J. R. Quine

Transmembrane Domain of M2 Protein from Influenza A Virus Studied by Solid-State 15N Polarization Inversion Spin Exchange at Magic Angle NMR

The M2 protein from the influenza A virus forms a proton channel in the virion that is essential for infection. This tetrameric protein appears to form a four-helix bundle spanning the viral membrane. Here the solid-state NMR method, 2D polarization inversion spin exchange at magic angle (PISEMA), has been used to obtain multiple constraints from specifically amino acid-labeled samples. The improvement of spectral resolution from 2D PISEMA over 1D methods and 2D separated local field methods is substantial. The reliability of the method is validated by comparison of anisotropic chemical shift and heteronuclear dipolar interactions from single site labeled samples. The quantitative interpretation of the high-resolution constraints confirms the helix tilt to be within the range of previous experimental determinations (32 degrees -38 degrees ). The binding of the channel inhibitor, amantadine, results in no change in the backbone structure at position Val(27,28), which is thought to be a potential binding site for the inhibitor.

Protein structure in anisotropic environments: development of orientational constraints

Solid state nuclear magnetic resonance has the potential to characterize membrane protein structures at very high resolution. This paper focuses on the use of ori- entational constraints for this purpose. These constraints are based on observing the orienta- tion dependence of nuclear spin interactions. The interpretation of these observations is based on accurate knowledge of the spin interaction tensors — the tensor element magnitudes and their orientation to the molecular frame. Numerous possibilities exist for developing these ori- entational constraints, each with their own strengths and weaknesses. This first of two papers focuses on the observation of these constraints and their inherent error, while the second pa- per demonstrates the utility, advantages, and disadvantages of these constraints for solving protein backbone structure.

SOME SERIES INVOLVING THE ZETA FUNCTION

JUNESANG CHOI, H.M. SRIVASTAVA AND J.R. QUINELots of formulas for series of zeta function have been developed in many ways.We show how we can apply the theory of the double gamma function, which hasrecently been revived according to the study of determinants of Laplacians, toevaluate some series involving the Riemann zeta function.1. INTRODUCTION

Protein structure in anisotropic environments: Unique structural fold from orientational constraints

The mathematical foundation of the determination of protein structure from orientational constraints is described. The tools used are vector algebra, gram matrices, and determinants. The discussion begins in the general abstract setting and proceeds to a discus- sion of how the methods can be applied to the determination of protein structure using solid state nuclear magnetic resonance. Examples are given relating to the structure of the peptide gramicidin A.

A global theorem for singularities of maps between oriented 2-manifolds

Let M and N be smooth compact oriented connected 2-manifolds. Suppose/: M-*N is smooth and every pointp € M is either a fold point, cusp point, or regular point of / i.e., / is excellent in the sense of Whitney. Let M + be the closure of the set of regular points at which / preserves orientation and M the closure of the set of regular points at which / reverses orientation. Let />„ ..., p„ be the cusp points and ¡i(pk) the local degree at the cusp pointy. We prove the following: X(M) IxfM) + 2 p(pk) (deg/)X(AO where x is the Euler characteristic and deg is the topological degree. We show that it is a generalization of the Riemann-Hurwitz formula of complex analysis and give some examples. Proof of the theorem. Let C°°(A/, N) be the set of smooth maps from M to N topologized in the usual fashion, where two functions are close if all of their derivatives are close. Let/ G CM(M, N). We sayp is a regular point off if we can find local coordinates (x,y) nearp and (w, v) near f(p) withp and f(p) having coordinates (0, 0) such that/is given by u = x, v =* y. Sayp is a fold point if the equations can be put in the form u = x, v = y2. Sayp is a cusp point if the equations can be put in the form u = x, v = y3 — xy. It is well known for these dimensions that these three types of local maps are the only stable types and that the stable types are dense. (See Whitney [17] or Callahan [2].) Say/is excellent if every pointp G M is one of the above types for/. In the following discussion, we will assume that/ is excellent and that the images of the cusp points are distinct. We will find it convenient to use a different form for the equation at a cusp point. Let x = 3/2 + s,y = t. This clearly defines aC°° homeomorphism and the cusp equation becomes u — 3t2 + s, v = -2r3 st. The general fold is now along the / axis. If we want the ordered pairs of vectors (3/3i, 3/3/) and (3/3m, 3/3t?) to be positively oriented in the tangent spaces TM and TN Presented to the Society, November, 6, 1976; received by the editors May 18, 1976 and, in revised form, August 22, 1976. AMS (MOS) subject classifications (1970). Primary 57D35, 57D45, 58C25; Secondary 57D25, 30A90.

