Entropic elasticity and dynamics of the bacterial chromosome:
a simulation study
We study the compression and extension dynamics of a DNA-like polymer interacting with non-DNA binding and DNA-binding proteins, by means of computer simulations. The geometry we consider is inspired by recent experiments probing the compressional elasticity of the bacterial nucleoid (DNA plus associated proteins), where DNA is confined into a cylindrical container and subjected to the action of a “piston” – a spherical bead to which an external force is applied. We quantify the effect of steric interactions (excluded volume) on the force-extension curves as the polymer is compressed. We find that non-DNA-binding proteins, even at low densities, exert an osmotic force which can be a lot larger than the entropic force exerted by the compressed DNA. The trends we observe are qualitatively robust with respect to changes in protein size, and are similar for neutral and charged proteins (and DNA). We also quantify the dynamics of DNA expansion following removal of the “piston”: while the expansion is well fitted by power laws, the apparent exponent depends on protein concentration, and protein-DNA interaction in a significant way. We further highlight an interesting kinetic process which we observe during the expansion of DNA interacting with DNA-binding proteins when the interaction strength is intermediate: the proteins bind while the DNA is packaged by the compression force, but they “pop-off” one-by-one as the force is removed, leading to a slow unzipping kinetics. Finally, we quantify the importance of supercoiling, which is an important feature of bacterial DNA in vivo.
The genome of living organisms, from bacteria to humans, is under remarkable confinement in physiological conditions Alberts et al. (2002); Marenduzzo, Orlandini, and Micheletti (2010). For instance, the circular chromosome of E. coli would be over 1 mm long if stretched out, yet it needs to fit within the bacterial cell which is a m ellipsoid. Likewise, there is about m of DNA in a single human nucleus, whose typical size is only about 10 m. Another useful way to quantify the degree of confinement of the bacterial DNA is to estimate its gyration radius, , at equilibrium, which is about 5 m; since this is larger than the bacterial cell, the bacterial genome is in the “semi-dilute” regime of polymer physics Doi and Edward (1986).
There are at least four mechanisms through which the bacterial chromosome is compacted within the cell Alberts et al. (2002); Joyeux (2015). First, and most obviously, it is confined within the cell wall; note however that the chromosome does not occupy the entire cell so it must be compacted further. Second, there is a depletion attraction between genome segments induced by the crowding of non-DNA-binding macromolecules de Vries (2012). Third, the genome is associated with a number of architectural or nucleoid associated proteins (NAPs) Browning, Grainger, and Busby (2010); Song and Loparo (2015) which can bind the DNA at more than one point, creating effective DNA-DNA attractive interactions which help to reduce the space occupied by the chromosome. Fourth, the bacterial chromosome is a supercoiled loop: i.e., the helical pitch is different from the one favoured thermodynamically – 10.5 base pairs (bps) for B-DNA Alberts et al. (2002). In practice, bacterial DNA is negatively supercoiled Bates and Maxwell (2005), so the helix is slightly underwound. The degree of negatively supercoiling is about 5, which means that in a length of DNA, which would have 20 turns at thermodynamic equilibrium, there are only 19 turns. The twist deficit can be converted into negative writhe, which creates a local folding of the DNA, again favouring compaction.
As a first approximation, one may consider only the first of these compaction mechanisms and view the bacterial chromosome as a biopolymer under tight confinement Jun and Wright (2010). According to this model, there is a large decrease in entropy when the DNA is within the cell, and one expects this to create an entropic pressure, or force, on the confining walls Pelletier et al. (2012); Jun and Mulder (2006); Jun, Thirumalai, and Ha (2008); Jung et al. (2012). This naturally explains why the bacterial chromosome tends to expand when the confining cell wall is removed. The entropic force exerted by the DNA was measured in an interesting experiment by Pelletier et al. Pelletier et al. (2012). In that work, single bacterial cells were trapped in an array of cylindrical cavities, with diameter just larger than the width of the bacteria, and the height much larger than its length. After the cells were trapped, their walls were lysed so that the enclosed DNA was free to expand and to increase its conformational entropy. The DNA was found to reach a height about ten times larger than that of the bacterium. The entropic force exerted by the expanding chromosome on a colloidal bead placed in the channel on top of the DNA was measured by means of optical tweezers, and it was found that this force was much larger than expected on the basis of a simple theory (reviewed in Section III) for Gaussian – i.e., infinitely thin – polymers, with the known persistence length of DNA, 50 nm.
In this work, we present a simulation study of a situation closely related to the experiment discussed above Pelletier et al. (2012). With respect to previous theoretical work on the bacterial chromosome based on polymer physics Jun and Mulder (2006); Jun, Thirumalai, and Ha (2008); Junier, Boccard, and Espéli (2014); Le et al. (2013); Benza et al. (2012); Scolari and Lagomarsino (2015); Jung et al. (2012, 2013); Kim et al. (2015); Shendruk et al. (2015); Wegner et al. (2016), the main novelty here is that we include in our simulations the effects of bacterial proteins, whether binding to DNA or not, and we quantify their effect on the compression elasticity of DNA and its dynamics.
More specifically, we analyse the entropic compression elasticity and expansion dynamics of a model bacterial DNA confined in a thin cylinder and subject to the action of a “piston”, a colloidal bead under the influence of an external force. This set-up allows us to measure the force experienced by the piston as a function of DNA compression. We find that steric effects and the presence of proteins strongly affect both the entropic elasticity of the DNA under compression, and also the dynamics of its extension once the piston is removed. In particular, non-DNA-binding proteins exert an osmotic pressure which can be at least as large as the entropic force exerted by the DNA; the macromolecular crowding they introduce is also important in determining the polymer dynamics. DNA-binding proteins can further lead to interesting “popping-off” kinetics, when the thermodynamic interactions with the DNA is tuned such that it leads to stable binding in confinement, but not in solution. Finally, we find that supercoiling and DNA topology have a relatively little effect on force-extension curves, but can affect the compression dynamics significantly.
