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Journal of Biomolecular Structure & Dynamics, ISSN 0739-1102
Volume 6. Issue Number 2 (1988), i>Adenine Press (1988).
A Theoretical Study of Formation of DNA
Noncanonical Structures Under
Negative Superhelical Stress
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V.V. Anshelevich, A.V. Vologodskii
and M.D. Frank-Kamenetskii
Institute of Molecular Genetics
USSR Academy of Sciences
Moscow
Abstract
The development of statistical mechanical models of the formation of noncanonical structures
in circular DNA and the finding of the energy parameters for these models made it possible to
predict the appearance of such structures in a DNA with any given sequence. It does not seem
feasible, however, to perform such calculations for DNA sequences of considerable length by
allowing for all the possible states. We propose a special algorithm for calculating the thermodynamic characteristics of various conformational rearrangements in DNA that occur
under negative supercoilings, allowing for several possible states of each base pair in the
chain. Calculations have been performed for a number of natural DNAs. According to these
calculations, the most likely noncanonical structures in DNA under normal conditions are
cruciform structures and the Z form. The results of the calculations are compared with the
experimental data reported in the literature. State diagrams have been computed for a number
of inserts in circular DNA that can adopt both the cruciform conformation and the lefthanded helical Z form.
Introduction
Considerable progress has been made to date towards the understanding of how the
left-handed helcial Z form appears in DNA under the influence of negative supercoiling. First of all, it was demonstrated that the B-Z transition can occur in DNA
segments with any sequence provided the torsional stress is big enough (1-4). A
statistical mechanical model of the transition was developed (5,6) and its energy
parameters were determined (3,6-8). As a result it became possible, in principle, to
predict by virtue of theoretical calculations the characteristics of Z form appearing
in DNA with any given base sequence. As regards transitions in specific DNA
segments of several dozen base pairs, such calculations can easily be performed by
allowing for all the states that contribute in an appreciable manner to the partition
function (3,7,9-11). However, this approch proves unacceptable for long enough
DNA chains, as an excessive volume of calculations would be required. Special
algorithms need to be developed.
247
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248
Anshelevich eta/.
The problem of calculating the equilibrium conformation of circular DNA with
allowance for all the possible base-pair states is essentially different from the
analogous problem for the linear molecule. The point is that the topological constraints imposed on the possible conformational rearrangements in circular DNA
give rise to an unlimited range ofinter-residue interactions. The most rigorous solution
ofthis problem was proposed in (12). However, that paper considered only two possible
states of each residue. This is quite insufficient for a consistent analysis of conformational rearrangements in circular DNA because of the influence that the various
local transitions have on each other. A consistent analysis must allow for at least the
mian state of the chain links (B form) and the possibilityofZ form, open regions and
cruciform structures arising. As follows from the structure and properties of the Z
form, any analysis of the B-Z transition for an arbitrary DNA sequence must allow
for three states of every residue, even if we do not consider any other conformational
states (5,6). Another shortcoming of the algorithm proposed in (12) was its inability
to find local conformational characteristics ofcircular DNA. such as maps describing
the distribution of conformations along the DNA chain. Hence the need for the
algorithm to calculate both global and local characteristics of circular DNA allowing
for several possible states of each base pair. That is what the present work is all
about We have developed the algorithm in question and used it for a number of
calculations, comparing the results with the available experimental data. Before we
go on to present the results, we shall briefly recapitulate the previously published
formulation of the model of the B-Z transition and cruciformation under the
influence of negative superhelicity (5,6,13) and discuss the parameters of this model.
