Bulletin of Mathematical Bioloyy Vol. 54, No. 5, pp. 727-732, 1992. Printed in Great Britain.

0092-8240/9255.00+ 0.00 Pergamon Press Ltd

© 1992 Society for Mathematical Biology

ISING M O D E L FOR B - Z TRANSITION IN SUPERCOILED DNA

SUBUENDU GHOSH Department of Biophysics, University of Delhi South Campus, New Delhi-110021, India

The possible existence of nucleic acids in right-handed and left-handed helical forms is considered. A statistical mechanical model is developed to obtain an expression for a change in twist during helical transformation in terms of corresponding free energies and linking for a supercoiled DNA. The theoretically predicted values are compared with those determined experimentally. The physico-chemical significance of the parameters is discussed.

Introduction. The existence of nucleic acids in different helical forms, especially the left-handed Z DNA structure in vitro has been well documented (Zimmerman, 1982; Dickerson et al., 1982; Rich et al., 1984; Zhong and Johnson, 1990; Ho et al., 1990; Lamba et al., 1989). A variety of conditions such as salt concentration, temperature, presence of ethanol and some specific proteins (Umehara et al., 1990; Gessner et al., 1989) and methylation (Doefler, 1983) are known to influence the conformational state of DNA. The latter also depends on the base-sequence (Gessner et al., 1989). However, the possible existence of left-handed DNA in vivo and its physiological significance are still widely debated. B to Z helical transformation can be induced in supercoiled DNA by untwisting the macromolecule (Hao and Olson, 1989; Benham, 1988; Singleton et al., 1982; Nordheim et al., 1982; 1983; Haniford and Pulleyblank, 1983; Wang et al., 1982). The thermodynamics of B - Z transition in supercoiled DNA has been considered by Sen and Majumdar (1987). Since these transformations are co-operative in nature (Ivanov et al., 1982) statistical mechanical methods offer a convenient approach to understanding the underlying physicochemical basis. Benham (1980, 1981, 1982, 1988) dealt with the problem of conformational transitions in DNA from a theoretical (statistical mechanical) point of view. Wang et al. (1982) and Haniford and Pulleyblank (1983) considered the B - Z transition in a set of plasmids containing d(pCpG), d(pCpG), as a function of supercoiling. A statistical mechanical model with all or none sequence assumption was developed by Peck and Wang (1983). In the present paper, we present an Ising model (one- dimensional) for the system of DNA with left-handed and right-handed helical forms existing in equilibrium.

727

728 S. GHOSH

The Model. The proposed model essentially belongs to a class of models considered by Z imm and Bragg (1959). In this, the helical states of D N A are considered, as a first approximat ion, to exist in r ight-handed (R) and left- handed (L) form in equilibrium with each other, i.e. R ~ - L .

Considering nearest neighbour interaction, the state of a complementary pair of nucleotides may be expressed as:

i-1 i R L

R 1 a

L a s (1)

where the relative weight factor of an R state with a neighbouring R state is 1, that with a neighbouring L state is a, and the relative weight factor of an L state with a neighbouring L state is s.

The eigenvalue of the matrix is given by:

1--2o_ s a 2 =0" (2)

1 Hence, 2 = ~ {(1 + s) _+ x/(1 -- s) 2 + 4a 2} (3)

The part i t ion function of the nucleotide chain of length N is given by Z(N) = C 2 m a x N , where C is a constant.

Hence, the fraction of nucleotides in the L state is:

1 0 In Z(N) s {1 - ( l - s ) O L = N 0 1 n s - -22 x / (1 - - s )2+4a2J" (4)

We express the weight factors as multiples of two separable functions, comprising of free energy (AG) of transition, and the linking difference ( a - %) as a = f l ( A G s ) f 2( c~ - ao), s =fx (AGR--,L)f 2(c~ -- %), where AG R_~L refers to the free energy of the r ight-handed to left-handed helical transition, and AGj refers to the free energy of left-right junc t ion formation.

To be more explicit we write:

o- = exp{ - A G s / R T + K(c~ -- 0%) 2 }

s = exp{ -- A G a _ , L / R T + K(c~ - %)2} (5)

where K is a constant parameter for a given nucleotide sequence.

