Small DNA circles can occur in Nature, for example as protein-constrained loops, and can be synthesized by a number of methods. Such small circles provide tractable systems for the study of the structure, thermodynamics and molecular dynamics of closed-circular DNA. In the present article, we review the occurrence and synthesis of small DNA circles, and examine their utility in studying the properties of DNA and DNA–protein interactions. In particular, we highlight the analysis of small circles using atomistic simulations.
Although the role of DNA in genetic information storage is well established, there are many complexities that remain to be understood about the regulation of its action and the interaction of DNA with other biomolecules. One prominent area of complexity involves the topological features of DNA: supercoiling, catenation and knotting, which arise from the double-helical structure. Transcription leads to the build-up of positive and negative supercoiling in front of and behind RNA polymerase, whereas replication similarly involves the induction of positive supercoiling, and ultimately leads to the catenation (linking) of daughter replicons [1,2]. Other processes, including recombination, can lead to the formation of knotted DNA. The DNA topoisomerase enzymes have evolved to control these properties; they unlink catenanes and knots and modulate supercoiling, ubiquitous features of DNA in most organisms. Supercoiling is known to act as a global regulator of gene expression, whereas negative supercoiling is a prerequisite for the initiation of many processes that involve the unwinding of the DNA helix, since negatively supercoiled DNA has a higher free energy than relaxed DNA, facilitating the unwinding of the helix in response to protein binding.
DNA topology is formally a property of closed double-stranded DNA circles, although the same effects can be manifested in looped regions and domains of circular or linear chromosomes. Our understanding of DNA topology and its effects has been based predominantly on the use of model circular DNA molecules, such as circular bacterial plasmids, and virus and bacteriophage genomes. Analysis of the behaviour of circular viral genomes led to the appreciation that circular DNA could adopt a supercoiled conformation when closed in an underwound or overwound state . The concepts of linking number (Lk), twist (Tw) and writhe (Wr) were adopted from mathematics to quantify this effect and describe the geometrical properties of supercoiled DNA. The linking number of a non-supercoiled, or ‘relaxed’, DNA circle is an integer corresponding to the number of double-helical turns in a DNA molecule of that size. Underwinding of DNA corresponds to a reduction in linking number (negative ∆Lk); this perturbation leads to the introduction of writhe, corresponding to a coiling of the DNA helix around itself, and an untwisting of the helix itself, described by the twist. Crucially, the sum of the twist and writhe of a closed circular DNA sums to its linking number (Lk=Tw+Wr).
An understanding of these effects has been facilitated by the use of plasmid DNAs of a few thousand base pairs. However, their relatively large size presents difficulties, for example in analysing structure, thermodynamics and MD (molecular dynamics) of closed-circular DNA; smaller circles are more appropriate for such analyses.
Small DNA circles and loops occur in a number of situations in vivo. For example, kDNA (kinetoplast DNA), the mitochondrial DNA of trypanosomatid parasites, consists of a network of several thousand so-called minicircles (0.5–10 kb) and a few dozen maxicircles (20–40 kb) . Another example is protein-constrained loops , which are involved in the regulation of cellular processes including transcription, replication and recombination, allowing distal regions of DNA to contact and affect each other. There are many proteins whose action involves looping, including the Lac repressor, λ repressor and the retinoid X receptor . The looped DNA is generally a few hundred base pairs in length, and DNA minicircles of these sizes provide plausible models for the behaviour of these loops; supercoiling is a key component of their behaviour.
There are several plausible definitions of DNA minicircles. The term minicircle, as applied to kDNA (above), encompasses circles up to 10 kb in size, larger than many of the plasmids that have been used to model the behaviour of much longer DNA, thus this definition is probably inappropriate. Most readily available plasmids are above 2 kb in size, and the smallest known are approximately 1 kb, such as the cryptic miniplasmid from Thermotoga sp. (846 bp)  and the small plasmid vector ΠVX (902 bp) . A minicircle could thus be defined as a circular DNA molecule that is smaller than a typical plasmid, for which specific synthetic methods must be adopted, such as the ligation of a linear DNA into a circle.
The ligation of linear DNA reveals interesting properties as the size of the DNA decreases below 1000 bp. Generally, for a flexible molecule, the efficiency of closure into a ring increases with decreasing size, as the probability of the ends coming together increases. However, the bending and torsional stiffness of the DNA helix dominate at smaller sizes, and the ligation probability decreases sharply and becomes periodic with length below ~500 bp , as the energy required to bend the DNA and to twist the ends into register becomes significant (Figure 1). Thus a good physical definition of a DNA minicircle would be under 500 bp.
