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J . Phys. Chem. 1988, 92, 6482-6485
helical RNA poly(rU).poly(rA).poly(rU), since its structure resembles a regular A-form double helix with an extra A-form strand inserted in the major groove (some small structural change is involved, mainly in the b a ~ k b o n d ) . The ~ ~ 70-cm-l mode observed in all A-form double helices splits into two narrower modes, and the other two modes (near 30 and 120 cm-]) shift by just a few cm-’ compared to double-helical-RNA sequences. These changes and splittings are similar to those observed by Eyster and Prohofsky which suggests that stacking of bases belonging to different strands and the formation of hydrogen bonds have a small influence on the phonon spectrum The observed modes are therefore likely to occur also in 1 1-fold single-helical DNA. The differences in the spectra are most likely due to differences in the secondary structures of the molecules. Poly(dA).poly(dT) belongs to the B-form family but shows some distinct features, e.g., a hydration spine and a much narrower minor groove compared with “normal” B-DNA.26,27 Indeed, the low-frequency Raman spectrum of poly(dA).poly(dT) is anomalous compared with the other B-form spectra. This is not due to the high AT content since the spectrum of poly(dA-dT).poly(dA-dT) is like that of any other B-form double helix. The 63-cm-] mode in the B-form polynucleotides shifts to about 70 cm-’ and becomes considerably more intense whereas the other modes do not shift considerably compared to spectra of “normal” B-DNA. This effect cannot be explained with the narrowing of the minor groove alone since C-DNA spectra are nearly identical with those of B-DNA,29 although the minor groove has n a r r ~ w e d . ~ ’ Another example for the influence of the secondary DNA structure on the low-frequency vibrations is poly(rC). At pH 7 poly(rC) forms a single helix with 6 bases per turn, whereas at pH 4 it forms a parallel double helix (probably with 12 base pairs per turn). The most prominent mode in the double-helical form is at 147 cm-I, whereas in the single-helical form it is well below 100 cm-I. These differences are not due to the formation of hydrogen bonds (see above) and must therefore represent the changes in RNA conformation. These observations can, so far, not be related to specific interactions within the DNA, or to “simple” motions of the single or the double helix. Torsional modes, for example, should be
-
(30) Arnott, S.; Bond, P. J.; Selsing, E.; Smitch, P. J. C. Nucleic Acids Res. 1976, 3, 2459.
observed at lower frequencies4 and do not have the right sequence dependence31to explain any of the observed modes. Recent results by Urabe et a1.I0 indicate that base stacking plays an important role in the “lOO-cm-l mode” of DNA (95 cm-’ in B-form, 117 cm-’ in A-form), whereas the backbone should only play a minor role. Base stacking energies depend very strongly on the AT and GC content, whereas the 100-cm-’ mode only shows a very small dependence on the sequence. Also, the most prominent mode in the poly(rC).poly(rC-H+) appears at 147 cm-I, compared with 95-120 cm-I in DNA and stacks of guanosine and monop h o s p h a t e ~ . ~These ~ * ~ ~results suggest that base stacking interactions only have a minor influence on the frequency, but a large influence on the intensity of this mode. The relatively large intensity of this mode suggests that it might be a libration of the bases.
-
Conclusions We have performed low-frequency Raman spectroscopy on concentrated solutions of polynucleotides with different sequences. The Raman spectra depend very strongly on secondary structure, but, within one conformation, the modes depend only slightly on sequence. The bridging hydrogen bonds between the strands has a small influence on the modes. The observations on poly(dA).poly(dT) show that even relatively small changes in the secondary structure can lead to relatively large frequency and intensity changes in some modes. A study of the low-frequency vibrations of other poly- and oligonucleotides with well-understood structures should lead to a better understanding of the intrahelical interactions of DNA and RNA. Acknowledgment. We thank Prof. W. L. Peticolas, Prof. E. W. Prohofsky, Prof. J. W. Powell, Prof. S. A. Lee, Dr. A. E. Garcia, Dr. G. A. Thomas, Prof. R. M. Wartell, and Prof. L. L. Van Zandt for helpful suggestions, discussions, and sharing of unpublished results. This work was supported in part by O N R Contract No. NOOO14-87-K-0478 and N S F Grant BBS 8615653. T.W. gratefully acknowledges an Arizona Board of Regents graduate scholarship. (31) Millar, D. P.;Robbins, R. J.; Zewail, A. H. J . Chem. Phys. 1981, 74, 4200. (32) Nielsen, 0 . F.; Lund, P. A,; Peterson, S. B. J. Am. Chem. Soc. 1982, 104. 1991.
