Biophysics of the DNA molecule

s __
Physics Reports 288 (1997)
Biophysics of the DNA molecule
Maxim D. Frank-Kamenetskii
Cenier jtir Adwncrd
Biotechrwlogy und Department oJ’ Biomrdical Engineering,
36 Cunmin~gton St.. Boston, MA 02215, USA
87.15.-v; 87.15.Da; 87.15.He; 87.15.Kg
DNA; Topology;
Gel electrophoresis;
Knots; PNA
1. Introduction
DNA plays a crucial role in all living organisms because it is the key molecule responsible
for storage, duplication, and realization of genetic information. DNA is a heteropolymeric
consisting of residues (nucleotides) of four types, A, T, C and G. Fig. 1 shows the chemical structure
of the DNA single strand and the complementary base pairs.
The genetic message is “written down” in the form of continuous text consisting of four letters
(DNA nucleotides A, G, T and C). This continuous text, however, is subdivided, in its biological meaning, into sections. The most significant sections are genes, parts of DNA, which carry
information about the sequence of amino acids in proteins.
The importance of the DNA molecule cannot be overestimated. It is therefore natural that the
molecule has been attracting attention not only of biologists and physicians but also of chemists
and physicists, even theorists (for a popular introduction into the field of DNA science, see, e.g.,
1993; 1997). For more than forty years already, the DNA molecule has been a
subject of biophysical studies. Many outstanding physicists, who made their names in various areas
of traditional physics, mostly in solid state physics, contributed by studying DNA. I.M. Lifshitz
did not publish many papers about DNA. Nevertheless, his role in directing attention of physicists
toward DNA biophysics was very significant, especially in the USSR.
Although I had been already in the field when Lifshitz stormed it in mid-1960s I also experienced
profound influence of his personality and style. Due to his enthusiasm and indisputable reputation
among Soviet physicists, DNA and protein biophysics temporarily became a focus of attention of
the Soviet physics community. This community was a unique phenomenon in the world of science.
Due to I.M. Lifshitz, I had an opportunity to present my work on DNA topology at the famous
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1997 Elsevier Science B.V. All rights reserved
Fig. 1. (a) DNA single strand and (b) the Watson-Crick
base pairs.
Landau seminar at the Kapitza Institute, an unforgettable experience by itself. By that time (it was
in 1975, 1 believe), Lifshitz had replaced Lev Landau as head of Theoretical Department of the
Vavilov Institute of Physical Problems (the official name of the Kapitza Institute; physicists called
it either “Physproblems” or “Kapichnik”) and he led the seminar. When I tried to start my talk, the
participants did not give me the opportunity to say a word shouting: “What is he going to speak
about?“, “He must first say what he is going to speak about”. The noise was really terrible, and I was
confused. It lasted for a while until Lifshitz stood up and said, not loudly, actually: “Stop screaming.
Let him get started”. Magically, these quiet words calmed everybody down and I started my talk. Of
course, I was interrupted many times during the talk, but these were more or less usual questions.
In this article, I give an overview of the area of DNA biophysics in retrospective. The field is
now so big that it would be impossible to a single person to cover all aspects of DNA biophysics.
The choice of topics and their coverage will inevitably reflect the writer’s personal taste and interest.
2. Major structures of DNA
In spite of the enormous versatility of living creatures and, accordingly, variability of genetic texts
that DNA molecules in different organisms carry, they all have virtually identical physical, spatial
structure: the double-helical B form discovered by Watson and Crick (1953). Sequences of the two
strands of the double helix obey the complementarity principle. This principle is the most important
law in the field of DNA, and, probably, is the most important law of the living nature. It declares
that, in the double helix, A always opposes T and visa versa, whereas G always opposes C and visa
versa (see Fig. 1).
M. D. Frank-Kanwnetskii
I Physics Repouts 288 (I 997) 13ST)
Fig. 2. B-DNA. The major and minor grooves are indicated.
2.1. B-DNA
B-DNA (see Fig. 2) consists of two helically twisted sugar-phosphate backbones stuffed with base
pairs of two types, AT and GC. The helix is right-handed with 10 base pairs per turn. The base
pairs are isomorphous: The distances between glycosidic bonds, which attach bases to sugar, are
virtually identical for AT and GC pairs. Because of this isomorphism, the regular double helix is
formed for an arbitrary sequence of nucleotides and the fact that DNA should form a double helix
imposes no limitations on DNA texts. The surface of the double helix is by no means cylindrical.
It has two very distinct grooves: the major groove and the minor groove. These grooves are extremely
important for the functioning of DNA because, in the cell, numerous proteins recognize specific sites
on DNA via binding with the grooves.
Each nucleotide has a direction and therefore the chemical direction is inherent in each of DNA
single strands. In the B-DNA double helix, the two strands have opposite directions. In B-DNA,
base-pairs are planar and perpendicular to the axis of the double helix.
Under normal conditions in solution, often referred to as “physiological” (neutral pH, room temperature, about 200mM NaCl), DNA adopts the B form. All available data indicate that the same
is true for the totality of DNA within the cell. It does not exclude, however, the possibility that
separate stretches of DNA carrying special nucleotide sequences would adopt other conformations.
2.2. B’-DNA
Up to now, only one such conformation is demonstrated, beyond any doubts, to exist under
physiological conditions. When several A residues in one strand (and, accordingly, several T residues
in the other DNA strand) occur, they adopt the B’ form. In many respects, the B’ form is similar to
the classical B form but there are also significant differences. The main difference consists in the fact
that base pairs in B’-DNA are not planar: They form a kind of propeller with a propeller twist of 20”.
hf. D. Frank-Kamenetskii
I Physics Reports 288 (I 997j 13-60
Stretches of A residues produce bends in the double helix (reviewed by Sinden, 1994). Such bends
play a very important role in DNA functioning. Although the structural basis of these bends is not
fully understood, the involvement of the B’ form in the DNA bending is very probable. In spite
of its importance, the B’ form does not differ dramatically from the B conformation. Other helical
conformations have been found in the course of DNA biophysical studies, which are significantly
different from B-DNA.
2.3. A-DNA
Similarly to B-DNA, the A form can be adopted by an arbitrary sequence of nucleotides. Like in
B-DNA, in A-DNA the two complementary strands are antiparallel and form right-handed helices.
DNA undergoes transition from the B to A form under dehydration conditions (reviewed by Ivanov
and Krylov, 1992). In A-DNA, the base pairs are planar but their planes make a considerable angle
with the axis of the double helix. In doing so, the base pairs shift from the center of the duplex
forming an empty channel in the center.
If any, A-DNA plays a rather modest role in DNA functioning. There are data indicating that
some proteins induce transition from B to A form. The reason for this remains to be elucidated.
2.4. Z-DNA
Z-DNA presents the most striking example of how different from the B form the DNA double
helix can be. Although in Z-DNA the complementary strands are antiparallel like in B-DNA, unlike
in B-DNA, they form left-handed, rather than right-handed, helices. There are many other dramatic
differences between Z- and B-DNA (reviewed by Dickerson, 1992).
Not any sequence can adopt the Z form. To adopt the Z form, the regular alternation of purines
(A or G) and pyrimidines (T or C) along one strand is strongly preferred. However, even this
is not enough for Z-DNA to be formed under physiological conditions. Nevertheless, Z-DNA can
be adopted by DNA stretches in cell due to DNA supercoiling (see Section 4.3.1). The biological
significance of Z-DNA, however, remains to be elucidated.
2.5. ps-DNA
The complementary strands in a DNA duplex can be parallel. Such parallel-stranded
(ps) DNA
is formed most readily if both strands catty only adenines and thymines and their sequence excludes formation of the ordinary antiparallel duplex (reviewed by Rippe and Jovin, 1992). If these
requirements are met, the parallel duplex is formed under quite normal conditions. It is right-handed,
but the AT pairs are not the usual, Watson-Crick ones, but rather the so-called reverse WatsonCrick.
Some other sequences also can adopt parallel duplexes. For instance, at acidic conditions two
strands carrying only C residues form parallel duplex consisting of protonated CC+ base pairs (see
Section 2.7).
M.D. Frunk-Kamenetskii
I Physics Reports 288 (I 997) 13~60
Fig. 3. The structure
triple helix is made.
of (a) pyrimidine
(TAT and C+GC) and (b) purine (AAT and CCC) base triads of which DNA
If DNA carries a homopurine-homopyrimidine
tract, a homopyrimidine
oligonucleotide can bind
to this tract lying in the major groove and forming Hoogsteen pairs with DNA bases (Moser and
Dervan, 1987; Le Doan et al., 1987; Lyamichev et al., 1988; reviewed by Frank-Kamenetskii
Mirkin, 1995; Soyfer and Potaman, 1996). The canonical base-triads thus formed are shown in
Fig. 3. In recent years, the variety of sequences, which have been found to be capable to form
triplexes, has been significantly enlarged (reviewed by Frank-Kamenetskii
and Mirkin, 1995; Soyfer
and Potaman, 1996). In addition to intermolecular triplexes, intramolecular triplexes or H-DNA can
be formed under certain conditions (see Section 4.4.3).
M. D. Frank-Kummetskii
I Physics Reports 288 (1997)
Fig. 4. G quadmplex.
2.7. Quadruplexes
Of all nucleotides, guanines are the most versatile in forming different structures. They may
form GG pairs but the most stable structure, which is formed in the presence of monovalent
cations (especially potassium), is G4 quadruplex (see Fig. 4). G-quadruplexes
may exist in a
variety of modifications: all-parallel, all-antiparallel and others (reviewed by Sinden, 1994). As a
result, G-quadruplexes are easily formed both inter- and intramolecularly,
again with a variety of
A totally unusual quadruplex structure was discovered by Gehring et al. (1993). It contains two
hemiprotonated parallel-stranded
duplexes consisting of CC+ pairs. The two parallel-stranded
duplexes are associated in a mutually antiparallel manner so that CC+ base pairs from one duplex are
“layered” by CC+ pairs from the other duplex, thus alternating along the structure.
3. Methods to study DNA (General)
The whole arsenal of physical methods, which are generally used to study molecular structures,
is applied to studying DNA. In this section we will briefly consider the most important of these
methods emphasizing their role in the field of DNA biophysics.
3.1. X-ray analysis
As in other fields where molecular structure is essential, X-ray analysis occupies a unique position
among methods to study DNA structure as the only direct method, which permits to elucidate the
structure in all details. X-ray crystallography is absolutely indispensable in the study of the detailed
structure of complexes of DNA with proteins, which is most essential for understanding how DNA
molecules function in the cell.
Reports 288 ilW7)
However, for a long period of time the whole edifice of molecular biology relied on one of the
indirect versions of the X-ray analysis, fiber diffraction, rather than on classical X-ray crystallography.
3.1. I. Fiber d$%action
Long DNA molecules cannot be crystallized. As a result, from early 1950s till late-l 970s only
X-ray diffraction from DNA fibers was used to elucidate the DNA structure. Such data were used by
Watson and Crick (1953) to propose structures for B- and A-DNA. Fiber diffraction is an essentially
indirect method, and, to elucidate structure from the data on fibers, one should heavily rely on
theoretical approaches, such as conformational analysis.
3.1.2. X-ray crystallography
First DNA crystals became available only in the late 1970s after remarkable progress in chemical
synthesis of short DNA pieces had been achieved. This led to many discoveries, first of all of
left-handed Z-DNA by Rich and co-workers (Wang et al., 1979).
In recent years, a lot of very detailed data on the structure of DNA have been obtained by X-ray
crystallography, including detailed study of the B, B’, A and Z forms (reviewed by Dickerson, 1992).
Among the most recent achievements of the method is solution of the structure of G-quadruplex
(Kang et al., 1992) and of C-quadruplex (Chen et al., 1994).
