Table Of ContentMOLECULAR BIOLOGY
An International Series of Monographs and Textbooks
Editors: BERNARD HORECKER, NATHAN O. KAPLAN, JULIUS MARMUR, AND
HAROLD A. SCHERAGA
A complete list of titles in this series appears at the end of this volume.
Conformational Properties
of
Macromolecules
A. J. Hopfinger
Department of Macromolecular Science
Case Western Reserve University
Cleveland, Ohio
1973
ACADEMIC PRESS New York and London
A Subsidiary of Harcourt Brace Jovanouich, Publishers
COPYRIGHT © 1973, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR
TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC
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ACADEMIC PRESS, INC.
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United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
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Library of Congress Cataloging in Publication Data
Hopfinger, A J
Conformational properties of macromolecules.
(Molecular biology; an international series of
monographs and textbooks)
Includes bibliographical references.
1. Macromolecules. 2. Conformational analysis.
I. Title. II. Series. [DNLM: 1. Macromolecular
systems. QU 55 H792c]
QD381.H66 539'.6 72-9328
ISBN 0-12-355550-7
PRINTED IN THE UNITED STATES OF AMERICA
To my mother, LaVerne
Preface
The application of empirical potential functions to determine the spatial
properties and, in a few instances, the thermodynamic properties of macro-
molecules has been the source of considerable controversy among macro-
molecular scientists. On the positive side, theoretical conformational analysis,
as this area of research has come to be known, has provided considerable
information about the interrelationship between nonbonded molecular
energetics and the spatial organization of macromolecules. From the negative
point of view, theoretical conformational analysis lacks a solid foundation in
that the essential potential energy functions cannot be ascertained as being
reliable. Regardless of one's position on the merits or faults of theoretical
conformational analysis, the fact remains that since 1962, when Liquori and
co-workers carried out what can be considered the first conformational
energy calculation on a polymer-chain molecule, over one hundred and thirty
papers have been published involving the use of theoretical conformational
energy calculations.
Theoretical conformational analysis has become a popular technique for
investigating macromolecular structure. Therefore, it is most appropriate
that a comprehensive discussion of this topic be available for reference and
review by experienced researchers in the field, and to provide an inclusive
guide to those workers just entering the field. This particular book provides a
wealth of molecular parameters very often needed by structural chemists,
X-ray crystallographers, biophysicists, physical chemists, and macromolec
ular scientists. Thus, while the central topic of the book, conformational
analysis of macromolecules, is of explicit interest to only a select group of
researchers, the data contained in the text is of general interest to most
physical and biological scientists who are concerned with molecular structure.
The application of theoretical conformational analysis to the study of the
structural properties of polypeptides is given considerable emphasis in the
text. There are two reasons for this. First, the structural properties of poly
peptides are of special interest to the author. Second, a large portion of the
research which has been performed using conformational energy calculations
has been concerned with the spatial organization of polypeptide chains.
Notable are the studies of Scheraga and Ramachandran. Those readers who
do not work in protein chemistry should consider the structural studies of
ix
X Preface
polypeptide chains as examples of how conformational energy calculations
can be applied. The conformational potential energy functions, geometric
algorithms, transition theories, and fluctuation models presented in the text
are not restricted to polypeptides, but, in general, are applicable to most
macromolecules. At the same time, however, some of the discussions dealing
with polypeptide chain structure may, in fact, be of special interest to protein
chemists.
The book is divided into five chapters. Chapter 1 deals with the classification
and characterization of macromolecular structure. Application of linear
algebra to generate macromolecular conformations is discussed extensively.
Chapter 2 deals with molecular energetics. Special consideration is given to
the various types of semiempirical potential functions used in conformational
energy calculations. All significant contributions to the total conformational
free energy are discussed in detail. Chapter 3 presents the results of a number of
significant studies involving the use of theoretical conformational analysis.
It is left to the reader to judge, from the reported investigations, how reliable,
and useful, conformational energy calculations might be for a specific type of
structural investigation. Chapter 4 discusses structural transitions in macro-
molecules. Specific emphasis is given to the description of how conformational
energy calculations might be useful to help describe the nature and properties
of such transitions. Chapter 5 presents a number of research topics where the
application of conformational energy calculations might be beneficial.
