physics of dna: unravelling hidden abilities encoded …...between two dna molecules—is one of the...
TRANSCRIPT
This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.
Physics of DNA: unravelling hidden abilities encoded in the structure
of ‘the most important molecule’wzAlexei A. Kornyshev*
Received 9th March 2010, Accepted 25th June 2010
DOI: 10.1039/c004107f
A comprehensive article ‘‘Structure and Interactions of Biological Helices’’, published in 2007 in
Reviews of Modern Physics, overviewed various aspects of the effect of DNA structure on
DNA–DNA interactions in solution and related phenomena, with a thorough analysis of the
theory of these effects. Here, an updated qualitative account of this area is presented without any
sophisticated ‘algebra’. It overviews the basic principles of the structure-specific interactions
between double-stranded DNA and focuses on the physics behind several related properties
encoded in the structure of DNA. Among them are (i) DNA condensation and aptitude to pack
into small compartments of cells or viral capcids, (ii) the structure of DNA mesophases, and (iii)
the ability of homologous genes to recognize each other prior to recombination from a distance.
Highlighted are some of latest developments of the theory, including the shape of the ‘recognition
well’. The article ends with a brief discussion of the first experimental evidence of the protein-free
homology recognition in a ‘test tube’.
The discovery of the structure of DNA five decades
ago157–159 has revealed an ingenious built-in mechanism for
the storage and replication of genetic information. Various
aspects of DNA functionality were discovered throughout the
following years, most of them based on DNA unzipping and
‘reactions’ of individual strands. Unzipped double stranded
(ds-)DNA was viewed as a ‘hard disc’ storing all the genetic
information, which is released for protein synthesis or
replication under the action of specific proteins. So, the
common line of thought was to consider DNA existing in
two main forms: an inactive archival, double-stranded one, and
an active (taking part in replication or translation) single-
stranded form. But does the Watson and Crick structure of
ds-DNA have any other functionality in its zipped form? In
fact, the dogma that it does not, may be wrong. There is a
wealth of experimental data suggesting that there are other
functional properties encoded in the structure of ds-DNA, and
their realization does not cause DNA to unzip.
The structure of ds-DNA affects interactions between DNA
molecules, and in turn—the interaction affects their structure.
This influences packing of genetic material, be it in
chromosomes or viral heads. Sequence-dependent ability to
bend affects DNA interactions with nucleosomes: DNA wraps
on histones not randomly but with defined sequence tracks.
Packing in chromatin is not random; it reveals strand-to-grove
alignment between DNA and hierarchical structures at a
higher level of organization. In phage heads, cholesteric
liquid-crystal type ordering of DNA reveals strong bi-axial
correlations, a consequence of the chiral structure of ds-DNA.
Last but not least, as it became clear very recently, the function
of pairing of homologues prior to genetic recombination may
also be encoded in the DNA morphology.
Forces acting between ds-DNA in solution were measured
in the pioneering works of Rau and Parsegian (1980).162
Investigations of that kind have continued since then—so far
on the level of studying DNA assemblies (but we are at the
verge of massive penetration of single-molecule techniques
into this area and can witness already some initial results).
The discovered forces appeared to have very short decay
lengths (2–4 A) but huge pre-exponential factors, amplified
by the length of the molecules. It has been made clear that
these forces are responsible for the structure of DNA
Department of Chemistry, Faculty of Natural Sciences,South Kensington Campus, Imperial College London, SW7 2AZ, UK.E-mail: [email protected]; Tel: +44 20 7594-5786w This paper is dedicated to Adrian Parsegian on the occasion of hisrecent 70th anniversary, whose pioneering work on macromolecularinteractions opened new routes to quantitative characterization of theforces acting between DNA and the properties of DNA liquid crystals.z Electronic supplementary information (ESI) available: Supplementarynote. See DOI: 10.1039/c004107f
Alexei A. Kornyshev
Alexei Kornyshev is Professorof Chemical Physics atImperial College London(http://www3.imperial.ac.uk/people/a.kornyshev). A theo-retical physicist by back-ground, he is renowned forhis works at the interface ofphysics, chemistry andbiology. An author of >200original papers and 30 reviewarticles, he was a recipient ofthe 1991 Humboldt Prize,2001 Royal Society WolfsonAward, 2003 SchonbeinContribution-to-Science medal,
2006 RSC Barker Medal; he is an elected Fellow of the IOP,IUPAC, ISE, and Foreign Member of Royal Danish Academyof Science. This year he received an RSC Interdisciplinary Prizeand lectureship; this article covers the material of one of thelectures.
PERSPECTIVE www.rsc.org/pccp | Physical Chemistry Chemical Physics
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
Phys. Chem. Chem. Phys. This journal is c the Owner Societies 2010
aggregates. Thus, unzipped DNA molecules ‘feel’ each other
from a distance when they pack and condense. Their chirality
is important in the formation of cholesteric liquid crystals,
where DNA molecules are separated by several layers of
water. So what are the forces that propagate that ‘chiral
message’ between them? How do two molecules recognize
each other’s base pair texts, i.e. detect the degree of their
homology from a distance without unzipping?
Answering the latter question is crucial in the context of one
of the most intriguing puzzles of molecular genetics. Homo-
logous recombination—the shuffling of homologous genes
between two DNA molecules—is one of the key processes in
living matter. It warrants genetic diversity and speeds up
evolution; it is also instrumental in DNA repair. It is believed
that this process begins with strand exchange between two
paired DNA molecules. This and subsequent stages of
homologous recombination are promoted by recombination
proteins, and are relatively well studied. However, mutual
recognition of homologues before the strand exchange, which
could be behind their pairing, for years remained a mystery.
Extensive biological literature on the subject has been
surveyed recently in two major reviews (refs. 1 and 2).
However, some findings, published in physical journals, were
not covered there.
In 2001, a physical mechanism was suggested for intact
ds-DNA sequences to recognize each other from a distance in
electrolytic solution.3 This mechanism provides a ‘snapshot’
recognition without any help of proteins and without
unzipping. Based on a theory of electrostatic interactions
between helical molecules, a difference in the electrostatic
interaction energy between homologous duplexes and that
between non-homologous duplexes, called the recognition
energy, was calculated. It was shown that the so-called
electrostatic zipper responsible for possible DNA–DNA
attraction4 can work for homologous sequences, but breaks
down for non-homologous ones, if they are long enough. This
mechanism of homology recognition comes out as an innate,
protein-independent, built-in, ‘well thought through’
consequence of the ingenious Watson–Crick structure of
DNA. However, the electrostatic recognition mechanism is
expected to be efficient between bare DNA molecules at much
smaller distances than those between paired chromosomes; it
remains to be understood how it can be utilized in vivo.
At the same time, the field of DNA–DNA interactions as a
whole, their theory in particular, has matured during the last
decade, independently of the fact whether one of its
predictions—recognition of homology at the DNA–DNA
level—is realized in the cell machinery or not. In this article
we will therefore overview the development of this theory for
physical scientists, chemists and physicists, in qualitative terms
and compact format, avoiding any complicated algebra. In
this respect alone this article differs from an ‘all-encompassing’
and mathematically detailed article in ref. 5. Similarly,
complex biological terminology will be avoided, in an attempt
to discuss every issue in simple physical terms. Furthermore
the material presented below will cover several of the most
recent developments and our progress in understanding the
problems under study. Last but not least, instead of the
comprehensive approach taken in ref. 5, I decided to ‘violate’
the logical and historical sequence here. To make it, in my
opinion, more interesting, I will start the story from its most
risky ‘end’—from gene–gene recognition.
The views expressed in this article are based on 15 years of
joint work and discussions of every aspect of this area with
Sergey Leikin (NIH). Many of the underlying ideas belong to
both of us or were generated by him. Unfortunately other
commitments did not allow him to take active part in
preparing or revising the text, which therefore represents to
a high extent my personal vision of the topic.
Homologous recombination for gene shuffling and
DNA repair
Genetic recombination
Recombination of genes is a process in which sequences are
exchanged between two DNA molecules. The process involves
several steps of breaking and rejoining of DNA. The existence
of recombination was central already to Mendel’s theory of
heredity in 1886, long before the discovery of the physical
nature of genes. In formal genetics the efficiency of recombi-
nation is characterized by the recombination frequency:
rf ¼ R1þR2P1þP2þR1þR2
, where R1 and R2 are the number of
recombinants and P1 and P2 the number of parental
characters that can be directly observed.6 In homologous
recombination, the fragments of genes of the same homology
are swapped, i.e. those that have the same genetic function, the
‘ATGC-texts’ that are almost identical (as biologists use to
say, ‘conserved’). This process makes possible gene shuffling
between two parental copies of DNA, crucial for evolution
and genetic diversity. A similar phenomenon is utilized in
DNA repair, when the cell uses a back-up copy of the genome
as a template for repairs. Most of the stages of recombination
machinery are well studied and understood.6–10 However,
homologous recombination still ‘keeps few secrets’, and
understanding of all its aspects is regarded as one of the key
challenges of the ‘‘post-genomic era’’.11
It is worthwhile stressing the vast biological importance of
the recombination-involving processes.
The repair of the double-strand breaks in DNA, caused by
possible side effects of normal metabolic activity, imbalance of
biochemical reactions or radiation damage, takes place
continuously in the cells, and it is the process responsible for
the robustness of life. Failure in repair can be random,
programmed, or caused by external factors and has a set of
consequences: (i) the cell and the whole organism aging, (ii)
apoptosis (programmed cell death), and (iii) tumor-forming
accelerated cell division. Thus, the ability to repair DNA
properly is vital for maintaining the integrity of the genome
and thus the normal functioning of the cell and the whole
organism. In rare cases a failure to repair can have another
effect: the mutations produced by replication of non-repaired
or wrongly repaired genes can give rise to a new direction or
acceleration of evolution.
Production of new combinations of nucleotide sequences
during chromosomal crossover in meiosis—the latter term
stands for a process of reductional division in which the
number of chromosomes per cell is cut twice, the process used
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.
for sexual reproduction in eukaryotes—promotes genetic
variety in a ‘regular way’. Put it simply, meiosis and fertiliza-
tion generate distinct and unique individuals in populations
through the stage of homologous recombination. If ‘properly’
mixed, most of the new combinations of alleles will not give
rise to anomalous or unhealthy off-spring, but some of the
resulting populations may appear to be more capable than
others to adapt to changing external conditions. This option is
crucial, as it allows evolution and genetic diversity not to rely
exclusively on rare beneficial mutations.
As the process of homologous recombination is promoted
by specific recombination proteins there is a great interest in
understanding the biochemistry and biomechanics of their
influence on recombination. This knowledge is instrumental
for future therapies that may ensure ‘smoothly running’
recombination, as well as the bioengineering of targeted
recombinants. Efforts in this direction were celebrated by the
awarding of the 2009 Nobel Prize to Blackburn, Greider and
Szostak ‘‘for the discovery of how chromosomes are protected
by telomeres and the enzyme telomerase’’. The studies of
recombination-proteins—DNA interactions in chromosomes
are currently the mainstream research in recombination
science. In the present article, we will take a side track and
focus on one missing, but essential link in the chain of
recombination.
The stages of homologous recombination are similar in
DNA repair and in meiosis. In meiosis the first step is a
double-strand break, while in DNA repair it may exist already
as a result of DNA damage (segments of DNA around the
break on the 50 end of the damaged chromosome are
removed). A textbook (and very much simplified!) sketch of
the main steps is shown in Fig. 1, known under the name of the
Holliday Junction Mechanism. Depending on how the two
junctions are resolved, there can be two outcomes: the meiotic
version results in either chromosomal crossover or, formally,
a non-crossover (the right and left paths in Fig. 1).
However even the left path may still not lead to a dead-end.
Indeed, resulting heteroduplex DNA will contain mismatched
base pairs. This will happen because the two recombining
DNA molecules are homologous, but not identical. Indeed, if
the paired chromosomes are different copies of one ancestral
molecule or are coming from independent sources, such as
mother and father, they will differ in a small percentage of
their nucleotides. As healthy product DNA molecules are not
supposed to have mismatched pairs this situation may be
cured by repair enzymes (immediately upon formation of
non-recombinant tracks or left to be done in the course of
the subsequent DNA replication). They will replace one
of the ‘wrong’ bases in a mismatched pair with the correct,
complimentary one. Depending on which of the base pairs is
replaced, this procedure will result in either spontaneous
restoring of the initial sequences, or stabilization of the
recombinant sections. It is often assumed that there is no
‘vector’ in such process, i.e. both variants of repair are equally
probable (quoting ref. 12, ‘‘repair enzymes do not have
intelligence’’). It is therefore not surprising that the repair
here produces on average 50% of recombinant base pair
tracks of rather random length and distribution along the
DNA molecules.
As a result of just these two scenarios we get all kinds of
recombinants that all contribute to the genetic variety of the
off-spring. But there is, of course, much more to it, and for a
description of recombination in all its complexity we refer
the reader to specialized literature.6–12 In this article we
concentrate on the putative ‘zero’ stage, called pairing.
The pairing enigma
The key point in homologous recombination is the swapping
of correct genes: only regions with homologous sequences
should be exchanged or used as a template for repair.
Recombination mistakes in meiosis lead to similar
consequences as those which occur in DNA repair: they either
cause severe genetic diseases (including some forms of cancer,
Alzheimer’s disease, color blindness, etc.)13 or contribute to
ageing.14 Fortunately, such errors are rare. The recognition of
sequence homology occurs with miraculous precision. In
site-specific recombination, the exchange happens at specific,
designated loci recognized by the complex recombination
machinery of the cell (involving multiple proteins). In
homologous recombination the exchange can occur anywhere.
It has been established that at least 50–100 bp homology is
required.15–17 This ensures that the fragments belong to two
alleles of the same gene rather than to different genes.
Although, as mentioned, errors in this process are
infrequent, one may envisage that in the future we might find
the means of assisting nature in reducing them even further,
diminishing the unhealthy consequences of errant recombina-
tion. But for this we need to understand the recognition
mechanism in depth.
Textbooks tell us that ‘‘we know only one mechanism for
nucleic acids to recognize one another on the basis of
sequence: complementarity between single strands’’.13 With thatFig. 1 A textbook sketch of the stages of homologous recombination
(after ref. 13).
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
Phys. Chem. Chem. Phys. This journal is c the Owner Societies 2010
mechanism in mind, the recognition is often attributed to (i)
the breakage of ds-DNA and the formation of single strands
mediated by specialized proteins (e.g., the RecA family),13
followed by (ii) a single strand recognizing and invading a
homologous double helix through hydrogen-bond formation
with bases.
Historically, this idea goes back to the ‘‘unpairing hypothesis’’
of Crick,18 who proposed that the strands would unpair so
that the ‘‘top’’ strand of one homologue could pair with the
‘‘bottom’’ strand of the other. Thus the ‘‘bubbles’’ that
resulted from unpairing were supposed to rapidly adopt
stem-loop configurations and the interaction would then have
begun with ‘‘kissing’’ interactions between the loops. Various
adaptations of Crick’s hypothesis were offered in the
1970s.19–23
Within the scope of these ideas, recognition should take
place at the stage of unpaired or broken strand exchange
(Fig. 1). But if so, it would most likely achieve high efficiency
only for about 10 base fragments.24 If this were the only
recognition mechanism, frequent mistakes would be
inevitable. So, might there be an initial, ‘‘snap-shot’’ recogni-
tion stage of recombination, during which long ds-DNA
tracks recognize each other as a whole from a distance?
‘‘Decades of research into homologous recombination have
unravelled many of the details concerning the transfer of
information between two homologous sequences. By contrast,
the processes by which the interacting molecules initially
co-localize are largely unknown. How can two homologous
needles find each other in the genomic haystack?’’. This was
the key phrase of the 2008 review article by Barzel and
Kupiec,1 followed by another question: ‘‘is homologous pairing
an innate general characteristic of the genome?’’ Similar queries
were also raised and discussed by Zickler in an earlier review2
on the same subject.
In these two articles one finds an overview of an extensive
list of publications in which this problem was addressed by
biologists. That quest began long before the discovery of DNA
structure and function, i.e. molecular understanding of genes
(cf. McClintock,25 in which the author stated that ‘‘there is a
tendency for chromosomes to associate 2-by-2 in the prophase of
meiosis’’). We will not list all those papers here, well covered in
the above-mentioned review articles, but point out just a few
milestones. For example, Henikoff speculated about the
existence of ‘‘some form of communication between homologous
DNA sequences outside of the recombination process’’26 and
Keeney and Kleckner27 hypothesized that ‘‘homology is sensed
directly at the DNA level, guided by direct physical interactions
between DNA duplexes in accessible regions . . .such as
nucleosome-free regions.’’2 Further experiments of Kleckner
and co-workers28,29 gave indirect evidence which indicated
pairing of seemingly intact homologous ds-DNA in the
absence of known recombination proteins, as assumed by
the authors (for a review see also ref. 30). Based on those
observations, they concluded that transient pairing of large
homologous fragments should be an initial, coarse recognition
step. The double helix breakage, single-strand formation and
fine recognition, were assumed to occur as subsequent steps.
Kleckner attributed the recognition and pairing of intact
double helices to some ‘‘unspecified DNA–DNA interaction’’,28
which, since it is not site-specific and involves long stretches of
DNA, cannot be provided solely by proteins.
But what is the physical nature of this interaction?31 Barzel
and Kupiec sum up many attempts to unravel the enigma of
homology recognition, but in the end of the review they
conclude: ‘‘After a long journey we are back at the starting
position. The mechanism of homologous pairing has so far
resisted our survey of possible explanations.’’1
Possible answer
Both review articles1,2 were addressed to a biological audience
and did not cover pertinent research presented in physical
journals. In 2001, a simple, but nontrivial electrostatic
mechanism of homology recognition of intact DNA duplexes
without any assistance of proteins was suggested in Physical
Review Letters.3 This mechanism resulted from a theory of
electrostatic interaction of biomolecules with helical charge
patterns in solution.4,32–34 That theory itself was able to
explain a number of observations of DNA aggregation and
poly- or meso-morphism. Its prediction of electrostatic
recognition of homology at the ‘bare’ DNA level, has
prompted a highly speculative, yet promising hypothesis, that
this physical effect might be behind the putative recognition
stage of homologous recombination. But until recently this
prediction was not supported by any direct experiments. In
ref. 3 the difference between the energy of electrostatic
interaction of two DNA duplexes with identical sequence text,
in parallel juxtaposition, and that of the interaction of
duplexes with unrelated (non-homologous) sequences was
calculated. A relatively simple analytical formula was obtained
for this difference, termed the recognition energy. This formula
revealed a dependence of recognition energy on the length, L,
of the interacting duplexes (which was equal, in that deriva-
tion, to the juxtaposition length of the molecules) and the
interaxial separation, R, between the duplexes. The interaction
and recognition energies decay exponentially with R, but the
pre-exponential factor scales up with the juxtaposition length.