The ligation probability of small DNA circles
A rigorous definition of DNA minicircles is probably not necessary, but from these considerations, we suggest that minicircles are molecules less than 500–1000 bp in size that cannot replicate independently.
Synthesis of small DNA circles
The obvious method for making DNA minicircles is by ligation of linear molecules, but, as illustrated in Figure 1, this is inefficient for circles <500 bp. However, ligation efficiency can be enhanced by the presence of certain proteins and specific DNA sequences. Minicircles in the range ~100–250 bp can be made using DCLCE (DNA circularization by long cohesive ends) , a strategy involving annealing and ligating long synthetic single-stranded DNAs. This method can be refined further using the PCR-based methodology LAMA (ligase-assisted minicircle accumulation) to optimize the quality and yield of the products . An earlier study demonstrated that small DNA fragments of >98 bp cyclize rapidly by ligation in the presence of the histone-like protein HU, thus indicating that the protein can mediate tight curvature of bound DNA molecules . Similarly, Kahn and Crothers  reported the preparation of minicircles of 150–166 bp via T4 ligase-mediated DNA cyclization following the binding of CAP (catabolite activator protein). It is well known that phased A-tracts in DNA lead to curvature  and this can be exploited to facilitate ligation of small DNA circles; indeed, circles as small as 105 bp have been prepared using this method .
A double-stranded circle of 42 bp has been constructed using synthetic oligonucleotides which form a ‘dumb-bell-shaped’ structure that is able to undergo single-strand ligation ; in principle, such circles can be synthesized in amounts suitable for high-resolution structural studies. Brown and colleagues have developed a synthetic approach for the preparation of minicircles of <100 bp with high efficiency . The methodology uses a click ligation reaction for the intramolecular circularization of a single-stranded oligonucleotide that is used as a template for the synthesis of a covalently closed double-stranded circle. In this case, the site of ligation consists of an unnatural DNA backbone linkage. Site-specific recombination reactions have also been used to excise small circles from plasmid substrates. In vitro loxP-Cre reactions were used to produce circles of 200–400 bp . More recently, λ-Int-mediated recombination in vivo in Escherichia coli  has also been used; this is particularly well suited to circles in the size range 250–700 bp. Fogg et al.  have shown that milligram quantities of supercoiled minicircle DNA can be made using this method. The λ-Int recombination technology is not ideal for circles <250 bp.
Use of small DNA circles for analysing the physical properties of DNA
The probability of ring closure for a circular polymer is frequently described in terms of the J factor , which is equivalent to the concentration of one end of the polymer in a location and conformation able to join the other. This formulation was used by Shore and Baldwin  in experimentally determining the ligation probability of short DNA fragments, demonstrating the periodic dependence on length (see above) in the range 235–255 bp (Figure 1). This periodicity corresponds to the helical repeat of the DNA as, if the length is not a multiple of the helical repeat, torsional energy is required to align the ends for ligation. These experiments allow the determination of the bending and torsional rigidity and the helical repeat of DNA. This approach was extended to include larger circle sizes by Taylor and Hagerman  and the effect was modelled theoretically using Monte Carlo simulations by Shimada and Yamakawa , and Levene and Crothers .
However, more recently, Cloutier and Widom [22,23] used cyclization measurements on very short DNA loops (~90 bp), to propose that DNA is up to 10000 times more flexible than would be predicted by the simple polymer theories used in the simulations above (Figure 1). These controversial results were later disputed in a biochemical study by Du et al.  using circles of similar sizes (Figure 1), and in a complementary cryo-EM (electron microscopy) study of the shape of the circles by Demurtas et al. . In addition, more fine-grained simulations by Czapla et al. , which were able to take account of sequence-dependent effects, including intrinsic bending and anisotropic flexibility of the DNA, suggested that these effects might contribute to the apparent enhanced flexibility of some of these short sequences, which contain nucleosome-positioning sequences, with some repetitive motifs. In fact, it was already known that intrinsic bending and anisotropy of DNA can facilitate cyclization of very short DNA circles; Ulanovsky et al.  efficiently produced circles of 105–168 bp by ligation of intrinsically bent 21-mers , and Drew and Travers  produced a 169 bp circle containing nucleosome-positioning sequences, whose anisotropic flexibility caused the circle to have a defined inside/outside conformation. This debate also inspired a number of atomistic MD studies (see below), which have addressed the detailed structure of very small circles. In addition, recent single-molecule experiments involving a fluorescence-based protein-free assay for analysing cyclization  showed that DNA molecules of <100 bp had a much higher bendability than had been predicted from traditional worm-like chain models.
