Coil Transitions in

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Biomacromolecules 2000, 1, 597-603

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Interplay between Folding/Unfolding and Helix/Coil Transitions in Giant DNA Sergey V. Mikhailenko,† Vladimir G. Sergeyev,*,‡ Anatoly A. Zinchenko,‡ Marat O. Gallyamov,§ Igor V. Yaminsky,‡ and Kenichi Yoshikawa*,| Graduate School of Human Informatics, Nagoya University, Nagoya 464-8601, Japan; Department of Polymer Science, Faculty of Chemistry, Moscow State University, Moscow 119899, Russia; Faculty of Physics, Moscow State University, Moscow 119899, Russia; and Department of Physics, Graduate School of Science, Kyoto University & CREST, Kyoto 606-8502, Japan Received April 7, 2000; Revised Manuscript Received September 8, 2000

It has been well established that double-stranded DNA undergoes a melting, or helix/coil, transition into a single-stranded coil state with an increase in temperature. On the other hand, it has recently been found that, at a fixed temperature, long DNA, larger than several kilobase pairs, exhibits a discrete transition, or switching, between elongated and folded states, preserving its double-stranded structure, with the addition of various condensation agents, such as alcohol, hydrophilic polymer, multivalent cation, and cationic surfactant. In the present study, we examined the interplay between the folding/unfolding transition and the helix/coil transition in individual giant DNA molecules, by observing the conformation of single molecular chains with fluorescence microscopy. The results indicate that the helix-to-coil transition tightly cooperates with the unfolding transition in DNA. Introduction The helix/coil transition in DNA chains has currently been the subject of intensive experimental and theoretical investigations.1,2 This problem is interesting because of its relation to the process of DNA transcription, in which the two strands have to separate from each other to allow the reading of the genetic information. The temperature-induced helix/coil transition in DNA chains has also been successfully applied in several biochemical procedures, such as Southern blotting3 and the polymerase chain reaction (PCR).4 On the other hand, it has recently been found that long DNA chains undergo abrupt switching, or a discrete transition, between an elongated state and a collapsed compact state.5-7 In both prokaryote and eukaryote cells, DNA chains with lengths in the millimeter to centimeter range are folded in a narrow space on the order of several micrometers,8,9 forming highly condensed structures.10 Thus, it is thought that the manner of the helix/coil transition is strongly dependent on the higher order structure, either in the elongated state or the compact state.11 Previous physicochemical studies on the helix/coil transition have been performed mostly with short DNA chains, with less than a few hundred base pairs. Since the persistent length of a DNA molecule is around 500 Å, DNA chains on the order of 100 base pairs, or 300 Å, cannot fold by themselves into a compact state. With the addition of condensation agents, * Corresponding authors. E-mail: [email protected] (V.G.S.) and [email protected] (K.Y.). † Nagoya University. ‡ Faculty of Chemistry, Moscow State University. § Faculty of Physics, Moscow State University. | Kyoto University & CREST.

these short DNA molecules tend to aggregate with each other, instead of a folding into a compact conformation in the individual chains. This means that the interplay between the folding/unfolding transition and helix/coil transition in a long DNA molecule remains an important unexplored problem in biological science.12 Experimentally, the helix/coil transition, or so-called melting transition, is usually induced by exposing DNA solution to an elevated temperature. It has been well established that melting is the cooperative process of thermal disruption of the interbase hydrogen bonds, resulting in the gradual unstacking of DNA strands and denaturation from a double-stranded into a single-stranded state.1 In other words, the helix/coil transition in long DNAs proceeds gradually over a temperature range of a few tens of degrees Celsius. The addition of alcohol affects the melting temperature (Tm) of DNA molecules.13-16 Generally, an increase in the alcohol concentration tends to decrease the melting temperature. It is also known that DNA molecules aggregate and precipitate with the addition of alcohol.17 This process belongs to the phenomenon usually called DNA condensation.9 Precipitation and purification of DNA molecules with alcohol are described as the protocol frequently used in experiments in molecular biology.18 It is to be noted that previous studies on DNA denaturation have usually been carried out by using methods such as UV, CD, and fluorescence spectroscopies.19-21 These methods provide information on the conformational changes in the ensemble of DNA molecules, implying that the conformational changes in the individual macromolecules are averaged out in the quantities observed by these methods. Even in the case where individual macromolecules undergo a discrete conformational change,