The recognition energy was found to have values Z 1kBT
for sequences with more than 50–100 base pairs at a 1 nm
surface-to-surface separation.
Notably, the latter correlates with the above-mentioned
observations of an extremely low frequency of recombination
events for sequences having this many or fewer base pairs.15–17
In ref. 35, Leikin commented on the impact of the suggested
recognition mechanism: ‘‘DNA in a cell may find its match by a
two-step process: first it locates a 100- to 200-base-long section
that is perhaps 90% identical. Then the protein-mediated
process binds tightly to a roughly 10-base-long stretch of
perfectly matched DNA. Like zooming-in on a best part of a
microscope slide, first you see a coarse grain mechanism, then
fine tuning.’’ Nowadays, this statement can be strengthened,
as, strictly speaking we see no reason why at the first step the
electrostatic snapshot recognition would not be able to locate
substantially longer tracks.
Note that the values of the recognition energy, provided in
ref. 3, were calculated for torsionally-rigid molecules and
incorporated only the innate twist-angle distortions of the
double helical sugar-phosphate backbone.36 Extensions of this
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.
theory,37,38 which have included various effects of DNA
elasticity, have led to somewhat smaller values of the recognition
energy, but did not change the effect qualitatively.
We don’t know the length of two bare DNA tracks that
could reasonably be placed in juxtaposition in the cell, how
close its fragments can come to each other, and what their
ability is to undulate to provide the proximity of the DNA
tracks to be recognized. This depends on the available space
and, presumably, the length of the nucleosome-free stretches
or the machinery of getting off the histones. Nothing seems to
forbid us to speculate about a juxtaposition of several
thousand base-pair tracks, hence providing a much larger
recognition energy. And for such large lengths of juxtaposi-
tion, the recognition energy still may comprise the same few
kBT but at larger interaxial separations, i.e. longer tracks can
recognize each other from greater distances.
The latter is a special aspect of the problem, in view of
spooky assumptions that in order to warrant chromoseome
pairing, DNA molecules must be able to ‘see’ the mutual
sequence homology at 100 nm inter-chromosomal distances.
Helix-specific electrostatic forces between straight DNA
molecules could hardly explain this. As already mentioned
and described further below, the interaction between two
parallel DNA molecules decays exponentially with interaxial
separation. The decay length of that interaction is several A
(according to calculations and experiments, E4 A). At
separations much larger than that decay length, all such
interactions will be ‘dead’, even when amplified by a very large
pre-exponential factor, which scales up with the juxtaposition
length. Somehow, for the snapshot electrostatic recognition to
get utilized in the cell, the two DNA molecules must be able to
find a way to come close to distances of at least several
nanometres. As we will discuss below, this mystery may
partially be resolved by taking into account DNA undulations.
The full picture of the helix-specific DNA–DNA interaction
and related phenomena developed in the last decade has been
reviewed in ref. 5 (see also its recent implications for the
structural differences of DNA in crystals and solutions,39 to
which we will refer once more later in this paper). Below we
will explicate pertinent elements of the theory of helix-specific
DNA–DNA interaction and DNA aggregation in electrolytic
solutions, and then concentrate on the homology recognition
mechanism, as evolved from that theory. We will, however,
need to discuss first what is known about the patterns of
counterion distribution around DNA, which are critical for
modeling DNA–DNA interactions.
Counterions and their role in DNA aggregation:
facts and speculation
DNA packing
Some counterions, when added to a solution containing
ds-DNA molecules, induce DNA aggregation. Molecules
longer than 400 base pairs will condense primarily into dense
toroidal or spheroidal structures or, less often, into rod/fibre-
like structures.40–46 Double-stranded fragments shorter than
the DNA bending persistence length condense at high
concentrations or under osmotic stress into liquid crystalline
phases.47–57 No matter what the overall morphology of the
aggregate, locally, DNA packing follows a columnar or
cholesteric ordering.
This counterion-induced condensation of DNA is readily
reproduced in a test tube and yet is one of the most
fundamental processes crucial to our very existence.
Protamine, a basic polypeptide acting as a DNA counterion,
binds and condenses DNA into compact toroidal subunits in
the sperm of most vertebrates, inactivating and packaging
centimetres of DNA in a micron-size sperm head until it is
reactivated after fertilization. Similar packing takes place in
phage heads; moreover the patterns of packing can be
manipulated by injection of spermine.57 Counterion-induced
DNA condensation is therefore one of the most studied, best
surveyed subjects in the DNA biophysics. Not only it is
relevant for the understanding of how DNA is stored in sperm
heads or viral capsids, but there are also speculations that the
nature of condensed phases of DNA is important for the
function of nucleic acids.50,58–60 For the most updated detailed
review of the counterion-induced condensing effects we
address the reader to Chapter 7 of ref. 5 and the references
quoted in it, as well as to an earlier renowned review61
(for a recent account of peptide-induced DNA condensation
in gene therapies and biotechnology see ref. 62). Nevertheless,
we briefly discuss these effects and views about them, as related
to the subject of this perspective.
Cations that condense DNA
The popular term ‘‘DNA condensation by multivalent
cations’’,61 must not be misread as if DNA is condensed by
small, point-like ions with Z 3+ charge, although some
interpretations have been built on this image. In fact, none
of the commonly used DNA-condensing cations with Z 3+ is
point-like. Certainly, spermine (Sp), spermidine (Spd),
protamine, polylysine and other poly-cations that condense
DNA in vivo and in vitro cannot be approximated as such. The
distance between ionic groups in these polyions is comparable
with the pertinent lengths in the problem. It would, thus, be
better to picture them by a flexible chain of 1+ charges.
Cobalt hexamine (Co-hex, Co[NH3]63+) is the only commonly
used multivalent cation that is not a polymer, but it is also
fairly large, with its diameter B6 A.63 The only ‘‘point-like’’
cations that condense B-DNA are divalent Mn2+ and
Cd2+.64,65 Unlike Sp, Spd and Co-hex, Mn2+ condenses
DNA more efficiently at elevated temperatures, e.g., MnCl2condenses DNA only above 40–45 1C;65 in 150 mM
Mn(ClO4)2, DNA is condensed already at 5 1C, but the
strength of the attraction between DNA still increases with
increasing temperature. The latter is indicated by decreasing
interaxial spacing65 and measured intermolecular forces.66
Note that Ca2+ and Mg2+ do not cause DNA condensation
under the same conditions.61,64,65
One can draw many examples of cation specificity of DNA
condensation. For instance: (i) for several polymeric di-amines
with different spacers between their two charged amine
groups, a few were able to condense DNA and few were
not;67 (ii) the condensing effect of spermine homologues
with different spacers between two central amines shows
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
Phys. Chem. Chem. Phys. This journal is c the Owner Societies 2010
non-monotonic dependence on the spacer length;68 (iii) DNA
condensation by triply charged cobalt amines [Co-hex,
tris(ethylenediamine), cobalt (Co-en) and cobalt sepulchrate
(Co-sep)] is different for each of these cations:63 a much lower
concentration of Co-sep3+ was needed to condense DNA than
that of Co-en3+ or Co-en3+. (iv) Stereoisomers of Co-en3+
have different condensing effects.
For all those ions, however, a similar extent of DNA charge
neutralization was found at the onset of condensation
(ca. 80–90%).60,69–72 Although DNA-condensing counterions
may be different they often produce similar effects. They cause
similar hexagonally ordered (at short range) packing in toroids
and other forms of aggregates51,57,73,74 with almost the same
interaxial spacing of R B 30 � 3 A.55,75 Similar structures
were observed upon osmotic compression of aggregates
facilitated by different counterions.65 These similarities suggest
that common forces may be behind DNA condensation.
A desire to establish a unifying physics of such effects was
heated up by an earlier discovery of attraction of like charged
colloids (for a review see ref. 76). Together with the renowned
polyelectrolyte model of DNA, which envisions DNA as a
homogeneously charged rod,77 it triggered a flux of works on
the theory of counterion-induced electrostatic attraction
between like charged rods. A variety of models, all predicting
such attraction in a certain critical range of parameters have
been developed; for a review see ref. 78. A challenge remains to
reconcile the predictions of these models with understanding
why some counterions condense DNA better than others, and
‘map the reality’ on such models.
No matter how strong many of us believe in the reductionist
power of physics, common sense advises us that the chemistry
in these phenomena cannot be ignored. Furthermore, this
chemistry is not supposed to be so complex that its con-
sequences could not be formulated in simple physical terms.
In the first place, this chemistry is reflected in different binding
constants of counterions with respect to various types of sites
on DNA. The resolubilization of DNA aggregates60,79,80 with
different counterions may be an illustration of the latter point.
It is often accompanied by charge inversion: the reversal of
DNA charge from negative to positive, observed, e.g. in DNA
aggregates at high concentrations of Sp and Spd. A strong
binding (‘chemisorption’) of multiply charged counterions
may produce a larger charge in the adsorbed layer than is
needed just to neutralize the charge of phosphates, thereby
resulting in effective charge reversal. It is easy to show that just
a several kBT deep binding potential will warrant this. Most
importantly, as we will discuss in the next section, specific
counterion binding can never lead to homogeneous charge
distribution if one takes into account the helical charge
distribution of phosphates. Recent synchrotron X-ray data
has allowed unambiguous detection of the positions of
polyamines in the grooves of DNA.81 These results are also
supported by using entirely different methods, such as
combination of capillary electrophoresis, FTIR and circular
dichroism,82 Fourier transform Raman spectroscopy,83 and
old but neatly performed and still credible NMR studies84
(see also a review of related work in ref. 81).
A number of physical theories were aimed at demonstrating
that for highly charged surfaces and multivalent point charges
charge reversal will take place even without chemisorption.85,86
But as we discuss below, the important chemical aspects
missing in such descriptions when applied to real systems
may, in fact, have important consequences for the physics of
these systems. As well as the fact that Sp and Spd cannot
be considered as single point charges, at concentrations
(Z 100 mM) required for DNA resolubilization they might
not even be fully dissociated, e.g., SpdCl3 solution might
contain not only Spd3+ and Cl� ions, but also SpdCl2+ and
even SpdCl2+. Incompletely dissociated ions may compete for
binding sites on the DNA. Displacing the fully dissociated
ones, they will reduce the degree of charge neutralization. This
option was demonstrated in the detailed experimental
studies of the structure of DNA aggregates condensed with
different Spd salts.87 That reference suggested that it is under-
neutralization due to preferential binding of the partially
dissociated species rather than charge reversal that underlies
the weakened attraction between DNA at high Spd concentration.
What these counterions actually do? ‘Bridges’, Wigner crystals,
chemisorption-based patterns
There is a conjecture that spermine or spermidine molecules
may bridge two DNA molecules across the water
gap.60,74,79,88,89 Such configurations should be expected when
these ions are ‘unhappy’ in the grooves of the DNA. Even so,
bridging may be considered only for DNA condensation by
long polyamines: it cannot explain the effect of compact ions,
e.g. Mn2+ and Cd2+. It seems unlikely to be a general or even
habitual mechanism for DNA condensation.
In another extreme, in the so called counterion-correlation
models of DNA condensation, DNA is approximated as a
homogeneously charged rod and condensed counterions as
point-like charges. Juxtaposition of positive and negative
charges occurs due to alignment of cations on one rod
opposite to ‘‘correlation holes’’ (spaces between cations) on
the opposing rod.90,91 Although the correlations between
positions of condensed counterions are generally
liquid-like,92,93 it was argued that charges Z 3+ might begin
organizing themselves into a quasi-crystalline lattice,94 akin to
the Wigner crystal.78,85,95
A brief digression for readers less familiar with the Wigner
crystallization.
This concept emerged originally from the theory of electron
gasses in a uniform neutralizing background at low electron
densities.96 It predicted the formation of electron ‘lattice’-like
ordering in the ground state, both in 3D and 2D systems, when
the electron density is lower than a certain critical value. For
instance, in 2D systems, the critical Wigner–Seitz radius
(average interparticle spacing in the units of Bohr radius)
should be larger than 35.97 This occurs because in a quantum
electron gas the potential energy dominates the kinetic energy
at low densities. 2D electronic Wigner crystals have been
observed in metal/insulator/semiconductor field-effect transistors
and levitating electrons above liquid helium stabilized by
electric fields; for a comprehensive review see ref. 98.
The Wigner crystal concept was later extended to classical
systems (for a review see ref. 76)—some examples of them are
2D-like adsorbed ionic layers at electrodes99 or colloidal
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.
crystals at liquid–liquid interfaces.100 In classical systems the
criterion is different: what matters here is the ratio a = LB/rswhere LB is the Bjerrum length, and rs is the average distance
between the particles. Reminder: the Bjerrum length is the
distance between two charges in a solvent for which their
Coulomb interaction energy is equal to the thermal energy,
kBT, i.e. LB = q2/ekBT (in Gaussian units) where q is the
charge of the particles and e is the dielectric constant of the
medium in which they interact. If one takes the dielectric
constant of water e = 80 and singly charged ions, LB = 7 A.
Wigner transitions occur when a c 1 because then the
Coulombic interactions between the ions dominates the ther-
mal energy. For 2D systems estimates show that it must be
>130.76 Even if one takes an average distance between ions, in
a very concentrated limit, also as 7 A, one needs a charge equal
to 12 (!) to satisfy the ‘130 criterion’. Note that this estimate
requires no Debye screening of the ion–ion interaction in the
2D layer by ions of the background electrolyte, because
otherwise the pair interaction between ions will be even weaker
and could not beat kBT. Last but not least, according to
Mermin–Wagner theorem,101,102 there could be generally no
phase with spontaneous breaking of a continuous symmetry for
T > 0, in D o 2 dimensions. Long-range order will be
destroyed by fluctuations. But short-range order may exist,
and the 2D Wigner crystal, if it emerges, will be characteristic
of exponentially decaying correlations.
To the author’s knowledge, the necessary mapping of the
Wigner crystal concept onto a cylindrical surface has not yet
been reported. In that geometry, the criterion for a, may be
slightly relaxed, although the ordering should be even of
shorter range, as this case is intermediate between 1D and 2D.
Computer simulations of rods with DNA surface charge
density in a medium with the dielectric constant e = 80
showed no evidence of crystalline-like organization of trivalent
point charges, dislocated on rods.103 Still, if the effective ewithin the layer of condensed counterions is much smaller
than 80, the effective coupling parameter may grow up
significantly and the correlations might become strong enough
for the formation of quasi-crystalline domains for already 3+
point charges. This could, in principle, rehabilitate the idea
about Wigner-like structures formed by some hypothetical
highly charged point-like counterions around DNAmolecules.
The stability of Wigner crystalline domains with regards to
thermal fluctuations remains to be studies, as such domains may
be rather fragile due to the quasi-1D nature of the DNA
cylindrical geometry. In principle, the latter become more robust
with decreasing temperature, due to the corresponding increase
of the Bjerrum length (diminishing thermal fluctuation).76 But in
electrolytes of physiological concentration, the temperature range
is very limited by the relatively high freezing point.
In any case, the DNA condensing counterions will
experience the highly inhomogeneous field of a helical charge
distribution of phosphates, and many of them will simply
chemisorb into the grooves of the DNA molecule. In that
case, counterions will rather follow the patterns imposed
by the structure of the DNA backbone, than form an
‘incommensurate’ hexagonal Wigner lattice on a cylinder.
Whatever the conditions are, the counterion-correlation
forces are not strong enough to explain the induction of
DNA condensation by Mn2+ and Cd2+. Selectivity
(condensation by Mn2+ and Cd2+ but not by Mg2+ or Ca2+)
and temperature-favored DNA condensation by these divalent
ions are inconsistent with the counterion-correlation model.
For spermine, spermidine and other cationic polyions, these
forces should be even weaker, because, as discussed, these
polyions are not point-like. Taking the form of flexible chains
of monovalent ions, they are expected to behave more like
monovalent rather than multivalent ions, as the distances
between the ionic groups are longer than between the
neighboring phosphates of DNA.
Can counterion-correlation forces stand behind DNA
condensation by Co-sep3+? The larger Co-sep3+ ion may be
expected to have a lower binding constant, in comparison to
Co-hex, and thus be floating around DNA in a quasi-free
fashion; having the charge 3+ it will then bear a propensity to
form Wigner crystals. The experimental observations are
exactly opposite at least for one of the most well studied
stereoisomers of Co-en3+; they suggest the importance of
preferential binding at some specific DNA sites, which can
be rationalized only if one incorporates the DNA structure
into the theory.
Taking the latter route, it is quite natural to describe, in the
first approximation, the forces between DNA as emerging in a
system of charges comprising phosphate charges running on
the strands and adsorbed counterions disposed predominantly
in the grooves. These adsorbed counterions will be considered
as ‘belonging’ to DNA and fixed. In many cases the latter will
be a good approximation.
However, for some systems it will only be a first approxi-
mation to (i) build the corresponding DNA–DNA interaction
Hamiltonians and then (ii) optimize the disposition of the
adsorbed counterions subject to the Hamiltonians describing
the interaction of the adsorbate ions with each other and with
the DNA. As a result, the distributions of adsorbed counter-
ions on each DNA molecule may get affected by the density of
DNA aggregates: they will adjust each other to minimize the
interaction energy. This approach requires construction of the
Hamiltonians describing ions on the surface of DNA. A first,
simplified version of such an approach has been successfully
applied to describe temperature-dependent DNA condensa-
tion in the presence of Mn2+ ions.104
What about ions such as Na+ or K+ of an ordinarily 1:1
electrolyte? They are not expected to form Wigner crystals.
They are neither expected to chemisorb on DNA, and they are
known not to be able to condense DNA without some osmotic
agent. Do they themselves condense near DNA, becoming ‘a
part of it’, or are they free in the ionic atmosphere around it?
According to theManning theory of counterion condensation,105
as developed for homogeneously charged rods,77 for the
charge density of phosphates on the bare DNA and no buffer
electrolyte, 70% of such ions will ‘condense’ in the vicinity of
DNA smeared along its surface. This amount of charge
compensation with account for the helix-specific forces
(see the subsequent paragraphs) is just under the amount
needed to provide attraction between DNA; the presence of
a buffer electrolyte will further reduce this value. But perhaps
to minimize the energy of the two molecules a bit more
counterions will come to DNA? As calculations show, already
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
Phys. Chem. Chem. Phys. This journal is c the Owner Societies 2010
80% compensation would have provided attraction.4 We will
come back to this question at the end of the article in view of
the results of new experiments.