ISING MODEL FOR B Z TRANSITION IN SUPERCOILED DNA 729

The rationale for choosing a function fz(c~-c%)=exp{K(e--c%) 2} is as follows. For a closed circular DNA the probability of transition would increase with an increase in supercoiling energy and the latter is given by a factor, constant (ct-c%) 2 (Camrini-Otero and Felsenfeld, 1978).

We note that the change in twisting can be written as:

--(ATw)=NTO (6)

where Nis the number ofnucleotide pairs in the DNA and 7 the change in twist per nucleotide.

Substituting expression (6) into equation (4) we obtain:

- - (ATw)=NY~ 1 -- x / ( l _ s ) 2 + 4 o 2 j (7)

Results and Discussion. In order to test the model the experimental viability of equation (7) has been established. The accepted range of values of free energy term and linking difference have been discussed by Nordheim et al. (1982; 1983) and Peck and Wang (1983). Following this the to ta l change in twist (-ATw) has been computed for a range of values of linking difference (c~-c%) with the help of equations (5) and (7). We observe that the model gives sigmoidal curves, the exact pattern of the experimental plots of (ATw) versus (e -e0) . The computed graphs are fitted with the experimental points (Peck and Wang, 1983), as shown in Fig. 1. The parameters (7 and K) for the best fit for different inserts n (the length of dG-dC inserted) into pBR 322 DNA are given in Table 1. The corresponding best choice of free energy values are AGj = 2.4 Kcals/mole and AGR_, L = 2 Kcals/mole.

Table 1 shows that for DNA with G-C base pair containing polynucleotide inserts of different chain length the 'K' and '7' values are different (Fig. 2). As inferred from equation (5), parameter Kdenotes the ability of a nucleotide site to undergo helical transition for a given twisting change. It depends on the sequence and structure of the DNA as well as that of the insert. This implies that the probability of right-handed to left-handed helical transition increases with an increase in the length n. Further, the change in twist per nucleotide pair as defined in equation (6) also increases with increase in insert length (n).

The increase in the length ofnucleotide insert (n) could raise the possibility of a greater number of neighbours interacting with each other. This leads to a higher degree 6f co-operativity and an increased B-Z transition probability which is reflected in the increase in K with n.

The interpretation of the factor '7' is not a straightforward job. As per our results a nucleotide pair undergo greater helical change with insertion of longer

730 S. GHOSH

10

o n=21

n=16

A n=12 ~ 5 o

o I =

0 ] 10 15 20

Figure 1. Negative supercoiled induced B - Z transition in plasmids containing d(pCpG)n d(pCpG), inserts. The experimental points, taken from Peck and Wang (1983), represented by the symbol 0. The theoretical graphs computed on the basis

of the proposed model are represented by the continuous lines.

Table 1. Best fit values

n 7 K

8 9.8 × 10 - 4 2.35 × 10 -3 12 15.2×10 -4 2.8 ×10 3 16 18.7× 10 -4 3.00× 10 -3 21 22.13×10 -4 3.15x 10 -3

G - C stretch. It m a y be men t ioned here tha t po lynuc leo t ides can assume m o r e t ha n a single lef t -handed con fo rma t ion . Wi th a change in the n u m b e r of nuc leot ide inserts (n) the ra t io of nucleot ides in different lef t -handed helical forms could vary, leading to a difference in 7 values.

In the end, it m a y be men t ioned tha t all the inserts had h o m o g e n e o u s base sequences (G-C) . It will be interes t ing to k n o w h o w the values o f K a n d 7 would change with a change in sequence. H o w e v e r , this could no t have been de te rmined due to lack of exper imenta l data .

ISING MODEL FOR B Z TRANSITION IN SUPERCOILED DNA 731

"7,

25

20

15

10

0 I i I i 6 10 14 18 22

H

3.3

3.1

2.9

o

2.7 .-~

2.5

2.3

r 2.1 26

Figure 2. Variation of the parameters K and ~ with the length of inserts (n) in pBR322 plasmid DNA.

LITERATURE

Benham, C. 1980. Theoretical analysis of transitions between B and Z conformation in torsionally stressed DNA. Nature 286, 637-638.

Benham, C. 1981. Theoretical analysis of competitive conformational transitions in torsionally stressed DNA. J. molec. Biol. 150, 43-68.