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Figure 1. (a) Long-axis length distribution of DNA molecules at different tert-butyl alcohol concentrations at 20 °C, obtained from measurement by fluorescence microscopy. At least 40 DNA molecules were measured at each alcohol concentration, and at least 80 were studied in the regions of coexistence of elongated and compact molecules. (b) Dependence of the long-axis length of T4 DNA molecules on the concentration of tert-butyl alcohol at 20 °C, where the open circles are the mean values and the bars are the standard deviation of the size distribution.

or switching, the discrete character disappears in the ensemble.22 Furthermore, the concentration of DNA used in these experiments has often been relatively high. Therefore, to analyze the behavior of DNA molecules in the dissolved state, special precautions must be used to avoid aggregation. Although the enhancement of the thermal stability of aggregated DNAs has been reported,23-25 unfortunately, in these past studies the experimental distinction between the effects of interchain aggregation and those of intrachain folding has been rather difficult. In the present study, we examined the temperature-induced helix/coil transition in DNAs in an aqueous solution of tertbutyl alcohol using fluorescence microscopy (FM), which

allows us to monitor the conformational changes in long T4 DNA molecules at the single-molecule level. The results of the FM observations are discussed together with DNA melting curves obtained by UV spectroscopic measurements. Results To identify suitable experimental conditions for measuring temperature-dependent conformational transitions, we performed single-chain observation by fluorescence microscopy to examine the dependence of the T4 DNA conformation on the tert-butyl alcohol concentration. Figure 1 shows the results of the fluorescence microscopic observation for individual T4 DNAs at different tert-butyl

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Figure 2. AFM picture of T4 DNA molecules in 55% (v/v) tert-butyl alcohol. Elongated state with partially segregated regions and compact state are noted in the same picture.

alcohol concentrations. The long-axis length, L, was measured directly from the videotapes, and the apparent values are given in Figure 1 without correction for the blurring effect. We have already confirmed that, in the measurement of individual DNA molecules by fluorescence microscopy, the blurring effect is on the order of 0.3 µm.5-7,26 Thus, the actual long-axis length can be roughly estimated by subtracting 0.6 µm from the apparent long-axis length. Figure 1a shows the apparent long-axis length distribution at five characteristic alcohol concentrations, together with the fluorescent images of DNA molecules in the unfolded (10% tert-butyl alcohol), elongated with apparent intrachain segregation (50% tert-butyl alcohol) and fully folded compact (70% tert-butyl alcohol) conformations. At intermediate alcohol concentration (54% (v/v)), unfolded and folded molecules coexist. The exact size of individual DNAs was previously evaluated as the hydrodynamic radius from the quantitative measurement of Brownian motion.7,27 The hydrodynamic radius of the completely folded state was found to be ca. 0.1 µm, indicating that the decrease in relative volume from the elongated state into the compact state is 10-4-10-5. Figure 1b shows the long-axis length distribution of DNAs plotted as a function of the tert-butyl alcohol concentration. These results indicate that the addition of tertbutyl alcohol induces the folding transition in DNA molecules in the region of around 53-57% (v/v) alcohol, where elongated and compact DNA chains coexist. From the timesuccessive images of the individual DNA molecules, the elongated and the compact DNAs are distinguished without any ambiguity. Thus, individual DNA molecules undergo an all-or-none type transition as judged from the switching of the apparent conformation, whereas the ensemble of molecules shows continuous “cooperative” behavior. Such switching of the conformation from an elongated state into

Figure 3. Left: Fluorescent images of T4 DNA molecules observed by FM in water-tert-butyl alcohol media (50:50 (v/v)) at different temperatures. Middle: Quasi-3D distribution of the fluorescence intensity on corresponding photographs. Right: Schematic representation of the correspondence between the actual conformation and the fluorescence image of individual DNA.

compact state corresponds well with the previous studies28-30 on the effect of various alcohols on single DNAs. Precisely speaking, as noted in Figure 1, DNA chains gradually shrink with an increase in the alcohol concentration even before the switch into the completely folded state. From careful observation of the time-dependent change in morphology of individual DNA chains, we can conclude that at an alcohol concentration of 20%-50% DNAs exhibit a partially folded conformation; i.e., segregation between folded and unfolded parts occurs in individual chains.28,30 Thus, the folding transition as shown in Figure 1b may better be classified as a change between the partially unfolded state and the fully folded state.31 However, since we would like to focus on the interplay between the switching of the higherorder structure and the helix/coil transition, in the present study we will skip the delicate distinction between the fully unfolded and partially folded states. Instead, the long-axis length will be considered to characterize the switching in the higher-order structure. Thus, we will use the term of folding transition to describe the discrete transition from the

