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Kinetics of Cooperative Conformational Transitions of Linear Biopolymers.

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Kinetics of Cooperative Conformational Transitions
of Linear Biopolymers
By Gerhard Schwarz and Jiirgen Engel"]
Cooperative conformational transitions of proteins and nucleic acids are of decisive importance to many processes of molecular biology, and particularly to their regulation. They
proceed via numerous interdependent elementary processes, and their kinetics are therefore
often complicated. They are frequently also very fast. However, kinetic analyses can be carried
out by chemical relaxation methods. The theoretical interpretation is comparatively simple
in the case of linear biopolymers. When the linear Ising model extended for kinetics was
applied to model peptides and polynucleotides, it provided an insight into the fundamental
principles of cooperative transformations.
1. Introduction
The biological function of nucleic acids and proteins is
largely determined by their specific conformation. This
function can be modified by a change in external conditions.
Conformational transitions are in fact important steps in
many reactions of molecular biology (examples: allosteric
regulation of enzyme activity", '1, replication of nucleic
acidsr3]).An important requirement here is that small
effects should induce large changes. This can be achieved
only by a positively cooperative interaction between the
elementary steps of the transformation, as a result of which
an elementary change that has already occurred at one
part of the molecule strongly promotes other elementary
changes that have not yet taken place.
We have alreadyf4]discussed the general characteristics of
cooperative conformational transitions of biopolymers for
the case of linear systems, since these present comparatively
simple conditions both theoretically and experimentally,
and so form a good basis for the interpretation of more
complicated systems. Equilibrium data on the helix-coil
transitions of some synthetic polypeptides or oligonucleotides built up from identical subunits and on the structural
transition between two conformations of poly-L-proline
were analyzed by means of the linear king model. The
experimental and theoretical foundations discussed in this
connection form the basis of the present progress report.
To obtain a complete understanding of cooperative conformational transitions, one must consider the kinetics of
such processes. Even in the case of linear model systems
consisting of chemically identical residues, however, one
may expect from the outset a very complicated reaction
course. As an example, let us consider the a-helix-coil
transition of a polyamino acid with N possible hydrogen
bonds between the CO and NH groups of complementary
peptide linkages. The transition to the random coil in this
case obviously requires N elementary steps, i. e. N breakages
[*I Prof. Dr. G. Schwarz and Prof. Dr. J. Engel
Abt. Biophysikalische Chemie, Biozentrum der Universitat
CH-4056 Basel, Klingelbergstrasse 70 (Switzerland)
568
of hydrogen bonds. Owing to the cooperativity of the
transition, these individual steps must proceed at different
rates according to whether or not adjacent hydrogen bonds
have already been broken.
Meaningful interpretations and evaluations of kinetic
experiments on systems involving very many reaction steps
cannot be based directly on the classical methods of
kinetics. The first or second order plots that are often
found in the literature are therefore of doubtful value
unless backed by theory.
Advantages are offered by the relaxation method, which
involves the kinetic study of the approach to a new equilibrium after a generally small disturbance of the old
equilibrium (e.g. by a temperature change). Relaxation
curves obtained in this way reflect in principle not only the
kinetics of the slowest unit process but also those of the
other intermediate reactions. Their theoretical calculation
for a postulated mechanism is facilitated by the fact that
the kinetic differential equations can be linearized for small
disturbances. However, the evaluation methods that have
been found suitable for simple reactions with one or two
stepsL5]
frequently cannot be used. It is nevertheless possible,
even in complicated cases, to establish the desired quantitative relation between a reasonable reaction mechanism
and the experimental curves by measurement and calculation of mean relaxation timesC6*'1.
Conformational changes may be so fast as to necessitate
the use of relaxation methods with very high time resolution. Thus in the case of the a-helix-coil transition, the
temperature jump method revealed only that the transition
time is evidently shorter than the heating time of the
measuring cell (approximately 1 microsecond)18.'I. Only
by still "faster" methods was it possible to determine
transition times in the region of lo-' to
seconds (see
Section 3.1).
Other transitions can be followed by more conventional
methods. Thus the denaturing time of very long-chain
deoxyribonucleic acids are in the region of seconds to
minutes, and the helix-I+heIix-II transition of poly-Lproline is even slower.