Third-order effect in solid-state NMR of quadrupolar nuclei

Abstract The theory and experimental observation of the third-order effect in solid-state NMR of quadrupolar nuclei are presented. The third-order effect consists of spherical harmonic terms up to rank l =6 and shifts NMR frequencies between two spin states that are not symmetric such as satellite transitions. Two-dimensional satellite transition magic-angle spinning experiment averages both the first and the second-order quadrupolar interactions making the quantitative measurement of the third-order effect possible. The third-order quadrupolar effect in andalusite has been measured at 11.7 T and its powder patterns are fitted with numerical simulations.

Atomic refinement with correlated solid-state NMR restraints

The orientation data provided by solid-state NMR can provide a great deal of structural information about membrane proteins. The quality of the information provided is, however, somewhat degraded by sign degeneracies in measurements of the dipolar coupling tensor. This is reflected in the dipolar coupling penalty function used in atomic refinement, which is less capable of properly restraining atoms when dipolar sign degeneracies are present. In this report we generate simulated solid-state NMR data using a variety of procedures, including back-calculation from crystal structures of alpha-helical and beta-sheet membrane proteins. We demonstrate that a large fraction of the dipolar sign degeneracies are resolved if anisotropic dipolar coupling measurements are correlated with anisotropic chemical shift measurements, and that all sign degeneracies can be resolved if three data types are correlated. The advantages of correlating data are demonstrated with atomic refinement of two test membrane proteins. When refinement is performed using correlated dipolar couplings and chemical shifts, perturbed structures converge to conformations with a larger fraction of correct dipolar signs than when data are uncorrelated. In addition, the final structures are closer to the original unperturbed structures when correlated data are used in the refinement. Thus, refinement with correlated data leads to improved atomic structures. The software used to correlate dipolar coupling and chemical shift data and to set up energy functions and their derivatives for refinement, CNS-SS02, is available at our web site.

Satellite transition rotational resonance of homonuclear quadrupolar spins: magic-angle effect on spin-echo decay and inversion recovery

Abstract Satellite transition rotational resonance and magic-angle effects on central-transition spin-echo decay and inversion recovery of half-integer quadrupolar nuclei are presented. Small magic-angle offsets can change the satellite transition rotational resonance condition and consequently lead to dramatic effects on line width, spin-echo decay and inversion recovery. An half degree magic-angle offset lengthens α-Al 2 O 3 spin-echo decay by 50% and inversion-recovery by nearly two orders of magnitude. The theory of satellite transition rotational resonance is presented and the origin of these magic-angle effects is explained.

Protein structural analysis from solid-state NMR-derived orientational constraints

High-resolution orientational constraints from solid-state NMR spectroscopy of uniformly aligned biological macromolecules provide a great structural analysis problem. Several approaches to this problem have been made in the past. Here a vector algebra method is developed that provides analytical solutions for the torsion angles and a concise and simple view of the structural possibilities. Numerical instabilities in this approach are easily predicted. Insight into how the structural ambiguities arise in the first place and how they can be reduced in number is demonstrated with this new approach.

HELIX PARAMETERS AND PROTEIN STRUCTURE USING QUATERNIONS

Abstract The mathematics of helices is important for understanding protein secondary and super-secondary structure, since every regular protein backbone structure can be considered as a helix. This paper presents a mathematical approach to helices using geometric algebra in the form of quaternions. The motivation is to make it more convienient to compute a solid-state NMR picture of the protein using orientational constraints. In terms of two parameters specifying a helix, formulas are given for the various helical parameters of interest in considering protein structure: residues per turn, pitch, radius and helix axis direction. Helices with period more than one residue are also considered, depending on more parameters and giving more complicated formulas. Applications to determining protein structure using solid-state NMR are considered.