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Figure 4. Long-axis lengths of T4 DNAs in water-tert-butyl alcohol media at different temperatures: (a) in water:tert-butyl alcohol ) 50: 50% (v/v); (b) in water:tert-butyl alcohol ) 75:25% (v/v).

elongated state with the apparent long-axis length over 1.5 µm into the completely folded conformation with L smaller than 1.0 µm, as shown in Figure 1b. To get insight into the microscopic structure of T4 DNA in water-tert-butyl alcohol solution, we have performed atomic force microscopic measurements.32-35 The experimental approach used here was to treat DNA molecules by gradually increasing the concentration of alcohol. In the range of 25-50% (v/v) of alcohol T4 DNAs were not fixed onto the mica substrate. However, at 55% (v/v) of alcohol, a number of distinct morphologies of DNA molecules were observed, where both the elongated and folded compact chains, the latter mostly with the morphology of a toroid, are noticed (Figure 2). These observations correspond well to the coexistence of elongated and compact DNAs in the region of 53-57% (v/v) of tert-butyl alcohol observed by fluorescence microscopy (Figure 1b). Next, we performed fluorescence microscopic experiments on temperature-induced transitions at fixed alcohol concentrations with solvent mixtures of the following compositions: (a) water:tert-butyl alcohol ) 50:50% (v/v); (b) water: tert-butyl alcohol ) 75:25% (v/v). Figure 3 shows typical images of T4 DNA molecules in water-tert-butyl alcohol solution (50:50% (v/v)) at different temperatures. Interestingly, DNA chains exhibit a recurrent folding/ unfolding transition (Figure 3a-c). These observations were carried out with the temperature increased in five-degree steps, and the sample was kept for 30 min at each temperature before measurement. At low temperatures, below 40 °C, individual elongated DNA molecules exhibit significant intrachain and translational Brownian motion. This means that at low temperatures the solvent quality is good for DNAs. With an increase in temperature up to 45 °C,

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elongated DNA molecules adopt the folded conformation with the apparent long-axis lengths markedly dropping from around 2.0 µm to less than 0.9 µm (Figure 4a). It is to be noted that almost all of the folded DNAs are located on the glass surface, indicating that the solvent quality is poor at the intermediate temperatures (45-50 °C). A further increase in temperature up to 60 °C restores the elongated conformation of DNAs, causing a drastic change in the solvability, i.e., DNA molecules again exhibit vivid intrachain and translational Brownian motion in bulk solution. The transition profile is very sharp in both cases, and with the stepwise temperature change of 5 °C, no coexistence of elongated and compact DNAs is found. Finally, when the temperature reached 72 °C, the formation of a networklike structure near the glass surface, where the bright folded particles are connected by thin threads, is observed (Figure 3d), suggesting that the segments in interchain and intrachain contacts tend to stick to each other. For comparison, we carried out the measurements of the conformation of DNA chains with fluorescence microscopy in a system with lower alcohol content (25% (v/v)). In this case, as shown in Figure 4b, the folding transition occurs between 50 and 55 °C, and the unfolding transition is observed between 60 and 65 °C. Here again, the unfolded elongated DNAs are freely moving in bulk solution, and the folded compact DNAs are attached to the glass surface. In other words, with an increase in temperature the solvent quality changes from good to bad and then from bad to good. Network formation is noticed above 70 °C. Thus, similar profiles in the re-entrant conformational changes are observed in both 25 and 50% (v/v) alcohol, and the transitions in the solution with the lower alcohol content are generated at the lower temperatures. To get information on DNA melting transition, UV spectroscopy measurements were also performed in the system with the same solvent composition as that for the fluorescence microscopic measurements. The melting curves obtained for free DNA and DAPI-labeled DNA are presented in Figure 5, where the vertical axis is the change in UV absorption at 260 nm. In the case of 50% (v/v) alcohol (open triangles), there is almost no melting below 45 °C. The melting transition is initiated at around 55 °C and is completed around 90 °C. In 25% (v/v) alcohol (open circles) heat denaturation starts at ca. 40 °C, being completed at 80 °C, indicating that the profile of melting transition shifts toward a lower temperature with a decrease in the alcohol concentration. It is to be noted that the absorption curves dramatically change in the presence of DAPI (filled symbols), and the most significant difference is the continuous decrease in absorption intensity up to 75 °C. With the further increase in temperature, absorption increases, indicating the occurrence of DNA melting. Discussion In comparing the data obtained by UV spectroscopy and FM observations (Figures 4 and 5), the following important items are noted: (1) The melting, or helix/coil, transition