In spite of remarkable accomplishments,
serious limitations are inherent in the method. It is extremely hard to obtain good crystals of DNA even if it adopts only a unique structure. Even if this
difficulty is overcome, it sometimes appears that the structure in crystal is significantly perturbed
by interaction with neighboring molecules. It is especially true of such subtle, but very important
from a biological viewpoint, deformations as bending of the double helix. Thus, the data obtained
by the methods of X-ray crystallography
should always be correlated with the results of indirect
methods, which permit to study DNA in solution. X-ray crystallography is totally helpless to study
such biologically significant problems of DNA biophysics as DNA supercoiling (see below).
3.2. Nuclear magnetic
The role of NMR constantly increases, and, in recent years, it has even started to compete successfully with X-ray crystallography in the field of DNA biophysics. This has become possible as
a result of development of two-dimensional proton NMR techniques, especially nuclear Overhauser
effect spectroscopy (NOESY). The great advantage of NMR, as compared with X-ray crystallography, consists in the fact that it does not require crystals. As a result, some DNA structures, that
resist crystallization, like DNA triplexes, have become the subjects of very fruitful study by NMR
(Rajagopal and Feigon, 1989).
A great advantage of NMR consists in the possibility of studying structural fluctuations, or
“breathing”, of the DNA double helix by following proton exchange in DNA bases. Such studies made it possible to find out important characteristics of base-pair fluctuational openings in DNA
(see Section 7.2).
Turning to the limitations of the method, it should be emphasized that resolution of even the
most powerful NMR spectrometers permits the study of only short DNA molecules containing
about a dozen of distinguishable nucleotides. Although crystals are not needed, only very concentrated solutions can be studied. Therefore, like X-ray crystallography, NMR is useless for studying
M.D. Frank-Kamenetskiil
Physics Reports 288 (1997)
many biologically relevant problems of DNA structure. In spite of its limitations, proton NMR has
firmly occupied the second position, after the X-ray crystallography, among methods to study DNA
3.3. Microscopic
Although microscopy looks like the most direct way to visualize structure, in application to DNA
it has too numerous limitations to occupy a position ahead of X-ray crystallography
or NMR.
Nevertheless, in recent years the role of microscopy in the field of DNA has significantly increased
due to progress in regular electron microscopy of DNA and its complexes and the development of
new techniques, cryoelectron microscopy and scanning force microscopy.
3.3.1. Regular electron microscopy
In regular transmission electron microscopy, DNA molecules are placed on the grid, dried and
contrasted by one or another method. In recent years the most popular technique of contrasting has
become staining with uranyl acetate. As a result, duplex DNA molecules and proteins attached to
them are clearly seen (see, e.g., Chemy et al., 1993a).
Regular electron microscopy permits one to see complexes of DNA with proteins. Poor resolution
usually does not permit one to observe the internal structure of the complex but permits mapping
of the location of the protein on DNA.
A limitation of regular electron microscopy stems from relatively poor resolution and possibility
of significant perturbation of DNA structure in the process of sample preparation. And still, in many
cases the method provides the most convincing evidence.
3.3.2. Cryoelectron microscopy
The method is based on obtaining vitrified water solutions via very quick cooling of extremely
thin (in the submicron range) samples. As a result, the DNA molecule is “frozen” in the state it
adopted in solution before cooling. In recent years, the method has received numerous applications
in the field of DNA and its complexes with proteins (Dubochet et al., 1992).
The great advantage over regular electron microscopy consists in avoiding the harsh procedures
of sample preparation, which strongly limits the value of the data obtained by regular electron
A major problem of cryoelectron microscopy stems from the low contrast of DNA molecules.
Without staining or other contrasting procedures they are barely visible in an electron microscope.
Nevertheless, DNA molecules and their complexes with proteins are extensively studied by the
method (reviewed by Dubochet et al., 1992).
3.3.3. Scanning jbrce microscopy
Scanning (or atomic) force Microscopy (SFM or AFM) provides reliable images of DNA
molecules and their complexes with proteins (reviewed by Shao et al., 1995). At present, the resolution of this method is not much higher than that of regular electron microscopy. However, there
is reasonable hope that the resolution will be significantly improved in the course of further development of the method.
M.D. Frank-Kanwnrtskiil
Physics Reports 288 (1997)
3.4. Optical methods
All optical methods that are traditionally applied to study molecular structures, are widely used to
study DNA. They are indispensable in routine investigations because they are cheap and quick. The
great advantage of those that use light in the visible and the UV region consists in the possibility of
studying very dilute DNA solutions, for which intermolecular interaction can be completely neglected.
On the other hand, all these methods are essentially indirect and provide any structural information
only after a careful assignment of particular spectra or spectra1 changes with the help of the more
direct methods described above.
3.4. I. U If spectroscopy
DNA bases absorb UV radiation around 260nm. The intensity of this absorption, which is easy to
measure with regular spectrophotometers,
changes when, for instance, the DNA double helix melts
(i.e. the complementary
strands separate at heating, see Section 6). Hence, UV spectroscopy has
been extensively applied to study DNA melting. Changes of UV absorption are too small for B-to-A
and B-to-Z transitions.
3.4.2. Ciuzlar dichvoism (CD)
CD spectra in the vicinity of 260nm are much more sensitive to DNA helical structure than UV
absorption. B-DNA, A-DNA, and Z-DNA have characteristic and very different CD spectra (Johnson,
1990), and this fact is extensively used in the study of structural transitions in DNA between different
helical structures (Ivanov and Krylov, 1992).
3.4.3. Inji-ared and Raman spectvoscop_v
Infrared and Raman spectra are sensitive to DNA structure. Correspondingly,
IR and Raman
spectroscopies are used to study DNA. However, the main limitation of these methods stems from
the fact that they require high concentrations of DNA. As a result, these methods are less popular
than UV and CD spectroscopies.
3.4.4. Fluorescent methods
DNA molecules practically do not emit absorbed radiation. Fluorescence methods are used via
binding to DNA of strongly fluorescent dye molecules, such as ethydium.
Fluorescence sensibilization and quenching due to excitation energy transfer between the donor
molecule of electronic excitation and the acceptor molecule of the excitation are also used (reviewed
by Clegg, 1992).
3.5. Theoretical methods
The paper that signified the beginning of extensive studies of DNA and its biological role, was
purely theoretical (Watson and Crick, 1953). Since then, theory has been playing a very important
role in study of DNA.
Fig. 5. Theoretical
models of DNA: (a) elastic-rod
model. (b) helix-coil
model. (c) polyelectrolyte
3.5.1. Conjbrma tional analysis
Conformational analysis was especially important during the era of fiber diffraction. In fact, what
Watson and Crick did in their classical paper (Watson and Crick, 1953) was a very simple, but
exceptionally efficient, variety of conformational analysis. Since then, the method has been extensively used to refine structures solved by X-ray crystallography (Dickerson, 1992). The method is
also often used to predict new structures. For instance, parallel-stranded DNA duplexes were first
predicted theoretically and then found experimentally.
3.5.2. Theoretical models
Like in the study of any important physical object, a number of simplified theoretical models of
DNA exist, different models being used to analyze different properties. Fig. 5 presents schematics of
some of these models.
The DNA double helix may be treated as an isotropic elastic rod (Fig. 5(a)). In the framework
of this model, the DNA molecule is described by only three parameters: bending and torsional
rigidities and the diameter. The model has proved to be extremely useful to analyze hydrodynamic
and other properties of linear DNA, when it behaves as a usual polymeric molecule. It also permitted
comprehensive theoretical treatment of DNA topology of both levels - knotting and supercoiling.
We discuss this model at some length in Section 4.
A quite different, but also very successful, model treats the DNA double helix as consisting
of base-pairs of two types: closed and open (Fig. 5(b)). This is the helix-coil model, which has
permitted to explain quantitatively all major features of DNA melting. We discuss this model in
Section 6.
The polyelectrolyte
model (Fig. 5(c)) treats DNA itself just as a charged cylinder but allows for
the mobile counterions surrounding the double helix. We discuss this model in Section 8.
M.D. Frunk-Kumenet.~kii/
Physics Reports 288 (1997)
4. Global DNA conformation
4.1. Elastic rod rnodeE of DNA
DNA behaves as an almost ideal polymer chain. No other polymer molecule is closer to the ideal
polymer chain than the DNA double helix. Due to unusually high bending rigidity of DNA, the ratio
of its persistence length, a, to its diameter, d, is very high. This leads to very small, sometimes
negligible, excluded volume effects under a variety of ambient conditions, not only at the O-point,
like with ordinary polymers (see, e.g., Grosberg and Khokhlov, 1994). This unusual rigidity stems
from the fact that DNA consists of two, rather than one, polymer chains. A common mechanism
of polymer flexibility, due to rotation around single bonds, is excluded for the double helix. It
exhibits bending flexibility only due to accumulation of small changes of angles between adjacent
base pairs. As a result, the DNA double helix is best modeled as an elastic rod (see Fig. 5). Within
first approximation, one can neglect the sequence dependence of the DNA bending and torsional
rigidities and treat DNA as a homogeneous and isotropic elastic rod. This model proved to be a
remarkably good first approximation to treat global DNA macromolecular properties.
Within the framework of this model, the DNA chain is characterized by three parameters:
The bending rigidity, measured in terms of the persistence length, a, or the Kuhn statistical length
(h = 2~); the torsional rigidity C; the DNA effective diameter, d. Numerous properties of linear and
circular DNA molecules can be quantitatively understood in terms of the elastic rod model and the
same set, under given ambient conditions, of the above three parameters.
Ambient conditions, especially the concentration of counterions in solution, may significantly affect
some of DNA parameters. This is the case for the DNA effective diameter. Because DNA is a
highly charged polyion, the excluded volume effects strongly depend on the screening of Coulomb
interaction between DNA segments approaching each other (see Section 8). As a result, the DNA
effective diameter significantly exceeds its geometrical diameter of 2 nm at a low concentration
of counterions in solution. In contrast, DNA bending and torsional rigidities are ionic-strengthindependent within a wide range of ambient conditions (see also Section 8.2).
For the theoretical treatment of statistical-mechanical
properties of DNA within the elastic-rod
model, a Metropolis-Monte-Carlo-type
approach was elaborated by Frank-Kamenetskii et al. (1985a).
In this approach, the DNA chain is modeled as a series of straight segments so that each Kuhn length
contains k such segments. The total elastic energy is the sum of terms, each of which corresponds
to a pair of adjacent straight segments and quadratically depends on the angle between them (see
et al., 1985a; Vologodskii and Frank-Kamenetskii,
1992 for details). The final
results are obtained, within the framework of the model, as asymptotic ones for the large k values.
Fortunately, all characteristics we studied leveled off very quickly with increasing k so that k = 10
proved to be a quite sufficient value to get very reliable quantitative asymptotic results (see Fig. 6).
this model corresponds to the elastic-rod model of the polymer chain (it is also
often referred as the worm-like model; see, e.g., Grosberg and Khohklov, 1994).
4.2. Linear DNA
DNA is a unique object for experimental studies of a virtually ideal macromolecular
coil. In
addition to the already mentioned fact of an exceptionally high a/d ratio, DNA samples are strictly
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Physics Reports 288 11997) 13-60
Fig. 6. Typical results of MetropolissMonte
Carlo calculations on the dependence on the number of straight segments per
Kuhn length, k, of a mean quantity (the mean writhing number, see Section, in the particular case) for a closed
polymer chain. The data are from Vologodskii and Frank-Kamenetskii
monodisperse and the length of the molecule can be varied in a very wide range: From below one
persistence length up to hundreds of persistence lengths. Moreover, recently developed techniques
make it possible to perform quantitative studies of single DNA molecules (Smith et al., 1992,
1996; Strick et al., 1996). In particular, Smith et al. (1992) performed remarkable measurements
of strain/extension
relationship on single DNA molecules. Bustamante et al. (1994) showed that
experimental data agree with theoretical predictions obtained within the framework of the elastic-rod
model. After the DNA molecule was fully extended, further increase of force led to a sharp transition
to a more extended DNA conformation, in which the average distance between adjacent base pairs
was 1.6 times larger than in the normal B-DNA (Smith et al., 1996; Strick et al., 1996).
