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Cooperative Conformational Transitions of Linear Biopolymers.

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J U N E 1970
P A G E S 389-472
Cooperative Conformational Transitions of Linear Biopolymers
By Jiirgen Engel and Gerhard Schwarzr*j
Conformational transitions in proteins, nucleic acids, and other biopolymers evidently
play a decisive role in many biological processes, particularly in control processes. They
often proceed cooperatively, i.e. the elementary process of the transition of an individual
segment of these macromolecules is influenced by the state of other segments via intramolecular interactions. In general, the segments favor the same state as their neighbors. The
resulting equilibrium properties of cooperative systems, e.g. the sharpness of the transitions and their dependence on the chain length, can be quantitatively explained for linear
systems by the linear k i n g model. The molecular causes of the cooperativity can be explained for simple model polymers.
1. Introduction
Nucleic acids contain chains usually made up of at
most four structural units, the nucleotides [see (2)].
Macromolecules are important structural units of
organisms, each by itself often exercising surprisingly
complicated functions. These biological macromolecules consist chemically o f chains built up from a
relatively small number of types of monomer units.
The most important of these macromolecules are the
proteins, which include the enzymes that catalyze and
regulate the metabolism, and the nucleic acids, which
carry genetic information in the form of a control
program for the biosynthesis of the proteins.
Proteins consist of polypeptide chains built up from a
maximum of about 20 amino acids [see ( I ) ] .
I Bise 2
Base 1
ribose in ribonucleic acid (RNA)
deoxyribose in deoxyribonucieic acid (DNA)
The nucleotides differ essentially in their bases. In
R N A these are cytosine (C), guanine (G), adenine (A),
and uracil (U), while in DNA, U is replaced by
thymine (T).
Biologically important chain molecules of this nature
are referred to by the general term “biopolymers”.
The decisive difference between similar polymers lies
in the sequence of the monomer units:
SIS2S3 ..... si
.. . . S N
A m i n o a c i d r e s i d u e ( = S e g m e n t of t h e c h a i n )
R’, RZ, R3. . . . = side chains, by which the amino acid residues differ
from one another. The peptide linkage has partial double-bond character, which prevents free rotation.
[*] Prof. Dr. J. Engel
Max-Planck-Institut fur Eiweiss- und Lederforschung
8 Miinchen 15, Schillerstrasse 46 (Germany)
Prof. Dr. G . Schwarz
Physikalisch-Chemisches Institut d e r Universitat
CH-4056 Basel, Klingelbergstrasse 80
Angew. Chem. internat. Edit. 1 Vol. 9 (1970)
1 No. 6
where the N chain members are numbered from left to
right and each S is a symbol for a monomer unit
(e.g. C, G, A, o r U in R N A chains).
Very simple sequences have been prepared synthetically, particularly polyamino acids, polypeptides with
a periodically repeating sequence of a few amino acid
residues, and analogous polynucleotides (e.g. poly-A).
These synthetic biopolymers are useful as model
substances, since the results of measurements are
easier to interpret than for the much more complicated
natural biopolymers.
Under suitable conditions, most biopolymers have a
specific steric structure (conformation), which is
stabilized by noncovalent (“weak”) interactions (such
as van der Waals forces, hydrogen bonds, hydrophobic
and electrostatic interactions) between the segments.
Native nucleic acids and synthetic polynucleotides
frequently have relatively simple helical-linear conformations, the basic structure of which does not
depend on the sequence of the segments. The best
known is the double helix discovered by Watson and
Crick (Fig. 1).
Fig. 1. Schematic representation of the D N A double heiix after
Watson and Crick [l]. The bands represent the sugar phosphate chains,
which are joined in the helix via the bases A, C, G, T [see formula (211
by hydrogen bonds ( - - ). A coil region is shown underneath.
Proteins generally have much more complicated, often
globular arrangements of the polypeptide chains
(“tertiary structure”), as might be expected from the
great variety of possible sequences. A few basic
structures (“secondary structures”) are often found in
limited regions of the protein; examples are the ahelix (Fig. 2), the pleated sheet (or @)structure, and
the polyproline I1 helix (see Section 3.2). Since such
structural types can be found in pure form in polyamino acids, these may be regarded as simple protein
The biological function of a biopolymer is usually
confined to its structured (native) state. The enzymes
are good examples of this. In their active sites, segments from different regions of the chains are brought
close to one another by chain folding in such a way
that they can cooperate in the catalysis of a metabolic
reaction 131. Even small changes in conformation
generally lead to a large change in the enzymatic
It has been found in a number of biological reguIation
phenomena that enzymes involved can assume conformations with different activities, between which
reversible transitions occur. There is much to indicate
that control processes of this nature are extremely
[l] J . D . Watson and F. H . C . Crick, Nature (London) 171,137,
964 (1953).
[21 R . B. Corey and L. Pauling, Proc. Roy. SOC.(London),
Ser. B 141, 10 (1953).
I31 D . C . Philips, Sci. American 215, 78 (1966).
Fig. 2. Schematic representation of the a-helix after Paulrng and
Corey 121. The band represents the polypeptide chain [see formula ( I ) ] ,
in which each amino acid residue is represented by its peptide CO
group. In the helix, hydrogen bonds (arrows) connect the peptide N H
group of each residue to the CO group of the third nearest residue.
A helix nucleus (top) in a coil region has only one such hydrogen bond,
but this fixes three residues in one turn of the helix.
important and widespread in molecular biology. An
example is provided by the allosteric regulation process
in enzymes consisting of several protein subunits
(“protomers”). A regulating agent (substrate, activator, or inhibitor) becomes bound to an active site
of the protein, as a result of which the activity of
other sites in the same moiecuIe are changed r4,sJ.
This cooperative interaction is evidently due to induced
conformational transitions of the protomers. Similar
processes are postulated for the regulation of the
function of repressors and biological membranes [6,7J.
There has been interesting speculation concerning its
possible role in the short-term storage of information
in the brain [81.
Nearly all biopolymers can change into a disordered
(usually inactive) state of higher entropy. One example
is the denaturation of proteins by changes in the
141 J . Monod, J . Wyman, and J . P . Changeux, J. molecular
Biol. 12, 88 (1965).
151 M . Kirtlry and D . E . Koshland, J. biol. Chemistry 242, 4192
[6] J . P . Changrux, J . Thiery, Y. Tung, and C. Kittel, Proc. nat.
Acad. Sci. USA 57, 337 (1967).
[7] G. Adam, Z.Naturforsch. 236, 181 (1967).
[8] A . Katchalsky and A. Oplatka, Neuroscience Res. Progr.
Bull. Suppl. 4,71 (1966).
