9396
J. Phys. Chem. B 1997, 101, 9396-9401
Discrete Coil-Globule Transition of Single Duplex DNAs Induced by Polyamines M. Takahashi,† K. Yoshikawa,*,† V. V. Vasilevskaya,‡ and A. R. Khokhlov‡ Graduate School of Human Informatics, Nagoya UniVersity, Nagoya 464-01, Japan, and Institute of Organoelement Compounds, Russian Academy of Sciences, VaViloVa st. 26, Moscow 117813, Russia ReceiVed: May 16, 1997; In Final Form: August 18, 1997X
Recently, it has become clear that single, long duplex DNAs exhibit a large discrete transition between elongated coil and compacted globule states. To obtain further insight into this phenomenon, in the present study we observed individual DNA chains in an aqueous environment by fluorescence microscopy. The long-axis lengths of individual T4DNA (166 kbp) were calibrated to obtain a size distribution. The main purpose of the present study was to determine the effect of the valence of cations on the coil-globule transition. We used the following multivalent cations to induce the compaction of long DNA chains: 1,3-diaminopropane (bivalent), spermidine (trivalent), and spermine (tetravalent). Our results showed that the collapse of isolated DNA chains induced by either bivalent or multivalent cations is discrete. The critical concentration of cation for inducing the transition was lowest for the tetravalent cation and highest for the bivalent cation. We also compare the properties of the transition observed experimentally with a theoretical calculation including the effects on condensation of multivalent cations and ion-exchange reaction.
1. Introduction 102-104
µm In biological systems, DNAs on the order of long are usually packed in a narrow space on the order of only 0.1-1 µm, i.e., bacteriophage head, cytoplasmic space in prokaryote, and nucleus in eukaryote.1 On the other hand, DNA chains exhibit a highly elongated coiled state in aqueous solution in the absence of condensation agents. Thus, the study of the collapsed and decollapsed transition of long duplex DNAs2-8 is expected to shed light on the dynamic change in the state of DNAs in the living cellular environment. Various chemical species, such as histone proteins, metal cations, and polyamines, are known to induce the compaction of long DNA chains. Among these, polyamines are widespread in both prokaryote and eukaryote cells and possess various biological effects. For example, it is known that λ phage is not generated in polyamine-required mutant E. coli.9 In eukaryote cells, polyamines play an essential role in the growth of both normal and neoplastic tissues.10,11 Several experimental studies have examined the interplay between polyamines and DNA.12-17 Theoretical investigations have also been performed accompanied by the development of the theory on polyelectrolytes18-20 and polymers in general.21-24 However, it has been difficult to obtain fully conclusive results from experiments on the physicochemical properties of the coilglobule transition in single DNA chains, since competition is always present between single-chain events and the aggregation of a number of chains under the usual experimental conditions. Actually, single-chain observation in aqueous solutions, in its strict sense, has been impossible with conventional experimental methods such as light scattering, X-ray analysis, and sedimentation. These methods require a relatively high concentration (more than about several µg/mL or 10 µM in base pair concentration) to obtain adequate sensitivity. In addition, these experimental methods provide information essentially only to the characteristics of the ensemble of polymer chains in solution. * To whom correspondence should be addressed. E-mail: f43943a@ nucc.cc.nagoya-u.ac.jp. Fax: +81-52-789-4808. † Nagoya University. ‡ Russian Academy of Sciences. X Abstract published in AdVance ACS Abstracts, October 1, 1997.