Normally, linear DNA is in the B-form. Numerous studies have made it possible to determine an
accurate value of the DNA persistence length, a, which proved to be very close to 50nm. Therefore,
the Kuhn statistical length for DNA is equal to 100nm.
4.3. DNA
It was unexpectedly found in 1963 that DNA exists in certain viruses in a closed circular (cc)
form. In this state, the two single strands of which the DNA consists are each closed on themselves. Fig. 7 schematically illustrates ccDNA. One can see that the two complementary
in ccDNA proved to be linked. They form a high-order linkage (of the order of Nl;I,, where
N is the number of pairs in the DNA and y. in the number of base pairs per turn of the
double helix). Initially, the discovery of circular DNA was not seen to be very significant, since
this form of DNA was regarded as exotic. However, in the course of time, the cc form of DNA
was discovered in an even greater number of organisms. Currently, it is generally acknowledged that
precisely this form of DNA is typical of the simplest DNAs, and also of the cytoplasm DNAs of
animals. Also most virus DNAs pass through a stage of the cc form in the course of infection of
M.D. Frank-Kamenetskii
I Physics Reports 288 (I 997) 13-60
Fig. 7. In a circular closed DNA, two complementary
strands form linkage of a high order.
The discovery of ccDNA has led to the formulation of fundamentally new problems, since it
turned out that many of the physical properties of the closed circular form differ radically from
those of the linear form. The difference between the properties of these two forms of DNA is not
at all due to the existence of end effects in the one case but not in the other.
There are two levels of DNA topology. First, ccDNA as a whole can be unknotted (form the
trivial knot, or unknot) or form knots of different types (see Fig. 8). Secondly, two complementary
strands in DNA are linked with each other topologically (Fig. 7).
4.3.1. Knots
The first problem that arises in theoretical analysis of ring polymer chains, including ccDNA,
is formulated in the following way. Let a ring molecule be formed by fortuitous closure of a
linear molecule consisting of n segments. What is the probability of forming a knotted chain, i.e.,
a nontrivial knot? This problem has been clearly formulated by Max Delbriick and solved by our
group (Vologodskii et al., 1974; Frank-Kamenetskii
et al., 1975). Statistical mechanics of knots. To solve the problem of statistical mechanics of knots, one
needs, first of all, a knot invariant. Indeed, a closed chain can be unknotted or can form knots of
different types. The very beginning of the table of knots is shown in Fig. 8. However, an analytical
expression for the knot invariant is unknown. Therefore, we had to use a computer and an algebraic
invariant elaborated in the topological theory of knots. We found that the most convenient invariant
was the Alexander polynomial (reviewed by Frank-Kamenetskii and Vologodskii, 1981; Vologodskii
and Frank-Kamenetskii,
The next problem consisted in generating closed polymer chains. In our first calculations, we
simulated DNA as a freely-joint polymer chain. Several methods exist to generate exclusively
closed chains for this model (Frank-Kamenetskii
and Vologodskii, 1981; Vologodskii and FrankKamenetskii, 1992). Using these methods and teaching the computer to calculate the Alexander
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Phy.Crs Reports 288 (1997)
Fig. 8. Knots.
polynomials and therefore to distinguish the knots of different type, we could calculate the knotting
have been performed
later by other researchers
and Vologodskii, 1981). The data on the relationship between the probability
of knot formation and the number of Kuhn lengths in the chain are collected together in Fig. 9.
We see that the results obtained by various authors agree very well with each other. This is not
surprising, since, in spite of a certain difference in the polymer models employed, to which certain
differences in the results are due, the presented data in all cases fit the model of an infinitely thin
polymer chain.
One can see from Fig. 9 that the probability of knot formation has an evident tendency to approach
unity as y1increases, though it was possible to perform the calculations only up to n values such that P
barely exceeds 0.5. Very recently, these calculations have been significantly extended using Vassiliev
invariants of knots (Deruchi and Tsurusaki, 1993a, b, 1994). These authors extended calculations up
to n = 1600. Remarkably, the data are well approximated by a simple equation:
P(n) = 1 - exp(-k-n)
where ~=3
x 10d3.
M.D. Frank-Kamenetskiil
Reports 288 (1997)
Fig. 9. Probability of knot formation, P, as a function of the number n of Kuhn statistical lengths for an infinitely thin
polymer chain. Different symbols correspond to results obtained by different authors (the data from Frank-Kamenetskii
and Vologodskii, I98 I ).
The above calculations were performed under the assumption that the polymer chain under consideration has zero diameter. In the very early stage of our study of knots we already realized that
the excluded volume effects should significantly decrease the knotting probability (Vologodskii et al.,
1974, Frank-Kamenetskii
et al., 1975). However, the knotting probability proved to be even more
sensitive to the excluded volume effects than we originally anticipated so that these effects could not
be neglected even in the case of DNA.
We arrived at this conclusion using the Metropolis-Monte
Carlo approach to calculate DNA
topological characteristics within the framework of the elastic-rod model (Frank-Kamenetskii
et al.,
This approach made it possible to simulate the behavior of DNA molecules allowing for excluded
volume effects (Klenin et al., 1988). So we arrived at quantitative predictions about the dependence
of knotting probability on the DNA effective diameter, d. Fig. 10 shows the results. One can see
a dramatic dependence of the P value on d. Even in case of DNA geometric diameter, which
corresponds to d = 0.02 in Fig. IO, the knotting probability is already significantly lower than for
d = 0. However, in reality the effective diameter of DNA noticeably exceeds its geometric value
due to the excluded volume effects, which are determined by the screened electrostatic interactions
between highly charged DNA segments (see Section 8). Therefore, the d value can be varied by
changing the ionic strength of the solution. Our theoretical predictions have been recently checked
experimentally (see Section Knotted DNAs.
more than hundred years.
has been raised, at least,
the discovery of closed
As mathematical objects, knots and links have been studied already for
The question of possible existence of such topological states in molecules
since late-1940s (see Frisch, 1993). It has acquired special interest since
circular DNA molecules. The calculations of the probability of knot
Fig. 10. Dependence of the equilibrium fraction of knotted molecules on DNA effective diameter, d, for closed DNA
containing 14 Kuhn lengths (lower curve), 20 Kuhn lengths (middle curve) and 30 Kuhn lengths (upper curve). The data
arc from Klenin et al. (1988). The diameter is measured in Kuhn lengths; so, to obtain the d value in nanometers one
has to multiply the figures on the abscissa by the factor of 100.
formation upon closing a polymer chain (see Section have posed the problem of the possible
existence of knotted DNAs. The results indicated that the equilibrium fraction of knotted DNAs must
be appreciable for circular DNAs containing more than about lo3 base pairs (30 Kuhn lengths).
In most cases, DNA molecules have even greater length, and the hypothesis has been put forward of the existence in the cell of special mechanisms that prevent the formation of knotted
DNAs (Frank-Kamenetskii
et al., 1975). In fact, in the course of replication of a knotted chain
(at least for some types of knots) the daughter strands cannot separate. That is, the replication of
knotted DNAs involves serious problems.
Knotted molecules were first detected in preparations of single-stranded circular DNAs after they
had been treated under special conditions with a type I topoisomerase (Liu et al., 1976). This was
the first case when a knotted molecule was observed. However, the problem of knotting of normal,
double-stranded DNAs continued to be very intriguing. It turned out that there is a special subclass
of topoisomerases called type 11 topoisomerases, which are capable of untying and tying knots in
ccDNAs. Moreover, these enzymes catalyze the formation of catenanes from pairs or from a larger
M.D. F~nnk-Kunlc~n~tskii I Physics Repouts 288 (I 997) 13-60
number of molecules of ccDNA. Here entire networks are formed, similarly to those observed
in vivo in kinetoplasts.
In contrast to type I topoisomerases, type II topoisomerases break, and then rejoin both strands of
DNA molecules. It has been shown that the enzyme “draws” a segment of the same or of another
molecule lying nearby through the “gap” that is formed in the intermediate state between the ends that
arise through breakage. Thus, the type II topoisomerases catalyze the process of mutual penetration
of segments of the double helix through one another. This process has been elaborated in details by
Wang and his collaborators in their remarkable studies of the enzyme by various methods including
X-ray crystallography (Berger et al., 1996; Wang, 1996). Consequently, these topoisomerases must
lead to the establishment of complete topological equilibrium (i.e., to a distribution of molecules
over the topological states that would correspond to freely permeable strands).
As we have noted above, DNA molecules need not be very long for a reliable proof of the detection of knotted molecules, but then the fraction of knots, as our calculation showed, must be small.
Liu et al. (1980) were able to overcome this contradiction by using topoisomerase II in very large
concentrations in which it substantially changed the macromolecular properties of the DNA itself.
Moreover, they did not add ATP to the enzyme, which is necessary for its normal operation. Precisely
under these extreme conditions, they found even in short DNAs having N = 4.5 x lo3 a considerable fraction of knotted molecules. They were able to detect them initially from the appearance of
new bands in the gel electrophoregram
that corresponded to a greater mobility. The study of the
properties of these fractions by various methods including electron microscopy has made it possible
to show that they correspond to knots of various types. If topoisomerase II in the normal amount
and ATP were added to a purified preparation of knotted molecules, rapid untying of the knots
took place (Liu et al., 1980). That is, the system rapidly relaxes to the equilibrium state for pure
DNA molecules, in which, as our calculations predicted, there should be practically no knots for
the given length. As to the reasons why the enzyme in high concentration sharply shifts the equilibrium toward knot formation, the most likely explanation is that the protein in high concentration
decreases the dimensions of the polymer coil of DNA by changing the character of the interaction
of remote segments along the chain. As our calculations showed (Frank-Kamenetskii
and Vologodskii, 1981) even a small change in the dimensions of the polymer coil sharply increases the
equilibrium fraction of knots. Knotted molecules of DNA (and also catenanes) were obtained also
by sophisticated methods employing various enzymes of DNA site-specific recombination (Spengler
et al., 1985).
Although the above experimental observations did not contradict our theoretical expectations, the
question about quantitative validity of the theory remained open. Almost 20 years after we first
published theoretical estimations of the probability of DNA knotting (Vologodskii et al., 1974; FrankKamenetskii et al., 1975), quantitative experimental data have been reported (Rybenkov et al., 1993;
Shaw and Wang, 1993) which fully agreed with the theory. In these experiments, the equilibrium
fraction of knotted DNA molecules at various ionic conditions was quantitatively measured while
molecules carrying “cohesive” ends randomly closed, in the absence of any proteins. Comparing the
fraction with theoretical predictions of Klenin et al. (1988) the value of DNA effective diameter,
d, was determined as a function of salt concentration.
The obtained dependence proved to be in
complete quantitative agreement with theoretical predictions of Stigter (1977) which were based on
the polyelectrolyte
theory (see Section 8).
M.D. Frunk-Krmlenetskiil
Physics Repouts 288 (1997)
4.3.2. Torus links and ribbons
From the schematics in Fig. 7 it is clear that the two complementary strands of DNA form a link,
in the topological sense. One can present a table of links similar to the table of knots in Fig. 8
(see Frank-Kamenetskii
and Vologodskii, 198 1). However, because the two complementary strands
of DNA are attached to each other forming the double helix, the links which DNA can form, belong
to a subclass of all possible links. Namely, they form a class of the so-called torus links because
the two strands could be put into a torus. For torus links, the well-known Gauss integral, which
defines the linking number value, Lk, is a strict topological invariant (see Frank-Kamenetskii
Vologodskii, 198 1).
There is another viewpoint on the torus links. The two strands in this case could be treated as the
edges of a ribbon. Therefore, the topological theory of torus links is actually the theory of ribbons. DNA supercoiling. The application of the topological ideas to studying the properties of
ccDNA was started by Fuller (197 1) when he applied the results of the ribbon theory to analyzing
the properties of these molecules. According to this theory (White, 1969; a simple derivation can
be found in Frank-Kamenetskii
and Vologodskii, 1981), besides the topological characteristic of a
ribbon, the Lk value, two differential-geometric
characteristics play an important role, the twist, Tw,
of the ribbon, and its writhing, WY. All three characteristics are interrelated by the condition:
Lk = Tw + Wr .