Counterions in the virtual reality
In view of all said, there was a strong incentive to provide an
independent insight into the distribution of ions about DNA
in aqueous solutions using an in silico ‘computational
microscope’, i.e. by means of atomistic molecular dynamics
or Monte Carlo computer simulations. We are not able to list
here all the reports of that kind, mentioning just a few.
Refs. 106 and 107 clearly predicted localization of poly-
amines in the grooves of DNA, predominantly in the major
groove. Ref. 106, in particular, was targeted to determine the
binding sites of putrescine, cadaverine, spermidine and
spermine on A- and B-DNA. The simulations either contained
no additional counterions or sufficient Na+ ions, together
with the charge on the polyamine, to provide more than 70%
neutralisation of the charges on the DNA phosphate. The
calculated stabilisation energies of the complexes indicated
that all four polyamines should stabilise A-DNA in preference
to B-DNA (cf. with the theory of ref. 34) which is in agreement
with experiment in the case of spermine and spermidine
(but not in the case of putrescine or cadaverine). The major
groove is the preferred binding site on A-DNA of all the
polyamines. Putrescine and cadaverine tend to bind to the
sugar-phosphate backbone of B-DNA, whereas spermidine
and spermine occupy more varied sites, including binding
along the backbone and bridging both the major and minor
grooves. At least for B-DNA all these results are in line with
experimental data.81–84
The various results of the Swedish–Singapore consortium108–113
were often at odds with experimental data, giving rather
controversial predictions on the preferential localization of
polyamines and other ions on DNA. For instance they do not
predict predominant adsorption of spermine and spermidine
in the major groove.
It is clear that one should take predictions of the current
computer simulations with a pinch of salt. Indeed, counterion
binding to DNA is strongly affected by the coordination
chemistry of the ions, water and DNA. Modeling of it may
require a full quantum mechanical description of electron
clouds of all relevant atoms. Furthermore, since counterion
binding affects interactions and conformation of DNA in
hexagonal aggregates, the converse must also be true; local
counterion binding may depend on collective effects concerned
with the large scale variations in the DNA interactions and
conformation. In other words, modeling of counterion–DNA
interactions may require ab initio simulations of millions of
interacting atoms over long time intervals and large conforma-
tional space, which are still beyond the reach of the available
computational technology.
Helix-specific electrostatic interactions between
DNA molecules: from interaction to recognition
As mentioned, an electrostatic mechanism for DNA–DNA
interaction4,32–34 and recognition of homology3 was discussed
in a number of papers;37,38 for a comprehensive summary, see
ref. 114. Fig. 2 and 3, borrowed from the indicated references,
and their captions, give the reader a snapshot of the recognition
principle, but we also expand on it below.
‘Electrostatic zipper’
Double-stranded DNA has negatively charged helical motifs
running on the two phosphate strands. These are counter-
balanced by specifically adsorbed counterions. If the latter
reside in the grooves or are condensed at/smeared along the
DNA surface, the compensation of charge can be complete or
partial, but in both cases it results in two separated motifs of
positive and negative charges on DNA. Thus formed, the
counterion adsorption/condensation charge patterns bear the
basic helical symmetry of the DNA. Two such DNAmolecules
in parallel juxtaposition could attract or repel each other,
depending on the mutual azimuthal orientation. The favorable
one effectively positions the negatively charged phosphate
strands closer to the adsorbed/condensed counterions, which
may result in attraction between DNA. For the attraction to
emerge, it does not take 100% compensation of charge of
phosphates by counterions; calculations has shown that if the
ions are predominantly adsorbed in the major groove; already
80% compensation can provide attraction.4
Interaction of ideal double helices in parallel juxtaposition and
columnar assemblies
Referring to the notations of Fig. 2, one can write the
interaction of two ideal double helices for a mutual azimuthal
orientation F = F1 � F2 as4,5,32
E = L{a0(R) � a1(R)cosF + a2(R)cos2F} (1)
where L is the juxtaposition length along which the molecules
can be considered parallel, and a0, a1, and a2 are the
coefficients (with units of energy per unit length) which depend
Fig. 2 Mutual alignment of two DNA molecules with parallel axes
(the picture is borrowed from the supplementary material of ref. 115).
The azimuthal orientation of each molecule Fn is shown by a bold
arrow in its top cross-section (drawn from the axis to the middle of the
minor groove of the molecule). The alignment with strands of one
molecule facing the grooves of the other one (a) leads to inter-
molecular attraction (or reduced repulsion)—zipped state, as
compared to relative orientations with strand-to-strand alignment
(b)—unzipped state. To realize the corresponding energy gain of
favourable juxtaposition, the strands must stay in register with the
grooves over the whole juxtaposition length.161
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.
on the interaxial separation between the molecules, R, on the
population of major and minor groves by adsorbed counter-
ions, and on the concentration of background electrolyte.
The values of a0, a1, and a2 all decay exponentially with R
but their decay ranges are different. We will not present
explicit expressions for a0, a1, and a2, as this has been done
many times3,4,38); for definiteness we refer the reader, to
eqn (A1) and (A2) of ref. 38. But it is worth discussing their
meaning in detail.
The first term in the brackets of eqn (1) consists of two items.
The first item is the classical, described-in-text-books inter-
action of noncompensated residual charges on the DNA—the
negative charge of phosphates and the positive charge of
adsorbed counterions. As mentioned, their sum must not
necessarily be equal to zero: there could be undercompensation
(net negative charge) or overcompensation (net positive
charge). This item is proportional to the product of the net
charge on the molecule, and is of course absent if the latter is
zero. Ordinarily, if there is no charge inversion due to
ion correlations in solutions of multivalent counterions
(the situation that we will discuss later in some detail), the
charge on the both molecules is compensated in the same way,
and this item is repulsive. When it is non zero, its decay range
is given by the Debye screening length:
�Rcl = k�1 (2)
The second item is due to image forces: the repulsion of
charges on one DNA molecule from the low dielectric
constant core of another DNA molecule.
A brief reminder about the concept of image forces and an
explanation how they materialize in the context of DNA–DNA
interactions. Independently on its sign, a point charge q in a
dielectric near its interface with a metal is attracted to its
mirror image in the metal. The attraction force is pq2, and at
a flat interface this force it is inversely proportional to the
distance to the interface. This is what usually people remember
from their undergraduate physics course.116 As is commonly
said, the mirror image is due to free electrons of the metal that
come from the metal bulk, in response to the external charge,
if the latter is positive, or move away to the bulk when it is
negative, in order to screen it. In fact, it is a convenient analogy
on which the effect of the polarizability of metal can be mapped.
Such an image charge always has the opposite sign to the external
charge, qimage = �q, and the charge is attracted to the metal.
However, an external point charge will get repelled from its
image near an interface with a dielectric, whose dielectric
constant is lower than that of the medium in which the
external charge is embedded. Generally, for a flat interface,
qimage ¼e� e0eþ e0
q
where e is the dielectric constant of the medium in which the
charge is located, and e0 is the dielectric constant of the
neighboring medium. When e > e0 image charge has the same
sign as the external charge; moreover when e c e0, qimage E q.
Thus a given point charge on a ‘first’ DNAmolecule sees the
image charge in the low dielectric constant cylindrical core of
the opposing, ‘second’ DNA. This image charge has the same
sign and approximately the same value. Our point charge also
gives rise to a similar image charge in the core of the DNA on
which it sits, but this just effectively doubles the charge value.
Note that for the DNA in an electrolytic solution the
situation is complicated by two factors: (i) we have to deal
with image charges of both phosphates and adsorbed counter-
ions, and (ii) because all electric fields are screened by
electrolyte. These two factors will cause quasi-exponential
decay of image repulsion with decay lengths determined by
the Debye screening length in the electrolyte and the helical
pitch—the parameter that characterizes the helical periodicity
of the charge distribution on the DNA molecule (cf. Fig. 2).
So, by its nature the image term is always repulsive. Its
decay is exponential at large interaxial separations; the decay
range given by
�Rrep ¼1
2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ g2
p ð3Þ
where k is the inverse Debye length in the electrolyte, and g =
2p/H is the ‘reciprocal pitch’ (H is the helical pitch, c.f. Fig. 2).
The other two terms in eqn (1), the azimuthal dependent
ones, describe the direct electrostatic interaction (Debye-
screened by the electrolyte) of the helical patterns of charges
of the two DNA molecules. These are comprised of charge
distributions of phosphates on the helical strands and of
adsorbed counterions that lie along the grooves or are smeared
homogeneously along the DNA surface. These terms are often
referred to as helix-specific (the image term is also helix
specific, but it has no azimuthal dependence). For an optimal
azimuthal orientation of the two molecules the effect of these
two terms is attraction. The decay range of a1 is
�Rð1Þhelix ¼
1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ g2
p ð4aÞ
and of a2,
�Rð2Þhelix ¼
1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ 4g2
p ð4bÞ
These two decay lengths correspond to two senior
cylindrical harmonics allowed by helical charged distributions
(for more about the selection rules that outline the corresponding
‘helical harmonics’ and determine in the end DNA–DNA
interactions see ref. 33).
Thus, the hierarchy of decay lengths is
�Rrep o �R(2)helix o �R(1)
helix o �Rcl (5)
For B-DNA H = 35.7 A, so that g = 0.176 A�1. For an
electrolytic solution of a physiological concentration k =
0.14 A. This gives �Rrep = 2.2 A, �R(2)helix = 2.63 A, �R(1)
helix =
4.4 A, �Rcl = 7 A Experimental measurements of the attractive
and repulsive forces in DNA–DNA interactions (for a review
see ref. 5 and for the latest data see ref. 117) seemed to reveal,
as we believe, �Rrep- and �R(1)helix-determined contributions. The
coincidence of experimental and theoretical results looks
amazing—the experimentally assessed decay range of
repulsion is 2.3 � 0.1 A and of attraction 4.8 � 0.5 A. But
this success of the theory has a solid reason. The characteristic
decay ranges are determined not by the details of the model,
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
Phys. Chem. Chem. Phys. This journal is c the Owner Societies 2010
but by constraints imposed by the helical symmetry of the
DNA,33 and this is why the obtained values are so robust.
If we compare eqns (3) and (4), we also see here another
general, model-independent result: the decay range of
attraction is two times longer than that of repulsion, and exactly
this is observed. This has a clear physical reason: the force
lines that provide interaction with its own image are two times
longer that those of direct interaction. The corresponding
effective doubling of distance R in the exponential decay law
R, as shown in Fig. 3 is here perceived as two-times shortening
of the decay length.
Trivial minimization of the energy given by eqn (1) in Fgives its optimal value
�F ¼ � arccosf a1ðRÞ=4a2ðRÞg ; a1 � 4a20 a1 � 4a2
�ð6Þ
Since a2(R) is a faster decaying function then a1(R), there is a
frustration point, R*, at which a1(R*) = 4a2(R*), such as at
R Z R*, a1 Z 4a2(R*) and thus �F = 0. For more details
about this effect see the original paper where it was
discovered32 or a review;5 3D pictures of the shape of this
frustrated potential as a function of F and R can be found in
Fig. 2 of ref. 38 or Fig. 9 of ref. 5. Altogether, this effect leads
to azimuthal pinning of two DNA molecules in parallel
juxtaposition, which vanishes either at large R, where the
interactions are weak or at the frustration point, near which
the potential as a function of F is flat and any azimuthal
orientation is possible (as we will see the latter is true only for
ideal helices of any length and for short fragments of
real DNA).
It is now worth inserting a note on a useful terminology,
which exploits the analogy of our pair potential with other
models of statistical physics. The angular-dependent part of
eqn (1) reads as E = L{�a1(R)cosF + 2a2(R)[cosF]2}. Let us
introduce vector s(z) pointing from the axis of a molecule to
the middle of the minor groove at an ‘altitude’ z. Two such
vectors (see arrows in Fig. 4), call them ‘spins’—each for each
molecule—taken at the same altitude, s1(z) and s2(z), will have
an angle F, so that cos F = s1(z)s2(z). For ideal rigid double
helices in the ground state the results will not depend on z, so
that we can simply write that cosF = s1�s2, keeping in mind
that the two vectors are taken at the same altitude.
Thus the angular-dependent part can be written as E =
L{�a1(R)s1s2 + 2a2(R)[s1s2]2}. Everyone familiar with the
models of magnetism118 will notice that, as both a1(R)
and a2(R) are positive, the first term is analogous to a
ferromagnetic-type interaction, as it tends to put the two spins
parallel to each other. The second term is unusual for magnetic
models. It will have a trend to put the spins perpendicular to
each other with the angle �901. As a1(R) dominates at
distances above the frustration point, one may expect a
‘ferromagnetic order’ of our fictitious ‘spins’ there, but below
the frustration point the angle would be non zero, having a �value, approaching �901 at very short interaxial separations.
Right at the frustration point all mutual orientations of ‘spins’
in such a system will be energetically equivalent. In a columnar
assembly of DNA, such a pair potential will favour
‘ferro’-ordering of spins consistent with the hexagonal
positional order, at interfacial separations above the frustration
point, whereas one cannot expect both ferro-order of spins and
hexagonal positional order below the frustration point.
A ground state,119 as well statistical theory120 of columnar
aggregates with azimuthal correlations has been developed. It
predicts the whole set of azimuthal ordering corresponding to
hexagonal or rombohedric positional order of molecules in the
assembly, i.e. phases, which emerge for interaxial separations
below the one at which any two molecules would have a
frustration point. We will not go into the details of that
sophisticated theory, but refer the reader to the original
papers. Note only, that as discussed in ref. 120 some of these
phases have been observed, whereas some remains to be
discovered (or disproved!).
Azimuthal correlations
The effect of azimuthal correlations was invoked to explain a
number of phenomena in DNA assemblies. First it helped
to explain poly-and meso-morphism in very dense
poorly hydrated assemblies;34 then it facilitated the
interpretation54,121,122 of a giant cholesteric pitch dependence
of cholesteric liquid crystals of DNA123 (see below). The next
evidence was obtained via a new look at old data for X-ray
Fig. 4 The concept of azimuthal orientation of an ideal double helix.
Red spots are the points where the lines drawn through the phosphate
strands hit the cross-section plan at a chosen altitude. Arrows (vectors)
drawn from the axis of each double helix to the middle of its minor
groove symbolizes the azimuthal orientations of the double helices. If
the two parallel double helices are ideal, the angles between two such
vectors will be the same at any altitude.
Fig. 3 Effecting doubling of the interaction distance for image forces
is perceived as two times shorter decay length.
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.
scattering from wet DNA fibers: this was the detection of the
influence of interaxial distance between the DNA molecules
(i.e. the density of the fibres) on the position of diffraction
maxima of the so called non-zero order layer lines.124 The
signatures of azimuthal correlations have been recently
discovered in the structure of DNA toroids condensed by
tetravalent spermine in bacteriophage capcids.57 This was
manifested with the rotation of the hexagonal lattice in the
toroidal bundle arising from DNA–DNA interactions. Here, it
will be timely to proceed to the interaction of skewed DNA
and the chiral torques.
So, azimuthal correlations are not ‘science fiction’, but
reality. They do not extend far beyond nanometre distance
of surface-to-surface separation between DNA molecules, but
were they are in place, they seem to play an important role in
the structure of DNA mesophases.
Straight DNA molecules in non-parallel juxtaposition:
cholesteric liquid crystals
A mathematically sophisticated theory of electrostatic
interaction of two molecules with helical charge motifs, whose
main axes are skewed with respect to each other, was first
developed in ref. 121. It was applied to DNA and was
used in speculations about the origin and properties of
cholesteric DNA liquid crystals. Indeed, the upshot of that
theory was that the structure of DNA determines the chiral
torque.
The value and the sign of the torque depend on the mutual
azimuthal orientation and the shortest distance between the
molecules. System parameters, such as the net charge of
adsorbed counterions and their distribution between the major
and minor grooves also affect the torque. The equilibrium
skew angle depends on this torque and the ‘returning force’,
which tends to put molecules into parallel juxtaposition. For
two molecules, such force comes out from the overall
attraction of the molecules if the charge of phosphates is
substantially compensated by counterions: the one we
obtained in the theory of interaction of parallel molecules.
Indeed, the moment the molecules get skewed, because the
interaction of sections of these molecules that are away from
the point of closest juxtaposition will be rapidly screened by
the electrolyte, that attraction will be lost. As the attraction of
parallel molecules scales up with the length of the molecules,
the equilibrium angle will be a decreasing function of the
length of the molecules.
In a dense assembly of the molecules, the returning force
may alternatively originate from repulsion due to non-
compensated charges on the molecules, but again the effect
will scale up with the length. Ref. 122 extended the theory of
interaction of two very long molecules on the theory of
interaction of triads of molecules of finite length, in which
two stay parallel and a third one is skewed in a plane parallel
to those of the two molecules. The model studied there was
specially designed for a description of the ground state of a
cholesteric liquid crystal of DNA, in which such triads
mimicked the ‘unit cell’’ of a layered cholesteric order. A
number of properties that have been observed experimentally
for DNA cholesterics have been successfully described there
including the dependence of the giant cholesteric pitch on the
average distance between the molecules.
Qualitatively the loss of the cholesteric order in favour of
line hexatic order at high densities of liquid crystals was
associated with azimuthal correlations that at short interaxial
separations gives rise to complicated ‘spin structures’ which
cannot warrant a systematic sign of the chiral torque. Before
that there is a lot of azimuthal fluctuations at the frustration
point and near it, where the orientations of the ‘spins’ are
energetically equivalent. These two trends essentially wash out
the tendency for the cholesteric order, as torques either
fluctuate or cannot maintain a systematic sign throughout
the lattice. The ground state, ‘triad’ level of analysis allows us
to rationalize a set of the trends reported in the literature,
whereas some of the predictions remain to be verified: (i) the
parabolic growth of the cholesteric pitch with the length of
the molecules, (ii) the effect of the adsorbed counterions on the
handedness of the cholesteric order, etc.122
A full statistical theory of cholesteric liquid crystals has not
yet been developed, but the mentioned works pave the way to
it. However, before getting involved in this, we still need to
sort out several issues. The analysis of Ref. 121, 122 resulted in
a suggestion of the same handedness rule: right-handed helices
form a cholesteric liquid crystal with a right-handed chiral
twist. But that analysis did not take into account the image
forces. As it has been recently noticed by Leikin and Lee, these
forces can also contribute chiral torques. A study is in progress
to find out whether the corresponding torque will have
opposite handedness, with an opposite trend in the handedness
of the cholesteric twist – the left handed twist, or not. If the
former was the case, the story of the sense of the cholesteric
twist and its inversion with a change of environment would
take a new turn.