Shore and Baldwin  also investigated the formation of supercoils in DNA circles of 235–255 bp by ethidium bromide titration and topoisomer analysis, and Horowitz and Wang  used measurements of the ratios of topoisomers in circles of closely similar size to determine the free energy of supercoiling down to 210 bp. They found that the free energy of supercoiling per unit length, which is independent of length above approximately 2 kb, increases almost 3-fold as the size decreases to 200 bp. They suggested that, at these small sizes, low levels of supercoiling are partitioned almost entirely into twist; the energy of writhing increases at small circle sizes, since the energy required to bend the DNA into a writhed figure-of-eight structure is increased when the DNA is already tightly bent. In fact, simulations by Le Bret [30,31] had already revealed this behaviour, and suggested that two supercoils (corresponding to a high density of supercoiling at these sizes) would be required to produce a figure-of-eight writhed molecule. This contention was later confirmed experimentally by Bednar et al. , who showed, using cryo-EM and gel electrophoresis, that a 178 bp circle forms an interwound figure-of-eight conformation only at a ∆Lk of −2, even under conditions where the ionic strength of the solution allows the close approach of the two helices at the crossover point of the figure-of-eight. MD simulations have also been used to analyse this behaviour in more detail (see below).
Atomistic simulations of small DNA circles
The interpretation of biophysical experiments on DNA circles is hampered by the lack of atomistically detailed information about their structure. However, this can be obtained using computer simulations of DNA minicircles performed at the atomic level. Such simulations are particularly expensive computationally, because even the smallest circles immersed in solvent molecules require simulation cells containing many more atoms than the short linear DNA sequences more commonly studied computationally . The debate surrounding the Cloutier and Widom [22,23] measurements (discussed above) stimulated an atomistic MD study by Lankas et al. , which showed that DNA bending can induce the formation of distinct ‘kinks’ in the DNA that might explain its enhanced flexibility. Biochemical evidence for the formation of kinks in smaller (<70 bp) circles was subsequently obtained by Du et al.  who systematically probed circles of sizes between 63 and 205 bp for disruptions using endonucleases that digest non-canonical DNA. However, kinks were not detected due to bending stress alone in circles of ~90 bp. These experiments also showed that supercoiling very small DNA circles can lead to the formation of regions of denatured DNA. Subsequently, Mitchell et al.  performed a series of atomistic MD calculations for these circles to obtain detailed atomic-level insight into the nature of these large and small disruptions in the DNA structure due to supercoiling. Their calculations revealed a rich repertoire of non-canonical DNA structures that occur at different levels of torsional stress, including bubbles of denatured DNA at the highest levels of specific linking difference (σ) investigated (σ=−0.16). Similar results showing the denaturation of DNA in response to negative supercoiling were also obtained for linear DNA sequences . Clearly, experiments and computer simulations on small DNA circles are able to reveal that, given sufficiently high levels of bending or torsional stress, the DNA exhibits behaviour that cannot be captured by simple models that treat the DNA as a homogeneous elastic rod.
Atomistic MD simulations have also been used to determine the global shape of supercoiled DNA, in particular how ∆Lk is partitioned between twist and writhe, for circles of sizes between 90 and 178 bp and at different salt concentrations . The calculations reveal a distinct asymmetry in the response of DNA to over- and under-twisting. Although circles of only 90 bp were able to form significantly writhed loops when positively twisted, negatively writhed DNA was only observed for circles of more than 148 bp and when the simulations were performed using high salt conditions. This mechanical asymmetry reflects the internal structure of duplex DNA. Positive torsion forces the tightly stacked DNA bases to clash, which then results in an abrupt buckling transition into a writhed structure. Conversely, negative torsion pulls the stacked bases apart, which weakens the double-helical structure of the duplex, and results either in denaturation or negative writhing depending upon the levels of superhelical stress imposed. The salt concentration also influences the partitioning of twist and writhe, because higher salt conditions screen the electrostatic repulsion within the charged DNA backbone more effectively, and thereby allow the DNA to adopt compact conformations. More recently, we have investigated twist–writhe partitioning using atomistic MD simulations for larger (260 bp) DNA circles, as shown in Figure 2.