Angew. Chem. internat. Edit. 1 Vol. 9 (1970) 1 No. 6
temperature, the pH, or the solvent. In transitions of
this type, the weak stabilizing interactions and the
associated blockage of free rotation about chemical
bonds are largely eliminated. In solution, the chains
(like other filamentous molecules) take the form of a
random coil. This will be referred t o briefly as the coil
state. The transition in the opposite direction (coil +
specific conformation) is a decisive step in the biosynthesis of On the other hand, a
process of the type double helix + coil must be
assumed in the duplication of DNA with its genetic
Conformational transitions often proceed cooperatively, i. e. the elementary processes in the individual
segments are influenced by the state of other chain
segments. There is usually a strong tendency to assume the state of the neighboring segment. Consequently, even relatively small variations in the external
parameters (e.g. temperature, pH, solvent composition)
can lead to practically complete conformational transition. This property is of fundamental importance
to the physiological control functions mentioned
A second characteristic of cooperative reactions is the
dependence of the transition on the chain length. The
sharpness of the transition generally increases with the
size of the molecule. This effect was first observed on
investigation of linear synthetic model substances,
since it is only here that the size of the molecute can
be systematically varied. Transitions that have been
fairly extensively investigated and that can be described theoretically are a-helix + coil for polyamino
acids, helix I + helix I1 for polyproline, and double
08 -
Fig. 4. Solvent-induced transition between two types of helix (I and 11)
of poly-L-proline [see formula (311 having chain lengths N = 12 ( 0 ) .
N = 30(A),and N = 132
[14]. The degree of transition 0 = number
of residues in the I stateltotal number of residues was determined polarcoil transiimetrically by a method similar to that used in the a-helix
tion (see caption of Fig. 3). The curves are calculated (see Section 3.2).
Abscissa: n-butanol in trifluoroethanol, v01.- %.
& 04
5 0 3
Fig. 3. Temperature-induced a-helix + coil transition of poly-ybenzyl L-glutamate [R in formula ( I ) = CHz-CH2-COO-CHf2oHsl
in dichloroacetic acid/dichloroethane for various chain lengths, N .
I: N = 1500;, 2: N = 46; 3: N = 26. The degree of transition Q was
determined from the optical rotation a by means of the equation
Residues in the helix state - a ac
Total number of residues
a H - ac
Fig. 5 . Temperature-induced double helix
coil transition of polyadenylic acid [all bases = A in formula (211 having various chain
lengths N (after Applequisf and Dumle 1151). The hypochromism I161
(decrease in UV absorption on helix formation) used as the measured
quantity does not give the degree of transition 0 directly, since it depends on the length of the helix section. This dependence was taken into
account in the calculation of the curves shown 1151.
0 :N
0 :N
2; 6:N
9; c):N
3; 0:N = 4; 8 : N
10; 0: N = 11.
9: N
6 ; @: N
= 8;
where a H and a c are the rotations of the pure helix form and of the
pure coil form. The values indicated by circles were given by Doty and
Yung [ I l l , and all the other values by Zimm, Doly, and Jso [121. The
continuous curves were calculated [ l i . 131 (see Section 3.1).
[9] C. B. Anfinsen, Brookhaven Sympos. Biology 15,184 (1962).
[lo1 J.EngeZ and G.Beier,Kolloid-Z., Z. Polymere 197,7 (1964).
[11J P . Doty and J.T. Yang, J. Amer. chem. SOC.78, 498 (1956).
[12] B. H . Zimm, P . D o f y , and K . Iso, Proc. nat. Acad. Sci.
USA 45, 1601 (1959).
[131 B. H . Zimm a n d J . K . Bragg, J. chem. Physics 28, 1246
(1958), 31, 526 (1959).
Angew. Chem. internal. Edit.
VoI. 9 (I9701 / NO. 6
helix + coil for polynucleotides (see Sections 3.1-3.3).
At this stage we shall merely show a few typical transition curves (Figs. 3-5) based on optical measurements. The above-mentioned characteristics of cooperative transitions are clearly recognizable.
[14] V. Gamer, J . Engel, D . WinkImair, and G . Krause, Biopolymers 9, 329 (1970).
1151 J . Applequist and V. Damle, J. Amer. chem. SOC.87, 1450
I161 J . R . Fresco and P. D o f y , J. Amer. chem. SOC. 79, 3928
39 1
2. Theoretical PrinciplesI17J
The representation of the sequence of a biopolymer chain as a
sequence of symbols for the segments can be extended to a
formal description of the conformation. We make use of the
fact that the conformation is determined by interactions
between the monomer units. Thus the i-th segment Si can
conceivably have the states Aj, Bi, Ci,etc. Each sequence of
possible states then represents a certain conformational state
of the entire chain. In general, there will be a very large
number of such structural forms, but only those with minimal
free energies will be really stable.
These forms can in principle be predicted from theory if the
energy interactions of the segments are known. However,
such calculations are usually possible only with powerful
computers. Very successful investigations of this nature have
been possible for simpler structures 118-201. Further developments in this field should one day make calculation of the
conformation of a given sequence feasible.
The stability of a conformation naturally depends on the
external conditions (e.g. temperature, pressure, solvent).
Conformational transitions may therefore occur when these
conditions are changed.
We have found that interactions occur between various
segments within the macromolecule. Thus it will be readily
understood that the free energy of a segment often depends
not only on the external conditions and its own internal
structure, but also on the states of other parts of the chain.
Consequently, the thermodynamic (and kinetic) properties
of the elementary processes of the change in state are influenced by the situation prevailing at that time in more or
less distant segments. This is the same as saying that the
conformational transitions proceed cooperatively. If the
process were non-cooperative, the transitions of single
segments could be dealt with individually, and the complete
transition would not depend on the molecular weight and
would not exhibit abnormal sharpness.
It is convenient in theoretical studies to consider the simplest
possible linear biopolymers first. Chains built up from only
one type of monomer unit (i.e. in which all the Si are the
same) are particularly suitable. This is the case for the transitions illustrated in Figures 3-5. Another fact that can be
used here is that each segment has only two possible states;
let these be denoted by A and B [e.g. in polyproline, A =
trans configuration of the peptide linkage, B = cis configuration (see Section 3.2)J. One extreme form of the conformation
will then be described as a pure sequence of A states (in our
example, this corresponds to the polyproline I1 helix) and
the other as a pure B sequence (polyproline I helix). There
are, in principle, also many mixed conformations, e.g.
A A B B ..... B A B B .... A B B
1 2 3 4
the total number being 2N. With only 100 chain segments,
therefore, we must expect about 1030 different states.
With cooperative interactions, the properties of the elementary process
become more or less complicated. A particularly simple
situation evidently arises if cooperative interactions occur
only between immediately adjacent segments. This corre[17] The kinetics of cooperative transitions will be dealt with
in a future paper.