Angew. Chem. internat. Edit. 1 VoI. I 1 (1972) 1 No. 7
2. Theoretical Principles
The situation is simple in a one-step process, e. g .
2.1. Execution and Evaluation of Chemical Relaxation
Experiments
A
A kinetic analysis by chemical relaxation experiments
involves essentially three steps, i. e. the disturbance of a
stationary state of the reaction system (usually of the
chemical equilibrium),measurement of the time dependence
of the subsequent change in a quantity P that depends on
the concentration of the reactants (e.g . optical density,
optical rotation), and finally the interpretation of the data
in the context of an overall reaction. Whereas the first two
steps present at most technical and experimental problems,
complicated questions of mathematical theory may arise
in the third for a reaction consisting of many unit steps.
As a concrete example, let us consider an experimentally
fairly clear and versatile method, i. e. the temperature jump
method. On the basis of the well-known van? Hoff equation
for the temperature dependence of chemical equilibrium
constants, a chemical equilibhum is displaced in a definite
manner by a temperature change. This can be brought
about experimentally in approximately a microsecond,
e. g . by electric heating[’]. If the new equilibrium is established more slowly, the change in the quantity P as a function
of time can be followed as a relaxation curve on the screen
of an oscilloscope.
Similar curves are obtained with sudden changes in other
parameters that influence the position of the equilibrium,
such as pressure, electric field, solvent composition, and
pH. As shown in Figure 1, a relaxation function @ ( t )that
can be found directly by experiment is thus defined. This
must now be correlated quantitatively and unambiguously
with the underlying reaction mechanism.
k
2
k21
B where
dc
a
= k,,c,-k,,c,
dt
Because of the conservation of mass, c,+c,
is always
constant, so that only one independent concentration
variable is involved. This may be taken as e.g. cP The
variation of any concentration-dependent parameter with
time can then be calculated by solution of eq. (1b). We find
here
The relaxation time T, as shown in Figure 1, can be easily
determined graphically from the measured relaxation
function @(t).If the equilibrium constant K = k , , / k , , is
known, the individual rate constants can be obtained
directly from eq. (2b).
In more complicated reactions with more than one step,
the variation of the concentration of each reactant can still
be described by a differential equation. This includes the
rate contributions of all elementary processes that lead
to a change in the chemical state in question.
If the difference between the number of reactants and the
number of mass conservation conditions is n (which
corresponds to the number of “independent” reaction steps
obtained when all the steps that lead to reaction cycles
are struck out), the overall reaction is completely described
by n kinetic differential equations. Naturally, these may be
extremely complicated. However, they can be linearized
(elimination of square and higher concentration terms) for
small deviations from a time-independent reference state
(small perturbations). A closed mathematical solution of
the problem is now always possible in principle[61.This is
characterized by n time constants, the relaxation times
z z2, z3...5,. These constants depend in a definite manner
on the reaction mechanism, the kinetic constants, and the
concentrations of the various reactants. It should be noted,
however, that the individual relaxation times are not directly correlated with individual reaction steps. In particular,
for the relaxation function @ ( t )of a sudden disturbance of
the equilibrium, the theory gives
,,
:*
-o
UI
\
t-
\
Fig. 1. Schematic representation of relaxation curves in jump experiments.
a) After a change in temperature, pressure, or other external’parameter
that is fast in relation to the reaction time, the initial equilibrium
(measured quantity Po)is replaced by a new equilibrium (measured
quantity P).@(r) = 6P/6Po defines a (normalized) relaxation .function,
which is directly obtainable by experiment. The tangent tofO(t)at r = O
cuts the asymptote at t = ~ ’[see eq. (4)]. b) When log Q ( t ) (or log 6 P )
is plotted against time, the resulting graph in the simplest case is a
straight line with slope = 0.434.1/~,when @ ( t ) consists of a single exponential function. c) If there are two exponential functions with
sufficiently different values of i i and 7* (broken line), a curve with a
linear tail is observed. This curve can still be satisfactorily broken down
into its components. d) For several overlapping relaxation terms, an
unambiguous breakdown is no longer possible, but the average relaxation time T’ can be determined from the initial slope (dotted).
