Surface-Induced DNA Superhelicity - American Chemical Society

Carlo simulation, we studied the adsorption process of circular semiflexible ... weak twisting adsorbed on the surface assumes higher superhelical twi...
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Biomacromolecules 2000, 1, 459-465

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Surface-Induced DNA Superhelicity Yuri S. Velichko Graduate School of Human Informatics, Nagoya University, Nagoya 464-8601, Japan, and Department of Physics, Moscow State University, Moscow 117234, Russia

Kenichi Yoshikawa* Department of Physics, Graduate School of Science, Kyoto University and CREST (CoreResearch for Evolutional Science and Technology) of Japan Science and Technology Corporation, Kyoto 606-8502, Japan

Alexei R. Khokhlov* Physics Department, Moscow State University, Moscow 117234, Russia Received March 13, 2000; Revised Manuscript Received July 3, 2000

Certain biopolymers, such as DNA, have a double-stranded twisted structure and frequently exhibit a supercoiled conformation. Over the past decade, extensive conformational analyses of different biopolymers have been performed using atomic force microscopy. In this technique, a necessary step in sample preparation is the adsorption of molecules on the surface, which could affect the chain conformation. Using a Monte Carlo simulation, we studied the adsorption process of circular semiflexible twisted double-stranded polymer chains on a solid surface with an emphasis on the conformational properties. We found that the conformation of an adsorbed chain strongly depends on the number of double-helical turns of the chain. Chains with weak twisting adsorbed on the surface assumes higher superhelical twisting, whereas the coiled state is seen in the bulk solution. After the adsorption, double-helical turns are accumulated in the adsorbed (“train”) sections giving more conformational freedom to the nonadsorbed (“loop”) sections. Chains with strong twisting show small conformational changes with adsorption. In both cases, superhelicity shows the opposite sign of writhing. 1. Introduction The adsorption of linear homo- and heteropolymers onto a surface has been actively studied by many researchers1-11 over the past several decades. Interaction between such polymers and a surface plays a significant role in biological systems. Some biopolymers, like double-stranded DNA, have a multistranded twisted structure, and even in bulk solutions, they exhibit marked conformational changes in their higherorder coiling structures depending on the degree of supercoiling. In particular, the supercoiled structures of circular DNA molecules have attracted considerable attention.12-17 The development of the methods of visualizing of macromolecules has stimulated the studies on the conformation analyses of large polymer molecules, especially DNA. Using fluorescence microscopy18,19 (FM), the actual changes in a single-chain molecule in solution have been demonstrated. Changing the molecular environment leads to an easily observed an actual change in the conformation in giant DNA in coil-globule transition. However, strong fluctuations and unclear images in the FM observation make the analysis of higher-order structures of chain rather complicated. Atomic force microscopy17,20-23 (AFM) makes it possible to see the * To whom correspondence should be addressed. E-mails: yoshikaw@ scphys.kyoto-u.ac.jp and [email protected].

exact conformation of a molecular chain. In preparing a sample for AFM observation, a drop of DNA solution is placed on the solid substratum and allowed to adsorb. After an incubation period, the samples are rinsed in water and blown with a dry gas. There are various methods for sample preparation, but all of them17,20-23 include a step in which a drop of solution is placed on a surface. Since molecules adsorb to the substratum, adsorption is a necessary step in AFM visualization. The deformation of molecules and polarization of the surface is inevitably generated during adsorption and strong polarization should occur in the case of polyelectrolytes. Such changes should be taken into account in the studies of the conformation of polymer molecules in AFM observation. An important problem in the experimental analysis of the AFM images is to clarify the effect of adsorption on the conformation of circular supercoiled DNA molecules. A circular DNA chain is characterized by a topological constant, i.e., the linking number Lk or number of doublehelical turns, which can be changed only by breaking the DNA strands. DNA molecules have relaxed structures with about 10.5 base pairs per double-helical turn.12,14 Changes in the linking number affect the torsional tension of DNA molecules and, as a result, the entire conformation of the