[18] G. N . Ramachandran and V. Sasisekharan, Advances
Protein Chemistry 23, 284 (1968).
[19] P. DeSantis, E. Giglio, A . Liquori, and Ripamonti, Nature
(London) 206, 465 (1965).
[20] D . Poland and H . A . Scheraga in G. D . Fasman: Poly-aAmino Acids. Marcel Dekker, New York 1967, p. 391.
sponds to the model of a linear lattice discussed by Ising in
1925 in connection with a theory of ferromagnetism [211.This
author was able to show that all the thermodynamic properties could be calculated relatively simply in this case. The
two-dimensional and three-dimensional models that were of
interest for his own purposes, however, led to extreme
mathematical difficulties, and further work in this field was
suspended for a long time.
With the growth of interest in cooperative transitions of
biopolymers, the linear Ising model acquired fundamental
importance for the theory of such processes I13,22-301. The
quantitative relationships can be represented particularly
elegantly by the matrix method (introduced by Kramers and
Wunnier[3*1).This method was first applied to a conformation transition by Zimm and Bragg[131, the particular case
being the helix + coil transition of polypeptides (see Section
3.1). In its simplified form, the theory can be extensively
generalized. It describes the essential properties of cooperative
transitions by means of the fundamental thermodynamic
parameters s and 0.
The parameter s is the equilibrium constant for the process
(or for BAA + BBA), which obviously represents the growth
of existing sequences of B or A states, since the reacting
(middle) segment is immediately adjacent both to an A state
and to a B state.
on the other hand, a new B sequence is started or an existing
one disappears. Let the equilibrium constant of this “nucIeation process” (for B) be GS.
The parameter 5 defined in this way is a measure of the cooperativity of the transition. If 5 < 1, the segments tend to
assume the same state as their neighbors, i.e. pairs of the
type AA or BB are more probable than pairs AB or BA
(positive cooperativity). If o = 1, there is no cooperativity,
since the equilibrium constants of equations (3) and (4) are
equal (i.e. they do not depend on the states of the neighbors). If o > 1, adjacent segments tend to assume different
states. This negative cooperativity or anticooperativity has
not yet been detected with certainty in biopolymers.
On grounds of microscopic reversibility, the nucleation rate
of an A state in a B sequence is described by the same 0.The
equilibrium constant for
is 5x 11s. The difficulty of nucleation can be different for A
and B states only at the chain ends. The terminal segments
(with the numbers 1 and N ) are in fact characterized in that
they have only one neighbor. Zimm and Bragg made the
simplifying assumption that the formation of nuclei of the
cr-helix at the ends of a polypeptide chain is also described
by the parameter G. This can be easily justified for helix + coil
1211 E. Ising, 2. Physik 31, 253 (1925).
[22] J. A . Schellmann, C. R. Trav. Lab. Carlsberg, Sect. chim.
29, No. 15 (1955); J. physic. Chem. 62, 1485 (1958).
[23] L . Peller, J. physic. Chem. 63, 1194 (1959).
I241 J . H . Gibbs and E. A . DiMarzio, J. chem. Physics 28,1247
(1958); 31, 526 (1959).
[25] K . Nagui, J. physic. SOC.Japan IS, 407 (1960).
[26] S. Lifson and A . Roig, J. chem. Physics 34, 1963 (1961).
[27] J . Applequist, J. chem. Physics 38, 934 (1963).
[28] B. H. Zimm, J. chem. Physics 33, 1349 (1960).
[29] S. Lifson and B. H. Zimm, Biopolymers I , 15 (1963).
1301 D. M . Crothers and B. H . Zimm, J. molecular Biol. 9, 1
[31] H . A. Kramers and G. H . Wannier, Physic. Rev. 60, 252
Angew. Chem. internat. Edit.
1 Vol. 9 (1970) 1 No. 6
transitions (see Section 3). We shall introduce this assumption
for B first, and we can then describe the equilibrium properties of the transition completely by means of s and a.
To demonstrate this, let us calculate the equilibrium concentrations of each chain state for the case N = 4. If we start
with the pure A state, then e.g . the pure B state can be
reached in four steps by the following processes:
The first step here is associated with nucleation, and this is
followed by three growth steps. The equilibrium concentrations are accordingly given by
where A H u is the molar enthalpy of transition for the elementary growth process (3) and
No = l/o’I2
is the cooperative length. The name refers to the fact that
for 0 = l j 2 and N + m, the average length of unbroken A
or B sequences is exactly NO+ 1. We now see from eq. (lob)
that the apparent molar enthalpy of transition (after van’t
Hof) is greater by the factor No than the value found from
the elementary process.
N o . A H u in eq. (lob) is often referred to as the van’t Hoff
enthalpy AHvan,* ~ ~since
8 its
, value is obtained from a
“van’t Hoff plot” (In Kagainst l/T). It follows from equations
(lob) and (11) that
NO= AHvan’t Hoe/A H u
For as4 % 1, therefore, the two extreme forms would be
present in roughly equal quantities. If, moreover, the cooperativity is so strong that
(AHuIAHvan’t H~IT)’
This behavior of apparently infinitely long chains is found
N 3 No
we find that the concentrations of all the intermediate forms
are small in comparison with those of the extreme forms
(e.g. for a = 10-4: ZBBBA = 0.1 Cgggg; &BAA = 0.01 ZBBBB,
and FBAAA = 0.001 ZBBBB). In this case the transition in
a given chain proceeds practically to completion or not at all.
This “all-or-nothing” behavior is evidently to be expected
whenever the chains are very short and the cooperativity is
sufficiently strong. The quantitative condition is found to be
The degree of transition 0 , i.e. the fraction of B states
formed, is then
The double helix + coil transition of oligonucleotides has
been very satisfactorily described in this way [I59 321.
As with very short chains, relatively simple behavior is also
found in the limiting case of very long chains. The degree of
transition then becomes
so that the transition no longer depends on the chain length.