Angew. Chem. internat. Edit. 1 Vol. I 1 (1972) 1 No. 7
i.e. a sum of n exponential functions with the T~ as time
constants. The corresponding aplplitude factors pi do not
contain any direct kinetic quantities. They are, however,
dependent on the quantities measured and on the disturbing parameter. The pi therefore generally change if e. g. the
variation of the optical density with time is followed at a
different wavelength or if a sudden change in pressure is
used instead of a temperature change. However, such
manipulations have no influence on the zi.
The complete set of relaxation times q together with the
corresponding amplitude factors pi is known as the relaxation spectrum. A definite assignment of a relaxation
spectrum to an experimental curve is practically impossible
unless there are only a few T~ with very different orders of
569
magnitude and comparable pi (see Fig. 1 b-Id). If this
is not so, a kinetic analysis can be carried out with other
parameters that are directly obtainable by experiment. A
parameter that is particularly suitable for this purpose is
the mean (reciprocal) relaxation time T’I~], which is defined
by the following relation :
describe the growth of existing A and B sequences, the
equilibrium constant s being identical with the corresponding parameter of the Zimm-Bragg theoryI4I. In the nucleation and decomposition steps
..AAA..
(FA . . A B A . . w h e r e h = o . s
k,A
kDA
and
(4)
BAB.
(The relation with 0 is easily verified from eq. (3a).) As
indicated in Figures 1a and 1d, 7’ is given by the tangent
to the relaxation curve at time t = O (instant of the disturbance). Because of this relation, z* is theoretically also
relatively easy to obtaint6’, provided that an equilibrium
state of the system exists at t = O (as is usually the case).
Apart from a knowledge of the elementary processes and
their kinetic constants, one then requires only equilibrium
properties for the calculation of 7*. There is thus no need
to solve differential equations.
Very fast relaxation processes (7 5 1 psec) have so far been
investigated only by methods that make use of a periodic
disturbance in the form of a sine function, e.g. a sound
wave or an alternating electric field. In this way, small
fluctuations of the equilibrium state of the reaction system
are produced. If the vibrations are fast enough in comparison with the reaction time, the actual change in concentration of the reactants with time will lag behind its
equilibrium value. The theory then leads to an absorption
of energy in the region around the relaxation frequency
vR’ 1/(2m) (e. g. sound absorption when ultrasonic radiation is used, and ohmic or dielectric losses in an alternating
electric field).The chemical relaxation times can be obtained
by evaluation of corresponding measurements[61.
k
2 . . B B B . . where
k DB
b=s
kDB
the equilibrium constants 0 s and SJO are fixed by the
equilibrium theory, the cooperative parameter 0 < 1 showing how much more difficult nucleation is than growth
(a small 0 indicates great difficulty of nucleation, i. e. high
cooperativity). Since only the ratio of the equilibrium
constants is fixed by thermodynamics, one kinetic parameter remains open for each of the processes. In the general
case, special nucleation and decomposition steps for the
ends of the molecule must also be taken into account, since
only one nearest neighboring state exists in these positions“’].
In a chain with N segments, the sum of the molecules in
the up to 2Ndifferent conformational states must naturally
remain constant. Since this at first sight is the only mass
conservation condition, one should expect a practically
continuousspectrum 0 f 2 ~ -1relaxation times. Fortunately,
the problem can be further simplified to a greater or lesser
extent according to the particular conditions. For example,
the mean relaxation time can be calculated without special
difficulties with the aid of the equilibrium theory and
compared with experimental data.
In the common limiting case of very long chains with high
cooperativity, i. e. for
N b cooperative length N o
=
2.2. The Kinetics of the Linear Ising Model
As described earlier in some detailL4],the thermodynamic
behavior of cooperative conformational transitions of linear
biopolymers can be quantitatively understood with the
aid of the linear king model. In this connection, the elementary states of the individual segments corresponding
to the two extreme conformations are designated A and B.
As kinetic elementary processes of the transition, therefore,
we find reactions of the type
..XAY..
+ ..xnY..
(5)
at any point on the chain molecule; according to the basic
assumption of the Ising model, the rate constants also
depend on the two nearest neighboring states X and Y
each of which may also be equal to A or B. We must therefore distinguish between two types of elementary reactions.
The growth steps
..AAB..
. . B A A ..