In writing this book I have endeavored to assemble a comprehensive
ensemble of information about conformational energy calculations. I have
tried to point out all of the tenuous assumptions which have been made in
this area of research. Keeping this in mind, I believe this area of study has its
greatest days yet ahead. There are several reasons for this optimism. The two
most significant factors are (a) advances in computer technology and applied
mathematical methods, and (b) the high quality of the researchers interested
in structural calculations of macromolecules.
Last, I would like to thank my wife, Kathleen, for her long hours of patient
proofreading, and my secretary, Ina Cael, for her perseverance in typing and
proofreading the manuscript.
Chapter i | Macromolecular Geometry
I. Introduction
The architecture of macromolecules is as fascinating as it is complex. While
the basic building blocks, the atoms, have rather simple bonding geometries,
the versatility in the number of ways they may be combined coupled with the
large numbers of atoms in macromolecules results in virtually an unlimited
number of different spatial macromolecular assemblies. Some linear poly
meric molecules adopt highly symmetric spatial structures such as helices.
Other linear polymeric molecules, most notably the globular proteins, adopt
spatial structures reminiscent of a discarded length of used baling wire.
Perhaps even more incredible than the variations in known macromolecular
structures is the fact that each globular protein has its own unique, yet
seemingly random, gross structure. This would suggest that the spatial
structure of macromolecules can be extremely sensitive to the functional
needs of the molecule.
The flexibility of many macromolecular chains give rise to an ensemble of
spatial geometries not having analogs in the world of "micromolecules."
Among these phenomena are (i) chain folding in which a long polymeric
molecule folds back on itself to form a minicrystal, (ii) interchain supercoiling
in which two or more macromolecules wrap around one another to generate
a braided-rope structure, and (iii) extended aggregation in which polymeric
chains are cross-linked to form two-dimensional networks which act as unique
molecular entities.
Much of the commercial value of polymers derives from the geometric
versatility of these molecules. The question to which we will address ourselves
in this chapter is, "How can we classify and generate macromolecular
structures?" We will say nothing about which geometric structure is most
probable for a given macromolecule, but rather characterize the nature of
those structures which are possible.
1
2 1. Macromolecular Geometry
II. Classification of Macromolecular Structure
A. General Comments
It is very difficult to describe precisely macromolecular geometries because
of the immense size of these molecules and the large number of possible degrees
of structural freedom. In order to have the capability of describing the spatial
properties of any linear macromolecule four levels of structural specification
have been defined. In the next section these four levels of structural identity
are discussed in detail.
B. Levels of Structural Identity
1. Primary Structure
This is the structural formula of the monomer units, including the bond
lengths and bond angles of the chemical bonds, and a list as to how the
monomer units are joined to one another. Ultimately the primary structure
dictates the resultant spatial geometry of the macromolecule. That this is the
case is conceptually clear when one considers the possible change in total
macromolecular geometry that can accompany a change from, say, tetra-
hedral to trigonal symmetry in one or more backbone bond angles. An even
more obvious factor which can modify the total macromolecular geometry is
interchanging different types of monomer units along the macromolecular
chain.
Of fundamental importance in being able to predict macromolecular
structure, or even rationalize experimental findings, is the availability of a
reliable set of values for bond lengths and bond angles as well as the deviation
values. The sensitivity of total macromolecular geometry to small changes in
the values of bond lengths and bond angles (of the magnitudes of the observed
deviations) is, as one would expect, specific to the particular macromolecule.
A limited number of studies indicate that the resultant macromolecular
geometry is rather insensitive to small changes in the values of bond lengths
and bond angles. Table 1-1 contains a list of bond lengths and bond angles.
2. Secondary Structure
This is the manner in which monomer units are folded to generate simple
geometric structures. For linear nonbranching macromolecules, which are
II. Classification of Macromolecular Structure 3
the only type considered in this book, virtually all secondary structures
possess cylindrical symmetry. In fact, nearly every secondary structure is
helical. There is some debate as to whether or not ß and zigzag structures are
really helical. Mathematically, these structures can be considered helices
having elliptical cross sections. Thus a discussion of secondary structure must
include a description of the properties of helices.