As can readily be seen, the midpoint of the transition (0 =
112) for any a always corresponds to s = 1. However, the
sharpness of the transition will increase as the value of the
cooperativity parameter a decreases. This leads inter alia to a
characteristic increase in the apparent molar enthalpy of
transition. Thus using the “apparent equilibrium constant”
which can generally be readily obtained by experiment, the
temperature coefficient (disregarding the temperature dependence of a, which is permissible at least when the cooperativity is strong) becomes
In particular, at the midpoint of the transition
Angew. Chem. internat. Edit./ Vol. 9 (1970)1 No. 6
Chains of medium length with
log o
can generally be described only by fairly complicated expressions. Typical features are the dependence on chain
length and the fact that each chain practically contains only
one to two nuclei (suppressed nucleation). The special end
effects, which are disregarded in the simplified ZimmTBragg
mode1 discussed so far, play a decisive role here and in the
“all-or-nothing’’ range. It is found that whereas the dependence of helix + coil transitions on the chain length is
described very satisfactorily by the Zjmm-Bragg model the
intersecting transition curves of polyproline (Fig. 4) cannot
be explained on this basis. It has been shown that this phenomenon can be explained only if nucleation parameters for
each of the two states are consideredI331. It is difficult to
believe that nucleation at the chain ends is hindered in only
one of the two helix forms of polyproline. These considerations have led to a general theory of cooperative
transitions of linear biopolymers, in which the interactions
are still restricted to immediate neighbors, but where no
other special assumptions are necessary 134.351. Four independent thermodynamic parameters are required in this
From the detailed presentation of the general theoryr351, we
find that if we introduce the nucleation parameters ai,of,
(at one end of the chain) and o x , a; (at the other end),
agrees with the Zimm-Bragg a value. It is convenient to
define two end effect parameters
= C~;/CS~
p’ and p” can be expressed (though less concisely) in terms
of the free energies of the cooperative interactions, as
1321 D. Porschke, Diplomarbeit, Universitat Gottingen 1966;
Dissertation, Techn. Hochschule Braunschweig 1968.
[33] G. Schwarz, Biopolymers 5 , 321 (1967).
[34] J . Applequist, Biopolymers 6, 117 (1968).
[35] G. Schwarz, Biopolymers 6, 873 (1968).
The terms Ga, GL, GX, and Gf; denote any additional
free energies of the end states that might be caused by the
absence of the neighbor on one side, by the special chemical
structure (end groups), and by special interactions with the
Similarly, CT can be expressed by
RTln a= RTln as-RTIns= - ( A G K - A G ~ )
where AGK and AGw are the free energies of the nucleation
step and of the growth step respectively.
Using the matrix method, the properties of the conformational
equilibrium can now be calculated as a function of the chain
length N , the Zimm-Bragg parameters s and c, and the
quantities p’ and p”. The influence of p’ and p” decreases
with increasing chain length. Finally, when N 9 NO,the end
effects can be completely ignored, so that irrespective of the
particular p’ and p” values, the limiting case discussed in
connection with the Zimm-Bragg model is realized. In
general, therefore, the theory permits the representation of
the transition curves in the form
@= f (s, 5 , p’,
where the four thermodynamic parameters must be regarded
as a function of the variables of state. With stronger cooperativity and values of N that are not too small, however,
the degree of transition 0 is influenced mainly by changes
in s, since s, unlike c,p’, and p”, occurs in high powers (up
to N ) in eq. (20). 0 can then be regarded, to a first approximation, as a function of s alone (the other parameters being
treated as constants). If the dependence of the latter on the
experimental conditions (such as temperature, pressure,
solvent) is known, the transition curves to be expected experimentally can be calculated from eq. (20). Since s is the
equilibrium constant of the growth process, the required
relations can often be deduced from known thermodynamic
relations; for example, the temperature dependence can be
given by the van’t Hoff isochore
as used earlier in eq. (lob). (For limitations see Section 3.2.)
The linear king model with interaction of nearest neighbors
can in principle be extended e . g . to include cases where cooperative interactions extend beyond the nearest neighbors.
However, this leads to much more complicated calculations,
which are often not worth the trouble in any case, since if the
cooperativity is strong and the chains are not too short, the
actual interactions can be formally grouped together into
interactions of nearest neighbors. For this reason, the theory
discussed may be regarded as a useful basis, to a first approximation at least, for cooperative transitions of linear
biopolymers that are of practical interest.
3. Experiments on Simple Linear Model Systems
Investigations on simple linear biopolymers are very
important for a n understanding of t h e principles of
cooperative transitions. I n these cases one c a n expect
a theoretical description with few parameters and a
molecular interpretation of the experimentally determined parameters t o be possible.
A general comment is necessary concerning the number of these parameters. If t h e experimental transition
data can be described t o within the limits of error by
a very small number of parameters (e.g. by s and o),
this does not necessarily mean that the theoretical
model selected (in this example t h e Zimm-Bragg
model) is strictly correct. I t does, however, mean that
additional parameters of a possibly more valid theory
(e.g. (3‘ and $”, or parameters that take long-range
interactions a n d a complicated sequence into account)
cannot b e determined without further experiments. It
is therefore also necessary in practice to apply t h e
simplest possible theory that fits t h e main properties
t o the simplest possible systems.
For a n understanding of the molecular causes of
cooperativity, it has been found convenient to describe the experimental data by means of the theories
discussed above, the parameters (e.g. o) found by
matching then being explained on the basis of t h e
molecular facts of the case in question (e.g. o by
unfavorable interactions that arise in nucleation). A
statistical thermodynamic treatment based directly o n
molecular data is less general and is also difficult,
since the interatomic interactions in biopolymers and
the polymer-solvent interactions are only approximately known.
3.1. The a-Helix
+ Coil Transition
Most of the investigations on this important transition have been carried o u t with poly-L-glutamic
acid a n d its derivatives. A pioneering study was
the theoretical description by Zimm and Bragg[131
of t h e temperature-induced rearrangement of polyy-benzyl L-glutamate [R in formula (1) =
CHz-CH2-COO-CHzCsH51 (Fig. 3). The transition
is very sharp for long chains. T h e slope a n d position
of the curves depend o n t h e chain length.
For a quantitative description, Zimm a n d Bragg used
their simple Ising model (see Section 2), which contains
s and only one nucleation parameter o. They defined
a n amino acid residue as being in the B state (helix)
when its CO group is linked t o the NH group of t h e
third-nearest neighbor by a hydrogen bond of t h e
type found in the a-helix (see Fig. 2). A residue is in
the A state (coil) if its CO group is free. To fit t h e
theoretical transition curves to t h e experimental
values, it was assumed that s depends o n t h e temperature exclusively in accordance with the van’t Hoff
isochore [eq. (21)]. The enthalpy of transition AHu,
which was unknown initially, was varied together
with o until the best fit was found for all the experimental values according t o t h e criterion of least square
T h e values for the theoretical curves that describe the
results well (see Fig. 3) are: o = 2x10-4 and A H , = 990
cal/mole. T h e cooperative length NO is thus about
70 residues. T h e AHu value has also been confirmed
directly by calorimetric measurements. T h e most
Angew. Chem. internat. Edit.
1 Yol. 9 (1970)/ No. 6
reliable of the published results 136-391 is probably
AH, = 950 & 20 cal/mole, which was obtained by a
superior calorimetric method [361.