Cholesteric ordering is known to emerge at intermediate
densities of the liquid crystal, switching into a columnar state
at high densities and to a fully disordered (liquid) state at low
densities. It was thus assumed that the DNA–DNA pair
potential in the cholestric phase will be only weekly affected
by image forces, whose decay range is two times shorter than
those of the direct-interaction helix-specific forces. At the time
this assumption was a tentative measure. Indeed, to calculate
the pair potential of the helix specific attraction forces for
molecules with skewed axes was a major enterprise. But the
calculation of image forces for skewed helices is even more
difficult and cumbersome, and it was thus avoided in the ‘first
approximation’. With such an assumption, the nature of the
cholesteric phase was rationalized as follows:
1. The calculated chiral torque due to helix-specific direct
electrostatic interaction between the molecules is highly
sensitive to azimuthal orientations of the molecules.
Thus, their interactions become not only chiral but naturally
biaxial.
2. At short interaxial separations the complexity of
frustrated azimuthal orientations between the molecules
diminishes the possibility of definite chiral torques and the
cholesteric phase becomes impossible. It is destroyed even
earlier: at the frustration point the energy cost of azimuthal
rotations of the molecules vanishes and thermal fluctuations
will wash out chiral torques.
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
Phys. Chem. Chem. Phys. This journal is c the Owner Societies 2010
3. An ordered array of chiral torques, which permits the
formation of the cholesteric phase, is expected to subsist at
intermediate densities of the liquid crystal, i.e. at intermediate
distances between DNA molecules, where all of them
energetically prefer to be azimuthally oriented in the same way.
4. Increasing interaxial separation makes interactions
exponentially weaker. When they become substantially smaller
than the thermal energy, thermal fluctuations destroy the
cholesteric phase in favour of an azimuthally disordered
phase; but at these distances the liquid crystal order is also
destroyed.
5. For these reasons, the cholesteric phase exists at inter-
mediate distances, and the parabolic pitch dependence on the
density of liquid crystals receives explanation. The handedness
of this phase is determined by the handedness of the chiral
torque of the helix specific direct-interaction forces.
However if the image forces contributed a torque of
opposite handedness, this interpretation might not survive.
For that case, Lee and Leikin suggested an alternative
scenario. It rests on the fact that, as it was always clear, image
forces are azimuthally-independent. As in the previous
interpretation, at high densities of the crystal, helix-specific
attraction forces give rise to azimuthally frustrated phases and
cannot cause definite torques, but the local chiral torques that
are caused by them will be, in addition, compensated by the
torques of opposite chirality, coming from the image forces.
At lower densities the cholesteric phase will then emerge due to
the (yet putative) left-handed image force torques (!) but not
the right handed helix-specific direct-interaction torques.
Indeed, as image-torques are azimuthally-independent, they
cannot be washed out by thermal azimuthal fluctuations
(rotations of molecules about their main axes). On the
contrary, the helix-specific direct-interaction torques can be
substantially suppressed by these fluctuations; the larger the
interaxial separations, the stronger the suppression. In such a
scenario the sense of the cholestric phase will be dominated by
the torques caused by azimuthally-independent, non-biaxial,
image-forces. All torques will ‘die’ with the increase of the
interaxial separation, and thus the transition from the
cholesteric to liquid phase will take place, as in the previous
interpretation.
As will be discussed below the development of the theory,
which allowed for DNA undulations and vibrations of
molecules as a whole in a liquid crystalline aggregate, has
revealed a dramatic amplification of all shorter range compo-
nents of interactions. This includes the faster decaying higher
order helical harmonics in the direct interaction that favor
azimuthal frustrations, as well as the shorter-range image-
forces. All in all, whether the image-force contribution to the
torque is left handed or right handed, this newly discovered
role of undulations and vibrations pushes us to more seriously
consider the role of image forces in the theory of cholesteric
liquid crystals. Building such theory is in progress.
The effect of counterion population of the major and minor
groove and DNA polymorphism. DNA vs. RNA world
It has been shown that the higher relative portion of the ions in
the minor groove amplifies a1(R) relative to a2(R), as a result
the ‘frustration point’ moves to closer interaxial separations,
and the overall attraction increases. For A-DNA the wider
groove is wider and the minor group is narrower than for
B-DNA, and the effect of preferential counterion adsorption
in the major groove, would be even stronger, as such geometry
would allow even stronger separation of the positive and
negative charges. It was this physics that allowed a novel
explanation of the DNA transition from B to A form in very
dense aggregates.34,125 The estimated gain in the interaction
energy, as well as of the entropy of simple ferro-ordering, was
substantial enough to drive such polymorphic transitions.
Since the secondary structure of double-stranded RNA is
close to A-DNA, an interesting speculation was suggested
by Sergey Leikin. Why has the putative prehistoric
‘RNA world’126–130 turned with evolution into the current
DNA world? Obviously a ss-RNA molecule is too open and
vulnerable to sustain long-terms ‘chemical attacks’ on its base
pairs. But RNA can exist in a double-stranded form. What
was wrong with storing information within double-stranded
RNA, why was DNA needed? In the aforementioned
references various motifs were discussed. Leikin suggested a
new one, which follows from the theory of interaction of
helical molecules. In RNA bundles, if they are condensed by
counterions, the aggregates would be too strongly bound, and
it would be harder to unfold them upon request. Calculations
also show that it will be very difficult to condense RNA as long
as not all counterions condense in the major groove.
Condensation of double-stranded RNA is an interesting issue
which currently attracts much attention. Note that double
stranded RNA with counterions chemisorbed into the minor
groove might resist aggregation (c.f. ref. 34); theoretical and
experimental investigations are on the way to verify whether
this is true or not.
At the same time, most viruses have genomes in the form of
ss-RNA rather than ds-DNA and they eject them into the cell
and replicate and evolve faster than any non-viral organism.
But they are packed in viral capcids by special motor proteins,
which exert on them a force larger than 50 pN. This is
equivalent to 50 atm pressure on a viral capsid from inside,
when the ‘genie is inside the bottle’.131–133 Nothing of this kind
could be used for the storage of information inside a
normal cell.
For the same reasons Leikin suggested that DNA was
‘designed’ to be charged to prevent easy aggregation. Staying
in a double-stranded form still allows this aggregation to
happen by the counterions adsorbing into the grooves, as this
promotes DNA–DNA attraction. The thus caused attraction
can be large but not huge, and able to be turned off upon
request. When DNA is packed with the help of histones rather
than in free bundles, the role of ‘counter-cations’ is largely
played by the histones themselves, that are positively charged.
But histones can get stripped of DNA when the information
stored in it needs to be processed.
Real double helices: DNA–DNA interaction and accumulating
disorder. Recognition energy
Ref. 3 made a major step forward, by taking into account that
DNA is not an ideal helical staircase and that distortions of its
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.
helical structure (i) correlate with the sequence134 and (ii) lead
to variation of the local values of the helical pitch. The latter
suggested extending eqn (1) to3
E ¼ZL
0
fa0ðRÞ � a1ðRÞ cosFðzÞ þ a2ðRÞ cos 2FðzÞg ð7Þ
because mutual azimuthal orientations at different ‘altitudes’,
z, are no longer constant. If the molecules are rigid—assume
this first as a simplifying approximation—and if you know the
exact patterns of distortions of the twist angle on each DNA,
you can plug them into eqn (7) and calculate the interaction
energy. If the molecules are not rigid, you will need to
calculate the optimal values of F(z) self-consistently37,38
(to be discussed below).
As patterns of distortion for the same sequences are the
same, two homologous DNA molecules can stay in register
along the whole juxtaposition length and F(z) R 0. Incon-
trast, two non-homologous sequences, when even put in
register at one end will lose register after a certain distance,
as distortions of helicity accumulate in a random-walk
fashion, when averaged over long sequences.39 This situation
is sketched in Fig. 5. If the molecules are rigid, attraction will
then no longer be possible. For sufficiently long DNA tracks
there will be a remarkable difference in the energy of
non-homologous and homologous pairs, which may be
responsible for the pairing of homologues. That energy
difference is called the recognition energy. The formula
obtained for it3 shows its generally non-linear growth with
the juxtaposition length—first quadratic and then linear, with
an exponential crossover between the two limits.
In command: the helical coherence length, a new cumulative
characteristic of DNA
The characteristic length above which the difference becomes
noticeable, called the helical coherence length lc3 was found to
be equal to
lc = h/hdO2i (8)
where h is the helical rise (3.4 A) and hdO2i is, given in radians,
a mean square of twist angle deviation about the average twist
angle E341. A typical value of hdO2i1/2 E 61, gives lc E300 A. Remarkably, l c is determined by the statistically
averaged properties of each individual molecule; it bears no
information about interactions between them. It thus may be
considered as an innate cumulative parameter that in one
number characterizes the random-walk-like accumulation of
disorder along DNA.
The formula for the recognition energy3 suggests that
already at 1 nm surface-to-surface separation, it exceeds the
thermal energy for 50–100 bp duplex length (which is some-
how correlated with the minimum length of homology needed
for recombination24).
A more accurate expression for lc was obtained later upon
taking into account not only the variation in twist angles, but
also in other degrees of freedom, such as roll, propeller twist,
and rise, as well as torsional and stretching thermal
fluctuations.39 As a result, a new effective and more precise
value of lc is given by
1
lc¼ 1
lO;Oþ 1
lh;h� 2
lO;hþ 1
lpð8Þ
Fig. 5 Accumulating mismatch (taken from the original ref. 3). (a) B-DNA sketched as a stack of base pairs (disks). Each base pair has two
negatively charged phosphate groups. The base pair orientation at the altitude z is described by the azimuthal phase angle F(z) of the middle of the
minor groove. Each combination of adjacent base pairs has a preferred twist-angle O(z) = hOi + DO(z), where hOi = 34–351 andffiffiffiffiffiffiffiffiffiffiffiffiffihDO2i
q¼ 4�61.36 For rigid molecules the phase angle accumulates according to the preferred twist angles between adjacent base pairs, i.e.
F(z) =Rz0O(z0)dz0. The deviations of the phase angle from ideal helicity accumulate along the z-axis as a ‘random walk’ over a characteristic
length, called the helical coherence length lc = h/(DO)2 (DO given in radians3), beyond which non-homologous molecules cannot maintain
favorable juxtaposition. (b) The sequence-dependent twist modulation, O(z), leads to axial variation of the local helical pitch. As a result, only
homologous sequences can have negatively charged strands facing positively charged grooves over a large juxtaposition length. (For visualization,
the variation ofH(z) is strongly exaggerated). (c) Molecules with unrelated sequences have uncorrelated twist modulations; this results in the loss of
register between opposing strands and grooves, and a larger interaction energy. The loss of register takes place over the length lc, and it is this
quantity that determines the length of a sequence above which the double helices can sense the difference between the juxtaposition of homologous
and non-homologous tracks.
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
Phys. Chem. Chem. Phys. This journal is c the Owner Societies 2010
Here, the different coherence lengths, tagged by different
subscripts, are due variance in solely—twist angles (lO,O),vertical rise (lh,h), correlated twist-rise (lO,h), and torsion-
stretching elastic thermal fluctuations (lp). Expressions for
these values are somewhat more involved than the
‘super-simple’ eqn (7); the reader may find them in
eqn (7)–(11) of ref. 39. But they have similar meaning, in
respect of the degrees of freedom they describe. Knowing the
detailed DNA structure, the values of lO,O, lh,h, and lO,h canbe calculated; to estimate the value of lp we need in addition
the torsional and stretching elastic moduli of DNA for a given
temperature. This was done for the first time in ref. 39 (see its
Fig. 5) for DNA in crystals and in solution. In solution lO,h ischaracteristic of anticorrelations and is negative, so that its
contribution as well as of other degrees of freedom diminishes
the resulting value of lc. In crystals lO,h >0, but its absolute
value becomes very large and its contribution practically
disappears from eqn (8). Altogether, taking into account all
the terms in eqn (8) with parameters characteristic for free
DNA in solution, we get a substantially reduced value of lc(E100 A). Treatment of the data banks for DNA structure in
DNA crystals ends up with lc E1000 A.
This dramatic change of lc as well as the mentioned
theoretically estimated values is reproduced by the analysis of
experimental data from X-ray diffraction patterns of DNA in
crystals and from DNA in wet fibers.39 Such analysis was made
possible by the development of a new theory of X-ray
diffraction of DNA with account of random-walk-like accumu-
lation of distortions of ideal helicity.39,135 According to this
theory, these kind of distortions give rise to a Lorentzian
broadening of each nth layer line n in the diffraction pattern
with the line half-width n2
4lc. It is shown in ref. 39 that one can
best retrieve the value of lc from the 5th layer line.
Torsional and stretching elasticity and their effect on the
recognition energy
But what is the physical meaning of such a dramatic difference
of the helical coherence length in crystals and solutions
(wet fibers)? The answer was given in ref. 39, but we will also
briefly summarize it here, because this is just another mani-
festation of the recognition effect.
Indeed, the dramatic growth of lc means that the DNA
structure becomes closer to an ideal double helix. The impetus
to become ideal has a simple nature. In all the treated
diffraction experiments, on crystals or wet fibers, the DNA
molecules were not homologous; their texts were random to
each other. Since DNA double helices are not rigid construc-
tions, they can in principle adjust the ‘non-homology
mismatch’ in their surface charge pattern by adapting their
structure to minimize the mismatch. Two isolated molecules
adapating each other must not necessarily become ideal
double helices, as they may just have a compromise, by
adopting—as far as their elasticity allows—some kind of
similar patterns of distortions to stay in register over the whole
juxtaposition length. Indeed, all what they need is to minimize
a difference in the patterns of distortions. This is no longer
possible in an assembly of non-homologous DNA, in which
each molecule is surrounded by some six neighbors that
cannot be ‘satisfied’ individually. The adaptation takes place
as a collective effect in which all molecules tend to acquire the
identical structure. In a large DNA assembly it is impossible to
become identically randomly distorted, and the easiest way to
become identical is to become ideal. Social analogies of this
phenomenon are inherent to utopist societies and totalitarian
regimes, and partly in military regiments. But even there total
uniformity of individuals can never be achieved. In the same
way this works for DNA: even in crystals, the ‘uniformity’ is
only statistically achieved, as lc E 1000 A but not, say, 106 A.
However, strictly speaking, the detailed theory of adaptation
does not operate with an ‘adjustable’ lc. It is a bit more
complicated! One more, new characteristic length emerges in the
problem, determined by the DNA elasticity and the helix-specific,
azimuthal-dependent, zipper-type DNA–DNA interaction, with
lc characterizing the structure of an isolated DNA molecule.
The laws of adaptation and how the latter affects DNA–DNA
interaction energy are remarkably interesting, and we summarize
them here in some detail. The adaptation costs energy of elastic
deformation. It may be complete or incomplete, depending on
how ‘soft’ the molecules are and how strong the attractive
interaction between them is. Indeed, there will always be a
trade-off between what you want to gain in terms of electrostatic
interaction and what you are ‘ready to pay’ in the elastic energy
currency. For instance, if you are very rigid but the potential for
attraction upon adjustment is strong, your ‘payments’ may not
be evenly spread along the length of the molecules, but localized
in the form of torsional kink-solitons.38 The corresponding
elastic energy cost will contribute to the recognition energy,
but altogether the mere ability to adapt will reduce the total
(electrostatic+elastic) recognition energy. This means that
adaptation allows ‘mimicry’: due to their ability to adapt, two
non-homologous DNA molecules will feel less uncomfortable
near each other than the same molecules if they are rigid.
The calculation of recognition energy in ref. 3 was performed
under an assumption of rigid molecules, although the general
equation was derived also with account for the torsional elasticity
of the molecules (which can be as well extended to stretching
elasticity). The approximate (variational) solution of this
problem was reported later.37 The analysis performed there has
shown that the typical adaptability of DNA at room temperature
sits somewhere in between the rigid and soft limits.
The adaptation response is controlled by two characteristic
lengths: the helical coherence length, lc, and the helical adaptation
length, lh. Whereas lc is a parameter of DNA, lh, depends on the
strength of the attractive interaction between the molecules; it is
thus the function of (i) the distance between them, (ii) the
distribution of adsorbed counterions between the minor and
major grooves, and (iii) the screening properties of the electrolytic
solution. Generally, the stronger the interaction, the shorter lh.If lh { lc the problem can be solved exactly by the smallness
of parameter lh/lc. In this case the mismatch does not
accumulate but is relaxed continuously and in the same way
along the whole juxtaposition length. Ref. 37 extended the
analysis to the most interesting case, typical for the realistic
DNA parameters: lh \ lc, where the behaviour is generally
more complicated: it allows abrupt changes of lh, ‘first-order-transition-wise’. But what is important here, is that the
molecules still adapt their structures homogeneously along the
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.
juxtaposition length. For lh c lc38 the mismatch accumulates
along the two DNA molecules to a certain point but then it is
relaxed abruptly via the already mentioned torsional kink-
soliton. After this, the mismatch starts to accumulate again,
before the emergence of the next kink-soliton, after which it is
again relaxed, and off it goes, for long juxtaposition resulting in
a soliton lattice. . . The comparison of these two modes of
response also rings bells of interesting social analogies.zCalculations in ref. 37 show that the ability to adapt makes
the recognition energy approximately two times smaller, but
still not negligible.
The recognition funnel
Recently, this theory was extended to describe the shape of the
potential well in which two DNA molecules may be trapped
if they slide along each other at a certain interaxial
separation,136,137 being simultaneously free to minimize their
interaction energy azimuthally. Since the azimuthal orienta-
tion is adjusted at each Dz, the well is one dimensional: it is the
function of the axial shift, Dz. The well must be symmetric
about the minimum which corresponds to the juxtaposition
of homologous genes ‘face-to-face’, the point taken for zero,
Dz = 0. The depth of the well must be equal to the recognition
energy. It was interesting to find the shape of the well and
estimate the typical values of the force that returns the config-
uration of two molecules to the bottom of the well (Fig. 6).
It was clear from the beginning, that if even the recognition
energy is large but the ‘capture’ force is weak, i.e. the well is
too shallow, it could hardly be used for the formation of the
paired complex. Alternatively, if the force appeared to be too
high, the ‘recognition reaction’ still may not be irreversible,
because the decoupling after the completion of recombination
could also proceed via the perpendicular direction. But it was
critical to analyze the force in any case, with a view to the
forthcoming single-molecule experiments in which the
decoupling force could be measured.