10.1021/bm000021l CCC: $19.00 © 2000 American Chemical Society Published on Web 08/03/2000

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chains. It is well-known that DNA molecules with a disturbed linking number exhibit supercoiled structures.12-17 Long polymer chains dissolved in solution show scaling behavior.26-28 The mean square end-to-end distance 〈R2〉1/2 and mean square radius of gyration 〈RG2〉1/2 obey low power dependencies ∝ NMν, where ν is the fractal dimension. The fractal dimension is independent of the chemical structure of the polymer chain. For a long flexible polymer molecule with excluded volume, it is equal to ν ) 3/(dim + 2), where dim is the dimensionality of space. Spatial characteristics, such as mass distribution, are not completely determined by the fractal dimension. Fractals, characterized by the same fractal dimension, can differ by texture and are distinguished by their “lacunarity”.29 Lacunarity is a counterpart of fractal dimension and is closely related to the mass distribution. It is higher if free regions of the fractal structure include a large amount of space. It has been shown for a double-stranded polymer chain30 that twisting affects the mass distribution of the chain and changes the scaling properties by affecting the lacunarity of the chain: 〈RG 〉 ∝ (Lk/NM) NM 2

2γ

2ν

(1)

where ν is the “true” fractal dimension and γ is the twisting exponent to characterize how the chain size scales with the value of the linking number normalized by the chain length Lk/NM. The behavior of a polymer chain near the surface depends on the adsorption energy.2,6 When the energy of adsorption is less than a critical value, the number of monomer units located on the surface Nads is independent of chain length NM. On the other hand, when the energy of adsorption is larger than the threshold, adsorbed structures are characterized by binding of the chain with the surface, and Nads increases with chain length NM. At the adsorption threshold, the polymer chain consists of adsorbed parts (trains) and the other parts freely dissolved in the solution (loops). The distribution of trains and loops depends on the adsorption energy and solution properties.6 Another question connected with the adsorption process is the conformational behavior of a polymer chain. If a chain is a heteropolymer, there will be selective adsorption of its parts with the effect of translational entropy or differences in interaction with the surface. Similar adsorption behavior will be encountered for copolymers,9 stiff-flexible copolymers,10 or polymers with coiled and helical parts.11 If we compare these with a twisted homopolymer chain we can identify one principle distinction: double-helical turns are not connected with a chain sequence, and the chain cannot be mapped by twisting. Computer simulation techniques was perfectly developed during the last two decades and widely applied for studies of different molecular systems. Biopolymers are rather complex for theoretical and experimental studies. By the way, results from the computer experiment show good agreements with both theory and experiment. For example, morphological variation of DNA molecules in collapsed state was studied in detail24,25 and confirmed by experimental observation. Our goal is to clarify the effect of a surface on the conformational behavior of a double-stranded twisted poly-

Figure 1. Snapshot of a double-stranded polymer chain NM ) 100, Lk ) 10.

mer chain with different twisting value using a Monte Carlo simulation. 2. Model We studied a ladderlike chain model30 for a doublestranded twisted polymer chain (Figure 1). A chain is constructed of a pair of linear strands of length NM where each monomer unit is connected to three neighboring monomer units by two tangent bond vectors BtR(i) along the strand, where R is the strand index, and by one normal bond vector b n(i) with monomer units of another strand. Two additional tangent vectors connect the end pairs of the chain and form a circular or closed chain. We use the off-lattice model,31 where each monomer unit is represented as a sphere with radius rbead ) 1, connected in three-dimensional space by bonds, with various length lbond ∈ (2rbead,lmax) (for normal and tangent bond vectors), where lmax ) 2.8rbead is the maximum bond length. Bending stiffness is represented by tangent angle θR(i), which exhibits dynamic degrees of freedom with a temperature-dependent potential Ub(θ)/kBT ) kb(cos(θ(i)) - cos(θ0))2

