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 [I] J . MOnod, J . Wyman. and J . P. Changeux, J . MoI. Biol. I2,88 (1y65). [2] M . Kirtley and D. E. Koshland, J Biol. Chem. 242, 4192 (1967) [3] J . D. Warson: Molecular Biology ofthe Gene. Benjamin, New York 1965. [4] J . Engel and G . Schkarz, Angew. Chem. 82, 468 (1970); Angew. Chem. internat. Edit. 9, 389 (1970). [5] M . Eigen and L. DeMaeyer in S L. Friess, E . S. Lewis, and A . Weissberger: Techniques o f Organic Chemlstry. WiIey, New York 1963, Val. VIII/2, p. 895. [6] G. Schwarz, Rev. Mod. Phys. 40,206 (1968). [7] G. Schwarz. Habiiitationsschrift, Technische Universitiit Braunschwerg 1966. [8] R . Lumry. R . Legare, and W G . Miller, Biopolyrners 2,489 1196J1. [9] E. Hatnori and H . A. Scheraga, J. Phys. Chem. 71,4147 (1967). [lo] G Schwarz, Biopolymers 6, 873 (1968). [I 13 C . Schwnrz, Ber. Bunsenges. Phys. Cbem. 68,843 (1964). [I21 G . Schwarz. J. Mol. Biol. 11, 64 (1965). [13] 0.A. McQunrry, J . P. McTague, and H . Reiss, Biopolymers 3,657 (1965) [I41 J B Keller, J Chem. Phys. 37, 2584 (1962); 38, 325 (1963). [is] A. C Pipkin and J . H . Gibbs, Biopolymers 4, 3 (1966). [I61 M . E . Craig and D.M . Crothers, Biopolymers 6, 385 (1968) [I71 A . Silberberg and R . Simha, Biopolymers 6, 479 (19681. [I81 G. Schwarz, Ber. Bunsenges. Phys. Chem. 75, 40 (1971). 119) G . Schwarz, J . Theor. Biol., in press. [20] K . Bergmann, M . Eigen, and L DeMaeyer. Ber Bunsenges. Phys. Chem. 67, 819 (1963). [21] ?: K . Saksena, B. Michels, and R. Zona, J Chim. Phys 65. 597 (1968). [22] H . Noguchi and J . 7: Yang, Biopolymers I , 359 (1963). [23] G . G . Hamnws and P . B. Roberrs, J . Amer. Chem. SOC. 91, 1862 (1969). [24] G. Schwarz, J. Phys. Chem. 71,4021 (1967). 1251 G. Schwai-z and J . Seelrg. Biopolymers 6, 1263 (1968). [26] A Wada, Chem. Phys. Lett. 8,211 (1971). [27] E . M . Bradburj, C . Crane-Robinson, H . Goldman, and H . W E. Rarrle, Nature 217, 812 (1968). [28] J . A . Eerreiti and L. Paoliiio, Biopolymers 7, 155 (1969). [29] R. Ullmann, Biopolymers Y, 471 (1970). [30] J . A . Ferrerfi, E W Ninham. and V A . Parsegiun, Macromolecule 3, 34 (1970) [31] D. Winklmnir, J . Engel, and I.:Ganser, Biopolymers 10,721 (1971). [32] J . Engef. Biopolymers 4,945 (1966). [33] J . Engei, Proceedings of the Symposium on Pepiide Chemistry 1969, North-Holland Co., in press. [34] R . C. Neurnann, Jr. and I! Jonus, J. Amer. Chem. SOC. YO, 1970 (1968). [35] K.-l. I)ahtquist, S. Forsen, and ?: Afm,Acta Chem. Scand., in press. [36] P. A . lemussi, T. Tancredi, and F. Quadrrfogiio, J Phys. Chem. 73, 4221 (1969) [37] 7 Miyarawo, Bull. Chem. SOC. Japan 34,691 (1961). [38] T: Miyarawa in M . A . Srahfmunn: Polyamtno Acids, Polypeptides and Proteins. Wisconsin Press, Madison 1962, p 201 [39] D. Porschke, Diplomarbeit, Universitat Gottingen 1966. and Dissertation, Technische Universltat Braunschweig 1968. [40] D. Porschke and M . Eigen,K. Mol. Biol. 62,316 (1971). [41) M . Eigen and D. Plirschke, J. Mol. Biol. 53,123 (1970). [42] M. Eigen. Lecture delivered ar the conference “Biochemische Analytik”, Munchen 1970. 1433 M . Ergen, Nobel Symposrum 1967. 1441 D M Crothers, J. Mol. Biol. 9, 712 (1964). [45] H . C. Spatz and D. M . Crorhers, J. Mol. Biol 42, 196 (1969). [46] R . Lumrj, R . Eilton, and J F. Brandrs, Biopolymers 4.917 (L966) [47] F. PohL A w w . Chem., in press; Angew. Chem. internat. Edit.. in press 575

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