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Requirements for DNA-Bridging Proteins to Act as Topological Barriers of the Bacterial Genome

Bacterial genomes have been shown to be partitioned into several-kilobase-long chromosomal domains that are topologically independent from each other, meaning that change of DNA superhelicity in one domain does not propagate to neighbors. Both in vivo and in vitro experiments have been performed to question the nature of the topological barriers at play, leading to several predictions on possible molecular actors. Here, we address the question of topological barriers using polymer models of supercoiled DNA chains that are constrained such as to mimic the action of predicted molecular actors. More specifically, we determine under which conditions DNA-bridging proteins may act as topological barriers. To this end, we developed a coarse-grained bead-and-spring model and investigated its properties through Brownian dynamics simulations. As a result, we find that DNA-bridging proteins must exert rather strong constraints on their binding sites; they must block the diffusion of the excess of twist through the two binding sites on the DNA molecule and, simultaneously, prevent the rotation of one DNA segment relative to the other one. Importantly, not all DNA-bridging proteins satisfy this second condition. For example, single bridges formed by proteins that bind DNA nonspecifically, like H-NS dimers, are expected to fail with this respect. Our findings might also explain, in the case of specific DNA-bridging proteins like LacI, why multiple bridges are required to create stable independent topological domains. Strikingly, when the relative rotation of the DNA segments is not prevented, relaxation results in complex intrication of the two domains. Moreover, although the value of the torsional stress in each domain may vary, their differential is preserved. Our work also predicts that nucleoid-associated proteins known to wrap DNA must form higher protein-DNA complexes to efficiently work as topological barriers.


Statement of significance:
Although the question of independent topological domains in bacterial genomes has been addressed for nearly two decades, the nature of topological barriers is still elusive. Most plausible models for the formation of topological barriers are actively transcribing RNAP and the formation of DNA loops by certain nucleoid proteins. In the present paper, we focus on this latter mechanism and establish under which conditions DNA-bridging proteins may serve as topological barriers. We show that, contrary to popular belief, formation of a loop is not sufficient and that proteins must additionally block the diffusion of twist and prevent the relative rotation of DNA segments. These findings set strong theoretical limits on the ability of bridging proteins to create topologically independent genomic domains.

INTRODUCTION
The genetic information of most bacteria is encoded in a circular DNA molecule comprising up to several millions of base pairs (bp). Such closed molecules are topologically constrained, because they lack free ends capable of rotating and releasing the torsional stress (1). In vivo alterations of the torsional state of circular DNA is mediated by the recruitment of enzymes called topoisomerases, which can either increase or decrease the twist of the double helix by opening transiently one or two strands (1). The net result of the action of all types of topoisomerases is that the genomic DNA of most bacteria is significantly undertwisted (negatively supercoiled) and winds about itself to transfer part of the torsional stress to the bending degrees of freedom, thereby forming plectonemes (1). If the DNA were not subject to any additional constraint beyond closure, then one single nick of one strand or one single break of the two strands would suffice to release the torsional stress of the full molecule. It has however long be known that at least several tens of nicks are required to achieve this goal in Escherichia coli (2,3). This indicates that there exist barriers which block the diffusion of the torsional stress along the DNA molecule and organize the chromosome of E. coli into many independent topological domains, whose torsional state is not affected by the relaxation of other domains. It is currently estimated that the genomic DNA of bacteria like E. coli is composed of several hundreds of different topological domains with variable size (average size ≈10 kbp) and position (4). More generally, the partitioning of the bacterial chromosome into ≈10 kbp independent topological domains is believed to hold in most bacteria (5) and to be related, in part, to the insulation of fundamental co-expression units that are neighbors along the genome (6).
The work reported here deals with the topological barriers which make this partitioning possible, that is, the barriers which block the diffusion of torsional stress along the DNA molecule and are responsible for its division into independent topological domains.
Although this question has now been addressed for nearly two decades (7)(8)(9), the mechanisms underlying the formation of topological barriers remain mostly elusive. Most plausible models for the formation of topological barriers are (i) actively transcribing RNAP, which generate both positive and negative supercoils (10) and may consequently block the displacement and dissipation of plectonemes (11,12), and (ii) formation of DNA loops by certain nucleoid proteins, which may serve as topological barriers (13)(14)(15)(16). In the present paper, we focus on this latter mechanism and establish under which conditions DNA-bridging proteins may serve as topological barriers.
Several DNA-bridging proteins have been studied in some detail, including transcription regulators like the LacI repressor (17), H-NS (18) and H-NS-like proteins (19), Lsr2 (20), Fis (21) and Lrp (22). In their functional form, all of these proteins have at least two independent DNA-binding domains, so that they can interact with two DNA duplexes simultaneously and form a bridge between two sites that are widely separated from the genomic point of view. Most of these proteins also bind non-specifically but with high affinity to the DNA molecule. However, among all these DNA-bridging proteins, LacI is to date the only one that has been proved to work as a topological barrier (15,23,24).
The guiding line of the present paper is consequently the experimental demonstration in (15) that the binding of a DNA-binding protein to its recognition sites in two different locations on a supercoiled DNA molecule can confine free supercoils to a defined region and divide the DNA molecule into two distinct topological domains (15). In order to gain more detailed information on this mechanism, we determined the minimal physical properties that must be added to a standard model of circular DNA to reproduce the results described in (15).
The results presented in this article reveal that the formation of DNA loops is by no means sufficient to create topologically independent domains and that DNA-bridging proteins must exert rather strong constraints on their binding sites in order to act as topological barriers: They must block the diffusion of the excess of twist through both binding sites and must additionally block the rotation of one DNA segment relative to the other one.