The surprising positive sign of A H , (increase in the helix
content with rising temperature) can be explained by the fact
that the influence of the temperature on the stability of the
a-helix is mainly indirect. In the solvent used (80 parts of
dichloroacetic acid/20 parts of 1,2-dichloroethane), the dichloroacetic acid competes for the hydrogen bonds of the
helix by adding t o the free CO and/or N H groups of the
residues in the coil state [40,411. The temperature coefficient
of this binding process determines the temperature dependence of the helix + coil transition[13,4*1. This is supported by the fact that the transition can be induced by even
a small increase in the dichloroacetic acid concentration.
Since cs M 10-4, ASK-ASW M 18 e. u., in agreement
with the order of magnitude expected for the freezingin of the degrees of freedom of two amino acid residues 1131. Though interactions other than hydrogen
bonds (e.g. hydrophobic interactions of the side
chains and binding of the solvent) and the change in
entropy on solvation and desolvation are disregarded
in this simplified picture, it undoubtedly shows the
principal cause of the cooperativity of the u-helix +
coil transition.
This picture also explains the validity of the simple
Zimm-Bragg model with only one cs for the a-helix.
The Zimm-Bragg theory may be regarded as a special
case of the general theory with cs = ob = &.
If the calorimetric enthalpy of transition is known,
According t o eq. (15), therefore, ok = csz = 1. It is
cs values can be conveniently estimated by means of
equations (lob) and (11) or (12) from the slope of a . in fact t o be expected that the formation of a helix (B)
nucIeus a t the ends is a t least as difficult as in the
single transition curve. It is assumed in this methinterior
of the chain. The unfavorable entropy balance
od [38,431 that the Zimm-Bragg model is valid, that s
in both cases, and a turn at the end of the
depends in a simple manner on the temperature in
of the stabilizing hydrogen bonds. In
accordance with the van’t Hoff isochore, and that
of a coil (A) nucleus is much
NO. Whereas the last of these conditions can
in the interior, since four
usually be satisfied by the synthesis of high molecular
hydrogen bonds have to be broken in the interior to
weight polyamino acids, the other assumptions need
form a coil nucleus, whereas only one hydrogen bond
not be correct for every system (see Section 3.2).
is broken at the end of the chain.
For the molecular interpretation of CS, Zimm and
The validity of the simple entropy-based explanation
Bragg assumed that the difficulty of nucleation is
of cs can be checked experimentally. Thus according
mainly due to entropy. In the growth step, i.e. in the
to eq. (23), cs should be independent of the external
addition of a residue to an existing helix section (see
conditions, whereas it must depend on the temperature
Fig. 2, bottom), the binding enthalpy AHw of a
whenever an interaction enthalpy is involved (as in
NH...OC hydrogen bond is gained, and the energy
see Section 3.2). It has unfortunately
T A S ~must be supplied. ASw is essentially the
possible to separate the
entropy lost when an amino acid residue loses the free
G on the temperature,
rotation about the HN-C, and C,-CO bonds [see
the polymer concenformula ( I ) ] on transition from the coil to the helix
tration from one another and from possible systematic
conformation. On formation of the first hydrogen
errors in the determination of cs (see above and Secbond corresponding to those in an a-helix (see Fig. 2,
tion 3.2) 138,431. A second conclusion from the simple
top), the same enthalpy is gained as in growth:
entropy explanation is that cs should also be independent of the side chains of the polyamino acids forming
the u-helix. Several interesting results have been
obtained in this connection.
(The subscripts refer to nucleation and growth.) How-
ever, the entropy loss ASK is three times as large as
for growth, since three residues are fixed in a helix
conformation on nucleation. With eq. (22) it follows
from eq. (19) that
1361 T . Ackermann and H . Ruterjans, Z . physik. Chem. N. F. 41,
116 (1964).
1371 F. E. Karasz, J . M . O’Reilly, and H . A. Bair, Nature (London) 202, 693 (1964).
[38] F. E. Karasz and J . M . O’ReiIly, Biopolymers4,1015 (1966).
1391 C . Ciacometti, A . Turolla, and R. Boni, Biopolymers 6, 441
1401 W. E . Stewart, L. Mandelkern, and R. E. Click, Biochemistry 6 , 143 (1967).
[41] M . Stake and J . N . Klotz, Biochemistry 5 , 1726 (1966).
[42] G . Ciacometti in A . Rich and N . Davison: Structural Chemistry and Molecular Biology. Freeman, San Francisco 1968,
p. 67.
1431 T . Ackermann and
(I. E.
Neumann, Biopolymers 5 , 649
Angew. Chem. internat. Edit. J Vol. 9 (1970) J No. 6
The solvent-induced helix + coil transition of another
derivative of polyglutamic acid with uncharged side
chains, i. e. poly-Ns-( 3-hydroxypropyl)-~-glutamine
[R in formula ( I ) = (CH2)z-CO -NH-(CH&-OH],
in formic acidlwater could be described by a cs value
of 3x10-4[441. A similar D value (1.5~10-4) has also
been found for poly-E-benzoxycarbonyl-L-lysine[R =
( C H ~ ) ~ - N H - C O O - C H ~ C , ~ H1451.
S ] The good agreement of the cs values indicates that the Zimm-Bragg
hypothesis, according to which the cooperativity is
independent of the side chains, is correct. The solvent
evidently also has only a small effect on CS,since the
polymers mentioned were examined in very different
On the other hand, a much lower cooperativity was
found for polyglutamic acid [R = (CH2)2-COOHl,
[44] N . Loton, M . Bixon, and A . Berger, Biopolymers 5 , 69
[451 M . Cortijo, .4. Roig, and F. C . Blanco, Biopolymers 7 , 315
which contains free and therefore ionizable COOH
groups: G = 3x10-3, cooperative length Now16 [46,471.
At low p H values when the COOH groups are undissociated, the helix is stable. However, it is broken
up at higher p H values by electrostatic repulsion
between the COOQ groups. The entropy of the resulting more disordered state is nevertheless undoubtedly
lower than that of the uncharged coil, since repulsion
between the side chains greatly restricts the mobility
of the residues. Since the difference between ASK and
AS, is thus smaller, a lower cooperativity is to be
expected, in agreement with the entropy-based explanation of G [eq. (23)].
A high cooperativity of non-entropic origin is observed in a completely different type of transition
between two ordered helical conformations of the
polyamino acid poly-L-proline.
3.2. The Helix I
+ Helix I1 Transition of
Poly-L-proline (3) can exist in two forms, I and I1
(see Fig. 6), the conformations of which are known
from X-ray structure analyses148.491. Form I is a
and are shielded to some extent by the proline rings.
Form I1 is a steep left-handed helix with all the peptide
linkages in the trans configuration. The CO groups
are almost perpendicular to the axis, and are freely
accessible to the solvent.
Both forms are sterically stabilized by the bulky rings.