The ccDNA is generally not characterized
turns (the number of supercoils 7):
by the total quantity Lk, but by the number of excess
The number of base pairs per turn of the double helix, yo, is rigorously fixed under given ambient
conditions. However, upon changing the ambient conditions (temperature, composition of solvent,
etc.), it can vary. Therefore, the number of supercoils z, in contrast to the Lk value, is a topological
invariant of DNA only under fixed ambient conditions.
Very valuable information on the energy and conformation characteristics of ccDNA has arisen
from experiments in which the value of Lk could vary, and the equilibrium distribution of the cc
molecules over the Lk value was studied. The most convenient way to vary Lk is to employ special
enzymes we have already mentioned above, the topoisomerases. The studies under discussion employed type I topoisomerases, which alter the topological state of ccDNA by breaking and rejoining
only one of the strands of the double helix. The mechanism of action of these enzymes has recently
been elaborated in great details (Wang, 1996). These enzymes relax the distribution of the molecules
over the Lk value to its equilibrium state. The very sensitive gel-electrophoresis
method was used to
analyze the distribution of the ccDNA molecules over the Lk value. This method can easily separate
two molecules of ccDNA that differ in Lk just by one (see Section 5.1.3).
Naturally, the maximum of the equilibrium distribution always corresponds to z = 0 because this
minimizes the elastic energy. Note that, although the quantity z can only adopt discrete values that
differ by no less than unity, it is not required to be an integer. Therefore, as a rule, molecules
having z = 0 do not appear in a preparation. A distribution, in which the molecules having positive
and negative values of z are separated, is obtained when the electrophoresis
is performed under
M.D. Frunk-Kamenetskiil
Reports 288 (1997)
conditions differing from those under which the reaction with the topoisomerase is conducted. The
change in the conditions means that we must substitute some other value 7; instead of y. in Eq. (2)
without changing the Lk value. This means that the entire distribution is shifted by the amount of
67 = N[( llyo) - (l/$,)]. Then the molecules that had the z value in the original distribution will have
the values z’ = z + 67 in the new distribution. If the value 67 is large enough, all of the topoisomers
are well separated.
Experiments have shown that the obtained distribution is always normal (Depew and Wang, 1975;
Pulleblank et al., 1975; Horowitz and Wang, 1984). The variance, (TV), of this normal distribution
was measured for different DNAs. These experiments have played a very important role in studying
the physical properties of ccDNA. They made it possible to determine the free energy of supercoiling,
which is directly connected to the variance:
= llOOksT
where kB is the Boltzmann
N-’ r2,
T is the absolute temperature. Theoreticat understanding of DNA supercoiEiny. Quantitative explanation and prediction
of a variety of DNA topological characteristics, most notably the data on the equilibrium knotting probability and on the equilibrium distribution of ccDNA over topoisomers (see above and
Section 7.1), demonstrated a remarkable success of the DNA elastic-rod model. The model also
proved to be extremely successful in theoretical treatment of the phenomenon of DNA supercoiling.
In its traditional form, the Monte Carlo approach does not permit simulating highly or even moderately supercoiled molecules because the probability of their occurrence due to thermal motion is
negligible. We have extended our Metropolis-Monte
Carlo calculations (Frank-Kamenetskii
et al.,
1985a) to make it possible to generate supercoiled DNA molecules with arbitrary supercoiling
(Klenin et al., 1991; Vologodskii et al., 1992). In brief, our computational procedure is as follows
(the method is described at length by Vologodskii and Frank-Kamenetskii,
We consider a phantom closed chain, in which self-intersections are allowed. Elementary steps to
change the conformations are introduced. After each elementary step, the energy is calculated:
+ 2E*(C/hN)[r
where E({r}) is the elastic energy of the DNA chain, h is the distance between adjacent base
pairs along the DNA axis. Then the regular Metropolis-Monte
Carlo rules are applied: if the energy difference between the step under consideration and the previous energy AEg < 0, then the
new conformation is accepted; if AE, > 0, the new conformation is accepted with the probability
of exp( -AE,,ksT).
However, this is only a conditional acceptance. The new conformation needs
to meet two additional criteria. First, of all possible pairs of the straight segments none could
approach each other closer than a distance d. Secondly, the chain should remain unknotted as
a result of conformational
change. The knot checking procedure is carried out as in the case of
knotting probability calculations described in Section An ensemble of chains thus generated
is used to calculate different averaged characteristics of supercoiled molecules and enables one
to obtain theoretical images of supercoiled molecules. Fig. 1 I presents examples of such images.
Our theoretical predictions about the shape of supercoiled DNA molecules agree with most available
experimental data.
M.D. Frank-Kamenetskiil Physics Reports 288 (1997) 13WX
Fig. 11. Results of computer simulations of supercoiled
a = rye/N. The data are from Klenin et al. (1991).
DNA molecules
for different
values of superhelical
M.D. Frank-Kamenetskiil
Physics Reports 288 (1997)
Marko and Siggia ( 1994, 1995) developed an approximate analytical theory describing the structures of supercoiled DNA molecules. This theory provides insight into the role of entropic effects in
the shapes of supercoiled DNA molecules of the type shown in Fig. 11.
4.4. Breakdown of the elastic-rod model: DNA unusual structures induced by supercoiling
With increasing negative supercoiling, the elastic-rod model breaks down. This happens when the
elastic energy stored in the form of bending and torsional deformations exceeds the energy necessary
for local formation of unusual DNA structures. These unusual structures release superhelical stress
thus decreasing the total energy of the molecule. The competition between different unusual structures for the total pool of the superhelical energy dramatically depends on the presence of special
sequence motifs, which favor various unusual structures. Before these unusual structures (cruciforms,
Z-DNA, H-DNA) were discovered, the main reason for breakdown of the double helix was believed
to be the local melting (separation of DNA complementary
strands, see Section 6). Anshelevich
et al. ( 1979) and Vologodskii et al. (1979b) were the first to include DNA melting and cruciform
formation into comprehensive statistical mechanical treatment of supercoiled DNA. As other unusual
structures emerged and their energy parameter became available, the treatment has been modified
accordingly (Vologodskii and Frank-Kamenetskii,
1982; Frank-Kamenetskii
and Vologodskii, 1984;
Vologodskii and Frank-Kamenetskii,
1984; Anshelevich et al., 1988). These unusual structures are
briefly described below.
4.4.1. Z-DNA
Negative supercoiling mostly favors formation of left-handed Z-DNA (see Section 2.4) because,
in this case, the maximal release of superhelical stress per base pair adopting a non-B-DNA structure
is achieved. As a result, although under physiological ambient conditions the Z form is energetically
very unfavorable as compared with B-DNA, it is easily adopted in negatively supercoiled DNA by
appropriate DNA sequences (with alternating purines and pyrimidines).
Linear DNA with the appropriate sequence adopts the Z conformation at a very high salt concentration (about 3 M NaCl).
4.4.2. Cruciforms
Another structure readily formed under negative supercoiling is cruciform, which requires palindromic regions (see Fig. 12). To form a cruciform, a palindromic region should be larger than a
certain minimum. For example, six-base-pair-long palindromes recognized by restriction enzymes do
not form cruciforms under any conditions.
4.4.3. H-DNA
H-DNA forms a special class of unusual structures, which are adopted under superhelical stress
by sequences carrying purines (A and G) in one strand and pyrimidines (T and C) in the other, i.e.
by Mirkin and Frank-Kamenetskii,
and Mirkin, 1995; Soyfer and Potaman, 1996). The major element of H-DNA
is a triplex formed by one half of the insert adopting the H form and by one of the two strands
of the second half of the insert (Fig. 13). Two major classes of triplexes are known - pyrimidine-
M.D. Frank-Kamenetskiil
3’1--: AiiciAbAi&i
. ..j ii
Reports 288 (1997)
Fig. 12. A cruciform
formed in ColEl
DNA when the molecule
is in a superhelical
Fig. 13. H form structure of DNA. Two possible “isomer%? variants of the structure are shown. The Watson-Crick pairing
is designated by filled circles, while the GC Hoogsteen pairing, involving the presence of an extra proton, is designated
by plus symbols.
(PyPuPy) and pyrimidine-purine-purine
(PyPuPu). Fig. 3 shows the canonical
base-triads entering these triplexes.
Always two isomeric forms of H-DNA are possible, which are designated as H-y3, H-y5, H-r3
and H-r5, depending on which kind of triplex is formed and which half of the insert forms the
triplex (see Fig. 13). H-DNA may be considered as an intramolecular triplex (it is often referred
to in this way). Its formation under physiological ambient conditions occurs only under superhelical
The discovery of H-DNA (Lyamichev et al., 1985, 1986; Mirkin et al., 1987; reviewed by
Mirkin and Frank-Kamenetskii,
1994; Frank-Kamenetskii
and Mirkin, 1995; Soyfer and Potaman,
1996) stimulated studies of intermolecular triplexes, which may be formed between homopurinehomopyrimidine
regions of duplex DNA and corresponding pyrimidine or purine oligonucleotides
(see Section 2.6).
5. Special methods
In this section we consider the most important methods specially developed to study DNA. These
methods have been introduced relatively recently (in the past twenty years) but they in many cases
hf. D. Frank-Kumenetskiil
Physics Reports 288 (1997)
have pushed aside the traditional methods. They are widely used in genetic engineering and biotechnology. But they have also proved to be extremely useful tools in the field of DNA biophysics.
5.1. Gel electrophoresis
Gel electrophoresis is a simple technique, introduced in early 1970s which truly revolutionized
the studies of DNA and, subsequently, the whole field of molecular biology and biotechnology.
It is extensively used in biophysical studies of DNA. All experimental developments in this field
in the past twenty years are connected, directly or indirectly, with the gel electrophoresis method.
Gel electrophoresis has pushed aside ultracentrifugation
as a method to separate the DNA molecules.
5.1. I. Background
Gel electrophoresis differs from electrophoresis in solution only in the nature of the medium in
which molecules are separated by the electric field. In case of gel electrophoresis, the medium is a
gel, a polymer network. The most popular in the field of DNA are gels made of polyacrylamide or
agarose. Originally, the great advantage of using gels in separating DNA molecules was discovered
purely empirically. The understanding came later after some ideas of P.-G. De Gennes were borrowed
from polymer physics, namely, the notion of reptation of polymer molecules (see, e.g., Grosberg
and Khokhlov, 1994).
As in regular electrophoresis,
the electrophoretic
mobility, ,,Y, is defined as the proportionality
coefficient between velocity of movement, v, and the electric field, E:
The electric force applied to DNA of length L is proportional
a negative charge). Hence
to L (because
each residue carries
where D is the diffusion coefficient:
D = (X*)/T ,
z is the characteristic time of a DNA
mean-square shift of the molecule after
All movements other than within the
experiences the Brownian motion only
friction in the course of such movement
For the ideal polymer
molecule to go out of its original “tube” and (x2) is the
it went out of the tube.
“tube” are forbidden in the gel. As a result, the molecule
along its own axis (i.e., within the “tube”). Because the
is proportional to L,
(x2) CKL
and we finally obtain:
rux l/L,
M.D. Funk-Kammetskiii
Physics Reports 288 (1997)
whereas without the gel similar consideration would lead to the lack of dependence of p on L.
Eq. (9) explains why the gel is so efficient a medium to separate DNA molecules according to their
lengths during electrophoresis.
The consequences of Eq. (9) are really far-reaching. The entire idea of genetic engineering, i.e.,
reshuffling of DNA pieces extracted from different organisms, has become feasible only after two
major breakthroughs: the discovery of restriction enzymes, which cut long DNA molecules into
shorter pieces recognizing special short nucleotide sequences, and the implementation of gel electrophoresis to separate the pieces obtained after cutting. Each piece forms its own band in the gel
after the electric field is switched off. Then the gel is cut by an ordinary razor to obtain one unique
piece of DNA. Restriction enzymes and gel electrophoresis made it possible to obtain samples of
absolutely identical DNA molecules of practically any length for biophysical studies.