(2)

where T is the temperature, θ0 ) π/6 is the equilibrium bond angle, and kb is a stiffness parameter of the bending energy. Torsional stiffness of the chain is represented by torsional potential as a function of the angle ϑ(i) between neighboring normal vectors, and is described by a relationship similar to eq 2, with kt replacing kb as a parameter of torsion energy stiffness. In our simulation, we use stiffness parameters of kb ) kt ) 3. The forces in the interaction of monomer units with the surface are assumed to be short-range. The surface is modeled as a penetrable adsorption potential containing short-range repulsive, proportional to z-6, and van der Waals attractive, proportional to -z-3, parts:

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Figure 2. Snapshots of negatively supercoiled conformations obtained in the simulation with chain length NM ) 100, bending and torsional stiffness parameters kb ) kt ) 3, linking number Lk ) 8, and attractive potential C ) 7. Double-stranded chain is shown by the main axis.

Uads(z)/kBT )

{

∞ ze0 C[(rbond/z)6 - (rbond/z)3], z > 0

(3)

where C is an adsorption energy constant, which characterizes the dielectric properties of the surface material and the electric polarizability of the monomer unit, and z is a coordinate to determine the distance from the surface. A monomer unit can approach the surface at a minimal distance equal to rbrad while the adsorption energy Uads(r) reaches a minimum at distance z* ) 2-1/3rbead = 1.25rbead. At distance 3rbead, which corresponds to the next closely packed monomer unit in the z-coordinate direction, the adsorption energy is about 0.1Uads(z*). The system is exposed to thermal agitation by the standard method in a Monte Carlo simulation: i.e., random jumping of monomer units in random directions by displacements (∆x, ∆y, ∆z) chosen randomly from the interval [-0.4, 0.4] for all directions independently. The transition probability of a jump is accepted according to the standard Metropolis algorithm32 W ) exp(-∆E/kBT), where ∆E is the total change in potential energy. In each Monte Carlo update step, each monomer unit has one chance to move. Chain twisting is determined by the linking number Lk or by the number of times one strand of a chain is linked to the other strand.12,16 This parameter is fixed constant during each run of the simulation and describes the initial configuration. The linking number will give a different torsional stress for chains of different length NM. It is more useful to describe the chain in terms of normalized parameters independent of the chain length. The linking number density σ, used in our simulation, is the linking number Lk normalized by the chain length NM σ)

Lk NM

(4)

and is varied in our simulation over the interval σ ) [0, 0.22]. For a circular chain, the linking number Lk12,14,16 is a

Figure 3. (a) Dependence of writhing number normalized on the chain length Wr/NM on the linking number density σ for chains of length NM ) 100 and 200 in bulk solution and adsorbed on a surface with attractive potential C ) 7. (b) Dependence of writhing number Wr on the attractive potential C for chains of length NM ) 100 with linking number density σ ) (0.1;0.16;0.2).

topological constant and can be subdivided into two components, Lk ) Tw + Wr, where twisting Tw is the number of helical turns that one strand makes around the other and writhing Wr is a factor that describes the number of intersections of chain axes: Wr )

Br(s1) × B d r(s′1)]b r 12 1 II [d 4π C C′ |b r 12|

(5)

where B dr(si) ) (dr b(si)/ds), ds is a displacement along the ith strand, ri(s) is the position of the ith strand in the lab frame, rij ) ri(s) is the distance between the ith and jth strands, and the integral is taken along the helix axis C. We performed both lattice and off-lattice simulations (we present here only off-lattice model results) and obtained essentially the same dependencies for all characteristic values. 3. Results It has been indicated30 that twisting affects the conformation of a circular double-stranded polymer chain, such as in the formation of supercoiled structures, only when the linking number density reaches a critical value σS. The critical linking number density is associated with a kind of saturation of chains with linking number Lk. One full turn of the double-

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Figure 4. (a) Dependence of adsorption energy Eads on the attractive potential C for chains with linking number density σ ) (0/2;0.2). (b) Dependence of the location of the coil center of mass ZCM on the attractive potential C for chains with linking number density σ ) (0.1;0.2).

helix affects only the behavior of the part of the chain up to the persistent length (lP). Therefore, saturation occurs when L ∼ lP Lk