Intramolecular hydrogen bonds such as occur in the
cc-helix cannot occur here, owing to the absence of the
peptide N H groups. Which of the two forms is more
stable depends on the interactions with the solvent.
Thus form I is more stable in n-butanol, and form I1
in benzyl alcohol or trifluoroethanol. It has been
shown spectroscopically that all three alcohols form
hydrogen bonds with the CO groupstsol. The relatively
strong H-bond donors benzyl alcohol and trifluoroethanol prefer the open form II[51], and so displace
the I + I1 equilibrium in favor of this form.
The reverse transition I1 + I can be induced by
addition of n-butanol to benzyl alcohol or trifluoroethanol, and can easily be followed polarimetrically
(Figs. 4 and 7). We again find the dependence of the
transition on the chain length that characterizes
cooperative transitions, but a new feature, which was
not seen in the helix + coil transition, is an intersection of the curves (Fig. 4), which, as mentioned in
Section 2, cannot be explained by the simple ZimmBragg model. There is no definite intersection in benzyl
alcoholln-butanol (Fig. 7), but the Zimm-Bragg
theory does not lead to a satisfactory description even
shallow right-handed helix with all the peptide bonds
in the cis configuration. The CO groups in this compact helix are almost parallel to the axis of the helix,
n-C,HgOH I%]-
Fig. 7. Solvent-induced transition of poly-L-proline as in Fig. 4, but
in benzyl alcoholln-butanol (v01.-%). Experimental values for N = 14
N = 33 (A),
N = 90 (A),
and N = 217 (0).
The curves were
Fig. 6. Space-filling models of the I helix (top) and of the I1 helix
(bottom) of poly-L-proline. The small white spheres and the large gray
spheres are hydrogen and oxygen atoms respectively. The peptide nitrogen atoms appear as gray triangles. The carbon atoms are black [cf. (311.
[461 R. L. Snipp, W. G . Miller, and R. E. Nylund, J. Amer.
chem. SOC.87, 3547 (1965).
[47] D . S . Orlander and A . Holtzer, J. Amer. chern. SOC.90,
4549 (1968).
[4S] E . Katchalski, A . Berger, and I . Kurtz in G . N . Ramachandran: Aspects of Protein Structure. Academic Press, New York
1963, p. 205.
[49] L . Mandelkern in G. D . Fasman: Poly-n-Amino Acids.
Marcel Dekker, New York 1967, p. 675.
On the other hand, an attempt to describe the transition with the aid of the general equilibrium theory [351
(Section 2) was successful[141 .The I state of poly-Lproline was equated to the B state of the theory with
no restriction of generality. To match the results, in
contrast with the procedure described in Section 3.1,
no definite function was given for the dependence of s
on the composition of the solvent, but use was made
[50] H . Strassmair, J. Engel, and G. Zundel, Biopolymers 8 ,
237 (1969).
[ S l ] H . Strassmair, S . Knof, and J . Engel, 2. physiol. Chem.
350,1153 (1969).
Angew. Chem. internat. Edit.
VoI. 9 (1970)/ No. 6
of the facts that o, F‘, and P” should be equal in one
solvent system and that there is one s value for each
solvent composition q.
Theoretical transition curves that best describe all the
experimental values for a solvent system on the basis
of the least square deviation criterion can be found
by an electronic computer with variation of Q, p‘,
and P’’, using a special program (Gaushaus 1521). This
method of fitting has the advantage that the s function
not only may be unknown, but is actually obtained
from the experimental transition data (Fig. 8). The
Fig. 8. Dependence of the equilibrium constant for the growth step of
the I
IT transition on the volume fraction of n-butanol in trifluoroand benryl alcohol (A).
ethanol (0)
nucleation parameters determined by this method are
unaffected by possible errors in the selection of an s
function. The experimental results agree with theory
to within the limits of error (see Figs. 4 and 7). The
best values of o, p’, and P” for the theoretical curves
in Figures 4 and 7 are given in Table 1.
Table I . Nucleation parameters for the helix I
as found by fitting t o the experimental values.
Solvent system
Benzyl alcohol/
e helix I1 transition
I P”
(1.0 & 0.5) 10-5
(0.5 i 0.2) 10-5
+ 4) 1 0 - 4
(1.5 f 0.6) 104
The very small values of Q point t o a very high cooperativity. A nucleus is less probable by a factor of
about lo5 than an unbroken sequence having the
same conformational states. The cooperative length is
about 320. According to eq. (18), the unfavorable
(positive) free energy that occurs in addition to the
energy between the same states on nucleation is
Since o is the same for both solvent systems (Table l),
the explanation of AG, must be sought in the internal
properties of the polyproline chain. It may be assumed
1521 We a r e grateful t o the Computer Center of the University
of Wisconsin, Madison, Wisc. (USA), for kindly supplying t h e
Angew. Chem. internat. Edit.
that the entropy component in AGO is very small,
since the helices I and I1 and molecules containing
both forms are very rigid and do not differ greatly in
their degrees of freedom. Thus unlike in the helix +
coil transition, AG, = AH, must be interpreted here
as the enthalpy of the interaction between the residues.
A large part of AH, is due to repulsion between CO
groups, which occupy unfavorable positions in relation
to one another a t the boundary -.II-I--(the N-terminal
end of the chain being written on the left). The bulky
model of the I nucleus in a sequence of I1 states
(Fig. 9) shows, in addition to other contacts, a very
Vol. 9 (1970)
/ No. 6
Fig. 9. Polyproline I1 helix interrupted a t the very top of the drawing
by a single cis linkage. The imino end is on the left. On the left-hand
side of the nucleus, i.c. at the . . 11-1 . . boundary, electrostatic repulsion occurs between two CO groups indicated by crosses.
close proximity of two carbonyl oxygen atoms having
a negative charge of about 0.4 e. Calculation of the
electrostatic [14,531and van der Waals interactions [14J
at the boundary leads to a AH, value that agrees well
with the value calculated from G by means of eq. (18).
The electrostatic interactions make the larger contribution 1141, and have a considerable range. The
apparent conflict between this fact and the success of
the theory, which takes into account only interactions
between nearest neighbors, is resolved as follows.
Owing to the high cooperativity the length of the
unbroken sequences of the same state in a chain is
generally much greater than the range of the interactions. The latter can therefore be formally described
as interactions between the nearest neighbors at a
boundary between two sequences.
