melting of cross-linked dna v. cross-linking effect caused by local stabilization of the double...

Melting of Cross-Linked DNA V. Cross-Linking EffectCaused by Local Stabilization of

the Double Helix

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Abstract

DNA interstrand cross-links are usually formed due to bidentate covalent or coordinationbinding of a cross-linking agent to nucleotides of different strands. However interstrand link-ages can be also caused by any type of chemical modification that gives rise to a strong localstabilization of the double helix. These stabilized sites conserve their helical structure andprevent local and total strand separation at temperatures above the melting of ordinary AT andGC base pairs. This local stabilization makes DNA melting fully reversible and independentof strand concentration like ordinary covalent interstrand cross-links. The stabilization canbe caused by all the types of chemical modifications (interstrand cross-links, intrastrandcross-links or monofunctional adducts) if they give rise to a strong enough local stabilizationof the double helix. Our calculation demonstrates that an increase in stability by 25 to 30 kcalin the free energy of a single base pair of the double helix is sufficient for this “cross-linkingeffect” (i.e. conserving the helicity of this base pair and preventing strand separation aftermelting of ordinary base pairs). For the situation where there is more then one stabilized sitein a DNA duplex (e.g., 1 stabilized site per 1000 bp), a lower stabilization per site is sufficientfor the “cross-linking effect” (18 - 20 kcal). A substantial increase in DNA stability was foundin various experimental studies for some metal-based anti-tumor compounds. These com-pounds may give rise to the effect described above. If ligand induced stabilization is distrib-uted among several neighboring base pairs, a much lower minimum increase per stabilizedbase pair is sufficient to produce the cross-linking effect (1 bp- 24.4 kcal; 5 bp- 5.3 kcal; 10bp- 2.9 kcal, 25 bp- 1.4 kcal; 50 bp- 1.0 kcal). The relatively weak non-covalent binding ofhistones or protamines that cover long regions of DNA (20- 40 bp) can also cause this effectif the salt concentration of the solution is sufficiently low to cause strong local stabilizationof the double helix. Stretches of GC pairs more than 25 bp in length inserted into poly(AT)DNA also exhibit properties of stabilizing interstrand cross-links.

Introduction

Some of the compounds that form interstrand cross-links in DNA molecules areeffective antitumor drugs (1). DNA interstrand cross-links can be highly lethallesions because they can irreversibly halt DNA metabolic processes. In addition,interstrand cross-links are usually more stable against DNA repair systems in com-parison with other types of DNA chemical modifications (intrastrand cross-linksand monofunctional adducts) (2, 3). Correlation between cytotoxicity and abilityto form DNA interstrand cross-links in vitro has been demonstrated for severalcross-linking agents (4, 5).

Interstrand cross-links are usually formed by covalent or strong coordinationbidentate binding of cross-linking agents to nucleotides of different DNA strands.Interstrand cross-linking prevents complete strand separation but does not preventmelting of the double helix at sites of cross-linking as well as other regions of theDNA (Figure 1B.1 and 1B.2).

Journal of Biomolecular Structure &Dynamics, ISSN 0739-1102Volume 20, Issue Number 4, (2003)©Adenine Press (2003)

Alexander S. Fridman1

Viktor Brabec2

Samvel G. Haroutiunian3

Roger M. Wartell4

Dmitri Y. Lando1,*

1Institute of Bioorganic Chemistry

Belarus National Academy of Sciences

Kuprevich St. 5/2

220141 Minsk, Belarus2Institute of Biophysics

Academy of Sciences of the Czech Republic

Královopolská 135

CZ-61265 Brno, Czech Republic3Chair of Molecular Physics

Department of Physics

Yerevan State University

375049 Yerevan, Armenia4School of Biology

Georgia Institute of Technology

Atlanta, Georgia 30332-0230, USA

533

* Phone: 375-17-2648263Fax: 375-17-2648647Email: Lando@iboch.bas-net.by

Interstrand cross-links give rise to three effects that influence stability of the dou-ble helix:

1. A single interstrand cross-link is sufficient to prevent full strand sepa-ration (Figure 1B.1) and decrease the order of reaction of strand dissocia-tion from two to one. It makes melting independent of strand concentra-tion and stabilizes the entire double helix. The effect is pronounced in thecase of short DNA’s less than 100 bp in length (6, 7, 17, 18).

2. In addition to decreasing the order of reaction of strand dissociation, ωinterstrand cross-links cause the formation of (ω- 1) loops in a fully melt-ed state (Figure 1B.2). Loop formation decreases the entropy of the par-tially melted and fully melted states increasing the stability of the doublehelix. This effect gives rise to a strong increase in the melting temperatureif the average distance between neighboring cross-links is less 100 bp (14,15).

3. Besides these two effects interstrand cross-links can locally stabilize(24) or destabilize (6-8) the double helix.

However, there is another way to prevent strand separation without covalently linkingthe two DNA strands. Any chemical modification of DNA or irreversible ligand bind-ing to DNA can prevent complete DNA melting and strand separation if it causes a suf-ficiently strong local stabilization of the double helix. This type of modification mightdemonstrate biological and thermodynamic effects that are similar to covalent inter-strand cross-links. However ordinary interstrand cross-links generally give rise to a localdistortion and local destabilization of the double helix and cannot prevent full DNAmelt-ing (6- 8). The chemical modifications under consideration always give rise to a stronglocal stabilization and can be caused by any type of chemical modifications (monofunc-tional adducts, interstrand and intrastrand cross-links) as well as by very stable non-cova-lent irreversible ligand binding to DNA. Base pairs with all the types of chemical mod-ifications (even with interstrand cross-links) that do not cause strong local stabilizationare melted in the same temperature interval as ordinary AT and GC base pairs. They onlyincrease or decrease the overall stability of the double helix, but do not prohibit meltingof the chemically modified base pairs. In contrast, modified strongly stabilized basepairs conserve the helicity after melting of ordinary AT and GC base pairs and preventfull melting as well as full strand separation (Figures 1C.1 and 1C.2).

Calculations carried out in this work demonstrate that an increase in the free ener-gy of the helix-coil transition of a single chemically modified base pair by 25- 30

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Figure 1: An oligonucleotide duplex (A.1, B.1, C.1) and a longDNA fragment (A.2, B.2, C.2) at high temperature for which allthe base pairs are melted except base pairs that are strongly sta-bilized by any chemical modification (C1 and C2).