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Figure 5. Melting curves (relative optical density vs temperature) of T4 DNA obtained by UV spectroscopy measurements at 260 nm in water: tert-butyl alcohol ) 50:50% (v/v) (triangles) and in water:tert-butyl alcohol ) 75:25% (v/v) (circles) in the absence (open symbols) or presence (filled symbols) of DAPI.

occurs mostly in the region where DNA has an elongated conformation, in both 50 and 25% (v/v) alcohol solutions. Thus, there exists a significant cooperative effect between the melting and the unfolding transition in DNAs. (2) As indicated by UV spectroscopy, the melting transition in the solution with 50% (v/v) alcohol is induced at a higher temperature than with 25% (v/v) alcohol. It should be mentioned that the presence of DAPI causes an apparent effect on the conformational behavior of DNAs observed by fluorescence microscopy. DAPI is a DNAspecific probe which forms a fluorescent complex by attaching to the minor groove of the AT-rich regions in DNA.36 The binding of DAPI was reported to stabilize the double-stranded DNA molecule against thermal denaturation,37 being consistent with the results given in Figure 5. The decrease of UV-absorption intensity of DNA at the beginning of the DNA melting curve in the presence of DAPI is ascribed to the formation of the condensed phase in DNA.38 This correlates well with the results of the direct observation by the fluorescent microscopy. The folded conformation prevents the further melting transition. With an increase in temperature, the melting transition starts abruptly around 5560 °C, being accompanied by the unfolding conformational transition. This implies that the solvent quality is rather good for the melted single-stranded parts. Actually, with the initiation of intensive melting all of the DNA molecules dissolve in the bulk solution. The formation of the intrachainsegregated state as exemplified in Figure 3c is also attributed to the effect of partial melting of a double-stranded structure in the soluble elongated DNAs. With a further increase in temperature, the single-stranded parts tend to stick to each other with an increase in the ratio of the melted base pairs, finally forming the network structure. The profile of the DNA melting curve in 25% (v/v) alcohol is similar to that in 50% (v/v) alcohol. Around 50-60 °C,

DNA molecules are affected by the poor solvent quality, forming the folded compact structure. Then, similar to the unfolding transition in the system with 50% alcohol, with an increase in temperature up to around 63 °C the solvent quality becomes good for the DNA molecules where the melted single-stranded part dominates the unmelted doublestranded part. With a further increase in temperature, network structure is formed due to sticky interactions between the single-stranded segments. Next, we should clarify the unexpected dependence of the melting behavior of long DNAs on the composition of solvent mixture, observed by the UV measurements, namely, the initiation of the melting transition at a lower temperature in the solution with lower alcohol content. As pointed out in the Introduction section, the addition of alcohol was known to decrease the melting temperature.24,25 However, the melting curves presented in Figure 5 reveal the opposite dependence. We attribute it to the effect of the alcoholinduced intrachain segregation, resulting in the increase of the share of the folded part in a DNA chain, which was shown to be more thermally stable than the elongated one. Therefore, the melting transition in the 50% (v/v) alcohol solution starts at a higher temperature compared to the 25% (v/v) solution. The decrease in optical density of the DAPIlabeled DNA in 50% (v/v) alcohol is much more distinctive than in 25% (v/v) alcohol. This fact is also related to higher condensing ability of the more concentrated alcohol solution. In the past studies from our research group,39-41 the folding transition of long DNAs is classified as a first-order phase transition. It is expected that the all-or-none character of the folding transition is related to the unique aspect of the interaction between segments in a DNA chain. That is, when the folded and elongated DNAs exhibit distinct double minima in the free energy profile, the all-or-none character of the transition between the two states is enhanced. In terms