S1089-5647(97)01639-8 CCC: $14.00
Recently, we showed that fluorescence microscopy is useful for observing single molecules of long duplex DNA chains and reported that individual DNA molecules undergo a first-order transition between an elongated coil state and a compacted globule state with the addition of various kinds of condensing agent, such as neutral flexible polymer,25 cationic and neutral surfactants,26 alcohol,27 polyamine spermidine,28 and inorganic metal cation.29 The purpose of the present study was to clarify the mechanism of the compaction of single DNA molecules, and also polyelectrolytes in general, induced by multivalent cations. We observed changes in the higher-order structure of T4DNA by fluorescence microscopy upon the addition of polyamines with different valences: spermine (SPM) with four positive charges, spermidine (SPD) with three positive charges, and 1,3-diaminopropane (DA3) with two positive charges. Here, since one of our previous studies reveals that diaminoalkanes with three or five methylene carbons are effective agents for compacting DNA molecules,30 we used diaminopropane as the bivalent cation. The experimental details and results will be presented in sections 2 and 3, respectively. In section 4, the experimental results will be considered theoretically, where the mechanism of the compaction of single molecules induced by multivalent cations is discussed in relation to the bimodality of the freeenergy profile of single macromolecules. Section 5 is the conclusion of the present study. Theoretical detail is given in the Appendix. 2. Experimental Section 2.1. Materials and Methods. Bacteriophage T4dC DNA (166 kbp) was purchased from Nippon Gene (Tokyo, Japan). The fluorescent dye 4′6-diamidino-2-phenylindole (DAPI) and 2-mercaptoethanol (2-ME) were obtained from Wako Pure Chemical (Osaka, Japan). Spermine tetrahydrochloride (SPM) and spermidine trihydrochloride (SPD) were obtained from Nacalai Tesque (Kyoto, Japan). 1,3-Diaminopropane dihydrochloride (DA3) was purchased from Tokyo Kasei (Tokyo, Japan). The structures of DA3, SPD, and SPM are shown in Chart 1. © 1997 American Chemical Society
Collapsed Transition of Single Duplex DNA CHART 1: Structural Formulas of Polyamines
J. Phys. Chem. B, Vol. 101, No. 45, 1997 9397 l, which was defined as the longest distance in the outline of the DNA image. Owing to the technical characteristics of the SIT camera and the resolution limit, there is a blurring effect of about 0.3-0.4 µm. Thus, the observed DNA image is slightly larger than the actual size of the DNA.27-31 3. Results
T4DNA was dissolved in 30 mM Tris-HCl buffer solution at pH 6.0 with 10 mM NaCl. To prevent aggregation, the final concentration of DNA was very dilute: 0.10 µM in nucleotides. As the fluorescent dye, 0.10 µM DAPI was used. It has been confirmed that the persistence length (ca. 600 Å), together with the contour length (ca. 57 µm), of T4DNA remains nearly constant before and after the addition of DAPI.31 To prevent the fluorescence from fading owing to illumination, 4% (v/v) 2-ME was added as an antioxidant. We examined the timedependent changes in the size distribution of DNAs and found that equilibrium is attained at least 2 h after sample preparation. In the present study, we will only show the results after equilibration. To minimize the adsorption of DNAs onto the glass surface, special care was taken to clean the glass microscope slides and cover slips thoroughly before the observations. They were soaked in acetone for 30 min and then in hydrogen peroxide for at least 2 h. They were washed repeatedly with distilled water and ethanol. Finally, they were dried at 35 °C for 30 min. 2.2. Fluorescence Microscopy. The sample solution depth was relatively large, ca. 120 µm, to avoid the surface effect of the glass plates. With this treatment, it is possible to observe individual DNAs in the aqueous environment, avoiding the effect from the glass surface. The samples were illuminated with UV light (365 nm). The fluorescence images of DNA molecules were observed using a Carl Zeiss microscope, Axiovert 135TV, equipped with a 100× oil-immersed objective lens. They were recorded on videotape with a high-sensitivity Hamamatsu SIT TV camera. The video images were analyzed with an image processor (Argus 50, Hamamatsu Photonics). The observations were carried out at room temperature (20 °C). To characterize the size of DNA, we measured the long-axis length
