Influencing the B-Z switch in supercoiled DNA

Biophysical Chemistry, 39 (1991) 85-90 85 Elsevier BIOCHE 01528 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Influencing the B-Z switch in supercoiled DNA Ansuman Lahiri Theoretical Nuclear Physics Dwsion, Saha Instrtute of Nuclear Physrcs, 92 Acharyn Prafulla Chnndra Road, Calcutta 700009, India Received 2 May 1990 Accepted 2 August 1990 B-Z transition; DNA, supercoiled; DNA, Z-; Antibody binding; Binding polynomial The effect of Z-binding ligands on the supercoiling threshold in the supercoil-induced B-Z transition has been examined from the point of view of a two-state model. Expressions have been derived for the determination of the shift in wit& supercoil density in terms of the physical parameters of the DNA-ligand system. Representative calculations indicate that the stabilizing action of Z-binding ligands on the Z conformation tn closed circular DNA depends largely on the binding characteristics of the l&and. Application of the theoretical data has been demonstrated zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE using the experimentalresultsreportedby Lafer et al. (J. Biol. Chetn. 261 (1986) 6438) 1. Introduction taking into consideration the energetics of ligand binding. The types of ligands which are known to In circular plasmids, negative supercoiling has bind Z-DNA include anti-Z antibodies [8], and been shown to facilitate B-Z transition in se- other Z-binding proteins [9], reel and recA pro- quences having defined periodicity [l]. A recent teins [lO,ll], cations which stabilize the Z-form study has demonstrated the production and sta- (like the divalent cation hexaaminecobalt chloride bilization of short stretches of Z-DNA in vivo by (HCC) and polyamines such as spermine [12]). supercoiling generated by biological processes [2], The relevance of this study rests on the fact that in thereby contributing evidence to support the hy- living cells such ligands are likely to be present pothesis that such transitions have biological im- and thus influence the value of the critical super- plications. Theoretical studies have made it possi- coil density needed to induce the B-to-Z transition ble to calculate the supercoil densities at which as reported in recent in vitro studies [13,14]. It is such B-Z transitions (which are, as a rule, highly also of importance to examine whether the use of cooperative) would take place quite accurately by Z-specific ligands (e.g., anti-Z antibodies) to de- considering the detailed energetics of the transi- termine the occurrence of a B-Z transition under tion process [3-71. This enables us to estimate the superhelical stress significantly perturbs the B-Z variations of supercoiling density levels in plas- equilibrium itself, thereby distorting the results mids in vivo. However, to the author’s knowledge, obtained [ 131. no calculations of the effects of Z-binding ligands To this end we have exploited the observed on the B-Z equilibrium have been carried out. cooperative nature of the B-Z transition to con- The aim of this work is to study the B-Z struct a formalism which yields analytical expres- transition in the presence of Z-binding ligands by sions for quantitative calculations of the transition points in terms of the physical parameters of the DNA-ligand system. The relations obtained in Correspondence address: A. Lahiri, Theoretical Nuclear Physics Division, Saha Institute of Nuclear Physics, 92 Acharya section 2 have been applied to the analysis of Prafulla Chandra Road, Calcutta 700009, India. experimental observations of the shifts in B-Z 0301-4622/91/$03.50 0 1991 Elsevier Science Publishers B.V. (Biomedical Division) 86 A. L.uhiri/ Influencing the B-Z switch in supercoiled DNA transition points in a plasmid in the presence of shorter or multiple stretches of Z-transformed re- anti-Z antibodies [13]. The treatment of the pre- gions [3-71. sent problem is conceptually similar to the solu- In the absence of any ligand, we denote the tion of the problem of denaturation transition of B-DNA species as B,, and the molecules in which macromolecules in the presence of ligands binding Z form is present as Z,. Then, we can always differentially to the native and denatured states visuahse an equilibrium of the type zyxwvutsrqponmlkjih [15,16]. However, there are some differences be- tween the two systems. In the latter, the energy of B$Z, the junctions between the states does not occur explicitly as long as it is high enough. The system with an equilibrium constant L,. The