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of the criterion of Landau,42 such a system with doubleminimum free energy with respect to an order parameter is classified as a first-order phase transition. Generally, in a first-order phase transition the correlation length becomes longer (or shorter) when the free-energy barrier becomes higher (or lower). Since it is expected that the negative charge in DNA is decreased in a low-polar medium, the free-energy barrier between the elongated and compact states becomes lower for DNA in an alcohol solution. This means that in an alcohol solution the correlation length is shorter than the size of a DNA chain itself, and the intrachain coexistence, or intrachain segregation, is induced. In the present study, we have to consider one additional important factor to determine the properties of the folding/unfolding transition, namely, the melted, or single-stranded, part in DNA chain exhibits unique characteristics. The pairwise interaction between the segments with weak negative charge in the single-stranded part should be attractive in contacts. Thus, DNA segments tend to stick to each other during the intrachain and translational Brownian motion, in both interand intrachain interactions. The single-stranded segments induced by the melting transition are, therefore, expected to be sticky to each other in their contacts, whereas the longrange interaction between these segments is expected to be repulsive due to the like-charge effects. In other words, the repulsive Coulombic interaction between the remaining negative charges prevents the full collapse of DNAs. Thus, the network structure as shown in Figure 3d is generated. As an additional remark, it may be of value to mention the effect of the distribution of the GC and AT base pairs content along a DNA molecule. It is well-known that doublestranded DNA with high AT content tends to melt easier than that with high GC content.43 Thus, in the present study it is highly expected that the AT-rich part melts before the GC-rich region. In other words, in the partially melted state, the GC-rich part preserves double-stranded conformation, while the AT-rich part exists as single-stranded chains. To ascertain this hypothesis will be an interesting target for future research. Conclusion It was found that long DNA chains exhibit recurrent folding/unfolding transition in mixed water-tert-butyl alcohol solution with a change in temperature. It has been confirmed that the melting transition is suppressed in the folded conformation of DNA molecule. Experimental Section Bacteriophage T4dC DNA (166 kilobase pairs, contour length 57 µm5) was purchased from Nippon Gene. The fluorescent dye 4,6′-diamidino-2-phenylindole (DAPI), the antioxidant 2-mercaptoethanol (ME), and tert-butyl alcohol (spectral grade) were obtained from Wako Pure Chemical Industries Ltd. ME was used as a free-radical scavenger to reduce fluorescent fading and light-induced damage of DNA molecules. Solvent mixtures were prepared on a volume basis. Alcohol was added dropwise to a DNA solution under mild stirring and then stored for 24 h before observation.

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Fluorescence microscopic measurement was performed as in the previous studies,7,29 and here we give only brief description of the experimental procedure. The samples were illuminated with 365 nm UV light, and fluorescence images of DNA molecules were observed using a Zeiss Axiovert 135 TV microscope equipped with a 100× oil-immersed lens and recorded on S-VHS videotape through a high-sensitivity Hamamatsu SIT TV camera. The observations were carried out at 25 °C. The apparent length of the long-axis L, which was defined as the longest distance in the outline of the DNA image, was calibrated with an Argus 10 image processor (Hamamatsu Photonics). The blurring effect was estimated to be on the order of 0.3 µm,5-7,26 and the data for L are given in the text without correction. The sample solutions, microscope slides and coverslips were carefully prepared as in previous studies. DNA molecules were diluted with 0.5×TBE buffer solution containing 4% (v/v) ME. The final concentrations of DNA in nucleotides and DAPI were 0.6 µm. The temperature was regulated using a water jacket manufactured in our laboratory, connected to an Eyela UA100 thermostat system. The precision of the temperature control was (1 °C. Absorption spectra were obtained with a Specord M-40 spectrophotometer. The concentration of DNA in the solution was calculated by assuming that the molar extinction coefficient for native DNA in water is 6600 M-1 cm-1.44 Cells with an optical path of 1 cm were used for the UV measurements. The temperature was regulated using a water jacket cell connected to the thermostat system. AFM measurements were performed as follows. A NanoScope III (Digital Instruments) equipped with commercial AFM cantilevers with silicon tips operating in tapping mode was used for all experiments. Freshly cleaved mica (muscovite) was used as a substrate. The liquid cell (Digital Instruments) was flushed with about 5 mL of 25% (v/v) tertbutyl alcohol solution before the initial imaging. After that the cell was filled with 45% (v/v) tert-butyl alcohol solution containing T4 DNA with concentration 0.5 µm in nucleotides, and then the alcohol content was gradually increased up to 55% (v/v) for the observation. Thus, the DNA molecules attached to the mica surface were imaged without drying of the sample. Acknowledgment. This work was partly supported by Grant No. 99-03-33337 from the Russian Foundation for Basic Research and the Sasakawa Scientific Research Grant from the Japan Scientific Society. S.V.M. is also grateful to the Ministry of Education, Science, and Culture of Japan for the scholarship. The authors acknowledge Prof. A. B. Zezin (Moscow State University) for valuable discussions. References and Notes (1) Cantor, C. R.; Schimmel, P. R. Biophysical Chemistry, W. H. Freeman and Co.: San Francisco, 1980; Vol. III. (2) Breslauer, K. G. In Thermodynamic Data for Biochemistry and Biotechnology; Hinz, H., Ed.; Springer-Verlag: New York, 1986; pp 402-427. (3) Southern, E. M. J. Mol. Biol. 1975, 98, 503. (4) Saiki, R. K.; Gelfand, D. H.; Stoffel, S.; Scharf, S. J.; Higuchi, R.; Horn, G.; Mullis, K. B.; Erlich, H. A. Science 1988, 239, 487. (5) Yoshikawa, K.; Matsuzawa, Y.; Minagawa, K.; Doi, M.; Matsumoto, M. Biochem. Biophys. Res. Commun. 1992, 188, 1274.