Benham, C. J. 1982. Statistical mechanical analysis of competing conformational transitions in superhelical DNA. Cold Spring Harb. Syrup. quant. Biol. 47, 219-228.

Benham, C. J. 1988. Superhelical DNA. Comments molec, cell. Biophys. 4(1), 35-53. Camrini-Otero, R. D. and G. Felsenfeld, 1978. A simple model of DNA superhelices in solution.

Proc. natn. Acad. Sci (USA) 75, 1708-1712. Dickerson, R. E., H. R. Drew, B. N. Conner, R. M. Wing, A. V. Fratini and M. L. Kapka. 1982.

The anatomy of A-, B-, and Z-DNA. Science 216, 475-485. Doerfler, N. 1983. DNA methylation and gene activity. Ann. Rev. Biochem. 52, 93-124. Gessner, R. V., C. A. Frederick, G. J. Quigley, A. Rich and A. H. Wang. 1989. The molecular

structure of the left-handed Z-DNA double-helix at 1.0 A atomic resolution. J. biol. Chem. 264, 7921-7935.

Ivanov, V. I., L. E. Minchenkova, E. E. Minyat and A. K. Schyolkina. 1982. Cooperative transitions in DNA with no separation of strands. Cold Spring Harb. Syrup. quant. Biol. 47, 243-250.

Haniford, D. B. and D. E. Pulleyblank. 1983. Facile transition ofpoly [d(TG), d(CA)] into a left- handed helix in physiological conditions. Nature 302, 632-634.

Hao, M. H. and,W. K. Olson. 1989. Modeling DNA supercoils and knots with B-spline functior/s. Biopolymers 28(4), 873-900.

Ho, P. S., G. Zohu and L. B. Clark. 1990. Polarised electronic spectra ofZ DNA single crystals. Biopolymers 30(1/2), 151-163.

Lamba, O. P., A. H. J. Wang and G. J. Thomas, Jr. 1989. Low frequency dynamics and raman scattering of crystals, of B-, A- and Z DNA and fibers of C-DNA. Biopolymers 28(2), 667-678.

732 S. GHOSH

Nordheim, A., E. M. Lafen, L. J. Peck, J. C. Wang, B. D. Stollar and A. Rich. 1982. Negatively supercoiled plasmids contain left-handed Z DNA segments as detected by specific antibody binding. Cell 31,309 318.

Nordheim, A. and A. Rich. 1983. The sequence (dC-dA), (dG-dT), forms left-handed Z-DNA in negatively supercoiled plasmids. Proc. natn. Acad. Sci. (USA) 80, 1821 1825.

Peck, L. J. and J. C. Wang. 1983. Energetics of B to Z transition in DNA. Proc. natn. Acad. Sei. USA 80, 6206-6210.

Rich, A., A. Nordheim and A. H. J. Wang. 1984. The chemistry and biology of left-handed Z DNA. Ann. Rev. Biochem. 53, 791-846.

Sen, S. and R. Majumdar. 1987. Thermodynamics of B-Z transition in supercoiled DNA. Nucl. Acid Res. 15(14), 5863-5871.

Singleton, C. K., J. Khysik, S. M. Stirdivant and R. D. Well. 1982. Left-handed Z-DNA is induced by supercoiling in physiological ionic conditions. Nature 299, 312-316.

Umehara, T., J. Kuwavara, S. Mashimo and S. Yagihera. 1990. Dielectric study on hydration of B, A and Z DNA. Biopolymers 30(7/8), 649-656.

Wang, J. C., J. L. Peck and K. Becherer. 1982. DNA supercoiling and its effects on DNA structure and function. Cold Spring Harb. Symp. quant. Biol. 47, 8541.

Zhong, L. and W. C. Johnson, Jr. 1990. Poly [d(GmSC). d(GmSC)] can assume the Z-form: a CD study. Biopolymers 30(7/8), 821 828.

Zimm, B. M. and J. K. Bragg. 1959. Theory of the phase transition between helix and random coil in polypeptide chains. J. Chem. Phys. 31(2), 52(~535.

Zimmerman, S. B. 1982. The three dimensional structure of DNA. Ann. Rev. Biochem. 51, 395-427.

Rece ived 7 J u l y 1991