equilibrium effectively eliminates the junction by adopting constant is given by the free energy change of the either of the two possible states as a whole. Evi- transition AG when N, base-pairs transform from dently, in the supercoil-induced B-Z transition in the B to the Z form short stretches of DNA, junctions are present whenever there is a transition from the B to Z L, = exp[ - AG/R T ] (1) form. Moreover, the separation between the two where junctions, i.e., the number of base-pairs trans- formed at the critical point, depends on the energy AG = AG, + N,AG az + AG, (2) of the junction. Ligand binding, in general, alters these quantities. The consequences of these dif- the respective terms in the sum correspond to the ferences are discussed below. nucleation energy AG, arising from the deforma- tion at the two junctions separating the Z-stretch from the rest of the B form, the change in intrinsic 2. Theory free energy per base-pair AC,, resulting from the structural changes involved in going from the B to Let us consider that in a circular plasmid of N Z form, and the change in the supercoiling energy base-pairs and supercoil density cr, N, base-pairs AGs. On the basis of the elastic model of DNA, form a potential Z-stretch. Our basic assumption Sen and Majumdar [7] have derived an expression is that the B-to-Z transition in a short potential for the change in supercoiling energy when 2m Z-stretch takes place in an all-or-none manner at a base-pairs undergo the B-Z transition in a closed particular value of the supercoil density a if N, is circular molecule of N base-pairs. Substituting the not larger than a critical length NC determined by value of N, for 2m into this expression, we obtain the base sequence and other environmental fac- tors. The magnitude of the supercoil density (which is equal to the ratio of extra linking in the super- (3) coiled molecule to the number of linking in the relaxed state) at which such flipping of base-pairs where r = 2?r2bc/a( b + c) is a term involving the occurs is referred to as the critical supercoil den- lattice parameter a of the B-DNA double helix, sity a, which depends among other things on N,. the bending stiffness b of the DNA central axis If N, happens to be longer than N,, the minimum and the torsional rigidity c for twisting about that cooperative unit, the B-to-Z transition, proceeds axis, and $I = (A, +A,)/A,A, is a term involv- smoothly with increasing negative supercoil after ing A,, the number of base-pairs per turn in the first cooperative transition of NC base-pairs at B-DNA and A, the corresponding parameter for a reduced value of the critical supercoil density. Z-DNA. For the above case, the critical number Thermodynamic and statistical-mechanical studies of base-pairs NC undergoing cooperative transition of the transition reveal that the above assumption is given by is quite close to actuality because of the high energy costs of the intermediate species having N, = ( AGJ~) 1’2/+ (4)  A. L.ahiri/ Influencing the B-Z switch m supercoded DNA 87 which can be determined from the minima of the bind to the macromolecule when it undergoes the AG = 0 curve [7]. said transition. In the presence of a ligand X with differential In the absence of ligands, the binding poly- affinity for B and zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Z form, additional equilibria nomials are equal to unity and hence the condi- will be attained: tion L = 1 or AG = 0 gives $+XsB, z, + x SZ, uc = -&[~G,+A~G,,]-* (11) B, + X 4 B, z, + x 2 z, with the condition that Nz = N, if N, I NC and N, = NC if N, > NC. From eq. 11, one finds that the point of transi- Using (X) to denote the activity of ligand X and tion, a,, depends on the magnitudes of AG, and with &, &,.._ and {r, 12,. . . representing the cor- AG,,. One can well imagine a situation in which responding equilibrium constants for the series of decreases in value and becomes negative A% binding reactions shown above, the binding poly- (say, in high ionic strength solutions) to make the nomials for the B and Z form can be written as transition possible in the case of the relaxed state ( uC= 0) or of positive supercoils ( uC> 0). p,(x)=1+P,(x)+P,P2(x)*+... (5) In the presence of ligands, calculation of the P=(x) = l+{,(x) +T&(x)*+ “’ (6) fraction of molecules in the Z form is in general dependent on the detailed structure of the binding where the summations continue up to the terms at polynomials. In a manner similar to that described which saturation with ligands occurs. By using the above, the transition point is obtained for the binding polynomials the average fraction of Z- condition L = 1, or transformed species AG,, + N,AG,, + AG, ~=((z,)+(z,)+(z*)+...