A. The fully melted state of an oligonucleotideduplex (A.1) and long DNA fragment (A.2) withoutinterstrand cross-links and unmeltable stabilizedsites. After full melting, complete strand separationoccurs.B. The fully melted state of an oligonucleotideduplex (B.1) with a single ordinary interstrandcross-link (ω=1) and of a long DNA fragment (B.2)with four ordinary interstrand cross-links (ω=4).Full melting occurs but without complete strandseparation.C. The partially melted state of an oligonucleotideduplex (C.1) in which cross-linking is caused by asingle (ω=1) strongly stabilized unmeltable site andpartially melted state of a long DNA fragment (C.2)with four (ω=4) stabilized sites. Both full meltingand complete strand separation are impossible.

kcal prevents strands separation after melting of ordinary AT and GC base pairs(Figure 1C.1). A lower increase, 18- 20 kcal per modified site, is sufficient for a“cross-linking effect” if there are several (ω) modified sites in the DNA chain. ωmodified sites give rise to the formation of (ω- 1) loops after the melting of ordi-nary base pairs as in the case of ordinary covalent interstrand cross-links (Figure1C.2). These loops are formed by non-meltable base pairs stabilized by a chemi-cal modification. In the case of ordinary covalent interstrand cross-links, the loopsare formed by cross-linked nucleotides of melted base pairs (Figure 1B.2). It isshown in this paper that some metal base antitumor-compounds, stretches of GCpairs, as well as DNA regions covered by histones or protamines demonstrate prop-erties analogous to stabilizing interstrand cross-links.

Methods

Calculation Methods

Calculation of melting curves of non-modified DNA and chemically modifiedDNA with locally stabilized sites were made using the relations developed in ref-erence (12) which were slight modifications to the Poland (10) and Fixman-Freire (11) algorithms. For DNA with interstrand cross-links, the methodsdescribed in references (13- 16) were used. The relations published in thesepapers (10- 16) allow the calculation of a series of unconditional probabilitiesthat the base pair indexed as k is in the helical state (ρ(k), k = 1 to N where N isthe number of base pairs in the DNA chain). The fully melted state is not takeninto account during calculation of these probabilities. Eq. [1] gives ϑint, theaverage degree of helicity for all the states of the DNA molecule that contain atleast one helical unit (10).

Nϑint = N-1 · Σρ(k); [1]

k = 1

Poland’s Eq. [20] from paper (10) was used to take into account the fully melted stateand calculate the total degree of helicity averaged over all the states of DNA molecule:

ϑ = ϑint · ϑext [2]

where ϑext is the fraction of DNA molecules that contain at least one helical unit(1- ϑext is the fraction of fully melted molecules (without helical units)).

From relations developed in references (10) and (14), it follows that

ϑext = F + 1 - (2F + F2)0.5 for DNA without interstrand cross-links [3a]ϑext = [1 + β-1 · ρ(1) · γ(N)]-1 for DNA with interstrand cross-links [3b]

where

F = ρ(1) · γ(N) · (β · Ct)-1 for non-self-complementary strands; [3c]F = ρ(1) · γ(N) · (4 · β · Ct)-1 for self-complementary strands; [3d]

N-1γ(N) = δL(N) · r(N) · ∏ t(k); [3e]

k = 1

where δL(N) is the loop entropy factor of fully melted DNA (δL(N) = 1 for DNAwithout interstrand cross-links); β is the parameter describing the initial associationof strands; Ct is the total molar concentration of DNA strands; calculation of r(N),t(k), δL(N) is described in (10, 11, 14).

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From relations obtained by Poland (10) and Fixman and Freire (11) one can deriveEqs. [4] for the average number of internal melted regions (mint) and for the averagethe number of helical regions for the states that include at least one helical unit (nint):

N-1mint = σ · ∑ W(k) [4a]

k = 2

nint = mint + 1 [4b]

σ is the cooperativity factor of DNA melting (the statistical weight assigned to theboundaries of an internal melted region). Calculation of W(k) is described in (11, 14).

The fully melted state of a DNA molecule forms neither helical regions nor loops(Figures 1A.1 and 1A.2). Therefore the average number of helical regions perDNA molecule (n) can be found using Eq. [5]:

n = ϑext · nint [5]

The following parameter values were used to calculate DNA melting curves: totalnumber of base pairs, N = 5000 bp; fraction of GC base pairs, XGC = 0 or XGC = 0.5(for XGC = 0.5, the DNA sequence was produced with a random number generator);the loop entropy factor for a loop of k base pairs formed by an internal meltedregion,δ(k) = (k + 1)-1.7; the cooperativity factor, or statistical weight assigned to theboundaries of an internal melted region bordered by helical base pairs, σ = 5 · 10-5;strand association parameter, β = 5 · 10-5; enthalpy, entropy and melting tempera-tures of AT and GC base pairs, ∆HAT = 8.39 kcal/(mole bp), ∆HGC = 9.44 kcal/(molebp), ∆S = ∆SAT = ∆SGC = 24.8 cal./deg., TAT = 65.2º C and TGC = 107.8º C (9). Itwas assumed in all calculations that DNA strands are non-self-complementary.

Results and Discussion

Qualitative Comparison of Melting Behavior of Natural DNA, DNA that Contains Interstrand Cross-Links andDNA that Includes Locally Stabilized Sites

In a non-modified DNA that is partially melted, the helical regions prevent fullstrand separation. Only after melting of all base pairs in the DNA chain, the strandswill separate (Figure 1A.1 and 1A.2). Therefore the process of full strand separa-tion is equivalent to the transition from a partially (or fully) helical state to the fullymelted state as depicted in Figures 1A.1 and 1A.2. The separation transition isdependent on total strand concentration (Ct) and helix nucleation parameter (β). Itis characterized by the fraction of fully melted molecules that is equal to 1- ϑextwhere ϑext is the fraction of DNA molecules that contain at least one helical unit(0<ϑext<1). If T ~ Tm, i.e. ϑ ~ 0.5, a lot of internal melted regions (i.e. meltedregions bordered upon helical ones from both sides) arise in long DNA molecules.These regions form loops. At higher temperatures when ϑext and ϑ <<1, neitherinternal melted regions nor loops form (Figures 1A.1 and 1A.2).