570
..ABB..
k
. . B B A . . where k D
the effects at the ends of the molecule can be neglected141.
Moreover, in the transition range (where A and B are
present in roughly comparable quantities), the contributions of the processes (7) and (8) to the overall reaction become vanishingly small, since the “nuclei” ABA and BAB
exist in very low concentrations in comparison with AAB,
ABB, etc. The fraction of B segments, 0,will therefore
change almost exclusively by growth steps in accordance
with eq. (6), i. e.
where yAABand yABBare the fractions of A and B segments
respectively with the neighbors A (left) and B (right) (the
factor 2 arises because yAAB
=yBAAand yABB
=yBBA).
If the
equilibrium is suddenly disturbed at t=O, with the result
that s+s+6sand t h e e q u i l i b r i u m v a l u e 0 , - t ~ = 0 , + 6 ~ ,
then according to eq. (4) (with P = 0,
&Po= - 60)‘”-
=
Angew. Chem. internat. Edit. 1 Vol. I 1 (1972) 1 No. 7
(values with bars are equilibrium values). The thermodynamic theory enables us to express
and 6616s by 0
and O.Thus["]
The maximum relaxation time r&x thus occurs at the midpoint of the transition (0=0.5).
For N S N o , end effects and nucleation can no longer be
disregarded in the kinetics of the transition. With plausible
assumptions for the nucleation rates, T* is obtained as a
complicated function of the general form"01
steric position of the correct CO and NH groups must be
reached by suitable rotation about two intervening C-C
or C-N bonds (see Fig. 2). With an activation energy of a
few kcal/mole for the rotation, therefore, one may expect a
k , value of about 101o-lO1l s-'. Since the r~ values for
polypeptides are in the region of 10-4-10-2, depending
on the system, eq. (11 b) gives maximum relaxation times
of about
to
seconds for the helix-coil transitionlll, 121
//coilj
HpJ
(p', p" describe the end effects14.loJ).The function f
always
lies below the limiting function 4 0 (1- 0 )that is valid for
N b N o according to eq. (11a), i. e. the transition becomes
faster with decreasing chain length.
A calculation of the chemical relaxation behavior of cooperative transitions that goes beyond the average value
T* can be carried out relatively easily in some cases if use
is made of special properties of the system and special experimental conditions. Examples are the "all-or-nothing"
behavior of short-chain poly-t-prolines (see Section 3.2)
and oligonucleotides (see Section 3.3) and the "suppressed
nucleation" in poly-t-proline (see Section 3.2). More complete calculations of kinetic curves have also been attempted
for the case of long chains, with simplifying assumptions
for the kinetics of the elementary processes"3or by
computer calculationsI'6*l7).
It has recently become possible to place the theory on a
generally valid basis. It has been shown for any chain
length that the complete kinetics of the linear Ising model
are described by 4N - 5 independent, non-linear differential equations['81. For N - co (i. e. N + N o ) , the number of
equations decreases to four. Three of the relaxation times
defined in this way occur with vanishing amplitudes in the
transition range with sufficientlyhigh cooperativity ( N ob I),
so that the fourth must be identical with T*. In this limiting
case, which is of practical importance, the transition is
therefore described by a single relaxation time [as eq. (2a)],
despite the complicated reaction
3. Experiments on Simple Linear Model Systems
3.1. The a-Helix+Coil Transition
The experiments were carried out with sufficiently long
polypeptide chains ( N b N , ) to enable one to expect a
relaxation time in accordance with eq. (1I) in the transition
range. The essential elementary process of the transition
consists in the growth and degradation of helix fragments
within a polypeptide chain by formation and rupture of
hydrogen bonds at the junctions between coil and helix
regions. The rate constant k , describes the formation of
such a hydrogen bond. This formation does not require
any appreciable activation energyC2'], but a favorable
Angew. Chem. internat. Edit.
1 Vol. I1 (1972) 1 No. 7
Fig. 2. Transition of a peptide linkage from the coil state to the helix
state at a helix/coil junction with formation of a hydrogen bond characteristic of the a-helix between a NH and a CO group. The C-C and
C-N single bonds of the amino acid residue (indicated by arrows),
which allow free rotation in the coil state, are fixed by the formation of
this bond.