Theory
Model
Consider a DNA segment in which every base pair is in the Z form, flanked by BONA segments. As with other conformational transitions in DNA. the free energy
of this segment is counted off its free energy in the B form and is expressed as the
sum of the free energies of transition for individual base pairs, ~F, and the free
energy of the two boundaries separating this segment from the flanking B-DNA
segments, 2Ffz. However, the structural peculiarity of the Z form (14) requires that
the statistical mechanical model of the B-Z transition take into account the following
two factors that make it distinct from the models of other conformational transitions in DNA:
1. Each base pair in the Z form can be in one of two energetically non-equivalent
conformations. The most advantageous state of a pair corresponds to the syn
conformation of purine and the anti conformation of pyrimidine, with the free
energy ~F1 • The state corresponding to the syn conformation of pyrimiding and
the anti conformation of purine can also be in the Z form but is characterized by a
higher free energy ~F11 •
2. In a regular Z helix the syn and anti conformations of nucleotides in either DNA
chain must strictly alternate. Any perturbation of this alternation is associated
with additional "phase change" free energy Fjzz.
DNA and Negative Superhelical Stress
249
The values of ~F1 and ~F 11 are assumed to depend only on the type of the given base
pair and to be independent of the adjacent pairs. As follows from (6,9), this simple
assumption is consistent with the available experimental data. Thus a pair's transition
from B form to the Z form is characterized by four energy parameters: ~FlT• ~F~,
A
II
aFAT•
......II
~.t'GC.
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The free energy of any segment that is in the Z form and has a well-defined conformation of each base pair in it can be unambiguously expressed in terms of these six
energy parameters within the framework of the above model.
The formation of cruciform structures in palindromic regions is associated with a
free energy increase due to the four helix ends forming at the base of the cross and
loops forming in the hairpins. We believe that this increase of free energy F c upon
the linear-to-cruciform transition of a palindromic region does not depend on the
size of the palindromic, but only on the number and type of bases in the hairpin
loops. To allow for this dependence we assumed, as we did before (13), that the state
of unpaired bases in the loops is energetically equivalent to the state ofDNAlinks in
the melted conformation. Consequently, the value of Fe was expressed as
[1]
where~~ denotes the free energies of the melting of pairs forming hairpin loops, and F~
is a constant that does not depend on the sequence of bases in the palindrome.
Besides perfect palindromes, we considered palindromes having one mismatched
pair if the mismatched pair was removed from the palindrome ends by at least three
base pairs. When such a palindrome adopts the cruciform conformation, one mismatched pair may form in each hairpin. Four mismatched pairs are possible. The
results obtained in (15) show that the additional increase of the cruciform's free
energy is roughly the same for all four mismatched pairs. We assumed, in accordance with (15), that this free energy increase amounts to 4.6 kcal/mol regardless of
the mismatched pair.
The formation of Z-form segments and cruciform structures in circular DNA is
associated with a change of the supercoilingenergy ~G. In accordance with (6,7,13),
the value of ~G for a given microstate was determined by the formula
~G
=
llOORTM/is ((a
+ K8znsz/M + Kznz/M + ncfM)2 - a 2)
[2]
where R is the gas constant, Tis absolute temperature, M is the number ofbase pairs
in DNA, y8 is the number ofbase pairs per helix turn in B-form DNA, a is superhelix
density, the parameter K 8 z, which is equal to 4.2, characterizes the unwinding of the
double helix at the B-Z boundary (7), n 8 z is the number of boundaries, ~ is the
number of base pairs entering into cruciform structures. The parameter Kz, which
characterizes helix unwinding upon the trnasition from the B form to the Z form, is
expressed through the number of base pairs per double helix turn in B form and Z
form DNA
Kz = 1 + y 8 = 1.9
[3)
250
Anshelevich et al.
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Values of Parameters
Experimental data demonstrate that the superhelix density at which the B-Z transition
occurs in a given DNA strongly depends on the ionic conditions in solution (7 ,14,16).
This is also true of cruciform structures, though to lesser extent. The parameters of
the B-Z transition were evaluated in the TBE buffer (3,6,7), and all the calculation
results cited below refer to these conditions. We determined the value of the parameter
F~ characterizing cruciformation in this buffer on the bases of the data from (17).