In contrast to non-modified DNA, full strand separation is impossible in cross-linked DNA even after melting of all base pairs. As with a non-modified DNA, across-linked DNA does not form helical and internal melted regions at high tem-perature when ϑext and ϑ → 0 but it forms (ω- 1) loops where ω is the number ofinterstrand cross-links in its chain (Figures 1B.1 and 1B.2). Cross-linking decreas-es the order of the reaction of full strand separation from 2 to 1 and makes Tm aswell as melting behavior as a whole independent of DNA concentration (17, 18).However, for cross-linked DNA, strand separation (ϑext) is dependent on the helixinitiation parameter, β, as for DNA without interstrand cross-links.

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One can imagine another possible way of preventing strand separation without acovalent interstrand cross-link. If any chemical modification or irreversibly boundligand causes a strong local stabilization of one (L = 1 bp) or several (L >1 bp) neigh-boring base pairs, then full melting as well as full strand separation will not occurafter melting of the non-modified base pairs (Figures 1C.1 and 1C.2). Such modifi-cations might produce the same influence on melting behavior and biological effectsas ordinary covalent interstrand cross-links. Because of the prohibition of full melt-ing, this type of modification makes melting of the non-modified base pairs inde-pendent of helix nucleation parameter β in contrast to non-modified DNA as well asDNA with ordinary covalent interstrand cross-links. In this case, ϑext ≡ 1 until fullmelting of non-modified AT and GC base pairs. A computer modeling procedure isdescribed below for evaluating the minimum value of local stabilization of a DNAdouble helix that is necessary to prohibit full melting and full strand separation.

An increase in the free energy of the helix-coil transition of modified base pairscauses local stabilization of the double helix. We suppose that a chemical modifi-cation gives sufficient stabilization for the prohibition of local and total melting aswell as of strand separation if all the stabilized sites maintain helical base pairsafter melting the non-modified AT and GC base pairs. To illustrate this cross-link-ing effect let us consider the results of calculation for a long DNA that contains asingle strongly stabilized site (ω = 1) as well as several stabilized sites (ω >1).

“Cross-Linking Effect” for a DNA with a Single Strongly Stabilized Site (ω = 1)

Let δ(∆F) be the change in free energy of a single chemically modified stronglystabilized site. Strand separation occurs only after melting of this stabilized site.For a given δ(∆F), existence of a temperature interval for which ϑext is high enough(0.99 <ϑext <1) and ϑ is only slightly higher than the fraction of stabilized basepairs (L/N) demonstrates that the δ(∆F) value is sufficient for a cross-linking effect.The occurrence of a cross-linking effect with an increase in δ(∆F) for a single sta-bilized site located at base pair number 2500 is shown in Figure 2 for poly(AT) andin Figure 3 for a random sequence of AT and GC base pairs (N = 5000 bp).

For non-modified poly(AT) [δ(∆F) = 0] as well as for 0 <δ(∆F) <9 kcal per stabilizedsite, the differential melting curve, ϑ´T(T), and the curve of strand separation, ϑext(T),are located in the same temperature regions (Figure 2). If δ(∆F) >11 kcal, the dif-ferential melting curve and Tm are not changed significantly, but there is a strongincrease in the fraction of non-dissociated DNA molecules, ϑext(T). For δ(∆F) = 18kcal, there is a temperature interval (65.4º C <T <66º C) in which the fraction of notfully dissociated molecules, ϑext(T), is close to 1 but the fraction of helical base pairs,ϑ(T), as well as ϑ´T(T) is close to zero. This implies that full melting and full strandseparation do not occur after melting of ordinary (non-modified) base pairs.

Similar results were obtained for the heterogeneous DNA chain with a single sta-bilized site at the 2500th base pair (Figure 3). Tm, ϑext(T), and ϑ´T(T) are notmarkedly changed during an increase in δ(∆F) from 0 to 12 kcal. Further increas-es in δ(∆F) give rise to a strong elevation of ϑext(T) without a change in the differ-ential melting curve. For δ(∆F) >22 kcal, there is a temperature interval (90º C<T<91º C) in which the fraction of not fully melted DNA’s, ϑext(T), is closeto 1 but the fraction of helical base pairs, ϑ(T), is close to zero.

“Cross-linking Effect” for a DNA with Several Stabilized Sites (ω>1)

If a DNA includes more than one strongly stabilized site (ω >1), melting of the non-mod-ified base pairs at a temperature where all the stabilized sites remain helical gives rise to

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Figure 2: The illustration of the cross-linking effect forpoly(AT) with a single stabilized site (N = 5000 bp,ω=1). ϑext(T) and ϑ´T(T) for various δ(∆F) of base pairnumber 2500. δ(∆F) values are shown in the Figure. Ifδ(∆F) ≥ 18 kcal, strand separation does not occur aftermelting of ordinary base pairs.

Figure 3: The illustration of the cross-linkingeffect for a DNA with a single stabilized site anda random distribution of AT and GC base pairs (N= 5000 bp, GC = 50%, ω = 1). ϑext(T) and

ϑ´T(T) for various δ(∆F) of base pair number2500. δ(∆F) values are shown in the Figure. Ifδ(∆F) ≥ 22 kcal, strand separation does not occurafter melting of ordinary base pairs.

ω helical regions that form ω- 1 loops (Figure 1C.2). Therefore n = ω where n is the num-ber of helical regions. This equality indicates the prohibition of local strand separation atthe stabilized sites. If stabilized base pairs are absent or are not stable enough to prohib-it local strand separation then n ≅ 0 when the temperature is high enough (Figure 1A.2).

Figure 4 illustrates the occurrence of the cross-linking effect with increasing of δ(∆F)for a 5000 bp poly(AT). Alteration of n(T) and ϑ´T(T) caused by an increase in thefree energy of six separate base pairs (ω = 6) at positions 1, 1000, 2000, 3000, 4000,5000 is considered. Poly AT without stabilized sites [δ(∆F) = 0] melts from the endsand does not form loops. Therefore there is a single helical region (n ≈ 1) beforeDNA melting (T <Tm = 65º C). At higher temperatures after ordinary base pairs aremelted, n = 0. Elevation of δ(∆F) for each of the six base pairs from zero up to 9 kcalcauses a small monotonous increase in the melting temperature (Tm) but does notgive rise to loop formation (n is close to zero if T >Tm). Further elevation of δ(∆F)does not change Tm and differential melting curve but increases the number of heli-cal regions (n) and loops at high temperatures from n = 1 up to n = ω = 6. If δ(∆F)≥15kcal, local and total strand separation do not take place for 65.8º C <T <68º Calthough the degree of helicity is close to zero for this temperature interval.