5.1.2. Pulsed-field gel electrophoresis
Even gel electrophoresis
has its limitations. According to Eq. (9), with increasing length of
molecules their electrophoretic mobility decreases. Therefore, to separate very long DNA molecules
in a practically acceptable time scale, one needs to increase the electric field. However, the treatment in Section 5.1.1 is valid only for the case of very low electric fields, which do not deform
the molecules (otherwise Eq. (6) would fail). At high fields, the DNA molecule straightens along
the electric field. As a result, it moves not like a polymer coil but like a rod-like, straight object.
The electrophoretic
mobility of such a molecule does not depend on its length independently of
whether it moves in pure solvent or in gel, because, in this case, the friction and the driving force
are both proportional to L. Thus, in a strong field, separation with respect to length occurs only for
a short time before DNA molecules are straightened. It is totally senseless to conduct electrophoresis
longer than this time because all molecules, independently of their length, will just shift by the same
distance. Does this mean that long DNA molecules cannot be separated in gel?
Schwartz and Cantor (1984) found a simple way out of the deadlock. If, soon after molecules are
straightened, the direction of the electric field is significantly changed, then, before the molecules
are straightened in the new direction, they assume again the shape of a polymer coil and will be
separated for the same short time as while moving in the first direction. After straightening in the
second direction, the field again is switched to the first direction, etc. As a result of such cycles
or pulses, the molecules effectively move in the diagonal direction and the separation takes place
throughout the duration of the experiment.
Pulsed-field gel electrophoresis
dramatically increased the range of lengths of DNA molecules
that can be separated in gels. The method makes it possible to separate entire chromosomal DNA
molecules. Implementation of this method opened the way to such ambitious projects as the Human
Genome Project, which is designated to sequence the entire human genome.
5.1.3. Separation of DNA of difSerent topological jbrms
Gel electrophoresis created a kind of revolution in the field of DNA topology and supercoiling.
Although for a different reason than linear DNA, closed circular DNA molecules belonging to different topological classes move in a gel with different velocities. As a result, knots of different types can
be separated in a gel (Rybenkov et al., 1993; Shaw and Wang, 1993). The same is true for different
topoisomers, ccDNA molecules differing in the linking number Lk (see Fig. 14). It is not the Lk
M. D. Frank-Kanzenetskii
I Physics Reports 288 (1997)
Fig. 14. Separation of DNA molecules differing by the number of superhelical turns, done by the gel electrophoresis
technique. The experiment was conducted with DNA of a small pA03 plasmid, containing 1683 nucleotide pairs. Originally,
the molecules were put from the top, near the negative electrode (the place is not shown in the figure).
Fig. 15. A typical pattern of two-dimensional
in DNA.
gel electrophoresis,
during the formation
of an unusual structure
value according to which DNA molecules are actually separated, but the writhing number, Wr (see
Section More precisely, the mobility depends on the absolute value of writhing, 1Wrl.
5. I. 4. Two-dimensional gel elec trophoresis
The ordinary, one-dimensional
gel electrophoresis does not separate supercoiled molecules that
have the same absolute value of the number of superhelical turns, r, but different sign because
those molecules have the same absolute value of writhing, 1Wrl. When a structural transition into an
unusual structure occurs, although the Lk value does not change, both twisting and writhing change
(their sum, which is the linking number, remaining unchanged). As a result, a topoisomer carrying
an unusual structure may move in a gel with the same speed as another topoisomer without an
unusual structure. To avoid such confusion, two-dimensional gels are used.
A specially prepared mixture of different topoisomers of one and the same DNA, carrying an insert
capable of changing into an alternative structure, is placed in the left top angle of a quadrangular
gel plate (see Fig. 15). Then an electric field is applied to force DNA molecules to move from
top to bottom along the left edge of the plate. Following the separation of topoisomers in the first
direction, the gel is saturated with chloroquine molecules, which lessen the superhelical stress. The
M. D. Frclnk-Kumc~nrtskii I Physics
288 (I 997/ 13%50
chloroquine concentration is chosen in such a way as to make the superhelical stress insufficient
for the formation of an unusual structure. After that, the direction of the electric field is changed
to force the molecules to move from left to right. As a result, the sequence of spots in the second
direction corresponds to the topoisomers’ sequence.
The uppermost spot in Fig. 15 corresponds to zero topoisomer, i.e., to a relaxed and nonsuperhelical DNA. The spots coming clockwise from that correspond to positive topoisomers; those
going anticlockwise, to the negative. One can clearly see the mobility drop, observed in this case between - 10 and - 12 topoisomers. This means that in topoisomers - 12, - 13,. . . , an unusual structure
is present, while in topoisomers . . . , -9, - 10 it is absent. Topoisomer - 11 occupies an intermediate
position: it carries the unusual structure during, roughly, half the time of its movement in the gel.
5.2. Chemical, photochemical
and enzymatic
A large variety of special approaches have been attempted to study DNA structures. They are
based on different reactivity of DNA adopting different structures with respect to chemical, photochemical and enzymatic reactions. In many cases, these methods make it possible to arrive at very
specific conclusions about the structure of a particular region of DNA under conditions that totally
exclude application of not only X-ray crystallography
or NMR but even spectroscopy and other
indirect physical methods. Sometimes, the methods under consideration are applicable even in vivo.
To explain the general ideology underlying these methods, let us consider a specific example.
In Section 2.6 we mentioned intermolecular triplexes, which are formed between homopurine-homopyrimidine regions of DNA and the corresponding homopurine or homopyrimidine oligonucleotides.
If such a complex is actually formed, the reactivity of the N7 position of guanine (this is one of
the two nitrogens in the five-member ring of guanine) should dramatically decrease because this
nitrogen is sheltered in the triplex by the Hoogsteen pairing (see Fig. 3).
The chemical agent used is dimethyl sulfate (DMS), which reacts with the N7 position of guanine
alkylating it. This alkylation occurs in single-stranded as well as in duplex nucleic acids. However,
it cannot take place in triplexes. As a result, in the complex of duplex DNA with oligonucleotide,
which forms a triplex, all guanines in the duplex outside the triplex zone will be alkylated by DMS,
whereas guanines within the triplex zone will remain unmodified.
Then the DNA piece under study is end-labeled and subjected to hot piperidine treatment. Piperidine will convert the sites of alkylated guanines into chain breaks. Such breaks will never occur in
the triplex zone. Fig. 16 shows the pattern that is obtained after separation of the fragments in gel
and radioautography.
The above example is a specific case of the footprinting assay. Such assays can be applied, for
instance, to complexes of DNA with proteins to find out which sequences are recognized by the
proteins. Instead of DMS, DNAase I, which cuts the uncovered DNA duplex, is often used. The
yield of some photoproducts, which can also be converted to strand breaks, dramatically decreases
when a duplex region is covered by a protein or an oligonucleotide.
Hence the photofootprinting
assay is useful (Lyamichev et al., 1990, 1991).
Some chemical reagents, like diethyl pyrocarbonate, potassium permanganate, osmium tetroxide, do
not react with bases in the double helix but react with open bases. The products can be converted
into chain breaks. These reagents are widely used to detect open regions. Single strand-specific
nucleases, which digest single strands but do not digest duplex, are used in a similar way.
M.D. Frank-Karnenetskii I Physics Reports 288 (1997)
Fig. 16. The result of footprinting experiment with dimethyl sulfate of a complex of duplex DNA carrying homopurine-homopyrimidine
insert with corresponding pyrimidine oligonucleotide. The data are from Chemy et al. (1993a).
Chemical, photochemical and enzymatic probing is an extremely powerful method to detect unusual
structures, like Z-DNA, cruciforms, H-DNA, G-quadruplexes.
6. Melting
of DNA
Soon after the discovery of the double helix by Watson and Crick (1953) the phenomenon of
DNA melting was demonstrated experimentally. It was shown that when the DNA solution is heated,
M.D. Frunk-Kamenetskii
I Physics Reports 2118 (1997)
Fig. 17. Melting of DNA. (a) The helix-coil transition of a DNA molecule (intramolecular melting). (b) Typical DNA
melting profile. This curve is also often called the differential melting curve. The curve was obtained for DNA which has
the code name of ColEl and contains about 6500 nucleotide pairs.
the complementary
strands separate: instead of the regular double helix two single-stranded DNA
coils emerge (Marmur and Doty, 1962). This phenomenon is also called the helix-coil transition.
The DNA melting may be monitored by various techniques. Two most popular methods are UVspectrometry (see Section 3.4.1) and microcalorimetry
(reviewed by Breslauer et al., 1992). Instead
of exhibiting a phase transition, DNA melts gradually, in a wide temperature range (Fig. 17). DNAs
from different organisms differ in their melting profiles.
M.D. Frank-Kamenetskiil
6.1. Helix-coil
Physics Reports 288 (1997)
In attempts to understand the phenomenon of DNA melting, a simplified theoretical model was
elaborated (see Fig. 5) which treated DNA as a one-dimensional array of interacting spins. Each
spin corresponded to a DNA base pair. Spin up corresponded to the helical state while spin down
corresponded to the melted (open) state of the base pair. Two features made the problem much
more difficult and much more interesting than the one-dimensional Ising model well known in the
solid state physics. First, because open regions in DNA presented closed polymer chains, a longrange interaction between spins emerged. Secondly, because two base pairs in DNA (AT and GC)
have different stability, DNA had to be modeled as a linear array of spins under the influence of
disorder external magnetic field. Although irregular, the sequence is fixed so that the external field is
quenched. Therefore, the system is equilibrated with respect to the direction of spins (up and down)
but not with respect to the field (base pairs AT and GC). I.M. Lifshitz labeled such systems as having
linear memory. This second feature of the DNA helix-coil model presented a major challenge to
theorists and attracted considerable attention in 1960s and 1970s. Like DNA topology, DNA melting
belongs to biophysical problems, which are sometimes labeled as “biologically inspired physics”
(Peliti, 1990).
It is worth to mention that knots first emerged in the Russian biophysics community not in
connection with circular DNAs but in connection with closed circles of single-stranded DNA formed
in the process of DNA melting. I believe this question first attracted the attention of a wide audience
during I.M. Lifshitz’s brilliant, I would even say charismatic, lecture at one of the regular Winter
Schools on Molecular Biology in Dubna near Moscow (I guess it was in 1969). Speaking about the
possibility of diffusion of knots from the ends of linear DNA in the process of melting, I.M., by a
perfectly theatrical gesture, took out the belt from his pants and tied it into the trefoil knot in front
of a stunned audience of about 500 Russian molecular biologists and biophysicists. In literature, the
possible topological effects due to circular nature of DNA melted regions were first discussed by
Shugalii et al. (1969) and Vedenov et al. ( 197 1).
6.1. I. Theoretical development
In statistical-mechanical
terms, the second feature of DNA helix-coil model (the linear memory
due to the fixed sequence of DNA base pairs) means that one cannot average the partition function
over different sequences of AT and GC pairs even if one assumes that the sequence itself is totally
random. In reality, of course, the sequence is not random because it carries the genetic information.
However, at early stages of treatment of the DNA melting phenomenon, long before the first real
DNA sequences became available, the sequence was assumed to be random in theoretical studies.
This made it possible to apply not only numerical but also analytical tools to treat the problem.
The most elegant analytical approach was proposed by Lifshitz (1973). Among others, important
contributions of Vedenov et al. ( 1971) and Azbel ( 1972) are worth mentioning.
As to the numerical solution, the challenge was to reduce the problem of direct computation of the
partition function for a chain consisting of a very large number (N) of base pairs (“spins”), which
required exponentially large computer time, to a procedure, which required polynomial time N” with
as small an a as possible. Several rigorous algorithms were proposed (Vedenov et al., 1967, 1971;
Poland, 1974).