METHODS
The coarse-grained bead-and-spring model developed for the present study is described in detail in Model and Simulations in the Supporting Material. In brief, bacterial DNA is modeled, as in (30,36), as circular chains of n beads with radius 0 . As in (31,36), the torsional energy term was borrowed from (37) and requires the introduction of a body-fixed frame ( , , ) k k k u f v , where k u denotes the unit vector pointing from bead k to bead 1 k + . The torsional rigidity opposing rotation of ( , , ) ) was adjusted so that at equilibrium the writhe contribution accounts for approximately 70% of the linking number difference (38), as is illustrated in Fig. S1. The values of the bending and torsional rigidities are close together, in agreement with experimental results (38). It may be worth emphasizing that introduction of the body-fixed frame ( , , ) k k k u f v is crucial for modeling correctly the torsion of double-stranded DNA with a "single-stranded" bead-and-spring model. This procedure is quite realistic and has already proved very useful for studying, for example, the interplay of DNA demixing and supercoiling in nucleoid compaction (36), the buckling transition in double-stranded DNA and RNA (39), the influence of nucleoid-associated proteins on DNA supercoiling (40)

RESULTS AND DISCUSSION
As a starting point, we recall the experimental results in (15), where the torsional relaxation properties of 4100 bp plasmids with initial superhelical density 0.06 Of special interest to us were the experiments performed with plasmids containing tandem copies of the binding site of the sequencespecific DNA-binding protein LacI at two different locations, such that the plasmid was divided into two stable loops of respective length 2900 bp and 1200 bp upon binding of LacI.
The 1200 bp region moreover contained the recognition site for one nicking enzyme (Nt.BbvC1) and the 2900 bp region the recognition site for a second nicking enzyme (Nb.BtsI). In the absence of LacI, addition of Nt.BbvC1 alone or Nb.BtsI alone was sufficient to release the full torsional stress of the plasmid. In contrast, in the presence of LacI, addition of Nt.BbvC1 alone removed only the 7 negative supercoils of the 1200 bp region, while addition of Nb.BtsI alone removed only the 17 negative supercoils of the 2900 bp region (15).
Simultaneous addition of both enzymes was required to relax the full plasmid (15). This experiment demonstrates very clearly that pairs of LacI bridges block the diffusion of torsional stress and divide the plasmids into two independent topological domains.
We report below on our efforts to determine the minimal set of mandatory properties that allow molecular bridges to act as topological barriers and block the diffusion of torsional stress in out-of-equilibrium supercoiled chains.