Critical Slowing Down in Periodically Perturbed Chemical Oscillations G . Dewel*,+and P. Borckmanst Service de Chimie-Physique, CP 231 -Campus Plaine, UniversitP Libre de Bruxelles, B- 1050 Bruxelles, Belgium (Received: May 2, 1988; In Final Form: August 17, 1988)
We analyze recent experimentson the nonlinear relaxation (critical slowing down, phase slippage, frequency pulling) of chemical oscillators subjected to low-amplitude forcing in the vicinity of the transition from entrained behavior to quasi-periodic motion using a simple model. The phase slippage staircase is reinterpreted in terms of a plateau behavior analogous to that occurring near the hysteresis limits
Periodic perturbations of an oscillator system of the limit cycle type, with frequency coo, can give rise to various dynamical behaviors.lI2 According to the value of the perturbation amplitude ( e ) , frequency (w&, and initial conditions, the response approaches a periodic, quasi-periodic, or chaotic trajectory. When the amplitude is weak, only two kinds of situations are known to emerge. ‘Research Associates at the Belgium National Science Foundation (F. N.R.S.).
0022-3654/88/2092-6482$01.50/0
(1) Entrainment occurs around special frequencies satisfying wp/wo = R, where R is a rational number. In particular, the band corresponding to R = 1 is called the fundamental entrainment band. (2) Outside these bands, the response produces quasi-periodic variations of the state variables. ( I ) Rehmus, P.; Ross, J. In Oscillafionsand Trauelling Waues; Field, R. J., Burger, M., Eds.; Wiley: New York, 1985; Chapter 9, p 287. ( 2 ) Schneider, F. W. Ann. Reo. Chem. 1985, 36, 347.
0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 23, 1988 6483
Letters From theoretical works and numerical computations of simple critical slowing down (Le., the slow decay of transients) has been predicted to occur at the transition from entrainment behavior to quasi-periodic motion. This phenomenon was characterized experimentally for the first time in a gas-phase combustion reaction;6 it has also been observed in the BelousovZhabotinskii reaction' and recently in the case of a surface reaction.8 In this Letter we aim at putting the nonlinear relaxation near the edges of the fundamental entrainment band in the perspective of the relaxation near the hysteresis limits in multistable systems using a simple m ~ d e l . ~ Other , ~ properties of this relaxation are analyzed in relation with recent experimental work. Following ref 4 we first briefly recall the derivation of the basic kinetic equation of the model. We consider a chemical system with time evolution given by dX/dt = F(X) cP(wpt) (1)
Near the boundaries but inside the band up
By applying a standard multiple-scale method one seeks solutions of (1) close to the unperturbed limit cycle B. Thus we expand X as x = Xo(7,T) tXl(7,T) ... (2)
+
+
The solutions depend on both a long time scale T = tt/2 and a short time T = (wo t6)t with 6 = up- w 0 / c Substituting (2) into ( 1 ) and equating like powers of c yields the following solution for Xo
+
Xo(7,T) = B(7 - 44T)) (3) where the phase $(T) with respect to the periodic perturbation is determined at the next stage in the perturbation procedure. m results in the The requirement that X1 be bounded as 7 following slow-time equation valid in the limit of a strongly attracting limit cycle B339in the case of a sinusoidal perturbation d$/dT = -(COS$) - 2(wP - W O ) / € ~ (4)
-
with #( T) = $J(T ) - Q and where Q and y are model-dependent constants arising from the reduction of the description in terms of the phase alone. In the following for simplicity we take Q = 0 and y = 1 . The nature of the solutions of (4) greatly depends on the value of up;when the following inequality is satisfied wo - € / 2 < wp < wo + € / 2 (5) eq 4 admits two steady solutions
in the range [0,2a]:
&* = *cos-1 (2(w, - U p ) / € )
(6)
The stable solution satisfying sin $s < 0 gives the phase-locking angle of the entrained solution; the inequality (5) therefore determines the width of the fundamental band. In particular at the boundaries of this band one finds up = wB- = oo- t/2; &- = 0 (lower edge) wp = w ~ = + wo
+ c/2;
c$~+ =P
(upper edge)
(7)
The phase-locking angles at the edges of the band differ by a radians. This general property of simple sinusoidal perturbations'O has been verified in recent experiments.6v8 (3) Machlup, S.;Sluckin, T. J . Theor. Biol. 1980, 84, 119. (4) Rehmus, P.; Ross, J. J . Chem. Phys. 1983, 78, 3747. (5) Kevrekidis, I. G.; Schmidt, L. D.; Aris, R. Chem. Eng. Commun. 1984, 30, 323. (6) Pugh, S. A,; Schell, M.; Ross, J . J . Chem. Phys. 1986, 85, 868. (7) Hudson, J. L.; Lamba, P.; Mankin, J . C . J . Phys. Chem. 1986, 90, 3430. (8) Eiswirth, M.; Ertl, G.; Phys. Reu. Len. 1988, 60,1526. (9) Ermentrout, G. B.; Rinzel, J. J . Moth. Biol. 1981, 1 1 , 269. (10) Loud, S. Ann. Moth. 1959, 70, 490.