(6)

where L is the contour length of the chain and lP is the persistent length of the chain. Thus, the full region of variation of σ can be divided into two parts by the critical linking number density σS. At σ > σS, the chain forms supercoiled structures.30 The critical linking number density for the chain used in our simulation was found to be σS ) 0.13. Figure 2 shows snapshots of a chain with linking number density σ ) 0.08 on a surface with attractive potential C ) 7. An unsaturated chain with linking number (σ < σS) shows a supercoiled conformation characterized by negative writhing of about Wr ) -2. The chain forms some loopike structures (Figure 2, parts a and b), whereas in the bulk solution the chain is similar to a random coil and writhing is about zero. Figure 3a shows the dependence of writhing number Wr/ NM divided on the chain length on the linking number density σ for a chains in the bulk solution and at the surface with attractive potential C ) 7. The writhing number in the region of small linking number density, where the chain is unsaturated, varies with σ. On the surface, with small σ, a chain

exhibits negative superhelicity, whereas in the bulk solution at the same value of σ a chain behaves as a simple coil and the value of the writhing number is about zero. On the other hand, the difference between the writhing numbers of the chain in the bulk solution and on the surface is negligible at a high value of σ, where the chain in the bulk is saturated. Figure 3b shows the dependence of writhing number Wr on the attractive potential C for chains with different linking number density σ. Wr depends on the attractive potential C for chains with a linking number density lower than the critical value σ < σS, while the writhing of chains with a higher value of σ (σ > σS) does not vary with C. Writhing Wr of the unsaturated chain decreases to a negative value, indicating that the chain assumes a supercoiled conformation, while in the bulk solution the chain takes a relaxed conformation with Wr ) 0. Adsorption energy Eads (Figure 4a) shows monotonic behavior with an increase in the attractive potential C for chains with different linking number densities σ. We have found that the critical attractive potential CC, corresponding to the point when adsorption begins, depends on the linking number density σ. Chains with a higher linking number density begin adsorption with a smaller attractive potential C than chains with a lower linking number density σ. The behavior of the location of the center of mass (Figure 4b) also shows a different critical attraction potential CC for chains with a different linking number density. It is also noted that the adsorption is enhanced for less twisted chains as shown in the difference of slope for chains with different linking number density (Figure 4a). Owe to the limited accuracy in the calculation at the present, it may be safe to avoid concluding any exact dependencies. A change in chain twisting affects the torsional and bending angles, and thus the torsional and bending energies NM

Etors )

Ut (θ(i)) ∑ i)0

(7)

2 NM

Ebend )

∑ ∑Ub(ϑj(i)) j)0 i)0

(8)

where i and j are indices of strands and monomer units, and potentials Ut(θ(j)) and Ub(ϑi(j)) are calculated from eq 2. Parts a and b of Figure 5 show the effect of adsorption on the bending and torsional energies for a chain unsaturated with regard to linking number density. Torsional energy increases and bending energy decreases with increasing attractive potential C. We also found a negligible effect on a saturated chain (σ ) 0.16), when the linking number density approaches a critical value σS. With an increase in the linking number density, this effect becomes weaker and eventually disappears when the linking number density is higher than a critical value (σ ) 0.2). An increase in linking number density makes a polymer chain more compact due to the formation of some superhelical structures.30 Adsorption also affects the chain size. Figure 6 shows the dependence of 〈RG2/NM〉, i.e., the mean square radius of gyration divided by the chain length, on the attractive potential C for different values of linking

Surface-Induced DNA Superhelicity

Figure 5. (a) Dependence of bending energy Ebend and torsional energy Etors on the attractive potential C for chains with linking number density σ ) (0.1;0.16;0.2).

Figure 6. Dependence of mean square radius of gyration divided by the chain length RG2/NM on the attractive potential C for chains with linking number density σ ) (0.1;0.16;0.2).

number density σ. We found that the radius of gyration 〈RG2/ NM〉 increases as adsorption occurs, and a chain unsaturated with regard to linking number shows a stronger effect than a saturated chain. 4. Discussion We studied the effect of a surface on the conformational behavior of circular double-stranded polymer chains with different linking number density values.