Preparation of torsionally out-of-equilibrium circular chains.
Topological barriers are objects that block the diffusion of torsional stress and are able to maintain different torsional conditions on their left and right sides. Characterization of the underlying mechanism requires prior understanding of how torsional stress diffuses in out-ofequilibrium supercoiled chains. To prepare such chains, we first let circular chains with , which were chosen so that the distance between their centers is smaller than 10 nm. This step is illustrated in Fig. 1(a), which shows an equilibrated chain with where 12 r W quantifies the winding of one moiety of the chain around the other one, that is, loosely speaking, their intrication. The out-of-equilibrium chains prepared as described above The main purpose of this work is to understand how DNA-bridging proteins like LacI (15,23,24) can maintain such an unbalance when the whole chain is relaxed. Since DNAbridging proteins form DNA loops by dynamically cross-linking widely separated DNA sites, the question amounts here to determine the constraints that must be imposed to beads α and β to divide the circular DNA chain into two topologically independent loops. Note that DNAbridging proteins are not introduced explicitly in the model: Rather, constraints on beads α and β are meant to model the mechanisms that allow the proteins to act as topological barriers.
Twist equilibrates much more rapidly than writhe.
In order to understand in more detail how the diffusion of torsional stress proceeds, the out-of-equilibrium circular chains prepared as described above were allowed to relax without constraint and all relevant quantities were monitored. From a practical point of view, four different initial out-of-equilibrium conformations were prepared for each value of 0 k L ∆ and 100 different relaxation trajectories were integrated for 1 ms for each initial conformation.
, and r j W ( 1 j = , 2) was then averaged over these 100 trajectories. In the absence of any constraint on beads α and β, all conformations obtained at the end of the 1 ms integration time satisfy  Based on recent experimental results (12)(13)(14)(15)(16), one may wonder whether DNA loops are ipso facto independent topological domains, or whether some additional, more subtle properties of DNA-bridging proteins are required to achieve this goal. We emphasize that the word "loop" is used here according to its biological meaning and not the geometrical one.
More precisely, it does not refer to a closed curve, but rather to a conformation of the DNA chain, where two DNA sites that are widely separated from the genomic point of view are maintained close to each other by some other macromolecule, so that the segment of DNA located between these two sites looks like a loop when it is observed from a long distance or projected on a plane. However, the chain is not closed and torsional stress cannot diffuse directly from one extremity of the loop to the other one through the bridging macromolecule.
In order to ascertain this point, we launched a series of simulations with the crudest model for DNA-bridging proteins, namely a simple bond between beads α and β. The potential energy of the system was written as pot , where pot E is the energy of the naked circular DNA chain described in Eqs. (S1)-(S5) and BP E an additional term, which models the action of DNA-bridging proteins. For the simplest model, we used where d αβ is the distance between the centers of beads α and β and 0 4 d αβ = nm. For the sake of simplicity, we used the same value of the stretching rigidity as for the DNA chain, that is topological barriers is consequently related to some property, which is subtler than the mere connection between two genetically widely separated DNA sites.
Still, it is worth noting that the αβ bond has a significant effect on the time evolution of 12 r W , as can be checked in Fig. S5. The reason is the following. In order to compute the partial writhes r j W (see the Supporting Material), one has to introduce two "virtual" closed chains 1 C and 2 C . 1 C is composed of beads α to 1 β − , plus the "unphysical" segment between beads β and α, while 2 C is composed of beads β to 1 α − , plus the same "unphysical" segment between beads α and β. When beads α and β are bonded, they remain close to each other for 0 t > , as indicated by the positions of the two arrows in Fig. 3(a), so that the unphysical segment between beads α and β remains small and 1 C is a good representation of the initially torsionally relaxed DNA loop (and 1 r W is a measure of its writhe), 2 C is a good representation of the initially torsionally stressed DNA loop (and 2 r W is a measure of its writhe), and 12 r W is a measure of the intrication of the two loops. In this case, 12 r W remains close to zero at all times, as can be checked in Fig. S5(b). This indicates that, for bridged DNA, equilibration of the torsional stress is not accompanied by any significant intrication of the two circular moieties. In contrast, for the system without the BP E term, beads α and β do not remain close to each other for 0 t > , as indicated by the positions of the two arrows in Fig. 1(c), so that the unphysical segment between beads α and β becomes arbitrarily large and 1 C , 2 C , the r j W and 12 r W lose their physical meaning. In this case 12 r W varies quite widely with changing DNA conformations, as is illustrated in Fig. S5(a) for an initial ).