Note that, as shown in ( 18), the parameter F~ grows somewhat with increasing palindrome size. Therefore its value cited below is only adequate for palindromes whose
size is not greatly different from 40 bp. Nevertheless, we did not adjust the value of
F~ in our calculations, for the experimental data available to date do not warrant a
reliable assessment of the relation between palindrome size and F~. The parameters
~fAT and ~foe for the TBE buffer and 20°C were found on the basis of the DNA
melting data (19).
The parameters ofB-Z transition are listed in Table I. The value ofF~ was taken to be
12 kcal/mol, and the vlaues of ~fAT and ~foe were 0.86 kcal/mol and 1.87 kcal/
mol respectively.
Calculation of Thennodynamic Characteristics of Circular DNA with Several Competing
States
Consider a DNA segment consisting of residues number 1,2,... ,N, each of which can
be in one ofL+ 1 states O,l, ... ,L. As a particular case the segment may coincide with
the whole DNA We always consider one arbitrarily chosen DNA strand where each
residue corresponds to one of four nucleotides: A,C,G,T. We shall assume that the
residues bordering on our segment are fixed in state 0 (B form). Moreover, the
sequence is assumed to contain only non-overlapping palindromic regions.
The statistical weight of microstate in which the n-th residue is in state~ is defined
by the expression
[4]
Table I
The Set of Parameters of the B-Z Transition
Parameter
Quantity Kcal/mol
Reference
L\F~
0.33
(7)
AF~n
L\F::C
1.15
(6)
2.6
(3)
L\F~~
3.6
(3)
5.2
(7)
4.0
(6)
Ffz
Ff
251
DNA and Negative Superhelical Stress
where
N
L
L\Tw(ll, ...,lN) =
[5]
tn (ln,ln+l)
n=O
N
q(ll,... ,lN) =
TI qn (~,ln+l)
[6]
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n=O
The model formualted in the previous section takes into account four residue states.
We shall assign the index 0 to the B form, 1 and 2 to the Z form assuming the base in
the chosen chain is in the anti and syn conformations respectively (Z8 and Z 5 conformations), 3 to cruciform structure (Cr.). The function G(L\Tw) in [4] has the
form
G(L\Tw)/RT = (llOOM/y~) ((cr
+ y8 L\Tw/Mi- <1)
[7]
where M is the total number of residues in circular DNA The function ~(1,1') and
qn(l,l') are defined by the following matrices (1 is the line number, 1' is the column
number):
0
tnCU') =
Ys 1
qn(l,l') =
Ksz
+ Kz
Ksz
(n+l)
Kc
+ Kz
Ksz
Kz
Kz
(n+l)
Kc
Ksz
Kz
Kz
(n+l)
Kc
0
0
0
0
p<n+l)
a
1
0 BZ
0
sz
cr p<n+l)
ZZ a
0
sz
(n)
crc
p~n+l)
0
(n) (n+l)
c Pa
(n+l)
OszPs
p~n+l)
0
p(n+l)
zz s
(n) (n+ I)
crc Ps
0
(n)p (n+l)
c c
(n) (n+ I)
crc Pc
(n) (n+ I)
crc Pc
p~n+l)
where
K(n)
c
= {
N~l if the n-th residue is at the left border of the j-th palindromic region
0 in all other cases
[8]
[9]
252
Anshelevich et al.
N~) is the length of the j-th palindromic region; the values of p~n>, p~n) are expressed
in terms of the change of free energy aF corresponding to the transition of a given
pair into the assigned state in the Z form (Za or ZJ:
p = exp(-dF/RT)
where aF assumes one of four values aF~, aF~, aFlT• aF~T;
BZ
crsz = exp(-Fj /RT)
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O'zz = exp(-Ff/RT)
(n) _
°C
-
0 if the. n-th and (n+ 1)-th residue are within the palindromic region
exp(-Fg>;RT) if the n-th residue is not and the (n+1)-th residue is
within the j-th palindromic region
{
1 other cases
(cn) = {o1 if the n-th link is one of the links of palindromic region
P
otherwise
Algorithm
Let us choose a value 1c: > 0 and integers kz. ks and Icg> such that the following
equalities are satisfied with the necessary accuracy:
[10]
Then
[11]
where ~(1,1') are integers. Introduce the function g(k) with the help of the equality
g(k) =
G(~c:klrs)/RT
[12]
Let Zn(l,k) be the partition function for the first n residues under the condition that
the (n+ 1)-th residue is in state 1 and
N
L ki (li,li+ I) =
i=n
K
Then
[13]
and the following recursion equation holds:
[14]
253
DNA and Negative Superhelical Stress
where k = 0,1,2,... ,kmax• n = 1,2,...,N.