Figure 5 illustrates the cross-linking effect for a DNA with a random distribution ofbase pairs (N = 5000 bp, GC = 50%). As in the previous case, there are six stabi-lized base pairs (ω = 6) at positions 1, 1000, 2000, 3000, 4000, 5000. In contrast topoly(AT), a heterogeneous DNA forms many internal melted regions in the temper-ature interval that corresponds to DNA melting. Their number tends to zero aftermelting of ordinary base pairs for a DNA without stabilized sites as well as for thecase of 0 <δ(∆F) <12 kcal. Alteration of δ(∆F) for each of these base pairs in thisinterval slightly increases Tm and strongly changes the differential melting curve(DMC). A further increase in δ(∆F) does not change DMC, but for a high enoughtemperature when almost all ordinary base pairs are melted, n tends to change fromzero to the value that indicates prohibition of local and total strand separation (n =ω = 6). If δ(∆F) ≥ 18 kcal, local strand separation does not occur at the stabilizedsites (n(T) = ω = 6) although ϑ is close to zero (90º C <T <93º C).

Determining the Temperature at Which Almost All Ordinary Base Pairs are Meltedin a DNA with Stabilized Sites

The above analysis indicated that at a sufficiently high value of δ(∆F) gives rise tocross-linking effect. To determine δ(∆F)cr, the minimum δ(∆F) that prevents fullmelting and strand separation, an appropriate temperature must be found at whichalmost all ordinary base pairs are melted in the DNA that includes strongly stabilizedsites. In the general case, the total number of ordinary helical base pairs (noh) can becalculated as the difference between the total number of helical base pairs in the DNAchain (ϑ · N) and the number of chemically modified base pairs that are stabilized:

ω Lnoh = ϑ · N - ϑext · ∑ ∑ρ (ni + k - 1) [6]

i = 1 k = 1

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Figure 4: The illustration of the cross-linking effect forpoly(AT) with six stabilized sites (N = 5000 bp, ω = 6).n(T) (the number of helical regions in DNA chain) andof ϑ´T(T) for various δ(∆F) of base pairs at positions 1,1000, 2000, 3000, 4000, 5000. δ(∆F) values are shownin the figure. If δ(∆F) ≥ 15 kcal, strand separation doesnot occur after melting of ordinary base pairs.

Figure 5: The illustration of the cross-linking effect fora DNA with six stabilized sites and a random distribu-tion of AT and GC base pairs (N = 5000 bp, GC = 50%,ω = 6). n(T) and ϑ´T(T) for various δ(∆F) of base pairsat positions 1, 1000, 2000, 3000, 4000, 5000. δ(∆F) val-ues are shown in the figure. If δ(∆F) ≥ 18 kcal, strandseparation does not occur after melting of ordinary basepairs.

where ni is the position number of the base pair where the stabilized site (L = 1) orits left end (L >1) is located.

Eq. [6] is valid for an arbitrary δ(∆F) value and is independent of the occurrence ofthe cross-linking effect. However, this expression does not by itself indicate thetemperature that is suitable for evaluating δ(∆F)cr. It is possible to find this tem-perature using the results of our previous study (26) obtained for a DNA with non-meltable sites. For this type of DNA, it was shown that if a DNA includes stronglystabilized non-meltable sites, then at a sufficiently high temperature when ϑ is closeto zero, all conserved helical regions of ordinary base pairs are adjacent to the leftor right side of the stabilized sites (Figure 6). There is no fluctuational closing ofmelted regions in the central part of loops, although fluctuational opening of basepairs in helical regions takes place and markedly influences some thermodynamicparameters of the double helix (the average length of helical and coil regions (12)).

If there is a single non-meltable site in the DNA chain or the shortest distancebetween neighboring stabilized sites exceeds 50 bp, then the average number ofbase pairs in each of these regions of ordinary helical base pairs (ns) is independ-ent of the DNA length, of the total number and location of the stabilized sites. Forhigh temperature (ϑ → 0), ns can be given with a good accuracy by:

ns ≈ s/(1-s) [7]where

s = exp[∆S · (T∞ - T)/(RT)] [7a]

In equations [7] and [7a], s is the statistical weight of a helical base pair; ∆S is theentropy change associated with melting one base pair and assigned the value ∆S =24.85 cal/(mole. K) (9); T is the temperature in Kelvin; T∞ is the melting temperatureof a non-modified DNA of infinite length and has the value T∞ = TAT+X · (TGC-TAT)with X the fraction of GC base pairs. For poly(AT), T∞ = TAT = 65.2º C (9). R is thegas constant per mole.

If a stabilized site is located at the left (or right) end of a DNA, then a single regionof ordinary base pairs forms at its right (or left) side (Figure 6a). All the other sta-bilized sites distant from the DNA ends allow for the formation of regions of ordi-nary base pairs at both sides (Figure 6b). This means that the total number of ordi-nary helical base pairs (noh) at a high enough temperature (ϑ <<1) is the following:

noh = (2ω - a)ns = (2ω - a) · s/(1-s) [8]

where a is the number of stabilized sites located at the ends of DNA molecule (thenumber of closed ends). a can be equal to 0, 1, 2 (Figure 6). The approximateresults given by Eqs. [7], [8] are in a good agreement with the more rigorous resultsdetermined by Eq. [6] if the cross-linking effect occurs.

From Eq. [7] if T = T∞ + 20º C, then s/(1- s) ≈1. Thus the number of ordinary hel-ical base pairs (noh) in a DNA with ω strongly stabilized sites is equal to 2ω- a atthis temperature. The fraction of ordinary helical base pairs, [(2ω- a)/(N- ωL)], isvery small. However this fraction would be much lower if there were no stronglystabilized sites, i.e. if δ(∆F) is equal to zero or is not high enough to give rise to across-linking effect (Figures 8-11, curves 1a, 2a, 3a). Thus T = T∞ + 20º C can beconsidered as the temperature of “almost full melting” of ordinary base pairs forDNA with strongly stabilized sites. If the cross-linking effect does not occur, nohcan be calculated using the rigorous approach of Eq. [6].