This estimate agrees well with the experimental data obtained so far. Sound absorption measurements on polyglutamic acid (see Fig. 3) show the maximum effect at
0= 0.5 and 15 MHzC2l1.
According to the theory, the additional sound absorption coefficient a per wavelength h (in
aqueous solution) produced by the helix-coil transition
should be
(p = density, u, =velocity of sound in the solvent, A V=
change in molar volume in the coil-helix transition, co =
concentration of the polypeptide in moles of monomer per
unit volume[24]).From v,=15 MHz, rmaris found to be
1 x l o w 8s. With 0=3 x 10-3[41,therefore, eq. (11b) gives
k,= 8 x 10' s- '. Moreover, from the maximum value of
ah, eq. (13) gives AV=0.5 cm3/mol, in agreement with
direct measurements of A V(221.
Similar experiments with poly-~-ornithine[~~~
also gave
T Lz~ ~ s. The measured dependence on the degree of
transition 0 confirms the formula (11a).
Measurements on poly-(y-benzyl-L-glutamate) were carried
out with the aid of a new chemical relaxation principle,
which is based on the displacement of achemical equilibrium
in the direction of stronger dielectric polarization by an
electric field["]. If the associated chemical relaxation is
much faster than the rotation of the dipoles involved in the
reaction, in addition to the usual dielectric relaxation
caused by rotation diffusion there is a further relaxation
effect at higher frequencies which reflects the chemical
571
O = O l 0.50.9
I
found that k , is 1.3 x 10'' s-', which is still within the
region of our earlier estimate. Another important result of
these experiments is the observation that an electric field
can directly influence a conformational transition. Calculation shows that a field strength of 200 kV/cm (such as
occurs on biological membranes) can cause an almost complete structural transition in the system discussed here in
times of about 1 psec. It is therefore conceivable that e.g.
nerve impulses may act directly on molecular processes by a
cooperative conformational change.
! I
I
L
I
1
L
5
6
7
PH
8
9
1
0
+
1
7
bl
t 5
Oh -
0 I
w
.
02 L
m
2
5
10
10'
v [MHzl-
Fig. 3. Ultrasonic measurements of the helix-coil transition of polyglutamic acid in aqueous solution (after Saksena et a[. [21]).The transition i s brought about (reversibly) by pH changes in the region p H = 4
to 6. The midpoint (0= 0.5) is situated a t about pH = 5.1.
a) According to eq. (13), Aa/v2 (Aa=sound absorption coefficient of the
solution minus that of the solvent, v=sound frequency) should have a
maximum at 0=0.5. The experimental data plotted show this clearly
for the frequencies 5.08 MHz and 26.2 MHz, whereas at 99.6 MHz, the
accuracy of the measurements is evidently no longer suffcient to show
the maximum. b) The sound absorption per wavelength (ah),, resulting
from the helix-coil transition (at 0= 0.5) is plotted on a double logarithmic diagram against the frequency. According to eq. (13) one should
obtain a characteristic curve fixed by only two experimental points in
l
( w = 2 ~ v A=constant
,
accordance with the function A w ~ / ( +w2?)
factor), which has a maximum at w = 2 x v R = I/T.
The values found here
for vRand A / 2 are indicated. They give 5 and, with the aid of eq. (13),the
reaction volume A 1/:
relaxationrz4! These conditions are satisfied for poly-(ybenzyl-L-glutamate). In the transition range, the polypeptide chains consist of a sequence of coil regions and
helix fragments, the latter having a considerable electric
dipole moment. The applied electric field displaces the
equilibrium between helix and coil regions in such a way
that the helix dipoles increase in the field direction, while
they decrease in the opposite direction. Since the rotation
diffusion of the helices, because of their length, is much
slower than the chemical relaxation of the helix growth,
a relatively fast dielectric polarization occurs in addition
to the rotational relaxation process. As expected, this effect
is found to reach a maximum at the midpoint of the transition (see Fig. 4), as in the case of the sound absorption. The
@-dependence,which was also measured, agrees well with
eq. (11af'251.The effect was recently also found for the
similar system poly-(P-bemybaspartate) in rn-cresol[26!
s found for poly-(y-benzyl-LFrom the rfmaxof 5 x
glutamate) and the value 0=0.4 x
that is valid for
this polypeptide at the high concentrations used[251,it is
572
0
02
0.4
0.6
08
1
0Fig. 4. Increment (maximum) of the dielectric constant (E!&) due to the
helix-coil transition of a concentrated solution of poly-(y-benzyl-Lglutamate) as a function of the degree of transition 0 (after [25]). The
values are calculated from the dielectric data. The curves shown
correspond to the calculated course.