M.D. Frank-Kanzenetskii
I Physics Reports 288 (1997)
However, an efficient way of solving the problem, which allowed for both the above features
of the DNA helix-coil model, was not available until Fixman and Friere (1977) proposed their
algorithm. In so doing they heavily relied on the Poland (1974) algorithm and some of our results
and Frank-Kamenetskii,
1969; Lukashin et al., 1976). Theoretical development
of the helix-coil model has been extensively reviewed by Vedenov et al. (1971), Wada et al. (1980),
Wada and Suyama (1985), and Wartell and Benight (1985).
It is worth mentioning that the helix-coil model without long-range interactions found applications
far beyond the area of DNA biophysics. Among other applications, the model has been extensively
used to study cx-helix-coil transition in polypeptides and most recently it was used by Selinger and
Selinger (1996) to explain experimental data on chiral order in random copolymers consisting of
two enantiomers.
6.1.2. Comparison with experiment
When the very first full DNA sequence appeared in 1977 (of bacteriophage #X174), DNA biophysicists were well equipped to compare quantitatively experimental DNA melting profiles with
theoretical predictions. It was first done by Lyubchenko et al. (1978). Essentially, it was the beginning of the end of the theme of DNA melting in DNA biophysics because theoretical prediction
correlated with experiment sufficiently well. Even more direct comparison was done by Kalambet
et al. (1985) using electron-microscopy
visualization of the melted regions in DNA with the known
sequence on different stages of the melting process. Such comparisons and similar studies (reviewed
by War-tell and Benight, 1985; Wada and Suyama, 1986) left no doubts that we correctly understood
in quantitative terms major features of the phenomenon of DNA melting.
6.1.3. Heterogeneous stacking
A theme that dominated the field after the first demonstration of a success of the theory in
achieving quantitative explanation of experimental data for DNAs with known sequences, was the
so-called heterogeneous stacking. In the original helix-coil model, the external field could acquire
only two values, corresponding to AT and GC pairs. This meant that interaction between all possible
combinations of near neighbors along the DNA chain was assumed to be the same. Of possible 16
types of nearest neighbors, or stacks, only 10 are different because of the complementarity
It was quite natural to attribute some remaining differences between theory and experiment to the
fact that these 10 parameters are different, i.e., to the effect of heterogeneous stacking. However,
the very fact that the original model, which ignored the difference, worked well, indicated that the
deviations from the mean interaction energy between adjacent base pairs were small as compared
with the energy itself. In other words, these data indicated that the heterogeneous stacking was a
small parameter.
In the first paper where heterogeneous stacking was allowed for, Gotoh and Tagashira ( 198 1)
overlooked the fact that of 10 parameters of heterogeneous stacking only 8 of their combinations
(invariants) actually determine the behavior of long DNA chains. When they adjusted all 10 parameters of heterogeneous stacking by comparing theory with experiment, a great confusion occurred
because, unexpectedly, the effect of heterogeneous stacking proved to be very large. Vologodskii
et al. (1984) dispelled the confusion adjusting 8 invariants, not 10 parameters, by comparing theory
with experiment. As a result, a reasonable set of relatively small parameters of heterogeneous stacking
M.D. Frank-Kamenetskii
I Physics Reports 288 (I 997) 13-60
emerged (Vologodskii et al., 1984). Although some uncertainty in the exact values of parameters of
heterogeneous stacking still remains (Doktycz et al., 1992; Doktycz et al., 1995; SantaLucia et al.,
1996), the problem is mostly solved.
6.2. Slow relaxational
The remarkable success of statistical-mechanical
theory in explaining the phenomenon of DNA
melting overshadowed some significant limitation of the approach. For a long time experimental
observations of hysteresis phenomena in DNA melting were largely ignored. However, when comparison of theory and experiment reached a high precision, kinetic effects in DNA melting could not
be ignored any longer.
A comprehensive
analysis of slow relaxation processes in DNA melting was performed by
Anshelevich et al. (1984a). The hysteresis phenomena in DNA melting are a direct consequence
of the fact that very long regions are melted out cooperatively in the course of the process. The
characteristic time of strands separation for a helical region consisting of m base pairs may be
roughly estimated as (see Anshelevich et al., 1984a, for more accurate expressions):
where r. x lo-’ s and s is the stability constant for a base pair. Although s is very close to unity
within the melting range, because m is several hundreds the S* value may be extremely large. Hence,
very large r values and a significant contribution of kinetic effects.
Subsequent thorough experimental studies completely confirmed all major theoretical predictions
(Kozyavkin et al., 1984,1986; Wada and Suyama, 1986).
Concluding this section, we would like to stress that thorough experimental and theoretical studies
of DNA melting have led to a virtually full understanding of the phenomenon. It is a rare example
of a real insight into a complicated biophysical process.
7. Fluctuations
in DNA
Although under “physiological” conditions DNA occupies its “ground state”, i.e. it is predominantly in the B form, DNA functioning is impossible to understand without knowledge of its
“excited’ states. Transient occupation of these excited states explains all variety of reactions DNA
experiences during its functioning. Indeed, fluctuations are responsible for DNA adjustment to a
variety of proteins, which interact with DNA, including the key proteins like repressors, activators,
PNA polymerase. DNA damage by radiation and chemical agents, including carcinogens and mutagens, is often possible due to fluctuations, which make the reaction groups, normally buried within
the double helix, accessible to chemical and photochemical modification. Therefore, the question of
most probable DNA excited states is of paramount importance. The answer to this question critically
depends on the topological state of DNA. In this section we mostly concentrate on the basic problem
about fluctuational motility of the double helix itself, without additional factors like supercoiling.
We briefly touch the effect of supercoiling at the end of the section.
M.D. Flunk-Kunlrnrtskiil
7.1. Bending and torsional jluctuations
Physics Rqwrts
oj’ the double helix
Due to thermal motion, the DNA molecule experiences bending and torsional fluctuations. The
amplitude of these fluctuations is determined by the values of DNA bending and torsional rigidities,
respectively. We have already discussed the DNA bending rigidity, which is quantitatively characterized by the value of DNA persistence length, a (see Section 4). The value of DNA torsional
rigidity can be determined from the data on the variance of the equilibrium distribution of closed
circular DNA molecules over the linking number, (r’) (see Section
Under equilibrium conditions allowing breaks and rejoining of one of the DNA strands (i.e., in
the presence of topoisomerases) LIR, and PI+ are independent random quantities, while the resultant
quantity z equals their sum:
Evidently, the mean values are (A RY) = ( WY) = (r) = 0. H owever, the quantities ((nnr)‘),
and (r’) differ from zero. Then, because of the independence of the random quantities
Wr, one obtains (Vologodskii et al., 1979a):
(@)= (<nTN’)2)+ (( Wr)2) .
~TII: and
The quantity (r2 ) is known from experiment, and (( WF)~ ) could be calculated by computer
simulation because the ribbon theory offers a simple analytical formula for the value provided that
the shape of the chain is known (see Frank-Kamenetskii
and Vologodskii, 198 1; Vologodskii and
1992). (Note that in so doing we extensively used our method of discrimination
of knotted and unknotted chains, see Section 4.3). To compare the values of (T’) and ((WY)‘) for
the same DNA we need to know the quantitative value of the DNA Kuhn length, which is known
with a good accuracy to be equal to 100 nm (see Section 4). Therefore, we could find ((A Trr)’ )
as a function of the DNA length by subtracting the calculated ((WY)“) value from the experimental
(2’ ) value. On the other hand, ((~Inr)’ ) is directly related to the torsional rigidity of the double
helix, C:
where h and 4 are the distance along the axis and the rotation angle between two adjacent base
pairs in the double helix, respectively.
In full agreement with this equation, ((ATw)~) proved to be strictly proportional to N. The slope
of the straight line made it possible to determine C, which proved to be 3.0 x lo-l9 erg cm (Klenin
et al., 1989).
This value of the torsional rigidity of DNA corresponds to a root-mean-square
amplitude of
thermal fluctuations in the value of the angle between adjacent base pairs of 4-5”. The obtained
results indicated that in sufficiently long supercoiled DNA one third of the superhelical energy is
stored in twisting and two thirds are stored in writhing. Thus, the analysis of the experimental
data on circular DNAs employing the topological approach made it possible to estimate one of the
fundamental characteristics of the double helix. These estimations agree with the results obtained by
other methods (see Taylor and Hagerman, 1990, and references therein).
hf. D. F~arzk-Kumenetskiil
7.2. Fluctuational
Pl1wic.r Reports 288 (1997)
openings of base pairs
Bending and torsional fluctuations in DNA are the result of accumulation of small displacements
from equilibrium positions of DNA base pairs. Extrapolating the theory of DNA melting (Section 6)
to temperatures well below the DNA melting range one concludes that a tiny fraction of base pairs
should be open at physiological temperatures. According to theoretical predictions, the double helix
should be interrupted by solitary open base pairs on average every IO5 base pairs (Frank-Kamenetskii
and Lazurkin, 1974; Frank-Kamenetskii,
1981, 1985). However, how reliable were the conclusions
based on such long extrapolations?
Certainly, experimental estimations of this fundamental characteristic of the double helix were
needed. However, for many years the question of base-pair opening probability, and the related
question of life-time of a base pair in closed state, were the subject of sharp controversy because
different methods led to quite different conclusions.
The most direct approach consisted in monitoring the kinetics of exchange of exchangeable
hydrogen atoms, which are buried within the double helix if the base pair is closed. Such protons can exchange only while the base pairs are open. Such data led to the opening probability,
which was three orders of magnitude higher than the above theoretical estimate (Mandal et al.,
1979; Cantor and Schimmel, 1980). We used another approach for probing the open base pairs.
We analyzed both experimentally
and theoretically the kinetics of DNA reaction with formaldehyde. Like in the case of hydrogen exchange, formaldehyde can react with bases only if they
are open. Our comprehensive
analysis led to an estimation of the base pair opening probability,
which was in full agreement with the theoretical prediction of 10P5 (Frank-Kamenetskii,
198 1, 1985,
The controversy was resolved by Gueron et al. ( 1987) who studied the hydrogen exchange kinetics
by nuclear magnetic resonance (NMR). They showed that in previous analysis of the hydrogenexchange data the intrinsic catalysis of the exchange had been overlooked. Quite unexpectedly,
exchange of exchangeable protons in open based pairs is catalyzed by the complementary bases. This
intrinsic catalysis overshadowed the effect of the external catalyst and led to a wrong conclusion
that the observed exchange rates were limited by the rate of base-pair opening. An analysis of
the NMR data (Gueron et al., 1987; Gueron and Leroy, 1995) gave the estimation of the opening
probability as 10P5, in full agreement with our theoretical expectation and the figure followed from
the formaldehyde data (Frank-Kamenetskii,
198 1, 1985, 1987).
The full picture of internal base-pair fluctuational opening emerged as follows. Base-pair
lifetime is 10e2 s, open base-pair lifetime is lo-’ s; and base-pair opening probability is lop5
1987; Gueron and Leroy, 1995). Of course, terminal base pairs have a much
shorter lifetime of lop6 s. Opening probability dramatically increases in the DNA “premelting”
Although well below the melting range base pairs open with very low probability, this fluctuational opening, or DNA “breathing”, plays an extremely important role. These fluctuations make
accessible for interaction and chemical reactions the active groups of DNA bases, which are
otherwise completely buried within the double helix. DNA modification by formaldehyde still remains the most thoroughly studied case, where DNA breathing plays a crucial role (Lukashin et al.,
1976; Frank-Kamenetskii,
198 1, 1985 ).
M.D. Frank-Kunzenetskiil
Physics Reports 288 (1997)
7.3. Fluctuutions in superhelical DNA
Negative supercoiling (see Section 4.3.2) highly increases the probability of fluctuational formation of different structures, which release the superhelical stress. This happens well below the
threshold superhelical density corresponding to the breakdown of the elastic-rod model discussed in
Section 4.4. A statistical-mechanical
treatment of these fluctuations is presented in Anshelevich et al.
( 1979), Vologodskii et al. (1979b), Vologodskii and Frank-Kamenetskii
( 1982), Frank-Kamenetskii
and Vologodskii (1984), Vologodskii and Frank-Kamenetskii
(1984) and Anshelevich et al. (1988).