Let us assume that a given δ(∆F) is sufficient to produce a cross-linking effect if allthe stabilized sites are not melted at this temperature (T = T∞ + 20º C). To find a suit-able criterion that reflects the prevention of full melting and strand separation at this

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Figure 6: Partially melted oligonucleotide duplexesthat include ten ordinary base pairs and one (a, b) or two(c) stabilized sites; four of ordinary base pairs are heli-cal and six are melted.

a) a single stabilized site (ω = 1) at one endof the helix (a single closed end, a = 1);b) a single stabilized site (ω = 1) in the cen-ter of the helix (no closed ends, a = 0);c) two stabilized sites (ω = 2) located at bothends of the helix (two closed ends, a = 2).

temperature, let us consider the two cases shown in the Figure 1C. For the first case(Criterion A), there is one stabilized site in the double helix (Figure 1C.1, ω = 1). Forthe second case (Criterion B) there are several stabilized sites (Figure 1C.2, ω = 4).

A. Criterion for Prohibition of Full Strand Separation for the Case of a SingleStabilized Site in the Double Helix (Figure 1C.1).

Let δ(∆F) represent an increase in the free energy of the helix-coil transition for asingle stabilized site in the DNA (ω = 1). The site can be considered to be a singlebase pair (L = 1 bp) or several neighboring base pairs (L >1 bp). Strand separationoccurs only after melting of this stabilized site. The maximum value of ϑext isequal to 1 and will be approached in only the limit δ(∆F) = ∞. Therefore, the con-dition ϑext ≡ 1 cannot be used for the determination of a minimum δ(∆F) necessaryfor a cross-linking effect. Let us suppose that the cross-linking effect occurs for agiven δ(∆F) if the ϑext value lies in the interval from 0.99 to 1 for T = T∞ + 20º C.This implies that the stabilized site is helical in more than 99% of DNA moleculesat the temperature of almost full melting of ordinary AT and GC base pairs. Thusthe following relations are considered valid when δ(∆F) is large enough to prohib-it strand dissociation:

ϑext ≥ 0.99 and ns(T) ~ 1 bp for T = T∞ + 20º C [9]

The high value for ϑext indicates an absence of strand dissociation, while the lowvalue of ns(T) ~ 1bp demonstrates almost full melting of ordinary (non-modified)AT and GC base pairs. If δ(∆F) value is not high enough to give rise to the cross-linking effect, then ns(T∞ + 20º C) <<1 (Figures 8 and 9). In this case ns is calcu-lated using Eqs. [6] and [10]:

ns=noh/(2 · ω- a) [10]

Let us denote the minimum δ(∆F) value necessary for the cross-linking effect asδ(∆F)cr, i.e. ϑext(T∞ + 20º C) = 0.99 for δ(∆F)cr.

The lower limit of ϑext corresponding to cross-linking (0.99) as well the tempera-ture of almost full melting of ordinary base pairs (T = T∞ + 20º C) are selected arbi-trarily to some degree. However, as will be shown below, a strong change in thesevalues only slightly alters δ(∆F)cr.

B. Criterion for Prohibition of Local and Full Strand Separation for the Case ofSeveral Stabilized Sites in the Double Helix (Figure 1C.2)

For the case of more than one stabilized site (ω >1, Figure 1C.2), an increase in thefree energy of the helix-coil transition, δ(∆F), is assumed to be sufficient for thecross-linking effect if all the stabilized sites include helical base pairs at T = T∞ +20º C, i.e. after melting of almost all the ordinary base pairs. Criterion A is neces-sary but not sufficient for DNA’s that include two or more stabilized sites becauseit indicates the existence of unmelted regions (or a region) but this criterion doesnot give their number. Therefore another more rigid criterion that reflects the pres-ence of helical base pairs for all of the ω stabilized sites is necessary.

The criterion follows from Figure 1C.2. It is seen that if at a given temperature i)almost all ordinary base pairs are melted; ii) all the ordinary helical base pairs areincluded in the regions adjacent to stabilized sites; iii) all ω modified (stabilized)sites are helical, then ω- 1 internal melted regions (loops) and ω helical regions areformed, i.e. n = ω where n is the number of helical regions in the DNA moleculecalculated using Eqs. [4] and [5]). If stabilized sites are absent or δ(∆F) is not largeenough to give rise to the cross-linking effect, then loops and helical regions areabsent (n ≈ 0) at this temperature (Figures 1A.2 and 4, 5). Therefore an equality

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between the number of helical regions n and ω at T = T∞ + 20º C (after melting ofalmost all ordinary base pairs) might be a sufficient criterion for the cross-linkingeffect. However, only the maximum value of n is rigorously equal to ω. This valuecan be approached only in the limit δ(∆F)→∞. Therefore, the condition n = ω can-not be used for determining the δ(∆F) value necessary for the cross-linking effect.

Let us suppose that the cross-linking effect occurs if the normalized number of hel-ical regions (nnor) lies in the interval 0.99 to 1 where nnor is given by:

nnor = n/ω [11]

It is assumed that a given δ(∆F) value is sufficient for the cross-linking effect if theresults from calculations demonstrate that 0.99 ≤nnor <1 at T = T∞ + 20º C. For thistemperature, nnor <<1 if a given δ(∆F) is not sufficient for the cross-linking effect(Figures 10 and 11).

Thus the criterion of cross-link formation for ω >1 stabilized sites is the following:

nnor = n/ω ≥0.99, ns(T) ~ 1 bp for T = T∞ + 20º C [12]

A value for nnor close to one demonstrates the absence of full melting and the helic-ity of all the stabilized sites. ns(T) ~ 1 bp implies almost full melting of ordinary(non-modified) AT and GC base pairs (i.e. 1 bp per stabilized site located at anyDNA end or 2bp per stabilized site distant from the DNA ends).

As indicated for the single stabilized site situation, the selection of 0.99 as thelower limit of nnor = n/ω and the designation of a temperature of T = T∞ + 20º Cfor almost full melting of ordinary base pairs are somewhat arbitrary. However, asshown below, a significant change in these values only slightly alters δ(∆F)cr.