Blocking diffusion of the twist is NOT sufficient to create independent topological domains
The reason why forming a loop is not sufficient to create independent topological domains is of course that the bond between beads α and β described in the previous subsection is not able to block the rapid diffusion of the excess of twist through beads α and β and the subsequent equilibration of the writhe. In order to form independent topological domains, the diffusion of the excess of twist must absolutely be blocked. Stated in other words, rotation of the internal basis ( , , ) k k k u f v around k u must be forbidden at , From ( , k α β = ), after each integration step and rotating k f and k v around k u by this angle k δ cos sin sin cos for , We note that the k δ remain small, and the corrections in Eq. (5) where α ξ denotes the angle formed by beads β, α and 1 α + , and β ξ the angle formed by beads α, β and 1 β + . Note that the bending rigidity for these two angles was assumed to be five time larger than the bending rigidity of the DNA chain, in order for α ξ and β ξ to deviate only moderately from / 2 π . In this second model, the action of DNA-bridging proteins is consequently modeled by the potential energy terms in Eq.  Fig. 4(b). It is stressed that this behavior has no precise physical meaning but arises instead from the mathematical definition of the total writhe r W in Eq. (S11) and its somewhat arbitrary partitioning into the sum of the two partial quantities r j W and 12 r W in Eq.
(S17). In particular, r W does not display any sharp jump along single trajectories.
As can be checked in Fig. 4, the second model of DNA-bridging proteins succeeds in preventing equilibration of the twist and the writhe in the two moieties of the chain. Indeed, 2 Fig. 4(b)). According to Eq. (1), this limit is equivalent to 12 r r W W = . ) superimpose at short times, which indicates that the relaxation speed is proportional to the total amount of torsional stress. However, for longer times and/or larger values of the linking number difference, a slower regime sets in, which is probably due to the resistance opposed by plectonemes against the reorganization of the chain and the intrication of the two moieties. Finally, we note that after full relaxation the writhe contribution accounts for approximately 90% of the linking number difference, which is significantly larger than the 70% contribution in the absence of domains (see Fig. S1).
In conclusion, this second model of DNA-bridging proteins indicates that blocking the diffusion of the excess of twist at the two DNA loci bound by the proteins is sufficient to maintain a constant difference between the linking number difference, excess of twist, and writhe of the two moieties, but that it is not sufficient to insure true topological independence.

Indeed, the DNA chain evolves towards very compact and intricated final conformations,
where the topology of each moiety is different from the initial one. The relaxation limit 12 r r W W = (or equivalently 1 2 r r 0 W W + = ) appears non trivial and intriguing to us. We suspect that the explanation involves subtle considerations that should deserve further work.

Topological barriers must block diffusion of the twist AND the relative orientation of the DNA segments
The question that remains to be answered is consequently: What additional constraint must the DNA-bridging protein exert on its binding sites in order to preserve not only the difference in torsional stress between the two moieties of the DNA chain, but rather the precise initial torsional stress in each moiety ? The feeling that emerges upon examination of conformations like the one shown in Fig. 3(b), is that the winding of the initially relaxed moiety around the initially stressed one probably requires the relative rotation of the two DNA segments around the αβ bond, and that no winding would be possible if this rotation were blocked. In order to check this conjecture, a third bending term was added to BP E , namely 0 2 2 2 0 2 BP , , relaxed moiety and one torsionally stressed moiety, as can be checked in Fig. 3(c). This third model of DNA-bridging proteins finally describes a topological barrier. In order to act as topological barriers, DNA-bridging proteins must consequently exert rather strong constraints on their binding sites: Indeed, they must block the diffusion of the excess of twist through both binding sites and must additionally block the rotation of one segment relative to the other one. It is worth emphasizing that this last condition is probably not satisfied by all DNAbridging proteins. For example, it is well known that H-NS dimers bind most often nonspecifically to the DNA, which means that the two C-terminal DNA-binding domains (59) located at the end of flexible linkers (60) (16). In this context, we surmise that the slow relaxation observed in (15) for single DNA-bridges may correspond to the slow intrication obtained with our second model for DNA-bridging proteins, while the more stable topological separation observed in (15) for bridges in tandem corresponds to true barriers obtained with the third model for DNA-bridging proteins. We finally note that the intrication properties discussed here are also expected to contribute to the formation of topological domains that are, in effect, larger than the domain delineated by the binding sites of LacI as recently observed in (16).