In spite of numerous efforts, the possible role of transient formation of various structures induced
by supercoiling in the cell still remains to be elucidated.
8. Polyelectrolyte
properties of DNA
One of the most striking features of the DNA molecule is that each base pair carries two elementary negative charges. As a result, the DNA molecule is characterized by an extremely high
linear charge density. This negative charge attracts small cations from solution (usually Na+), which
create a positively charged cloud around the DNA chain. Because of the electrostatic interaction
between the charged DNA molecule and the cloud of counterions, different important properties of
DNA prove to be strongly dependent on the salt concentration (usually NaCl). For many years,
the problem of quantitative understanding of DNA polyelectrolyte
properties has been presenting
a challenge for DNA biophysicists.
By early 1980s primarily due to a seminal paper by Fixman (1979) it became clear that, although
not rigorous, the Poisson-Boltzmann
(PB) equation is applicable for treatment of even a highly
charged polyelectrolyte
in a wide range of salt concentration (Fixman, 1979; Anderson and Record,
1982; Anshelevich et al., 1984b; Frank-Kamenetskii
et al., 1987). As a result, the PB equation has
become a major tool for theoretical treatment of DNA polyelectrolyte properties.
8.1. Cylinder model oj’ DNA
A simplified model, which has been most popular in the field, treats DNA as a uniformly charged
cylinder of diameter dg. (We denote the DNA geometrical diameter by dg to distinguish it from the
DNA effective diameter d of the elastic-rod model considered in Section 4). The negative charge
of phosphate groups is supposed to be evenly spread throughout the surface of the cylinder (see
Fig. 5). The Manning condensation theory (Manning, 1969, 1972; Cantor and Schimmel, 1980),
which treats a limiting case of the model corresponding to dp+ 0, has acquired enormous popularity
among experimentalists because it led to very definite and simple conclusions.
8.1.1. Manning’s condensation theory
This theory models DNA as an infinitely thin charged thread with uniformly distributed linear
charge density, 4. According to an idea of Onsager that was followed up by Manning (1969), the
141 value cannot exceed some critical value, lqcl, because if lql> jqcl, the statistical integral for
the infinitely thin thread diverges (see reviews by Manning, 1972; Anderson and Record, 1982;
et al., 1987). The theory predicts the mobile counterions to precipitate on the
hf. D. Frunk-Kamenetskiil Physics Reports 288 (1997) 13-60
thread decreasing the absolute value of its linear charge density down to the critical value, jqcI.
These arguments led Manning to numerous and very handy equations describing the dependence
of various DNA properties on ionic strength (reviewed by Manning, 1972; see, e.g., Cantor and
Schimmel, 1980). Because the theory proved to be remarkably successful in explaining some DNA
properties, it was tempting to believe in the reality of the phenomenon of counterion precipitation, or
condensation. However, the PB equation does not predict any separation of the pool of counterions
into two distinctive groups or any discontinuity in the counterion concentration around the charged
cylinder even in the limiting case of dg + 0. This was generally considered, before early 1980s
as a failure of the Poisson-Boltzmann
equation because this equation is derived within the selfconsistent-field-theory
After rehabilitation of the PB equation (Fixman, 1979; Anderson and Record, 1982; Anshelevich
et al., 1984b; Frank-Kamenetskii
et al., 1987) and due to the results by Gueron and Weisbuch
(1981), Ramanathan and Woodbury (1982) and Zimm and Le Bret (19X3), the situation was clarified. The separation of the pool of counterions into two distinctive groups or discontinuity in the
counterion concentration around the polyion does not actually take place. Instead, all counterions
form a cloud around the polyion with the counterion concentration smoothly decreasing with departure from the polyion. Manning’s condensation theory correctly predicts the counterion distribution
at veiy large distances from the polyion but gives a totally wrong picture of the counterion distribution in the vicinity of the polyion. This explains why the condensation theory predicts well some
of the DNA properties and fails to predict other properties (for more discussion see the review by
et al., 1987).
8.1.2. Poisson-Boltzmann
Since during the early 1980s it became clear that the PB equation provided a reliable ground
for theoretical studies of polyelectrolytes,
the PB equation has become a major tool for theoretical
treatment of DNA polyelectrolyte
properties. In this section we briefly overview the major results
of applying the PB equation to DNA within the framework of the cylinder model. DNA melting. The theory makes it possible to calculate the ionic-strength
dependence of
the DNA melting temperature (T,,,). To do so, one needs to calculate the electrostatic free energy of
DNA surrounded by small ions. We performed such calculations using the cylinder model for both
duplex DNA and separated DNA strands (Frank-Kamenetskii
et al., 1987). More recently, similar
calculations were performed by Bond et al. (1994), who extended them to the case of melting of
DNA triplexes. A major conclusion from these calculations is that the ionic-strength dependence of
7’, is very sensitive not only to the linear charge density of the duplex and single-stranded states
(that was the case, of course, for the condensation theory) but also to the diameter of the cylinder
assumed for both DNA states. Such a parameter just does not exist in the condensation theory.
Taking the helix parameter values from the geometry of B-DNA, we adjusted the parameters
for single-stranded DNA from the condition of the best fit between theoretical and experimental
dependence of T, on ionic strength (Frank-Kamenetskii
et al., 1987). The conclusion was that the
distance between adjacent phosphate groups in single-stranded DNA is the same as the distance
between projections on the DNA axis of adjacent phosphate groups of a DNA strand within the
B-DNA duplex. This conclusion was unexpected because, apparently, there are no sterical obstacles
M.D. Frank-Kamenetskiil
Physics Reports 288 (1997)
Fig. 18. Chemical structure of peptide nucleic acid (PNA) and DNA. B is one of the four canonical
the PNA terminal group.
DNA bases, R is
for the free DNA single strand to be significantly stretched out as compared to the regular helical
form it has within the duplex.
It should be emphasized that although Manning’s explanation of the T, dependence on ionic
strength was considered as a great success of the condensation theory and found its way into textbooks (e.g., Cantor and Schimmel, 1980) from the theoretical viewpoint the condensation theory
is totally inapplicable to calculate the electrostatic free energy (see Frank-Kamenetskii
et al., 1987;
Lukashin et al., 1991a). Therefore, the PB derivations described above were essentially the first
attempt of a consistent theoretical analysis of the problem.
Very recently, such analysis was extended to interpret melting experiments for peptide nucleic
acid (PNA) and its complexes with DNA. PNA is an artificial molecule, which consists of DNA
bases and a neutral polypeptide-like
backbone (see Fig. 18; we discuss PNA at greater length in
Section 9). Two complementary PNA molecules from PNA/PNA duplex and complementary PNA
and DNA strands form PNA/DNA heteroduplex. Tomac et al., 1996 compared the ionic-strength
dependence of T, for PNA/PNA, PNA/DNA and DNA/DNA duplexes. While for the PNA/PNA
duplex the T,,, value did not depend on salt concentration due to neutrality of the molecule, in
the case of PNA/DNA duplexes j”,, slightly decreased with increasing ionic strength. Tomac et al.
(1996) successfully explained this effect using essentially the same analysis as was done by FrankKamenetskii et al. (1987) and Bond et al. (1994) for DNA melting and adjusting the only unknown
parameter, the diameter of the PNA/DNA duplex. These data, together with the data for DNA
triplexes by Bond et al. (1994), demonstrate that the cylinder model in combination with the PB
equation proved to be a very efficient tool for quantitative description of a wide variety of melting
experiments. B-Z transition. Even before applying the cylinder model to DNA melting, we applied it to
the B-Z transition (Frank-Kamenetskii
et al., 1985b). Actually, the data on the dependence of the
B-Z equilibrium on ionic concentration provided the first clear indication from the experimental side
that something was fundamentally wrong with the condensation theory. Indeed, because the linear
charge density for Z-DNA is lower than for B-DNA, the condensation theory predicted increasing
M.D. Frank-Kamenetskiil Plzysics Reports 288 (1997) 13-60
relative stability of B-DNA with respect to Z-DNA with increasing ionic concentration. In reality,
high salt stabilizes Z-DNA relative to B-DNA.
We showed that the cylinder model and the PB equation resolve this controversy. Our calculations
predicted the effect of maximal relative stability of B-DNA at intermediate ionic strength (FrankKamenetskii et al., 1985b). In other words, the electrostatic free energy difference between Z- and
B-DNA exhibited a maximum at ionic concentration near 0.1 M of monovalent salt. The effect was
sensitive to the specific values of the parameters of the model. It remained unclear whether the effect
would remain for more realistic models of charge distribution in B- and Z-DNA. This question was
addressed recently by Demaret and Gueron (1993) and Misra and Honig ( 1996) who treated, using
the PB equation, more realistic models of the charge distribution.
The possibility of existence of Z-DNA at low salt, as well as at high salt, has been extensively
studied experimentally by Ivanov et al. (1993).
X1.2.3. DNA effeective u’iunzeter. Reliable computations of the screened Coulomb potential around
DNA within the framework of the cylinder model and the PB approach, made it possible to calculate
the excluded volume effects in DNA due to the electrostatic interactions. Such calculations, first
performed by Stigter (1977), yielded a theoretical prediction of the dependence on ionic concentration
of the DNA effective diameter, d. These theoretical results proved to be in complete quantitative
agreement with experimental data, after the d value was determined from the data on DNA knotting
probability (Rybenkov et al., 1993; Shaw and Wang, 1993; see Section
8.2. Other models
We have already mentioned more realistic models, which allow for some deviations from the
original cylinder model (Demaret and Gueron, 1993; Misra and Honig, 1996). Although these
papers consider more realistic charge distributions, they still treat DNA as a stiff construction with
straight axis. Le Bret (1982) and Fixman (1982) calculated the electrostatic contribution into the
bending energy. These data made it possible to explain the experimentally observed
dependence of persistence length of various polyelectrolytes
on ionic strength (Tricot, 1984).
Concluding this section, it should be emphasized that all models treat the water surrounding the
DNA as a continuous medium with a fixed value of dielectric permittivity. Lukashin et al. (1991b)
considered the consequences of possible deviations from this assumption. We found that the character
of dependence of the water dielectric permittivity near the DNA surface may significantly affect the
results. This question needs further studies.
9. Specificity of DNA interactions
In previous sections, we concentrated on some of traditional problems of DNA biophysics, which
have been attracting attention for many years. Some of them (like DNA melting), have been
essentially solved, others (like DNA topology or DNA polyelectrolyte
properties) although being
traditional, still attract a good deal of attention. In this section I consider a problem, which has just
emerged as a DNA biophysics problem.
M.D. Frank-Kamenetskiil
Physics Reports 288 (1997)
In principle, the problem has long been around in a wider field of the DNA science. This is
the problem of sequence-specific recognition of sites on DNA by short single-stranded DNA chains
or their synthetic analogs. In the simplest form, this is the problem of “annealing”
between a single DNA strand and an oligonucleotide complementary to a short piece of the DNA
strand. Such complexing, due to the complementarity
principle, is used in numerous techniques,
which have wide applications in molecular biology and biotechnology,
such as DNA sequencing,
hybridization (Southern blotting), polymerase chain reaction (PCR), etc. The central question for all
these applications is the relation between specificity and affinity of binding of the oligonucleotide to
single-stranded DNA. Similarly, homopurine or homopyrimidine oligonucleotide can bind to duplex
DNA via Hoogsteen pairing (see Section 2.6; reviewed by Frank-Kamenetskii
and Mirkin, 1995;
Soyfer and Potaman, 1996). Finally, an oligonucleotide analog, peptide nucleic acid (PNA), forms
the most unusual complex with duplex DNA consisting of two homopyrimidine PNA molecules (see
below). Binding of PNA to duplex DNA exhibits a remarkable combination of high specificity and
9.1. Relationship between specificity and afJinity at equilibrium
In the case of equilibrium binding, the problem is a very simple one, at least in principle. Indeed,
according to von Hippel and Berg (1986) (we will refer to the paper as vHB) the specificity, a,,
may be defined as the ratio of concentration, [PL], of complexes of the ligand, L, with its specific
site, P, and the sum of concentrations of complexes, [B,L], of the ligand with all non-specific sites
Simple arguments based on the mass action law led von Hippel and Berg (1986) to the final equation
for CI,:
where K,_ and Kni are equilibrium
specific sites, respectively.
of binding of the ligand to the specific and ith non-
9. I. I. General considerations
Eq. (15) leads to the intuitively obvious conclusion that high selectivity is achieved when KL
is much larger than all KS, taken together (under natural assumption that [P] and each of [Bi]
are of the same order of magnitude). For the sake of simplicity and without loosing generality,
we concentrate on only two types of complex: the specific or “correct” one and a non-specific
or “frustrated” complex, which is actually always a representative of a large number of possible
non-specific complexes. Within the framework of the vHB treatment, the above two complexes are
characterized by two equilibrium binding constants, KL and Kf, respectively. High specificity requires
a very strong inequality to be valid:
M.D. Frank-Kammrtskiil
Physics Reports 288 (1997)
Thus, within the framework of the vHB consideration, high selectivity of binding may only be the
result of a very large free energy gap between the correct and non-specific or frustrated complexes.