Procedure for Determining a Minimum δ(∆F) ValueSufficient for the Cross-Linking Effect

For DNA’s without strongly stabilized sites, ϑext = 0 and nnor = 0 at a sufficientlyhigh temperature (e.g., T > T∞ + 5º C). A change in ϑext (or nnor) from 0 to 0.95with increasing δ(∆F) is readily observed in Figures 2 to 5, and a δ(∆F) value cor-responding to a given ϑext can be readily determined from plots of ϑext versusδ(∆F). However further increase in δ(∆F) gives rise to a low absolute and relativeelevation in ϑext because its maximum value tends to 1 for δ(∆F)→∞. For this rea-son a plot 1- ϑext versus δ(∆F) is used to determine δ(∆F) corresponding to ϑext =0.99 or any other specific ϑext value because there is a larger relative difference in(1- ϑext) values. Therefore a logarithmic scale for (1- ϑext) allows one to follow itschange and the appearance of the cross-linking effect with increasing δ(∆F). Thesame approach can be used for nnor in the case of ω >1 because the normalizednumber of helical regions (nnor) tends to 1 if δ(∆F)→∞.

Figure 7 illustrates how one may determine δ(∆F)cr. For the example employed,δ(∆F)cr = 20 kcal is equal to the value corresponding to ϑext = 0.99 (or nnor = 0.99)and T = T∞ + 20º C where T∞ = TAT + x · (TGC - TAT).

Determination of δ(∆F)cr for Poly(AT) and DNA with a Single Stabilized Site(ω=1) One Base Pair Long (L = 1)

Let us determine δ(∆F)cr for poly(AT) with a single strongly stabilized site thatincludes a single base pair located in the center of DNA molecule (N = 5000 bp, T∞ = 65.2º C, ω = 1, n1 = 2500, L = 1 bp, a = 0). Appearance of the cross-linkingeffect with increasing δ(∆F) is observed in Figure 8 where the dependence of 1-ϑextand ns on δ(∆F) is given for the three temperatures: T = T∞ + 10 = 75.2º C,

541DNA Cross-Linking Caused

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Figure 7: The illustration of the procedure of determi-nation of minimum increase in the free energy of thehelix-coil transition (per stabilized site) necessary togive rise to cross-linking effect. As shown in the Figure,δ(∆F)cr = 20 kcal per stabilized site. It means that ϑext =0.99 (for ω = 1) or nnor = 0.99 (for ω ≥2) for this δ(∆F)at T = T∞ + 20º C where T∞ = TAT + XGC · (TGC - TAT).

T = T∞ + 20 = 85.2º C and T = T∞ + 30 = 95.2º C. For T = T∞ + 20 = 85.2º C ϑextand ns are close to zero for 0 <δ(∆F) <15 kcal. Further increase in δ(∆F) gives riseto the cross-linking effect that causes a strong increase in ϑext and ns. ϑext = 0.99at T = T∞ + 20 = 85.2º C if δ(∆F) = 24.4 kcal. For the temperature and δ(∆F) val-ues the fraction of ordinary helical base pairs is very low (~2/4999) because ns = 1bp. Thus, δ(∆F)cr = 24.4 kcal according to Criterion A.

Criterion A is relatively stable to the values of ϑext and T. A requirement for ϑextto be equal to 0.9 or 0.999 instead of ϑext = 0.99 only slightly changes δ(∆F)cr by± 2.5 kcal, or ~10%. A change in the temperature selected for the evaluation(±10ºC) causes only a 5% change in δ(∆F)cr. It is also seen from Figure 8 thatchanging the temperature by ±10ºC in the determination of the cross-linking effectdoes not make the fraction of helical ordinary base pairs comparable with the frac-tion of melted ones (ns = 2.4 bp for T = T∞ + 10º C, ns = 1 bp for T = T∞ + 20º Cand ns = 0.6 bp for T = T∞ + 30º C). For all temperature values, the fraction of ordi-nary helical base pairs is less than 0.1%. Thus a change in the parameter valuesused for determining δ(∆F)cr only slightly influences the final δ(∆F)cr.

A similar evaluation of δ(∆F)cr was carried out for DNA with a random sequence ofAT and GC base pairs, and a single strongly stabilized site in the center of DNA mol-ecule (T∞ = 86.3º C, GC = 50%, N = 5000 bp, ω = 1, L = 1 bp, a = 0). The appear-ance of the cross-linking effect with increasing δ(∆F) is shown in Figure 9 for thethree temperature values: T = T∞ + 10º C, T = T∞ + 20º C and T = T∞ + 30º C. Asfollows from Figure 9, ϑext = 0.99 and ns = 1.2 bp at T = T∞ + 20 = 106.3º C if δ(∆F) = 27 kcal. The value of δ(∆F)cr is close to the result obtained for poly(AT).It is seen from Figure 9 that δ(∆F)cr is not markedly changed if other values of ϑext(0.9 or 0.999) or T are used for its determination.

Determination of δ(∆F)cr for Poly(AT) and DNA with Several Stabilized Sites (ω >1)Each a Single Base Pair (L=1bp)

If DNA includes more than one stabilized site (ω >1), Criterion B is used fordetermination of δ(∆F)cr. Melting of almost all ordinary AT and GC base pairswhile maintaining helical base pairs at all the stabilized sites gives rise to forma-tion of ω helical regions and ω - 1 internal melted regions. There is no fluctuationalclosing of loops by formation of short helical regions located between strongly sta-bilized sites and not adjacent to these sites (26) (Figure 1C.2).

Let us determine δ(∆F)cr for homogeneous and heterogeneous DNA’s with 6 sta-bilized sites. As in the previous example, the analysis for determining the condi-tions for the cross-linking effect was carried out at the temperatures T = T∞ + 10º C,T = T∞ + 20º C and T = T∞ + 30º C. The dependence of ns on δ(∆F), the parame-ter that reflects “almost full melting” of ordinary base pairs, and of nnor, the char-acteristic of local strand separation at stabilized sites, are shown in Figure 10 forpoly(AT) and in Figure 11 for a heterogeneous DNA (GC = 50%).

For poly(AT), there is no cross-linking effect for 0 <δ(∆F) <12 kcal at all threetemperatures (nnor = 0). nnor increases with increasing δ(∆F) and δ(∆F)cr = 18 kcal(nnor = 0.99 and ns = 1 bp) for T = T∞ + 20º C. Similar results were obtained forthe two other temperature values (±10º C) and for nnor = 0.9 or nnor = 0.999. Thus,δ(∆F)cr value is only slightly dependent on nnor and the temperature selected for itsevaluation (Figure 10). Similar results have been obtained for DNA with a randomdistribution of AT and GC base pairs (Figure 11): δ(∆F)cr = 19.7 kcal, ns = 1.3 bp.