Figure 1 shows fluorescent images of T4DNA: (A) coil state with [SPD] ) 10 µM; (B) coexistence of the coil and globule states with [SPD] ) 90 µM; (C) coexistence of the coil and globule states with [DA3] ) 2 mM; (D) globule state with [SPM] ) 50 µM. From top to bottom, the figures show black-andwhite (binary) pictures of the fluorescence images of T4DNA, fluorescent light intensity distributions, and the corresponding schematic representations. In the absence and even in the presence of minute amounts of multivalent cation, individual DNA molecules assume the elongated random coil state, as shown in Figure 1A. With an increase in the concentration of multivalent cation, DNA collapses into the compact globule state. Between the two extreme states of pure coil and pure globule, the coil and globule states coexist. Parts B and C of Figure 1 show the coexistence of the coil and globule states in the presence of trivalent and bivalent cations, respectively. Figure 2 shows histograms of the distributions of the longaxis length l of T4DNA in the presence of polyamines at various concentrations. [DA3], [SPD], and [SPM] show the molar concentration of 1,3-diaminopropane, spermidine, and spermine, respectively. At low concentrations of the polyamines, regardless of the valence, all of the DNA chains exist in the elongated coil state. For the bivalent cation, the histogram shows a bimodal distribution for the coil and globule states at [DA3] ) 0.4 mM. With an increase in [DA3], the fraction of molecules in the globule state increases, and finally, at 6 mM, all of the DNA molecules exist in the collapsed globule state. For the trivalent cation, coexistence of the coil and globule states occurs at [SPD] ) 80-100 µM. For the tetravalent cation, the coexistence region shifts to a still lower concentration (see Figure 2C). Figure 3 shows the dependence of the long-axis length l on the concentrations of DA3, SPD, and SPM. The shaded regions indicate coexistence of the coil and globule states, where the average values of l are given separately for the coil and globule states. In this figure, Ct corresponds to the concentration of polyamine at which the coil and globule state populations are
Figure 1. Fluorescence micrographs of T4DNA under various conditions: (A) coil state in 10 µM spermidine (SPD) solution; (B) coexistence of the coil and globule states in 90 µM spermidine aqueous solution; (C) coexistence of the coil and globule states in 2 mM 1,3-diaminopropane (DA3) solution; (D) globule state in 50 µM spermine (SPM). The top row shows fluorescence microscopic images. The middle row represents the spatial distribution of fluorescent light intensity. The bottom row shows the relationship between the conformation of the actual DNA chain and the corresponding fluorescence image. Owing to a blurring effect (a ≈ 0.3 µm), the fluorescence image is larger than the actual size of the DNA chain.
9398 J. Phys. Chem. B, Vol. 101, No. 45, 1997
Figure 2. Histograms of the distribution of the long-axis lengths of T4DNA molecules: (A) 1,3-diaminopropane solution (DA3); (B) spermidine (SPD); (C) spermine (SPM). One hundred T4DNA’s were analyzed for each condition. [DNA] ) 0.10 µM in units of base pair concentration. Each area of the histogram is normalized to be equal. The populations of molecules in the globule and coil states are indicated by different shading in the histogram. In the actual observation, it is easy to distinguish between the coil and globule states due to intrachain thermal motion, in addition to the size of the fluorescent images.
the same; i.e., the free energies of the coil and globule states are the same at Ct. Figure 3 reflects the following general trends for the effect of polyamines on the DNA collapse transition. (i) Regardless of the valence, the collapse of DNA is characterized by a discrete transition between the coil and globule states. (ii) As the valence of the polyamine increases, the coexistence region shifts to a lower concentration and the width of the coexistence decreases. (iii) With a bivalent cation, the sizes of molecules in the coil and globule states show a rather remarkable dependence on the concentration, especially for the coexistence region. On the other hand, the coil and globule states maintain essentially the same dimensions for trivalent and tetravalent cations. The general features of this transition are summarized in Table 1. These results also confirm the above characteristics i-iii. 4. Comparison with Theory and Discussion We have performed a theoretical analysis in order to explain the coil-globule transition induced by multivalent cations, taking into account of the following effects: the condensation of multivalent cations and an ion-exchange reaction. The essence of the theoretical approach is summarized in the Appendix.
Takahashi et al.
Figure 3. Change in the average long-axis length for multivalent cations with different valences: (A) bivalent cation (DA3); (B) trivalent cation (SPD); (C) tetravalent cations (SPM). The shaded area represents the region of the coexistence of the coil and globule states. C1 and C2 indicate the lower and upper limits of the concentration in the coexistence region. The transition point Ct indicates the point at which the coil and globule states exhibit the same populations. Their real values are shown in Table 1. The vertical bar indicates the standard deviation.