Let Pn(l,k) be the probability of the n-th residue being in state 1 and
L
i=n
~(li,li+ I)
= k.
According to the predefined boundary conditions
PN+ 1(l,k) = o(l)O(k)
[15]
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where o is the Dirac's function, and the following recursion equation holds:
Pn(l,k) =
L Pn(l,l',k)pn+l(l',k-~{1,1'))
[16]
I'
where
P{ll' k) = {Zn-l{l,k)qnQ,l')/Zn{l',k-kn{l,l')), if Z{~,k-k{ln,l')) = 0
' '
0 otherwise
[17]
is the conditional probability of the n-th residue being in state 1, provided the (n + 1)-th
residue is in state 1' and
L ~(li,li+ I) =
i=n
k.
Thus, the recursion procedure [13[-[14] makes it possible to calculate the values
Zn{l,k) for all n from 0 to N and, consequently, to obtain the conditional pro·
babilities P n(l,l',k) in accordance with [17]. After this the second recursion pro·
cedure [15]-[16] can be used to calculate the probabilities Pn{l,k).
Since we are considering a situation where the alternative structures in linear DNA
are less advantageous than the B form, the value of~ax can be chosen on the basis
of the relation
All the other thermodynamic values that may interest us are expressed directly
through P n(l,l' ,k) and Pn(l,k). For example, the probability of the n-th residue being
in state 1
Pn{l)
=
Lk Pn{l,k);
[18]
the probability of the n-th residue being in state 1, and the (n+ 1)-th link being in
state 1'
[19]
254
Anshelevich et a/.
the probability of the change of equilibrium twisting
~ Tw
being equal to kK/y 8
Pr(k) = p 0 (0,k);
[20]
the mean number of residues in state 1
N
Iii=
L
n=l
[21]
Pn(l);
the mean change of equilibrium twisting
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[22]
The number of operations required for calculations under this algorithm is proportional to k.naxN.
Results and Discussion
Figure 1 presents the results of calculations as to the formation of Z-form segments
and cruciform structures for <j>Xl74 DNA The calculations were performed for a
superhelix density of -0.06, which corresponds to the a value of this phage's replicative
~X114
0.6
0.1
0.4
0.1
40
10
1000
1000
3000
4000
5000
Figure 1: Dependence ofprobability ofZ-form segments on the chain coordinate for<j>Xl74 DNA at a = -0.06
(above). Shown below are data from (21) on anti-Z antibody binding on this DNA The ordinate here is
the percentage of molecules in which antibody binding occurred in this particular part of the molecule.
255
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DNA and Negative Superhelical Stress
DNA form isolated from the cell (20). Note first of all that at this superhelix density,
under the ionic conditions specified, only left-handed helical Z-form segments
arise with an appreciable probability. The probability ofcruciform structure appearing
in the main palindrome of this DNA is as low as 0.0013. Meanwhile the Z form arises
with a probability reaching 0.9 (in the segment with the TGCGTGTACGCGCAG
sequence). In several dozen sites the probability of Z-helix formation exceeds O.ol.
The results of these calculations maybe compared with the experimental data from
(21), where the binding sites of anti-Z antibodies on supercoiled <i>Xl74 DNA were
mapped. This comparison is semi-quantitative, for the probability of antibody
binding to a particular DNA site is hardly proportional, in any strict sense, to the
probability of the Z form arising at that site. Nevertheless, the comparison of our
calculations with the experimental results (Figure 1) clearly deomstrates an agreement
between theory and experiment.