< I.1 < 1
(8)
The steady solutions of (4) take the form
dSi - $Lt = f(2j~L)I/~ (stable) - & = = ~ ( 2 p ) ' / ~ (unstable)
(9)
In (8) and (9) the upper sign corresponds to the upper edge of the band. Every perturbation will come at rest at &+(modulo 2a). In the case of small perturbations and sufficiently far from the boundary of the band such that 14 - ~#4(21.1)~/~< 1, the relaxation is exponential with a diverging relaxation time trl
+
where X is the state vector, F is a vector function of X describing the chemical reactions, and CPis a small-amplitude periodic perturbation of period Pp = 2n/wp. When e = 0, the system admits an orbitally stable limit cycle with period Po = 2a/w0, Le., X = B ( t ) = B(t+Po)
- OB* = T€p/2; 0
as up
= (Elup - (L)B*l)-'/2
t,,
N
-
wB+
with z1 = -0.5
[up- w B * ~ "
(10)
where we have introduced in (10) a dynamical exponent z1 characterizing the singularity of the relaxation time near the boundary." Such a dramatic increase of the relaxation time has recently been observed near the upper edge of the fundamental band in a study of the effect of periodic perturbations on the oscillator combustion of acetaldehyde in a CSTR (cool flames).6 A similar prolonged transient behavior has also been reported when sinusoidal perturbations are applied (near the edge of the 2/1 superharmonic entrainment band) to the oxygen partial pressure in the oscillatory oxidation of CO on Pt(1 This slowing down can lead to an inaccurate experimental determination of the boundaries of the entrainment bands. Because of the generic character of eq 4 the exponent z1 should largely be independent of the details of the chemical mechanisms responsible for the instability. However, as in the problem of spinodal slowing down near the hysteresis limits of a bistable, the experimental determination of the exponent is d i f f i c ~ l t .Indeed ~~~~~ on approaching the boundary 1.1 0, the deviations from exponential behavior become increasingly important. As an example near but inside the upper edge, a direct integration of (4) yields
-
K I exp(-T(p(2 - P ) ) ' / ~ )9 1 K1 exp(-T(p(2 - P ) ) ' / ~ ) 1 where
and $o is the initial condition. Within the band the system exhibits excitability. For up< w ~ + if the initial phase c $ ~ is in the interval &+ < +o < &+ it gradually decays back to &+; however, if 4o < +u+, then the system must traverse the entire cycle of phases before coming at rest at &+. The excitability threshold goes to zero at the edge of the band. As a result the nonlinear relaxation strongly depends on the direction of the initial perturbation. As an example we consider the relaxation of the instantaneous period Pi. From (3), Pi is obviously given by Pi = Pp/(l
- €((d4/dT)/2wp))
(13)
Near but inside the upper edge, wp - wB+ = -e1.1/2 and (13) becomes
Pi = P p / ( l
+ €((COS4 + 1 - p)/2wp))
(14)
(11) Dewel, G.; Borckmans, P.; Walgraef, D. J . Phys. Chem. 1984, 88, 5442. (12) Ganapathisubramanian, N.; Showalter, K. J . Chem. Phys. 1986,84, 5427. (13) Laplante, J. P.; Borckmans, P.; Dewel, G.; Gimenez, M.; Micheau, J . C . J . Phys. Chem. 1987, 91, 3401.
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The Journal of Physical Chemistry, Vol. 92, No. 23, 1988
Letters Pi - Pp
I
P.I- Pp P-? Pi
(bl
T
2s
T
S
/" /
Figure 1. Relaxation of the intantaneous period
Piin the entrainment band after an in-phase (a) or P out of phase (b) perturbation c = 0.2,
p
= 0.1, P , = 7.35.
%..I%
'
ws,icLb
UP/
wo
Figure 2. Beat frequency wb for fixed (small) amplitude of perturbation as a function of the ratio of the driving frequency (up)to the autonomous
frequency (cog). The straight lines correspond to the beating frequency (ob')between two linear oscillators of respective frequencies upand w,. The differences wbl - wb constitutes the frequency pulling. where is given by (1 1). As shown in Figure 1 the relaxation of an in-phase perturbation (c$~) is different from that of a perturbation T out of phase. Such a dependence of the relaxation on the initial phase disturbance has been reported in ref 6. A similar effect has been predictedi4 and observedi2 in the vicinity of a nonequilibrium critical point. Near but outside the upper boundary of the band the phase will continue to decrease on the slow time scale according to dc#J/dT = up
-COS
4 - 1-p