The theoretical size distribution of DNA that depends on the concentration of multivalent cations is exemplified in Figure 4. In Figure 5 we summarize the theoretically calculated dependence of the swelling ratio R of DNA molecules on the concentration of multivalent cations. For the parameters chosen in Figure 5, in all cases the swelling ratio R has two stable values at some concentration of multivalent cation; i.e., the transition from a coil to globule state is a discrete first-order phase transition. The width of the region of the coexistence of the globular and coil states was calculated from theoretical probability distribution functions analogous to those shown in Figure 2. Actually, the coexistence region given in Figure 5 indicates the range of concentrations at which the relative ratio between the coil and globule states is within 1 × 10-2 to 1 × 102. Outside this region, the minor fraction (coil or globule) becomes less than 1%, which corresponds to the experimental trend that only either the coil or globule is observed outside the coexistence region. It is clear that the width of this coexistence region increases with a decrease in the valence of the cation and that the amplitude of the coil-globule transition (i.e., the degree of
Collapsed Transition of Single Duplex DNA
J. Phys. Chem. B, Vol. 101, No. 45, 1997 9399
TABLE 1: Polyamine Concentration To Induce the Collapsed Transition of T4DNAa
polyamines C1 (µM) C2 (µΜ) Ct (µM) C2 - C1 (µM) ∆l (µm) DA3 (M2+) SPD (M3+) SPM (M4+)
400 80 8
2000 200 20
900 90 9.5
1600 120 12
2.2 2.3 2.3
a C1, C2, Ct, and ∆l are given in the schematic figure above. The coexistence region is between C1 and C2. Ct corresponds to the same free energy between coil and globule. ∆l is the change of the longaxis length due to the first-order phase transition.
Figure 4. Size distribution for the swelling ratio R at three different concentrations of different multivalent cations. Concentrations of the multivalent cation are c ) Sz/(V1 + V2 ). In the middle row distributions, the ratios between coil and globule in coexistence states are 1:9, 1:1, and 1:6 for bivalent, trivalent, and tetravalent cation, respectively. The following parameters were used: N ) 100, b ) 100 nm, d ) 2 nm, Q ) 16, QN ) 480N, χ ) 1.2; volume fraction of DNA links within the field of DNA molecule in the ideal coil state φ0 ) 2.5 × 10-4 (φ0 ) Nbd2/V0); the ratio of polymer volume Vp ()Nbd2) to the volume of system Ω, r ) Vp/Ω ) 5 × 10-9, where Ω ) V1 + V2 .
Figure 5. Dependence of the swelling ratio R of a DNA molecule on the concentration of different multivalent cations, where R ) (V /V0)1/3) R/R0 and R and R0 are the average radii for a DNA chain and the corresponding Gaussian chain, respectively. c1 and c2 are the concentrations where the relative populations of coil to globule are 1 × 10-2 to 1 × 102, respectively. The physicochemical meanings of c1, c2, and ct are given in the footnote for Table 1. Parameters used in the calculation are the same as those in Figure 4. In the theoretical calculations, we have not tried to fit the absolute values of the concentrations c to those of C in the experiment (Figure 3).
the change in the size of DNA molecules between the coil and globule states at the transition point) also decreases. This trend agrees well semiquantitatively with the experimental results given in Figure 3. By the optimization of the “adjustable parameters” in the theory, the correspondence between the theory and experiment will be further improved.
