Physics of DNA: unravelling hidden abilities encoded …...between two DNA molecules—is one of the key processes in living matter. It warrants genetic diversity and speeds up evolution;

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

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http://dx.doi.org/10.1039/c004107f

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

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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).

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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

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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

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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

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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

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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

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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,

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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.

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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.

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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

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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.

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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

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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.

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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

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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

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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.

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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

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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)

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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

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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.

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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

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