TABLE 1-1
A General List of Acceptable Bond Length s and Bond Angles in Some Specific
Classes of Molecular Structure
I. Bond lengths
Bond length Bond length
Group (Ä) Group (A);
C—G G—O
(a) Any molecule 1.54 (a) Carbonyl 1.27
C=-C (b) Garboxyl 1.25
(a) Aromatic rings 1.38 G=0
(b) Olefinic linear chains L40 (a) Carbon near sp2 1.23
(b) Carbon near sp 1.16
C=C
(a) Any molecule 1.34 C—S
C=C (a) n-Paraffins 1.82
(a) Any molecule 1.21 (b) Cyclic molecules 1.75
G—N C—F
(a) Polyamides 1.49 (a) Any molecule 1.39
(b) All other molecules 1.47 G—Cl
0=N (a) Any molecule 1.77
(a) Polypeptides 1.33 N=0
(b) Five-and six-membered 1.36 (a) Any molecule 1.20
rings N—H
C=N (a) Any molecule 1.01
(a) Any molecule 1.27 O—H
G=N (a) Any molecule 0.96
(a) Any molecule 1.16 O—F
G—H (a) Any molecule 1.40
(a) n-Paraffin chains 1.09 O—Cl
(b) Ethylenic chains 1.06 (a) Any molecule 1.68
(c) Aromatic rings 1.08 H—S
G—OX (a) Any molecule 1.33
(a) X = hydrogen 1.33 S-S
(b) X% hydrogen 1.42 (a) Any molecule 2.05
4 1. Macromolecular Geometry
TABLE 1-1 (continued)
II. Bond angles
Bond angle Bond angle
Group (deg) Group (deg)
c—c—c C—N=0 112.0-116.0
(a) Short linear chains 109.5-110.5 O—N—O 122.5-135.0
(b) «-paraffin chain 111.0-115.0 Cl—N=0 113.2
polymers (backbone) O—N=0 113.5-119.6
H—C—H 109.5-112.0 C—O—C 110.0-114.0
H—C—F 111.6 Cl—O—O 105.6-109.3
H—C—Cl 111.8 H—O—C 106.7-109.0
F—C—F 108.0-111.0 Cl—O—Cl 110.8
Cl—C—Cl 109.5-113.0 F—O—F 103.2
H—C—C 110.2
F—C—C 109.4 O—P—O 101.0-103.0
Cl—C—F 100.0 o—p—cr 114.7-117.3
C—P—C 100.4
F—C(sp2)=C 125.2 O—P—S 116.5
N—C(sp2)=S 124.8
C—C(sp2)—S 125.0 H—S—C 100.3
N—C(sp2)=0 121.5 o—s—o 120.0
O—C(sp2)=0' 124.3 Cl—S—Cl 101.0
C—C(sp2)=0 122.0-124.5 Cl—S—S 105.0
F—S—F 93.5
In constructing Table 1-1 typical representative values for bond lengths and angles are
reported. Obviously there can be deviations in these "typical" values. This is especially true
for the bond angles where, in some cases, a range of values had to be given.
The simplest helix can be generated by uniformly winding a string around
a cylinder of constant radius as shown in Fig. 1-1. As indicated in Fig. 1-1
there are three fundamental parameters which uniquely describe such a
simple helix: r, the radius of the helix, /, the axial advancement of the helix
for one complete turn, and ω, the frequency constant of rotation. Using these
three parameters we can construct the parametric equations, in t, for such a
simple helix.
# = rsina>/, y = rcosa>/, z = lt (1-1)
The parameters ω and / can be defined, for a polymer composed of a sequence
of identical monomer units, in terms of n, the number of monomer units per
turn, ώ, the rotational advancement per monomer unit, and /, the axial
advancement per monomer unit.
ώ = ηω, I = nl
so that
x = rs'mna)t, y = rcosncot, z = nit (1-2)