Biomacromolecules, Vol. 1, No. 3, 2000 463

The formation of supercoiled structures begins when a polymer chain is saturated with double-helical turns.30 We observed conformational changes in the higher order coiling structures of a chain unsaturated with regard to linking number (Figure 3), from open to supercoiled, with an increase in the attractive potential C, while the conformational properties of saturated chain remain the same as in the bulk solution. The radius of gyration, together with torsional and bending energies, also shows a weaker correspondence than those in an unsaturated chain. It seems likely that a chain saturated with regard to linking number is more resistant to conformational changes than an unsaturated chain. The writhing number Wr shows independence on the chain length. All points (Figure 3a) of normalized writhing number Wr/NM lies on the same line that indicates formation of local structures. The number of loops increases with chain length NM and change chain superhelicity. Adsorption affects the chain conformation and makes it more flat. With rather strong attractive potential C polymer chain lies on the surface and shows properties closed with two-dimensional chain. The critical linking number density30 σC ∝ lP-1 becomes sensible to the changes of conformational freedom and differ on the surface and in the bulk solution. Figure 3a shows that growth of the writhing number Wr for adsorbed chain begins earlier than for dissolved polymer chain and σC(C ) 7)/σC(C ) 0) is closed to 0.5. By the way, the writhing number for adsorbed chains decreases until σ < σC and is negative whereas for the chains in the bulk solution it is closed to zero. Together with the effect on the critical linking number density, adsorption changes the relation between bending and torsional tensions. We found increase of torsional energy Etors and decrease of bending energy Ebead (Figure 5, parts a and b) with increase in the adsorption potential C for unsaturated with regard to linking number chains. Increase of torsional energy corresponds to increase of torsional angles or chain twisting. Calculation of changes of torsional and bending energies for adsorbed and dissolved in the bulk solution chains ∆Etors(C) ) Etors(C) - Etors(C ) 0)

(9)

∆Ebend(C) ) Ebend(C) - Ebend(C ) 0)

(10)

shows that ∆Etors(C) + ∆Ebend(C) < 0; i.e., energetically it is more advantageous to obtain conformation with high twisting. We found here that unsaturated with regard to linking number polymer chain forms supercoiled structures with negative writhing. We also evaluate the distribution of torsional and bending energies of bond pair as a function of distance from the surface H(z) ) 〈U(ϑ(i))〉

(11)

where H(z) is the average torsional or bending energy of bond pair in a layer of width ∆z, at a distance z from the surface, ϑ(i) is the angle between the ith and i + 1 bond vectors, which are located in the ∆z layer, and U(ϑ(i)) is the torsional or bending potential energy for each energy average, where the average is calculated for all different conformations. Figures 7 and 8 show the distribution of

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Figure 7. Distribution of average torsional energy per bond Htors on the distance z from the surface for chains with linking number density σ ) (0.1;0.2) and different attractive potential C.

torsional (Htors(z)) and bending (Hbend(z)) average energies for chains with linking number density σ ) 0.1 and 0.2 for different values of the attractive potential (C > CC). Note that z ) 1 corresponds to the minimal distance from the surface which the monomer unit can reach and that at z ) 21/3 ≈ 1.25 the adsorption energy potential in (3) has a minimum. When the chain lies flat on the surface, torsional energy has a minimal value, due to a smaller number of conformations with a large torsional angle in the ∆z layer. We can find a peak, which appears at z ≈ 2.5 from the surface on the curve of average torsional energy (Figure 7, C ) 7.5), whereas the distribution of bending energy (Figure 8, C ) 7.5) has a minimum. With an increase in attractive potential C, the peak intensity at Htors(z) grows, and the distribution outside the peak region tends to decrease with an increase in distance. The bending distribution also shows a peak, but its position is shifted (z = 4) from the peak in the distribution of torsional energy, and Hbend is minimal at about z = 2.5. An increase in the linking number density σ increases the average torsional energy and slightly decreases the bending energy per bond pair, and makes Htors(z) less dependent on adsorption energy. We found a difference in the behavior of Htors only at a distance greater than the average coil size (zdistinction = 7, zCM = 3) for attractive potential C > 7.5. With an increase in distance from the surface, the distribution of average bending energy (Figure 8, C ) 3.5) shows a peak, close to the surface, and outside the peak region, the distribution is practically constant. With an increase in attractive potential C, the peak intensity at Hbend(z) shifts from the surface. We suppose that an attractive surface accumulates chain fragments with a large number of double-helical turns due to lower translational entropy. A surface may affect the linking number density distribution along the chain, through