‘Rigid’ DNA
It was natural to start our analysis with the simplest case—
approximation of rigid molecules. In this case the solution is
exact, mathematically easy, fully analytical, and the most
transparent. Furthermore, and most importantly, it is expected
to give the deepest well and the strongest capture force. If in this
approximation the force at all conditions was found to be
negligible, we would not need to bother about more sophisticated
analysis invoking the DNA elasticity, end of story! The solution
of this problem was obtained in ref. 136. The potential well was
found to be quasi-exponential, its depth at the minimum equal to
the recognition energy, and the width equal to lc. The expressionfor the force that follows from the found shape is very simple but
only for interaxial separations above the frustration point, R*, i.e.
when a1(R) Z 4a2(R) [the functions a1(R) and a2(R) were
defined and discussed above in the section devoted to the
interaction of ideal helices].138 In that case it reads,
F ¼� sgnðDzÞ a1ðRÞe�jDzjlc � 4a2ðRÞe�4
Dzj jlc
h i
� L� 2jDzjlc
; DzoL=2 and F ¼ 0; Dz4L=2
ð9Þ
where the sign-function sgn(Dz) R Dz/|z|. About the bottom of
the well, Dz=0, the force changes its sign abruptly. Indeed, here
F jDz�0¼ �sgn(Dz)[a1(R) � 4a2(R)](L/lc). Thus, exactly at the
minimum of the well, the force is zero only at the frustration
point R = R* (Fig. 7).
The fact that at any other R the value of the force at Dz= 0
is not defined should not bother the reader: it is an artifact of
the model which does not take into account the finite size of
the phosphates. Once taken into account, the well rounds up
Fig. 6 A sketch of a juxtaposition window for the distance
recognition of homologous genes. Parallel (a) and (b) antiparallel
alignment of homologous genes. Dz is the relative homology shift
along the main axis of two DNA molecules in parallel alignment
within the juxtaposition window; the bottom of the trapping well is at
Dz = 0. In antiparallel alignment, interaction does not depend on Dzand the molecules if they are allowed to freely rotate about their main
axes will slide along each other ‘friction-free’.
Fig. 7 The sliding force between two parallel homologous sets of
genes (eqn (9)), calculated for the indicated values of the juxtaposition
length L/lc at about 10 A surface-to-surface separation. Parameters:
a1 = 0.6 pN a2 = 0.14 pN.3 As lc E 100 A, L/lc = 100 corresponds
to 2950 base pairs. The absolute value of the force will be several times
weaker with account for DNA elasticity, as discussed in the next
sub-section. For antiparallel alignment the force is zero.
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
Phys. Chem. Chem. Phys. This journal is c the Owner Societies 2010
at the bottom, and the force will not jump, but smoothly
change in sign.
Note that at the frustration point the force vanishes only for
Dz = 0, but its absolute value will grow with the increase of
|Dz|. The behavior of the force at interaxial separations belowthe frustration point is more complex.138 The absolute value of
the force increases monotonically with diminishing |Dz| butonly down to a certain point after which it starts to decrease
and then vanishes at even smaller |Dz|. Then it grows again
reaching its maximum at Dz = 0. The force, however, never
changes sign, except about the minimum of the well Dz = 0.
This behavior was first noticed and it physics discussed in
ref. 137. Of course the tendency for the force to become small,
remaining negative, simply means that the potential well here
becomes less steep as the force vanishes when the potential
becomes flat.
If the juxtaposition length is large, L c lc, which is quite
realistic to imagine, and we were interested to evaluate the
force somewhere near the half-width of the well, we get an
extremely simple purely exponential dependence:
F � � L
lca1 e
�jDzjlc : ð10Þ
An estimate of a1 for about 1 nm surface to surface separation
gives a1 = 0.6 pn.3 Thus, at the well half-width, |Dz| = lc,F � � L
lc0:2 pN. For 1 kbp long L and lc E 100 A,
F E �7 pN. This is a strong force, perhaps too strong, as
close to the bottom of the well it can even be larger.
Thus, this estimate encourages the development of a more
sophisticated theory that will take into account DNA elasti-
city, which expected to reduce the force.
Homology recognition with account for DNA elasticity
Torsional adaptation. A theory, which explicitly considered
torsional adaptation but is easily extendable to other
distortions and other adaptation modes (through renomaliza-
tion of lc), was developed in ref. 137. As expected, the well
appears generally shallower and less steep, its depth still
growing with the juxtaposition length. The absolute value of
the force is some four times smaller, but it also scales up with
the juxtaposition length. The behaviour of the force for the
interaxial separations below the frustration point has similar
non-monotonic features of Dz-dependence.That work has resulted in a concept of an adaptable
homology recognition funnel: two torsionally elastic DNA
molecules can slide along each other as well as come closer
moving in the normal direction, at the same time elastically
relaxing the structural mismatches (translated into the charge
pattern mismatches). In the language of chemical kinetics, they
will thus move simultaneously along three ‘reaction’ coordi-
nates: (i) axial shift, Dz (ii) interaxial separation, R (iii) and a
more complicated functional variable—the field of local
mutual azimuthal angles f(z).Assuming that torsional adaptations occur faster than
translational motions of the molecules, one can minimize the
interaction energy over the realizations of f(z) (this is a
functional minimization), to give the reaction’s potential
energy surface as a function of two coordinates, R and Dz.Typical plots of such surfaces are shown in ref. 137. We will
not reproduce them here, but just note that they (i) show
strong dependence on the population of adsorbed counterions
in the major and minor grooves of the interacting DNA
molecules, and (ii) demonstrate repulsion at short distances,
determined by the a0 term (the one which represents image
forces and the repulsion of non-compensated charges).
Thus the potential energy surface at substantial charge
compensation has a minimum along the R-direction and a
minimum at Dz = 0. This funnel will ‘suck’ the molecules into
a juxtaposition with Dz = 0.
Undulations. So far, this consideration has treated the
molecules as being straight within the length of the parallel
juxtaposition, which is certainly a strong idealization.
Undulations were a big issue in the theory of interacting
polyelectrolytes, considered as homogenously charged elastic
rods in a set of classical papers in that area (for the latest
review see ref. 139). In the theory of helix specific forces in
DNA assemblies they were initially considered to be of minor
importance, under an assumption that each molecule in the
assembly is confined within a unit cell, and this confinement
suppresses undulations. Furthermore, since their wavelength is
much longer than the helical pitch of DNA and the more so
longer than the much smaller decay range of helix specific
forces, undulations were assumed to influence only the
pre-exponential factor of the interaction. All these assump-
tions have appeared to be, at best, inaccurate!139,140
Indeed, it was shown that undulations of DNA in hydrated
aggregates strongly amplify rather than weaken the
helix-specific interactions. They have a much stronger effect
on the interaction modes that intrinsically have shorter decay
lengths, than those with longer decay lengths. Estimates have
shown that they increase helix-specific attraction twice and
image-force repulsion some 20 times, shifting the minimum of
interaction potential to larger interfacial separations.140 Inter-
estingly, now the decay length of the image-force repulsion
may no longer appear to be exactly 12of the decay length of the
direct helix-specific force (c.f. eqn (3) and (4a), Fig. 3). This
consequence has to be further investigated both theoretically
and experimentally
Introducing undulations in the context of homology
recognition, we may suppose that configurations, driven by
the helical electrostatic zipper (attraction) between the homo-
logous tracks, are provided by long wavelength bending
(‘bowing’) fluctuations. Such intermediate configurations are
dynamic, and short-lived, but as a result of them the molecules
can get trapped into the funnel. Indeed, undulations can
dramatically reduce the effective interaxial separation R, thus
making possible chasing each other’s homology between
chromosomes separated on average by much larger distances.
But in a trapped state the average interaxial separation may be
larger than calculated in the absence of undulations, because
undulations most strongly amplify image force repulsion. If all
this is true, large-scale bending fluctuations and smaller
amplitude and shorter wavelength undulations will be an
essential part of this mechanism.
Supercoiling. There is another aspect of an ability to bend. It
has been shown in refs. 34 and 122, that helix specific forces
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.
also produce a chiral torque the value and the sign of which
depends on the distance of the closest interaxial separation of
the skewed molecules and their azimuthal orientations, Debye
length and the distribution of adsorbed counterions. Such
chiral torques at the background of basic attraction will cause
a trend for molecules to supercoil if they can bend. This is itself
a very interesting problem for DNA–DNA interaction and
aggregation, as supercoiled structures are abundant in nature,
and the ‘supercoiling reaction’ is believed to be of profound
biological importance, see e.g. refs. 141–146. A sketch for a
description of supercoiled structures was pointed out in ref. 5
and its web-appended Supplementary Material. In the analysis
of recognition wells we have not yet considered supercoiled
configurations. Will they be important? We have not enough
information to answer this question. If the recognition
proceeds through the interaction of nucleosome-free regions,
the ends of such regions may not be free enough to let these
sections of DNA wind about each other. It could be different,
if the old 1906 hypothesis of Janssens6 that homologous
chromatids twist around each other was unambiguously
reproduced by later studies; but this is still uncertain. Thus a
quasi-collinear juxtaposition may still be habitual.
Whether it takes place or not in chromosomal pairing,
supercoiling in the interaction of free DNA or self-interaction
of a circle DNA is a typical situation and is expected to be such
in single-molecule experiments.
How do the two homologous DNA molecules decouple?
Will two DNA molecules slide along each other out of the
bottom of the well to decouple? As commented by Leikin, if
we account for bending degrees of freedom, bending fluctua-
tions may help to ‘unzip’ the pair rather than slide molecules
as a whole along each other (by unzipping here we do not
mean unzipping of each ds-DNA, but gradual unbinding of
the homologous pair in a direction perpendicular to their main
axis). The characteristic range of the forces in R-direction is
much shorter than in Dz direction, but, of course, the force ismuch higher. However, the pair does not need to unzip as a
whole at once but by small fractions. Somehow, this question
remains open, because bending fluctuations may not only help
to unzip, but equally zip-back the pair, and the total work for
pair decoupling is the same as the total work for sliding out of
the well. Most likely the decoupling will proceed on a
two-dimensional potential energy surface137—along and
perpendicular to the molecular axis, with undulations playing
an important role to be studied.
Juxtaposition length and the length of the genes
Can we now explain the mentioned effect of negligibly low
frequency of recombination observed for genes shorter than
50–100 base pairs?15–17 It is still not clear, strictly speaking.
The longer the juxtaposition length, the stronger the force
trapping the two DNA molecules in the configuration with the
confrontation of homologues. Each two opposing homo-
logous genes of the two DNA molecules will have their
respective neighboring genes that are ordinarily also homo-
logous to each other. As long as the latter are ‘seen’ in the
juxtaposition window, their presence will amplify the effect.
Thus the mentioned effect would be straightforward to explain
if the size of the juxtaposition window coincided with then size
of the gene. However, we see no ground for this conjecture.
Back to the foundations: beyond the mean field theory.
The effects of ionic correlations in the solution
This section contains further details for those experts who may
worry about the rigor of the theory as well as those who may
have concerns about the degree of importance of the so-called
correlation effects. Indeed, in terms of the treatment of
electrolyte screening, the electrostatic zipper theory are, essen-
tially, a mean-field one. It rests on the what Michael Fischer147
use to call the Debye–Bjerrum approximation: (i) all ions of
electrolyte that are non-linearly responding to the electric field
of phosphates or chemisorbed on the DNA are treated as a
‘part of the molecule’, and (ii) the remaining ions of the
solution are described within the framework of the linearized
Poisson–Boltzman approximation.
‘Verbal’ justification of this approximations was twofold. In
the case of multivalent counterions, if they are almost irrever-
sibly chemisorbed on a DNA surface, the net charge of
phosphates is neutralized by the chemisorbed cations. This
neutralization does not eliminate the electric field, if the
cations settle in the grooves. The field decreases with the
distance about DNA, and according to electrostatic zipper
theory will have the helical symmetry of the molecule. If the
singly charged cations do not chemisorb (and many of them
do not) but just condense near the DNA in a Manning–Osawa
fashion90 thus compensating about 70% of the charge of
phosphates in the ‘counterions-only’ case or less in the
presence of a background electrolyte, the helical geometry of
the field will still be maintained. In the simplest approximation
the condensed counterions may then be seen as roughly
smeared along the DNA surface. This picture was questioned
many times, see e.g. the discussion above of the ideas
of the Wigner crystal theory of Rouzhina and Bloomfield,
Shklovskii, and others.86 But systematic quantitative theore-
tical analysis of the situation, which could test how well
justified is this picture, has been performed only very recently.
In his rather sophisticated investigation, Lee148 has found
that if a minute amount of DNA-condensing counterions is
present in the solution, which practically all chemisorb on the
DNA, screening by the rest of the electrolyte can indeed be
fairly well described within the linearized Poisson–Boltzmann
approximation for the ions of the solution, at least at distances
not too close to the DNA surface. The same will remain valid
for the case of the condensed singly charged cations. However,
if the multiply charged ions do not chemisorb, but float
around DNA, dynamic patterns of electrolyte charge on the
surfaces of the opposing DNA can be built.
The physical picture of these patterns, as found by Lee, is as
follows. The total charge in the ionic atmosphere of each DNA
will be positive, ‘requested’ by the negative charges of
phosphates. But on top of the average excess of the positive
charge there will be patches rich in anions. The cations
expelled from those regions will form patches with positive
charge even higher than the average one. Together with the
charges of phosphates, such patterns about the opposing
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
Phys. Chem. Chem. Phys. This journal is c the Owner Societies 2010
molecules adjust their phase shift to minimize the energy.
Specifically, as expected, the regions rich in positive charge
near one DNA molecule will confront regions rich in negative
charge in the vicinity of the opposing molecule. Such disposi-
tion of charges may provide additional attraction between
the DNA.
The attractive force caused by an effect in which fluctuations
in the charge distribution about one molecule correlate with
fluctuations in the charge distribution about the other
molecule, is not a new concept. In colloid science this effect
was introduced long ago in the context of attraction of likely
charged macro-ions (for a review see ref. 76). It received the
name of ionic correlations. For polyelectrolytes this type of
force was first deduced by Oosawa149 (who considered an
idealized model of a polyelectrolyte: uniformly charged line
with counter-ions surrounding it). Lee’s theory has gone
beyond Oosawa’s in four ways. The first improvement is that
the finite radius of the macro-ion (DNA radius) is taken into
account. Secondly, a finite salt concentration is included
(i.e. not only the ‘native’ counterions but also the background
electrolyte is present in the system). Also, the cylindrical
regions occupied by DNA are taken to have a much lower
dielectric constant that the surrounding solvent. Lastly, and
most importantly, the influence of a helical charge distribution
(which is crucial for DNA rather than for a less structured
polyelectrolyte) on the force has been considered.
Strictly speaking, Lee’s theory does not specify whether the
ionic correlation patterns are dynamic or static, i.e. whether
we speak about a phase adjustment of two standing waves or
about correlations of fluctuating patches of charges on the
opposing molecules. One may term this new kind of alignment
as the correlation zipper. But how strong could this zipper be
for two interacting DNA molecules?
Lee’s analysis148 has shown that correlation structures will
yet be rather weak for doubly charged ions giving rise to
perturbations on top of the mean-field solution. Correlation
effects should become dramatic for triply charged ions (again
we speak only about small spherical-like ions, and such ions as
spermine or spermidine do not belong to that category).
Strictly speaking, Lee’s approach, as it stands, cannot be
extended there, because the correlations become too strong
and cannot be described within the version of his perturbation
theory; so that conclusion must be considered as a plausible
extrapolation. In all cases, the correlation structures and the
correlation-induced attraction become important only at small
and moderate interaxial separations between DNA.
Lee treated the case of moderate distances, when the
patterns of charge density waves do not overlap in radial
direction; a description how these layers fuse, when the
molecules come close to each other, is more difficult and is
yet to be developed.
The helical structure causes the attractive force to be
dependent on the rotations of the molecules about their long
axes; the force has a similar two cosine structure as the KL
theory but with some important differences. Firstly, the signs
of the coefficients of the cosine terms are opposite to those of
the mean-field theory. The reason why this occurs is rather
tricky. The propensity to forming positively and negatively
charged patches is enhanced at higher concentrations of
electrolyte. Consequently, the occurrence of the patches is
facilitated by a spontaneous fluctuation of the local number
density. The small ions not chemisorbed in the grooves will
accumulate in front of the negative phosphate charges and will
want to line up with regions of high number density on the
other molecule, but with patches of the opposite sign. This will
create a different helical alignment than for a zipper with
counterions predominantly sitting in the grooves, and,
correspondingly, a different optimal azimuthal orientation.
This interesting effect is also seen in the results of the strong
coupling theory of Kanduc et al.,150 which deals with a limit
opposite to the one studied by Lee in his perturbation theory.
That pioneering study150 was yet performed only for single-
stranded helices.
Ref. 148 studied in detail the character of the decay of the
correlation forces. The result here is more complicated and
cannot be adequately described in a few lines, because a
distribution of fluctuation wavelengths contributes to the
overall interaction energy. Still, in brief, the characteristic
decay lengths for these correlation terms appear to lie between
those of the direct mean-field electrostatic interaction and
those of the mean-field image charge interaction. In the case
of univalent ions, the contribution of correlation forces is
small (as expected), and can, in most cases, be neglected. For
divalent ions the contribution from these forces was found to
be slightly more significant, and it can be treated as a correc-
tion to the mean-field result.
Lee also encountered another contribution to the force,
which although interesting is not something unfamiliar. In a
way it is analogous to image charge force of the mean-field
theory. This force is again due to the molecular interfaces.
Discrete image charge effects and loss of the Debye atmo-
sphere about the small ions, due to the exclusion of ions from
the macro-molecule, both contribute to this force; making it
repulsive. Similarly to the image charge force, this force does
not depend on the azimuthal orientation of the molecule, but
still depends on its helical structure. This force can also be
noticeable for divalent ions.
Somehow, even if the charge density waves of free
conterions and background electrolyte ions do form at inter-
mediate interaxial separations, and exclusion and modified
image forces get more important at short separations, they all
practically disappear at larger distances, reproducing there the
Kornyshev-Leikin theory and its extensions. If the counterions
chemisorb in the grooves, the KL theory becomes generally a
good first approximation.
Summary and open questions
The task of this section is to summarize the main ideas of the
theory, to comment on how robust they are, and describe how
we see the future development of the theory.
Electrostatic zipper: principles and approximations
Our considerations were based so far on the following
principles:
1. The molecules are covered by so called DNA-condensing
counterions, that are predominantly adsorbed into the major
and minor groves.
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.
2. The net charge of DNA, a result of strong charge
compensation of the negative charge of phosphates by positive
charge of adsorbed cations, is Debye-screened by the free ions
of the electrolyte; as the compensation is substantial, linear
response screening is assumed.