TI
An interpretation of double signals of the C,H and NH
protons in the nuclear magnetic resonance as resulting from
a helixscoil exchange with a value of only I O - ' s for
T ~"I, ~which
~ contradicts
*
these results, has recently been
refuted by a different interpretation of the signal^[^^^ 301.
3.2. The Helix I+Helix II Transition of Poiy-L-proline
This transition between two ordered conformations is very
c~operative'~].
A nucleus of the I form (cis state of the
peptide linkage) inside a I1 helix (trans state of the peptide
linkages) is about lo5 times less probable at 0=0.5 than
an uninterrupted helix, and vice versa (0z lo-', N o z 300).
This is due to energetically unfavorable conditions at the
1/11and II/I junctions. These therefore occur only relatively
infrequently. For short chains, even the content of molecules containing only one junction is small in relation to
the concentration of molecules in the pure forms (without
such junctions).
Table 1 shows that this "all-or-nothing'' case is still largely
satisfied for N = 20. The probability that a junction occurs
somewhere in the chain naturally increases with increasing
chain length. It is found quantitatively (Table 1) that with
N = 100 (i. e. about 1/3 of the cooperative length N o ) , the
content of molecules having one junction can no longer be
disregarded, whereas the content of nuclei (two junctions)
Angew. Chem. internat. Edit. / Vol. I1 (1972) / No. 7
is still very small (“suppressed n~cleation”~~]).
This simplification finally disappears above about N = 200.
A relaxation process of the transition can be initiated by a
sudden change in the solvent composition. Because of the
fairly slow elementary steps of the cis$ trans isomerization
of the peptide linkage (with an activation energy of 20 kcal/
mole), the course of the transition with time can be measur-
Relaxation curves for chains of medium length (e.g.
N 100% N,/3), for which the “suppressed nucleation”
proceed formally
approximation is valid (see Table I),
according to a zero-order rate law for a certain time if the
initial equilibrium lies in the transition range (see Fig. 5 b).
Table 1. Fraction of polyproline molecules that contain 1/11 or II/I
junctions in relation to the fraction of the pure forms I.. . I and I1 ... I1 in
benzyl alcohol/n-butanol at 70°C and 0 - 0 . 5 (calculated from the
equilibrium data [4]),
Chain contains
20
only 1
one junction
two junctions
only I1
1
0113
0.003
1
Chain length N
100
200
1
0.6
1
1.22
0.09
1
0.37
1
I
0.05;
8
4
12
t tminl
1
I
16
20
ed easily and very accurately (by polarimetry). The system
thus becomes an ideal model for checking detailed predictions regarding the kinetics of cooperative transitions.
Some interesting special cases will be discussed first.
The kinetics of the “all-or-nothing” case are particularly
simple. Since the system consists almost exclusively of
chains with practically identical elementary states, we
formally obtain a one-step process with first order reactions, e. g. for N =4 :
I1 I1 I1 I1
*Xk I I I I
t
0.6;”
40
”
80
i [minl
(14)
This process must naturally proceed via intermediate states.
Only those with at most one junction need be considered
(since all others are even less probable). If we also assume,
for simplicity, that all growth steps (including those at the
ends) are described by the same rate constant, the complete
kinetic mechanism for (14) is found to be
1.0
I
t
’
120
’
”
160
Go-
cl
0
.
OD-%.
I
t
0
I
rn
0
.
0
.
-
I
I
I
10
20
t [hl
I
I
30
Fig. 5. Three limiting cases of the kinetics of the I e I I transition of polyL-proline (from [31]).
a) Semilogarithmic plot of two relaxation curves in the “all-or-nothing”
range N=18. Change from 90 to 60 vol.-% of n-butanol in benzyl
alcohol, whereupon 01=0.5 +0.1
and 0.1 -0.5 -A-).