Molecular dynamics simulations of circular DNA
It is surprisingly difficult to design circles in silico that have specific linking differences precisely matching those in the laboratory, because of the uncertainty in the exact value of the local twist at the level of the individual DNA base steps, and imperfections in the parameterization of MD forcefields. Cyclization measurements on random DNA sequences provide arguably the most accurate experimental method for determining the DNA helical repeat, which is averaged over hundreds of base pairs, giving values between 10.5 and 10.6 bp (or 34.3–34.0°·bp−1 respectively) . A single-molecule experiment has also given a value of 10.6 bp or 34.0°·bp−1 . However, structural studies on short DNA fragments, which typically contain only one or two helical turns, gave an average local helical repeat of 10.4 bp (34.5°·bp−1) and 10.1 bp (35.5°·bp−1) for NMR and X-ray crystal structures respectively . An additional uncertainty arises due to the precise manner in which twist is defined and applied to the various types of experimental data . A detailed analysis of the DNA sequences available within the PDB has shown that the local twist is also highly sequence-dependent, with average values as low as 28.8°·bp−1 for AG steps, but as high as 41.8°·bp−1 for GA steps . Current atomistic MD forcefields for nucleic acids  report slightly lower average local twists than might be expected from the values obtained from cyclization measurements . For example, using a new forcefield which includes a modification to the Chi backbone angle , we obtain an average local helical repeat of 10.8 bp (33.2°·bp−1). However, for a 260 bp circle, we estimate a value of 10.7 bp (33.5°·bp−1) if the sequence dependence of the local twist of the forcefield is taken into account, which is slightly different from the value that the forcefield predicts for a DNA sequence in which each base step occurs with equal frequency. In general, the fact that the local twist is so highly sequence-dependent implies that the smaller the minicircle, the more likely the DNA will have a relaxed helical repeat that deviates from that measured for random DNA by cyclization, which reports a global average measured over a large number of base pair steps, each of which may have a significant variation in local twist. We predict that this may well be detected experimentally for DNA sequences with particularly anomalous twists (e.g. long stretches of alternating AG steps). If no such deviations are detected, then the question of how the inhomogeneous local helical twist values observed in X-ray and NMR experiments (and mimicked by MD forcefields) give rise to such a well-defined macroscopic value (34.3–34.0°·bp−1) will need to be addressed by the community.
Small DNA circles as probes for protein–DNA interactions
Small DNA circles have been used in investigations of DNA–protein interactions. Prunell and co-workers have utilized small circles as model systems for chromatin reconstitution [45,46], enabling a detailed analysis of nucleosome characteristics and linking number changes. Bates and Maxwell  used DNA circles between 116 and 427 bp formed by loxP-Cre recombination  to probe the energetic and steric limits of DNA gyrase, which uses the free energy of ATP hydrolysis to introduce negative supercoiling into circular DNA, with a ∆Lk of −2. They showed that gyrase was capable of introducing two negative supercoils into a 174 bp circle, in a step estimated to require much of the free energy available to the enzyme from ATP hydrolysis, but that the enzyme was able to change the linking number of a circle as small as 116 bp under conditions where the energetic requirement had been relaxed. This corresponds to approximately the length of the DNA known to be wrapped around the enzyme in the gyrase–DNA complex. Further work showed that single-cycle supercoiling by gyrase in the presence of a non-hydrolysable ATP analogue was not detectable with a 174 bp substrate , showing that the efficiency of energy coupling was very low under these conditions.
Conclusions and perspectives
Given the immense significance of DNA to our understanding of biological systems, it is of paramount importance that we gain a better understanding of its structure and dynamics. The availability of small DNA circles presents opportunities for increasing our understanding of the structural and dynamic diversity of DNA structure and topology and how its properties modulate its roles in biological processes.
Topological Aspects of DNA Function and Protein Folding: An Independent Meeting held at the Isaac Newton Institute for Mathematical Sciences, Cambridge, U.K., 3–7 September 2012, as part of the Isaac Newton Institute Programme Topological Dynamics in the Physical and Biological Sciences (16 July–21 December 2012). Organized and Edited by Andrew Bates (University of Liverpool, U.K.), Dorothy Buck (Imperial College London, U.K.), Sarah Harris (University of Leeds, U.K.), Andrzej Stasiak (University of Lausanne, Switzerland) and De Witt Sumners (Florida State University, U.S.A.).
We thank John Ward (University College London, London, U.K.) for helpful advice.
This work was supported by the Biotechnology and Biological Sciences Research Council (BBSRC) [grant number BB/I019294/1], and the BBSRC and the John Innes Foundation [grant number BB/J004561/1].