3. Water also reacts with the electric fields of these charges
in a linear response fashion; in the first approximation—
through its macroscopic dielectric constant.
4. Positive and negative charge motifs follow the basic
double-helical symmetry of the DNA; any distortions of the
ideal double-helical structure are translated into the
distortions of the helical charge patterns.
5. DNA torsional and stretching elasticity can be consid-
ered within the continuum elasticity theory and the linear
response approximation. No sequence dependence of elastic
moduli or spontaneous curvature of interacting DNA has been
explicitly involved.
Comments
On point 1. For cobalt-hexamine and such polycations as
spermine and spermidine the latter has been established
experimentally; moreover recent synchrotron X-ray data,
compared with various quantum chemistry calculations,
unambiguously showed preferential condensation of such
polycations in the major groove.81
On point 2. There have been substantial efforts devoted to
taking into account non-linear screening effects by multiply
charged counterions. Expected to lead to Wigner-crystal
patterns of charge around DNA and charge density waves,78
they may also provide attraction. The criterion for the
formation of such patterns requires high charges of point-like
counterions. However, many DNA-condensing counterions
are polycations (such as spermine, spermidine and some other
polyamines) that are formed by a chain of charges that cannot
be modelled as point-like. So far, none of these approaches has
been able to incorporate the effects of the helical symmetry
into electrostatic interaction translated into the rather unique
electrostatic ‘snap-shot’ recognition of homology. This effect
goes hand-in-hand with the idea of strong azimuthal correla-
tions between DNA, and the latter has recently received strong
support, resulting from a new look at old data for X-ray
scattering from DNA fibers.39,124 It is crucial, however, to
understand the role of Coulomb correlations in the screening
of DNA by free singly charged ions in the case of no or very
low concentration of specifically adsorbing cations. The first
steps in this direction were made in refs. 148 and 150.
Extension of this approach beyond the approximation of ideal
helices will be most interesting, because the experimental
studies must not necessarily be limited to aggregation in
solution containing classical DNA condensers.
On point 3. Consideration of water on the molecular level
may reveal new features; the same refers also to the considera-
tion of ions. New effects may be expected to arise particularly
at high densities of DNA aggregates. Both can be achieved in
fully atomistic Molecular Dynamic simulations. These have
not yet been reported for interacting DNAs, as generally such
simulations would have to be exorbitantly time consuming.
The latter can be reduced, using the cell model, which has been
considered already in theoretical analysis to take into account
the effect of the Donnan equilibrium (increase of the local
concentration of ions between DNA in dense aggregates to
maintain local electroneutrality).5
On point 4. We don’t see much ground for concern here,
except for when the adsorbed counterions have some direct
effect on DNA structure and elasticity. This, however, is not
supposed to be strong for polyamines, although in can be
different for cobalt and manganese. The fact that some of the
polyvalent counterions may have a propensity for adsorption
near particular base pairs is not supposed to generally
diminish the recognition effect. This will only strengthen the
differences in charge patterns for non-homologous pairs. For
more detailed discussion of specific interactions between
polyvalent counterions and DNA, we refer the reader to
section 4C of ref. 5.
On point 5. The continuum approximation applied to
describe torsional and stretching elasticity as well as bending
is a natural starting approach for building theory of such
complex phenomena. In the continuum approximation, we
employ ‘mechanical’ properties averaged over long DNA
tracks. However, if the genes in question have sections
particularly rich in AT or GC base pairs, we may need a
corresponding extension of the theory, as such sections will
have different torsional and bending elastic moduli. In
addition, AT-rich tracks may lead to spontaneous curvature
of the DNA which may induce additional contribution to the
recognition energy. This effect is easy to incorporate into the
theory of interaction of two DNA molecules, but this may be
more difficult for DNA liquid crystals.
First experiments on double-stranded DNA
homology recognition
Liquid crystalline experiments
Homology recognition between intact DNA duplexes in
protein-free pure electrolytic solutions was experimentally
demonstrated by Baldwin et al.115 In that work an equal
mixture of two families of double-helical 298-base-pair long
DNA fragments in electrolytic solution, was studied. The two
families had identical nucleotide composition and length, but
different sequences. One DNA family was fluorescently tagged
by green dye, the other one labelled by red dye. The
spontaneous segregation of colors within liquid-crystalline
spherulites formed by these DNA under mild osmotic stress,
visible in a confocal microscope as quantifiable green and red
areas, revealed that nucleotide sequence recognition occurs
without unzipping of the double helices. Control experiments
with labeling duplexes of the same sequences with the same
two dyes in a 50–50 mixture revealed no spontaneous
segregation! Thus it was proved that segregation was due to
sequence differences independent of the dyes (Fig. 8).
Typical polarizing microscopy images of the DNA
spherulites revealed patterns characteristic for cholesteric
order. This has indicated that DNA duplexes in the spherulites
are separated by more than one nanometre of water, where the
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
Phys. Chem. Chem. Phys. This journal is c the Owner Societies 2010
cholesteric order is known to exist.51,123 Note that the homology
segregation in ref. 115 has been detected without the presence
of any structure-altering ions in solutions, just NaCl as an
electrolyte, a standard buffer, and mild osmotic stress. This
result has attracted great attention,114,151 because it has shown
that homology recognition capability may indeed be an innate
property of the structure of DNA.
We may refer to another work which may hold supporting
evidence of the existence of this effect. Inoue et al.152 reported
apparent facilitated DNA aggregation of homologous DNA
as compared to that of mixtures of DNA with different
sequences in aqueous electrolyte solutions (with physiological
concentrations of Mg2+) of nanomolar DNA concentrations.
This had been detected using electrophoretic measurements
studying gel retardation of different DNA mixtures.
Those experiments were, however, less decisive. It was not
obvious from the measurements what the nature of the
retarded complex was, since it did not exactly correspond to
the molecular weight expected of a multimer DNA assembly.
Also, some of the samples contained complementary
single-stranded ends, which could amplify the recognition of
homologous fragments via splicing of the single strands
of complementary sequences. Aggregation of DNA mole-
cules was demonstrated using AFM, yet there were no
additional experiments to demonstrate that aggregation was
dependent on DNA homology and/or not mediated by
interactions with the APTES-coated mica surface (which is
known to condense DNA).153 Lastly, the authors of ref. 115
interpreted their data in terms of a putative transient cross-
hybridization between single-stranded ‘kissing’ bubbles and
flipped-out bases.
But in spite of these reservations and different interpreta-
tions of the mechanism underlying homologous recognition,
the results of ref. 152 may well have a similar origin to those of
Baldwin et al.115 and may also be related with the effect
predicted in ref. 3.
None of these experiments have yet established unambigu-
ously the mechanism of segregation. Although the formation
of ‘kissing bubbles’ or spontaneous bending are very unlikely
for relatively short molecules, such as the 298-bp fragment of
the liquid crystalline experiments, these has not been experi-
mentally excluded, strictly speaking. An important point in the
experiments of ref. 115 was that the total amount of AT and
GC base pairs was the same in the two studied families. In this
way it was stressed that the segregation is not an issue of the
‘chemistry’ of the molecules but rather the physics of their
interaction. At the same time it was not excluded that a
spontaneous bending of the molecules due to some, although
minor, presence of AT-tracks could also somehow contribute
to segregation. Furthermore, the experiments were performed
just for two homologues, thus not proving yet the universality
of this phenomenon. In order to bring light on the mechanism
of segregation, experiments with different length of duplexes
need to be performed, as well as in the presence of different
counterions. Work in this area is in progress; some first results
are encouraging: segregation does scale up with DNA length,
but more ‘points on the graph’ are needed to verify the
predicted length dependence to make these results publishable.
Unfortunately, reaching equilibrium in the aggregation
experiments is a painful process. It requires ultrafine control
of the concentration of the osmotic agent—PEG—and its
variation with time. A gradual increase of PEG concentration
with sample drying appears to be a pre-requisite of smoothly
reaching the equilibrium. If segregation is much slower than
aggregation, it may take a very long time for the DNA families
to segregate within the formed spherulites. Indeed, uncontrol-
lably fast aggregation can stick the aggregates in intermediate
basins of attraction. This impedes the ultimate segregation:
from those traps as they may not escape for weeks or ever!
Furthermore, liquid-crystalline experiments are not feasible
for DNA molecules that are much longer than their
persistence length, say a kilobase long or longer, whereas the
case of long DNA is of main interest for homology
recognition. Thus, not undermining the pioneering character
of results of ref. 115, one should admit that single-molecule
experiments will likely be the main players in deciphering the
key ‘details’ of homology recognition.
Single-molecule experiments
The first experiments of this kind were recently reported by the
team at Harvard University, led by Prentiss and Kleckner.154
The idea of those impressive and elegant experiments is based
on application of magnetic beads, the principle of which is
sketched in Fig. 9. Applying this principle, the experiments of
this group have shown pairing between regions of homology
of ds-DNA of 5 kb or less. The pairing was detected under
physiological concentration of univalent salt and temperature,
and, as in ref. 115, in the absence of proteins and multi-valent
counterions. But furthermore, it was observed without
crowding agents; adding the latter was shown to increase the
pairing reaction rate. The detected pairs of all studied lengths
were stable not only against thermal forces, but also against
shear forces acting on magnetic beads up to 10 pN. The
estimates of the trapping force presented in this Perspective
Fig. 8 Examples of confocal cross-sections of self-assempled
(PEG assisted) liquid crystal spherulites, as observed by Baldwin et al,
that show (a) spontaneous segregation of homology of two different
families of DNA-fragments, one labeled by a green chromophore and
the other one by a red chromophore, and (b) no segregation of
identical families, labeled by these two chromophores in identical
proportions (for details see Ref. 115)
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.
gave larger values of the trapping force for such juxtaposition
length. But ‘larger’ is not ‘smaller’—it is easy to imagine a
number of effects that can reduce the estimated values:
1. The actual juxtaposition length in the magnetic experi-
ments of the Harvard group may be smaller than the full
juxtaposition of 5 kb.
2. Decoupling may not necessarily go via sliding but via
step-by-step pair ‘unzipping’.
3. The effect of undulation may give rise to slightly larger
equilibrium surface-to-surface separation R between the
paired DNA than 10 A, and the increase of R exponentially
diminishes the interaction.
A remarkable observation, which is fully in line with the
above discussed theory, is that addition of non-homologous
DNA in solution does not suppress the pairing of the homo-
logous ones. This is a control experiment. It demonstrates no
random pairing of non-homologous DNA, corroborating the
control experiment of Baldwin et al.115 which showed no color
segregation in aggregates of the same homology.
The Harvard group has probed also pairing of partially
homologous molecules, and detected unambiguously, that the
coupling takes place where the homologues overlap. They also
found that longer homologues give sharper distributions
characterizing pairing, which is also in line with the
above discussed theory, where the recognition energy and
the trapping force both scale up with the length of
juxtaposition.
Substantial effort in the Harvard experiments (as well as in
experiments of Baldwin et al.), was to exclude Watson–Crick
pairing. This was necessary, in order to ensure that paired are
intact ds-DNA. In ref. 154 the proof was achieved from the
‘opposite end’, by a kind of reductio ad absurdum: stimulating
DNA melting and subsequent testing whether such stimula-
tion in anyway amplifies the recognition effect (as the
mentioned ‘‘bubble-kissing’’ hypothesis would suggest). And
it did not!
The Harvard experiments probed the effects which are
dramatically amplified by the kb-scale of homologies, whereas
the Imperial–NIH experiments had to stick to short (300 bp)
DNA fragments for which the homology recognition effects
are much more subtle. Both groups have seemingly discovered
the signature of the same effect, but the Harvard group had
more options to study its various aspects. We thus may
consider the Harvard work as a second major, convincing
experimental demonstration of the recognition of homology
without DNA unzipping.
However, there are still a number of issues in these experi-
ments that remain to be better understood. One of them is why
the homologous ds-DNAmolecules are attracted to each other
without DNA-condensing counterions? The electrostatic
zipper is not expected to work here, and the absence of
attraction between DNA in such systems was tested in various
DNA-condensation experiments. Without DNA-condensing
counterions or other ‘condensers’61 only osmotic stress can
help molecules to crowd and come close together. But Harvard
experiments detected homologous pairing without any
crowding agents! There may be three answers to this puzzle:
(i) the solution in fact contains some impurity of multivalent
cations coming either from DNA samples, magnetic beads, or
setup hardware; (ii) the stretched pairs essentially form braids,
and in the braided state aggregation might be possible with
single-valence counterions (DNA condensation of long DNA
in toroidal structures does not happen in NaCl solution
without other condensing agents, but perhaps for braiding
this will be different?); (iii) the magnetic beads ‘catalyze’ DNA
condensation. It would not be improbable, however, if the
answer to this puzzle came from somewhere else!
The probability of the first scenario ought to be excluded by
specially performed chemical analysis of the actual solution.
However, some observations of the Harvard team show that
impurities may not be the issue. They have found that the
pairing depended on the concentration and type of the
monovalent salt. Although this fact per se does not exclude
the possibility that there were very small trace quantities of
contaminants of divalent ions in their solution, the mono-
valents must be playing some significant role, since at low
concentrations of monovalents no pairing was observed. If
only the trace divalents mattered, then the concentration of
monovalent ions would not have played such important role!
The second scenario is a plain conjecture, which is to be
verified by theoretical analysis, on which our group is
currently working; we expect to get the answer pretty soon.
The third scenario implies that DNA molecules of a paired
couple essentially wrap around their magnetic bead. Preliminary
experiments, performed at Imperial College by Arach Goldar,
have shown strong segregation of homologues of DNA when
Fig. 9 The principle of the Harvard experiment.154 Two sorts of
DNA molecules are considered: one attached to monolayer covered
surface (blue), the other—attached to magnetic beads (red).
(The colors are used here for better visibility; no fluorescent tags were
employed.) DNA molecules are kilobases long so that in its native
form they coil into globules. It is expected that if the two sets are non-
homologous, DNAmolecules of those sets will not pair, but if they are
homologous they will have an impetus to couple. Under the influence
of a magnetic field, repelling the beads from the surface, the paired
molecules will stretch. Sill, below a certain critical value of the field the
corresponding bead will remain in the volume adjacent to the surface
(above the critical value all the beads will disappear in the bulk). Such
beads can be seen in a microscope. If their number is remarkably
above the noise level, this will be a proof of homology recognition on a
single-molecule level (and this what has been observed!).154 The critical
shear force that uncouples the pairs can be measured with relatively
high precision; for 5 kb long homologues it was found to be about 10
pN.154
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
Phys. Chem. Chem. Phys. This journal is c the Owner Societies 2010
adsorbed onto glass substrates. Those results were never
published, and finally were not even mentioned in ref. 115 in
order not to confuse the message of that work: its goal was to
prove the physical effect of the direct attraction between
homologous DNA in a pure solution, without any mediators
of interaction, such as surfaces or colloidal particles. In fact
those preliminary observations revealed that DNA homology
segregation was taking place anywhere—on the surface, on
colloidal impurities, etc. Why could not it occur on magnetic
beads? Prentiss’ group challenged this concern by the follow-
ing argument. She and her team have studied pairing as a
function of time in experiments, in which dsDNA fragments
were first kept in the capillary without the beads, the latter
added later. They showed that the pairing depends on that
time, which would not have occurred if the DNA–bead
interactions were critical for that. Although it is still not
excluded that the beads may have a stimulating effect on
pairing, the pairing must have started without the beads, c.f.
Fig. 10. Furthermore, if DNA molecules were partially
adsorbed on the beads, the positions of the beads would have
been different from the estimated ones. Another important
observation briefly mentioned in ref. 154 was that the rate and
efficiency of pairing is not suppressed in more complex
environments. This was simulated by adding into the solution
0.1% (E15 mM) amount of BSA protein, which had no effect
on pairing.
Conclusion
According to the theoretical concepts described above,
homology recognition between two intact DNA as an innate
property of DNA structure is encoded through a set of
connected effects:
1. Correlation between the sequence and the pattern of
distortions of the double-helical structure.
2. Translation of this pattern of distortions into distortions
of the helical patterns of charge distributions on DNA.
3. Accumulation of distortions of these distributions along
each molecule.
4. Loss of register in electrostatic zipper between two non-
homologous DNA texts over the length of juxtaposition longer
than the characteristic length of accumulation of distortions
(the helical coherence length). This results in breakdown of the
electrostatic zipper, in contrast to the ability to maintain the
register over any length for a pair of homologous (almost
identically distorted) double helices that results in a working
zipper.
5. Formation of a potential well when two DNA copies
slide one along each other with a minimum at direct confron-
tation of homologous fragments and half-width equal to the
helical coherence length.
Whereas, these five components of the effect are logically
ordered, they of course emerge spontaneously, so that
recognition takes place in one shot, like ‘love on first sight’.
The latter does not dismiss that there could be a lot of
‘chasing’ associated with random motions before the two
molecules get captured into the recognition well. Like people,
DNA might choose to attend many ‘parties’ before they find
and recognize their match. Softness and adaptability makes
pairing easier, but weakens the ability to distinguish homo-
logy vs. non-homology; for realistic parameters of DNA
elasticity this still does not dismiss the recognition well
(for some people, to smile is a big deal, and this is so
for DNA).
The theoretically calculated capture force scales up with the
juxtaposition length. The calculated values can match and
explain the experimental values of the critical value of the
shear force that warrants un-pairing.
Fig. 10 Important details of the Harvard experiment (again, with
colors used for visibility; the cartoon courtesy of Mara Prentiss). The
colored labels are the terminal moieties attached to the ssDNA tails at
the ends of the dsDNA. The green diamonds are digoxygenin and the
red circles are biotin. The digoxygenin attaches specifically to the anti-
digoxygenin (‘forks’) on the glass capillary, and the biotin attaches to
the streptavidin on the beads (not shown). The DNA piece attached to
the beads can be taken shorter than its potential partner attached
to the anti-digoxygenin on the glass surface, but they will get paired
exactly where homologous sections overlap. How this is proved? The
digoxygenin is always attached to one end of a complete l-frag DNA
used in these experiments. The biotin is attached to the opposite end of
the sequence fragment of the same DNA, and thus the bead attaches to
the biotin at the end of this fragment. In this way not only the
homology is preserved but, eventually, also the proper direction of
homology (c.f. Fig. 6 (a)), for which and only for which the two
sequences are expected to get paired. (The reverse sequence pairing
was not observed: it would have put the bead at the opposite end of the
fragment.) Given that the length of a fully stretched complete l-DNA
is known, the location of the bead was calculated in agreement with
experimentally observed location. This proves pairing of only homo-
logous tracks. Experiments have also shown correlation between the
length of homology and the pairing strength.