}{(~,)+(B,) + RT ln[P,(x)/P,(x)] =0 02) +(I&) + ..* +(z,) + (Z,) + (Z,) + . ..>-’ which yields (7) can be expressed 0’c = - &[AGn + N,AG,,] - F Lo[~z~~PBb>l ?= l+L,[P,(x)/P,(x)] =_c lh (8) (13) where L = L,[P,(x)/P,(x)]. with the condition that N, = No if N, < NC’ and We define the critical point as the point where Nz = N,’ if N, > N,‘. N,’ is determined in the same I = l/2. This yields the condition L = I, or way as that for eq. 4 above. The value of the critical length in the presence of ligands may in Pa(x) Lo= Pz(x) general differ from the previous value due to the presence of the additional term involving the bind- Another useful relation which readily transpires ing polynomials in the free energy expression. In the case of ligands exhibiting greater affinity to- from the definition of the binding polynomial is zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF [15,161 wards the Z form as compared to the B form, the effect of ligand binding can only reduce NC. a In L a =nz-nB (10) Therefore, values of N, which were slightly smaller than NC in the absence of ligands may turn out to where the term on the right-hand side corresponds be larger than the critical length in the presence of to the average number of ligands that dissociate or ligands, with the result that complication of the 88 A. Lnhiri / Influencmg the B-Z switch in supercoiled DNA calculations occurs. However, situations are also plasmid and from - 0.060 to - 0.055 for the parent possible where the critical length would remain vector on changing the concentration from 6.4 to unchanged after the addition of ligands. Such will 64 nM. be the case when the ligand binding term can be For such a low concentration of ligands we use expressed as a linear function of Nz, i.e., as NzAG, an approximate form of eq. 15 zyxwvutsrqponmlkjihgfedc [ 151: where AG, does not involve Nz. To avoid unnec- essary complications in the relations derived, we aAuc - (* a,), - (“ ~ c), RTA, shall henceforth assume that the change in N, a In zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO x Wx ,/x ,) = ~(nz -h>. with the introduction of ligand is small and can be (1 6 ) neglected. A general expression for the shift in the point We further assume that Nz = 30 base-pairs for of transition in terms of the binding polynomials pDPL6 with an insert for the given experimental is given as conditions [7], and nB = 0 since antibody is Z- specific. Since we do not know Nz for pDPL6 Aa without an insert, calculation has not been carried A%=d-%---- zr,+N,RT In zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (14) out for data obtained from this particular system. The following typical values for other parameters We can also derive an alternative relation to eq. have been taken: A, = 10.4, A, = 12, a = 3.4 A, 10 for evaluation of the change in the average h = 1.26 X lo5 erg cm rad-* mall’, c = 1.8 X lo5 number of hound ligands occurring at the B-Z erg cm rad-z mall’ [7]. For the plasmid with an transition insert, one can readily determine the value of n7 at physiological temperature from ey. 16, namely, (15) n, = 6 (for the low-affinity antibody) and tiz = 17 (for the high-affinity antibody). Eqs 10 and 15 are likely to be useful for the In spite of the approximate nature of the calcu- experimental characterization of Z-specific ligand lations, the ahove results provide an indication of binding. the average number of antibody molecules bound per DNA molecule at the transition point. One can also perform an estimation of the 3. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Discussion order of magnitude of the shifts in the transition points if the explicit forms of the binding poly- To illustrate the applicability of the above rela- nomials are known. For example, if the ligands are tions in extracting information from experimental sizeable, we can assume that, at most, one ligand results, we analyse below recently reported data can bind to the potential Z-stretch in both the B on the effects of anti-Z antibodies on the B-Z and the Z form. The ligands we consider are also equilibrium [13]. The phenomenon studied is the assumed to have negligible affinity for the B form stabilization of Z-DNA by antibody binding in as compared to that for the Z-form, i.e., 0, * 5,. supercoiled plasmids pDPL6 (a 2.2 kh pBR322 In such a situation, the fraction of molecules in