cation. (4) The amplitude of this collapse transition is larger for cations with a higher valence. In previous studies, it has been frequently reported that bivalent cations compete with three or more valence cations and inhibit DNA condensation13 and that bivalent cations cause DNA condensation only in solvents with a low dielectric constant (e.g., in an alcohol solution).16,32 These earlier experimental results have been obtained by the usual methods of observation, such as light scattering or sedimentation.8 In contrast to these previous results, with the direct observation of single DNA molecules, it has been shown that even in aqueous solution, diaminoalkanes with three and five methylene groups induce collapse of single DNAs.30 In the present article, we have confirmed that the bivalent cation DA3 induces a discrete transition in single DNA chains at least in the buffer solution used in the present study, albeit DA3 is much less effective in compacting DNA chain than spermidine and spermine. With a bivalent cation, the DNA shrinks gradually,
5. Conclusion The following properties of the coil-globule transition of single long DNA chains have been confirmed both experimentally and theoretically. (1) The DNA collapse transition induced by both bivalent and multivalent cations is discrete on the level of individual chains. (2) The cation concentration necessary to induce DNA collapse decreases with an increase in the valence of the cation. (3) The width of the coexistence region between the coil and globule states is rather large for bivalent cations and decreases with an increase in the valence of the
9400 J. Phys. Chem. B, Vol. 101, No. 45, 1997
Takahashi et al.
even in the coil state, around the transition point, whereas with spermidine or spermine the size of the coil remains essentially constant. This difference in the size of DNA at the level of individual chains causes a significant difference in the properties related to the ensemble average of the chains. Therefore, the discrete character of the transition induced by bivalent cations seems to have been overlooked in previous experiments that only examined the ensemble averages of DNAs.6,16,32 Besides DNAs, it is expected that synthetic charged polyelectrolytes may also be compacted by multivalent cations. Although there have been extensive studies on the condensation of polyelectrolytes in solution with multivalent cations,33,34 there have been few observations of the coil-globule transition at the level of a single chain. It would be interesting to extend the idea reported in the present paper to other synthetic polymers such as polystyrene derivatives.35-37 Acknowledgment. This work was supported in part by a grant from the Russian Foundation of Fundamental Research and by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Culture and Sports of Japan. Appendix The theory of DNA compaction by multivalent cations is based on the following assumptions. The total volume of the solution is divided into two parts: the effective volume of DNA macromolecules and the external solution. The free energy F of the DNA portion is written as
F ) Fela + Fmix + Ftrans + Felec
(1)
For the electrostatic interaction, the typical length is about 1 nm, i.e., the Debye-Hu¨ckel length in the usual buffered DNA solutions. Thus, it is plausible that the electrostatic contribution is not important in the free energy of the elongated coil in long DNA molecule as in the present research. (As the typical size of the elongated coil is on the order of micrometers, the average length between the segments in a DNA chain is more than about 10 nm.) On the other hand, in the globule state it has been already noted that the electrostatic change in the DNA chain is almost neutralized accompanied by the collapsed transition.27 This suggests that Felec does not contribute much to the free energy in the collapsed state of the long DNA chain. For the sake of simplification, we have, therefore, neglected Felec in the approximation in the following discussion. The free energy of the elastic deformation Fela of the chain was written in the Flory approximation modified by Birshtein to take into account possible chain compaction:38
{( ) ( ) }
Fela 3 V0 ) T 2 V
2/3
+
V V0
Ftrans ) T
∑
i)1,2
Ftrans i T
∑ S free,i ln z
)
i)1,2
Szfree,i
∑ Qi
Qi-
+ Vi
ln
+ Vi
i)1,2
∑Q
Q+free,i
+ free,i
i)1,2
2/3
(2)
where V is the effective volume of one DNA coil, V0 is the DNA volume in the Gaussian state, and T is the temperature. The free energy Fmix of the interaction of DNA molecules with solvent is written as in the Flory-Huggins theory:
Fmix V ) 3{(1 - φ) ln(1 - φ) + χφ (1 - φ)} T d