The f
values differ because of the different starting conditions [corresponding
to different s values in eq. (16)J b) Course of the transition with time
(linear plot) in the case of suppressed nucleation ( N = 217, change from
0,=0.34 to 0.71). The reaction rate remains constant for some time.
c) Sigmoid build-up curve of the transition for @,=I
-0 at N = 113.
Experimental values 000,
calculated values -.
(-o-)
(a;,
o;’,
o;~
0;;
, =nucleation parameters at the chain ends[‘]).
Since all the intermediate states Z are present in very low
concentrations, the well-known steady-state condition
dcz/dt=O may be assumed. We thus find from eq. (15) for
the rate constants that occur in eq. (14)r311:
(s = kF/kD). According to eq. (2), the chemical relaxation of
eq. (14) should be described by a single relaxation time
T = l/(i
+k). Corresponding behavior was in fact found
experimentally for N <20 (see Fig. 5a).
Angew. Chem. inlernat. Edit. 1 Vol. 11 (1972) 1 No. 7
The lack of dependence on the concentration of the I or I1
states still present is explained by the fact that under the
above conditions, the number of actual reaction sites, i. e.
of 1/11 and II/I junctions, remains constant for some time.
The transition occurs essentially by a displacement of these
sites in growth steps. New junctions are formed and
disappear only very slowly and in comparatively small
573
equilibrium investigation^[^. 411, the “zipper model” (see
Fig. 6, also Fig. 10 inL4])proved useful for the interpretation
of the kinetic data. For the short chains chosen by Pijrschke
and Eigen ( N I 18), the all-or-nothing approximation is
valid :
numbers. This effect therefore becomes important only
toward the end of the reaction. The initial rate in Figure 5 b
is thus proportional to the product of k,- k,=(l - ‘ , s ) k ,
and the number of junctions. If the latter is known for the
initial equilibrium ( e . g . from Table 1) as well as s for the
final state, k , can be determinedr3’!
Another instructive special case is where the initial state
chosen for long chains is outside the transition range, so
that only one practically pure form exists at first (0=0 or 1).
The fast growth steps can now occur only after the first
junctions have been formed by nucleation, which is a very
much slower process. The result is sigmoid build-up kinetics
(see Fig. 5 c). Experimental verification proves the validity
of the special assumption in this case, namely that the rate
constant of the nucleation is considerably smaller that that
of the growth step. Note that the small equilibrium constant
of the nucleation, c.s, would also be explained by a correspondingly fast reverse reaction (destruction of a nucleus).
i,
A N + U N F$ (Au),
k
(Coil)
Only one relaxation time is in fact found, and this varies
with the concentration of chains in the coil form as follows
Whereas calculation of the total relaxation curve was at
least approximately possible in the special cases described,
the kinetic analysis was based in the general case on mean
.
1081 m o l - ’ s - ’
&-Coil
f
Us-Coil
2.1 0% I
(Double helix)
according to the relation^'^] for bimolecular single-step
reactions. As in the “all-or-nothing” case of polyproline
(see Section 3.2),i and can be expressed by the elementary
--
- z,
10’~-l
107~-1
2 2
2.10~~
I -
2 3
2.106s-‘
1
107~-l
__I
2.106s-
2 4
T---
(AU)5-Double helix
2.1 0%- I
Fig. 6. Top: Zipper model of the formation of a double helix from a poly-A and a poly-U strand for N = 5. Helices in which the base pairs
(..) have not yet all been formed occur in small concentrations as intermediates (2, to Z,) (“all-or-nothing” case).
Bottom: Numerical values of the rate constants.
relaxation time^[^'-^^'. As in the a-helixecoil transition,
the theoretically[’o1required maximum of T* in the region
of 0= 0.5 was found. It was possible to determine zkaxfrom
the dependence of z*on the chain length. From this and eq.
(11b), the rate constant for the growth of cis (helix I) sequences was found to be k,= 1.5 s-’ (at 70°C in 60 v01.x
n-butanol in benzyl alcohol), a value that agrees satisfactorily with that obtained by evaluation of the special
cases described above. Similar values have in fact also been
obtained completely independently for the cis+ trans
isomerization of the amide linkage in N,N-dimethylacetamide by measurement of the nuclear magnetic resonanCe134- 361 and from IR datar37r38!
kinetic constants of the scheme in Figure 6 assuming
steady-state conditions for all intermediate states. These
are thus obtainable experimentally by measurement of the
dependence of 2 and 5 on the chain length.