Melting of DNA’s that Contain a Single Block (ω=1) of Stabilized Base Pairs (L>1)

We now examine the case of a chemical modification that gives rise to a strong localstabilization to L neighboring base pairs. In this case, an increase in the free energy

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Figure 8: Determination of δ(∆F)cr for poly(AT) (N =5000 bp, ω = 1) with a single stabilized base pair num-ber 2500 from the dependence 1-ϑext[δ(∆F)] calculatedfor T = T∞ + 20º C (curve 1). The results of calculationfor T = T∞ + 10º C (curve 2) and T = T∞ + 30º C(curve 3) are given for comparison.The corresponding curves for ns (1a, 2a, 3a) demonstratethat more than 99.9% of ordinary base pairs are melted.

Figure 9: Determination of δ(∆F)cr for DNA with a ran-dom distribution of AT and GC base pairs (N = 5000 bp,GC = 50%, ω = 1) and a single stabilized base pair num-ber 2500 from the dependence 1-ϑext[δ(∆F)] calculatedfor T = T∞ + 20º C (curve 1). The results of calculationfor T = T∞ + 10º C (curve 2) and T = T∞ + 30º C (curve 3)are given for comparison.The corresponding curves for ns (1a, 2a, 3a) demonstratethat more than 99.8% of ordinary base pairs are melted.

of the helix-coil transition of each of these L base pairs is equal to δ(∆F)L/L, whereδ(∆F)L is a free energy increase for the whole stabilized site. The free energy of thehelix-coil transition of a chemically modified base pair (∆Fmbp) is given by Eq. [13]:

∆Fmbp = ∆F + δ(∆F)L/L [13]

where ∆F is the free energy of the helix-coil transition of non-modified (ordinary)base pair.

Calculations demonstrate that the cross-linking effect takes place if δ(∆F)L = L · δ(∆F)cr where δ(∆F)cr is an increase in the free energy of the helix-coil transi-tion determined above for poly(AT) with a single stabilized site of a single base pair(ω = 1, L = 1, δ(∆F)cr = 24.4 kcal). For this case, the free energy of each of the Lmodified base pairs is increased by this δ(∆F)cr value:

∆Fmbp = ∆F + δ(∆F)cr [14]

The cross-linking effect does not occur if δ(∆F)L = δ(∆F)cr, where the δ(∆F)cr deter-mined for L = 1 is uniformly distributed among L neighboring base pairs (L >1 bp):

∆Fmbp = ∆F + δ(∆F)cr/L [15]

However, if one doubles the value of δ(∆F)L to 50 kcal the cross-linking effectoccurs for L = 1 to 50 bp (Figure 12). Figure 12 illustrates the conservation of thecross-linking effect for poly(AT) (N = 5000 bp) for a single stabilized region (ω = 1)and a uniform distribution of δ(∆F) = 50 kcal over one to L neighboring base pairs.The effect occurs for an L value up to 50 bp. A large change in L (from 1 to 50 bp)only slightly alters the differential melting curve, and the ϑext value remains close to1 when almost all ordinary base pairs are melted. This demonstrates that a relative-ly slight increase (~1 kcal) in the free energy of each modified base pair in a longmodified regions is sufficient to give result in the cross-linking effect. The effect ofa relatively slight linear increase with L in δ(∆F)cr (per stabilized site) as well as ofa strong decrease in δ(∆F)cr/L (per base pair in stabilized site of L bp) is readilyobserved in Figure 13. One has the following relationships for δ(∆F)cr/L: 1 bp - 24.4kcal; 5 bp - 5.3 kcal; 10 bp - 2.9 kcal; 25 bp - 1.4 kcal; 50 bp - 1.0 kcal.

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Figure 10: Determination of δ(∆F)cr forpoly(AT) (N = 5000 bp, ω = 6) with six sta-bilized sites at base pairs at positions 1,1000, 2000, 3000, 4000, 5000 from thedependence 1-nnor[δ(∆F)] calculated for T =T∞ + 20ºC (curve 1). The results of calcula-tion for T = T∞ + 10º C (curve 2) and T = T∞ + 30º C (curve 3) are given for com-parison. The corresponding curves for ns(1a, 2a, 3a) demonstrate that more than 99%of ordinary base pairs are melted.

Figure 11: Determination of δ(∆F)cr for DNA with arandom distribution of AT and GC base pairs (N = 5000bp, GC = 50%, ω = 6) and six stabilized sites at basepairs at positions 1, 1000, 2000, 3000, 4000, 5000 fromthe dependence 1-nnor[δ(∆F)] calculated for T = T∞ +20º C (curve 1). The results of calculation for T = T∞ +10º C (curve 2) and T = T∞ + 30º C (curve 3) are givenfor comparison.The corresponding curves for ns (1a, 2a, 3a) demonstratethat more than 98% of ordinary base pairs are melted.

Figure 12: The illustration of conservation of thecross-linking effect for poly(AT) (N = 5000bp, ω = 1) with a single stabilized site (region) afteruniform distribution of δ(∆F) = 50 kcal along Lneighboring base pairs (L = 1 ÷ 50bp) located

around the 2500th base pair. Melting before (1)and after (2) modification that increases total freeenergy of the helix-coil transition of L neighbor-ing base pairs. ϑext(T) and ϑ´T(T) are not changedafter an increase in L from 1 to 50 base pairs.

Melting of DNA’s that Contain Several Blocks (ω>1) of Stabilized Base Pairs(L>1). Examples of Compounds that Give Rise to a Strong Local Stabilization ofthe Double Helix

When the number of modified base pairs (ω · L) is comparable with the length ofthe DNA molecule (N), a two step character of the helix-coil transition occurs forDNA with periodic blocks of ordinary base pairs and base pairs stabilized by achemical modification or ligand binding (Figure 14).