The DNA molecule possesses an extremely high, bare charge Qb (ca. 600 per Kuhn segment). However, owing to the condensation, the actual linear charge density of DNA is much lower. For example, the linear charge density of DNA in solution with Na+ ions is reduced by 76% from the bare charge.12,18 Since the neutralizing counterions QN reside within a thin surface layer,18 the following relationship will hold: QN ) 0.76QbN. In the solution of multivalent cations these counterions will be replaced by cations with higher valence because of the large gain in translational entropy due to release of monovalent cations. It has been well established that in the free DNA chain without condensation, QN depends mildly on the concentration of the cations. Thus, it is expected that the change in the translational entropy due to the change in QN is not significant. Strictly speaking, the charge density in the collapsed state would be less than that in the coil state. However, as a rough approximation, we will neglect the change of QN for simplification. In other words, the contribution due to the change of QN accompanied by the collapsed transition can be regarded as being included in the phenomenological parameter χ. The “free-moving” ions are distributed over the total volume of solution, i.e., the volume occupied by DNA molecules and the external solution. Under the conditions for Donnan equilibrium between the interior of DNA coil and external solution, it was taken into account the fact that not all the counterions inside the DNA coil are indeed free, even if they are not condensed in the Manning sense.18 The concentration of counterions close to DNA chains is higher than in the intracoil region far from DNA chains, and these counterions should be considered partly trapped. Therefore, the effective number of free intracoil counterions appearing under the conditions for Donnan equilibrium should be considerably diminished. Let us denote the number of such counterions as Q. The translational entropy Ftrans of small free-moving ions in the system (counterions of DNA, Q+free, multivalent cations with z charges, Szfree, and counterions of multivalent cations, Q-) is written as
(3)
where d is the size of an elementary lattice cell in the FloryHuggins model. Here, we assume that it is equal to the diameter of a DNA chain. φ is the volume fraction of DNA links inside a DNA coil (φ ) Nbd2/V), N is the number of Kuhn segments in one DNA molecule, and b is the length of the Kuhn segment.
ln
Vi
(4)
Here, index i refers to the DNA effective volume (i ) 1) and external volume of solution (i ) 2). The following relationships among parameters hold because of the particle conservation principle, (X)total ) X1 + X2 (here, X ) Q+free, Sz, Q-), (Sz)total ) z(Q-)total and the condition of total electroneutrality, zSzfree,1 + Q+free,1 ) Q + Q1-. The number of free counterions of DNA, Q+free, increases with the increase of concentration of multivalent cations that replace them at a surface layer with condensed counterions, and at high concentration of multivalent ions the following relationships are expected to be valid:
(Q+free)total ) QN + Q
(5)
Szfree,1 + Szfree,2 ) (Sz)total - QN/z
(6)
Collapsed Transition of Single Duplex DNA Equilibrium values of DNA size (swelling ratio R ) (V/V0)1/3) and the concentration of small ions in different parts of the system are determined by minimization of the total free energy subjected to common equilibrium conditions (equality of osmotic pressures and chemical potentials in the internal and external effective volume). References and Notes (1) Watson, J. D.; Hopkins, N. H.; Roberts, J. W.; Steitz, J. A.; Weiner, A. M. Molecular Biology of the Gene, 4th ed.; The Benjamin/Cummings Publishing Company, Inc.: Menlo Park, CA, 1987. (2) Lerman, L. S. Proc. Natl. Acad. Sci. U.S.A. 1971, 68, 1886. (3) Laemmli, U. K. Proc. Natl. Acad. Sci. U.S.A. 1975, 72, 4288. (4) Chattoraj, D. K.; Gosule, L. C.; Schellman, J. A. J. Mol. Biol. 1978, 121, 327. (5) Post, C. B.; Zimm, B. H. Biopolymers 1982, 21, 2123. (6) Widom, J.; Baldwin, R. L. Biopolymers 1983, 22, 1595. (7) Murphy, L. D.; Zimmerman, S. B. Biochim. Biophys. Acta 1994, 1219, 277. (8) Bloomfield, V. A. Curr. Opin. Struct. Biol. 1996, 6, 334. (9) Tabor, C. W.; Tabor, H. Annu. ReV. Biochem. 1984, 53, 749. (10) Porter, C. W.; Bergeron, R. J. Science 1983, 219, 1083. (11) Hung, D. T.; Marton, L. J.; Deen, D. F.; Shafer, R. H. Science 1983, 221, 368. (12) Bloomfield, V. A.; Wilson, R. W. Polyamines in Biology and Medicine; Morris, D., Marton, L. J., Eds.; Marcel Dekker, Inc.: New York, 1981; Chapter 10. (13) Braunlin, W. H.; Strick, T. J.; Record, M. T., Jr. Biopolymers 1982, 21, 1301. (14) Vertino, P. M.; Bergeron, R. J.; Cavanaugh, P. F., Jr.; Porter, C. W. Biopolymers 1987, 26, 691. (15) Baeza, I.; Iba´n˜ez, M.; Wong, C.; Cha´vez, P.; Gariglio, P.; Oro´, J. Origins Life EVol. Biosphere 1992, 21, 225. (16) Bloomfield, V. A.; Ma, C.; Arscott, P. G. Macro-Ion Characterization from Dilute Solutions to Complex Fluid; Schmitz, K. S., Ed.; American Chemical Society: Washington, DC, 1994; Chapter 15.