CONCLUSIONS
DNA-bridging proteins form DNA loops by dynamically cross-linking DNA sites that are widely separated from the genomic point of view (17)(18)(19)(20)(21)(22). They usually bind DNA nonspecifically but with high affinity. DNA-bridging proteins have now been investigated for more than two decades, but it is only very recently that it has been demonstrated that a DNAbridging protein like the LacI repressor is able to separate circular plasmids into two topologically independent loops (15,23,24). To the best of our knowledge, this is at present the only demonstrated case of a DNA-bridging protein acting as a topological barrier. In the present work, we aimed at going one step further and determining the conditions that DNAbridging proteins must fulfill to act as topological barriers. We showed that these proteins must exert rather strong constraints on their binding sites: They must block the diffusion of the excess of twist through the two binding sites on the DNA molecule and must simultaneously prevent the rotation of one DNA segment relative to the other one. These criterions should be of great help when investigating the (existence, or lack of) topological barrier properties of other DNA-bridging proteins, like for example H-NS, which has been predicted to participate in the organization of the genome into topologically independent domains (6,61). Indeed, it is probable that not all DNA-bridging proteins satisfy the second condition. If it is not satisfied, then intricate conformations are obtained, with a preserved difference in the torsional stress between the two domains, but different values of the stress in each domain. A rationale for the intriguing limit 12 r r W W = (or equivalently 1 2 r r 0 W W + = ) is however missing.
This work also brings some light on another mechanism that has been proposed to explain the formation of independent topological domains, namely two proteins wrapping the DNA in two different locations (Ref. (15), Fig. 6). There is experimental evidence that certain wrapping proteins, like GalR (15), the λ O protein (15), and Fis (62) Still, our work suggests that the conditions for topological separation may be met by molecular geometries that do not require that the boundaries of the topological domains be close to each other. For example, one can imagine a topological barrier formed of two different proteins, which both anchor the DNA molecule to the cell membrane. According to our work, the DNA segment located between the two anchoring proteins may be topologically independent, provided that each protein is able to block the diffusion of twist along the DNA molecule and that the proteins cannot rotate with respect to the membrane. If this latter condition is not satisfied, then some complex winding of the DNA around itself is likely to occur, like for DNA-bridging proteins.
Another model which is frequently proposed for the formation of topological barriers, namely actively transcribing RNAP (10)(11)(12), is more difficult to discuss in terms of the results described in this paper, because several time scales need be considered. Indeed, it is likely that the two parts of the DNA molecule located respectively upstream and downstream from the transcribing RNAP will tend to increase their intricacy, in order to reduce the torsional stress induced by the blocking of the diffusion of twist at the RNAP. However, the RNAP is itself moving forward along the DNA, so that the ability of transcribing RNAP to act as topological barriers probably depends on the relative speeds of the two mechanisms. More work is needed to clarify this point. Similarly, the case of DNA-bending nucleoid proteins, like HU (63), certainly deserves further investigations.
Finally, we note with interest that the intricate structure obtained with DNA-bridging proteins that block the diffusion of twist along the DNA but do not prevent the relative rotation of the DNA segments are highly compact. It turns out that the mechanism leading to the formation of the bacterial nucleoid is a longstanding but still lively debated question (31,32,(64)(65)(66)(67). The point that has puzzled scientists for decades is that the volume of the   (b) Representative snapshot obtained upon relaxation of the out-of equilibrium conformation shown in Fig. 1(b) for the second model of DNA-bridging proteins (Eqs. (5) and (6)). (c) Representative snapshot obtained upon relaxation of the out-of equilibrium conformation shown in Fig. 1(b) for the third model of DNA-bridging proteins (Eqs. (5) and (7)). The   5) and (6)). Each curve was averaged over 4 different trajectories starting from the same initial conformation.
where e is the absolute charge of the electron and 0.7 B = ℓ nm the Bjerrum length of water.
In Eq. (S7), / B e ℓ is the net linear charge density along a DNA molecule immersed in a buffer with monovalent cations derived from Manning's counterion condensation theory (5,6). Note that electrostatic interactions between nearest neighbours are not included in Eq. (S5) because it is considered that they are already accounted for in the stretching and bending terms.
The dynamics of the system was investigated by integrating numerically overdamped Langevin equations. Practically, the updated positions and torsion angles at time step i+1 are computed from the positions and torsion angles at time step i according to ( where ( ) i k f are vectors of inter-particle forces arising from the potential energy pot E , 298 T = K is the temperature of the system, ( ) (S17) r W , 1 r W , and 2 r W quantify the winding of each closed chain around itself. In contrast, 12 r W quantifies the winding of 1 C around 2 C , that is, loosely speaking, the intrication of the two loops when beads α and β remain close to each other.
Finally, the global torsional state of each moiety can be described by the value of the partial linking number difference It results from Eqs. (S12) and (S16)-(S19) that     (6)). The four curves correspond to four different trajectories starting from the same initial conformation.  (7)). Each curve was averaged over 2 different trajectories starting from the same initial conformation.