Velichko et al.

Figure 8. Distribution of average bending energy per bond Hbend on the distance z from the surface for chains with linking number density σ ) (0.1;0.2)) and different attractive potential C.

the decrease in the number of double-helical turns in loop parts and increase of them in adsorbed sections (train). Accumulation of double-helical turns in adsorbed sections may happened until linking number density in the train parts is smaller than critical value σ < σC. Since the linking number Lk ) Tw + Wr is constant, changes in twisting has also influenced the writhing number Wr. For a unsaturated chain decrease in linking number density in the loop parts may decrease, twisting to zero, or change Wr. Negative superhelicity appears as looplike structures develop, and is a result of the inhomogeneous distribution of double-helical turns along the chain. The sign of writhing Wr ) Lk - Tw is negative, because the twisting number becomes higher than the linking number, Lk < Tw. A chain saturated with regard to linking number has more double-helical turns than an unsaturated chain, and changes in the linking number distribution are more difficult to realize. Adsorption of unsaturated chain is accompanied by changes in the torsional and bending energies together with changes in the chain superhelicity, while the saturated chain shows strong resistance to these changes. During the adsorption chains become more flat, what is pointed out by growth of radius of gyration (Figure 6). Compared with unsaturated polymers, the saturated chain shows more weak growth of size together with conservation of torsional and bending energies (Figure 5, parts a and b). It means that “motion” in the phase space with changes in the adsorption potential C occurs between conformations with fixed torsional and bending energies. This indicates the important difference in the behaviors of saturated and unsaturated chains on the surface. Compaction of a chain30 with an increase in linking number density σ affects its properties. We found different

Surface-Induced DNA Superhelicity

critical attractive potential values CC when adsorption begins, for chains with a different linking number density. Chains saturated with regard to linking number adsorb with CC ) 2 (Figure 4), while unsaturated chains adsorb only with an attractive potential of CC ) 3. Furthermore, the region of the adsorption threshold (Figure 4) for a chain saturated with regard to linking number is about twice as wide as that for an unsaturated chain, due to a more compact state of highly twisted polymers. For this reason the inclination of adsorption energy Eads, in the dependence on the attractive potential C (Figure 4a), decreases with increase in linking number density. These results also reflect the resistance of the saturated chain to the conformational changes. 5. Conclusion We report the results of a Monte Carlo simulation study on the adsorption of a twisted double-stranded polymer chain. We found that such a chain becomes less sensitive to external perturbation with an increase in the linking number density. A chain with a small number of double-helical turns shows substantial conformational changes with an increase in the attractive surface potential, while a chain saturated with regard to linking number shows essentially the same writhing number before and after adsorption. Adsorption affects the relation of bending and torsional forces and the distribution of double-helical turns along the chain, concentrating them in close vicinity to the surface. We also found that the linking number density affects the critical attractive potential CC, indicating that a chain saturated with regard to linking number is adsorbed more easily. Acknowledgment. We are very grateful to Prof. K. Takeyasu for interesting and stimulating discussion. References and Notes (1) Bruch, L. W.; Cole, M. W.; Zaremba, E. Physical Adsorption: Forces and Phenomena; Clarendon Press: Oxford, England, 1997. (2) Eisenriegler, E. Polymer Near Surface; World Scientific: Singapore, 1993. (3) Sumithra, K.; Baumgaertner, A. J. Chem. Phys. 1998, 109, 1540.