However, the discussion here is rather different from that
in the case of polyproline. A satisfactory interpretation of
the “all-or-nothing” kinetics of poly-L-proline was achieved
by the assumption that in the interior of the chain, all
elementary steps have equal rate constants [ k , and k , in
eq. (15)], and that only one elementary step, the nucleation
step, has a differentconstant. On application of this simplest
assumption to the formation of the double helix, 2 would
be equal to the elementary rate constant of the nucleation
step k,,, since by analogy with eq. (16a),
3.3. The Double Helix+Coil Transitionof Oligonucleotides
The formation and decomposition of short double helices
of polyriboadenylic acid (poly-A) and of polyribouridylic
acid (poly-U), respectively, with poly-A was investigated
by Porschke and Eigenr39.401 by the temperature jump
method and by rapid mixing in a flow apparatus. As in the
5 74
as ~ $ 1under the experimental conditions chosen by
Porschke and Eigen. It was found, however, that eq. (19),
and hence the simplest reaction scheme on which it is
Angew. Chem. internat. Edit. 1 Vol. 11 (1972) 1 No. 7
based, cannot be valid in this case, since measurement of
the temperature dependence of the reaction gave a strong
negative activation energy E, for /?, whereas only positive
activation energies are possible for rate constants of elementary reactions, i. e. for k o l . An increase in the rate with
falling temperature points to one or more fast equilibria
that precede the slower subsequent steps. The AH values
of these equilibria (which are evidently negative here)
contribute to the measured activation energy. It was concluded from the quantitative data (for AH and E,) that 2 , :
the first intermediate complex of two chains (with one base
pair) though rapidly formed, also decomposes rapidly. The
next intermediate state with two base pairs was also found
to be labile. A stable nucleus results only for the next step
(k23)-
The rate constants for the poly-A/poly-U ~ysteni[~’]
at the
midpoint of the transition, which agree best with the experimental data, are entered in the lower line of Figure 6.
It can be seen that the formation and dissociation of a basc
pair takes place very rapidly (within
to l o d 6 s).Nevertheless, the dissociation of a long section of helix can take
considerable times (about I s for N = 15), since each opened
pair in the interior of the chain closes again on average
two to five times before the next pair is opened. This follows
from the ratio of the rate constants of the propagation steps
~=2--5.
Ezgen1431has pointed out that the mutual recognition of
the nucleic acid codon and anticodon of more that three
nucleotides would be much too slow for biosynthesis,
though increased specificity could be achieved i n this way.
The actual codon length of three units agrees with the
length found for a stable pair; the time for the formation
and redissociation is approximately
s.
4. Outlook for More Complicated Systems
The rate of coiI formation of natural nucleic acids, because
of their very great chain lengths, is determined by an effect
that is negligible for short chain lengths. Since the chains
in the helix are strongly twisted together, a considerable
hydrodynamic resistance must be overcome in the coiling
of long chains. C r ~ l h e r was
s ~ ~the
~ most
~
successful in the
theoretical treatment of this case, but because of various
complications, no final conclusion has yet been reached‘451.
The situation is even less clear for proteins, because of their
more complicated three-dimensional structure. With very
high cooperativity, one can expect an “all-or-nothing”
mechanism with only two states that are appreciably
occupied, and a correspondingly slow transit~on‘~’~.
The
reader is referred here to the progress report by P ~ h i [ ~ ” .
To conclude, the following observation is valid for all
cooperative systems. The greater the cooperativity of a
system. the greater Is the stabjlity e.g. o f a protein structure
due to interactions that are very weak in some cases, and
the more specific e. g. an enzyme can be. However, the rates
of cooperative processes, e. g. of a conformational transition
that is.important to allosteric regulation, decreases. It must
be assumed that in the evolution of biologicaI macromoieAngew. Chem. internat. Edrt. 1 Vol. 11 ( 1 9 7 2 ) 1 No. 7
cules, a compromise has been reached m many cases between stabihty and dynamics.
Received: May 21, 1971 [A 886 1E]
German version: Angew. Chem. 84,615 (1972)
Translated by Express Translation Service, London
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