J. Phys. Chem. B, Vol. 101, No. 45, 1997 9401 (17) Pelta, J.; Livolant, F.; Sikorav, J.-L. J. Biol. Chem. 1996, 271, 5656. (18) Manning, G. S. Q. ReV. Biophys. 1978, 11, 179. (19) Cruz, M. O.; Belloni, L.; Delsanti, M.; Dalbiez, J. P.; Spalla, O.; Drifford, M. J. Chem. Phys. 1995, 103, 5781. (20) Rouzina, I.; Bloomfield, V. A. J. Phys. Chem. 1996, 100, 4292, 4305. (21) Grosberg, A.; Rabin, Y.; Havlin, S.; Neer, A. Europhys. Lett. 1993, 23, 373. (22) Grosberg, A. Yu; Khokhlov, A. R. Statistical Physics of Macromolecules; AIP Press: New York, 1994. (23) Widom, B. J. Phys. Chem. 1996, 100, 13190. (24) Vasilevskaya, V. V.; Khokhlov, A. R.; Kidoaki, S.; Yoshikawa, K. Biopolymers 1997, 41, 51. (25) Vasilevskaya, V. V.; Khokhlov, A. R.; Matsuzawa, Y.; Yoshikawa, K. J. Chem. Phys. 1995, 102, 6595. (26) Mel’nikov, S. M.; Yoshikawa, K. Biochem. Biophys. Res. Commun. 1997, 230, 514. (27) Ueda, M.; Yoshikawa, K. Phys. ReV. Lett. 1996, 77, 2133. (28) Yoshikawa, K.; Takahashi, M.; Vasilevskaya, V. V.; Khokhlov, A. R. Phys. ReV. Lett. 1996, 76, 3029. (29) Yoshikawa, K.; Kidoaki, S.; Takahashi, M.; Vasilevskaya, V. V.; Khokhlov, A. R. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 876. (30) Yoshikawa, Y.; Yoshikawa, K. FEBS Lett. 1995, 361, 277. (31) Yoshikawa, K.; Matsuzawa, Y.; Minagawa, K.; Doi, M.; Matsumoto, M. Biochem. Biophys. Res. Commun. 1992, 188, 1274. (32) Ma, C.; Bloomfield, V. A. Biophys. J. 1994, 67, 1678. (33) Narh, K. A.; Keller, A. J. Polym. Sci., Part B: Polym. Phys. 1993, 31, 231. (34) Delsanti, M.; Dalbiez, J. P.; Spalla, O.; Belloni, L.; Drifford, M. Macro-Ion Characterization from Dilute Solutions to Complex Fluid; Schmitz, K. S., Ed.; American Chemical Society: Washington, DC, 1994; Chapter 29. (35) Swislow, G.; Sun, S.-T.; Nishio, I.; Tanaka, T. Phys. ReV. Lett. 1980, 44, 796. (36) Tanaka, F. J. Chem. Phys. 1985, 82, 4707. (37) Park, I. H.; Wang, Q.-W.; Chu, B. Macromolecules 1987, 20, 1965. (38) Birshtein, T. M.; Prymitsin, V. A. Vysokomol. Soedin. 1987, 29A, 1858.