Transitions in Giant DNA (6) Vasilevskaya, V. V.; Khokhlov, A. R.; Matsuzawa, Y.; Yoshikawa, K. J. Chem. Phys. 1995, 102, 6595. (7) Mel’nikov, S. M.; Sergeyev, V. G.; Yoshikawa, K. J. Am. Chem. Soc. 1995, 117, 2401. (8) Bloomfield, V. A. Biopolymers 1991, 31, 1471. (9) Bloomfield, V. A. Curr. Opin. Struct. Biol. 1996, 6, 334. (10) Sikorav, J.-L.; Pelta, J.; Livolant, F. Biophys. J. 1994, 67, 1387. (11) Sikorav, J.-L.; Church, G. M. J. Mol. Biol. 1991, 222, 1085. (12) Grosberg, A. Yu. Biophysics 1984, 29, 621. (13) Geiduschek, F. P.; Herskovits, T. T. Arch. Biochem. Biophys. 1961, 95, 114. (14) Levine, L.; Gordon, J. A.; Jencks, W. P. Biochemistry 1963, 2, 168. (15) Usatyi, A. F.; Shlyakhtenko, L. S. Biopolymers 1974, 13, 2435. (16) Kypr, J.; Vorlickova, M. Biochem. Biophys. Res. Commun. 1984, 123, 831. (17) Potaman, V. N.; Bannikov, Yu. A.; Shlyakhtenko, L. S. Nucleic Acids Res. 1980, 8, 635. (18) Geck, P.; Nasz, I. Anal. Biochem. 1983, 135, 264. (19) Herskovits, T. T.; Singer, S. J.; Geiduschek, E. P. Arch. Biochem. Biophys. 1961, 94, 99. (20) Marmur, J.; Doty, P. J. Mol. Biol. 1961, 3, 585. (21) Darzynkiewicz, Z.; Traganos, F.; Sharpless, T.; Melamed, M. R. J. Cell Biol. 1976, 68, 1. (22) Yoshikawa, K.; Takahashi, M.; Vasilevskaya, V. V.; Khokhlov, A. R. Phys. ReV. Lett. 1996, 76, 3029. (23) Rupprecht, A.; Piskur, J. Acta Chem. Scand. 1983, B37, 863. (24) Rupprecht, A.; Piskur, J.; Shultz, J.; Nordenskiold, L.; Song, Z.; Lahajnar, G. Biopolymers 1994, 34, 897. (25) Piskur, J.; Rupprecht, A. FEBS Lett. 1995, 375, 174. (26) Yoshikawa, Y.; Emi, N.; Kanbe, T.; Yoshikawa, K.; Saito, H. FEBS Lett. 1996, 396, 71. (27) Yoshikawa, K.; Matsuzawa, Y. Physica D 1995, 84, 220.

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