Figure 2 shows the results of similar calculations for the DNAs of plasmids pA03,
pBR322, and of the SV40 virus. In contrast with <i>Xl74 DNA, the formation of
cruciform structures in these three DNAs is characterized by the same or higher
probability as the appearance of Z-form segments. The probability of cruciformation
is especially high in pA03 DNA, which is a fourth part of the ColEl plasmid carrying
its main palindrome of 31 bp (22). The formation of Z-form segments in pBR322
DNA and SV40 DNA has been tackled experimentally in a number of studies (2326). The data reported in (23-24) for pBR322 are not characterized by a very high
level of accuracy and do not contradict the results of our calculations. According to
some reports (14,27), the most probable site of Z-helix formation in this DNA is in
pA03
0.2
I
500
p~
0.2
0.1 ..
.J
I
•
I I
J.
.l
1500
1000
!~I ,II, :L, ',
,! J ol
,.
1000
2000
p8R322
L I.
• I
a!_l
0
,000
1
lj
Figure 2: Dependence of the probability of Z-form segments ( - ) and cruciform structures (-----) on
the chain coordinate for pA03, SV40 and pBR322 DNA The calculations correspond to the superhelix
density of -0.06.
256
Anshelevich et a/.
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the region with the CACGGGTGCGCATG sequence (a). According to our calculations the same probability of transition to the Z form characterizes the region with
the CGCACGCGGCGCAT sequence (b) in which the Z-helix "phase change"
occurs. It was suggested in (28) on the basis of the chemical probing of Z-DNA that
the Z form may with an appreciable probability extend beyond the a region by more
than 10 bp. This is ruled out by our calculations: an increased probability of the Z
conformation is observed only for 2-3 base pairs adjacement to the a region. Nor do
our calculations confirm the conclusion made in (25-26) to the effect that in SV40
DNA the Z form must first of all appear in the region between residues 1 and 400
which contains the SV40 enhancers (Figure 2).
Calculations show that the probabilities ofthe Z-form and cruciform structures in
these four circular DNAs grow monotonically with increasing superhelix density.
In a number of cases, however, circular DNAs display a more complex pattern of
transitions with increased negative superhelicity. Some such cases involving competitive transitions in two spaced DNA segments have been treated theoretically
and experimentally in (10,11,29-30). Let us consider the competition between rearrangements in one and the same region of circular DNA that is capable of adopting different forms. A system of this kind has been considered earlier in (3). The results of
calculations regarding the equilibrium between the B, Z and cruciform structure in
purine-pyrimidine palindromic sequence are presented in Figure 3. As one can see,
a
100
80
'0
20
8
60
40
'0
20
20
2
6
-d·100
'
6
8
Figure 3: Conformational diagrams for segments with sequences d(CG), · d(CG), (a), d(AT), · d(AT),
(b), d(CATG). · d(CATG). (c) built into 4000-bp-long circular DNA The ordinate is the total segment
length. For the sequence d(CATG), · d(CATG), the same diagram was calculated for the 21()()-bp-long
DNA examined in (17,32) (----).
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DNA and Negative Superhelical Stress
257
not very long segments with such sequences can exist only in the B and Z conformations. The pattern grows more complex as the segment length increases. One can
observe, with growing superhelix density, consecutive B-to-cruciform and cruciformto-Z transitions. Figure 3 shows that (CATG)n is the most convenient sequence for
experimental studies of such transitions. For such inserts in circular DNA one
could observe two transitions for not too large nand a convenient superhelix density. It was such inserts, d(CATG)n, that were studied experimentally in (17 ,32). We
performed special calculations for the short plasmids used in those studies (broken
curve in Figure 3c). When comparing our calculations with the experimental data
one has to bearin mind the following facts. Almost in all the plasmids used in ( 17,32)
the length of the palindromic region exceeded that of the purine-pyrimidine region
by 4-6 bp owing to the flanks of the d(CATG )n inserts. This broadened somewhat the
stability range of cruciform structures and hindered the cruciform-to-Z transition.