In contrast with the difficulty of nucleation in the
interior of a chain (expressed by o), the ratio of the
difficulties of nucleation for the two forms at the
chain ends (P’ and F”) is strongly dependent on the
solvent (Table 1). A possible explanation of this is e . g .
that binding that depends on the conformation of the
terminal residue occurs between the solvent and the
amino or carboxyl end groups. Binding of this type
and any other interaction of the solvent with the
terminal residues influences (3’ and p” via the terms
G i , Cg, GX, and Gg in eq. (17). The difference
between the interaction energies a t the ..I-II.. and
..II-I.. boundaries, which is also involved here, as well
as the loss of stabilization energy in the terminal
[53] G. Holzwarth and K . Backman, Biochemistry 3 , 883 (1969).
residues as a result of the absence of neighbors, can be
calculated from the molecular data for poly-L-proline [14,53,541.
However, since no detailed information about the
influence of solvents is available, the p values cannot
yet be calculated. On the other hand, the dependence
of the s parameter on the composition of the solvent
can be quantitatively described with the aid of the
concept that the two forms are stabilized mainly by
bonding of the solvent components X (benzyl alcohol
IY 4
Scheme 1.
or trifluoroethanol) and Y (n-butanol) to the peptide
CO groups of the proline residues. The complete
binding system (Scheme 1) leads to the relation
In this equation, SO is the equilibrium constant of the
growth step of the free residues. This gives the ratio
of I to I1 in the absence of interactions with the solvent,
and has been estimated to be about 2 from calculations
of the conformational energies of the I and I1 helices.
K and a are the binding constants and the activities
of the solvents. The dependence of the activities on
the solvent composition and some of the binding
constants have been determined experimentally. The
experimental s functions can be satisfactorily described
by eq. (25). Thus molecular interpretations have been
found for all the theoretical parameters (a, p’, p”,
and s) in the case of poly-L-proline 1141.
It has already been mentioned that the use of a van’t
Hoff isochore for the temperature dependence of a
transition can lead to errors in the determination of
the nucleation parameters (see Section 3.1). Consideration of eq. (25) shows that for transitions involving binding processes (which also include the cc-helix +
coil transition), SO and all the binding constants may
depend on the temperature, with the result that the
dependence of s can no longer be represented by an
equation of the type (21). Matching of the parameters
without the use of assumptions is therefore also preferable in other cases.
The difference in the free energy that is required to
convert poly-L-proline completely from one form into
the other is also important. It can be seen from
Figures 4 and 8 that for N > 100, a change of about
20 % in s is sufficient for about 95 %transition. This
small change in the equilibrium constant corresponds
to an energy change of about 0.1 kcal/mole. Thus this
small energy difference is enough to stabilize one conformation o r the other. In the case of poly-L-proline,
1541 A . J . Hopfnger and A . G . Walton, J. macromolecular Sci.,
Physics B 3, 171 (1969).
it is supplied by the binding of the solvent. This strong
dependence of the conformation on the free energy
(over a small range of the external parameters) is a
general property of cooperative systems. Consequently
proteins and nucleic acids have a stable conformation
even when the energy of their individual segments is
only slightly more favorable than in the disordered
3.3. The Double Helix-Coil Transition of
Nucleic Acids
The DNA double helix (Fig. 1) is stabilized by hydrogen bonds between bases of different strands and by
the stacking energy A&. between the bases of each
strand [551. AG,t., which is particularly important in
aqueous solution, consists of a large hydrophobic
component and special interactions between the
aromatic systems of the bases1561. The double helix +
coil transition (Fig. 5), which can be induced e . g . by a
temperature rise and can be readily detected spectroscopically by means of hypochromism in the UV, has
a chain length dependence that is qualitatively similar
to that of the a-helix + coil transition.
A system that has been extensively studied is that of
polyadenylic acid (poly-A) [15,161, the double helix of
which is basically the same as the Watson-Crick
double helix 1571. The theoretical description of the
transition was possible for N < 11 by transfer of the
Zimm-Bragg model with only one nucleation parameter to the double helix. The nucleation step, with
the equilibrium constant 0-s, is bimolecular in this
“zipper model” (see Fig. 10). Even better results are
obtained by the “staggering zipper model”, in which
states with mutually staggered chains are also taken
into account. The partition function was obtained for
this model by combinatorial analysis, in which particularly unlikely states were omitted from the very
beginning for simplicity. By fitting the theoretical
curves to the experimental transition curves for N = 2
to 11 115,161, a a value of 2x10-3 I/mole was obtained,
taking into account the dependence of the hypochromism on the chain lengthr151. In this system we
again find a new molecular cause of the cooperativity.
The initial coming together of two bases of different
strands (nucleation) is more difficult by AGst. than
the growth step, since the stacking energy appears
only when a base pair is added to an existing base pair
(see Fig. 10). The contribution of the hydrogen bonds
is the same for the nucleation and the growth steps,
since the number of bonds formed is always the same.
The entropy loss probably also differs by only a small
amount of the order of RT In 2, this difference being
due to the bimolecularity of the nucleation. It must
otherwise be assumed that the number of confor[55] P. 0.Ts’o in B. Pullmann: Molecular Association in Biology. Academic Press, New York 1968, p. 39.
[56] H . DeVoe in S . N . Timashefland G . D . Fasman: Structure
and Stability of Biological Macromolecules. Marcel Dekker
New York 1969, p. 2.
1571 A . Rich, D . R. Davies, F. H . C . Crick, and J . D . Watson,
J. molecular Biol. 3, 71 (1961).
Angew. Chem. internat. Edit.
/ Vol, 9 (1970) / No. 6
Fig. 10.
Intermediate states
model of the double helix 72 coil transition of nucleic acids having a short chain.
. . . = hydrogen bonds, 111 = stacking interactions.
- base;
mational degrees of freedom frozen in is similar in
both elementary processes. Using these assumptions
and eq. (19), a stacking energy of about - 4.5 kcal/
mole can be estimated from the experimental rs value
for the short-chain poly-A. Values of -5 to -7 kcall
mole were obtained from the transitions of high
molecular weight polynucleotide systems r361 and of
natural DNA 1281.
At the large chain lengths of these last systems[z8,361,
as for multistrand helices in general [58,591, it is
necessary t o take into account a new type of statistical
parameter that does not occur in single chains. If N
becomes large, open rings of residues in the coil state
inside closed helical regions can no longer be disregarded (Fig. 11). The difficulty of nucleation then
depends essentially (apart from on the stacking energy)
on the probability of meeting of the two ends of a ring
7ii i i i i i
Fig. 11. Nucleation in a two-strand molecule having a long chain (see
caption of Fig. 10 and text).
(arrows in the figure). This is inversely proportional to
( j + l)a, wherej is the number of segments in the ring
(= number of unpaired residues in the two strands).