On the basis of vHB-type analysis, Eaton et al. (1995) formulated an important principle of drug
selection, according to which high-affinity binding (i.e., binding with a high value of the equilibrium
binding constant) entails highly specific binding.
Note that the protein folding process may be considered within the above general approach as an
intramolecular self-assembly. Eq. (16) indicates that high specificity of folding requires selection of
special protein sequences, for which a large free energy gap exists between folded and misfolded
conformations. This conclusion fully agrees with the results of Monte Carlo simulations of simple
protein models by Sali et al. (1994).
The above consideration and conclusions explain, at least qualitatively, remarkable selectivity
of biomolecular interactions in a variety of real situations, both in nature and in the course of
selection of new drugs. However, one should not overlook two essential assumptions underlying
the above reasoning: (i) equilibrium of binding of the ligand to various binding sites is achieved
and (ii) strong inequality is valid as in Eq. ( 16) for any non-specific site. These rather restrictive assumptions cannot be universally valid and cases certainly exist for which a vHB-type treatment is inapplicable and its conclusions fail. Such cases present special interest to us because
of DNA with oligonucleotides
and their
they are met in the field of interaction
9.1.2. DNA case
Specificity of interaction between nucleic acids plays a crucial role in fundamental biological
processes of replication, transcription, translation and genetic recombination. The ability of DNA to
form highly specific complexes underlie the most important biotechnological
applications of DNA,
such as various hybridization techniques, polymerase chain reaction (PCR), etc. In contrast to other
cases of biomolecular recognition (DNA/protein,
etc.), where there is no obvious
universal principle, in case of DNA such remarkably simple general principle, the complementarity
rule, is available. As a result, the case of DNA interactions may be subjected to a comprehensive
theoretical treatment within the framework of simple models, in which real molecules are stripped
of all unnecessary details and only features important for answering basic questions about specificity
of interaction are left.
We have recently treated such models using the kinetic Monte Carlo approach (Lomakin and
1997). Our main conclusions are as follows. We have found that by changing
the values of parameters of the model one can either achieve high affinity and poor specificity or
high specificity and poor affinity but never both. Therefore, in contrast to the predictions of the vHB
model, for which specificity correlates with affinity (von Hippel and Berg, 1986; Eaton et al., 1995;
see above), in case of binding of oligonucleotides to DNA via Watson-Crick
pairing, affinity and
specificity anti-correlate with each other. Obviously, the vHB model fails in this case because the
strong inequality in Eq. (16) is not valid.
Our conclusion apparently contradicts well-known facts of a great success of Watson-Crick
recognition in uncountable applications, such as hybridization techniques (Southern blotting, in situ
Sanger sequencing, PCR, etc. How could all these methods possibly work if the
Watson-Crick pairing were either very weak or non-specific? There is no real contradiction between
our findings and the success of the above techniques. To be successful, these techniques do not
A4. D. Flunk-Kumc~net.vkiil
Physics Reports 288 (1997)
require sequence recognition to be so stringent as we understand it. The target sites and corresponding oligonucleotides
are sufficiently long so that the probability to encounter sites with very few
mismatches is negligible.
Thus, in many practical cases of recognition of single-stranded DNA by oligonucleotides
Watson-Crick pairing, stringent sequence specificity is not necessary. However, other cases definitely
exist for which such stringent recognition is essential.
Similar conclusions are valid for recognition of sites on duplex DNA due to the Hoogsteen mode
of binding (the triplex formation, see Section 2.6).
9.2. Irreversible
The above consideration was grounded on the assumption that equilibrium is reached during the
time of experiment. This is not necessarily the case for interactions between DNA molecules, as we
already discussed at some length in Section 6.2. According to Eq. (lo), one can encounter large
relaxation times in two alternative cases. One can either deal with the s value close to unity but
with very large m values (several hundreds) (which is the case for DNA melting, see Section 6.2)
or with not that large an m value (of about 10) but with the s value significantly larger than unity.
The most striking example of the latter situation is presented in case of DNA interaction with DNA
artificial analog, the peptide nucleic acid (PNA).
9.2.1. Peptide nucleic acid
PNA is a DNA synthetic
usual DNA bases attached
DNA, backbone (Fig. 18).
negative charges.
analog, which was put forward by Nielsen et al. (1991). It consists of
to a totally different backbone, which reminds the protein, rather than
PNA backbone is neutral, in contrast to DNA backbone, which carries PNAIDNA
duplexes. PNA oligomers form duplexes with complementary PNA and DNA
chains (Wittung et al., 1994; Egholm et al., 1993). However, because of neutrality of the PNA
backbone, melting temperature (T,) of PNA/PNA duplexes does not depend on the ionic strength
(see Fig. 19). Interestingly, the T, value for PNA/DNA duplexes decreases with increasing ionic
strength, in sharp contrast the DNA/DNA duplexes (see Fig. 19). This unusual behavior of the
PNA/DNA duplex has made it possible to subject the DNA polyelectrolyte
theory discussed in
Section 8 to a new critical test. Tomac et al. (1996) have shown that this behavior cannot be
consistently explained by the Manning’s condensation theory but finds its explanation within the
framework of DNA cylinder model, like it was done for DNA/DNA duplexes by Frank-Kamenetskii
et al. (1987) and Bond et al. (1994) (see also Section Complexes of PNA with duplex DNA. Difference between PNA and DNA oligomers manifests itself most strikingly in their interaction with duplex DNA. Only homopyrimidine PNAs (containing two types of bases, T and C) are known to form stable complexes with duplex DNA.
However, whereas homopyrimidine DNA oligomers form (DNA)3 triplexes with corresponding sites
on duplex DNA (Section 2.6; reviewed by Frank-Kamenetskii and Mirkin, 1995; Soyfer and Potaman,
1966), homopyrimidine PNA oligomers form a totally different structure with the same DNA sites.
Physics Reports 288 (1997J
M.D. Frank-Kamenetskiil
log[Na+] M
Fig. 19. Salt dependence of melting temperature (T,,) of PNAjPNA (upper
DNA/DNA (lower curve) duplexes. The data are from Tomac et al. (1996).
Fig. 20. The P-loop formation
(see the text for explanation).
Two PNA oligomers form triplex with the homopurine strand of the double helix leaving the
DNA strand displaced in a single-stranded form (Fig. 20). Such structure consisting of (PNA),/DNA triplex and a displaced DNA strand (Cherny, 1993b; Demidov et al., 1995)
is called the P-loop.
A major element of the P-loop is of course the (PNA),/DNA triplex. The P-loop is formed contrary
to the formation of unfavorable helix boundaries only because the triplex is remarkably stable. Why
is this triplex much more stable than the canonical (DNA)3 triplex? Two factors contribute to
this effect. First, whereas in the (DNA)3 triplex three negatively charged strands experience strong
electrostatic repulsion from each other, in (PNA),/DNA triplex such repulsion is totally absent. The
second factor follows from an X-ray crystallographic
study of the triplex by Betts et al. (1995).
M.D. Frank-Karnenetskiil Physics Reports 288 (1997) 13WS
According to the structural data, in addition to Watson-Crick and Hoogsteen pairing the complex
is stabilized by hydrogen bonds between amid nitrogens of the backbone of the PNA strand, which
forms Hoogsteen pairs with the DNA strand, and the phosphate oxygens of the DNA backbone.
These additional hydrogen bonds, together with the lack of electrostatic repulsion within the triplex,
make the (PNA)JDNA
triplexes the most stable nucleic acid-like complexes known to date.
Because two PNA oligomers are involved in the recognition process, PNA clamps or bis-PNAs,
which carry two PNA oligomers connected by a flexible linker, are often used to target homopurine
sites on nucleic acids. Additional stabilization of the complex is achieved by incorporating positive
charges into PNA or bis-PNA oligomers (Demidov et al., 1996; Veselkov et al., 1996a, b).
9.2.2. Specijicity of PNA interaction u,ith duplex DNA
Thus, homopyrimidine PNAs exhibit exceptionally high affinity to their target sites on DNA due
to unique properties of the PNA backbone. At the same time specificity of PNA-DNA interaction is
governed by essentially the same factors as in the case of usual DNA recognition because specificity
is determined by the same Watson-Crick and Hoogsteen base pairing. One should therefore expect
PNA to exhibit very poor sequence specificity.
Amazingly, experiment shows that conditions exist under which exceptionally high affinity of
PNAs to its target sites on dsDNA is supplemented by remarkable specificity of
interaction (Demidov et al., 1995; Veselkov et al., 1996a, b). This finding led Demidov et al. (1996)
to the conclusion that in case of PNA-DNA complexes we encounter a new principle of biomolecular
recognition. We concluded that the affinity of PNAs to their DNA target sites is so high that the
binding should be considered as irreversible. If this is true, the vHB-type treatment obviously fails.
Demidov et al. (1996) concluded, however, that although the overall process is irreversible, its
first stage is highly reversible and consists in fluctuational opening of the DNA double helix (see
Section 7.2) and in transient formation of the Watson-Crick duplex between one PNA oligomer and
the complementary DNA strand. Only after the triplex is formed because of the association of the
second PNA oligomer, the complex becomes irreversible.
Thus, a two-stage mechanism of the complex formation between duplex DNA and homopyrimidine
PNA makes it possible to reach both, the unprecedented affinity and very high sequence-specificity
of the interaction. First, a highly reversible “search” stage takes place, which is followed by an
irreversible “locking” stage. Our theoretical analysis shows that in a wide range of parameters a
model behaves in general agreement with this simple description (Lomakin and Frank-Kamenetskii,
It should be emphasized that because of extremely high affinity of interaction of homopyrimidine
PNAs with duplex DNA, the final state for a target site containing a small number of mismatches
(one or two) as compared with the correct site will also correspond to its almost full occupation.
So to reach optimal sequence specificity one has to be very careful in choosing the appropriate time
of incubation of DNA with PNA, which should be sufficiently long to secure high occupancy of the
correct site but should not be too long to prevent the mismatched site to be significantly occupied.
In other words, in case of PNA/DNA interaction we deal with kinetic discrimination between correct
and mismatched sites. Demidov et al. (1997) have recently subjected this question to a detailed
theoretical analysis.
Extensive further studies are needed to prove the proposed mechanism of PNA interaction with
duplex DNA. However, the very fact that PNA binds to duplex DNA with extremely high affinity
M.D. Frank-Kumenetskii
I Physics Reports 288 (1997)
and specificity has been proved beyond doubts (Demidov et al., 199.5; Veselkov et al., 1996a, b).
The data of Veselkov et al. (1996a, b) are especially convincing because they demonstrate how
these features of PNA make it possible to develop a very efficient method for cleaving long DNA
molecules (consisting of millions base pairs) in very limited and fully specific sites.
This work was supported
in part by NIH grant GM52201.
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