Our calculations (Figure 3c) were performed for inserts in which the length of the
palindromic region exceeded that ofthe purine-pyrimidine region by 4 bp. Furthermore, the formation of cruciform structures in palindromes less than 25-30 bp long
is characterized by a lower energy F~ (see above). The latter fact broadens the
cruciform range presented in Figure 3c. Considering all this, one can say that the
experimental data reported in (17,32) are not at variance with the results of our
calculations, even though the experimenters failed to observe the cruciform-to-Z
transition. As seen in Figure 3, d(CATG)9 is the most convenient insert for the
experimental observation of this transition.
An even more complex pattern of transitions can be observed in inserts with some
special sequences. We considered an insert with the sequence d((TAh(GC) 8(TA)7]
built into a 4400-bp-long circular DNA This insert is a palindrome and can form a
cruciform structure. The central part of the insert can easily form a Z helix which,
under a much higher superhelix stress, must extend to the insert's flanks. In this case
two Z-Z boundaries must arise. Figure 4 presents the calculated pattern of conformational transitions in this insert. One can see that three conformational rearrangements occur in succession with increasing negative superhelicity. First the Z form
arises in the central part of the insert, then it is ousted by the cruciform structure, and
then, at a still higher superhelix density, the whole insert adopts the Z form.
In none of the calculations described here do we consider the formation of open
region or the H form. In actual fact in most of our calculations we did allow for the
possibility of open regions, but the probability of open regions forming under the
given conditions proved to be negligibly low as compared with the probabilities of
cruciform structures and Z-form segments. Note that at present we do not know well
enough how the supercoiling energy changes upon the formation of open regions.
Further experimental studies are needed to refine the theoretical conception of the
formation of such regions in circular DNA In our calculations we allowed for the
possible formation of open regions in accordance with the model described in (33).
Our data show that the relatively low probability of open regions obtained in (34) at
a = -0.08 for <!>X174 DNA is in fact even lower, for the authors of(34) failed to allow
for the possible appearance of a much more probable alternative structure, the lefthanded helical Z form.
258
Anshelevich eta/.
11Tw
8
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0.6
6
- tJ ·100
Figure 4: Probability of Z form (1) and cruciform structure (2) in the d[(fA),(GC)s(TAh] segment inserted
into 4000-bp-long circular DNA depending on superhelix density. Also shown is the overall change of
axial twisting, aTw, as a result of these transitions (-----).
In our calculations we did not allow for the possibility of the H form. Though the
energy characteristics of the H form are as yet unknown, experimental data show
that even in long homopurine-homopyrimidine regions at neutral pH this form
of DNA arises at -a > 0.06 (35).
The problem of theoretical description of the B-Z transition in an arbitrary-sequence
DNA has recently been considered by Ho et a/. (36). Our approach differs from
theirs in the following ways. First, we proceeded from the B-Z transition model in
which a base pair rather than two pairs (as in (36)) is the elementary structural unit
subject to transition. Ours is a more general model whose simplest version contains
fewer parameters. Secondly, in (36) the partition function is calculated by directly
allowing for all the states that make an appreciable contribution to it. This is a very
ineffective procedure. The authors of (36) had to partition the sequence into short
independent segment, thereby ruling out the calculation of a priori probabilities of
this or that DNA segment adopting the Z form. In our algorithm the time of calculating
the transition characteristics grows in proportion to the first power of the sequence
length. As a result calculations can be performed for long enough chains without
any additional simplifications.
DNA and Negative Superhelical Stress
259
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References and Footnotes
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8320 (1985).
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15. F. Aboul-ela, D. Koh and I. Tinoco Jr.,Nucl. Acids Res. 13,4811 (1985).
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Date Received: July 29, 1988
Communicated by the Editor V.I. Ivanov