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.
Where we are with this paradigm and how widely is it
accepted? In fact this field has reached a critical moment:
There are first experiments that indicate the existence of an
option for genes to recognize its match at the level of bare
DNA without unzipping.
There is a theory which presented a physical mechanism
for such recognition without assistance of any proteins. The
first version of this theory was published 8 years before the
first experiments were performed.
Although the existence of the effect has been demon-
strated, no systematic studies have yet been reported on the
detailed verification of all the predictions of the theory. Some
of them were, however, already approved: stronger pairing of
longer homologues and no pairing of anitiparallel homologues.
The first experiments have been performed in a test tube,
but yet not in vivo.
We need more and different kinds of single-molecule experi-
ments for different sequences to prove the universality of this
phenomenon. It is crucial to study various predictions of the
theory, such as dependence on electrolyte concentration and
the nature of counterions, quantitative effect of the length of
homology, systematically measure the coupling force, investi-
gate the role of undulations etc.
Subsequently, this principle must be demonstrated in action,
in vivo, where the environment is much more complex that in a
test tube, with a focus on answering the question: how close
the bare sections of ds-DNA could come to each other to
execute this recognition mechanism? Only then biologists,
who are ordinarily skeptical about physical experiments
(not speaking about physical theories!) might accept recognition
of homology as a built-in, innate property of the DNA structure,
as a new dogma of molecular biology. There is still a lot to be
checked for this to happen.
Biologists often say, ‘‘who has seen naked DNA inside the
cell? Test-tube experiments? Who is interested?’’ This reaction
is typical. However, with all the reservations about an
immediate success of physics in biology 155 and escapades like
those discussed in this article, one should not forget one key
point. The birth of modern molecular biology half a century
ago rested on physical, X-ray diffraction experiments of
Franklin and Wilkins with DNA in fibres and crystals, and
a mathematically sophisticated—for its time—physical theory
of Cochran, Crick and Vand156 of X-ray scattering from
helical macromolecules. That theory allowed Crick to decipher
Franklin’s data,157 and unravel, together with Watson, the
double-stranded structure of the DNA.158 That structure
enclosed a built-in principle of the storage and replication of
genetic information.141,159 Today, incorporation into the
theory of DNA–DNA interactions of the Crick–Watson
double-helical symmetry, together with the now-known
deviations from that symmetry, may become a key to another
amazing built-in ability of the ‘most important molecule’.
Now it is the gift to recognize from a distance similia similibus,
which could explain the secret of a perfect match.
Note added in proof
New experimental demonstration of homologue pairing now
in a supercoiled configuration has just been reported:
X. Wang, X. Zhang, C. Mao and N. C. Seeman, Double
stranded DNA homology produces a physical signature,
Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 12547–12552.
Note added after first publication
This article replaces the version published on 10th August
2010, which contained an error in paragraph two of the
introduction.
Acknowledgements
I thank Sergey Leikin (NIH) for many discussions, ideas, and
suggestions for this Perspective, and his hospitality during my
many stays at NIH, Bethesda, that were always giving
momenta to each next round of our joint work. I also wish
to thank other close colleagues of mine with whom I have been
working and continue to work on various problems in this
area (Geoff Baldwin, Dominic Lee, John Seddon, and Aaron
Wynveen), those with whom I had a pleasure to work in the
past (Cristos Likos, Harmut Lowen, Godehard Sutmann), as
well as my former (Andrey Cherstvy and Sergey Malinin) and
present (Rugero Cortini and Tim Wilson) PhD students—for
discussions and joint research that have also influenced the
views expressed in this article. I am thankful to Mara Prentiss
for sending me and Sergey the results of her measurements
before publication and illuminating discussions of the achieve-
ments of the Harvard team. Special thanks are to Tim
Albrecht for critical reading of the manuscript. Stimulating
conversations with Gert van der Heijden, Anatoly Kolomeisky,
Stuart Lindsey, Wilma Olson, Rudy Podgornik, Adrian
Parsegian, Rob Philips, Don Rau, Joachim Treusch, Andrew
Traverse, Eugen Starostin, Loren Williams, and Lynn
Zechiedrich, and correspondence with Donald Forsdyke are
also gratefully acknowledged. This article is based on several
lectures at physics and chemistry colloquia in Caltech, Kavli
Institute in Santa Barbara (UCSB), Ohio Math Bio Institute,
Max-Plank Institute in Dresden, at Oxford, Cambridge,
Exeter, Manchester, Royal Danish Academy of Science and
other places, and the follow up discussions there. Thanks are
due to the Leverhulme Trust (Grant F/07058/AE, AAK) for
the financial support of our research in this area in the past
and the present support of EPSRC (Grant EP/H010106/1).
I am also thankful to ICTP-IAEA-UNESCO, Trieste, for
hosting a conference ‘‘From DNA inspired physics to
physics inspired biology’’—http://cdsagenda5.ictp.trieste.it/
full_display.php?ida=a08164, which I had a privilege to
organize and direct together with Wilma and Adrian, as well
as to the Wellcome Trust for co-sponsoring it. Talks and
conversations at that conference made clear a number of issues
discussed in this article, as well as pushed me to find time to
write it. Finally, the heading of this article borrowed a motto
from the title of an inspiring ‘bestseller’ on DNA by Maxim
Frank-Kamenetskii.160
Notes and references
1 A. Barzel and M. Kupiec, Finding a match: how do homologoussequences get together for recombination?, Nat. Rev. Genet.,2008, 9, 27–37.
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
Phys. Chem. Chem. Phys. This journal is c the Owner Societies 2010
2 D. Zickler, From early homologue recognition to synaptonemalcomplex formation, Chromosoma, 2006, 115, 158–174.
3 A. A. Kornyshev and S. Leikin, Sequence recognition in thepairing of DNA duplexes, Phys. Rev. Lett., 2001, 86, 3666–3669.
4 A. A. Kornyshev and S. Leikin, Electrostatic zipper motif forDNA aggregation, Phys. Rev. Lett., 1999, 82, 4138–4141.
5 A. A. Kornyshev, D. J. Lee, S. Leikin and A. Wynveen, Structureand interactions of biological helices, Rev. Mod. Phys., 2007, 79,943–996.
6 V. V. Kushev, Mechanisms of Genetic Recombination, ConsultantBureau, New York, 1974.
7 The Recombination of Genetic Material, ed. K. B. Low, AcademicPress, San Diego, 1988.
8 D. R. F. Leach, Genetic Recombination, Blackwell Science,Oxford, 1996.
9 Genetic Recombination Research Progress, ed. J. H. Schulz, NovaSci. Pbls, New York, 2004.
10 Molecular Genetic Recombination, ed. A. Aquilera andR. Rothstein, Springer, Berlin, 2007.
11 D. J. Sherratand and S. C. West, Royal Society Discussion (2004)Replicating and reshaping DNA: A celebration of the jubilee ofthe double helix, Phil. Trans. Royal Soc. B, 2004, p. 359.
12 J. Ringo, Funamental Genetics, Cambridge University Press,Cambridge UK, 2004.
13 B. Lewin, Genes VI, Oxford University Press, Oxford, 1997.14 F. Madia, et al., Longevity mutation in SCH9 prevents recombi-
nation errors and premature genomic stability in a Werner/Bloommodel system, J. Cell Biol., 2008, 180, 67–81.
15 B. S. Singer, L. Gold, P. Gauss and D. H. Doherty, Determina-tion of the amount of homology required for recombination inBacteriophage-T4, Cell, 1982, 31, 25–33.
16 J. Rubnitz and S. Subramani, The minimum amount of homologyrequired for homologous recombination in mammalian cells,Mol. Cell. Biol., 1984, 4, 2253–2258.
17 V. M. Watt, C. J. Ingles, M. S. Urdea and W. J. Rutter,Homology requirements for recombination in Escherichia-coli,Proc. Natl. Acad. Sci. U. S. A., 1985, 82, 4768–4772.
18 F. Crick, General model for chromosomes of higher organisms,Nature, 1971, 234, 25–27.
19 H. M. Sobell, Molecular mechanism for genetic recombination,Proc. Natl. Acad. Sci. U. S. A., 1972, 69, 2483–2487.
20 G. G. Doyle, A general theory of chromosome pairing based onthe palindromic DNA model of Sobell with modifications andamplification, J. Theor. Biol., 1978, 70, 171–184.
21 R. E. Wagner and M. Radman, A mechanism for initiation ofgenetic recombination, Proc. Natl. Acad. Sci. U. S. A., 1975, 72,3619–3622.
22 D. R. Forsdyke, A stem-loop ‘‘kissing’’ model for the initiation ofrecombination and the origin of introns, Mol. Biol. Evol., 1995,12, 949–958.
23 D. R. Forsdyke, Evolutionary Bioinfiormatics, Springer,New York, 2006.
24 R. D. Camerini-Otero and P. Hsieh, Homologous recombinationproteins in prokaryotes and eukaryotes, Annu. Rev. Genet., 1995,29, 509–552.
25 B. McClintock, The association of non-homologous parts ofchromosomes in the mid-prophase of meiosis in, Cell TissueRes., 1933, 19, 191–237.
26 S. Henikoff, Nuclear organization and gene expression:Homologous pairing and long-ranged interactions, Curr. Opin.Cell Biol., 1997, 9, 388–395.
27 S. Keeney and N. Kleckner, Communication between homo-logous chromosomes: Genetic alterations at a nuclease-hypersensitive site can altar mitotic chromatin structure at thatcite both in cis and in trans, Genes Cells, 1996, 1, 475–489.
28 B. M. Weiner and N. Kleckner, Chromosome pairing via multipleinterstitial interactions before and during meiosis in yeast, Cell,1994, 77, 977–991.
29 S. M. Burgess, N. Kleckner and B. M. Weiner, Somatic pairing ofhomologs in budding yeast: existence and modulation, GenesDev., 1999, 13, 1627–1641.
30 D. Zickler and N. Kleckner, Meiotic chromosomes: integratingstructure and function, Annu. Rev. Genet., 1999, 33, 603–754.
31 Later in scientific tabloids this phenomenon, has received acontroversial but catchy label: DNA ‘telepathy’. This logo grabs
attention, but as quotes could be so easily and were often lost, itproved to be misleading for wide public.
32 A. A. Kornyshev and S. Leikin, Theory of interaction betweenhelical molecules, J. Chem. Phys., 1997, 107, 3656–3674.
33 A. A. Kornyshev and S. Leikin, Symmetry laws for interactionbetween helical macromolecules, Biophys. J., 1998, 75, 2513–2519.
34 A. A. Kornyshev and S. Leikin, Electrostatic interaction betweenhelical macromolecules in dense aggregates: An impetus for DNApoly- and mesomorphism, Proc. Natl. Acad. Sci. U. S. A., 1998,95, 13579–13584.
35 ‘‘DNA need not unzip’’ 2001 Physical Review Focus http://focus.aps.org/v7/st19.html.
36 W. K. Olson and V. B. Zhurkin, Modeling DNA deformations,Curr. Opin. Struct. Biol., 2000, 10, 286–297.
37 A. G. Cherstvy, A. A. Kornyshev and S. Leikin, Torsionaldeformation of double helix in interaction and aggregation ofDNA, J. Phys. Chem. B, 2004, 108, 6508–6518.
38 A. A. Kornyshev and A. Wynveen, Nonlinear effects in thetorsional adjustment of interacting DNA, Phys. Rev. E: Stat.,Nonlinear, Soft Matter Phys., 2004, 69, 041905.
39 A. Wynveen, D. J. Lee, A. A. Kornyshev and S. Leikin, Helicalcoherence of DNA in crystals and solution, Nucleic Acids Res.,2008, 36, 5540–5551.
40 J. Widom and R. L. Baldwin, Cation-induced toroidal condensa-tion of DNA. Studies with cobalt hexamin, J. Mol. Biol., 1980,144, 431–453.
41 D. Lang, Regular superstructures of purified DNA in ethanolicsolutions, J. Mol. Biol., 1973, 78, 247–253; D. Lang, T. N. Taylor,D. C. Dobyan and D. M. Gray, circular DNA-electron-microscopy of ethanol-condensed molecules, J. Mol. Biol., 1976,106, 97–107.
42 T. H. Eickbush and E. N. Moudrianakis, Compaction of DNAhelices into either continuous supercoils or folded-fiber rods andtoroids, Cell, 1978, 13, 295–306.
43 D. K. Chattoraj, L. C. Gosule and J. A. Schellman, DNAcondensation with polyamines, J. Mol. Biol., 1978, 121, 327–337.
44 K. A. Marx and G. C. Ruben, Evidence for hydrated spermidinecalf thymus DNA toruses organized by circumferential dnawrapping, Nucleic Acids Res., 1983, 11, 1839–1854.
45 P. G. Arscott, A. Z. Li and V. A. Bloomfield, Condensation ofDNA by trivalent cations. 1. effects of DNA length and topologyon the size and shape of condensed particles, Biopolymers, 1990,30, 619–630.
46 N. V. Hud and I. D. Vilfan, Toroidal DNA condensates:Unraveling the fine structure and the role of nucleation indetermining size, Annu. Rev. Biophys. Biomol. Struct., 2005, 34,295–318.
47 C. Robinson, Liquid-crystalline structures in polypeptidesolutions, Tetrahedron, 1961, 13, 219–226.
48 R. L. Rill, P. R. Hilliard Jr. and G. C. Levy, Spontaneousordering of DNA, J. Biol. Chem., 1983, 258, 250–256.
49 R. L. Rill, Liquid crystalline phases in concentrated aqueoussolutions of Na+ DNA, Proc. Natl. Acad. Sci. U. S. A., 1986, 83,342–346.
50 J. L. Sikorav, J. Pelta and F. Livolant, A liquid crystalline phasein spermidine-condensed DNA, Biophys. J., 1994, 67, 1387–1392.
51 F. Livolant and A. Leforestier, Condensed phases of DNA:structures and phase transitions, Prog. Polym. Sci., 1996, 21,1115–1164.
52 J. Pelta, F. Livolant and J.-L. Sikorav, DNA aggregation inducedby polyamines and cobalthexamine, J. Biol. Chem., 1996, 271,5656–5662.
53 R. Podgornik, H. H. Strey, K. Gawrisch, D. C. Rau,A. Rupprecht and V. A. Parsegian, Bond orientational order,molecular motion and free energy of high-density DNAmesophases, Proc. Natl. Acad. Sci. U. S. A., 1996, 93, 4261–4624.
54 H. H. Strey, J. Wang, R. Podgornik, A. Rupprecht, L. Yu,V. A. Parsegian and E. B. Sirota, Refusing to Twist: Demonstra-tion of a Line Hexatic Phase in DNA Liquid Crystals, Phys. Rev.Lett., 2000, 84, 3105–3108.
55 E. Raspaud, D. Durand and F. Livolant, Interhelical spacing inliquid crystalline spermine and spermidine-DNA precipitates,Biophys. J., 2005, 88, 392–403.
56 N. Sartori Blanc, A. Senn, A. Leforestier, F. Livolant andJ. Dubochet, DNA in human and stallion spermatazoa forms
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.
local hexagonal packing with twist and many defects, J. Struct.Biol., 2001, 134, 76–81.
57 A. Leforestier and F. Livolant, Structure of toroidal DNAcollapsed inside the phage capsid, Proc. Natl. Acad. Sci. U. S. A.,2009, 106, 9157–9162.
58 M. A. Krasnow and N. R. Cozzarelli, Catenation of DNA. Ringsby Topoisomerases, J. Biol. Chem., 1982, 257, 2687–2693.
59 J. L. Sikorav and G. M. Church, Complementary recognition incondensed DNA: accelerated DNA renaturation, J. Mol. Biol.,1991, 222, 1085–1108.
60 E. Raspaud, I. Chaperon, A. Leforestier and F. Livolant,Spermine-induced aggregation of DNA, nucleosome andchromatin, Biophys. J., 1999, 77, 1547–1555.
61 V. A. Bloomfield, DNA condensation, Curr. Opin. Struct. Biol.,1996, 6, 334–341.
62 P. Saccardo and A. Villaverde, Gonzalez-Montalban N Peptide-mediated DNA condensation for non-viral gene therapy, Biotechnol.Adv., 2009, 27, 432–438.
63 H. Deng and V. A. Bloomfield, Structural effects of cobalt-aminecompounds on DNA condensation, Biophys. J., 1999, 77,1556–1561.
64 D. A. Knoll, M. G. Fried and V. A. Bloomfield, Heat-inducedDNA aggregation in the presence of divalent metal salts, inStructure and Expression: DNA and its Drug Complexes,ed. R. H. Sarma and M. H. Sarma, Adenine Press, Albany,NY, 1988, pp. 123–145.
65 D. C. Rau and V. A. Parsegian, Direct measurement oftemperature-dependent solvation forces between DNA doublehelices, Biophys. J., 1992, 61, 260–271.
66 S. Leikin, D. C. Rau and V. A. Parsegian, Direct measurement offorces between self-assembled proteins: temperature-dependentexponential forces between collagen triple helices, Proc. Natl.Acad. Sci. U. S. A., 1994, 91, 276–280.
67 A. A. Zinchenko, V. G. Sergeyev, K. Yamabe, S. Murata andK. Yoshikawa, Stereoisomeric Discrimination in DNACompaction, ChemBioChem, 2004, 5, 360–368.
68 V. Vijayanathan, T. Thomas, A. Shirahata and T. J. Thomas,DNA condensation by polyamines: a laser light scattering studyof structural effects, Biochemistry, 2001, 40, 13644–14651.
69 R. W. Wilson and V. A. Bloomfield, Counterion-inducedcondesation of deoxyribonucleic acid. A light scattering study,Biochemistry, 1979, 18, 2192–2196.
70 J. Widom and R. L. Baldwin, Monomolecular condensation oflambda-DNA induced by cobalt hexamine, Biopolymers, 1983,22, 1595–1620.
71 D.Matulis, I. Rouzina and V. A. Bloomfield, Thermodynamics ofDNA binding and condensation: isothermal titration calorimetryand electrostatic mechanism, J. Mol. Biol., 2000, 296, 1053–1063.
72 D. Baigl and K. Yoshikawa, Dielectric control of counterion-induced single-chain folding transition of DNA, Biophys. J., 2005,88, 3486–3493.
73 N. V. Hud, Double-stranded DNA organization in bacteriophageheads-an alternative toroid-based model, Biophys. J., 1995, 69,1355–1362.
74 J. A. Schellman and N. Parthasarathy, X-ray diffraction studieson cation-collapsed DNA, J. Mol. Biol., 1984, 175, 313–329.