The exponent u was found by computer simulation to
be about 2C601. If the probability of ring closure is
taken into account, the theory predicts very sharp
transition curves even at relatively small N , and true
phase transitions of first and higher order occur for
long chains, does not exhibit a phase transition, but
gives a relatively flat melting profile. The reason for
this lies in the inhomogeneity resulting from the sequence [63,641. The double helix contains very stable
as well as less stable base pairs. It must be expected
in general that biopolymers in which different residues
occur in rotation will give flatter melting curves than
their conformational homologs built up from identical
3.4. Outlook for Complicated Systems
An example of a more complicated system is the triple
helix + coil transition of collagen1651. Collagen is a
relatively simple linear fibrous protein, in which three
peptide chains are twisted into a three-stranded helix,
for which simple model peptides with the same triple
helix conformation have been synthesized. The dependence of the conformational transition on the chain
length, which can never be determined in proteins, can
be studied in these peptides[66,67]. The example of
collagen is interesting, since here it is possible to
explain the cooperativity in a real protein.
In general, a detailed understanding of the conformational transitions of proteins will be difficult even in
cases where the steric structure in known. Since interactions occur in globular proteins in all directions
between the residues, the linear king model can no
longer be used, and the calculation of two-dimensional
o r three-dimensional models is possible only approximately and with a great deal of work.
A simplification results from the fact that most
protein transitions appear to be “all-or-nothing” processes [68,691. This conclusion was drawn in most cases
from the observation that for each degree of transition, all the experimental quantities that are influenced by the transition consist additively of the
2 2 [59-611.
Experimental evidence of phase transitions has actually been found for high molecular weight polynucleotides built up from chemically identical residues [61,621. Natural DNA, which has extremely
[58] J . Applequist, J. chem. Physics 50, 600 (1969).
[59] J . Applequist, J. chem. Physics 50, 609 (1969).
1601 F. T . Wall, L. A . HiNer, and W. F. Atchison, J. chem.
Physics 23, 2314 (1955).
[61] J . Applequist, J. chem. Physics 45, 3459 (1966).
[62] J . Applequist in G. N . Ramachandran: Conformation of
Biopolymers. Academic Press, New York 1967, VoI. 1, p. 403.
Angew. Chem. internat. Edit.
/ Vol. 9
1 No. 6
[631 D . M . Crothers and N . R . Kallenbach, J. chem. Physics 45,
917 (1966).
1641 T. Finch and D . M . Crorhers, Biopolymers 6 , 863 (1868).
1651 W. Grassmann, J . Engel, K . Hannig, H . Hormann, K . Kiihn,
and A . Nordwig in L. Zechmeister: Fortschritte d e r Chemie organischer Naturstoffe. Springer, Vienna 1965, p. 196.
[661 J . Engel, J . Kurtz, E . Katchalski, and A . Berger, J. molecular Biol. 17, 255 (1966).
[671 J . Engel in G. N . Ramachandran: Conformation of Biopolymers. Academic Press, New York 1967, VoI. 2, p. 483.
[681 R . Lumry, R . Biltonen, and J . F. Brandts, Biopolymers 4 ,
917 (1966).
[691 C . Tunford, Advances Protein Chemistry 23,121 (1968).
values for the two limiting structures (e.g. for the
completely ordered and the completely coiled state).
If intermediate states (e.g. semi-coiled molecules)
were present, this additivity would not be expected
for quantities that reflect complicated properties of
the molecules (e.g. for the viscosity).
The all-or-nothing behavior of the large protein molecules points to a very high cooperativity. The transitions of the natural nucleic acids can be described
better than those of proteins, since the nucleic acids
usually have linear structures. Even here, however,
their complicated and in most cases unknown sequence stands in the way of a quantitative understanding. On the other hand, the knowledge of the
fundamental properties of simple cooperative transitions will suffice in most cases at least for a qualitative understanding of complicated systems.
Received: August 14, 1969
[A 760 IE]
German version: Angew. Chem. 82, 468 (1970)
Translated by Express Translation Service, London
Pyramidal Inversion
By Arvi Rauk, Leland C. Allen, and Kurt Mislow [*I
Pyramidal inversion is discussed from the point of view of recent theoretical and experimental investigations in an attempt 10 provide a unified description of this process. Quantum mechanical studies of pyramidal molecules indicate that the origin of the inversion
barrier may be dependent on the degree of angular constraint. Effectsdue to the electronegativity of substituents on the inversion center, to the presence of adjacent lone pairs,
and to inclusion of d-type functions in the basis set are discussed. The utility and limitations of molecular orbital calculations, vibrational spectroscopy, microwave spectroscopy, direct kinetic measurements, and dynamic nuclear magnetic resonance (DNMR)
spectroscopy as means for determining barriers to pyramidal inversion are discussed in
context with a review of the highlights of experimental observations on the subject.
Ambiguities that arise in the interpretation of barriers determined by D N M R are explored in detail. Factors that afect the magnitude of inversion barriers are discussed
separately in four broad categories: steric effects; efects of conjugation (including
( p -d)n conjugation) and hyperconjugation; effects of angular constraint; and efects of
heteroatomic substitution. In the last category, critical reference is made to the question
of electronegativity vs. lone pair-lone pair repulsions, the problem of rotation vs. inversion,
and the role of d orbitals.
1. Introduction
A molecule that contains a tricoordinate atom whose
stable position is not in the plane defined by the three
atoms directly bonded t o it, may, in principle, exist in
t w o conceptually distinct conformations which are
related by a transposition of the tricoordinate atom
from one side of the plane to the other. We describe
such a molecule as being pyramidal and, without
specifying mechanism, define pyramidal inversion as
any process that effects the interconversion of the two
conformations. The conformers shall also be referred
to as invertomers.
We shall restrict discussion exclusively to those molecules in which pyramidal inversion occurs without
bond formation o r breaking, and which remain in
their electronic ground state throughout the process.
Dr. A. Rauk, Prof. L. C. Allen, and Prof. K. Mislow
Department of Chemistry, Princeton University
Princeton, N.J. 08540 (USA)
Two competitive modes occur. Classically, a vibrational mode leads to inversion through a coplanar or
near coplanar arrangement of groups about the inversion center; the observed rate of inversion depends
on the relative populations of vibrational energy levels
above and below the top of the potential barrier to
inversion. Non-classically, inversion can occur by
quantum mechanical tunneling: a particle in a potential well penetrates the walls of the well and therefore,
if the wall is of finite thickness, has a finite probability
of existing on the other side of the wall. The rate of
tunneling depends in a complex way on the mass of the
particle, the shape of the potential well, the height and
shape of the barrier and the energy of the particle [1,21.
I n practice, however, tunneling is a n important consideration only when a t least one of the ligands bound
to the central atom is hydrogen or deuterium, when
the invertomers are not diastereomers, and at tem[I] D . M . Dennison and G. E. Uhlenbeck, Phys. Rev. 41, 313
[Z] R. S.Berry, J. chem. Phys. 32, 933 (1960).
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