Using results from the melting of DNA-histone H1 complexes (19), one can showthat individual molecules of histone H1 form stabilized regions that behave like aninterstrand cross-link for free DNA regions if the ionic strength of the solution islow enough. In this work all chromatin proteins except H1 were dissociated fromDNA in such a way that H1 molecules conserved their periodic location alongDNA chain (one H1 molecule per 200 bp). As follows from the differential melt-ing curve shown in paper (19), the ratio of the areas of the melting bands corre-sponding to regions of free DNA (TDNA = 46º C) and regions covered with histone(Th = 77º C) is equal to 1/4. Therefore the length of the region covered by a mol-ecule of H1 is equal to 40 bp, and the length of free DNA region is 160 bp. Ourcalculations demonstrate that the regions covered by histone molecules behave likestabilizing interstrand cross-links because the covered regions do not melt until thefree regions melt. The melting of DNA complexes with other histones (20) andprotamines (21) provide other examples of similar behavior. Calculations alsodemonstrate that a block of 25 or more adjacent GC base pairs behaves like aninterstrand cross-link for neighboring blocks of AT base pairs.

There are three examples of metal-based cross-linking agents that highly stabilizetheir DNA binding sites: [{trans-PtCl(NH3)2}2H2N(CH2)4]Cl2, rb = 0.0161, δTm =9.3º C (22); [Ru(phen)2(OH2)2]2+, rb = 0.015, δTm = 12.9º C (23) and[{cis-PtCl(NH3)2}2H2N(CH2)4]Cl2, rb = 0.02, δTm = 15º C (24) where rb is thenumber of chemical modifications caused by these compounds per nucleotide. Ourcalculations demonstrate that the strong increase in the melting temperature due tothese compounds is equivalent to an elevation caused by strongly stabilized basepairs that are able to prevent strand separation after melting of ordinary base pairs.These compounds may increase the free energy of melting the modified base pairsto a value that is sufficient for the formation of stabilizing interstrand cross-links.

Another example of a strongly stabilizing agent is adriamycin. This molecule doesnot form an interstrand cross-link. However covalent linkage of this drug to one ofthe DNA strands gives remarkable stability to the DNA duplex due to its unusual-ly strong non-covalent interaction (25).

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Figure 13: Dependence of δ(∆F)cr and δ(∆F)cr /L on Lfor poly(AT) with a single stabilized region of L basepairs (N = 5000 bp, ω = 1).

Figure 14: Two step character of the helix-coil transition for an increase in L for DNAwith a periodical sequence of AT base pairsand base pairs stabilized after chemical modi-fication. “Cross-linking effect” occurs fornon-modified AT blocks. ϑext(T) ≈ 1 and n =ω until their almost full melting. N = 5000 bp;δ(∆F) = 50 kcal; ω = 50. L values are shownin the figure. Poly(AT) without stabilizedsites corresponds to L = 0.

Acknowledgments

This work was supported by Fund of Fundamental Investigations of the Republicof Belarus (grant #X02R-049), by the Grant Agency of the Czech Republic (grantno. 305/02/1552), the Grant Agency of the Academy of Sciences of the CzechRepublic (grant no. A5004101), NFSAT MB 078-02 # CRDF 12027 and by ISTCFoundation (grant # A302-2).

References and Footnotes

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1.2.

3.

4.5.

6.7.8.

9.10.11.12.13.14.

15.

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17.18.19.20.21.22.23.

24.

25.

26.

K. W. Kohn, Cancer Research 56, 5533-5546 (1996).M. Tomasz, R. Lipman, D. Chowdary, J. Pawlak, G. L. Verdine and K. Nakanishi, Science235, 1204-1208 (1987).R. Bizanek, D. Chowdary, H. Arai, M. Kasai, C.S. Hughes, A.C. Sartorelli, S. Rockwell andM. Tomasz, Cancer Res. 53, 5127-5134 (1993).J. Hansson, R. Lewensohn, U. Ringborg and B. Nilsson, Cancer Research 47, 2631-2637 (1987).S. R. Keyes, R. Loomis, M. P. DiGiovannna, C. A. Pritsos, S. Rockwell and A. C. Sartorelli,Cancer Commun. 3, 351-356 (1991).C. Hofr, and V. Brabec, J. Biol. Chem. 276, 9655-9651 (2001).A. E. Ferentz and G. L. Verdine, Nucleic Acids and Molecular Biology 8, 14-40 (1994).S. S. G. E. van Boom, D. Yang, J. Reedijk, G. A. van der Marel and A. H.-J. Wang, J. Biomol.Struct. Dynam. 13, 989-998 (1996).R. M. Wartell and A. S. Benight, Physics Reports 126, 67 - 107 (1985).D. Poland, Biopolymers 13, 1859-1871 (1974).M. Fixman and J. J. Freire, Biopolymers 16, 2693-2704 (1977).D. Y. Lando, and A. S. Fridman, Biopolymers 58, 374-389 (2001)D. Y. Lando, J. Biomol. Struct. Dynam. 15, 129-140 (1997).D. Y. Lando, A. S. Fridman, A. G. Kabak, and A. A. Akhrem, J. Biomol. Struct. Dynam. 15,141-150 (1997).D. Y. Lando, A. S. Fridman, V. I. Krot,. and A. A. Akhrem, J. Biomol. Struct. Dynam. 16,59-67 (1998).D. Y. Lando, A. S. Fridman, S. G. Haroutiunian, A. S. Benight and Ph. Collery, J. Biomol.Struct. & Dynam. 17, 697-711 (2000). L. A. Marky and K. J. Breslauer, Biopolymers 26, 1601-1620 (1987).K. J. Breslauer, Methods in Enzymology 258, 221-242 (1995).K. Hayashi, J. Mol. Biol. 94, 397-408 (1975). R. L. Rubin and E. N. Moudrianakis, J. Mol. Biol. 67, 361-374 (1972).S. S. Yu and H. J. Li, Biopolymers 12, 2777-2788 (1973). N. Farrell, Y. Qu, L. Feng and B. Van Houten, Biochemistry 29, 9522 - 9531 (1990).N. Grover, T. W. Welch, T. A. Fairley, M. Cory and H. H. Thorp, Inorg. Chem. 33, 3544-3548 (1994).J. Kasparkova, O. Novakova, O. Vrana, N. Farrell and V. Brabeck, Biochemistry 38, 10997-11005 (1999).S. M. Zeman, D. R. Phillips and D. M. Crothers, Proc. Natl. Acad. Sci. USA 95, 11561-11565 (1998).D. Y. Lando, A. S. Fridman A. S., and R. M. Wartell, J. Biomol. Struct. & Dynam. 20, 519-532 (2003)

Date Received: November 22, 2002

Communicated by the Editor Valery Ivanov