75 N. V. Hud and K. H. Downing, Cryoelectron microscopy ofl-phage DNA condensates in vitreous ice: The fine Structure ofDNA toroids, Proc. Natl. Acad. Sci. U. S. A., 2001, 98, 14925–14930.
76 Y. Levin, Electrostatic correlations: from plasma to biology, Rep.Prog. Phys., 2002, 65, 1577–1632.
77 M. D. Frank-Kamenetskii, V. V. Anshelevich andA. V. Lukashin, Polyelectrolyte model of DNA, Usp. Fiz. Nauk.,1987, 151, 595–618.
78 A. Yu. Grosberg, T. T. Nguyen and B. I. Shklovskii, The physicsof charge inversion in chemical and biological systems, Rev. Mod.Phys., 2002, 74, 329–345.
79 E. Raspaud, M. Olvera de la Cruz, J. L. Sikorav and F. Livolant,Precipitation of DNA by polyamines: a polyelectrolyte behavior,Biophys. J., 1998, 74, 381–393.
80 M. Saminathan, T. Antony, A. Shirahata, L. H. Sigal, T. Thomasand T. J. Thomas, Ionic and structural specificity effects ofnatural and synthetic polyamines on the aggregation andresolubilization of single-double- and triple-stranded DNA,Biochemistry, 1999, 38, 3821–3830.
81 C. Hsiao, M. Tannenbaum, H. Van Deusen, E. Hershkovitz,G. Perng, A. Tannenbaum and L. D. Williams, in Nucleic AcidMetal Interactions, ed. N. Hud, RSC, London, 2008, pp.1–35.
82 J. Ruiz-Chica, M. A. Medina, F. Sanchez-Jimenez andF. J. Ramırez, On the interpretation of Raman spectra of1-aminooxy-spermine/DNA complexes, Nucleic Acids Res.,2004, 32, 579–589.
83 A. Ouameur and H.-A. Tajmir-Riahi, Structural Analysis ofDNA Interactions with Biogenic Polyamines and Cobalt(III)-hexamine Studied by Fourier Transform Infrared and CapillaryElectrophoresis, J. Biol. Chem., 2004, 279, 42041–42054.
84 E. Wemmer, K. S. Srivenugopal, B. R. Reid and D. R. Morris,Nuclear magnetic resonance studies of polyamine binding to adefined DNA sequence, J. Mol. Biol., 1985, 185, 457–459.
85 B. I. Shklovskii, Wigner crystal model of counterion inducedbundle formation of rod-like polyelectrolytes, Phys. Rev. Lett.,1999, 82, 3268–3271; B. I. Shklovskii, Screening of a macroion bymultivalent ions: Correlation-induced inversion of charge, Phys.Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1999,60, 5802–5811.
86 T. T. Nguyen, A. Y. Grosberg and B. I. Shklovskii, Macroions insalty water with multivalent ions: Giant inversion of charge, Phys.Rev. Lett., 2000, 85, 1568–1571; T. T. Nguyen, I. Rouzina andB. I. Shklovskii, Reentrant condensation of DNA induced bymultivalent counterions, J. Chem. Phys., 2000, 112, 2562–2568.
87 J. Yang and D. C. Rau, Incomplete ion dissociation underlies theweakened attraction between DNA helices at high spermidineconcentrations, Biophys. J., 2005, 89, 1932–1940.
88 S. A. Allison, J. C. Herr and J. M. Schurr, Structure of viral DNAcondensed by simple triamines: A light-scattering and electron-microscopy study, Biopolymers, 1981, 20, 469–488.
89 V. A. Bloomfield, D. M. Crothers and I. Tinoco Jr, Nucleic Acids:Structures, Properties and Functions, University Science Books,Sausalito, California, 2000.
90 F. Oosawa, Polyelectrolytes, Marcel Dekker, New York, 1971.91 R. Marquet and C. Houssier, Thermodynamics of cation-induced
DNA condensation, J. Biomol. Struct. Dyn., 1991, 9, 159–167.92 J. J. Arenzon, J. F. Stilck and Y. Levin, Simple Model for
Attraction between. Like-Charged Polyions, Eur. Phys. J. B,1999, 12, 79–82.
93 B.-Y. Ha and A. J. Liu, Counterion-mediated, non-pairwiseadditive attractions in bundles of like-charged rods, Phys. Rev.E: Stat.Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1999, 60,803–813.
94 I. Rouzina and V. A. Bloomfield, Macroion attraction due toelectrostatic correlation between screening counterions. I: mobilesurface-adsorbed ions and diffuse ion cloud, J. Phys. Chem., 1996,100, 9977–9989.
95 M. S. Loth and B. I. Shklovskii, Non-mean-field screening bymultivalent counterions, J. Phys.: Condens. Matter, 2009, 21, 424104.
96 E. P. Wigner, On the interaction of electrons in metals, Phys.Rev., 1934, 46, 1002–1011.
97 B. Tanatar and D. M. Ceperley, Ground state of the two-dimensional electron gas, Phys. Rev. B: Condens. Matter, 1989,39, 5005–5016.
98 T. Ando, A. B. Fowler and F. Stern, Electronic properties of two-dimensional systems, Rev. Mod. Phys., 1982, 54, 437–672.
99 M. A. Vorotyntsev and S. N. Ivanov, Statistical mechanicsof an ion ensemble adsorbed at a meteal/dielectric interface,Zh. Exper.Theor. Fiz., 1985, 88, 1729–1737.
100 M. E. Leunissen, A. van Blaadern, A. D. Hollingsworth,M. T. Sullivan and P. M. Chaikin, Electrostatics at the oil–waterinterface, stability, and order in emulsions and colloids,Proc. Natl. Acad. Sci. U. S. A., 2007, 104, 2585–2590.
101 N. D. Mermin and H. Wagner, ‘‘Absence of Ferromagnetism orAntiferromagnetism in One- or Two-Dimensional IsotropicHeisenberg Models’’, Phys. Rev. Lett., 1966, 17, 1133–1136.
102 A. Klein, L. J. Landau and D. S. Shucker, On the absence ofspontaneous breakdown of continuous symmetry for equilibriumstates in two dimensions’’, J. Stat. Phys., 1981, 26, 505–512.
103 M. Deserno and C. Holm, Theory and simulations of rigidpolyelectrolytes, Mol. Phys., 2002, 100, 2941–2956.
104 A. G. Cherstvy, A. A. Kornyshev and S. Leikin, Temperaturedependent DNA condensation triggered by rearrangements ofadsorbed ions, J. Phys. Chem. B, 2002, 106, 13362–13369.
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
Phys. Chem. Chem. Phys. This journal is c the Owner Societies 2010
105 G. S. Manning, The molecular theory of polyelectrolyte solutionswith applications to the electrostatic properties of polynucleo-tides, Q. Rev. Biophys., 1978, 11, 179–246.
106 K Bryson and R. J. Greenall, Binding sites of the polyaminesputrescine, cadaverine, spermidine and spermine on A- andB-DNA located by simulated annealing, J. Biomol. Struct.Dyn., 2000, 18, 393–412.
107 B. G. Feuerstein, N. Pattabiraman and L. J. Marton, Spermine-DNA interactions: a theoretical study, Proc. Natl. Acad. Sci.U. S. A., 1986, 83, 5948–5952.
108 A. P. Lyubartsev and A. Laaksonen, Molecular DynamicsSimulations of DNA in Solution with Different Counterions,J. Biomol. Struct. Dyn., 1998, 16, 579–587.
109 N. Korolev, A. P. Lyubartsev, L. Nordenskiold andA. Laaksonen, Spermine: An ‘‘invisible’’ component in thecrystals of B-DNA: A grand canonical Monte Carlo and MolecularDynamics simulation study, J. Mol. Biol., 2001, 308, 907–917.
110 N. Korolev, A. P. Lyubartsev, A. Laaksonen and L. Nordenskiold,On the competition between water, sodium ions and spermine inbinding to DNA. A molecular dynamics computer simulationstudy, Biophys. J., 2002, 82, 2860–2875.
111 N. Korolev, A. P. Lyubartsev, A. Laaksonen and L. Nordenskiold,A molecular dynamics simulation study of polyamine and sodiumDNA. Interplay between polyamine binding and DNA structure,Eur. Biophys. J., 2004, 33, 671–682.
112 Yuhua Cheng, Nikolay Korolev and Lars Nordenskiold,Similarities and differences in interaction of K+ and Na+ withcondensed ordered DNA. A molecular dynamics computersimulation study, Nucleic Acids Res., 2006, 34, 686–696.
113 L. Dai, Y. Mu, L. Nordenskiold, A. Lapp and J. R. C. van derMaarel, Charge Structure and Counterion Distribution inHexagonal DNA Liquid Crystal, Biophys. J., 2007, 92, 947–958.
114 A. Falaschi, Similia similibus: pairing of homologous chromo-somes driven by the physicochemical properties of DNA, HFSPJ., 2008, 2, 257–261.
115 G. Baldwin, N. J. Brooks, R. Robson, A. Wynveen, A. Goldar,S. Leikin, J. M. Seddon and A. A. Kornyshev, DNA doublehelices recognize mutual sequence homology in a protein-freeenvironment, J. Phys. Chem. B, 2008, 112, 1060–1064.
116 L. D. Landau and E. M. Lifshitz, Electrodynamics of continuousmedia, Course of Theoretical Physics, Oxford, Pergamon,2nd edn, 1984, vol. 8.
117 B. A. Todd, V. A. Parsegian, A. Shirata, T. J. Thomas andD. C. Rau, Attractive forces between cation condensed doublehelices, Biophys. J., 2008, 94, 4775–4782.
118 Yu. A. Izyumov and Yu. N. Skryabin, Statistical Theory ofMagnetically Ordered Systems, Consultants Bureau, New York,1988, p. 295.
119 H. M. Harreis, A. A. Kornyshev, C. N. Likos, H. Loewen andG. Sutmann, Phase behavior of columnar DNA assemblies, Phys.Rev. Lett., 2002, 89, 18303–18207.
120 A. Wynveen, D. J. Lee and A. A. Kornyshev, Statisticalmechanics of columnar DNA assemblies, Eur. Phys. J. E, 2005,16, 303–318.
121 A. A. Kornyshev and S. Leikin, Electrostatic Interaction betweenlong, rigid helical molecules at all mutual angles, Phys. Rev. E:Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 2000, 62,2576–2596.
122 A. A. Kornyshev, S. Leikin and S. V. Malinin, Chiral interactionand cholesteric liquid crystals of DNA, Eur. Phys. J. E, 2002, 7,83–93.
123 C. B. Stanley, H. Hong and H. H. Strey, DNA cholesteric pitch asa function of density and ionic strength, Biophys. J., 2005, 89,2552–2557.
124 A. A. Kornyshev, D. J. Lee, S. Leikin, A. Wynveen andS. Zimmerman, Direct observation of azimuthal correlationsbetween DNA in hydrated aggregates, Phys. Rev. Lett., 2005,95(14), 148102–4.
125 L. Rudd, D. J. Lee and A. A. Kornyshev, The role of electro-statics in the B to A transition of DNA: from solution toassembly, J. Phys.: Condens. Matter, 2007, 19, 416103.
126 W. Gilbert, The RNA World, Nature, 1986, 319, 618.127 C. Woese, The Genetic Code, Harper & Row, New York, 1968.128 L. E. Orgel, The origin of life on the Earth, Sci. Am., 1994, 271,
76–83.
129 L. E. Orgel, The origin of life—a review of facts and speculations,Trends Biochem. Sci., 1998, 23, 491–495.
130 Ed. R. F. Gesterland, T. Chech and J. F. Atkins, The RNAWorld, III Edition, 2006, Cold Spring Harbour LaboratoryPress, New York.
131 W. M. Gelbart and C. M. Knobler, Pressurized Viruses, Science,2009, 323, 1682–1683.
132 A. Evilevitch, L. Lavelle, C. M. Knobler, E. Raspaud andW. M. Gelbart, Osmotic pressure inhibition of DNA ejectionfrom phage, Proc. Natl. Acad. Sci. U. S. A., 2003, 100, 9292–9295.
133 P. K. Purohit, J. Kondev and R. Phillips, Mechanics of DNApackaging in viruses, Proc. Natl. Acad. Sci. U. S. A., 2003, 100,3173–3178.
134 W. K. Olson and V. B. Zhurkin, Modeling DNA deformations,Curr. Opin. Struct. Biol., 2000, 10, 286–297.
135 A. A. Kornyshev, D. J. Lee, A. Wynveen and S. Leikin, X-raydiffraction from real DNA (in preparation), 2010.
136 A. A. Kornyshev and A. Wynveen, The homology recognitionwell as an innate property of DNA structure, Proc. Natl. Acad.Sci. U. S. A., 2009, 106, 4683–4688.
137 D. J. Lee and A. A. Kornyshev, Homology recognition funnel,J. Chem. Phys., 2009, 131, 155104; D. J. Lee andA. A. Kornyshev, Erratum, J. Chem. Phys., 2009, 131, 219901.
138 At interaxial separations shorter than R* one can obtain theexpression for the force by differentiation of eqn B.5 of ref. 136with minimization of the optimal azimuthal angle as describedthere [note a rather obvious misprint in eqn B5: the argument ofthe second cosine should have a factor of 2, lost in print]. Theresulting expression is too cumbersome to be drawn here.
139 D. J. Lee, S. Leikin and A. Wynveen, Fluctuations and inter-actions of semi-flexible polyelectrolytes in columnar assemblies,J. Phys.: Condens. Matter, 2010, 22, 072202.
140 D. J. Lee, A. Wynveen, A. A. Kornyshev and S. Leikin, Undula-tions enhance the effect of helical structure on DNA interactionssubmitted to, J. Phys. Chem. B, 2010, in press.
141 R. R. Sinden, DNA Structure and Function, Academic Press, SandDiego, 1994.
142 L. Postow, N. J. Crisona, B. J. Peter, C. D. Hardy andN. R. Cozzarelli, Topological challenges to DNA replication:Conformations at the fork, Proc. Natl. Acad. Sci. U. S. A., 2001,98, 8219–8226.
143 Z. Bryant, M. D. Stone, J. Gore, S. B. Smith, N. R. Cozzarelli andC. Bustamante, Structural transitions and elasticity from torquemeasurements on DNA, Nature, 2003, 424, 338–341.
144 B. J. Peter, J. Arsuaga, A. M. Breier, A. B. Khodursky,P. O. Brown and N. R. Cozzarelli, Genomic transcriptionalresponse to loss of chromosomal supercoiling in Escherichia coli,Genome Biol., 2004, 5, R87.
145 L. S. Shlyakhtenko, L. Miloseska, V. N. Potaman, R. R. Sinden andY. I. Lyubchenko, Intersegmental interactions in supercoiled DNA:atomic force microscopy study, Ultramicroscopy, 2003, 97, 263–270.
146 Z. R. Liu, R. W. Deibler, H. S. Chan and L. Zechiedrich, Thewhy and how of DNA unlinking, Nucleic Acids Res., 2009, 37,661–671.
147 M. E. Fisher, The story of coulombic criticality, J. Stat. Phys.,1994, 75, 1–36.
148 D. J. Lee, Charge renormalization of helical macromolecules,J. Phys.: Condens. Matter, 2010, in press.
149 F. Oosawa, A theory on the effect of low molecular salts on thedissociation of linear polyacids, Biopolymers, 1968, 6, 134–144.
150 M. Kanduc, J. Dobnikar and R. Podgornik, Counterion-mediated electrostatic interactions between helical molecules, SoftMatter, 2009, 5, 868–877.
151 Spooky attraction of DNA from a distance, , New Scientist, 2008,2641, 15; G. Crabtree, Paired pairs, Nature, 2008, 451, 609;P. Szuromi, DNA’s self regard, Science, 2008, 319, 879; Seekingrecognition, , Biopolymers, 2008, 89, 4; C. Q. Choi, Double-HelixDouble Up, , Scientific American, 2008, April, 30.
152 S. Inoue, S. Sugiyama, A. A. Travers and T. Ohyama,Self-assembly of double-stranded DNA molecules at nanomolarconcentrations, Biochemistry, 2007, 46, 164–171.
153 Z. Liu, H. Zhou, G. Wei, Y. Song and L. Wang, Immobilizationand condensation of DNA with 3-aminopropyltriethoxysilanestudied by atomic force microscopy, J. Microsc., 2005, 218,233–239.
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online
This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.
154 C. Danilovicz, C. H. Lee, K. Kim, K. Hatch, V. W. Coljee,N. Kleckner and M. Prentiss, Single molecule detection of direct,homologous, DNA/DNA pairing, Proc. Natl. Acad. Sci. U. S. A.,2009, 106, 19824–19829.
155 V. A. Parsegian, Harness the hubris: Useful things physicists coulddo in biology, Phys. Today, 1997, 50, 23–27; A. A. Kornyshev,How physics can inspire biology, Physics World, 2009, 22, 16–17.
156 W. Cochran, F. H. C. Crick and V. Vand, The structure ofsynthetic polypeptides. I. The transform of atoms on a helix, ActaCrystallogr., 1952, 5, 581–586.
157 R. E. Franklin and R. G. Gosling, Molecular configuration insodium thymonucleate, Nature, 1953, 171, 740–741;R. E. Franklin and R. G. Gosling, Evidence for 2-chain helix incrystalline structure of sodium deoxyribonucleate, Nature, 1953,172, 156–157.
158 J. D. Watson and F. H. C. Crick, Molecular structure of nucleicacids – a structure for deoxyribose nucleic acid,Nature, 1953, 171,737–738.
159 J. D. Watson and F. H. C. Crick, Genetical implications of thestructure of deoxyribonucleic acid, Nature, 1953, 171, 964–967.
160 M. D. Frank-Kamenetskii, Unravelling DNA: The Most ImportantMolecule of Life, VCH, Cambridge, 1993.
161 Sequence-dependent variations in the local helical pitch H willdisrupt such register in juxtaposition of molecules with differentsequences but not in juxtaposition of molecules with the samesequences as illustrated in Fig. 5.
162 D. C. Rau, B. Lee and V. A. Parsegian, Measurement of therepulsive force between polyelectrolyte molecules in ionicsolution: hydration forces between parallel DNA double helices,Proc. Natl. Acad. Sci. U. S. A., 1984, 81, 2621–2625.
Dow
nloa
ded
by U
nive
rsity
of
Cal
ifor
nia
- Sa
nta
Bar
bara
on
04 M
ay 2
011
Publ
ishe
d on
10
Aug
ust 2
010
on h
ttp://
pubs
.rsc
.org
| do
i:10.
1039
/C00
4107
FView Online