Biomacromolecules, Vol. 1, No. 3, 2000 465 (4) van Eijk, M. C. P.; Cohen Stuart, M. A.; Rovillard, S.; De Coninck, J. Eur. Phys. J. B 1998, 1, 233. (5) Kuznetsov, D. V.; Sung, W. J. Phys. II Fr. 1997, 7, 1287. (6) Semenov, A. N.; Bonet-Avalos, J.; Johner, A.; Joanny, J. F. Macromolecules 1996, 29, 2179. (7) Johner, A.; Bonet-Avalov, J.; van der Linder, C. C.; Semenov, A. N.; Joanny, J. F. Macromolecules 1996, 29, 3629. (8) Sumithra, K.; Baumgaertner, A. J. Chem. Phys. 1999, 110, 2727. (9) Zheligovskaya, E. A.; Khalatur, P. G.; Khokhlov, A. R. J. Chem. Phys. 1997, 106, 8588. (10) van der Linder, C. C.; Leermakers, F. A. M.; Fleer, G. J. Macromolecules 1996, 29, 1172. (11) Carri, G. A.; Muthukumar, M. J. Chem. Phys. 1999, 82, 5405. (12) Vologodskii, A. V. Topology and Physical Properties of Circular DNA; Nauka: Moscow, 1988. (13) Cozzarelli, N. R.; Wang, J. C. DNA Topology and its Biological Effects; Cold Spring Harbor: New York, 1990. (14) Bates, A. D.; Maxwell, A. DNA Topology; Oxford University Press: New York, 1993. (15) Klenin, K. V.; Vologodslii, A. V.; Ashkevich, V. V.; Dykhne, A. M.; Frank-Kamenetskii, M. D. J. Mol. Biol. 1991, 217, 413. (16) White, J.; Bauer, W. J. Mol. Biol. 1986, 189, 329. (17) Boles, T. C.; White, J. H.; Cozzarelli, N. R. J. Mol. Biol. 1990, 213, 931. (18) Yanagida, M.; Hiraoka, Y.; Katsura, I. Cold Spring Harbor Symp. Quantum Biol. 1983, 48, 177. (19) Minagawa, K.; Matsuzawa, Y.; Yoshikawa, K.; Matsumoto, M.; Doi, M. FEBS Lett. 1991, 295, 67. (20) Yang, J.; Takeyasu, K.; Shao, Z. FEBS Lett. 1992, 301, 173. (21) Tanigawa, M.; Okada, T. Anal. Chim. Acta 1998, 365. (22) Ye Fang Hoh, H. J. Am. Chem. Soc. 1998, 120, 8903. (23) Ye Fang Hoh, H. Nucleic Acids Res. 1998, 26, 588. (24) Noguchi, H.; Yoshikawa, K. J. Chem. Phys. 1998, 109, 5070. (25) Ivanov, V. A.; Paul, W.; Binder, K. J. Chem. Phys. 1998, 109, 5659. (26) Flory, P. J. Statistical Mechanics of Chain Molecules; Wiley: New York, 1969. (27) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1978. (28) Grosberg, Yu.; Khokhlov, A. R. Statistical Physics of Macromolecules; American Institute of Physics: New York, 1994. (29) Meakin, P. Fractals, Scaling and Growth Far from Equilibrium; Cambridge University Press: Cambridge, England, 1998. (30) Yu. Velichko, S.; Yoshikawa, K.; Khokhlov, A. R. J. Chem. Phys. 1999, 111, 9424. (31) Milchev, A.; Paul, W.; Binder, K. J. Chem. Phys. 1993, 99, 4786. (32) Metropolis, N.; Rosenbluth, A.; Rosenbluth, M.; Teller, A.; Teller, E. J. Chem. Phys. 1953, 21, 1087. (33) Bednar, J.; Furrer, P.; Stasiak, A.; Dubochet, J.; Egelman, E. H.; Bates, A. D. J. Mol. Biol. 1994, 235, 825.

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