derivative) with and without an insert ((dC-dA) . the Z state is determined by inserting the follow- (dG-dT)),, (called (CA),, or (CA)). Filter binding ing values into the expressions for binding poly- studies with low-affinity antibodies at a con- nomials: pi = 0 for all zyxwvutsrqponmlkjihgfedcbaZ i and cl(X) = 01, 5, = 0 for centration of 6.4 nM and at a lo-fold higher i = 2, 3,. . . The required shift in supercoil density concentration demonstrated shifts of the mid- is obtained from eq. 14 as: points of the transition from -0.050 to - 0.042 for pDPL6 containing (CA),, and from - 0.075 to da, = &RZ- zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR ln(1 + a) (17) -0.060 for pDPL6 without an insert. With a high-affinity antibody preparation the midpoints At (r = 0 we obtain our previous result that u,’ = a,. shifted from -0.042 to -0.02 for the (CA),, At (Y> 0 the shift is in the direction of positive  A. Lahiri/ Inji’uencing the B-Z swiich in suvercolled DNA 89 supercoils. For (Yvarying from 0.1 to 10 the value turbation depends on the binding characteristics of ACT,varies from 5.3 X 10m5 to 1.3 X 10m3, Thus, or, more specifically, on the binding polynomials. we find that for the given values of concentrations The shift is insignificant for large molecules which the shifts are quite insignificant compared to the bind to the stretch in small numbers even if they value of a, in the absence of ligands. possess very strong affinity for the Z form. The The situation is different when the ligands are shift becomes considerable only when the number of relatively small size and bind to the stretch in of ligands which can bind to the stretch is large. considerable numbers. Consider the case, as in the These results may have importance in considera- above example, where the ligands bind only to the tions about the stability of the Z form in natural Z form and the maximum number of ligands sequences in vivo. Moreover, the above results can binding to the Z-stretch is n. For this situation, be used to estimate the error in detecting the B-Z one can obtain an estimate of the order of magni- transition when Z-binding ligands are used for tude of the shift in the transition point by perfor- that purpose. The variation of the shift with ligand ming the following simple calculation. If the ligand concentration gives the change in the number of binds to the Z form non-cooperatively, the bind- bound ligands occurring as a result of the transi- ing polynomial for the Z-form Pz(x) = (1 + ~y)~. tion. This may serve as useful information in Since PB(x) = 1 characterising the binding mode of the ligand. It should also be noted that a similar formalism should be valid for investigations regarding cruci- AU, = &RT ln(l + cu)“. 08) form extrusion under negative superhelical stress in plasmids containing palindromic sequences For n = N,, i.e., when each base-pair binds a where the transition for short stretches is also ligand, we have calculated the values of Au, for strongly cooperative. the same conditions as given above and found that Expressions such as those described above are they vary from 1.6 X lop3 to 0.039 for OLvarying also applicable for cases where the ligands alter from 0.1 to 10. It is apparent that in this case the the net supercoiling of the DNA molecule by shifts quite rapidly become comparable in order of causing winding or unwinding of the DNA during magnitude with u= for increasirg ligand concentra- the process of binding. For such cases the equi- tion. One can clearly envisage a situation where librium constants of the ligand binding reactions the B-to-Z transition would be stabilized in the would contain extra factors involving the change relaxed state or even for the case of positive in supercoiling energy due to ligand binding to the supercoils. molecule for that state [18,19]. Finally, it should be mentioned that for binding of extended ligands like large proteins to DNA, a more appropriate description of binding is pro- Acknowledgements vided by the method of McGhee and Von Hippel [17]. In such cases, the binding energy term involv- The author is indebted to Mr S. Sen, Dr D. ing the binding polynomials should be replaced by Dasgupta and Dr N. Bhattacharya for valuable the binding energy term calculated from the ap- suggestions. propriate McGhee-Von Hippel isotherms. References 4. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Conclusion 1 C.K. Singleton, J. Klysik, SM. Stirdivant and R.D. Wells, Nature 294 (1982) 312. 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