra. This case would occur if ligand A, in the lowest example, is turned 180◦. The resulting spectrum would just look like the one on the top. Spin diffusion is neglected in this figure. If the mixing time, allowing for magnetization transfer, is chosen long enough all present protons will show cross peaks to each other. These peaks will be, in general, of different intensity, in contrast to the figure. The in-tensity of the peaks provides valuable gradual information on the magnetization exchange and can be used for structure determination. Adapted from a figure by Dr. Víctor Sánchez-Pedregal.
For the measurement of the INPHARMA spectrum a simple NOESY pulse sequence4 is sufficient. In fig. 1.4 slices of such spectra are shown10
In the following chapter ’Theory’ I will explain the theoretical basis of the INPHARMA method in greater detail. Then I will explain the theoretical and mathematical foundations for the analytical gradient based structural refinement of INPHARMA spectra. This theoretical approach can be di-rectly applied to other systems, being on a general level mathematically equivalent to INPHARMA, like transferred NOE, interligand Overhauser effect spectra, or NOESY spectra of single molecules undergoing confor-mational exchange. In the chapter ’XPLOR-NIH Implementation’ I will shorty explain the implementation details of the INPHARMA restraints into XPLOR-NIH.36, 37 Then, in the chapter ’Results’, the application of INPHARMA restraints in the context of various test systems will be exem-plified.
E-19
B-m B-p
B-o
E-21 E-27
B-16/17 E-23 E-22/25
2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0
7.2 7.4 7.6 7.8 8.0 8.2
1 H(ppm)
1H (ppm)
E-19
B-m B-p
B-o
E-21 E-27
B-16/17 E-23 E-22/25
2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0
7.2 7.4 7.6 7.8 8.0 8.2
1 H(ppm)
1H (ppm)
Figure 1.4: Slice of an experimental INPHARMA spectrum of the tubulin-epothilone A- baccatin III system. The upper measurement, using 20 ms mixing time, only shows intraligand trNOE peaks in green and blue. The lower mea-surements, using 70 ms mixing time, also shows INPHARMA peaks in red. Mea-surement by Dr. Víctor Sánchez-Pedregal, adapted from a figure by Dr. Víctor Sánchez-Pedregal.
Theory
2.1 Simulation of INPHARMA Peak Volumes
The peakvolumesAof INPHARMA spectra evolve according to the follow-ing differential equation as described previously:11, 12
dA
dt =−R∗ ·A=−(RNOE+K)A. (2.1)
The NOE relaxation matrixRNOEand the chemical exchange matrixKboth contribute in physically different but mathematically equivalent ways to the magnetization exchange. Their sum will be called the effective relaxation matix R∗.
The NOE relaxation matrix RNOE can be seen as a matrix of submatrices Rs and is of the general form
The chemical species are denoted with the index s, here A,B,TA or TB.
The non-diagonal elements of R∗s are Rsij =σijs = 1
Dipolar cross relaxation of a spin is induced by neighboring moving spins (due to Brownian motion) which create randomly fluctuating magnetic fields. These random fields induce random transitions causing relaxation.
15
For the simple model of a rigid molecule undergoing isotropic rotational diffusion the spectral densities Jij,ns are:
Jij,ns = τijs
1 + (nω0τijs)2. (2.4)
The cross relaxation rateσijs between two individual protonsiandj belong-ing to species s depends on their distance rijs, on the isotropic rotational correlation time τijs of the length of the internuclear vector, their Larmor-frequency ωo and so on the strength of the external magnetic field B0. For the model of a rigid molecule, used here, the internuclear vector’s rota-tional correlation time τijs and the isotropic rotational correlation time of the molecular speciesτcsare identical. The gyromagnetic ratio of the proton γH, the reduced Planck-constant ~ and the magnetic constant µ0 are well known natural constants.
The diagonal elements of the matrixRs are Rsii=X
j6=i
ρsij (2.5)
with
ρsij = 1 10γi2γj2
~µ0 4π
2
1 rijs6
Jij,0s + 3Jij,1s + 6Jij,2s
. (2.6)
The following kinetic model, visualized in (fig. 1.2), is used:
TA + Bk12
k21
TB + A (2.7)
The chemical equilibrium state of the system is determined by the initial concentrations [T]0, [A]0, and [B]0 and the kinetical rate constantsk12 and k21. These are input parameters of the simulation. All the equilibrium concentrations of all species[A],[B],[TA]and [TB]are calculated from them and the kinetic model (2.7). The complete derivation of the kinetic matrix K was exemplified in [9] and has been published in detail in [11–13]. The calculation is repeated in the appendix of this thesis (ch. 9.1.1). This kinetic
matrix is used in the following: Here, using the identity matrixI, the initial peak volume matrix is given:
A0 =
The efficient and fast calculation of eigenvalues and eigenvectors of a quadratic matrix requires symmetric matrices as input. Though the NOE contribution to the effective relaxation matrix is symmetric the kinetic matrix contribu-tion is not. The diagonal symmetrizacontribu-tion matrix C with Cii = 1/√
Kii is used for symmetrization.
Not all atoms of the molecule will have the same chemical shift, like pro-tons in methyl groups. All cross peak volumes of them will be actually the observation of the sum of many individual-proton cross peaks. This is con-sidered in the grouping matrix G, having one column for every observable resonance, a group. The entries of the column are "1" for atoms belonging to the group and "0" for all other atoms.
The solution for the time evolution of the grouped peak volumes11, 12 is:
Agm(τm) = GAm(τm)Gtr
=Gexp (−R∗τm)A0Gtr
=GCexp (−R∗sτm)C−1A0Gtr
=GCχ∗sexp (−λ∗sτm) (χ∗s)−1C−1A0Gtr
=GCχ∗sexp (−λ∗sτm) (χ∗s)trC−1A0Gtr.
(2.10)
The following symbols are used:
Agm Grouped peak volume matrix for the mixing time with indexm
Am Individual-atom peak volume matrix for the mixing time with indexm A0 Individual-atom initial peak volume matrix
G Grouping matrix
λ Diagonal matrix of the eigenvalues of the effective, symmetrized relaxation matrixR∗s
χ∗s Orthonormal matrix of eigenvectors of the effective, symmetrized relaxation matrix R∗s withχ∗str =χ∗s−1
C Diagonal symmetrization matrix
τm Mixing time of the NOESY experiment with indexm
For clarity I define L≡χ∗s and Λ≡λ∗s. Now the equation becomes Agm =GAmGtr
=GCLexp (−Λτm)LtrC−1A0Gtr (2.11)