Our results complement the simulations of Refs. [Kim et al., 2015; Shendruk et al., 2015] which study the compaction of a confined bacterial DNA due to non-DNA-binding proteins, and quantify how the polymer collapse depends on crowder size, concentration, and polydispersity – here we additionally quantify how crowding affects the out-of-equilibrium expansion dynamics. Other relevant, yet distinct, works are those of Refs. [Joyeux, 2015; de Vries, 2012; Wegner et al., 2016] which quantified the extent to which H-NS and other DNA-binding proteins compactifies DNA.
Ii Materials and Methods
ii.1 Model: set-up and potentials used
The system we consider consists of a bacterial DNA molecule confined in a cylindrical pore, that is compressed by a piston in the absence or presence of proteins. DNA is modelled as a linear (Sections IV.2-IV.5) or circular (Section IV.6) self-avoiding polymer composed of spherical beads with diameter . Our coarse grained model and force fields are similar to those used to study bacterial DNA, or DNA-protein systems in Refs. [Jun and Mulder, 2006; Kim et al., 2015; Brackley et al., 2013; Treut, Kepes, and Orland, 2016].
The interaction potential between monomers is defined by three contributions. First, two neighbouring monomers are bound via a finitely extensible non-linear elastic (FENE) spring given by the potential
where is the distance between the ith bead and its nearest neighbour (the (i+1)th) along the chain, is the maximal extent of the bond and is the bond energy. Second, there is a steric (excluded volume) interaction between all beads that is set by the Weeks-Chandler-Andersen potential, as follows:
for , and otherwise. Here is the thermal energy ( is the Boltzmann constant and is the temperature), the distance between the ith and jth beads and the mean of the diameters of the two interacting beads, i.e., . [Note that under this choice of FENE and WCA potentials, the DNA bond length is approximately equal to .] Third, the bending rigidity of the polymer is introduced by a Kratky-Porod potential for every three adjacent monomers
where is the angle between the three consecutive monomers and the bending energy. sets the flexibility of the polymer, since it determines the persistence length (in units of ): . We use the well characterised persistence length for naked double-stranded DNA nm.
In the simulations where supercoiling is included, a circular polymer has to be considered, and three more interaction potentials taken into account. DNA supercoiling results from over- or under-twisting the helical DNA (positive or negative supercoiling, respectively) relatively to its relaxed state. In a DNA loop the number of times the double strands wrap around each other is fixed. So, in order to alleviate the torsional strain, caused by over- or under-winding, the DNA writhes up on itself. Supercoiling can be modelled by considering twisting rigidity potentials that can be defined by assigning an orientation to each one of the DNA polymer beads. One way of doing this is to decorate the surface of each bead with three patchy particles, as in the model of Ref. [Brackley, Morozov, and Marenduzzo, 2014]. The positions of the patches are such that they establish three orthogonal unit vectors, which define the bead’s reference frame. With this it is possible to define three angles that give the orientation of a bead with respect to its neighbour: one angle that gives the orientation of the bead with respect to the polymer backbone, and two dihedral angles and that set the degree of twisting in the plane perpendicular to the backbone. Therefore, the orientation of the bead is kept aligned with the polymer backbone by defining the potential
and the twisting rigidity of the polymer is modelled by assigning to both dihedral angles the energy
where is a phase related to the twist. More precisely, determines the number of beads involved in half a turn of the ribbon – , where is the pitch, i.e., the length scale corresponding to one full twist in equilibrium.
Proteins are also modelled, for simplicity, as spheres of diameter equal to , so that all beads in the simulation have the same size; in the Discussion and Conclusions section we also present selected results where we have considered a protein diameter of which matches more closely the size of a typical bacterial protein Phillips et al. (2013). In the simulations, proteins interact sterically with each other via the WCA potential in Eq. (2). In this study we consider two kinds of proteins: non-DNA-binding and DNA-binding. The interaction between non-DNA-binding proteins and the DNA is purely repulsive and, again, described by the potential in Eq. (2). For the DNA-binding proteins, we assume that they are non-specifically DNA-binding, so can bind to any DNA bead. The attractive interaction is set by a Lennard-Jones (LJ) potential
The parameter controls the magnitude of the protein-DNA interaction, and is the range of the interaction. For the simulations where both kinds of proteins are present, and , leading to a weak-moderate attractive protein-DNA interaction. For the simulations where only DNA-binding proteins are present, and we choose values for in the range .
Note that we do not explicitly consider electrostatic interactions between proteins and/or DNA beads, again for simplicity, although these are charged in reality. This approximation is motivated by the fact that for a physiological 150 mM concentration of monovalent salt the Debye length is around nm Ando and Skolnick (2010), which is below the size of the proteins, so that electrostatic interactions are heavily screened in practice. We performed some simulations where we did consider charged particles (see Appendix B) and the results confirm that one can rely on this simplification.
The piston is modelled as a rigid sphere whose diameter is slightly larger than the confining cylindrical pore’s diameter, in order to prevent particles from escaping around the sides. The interaction between the piston and the confined particles is purely repulsive and again described by the WCA potential [Eq. (2)]. The piston moves under a uniform externally applied force, compressing the DNA and the proteins against the cylinder walls (see Fig. 1).
We simulate the system by using Brownian dynamics (BD) via the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) code in the BD mode. In other words, we use a molecular dynamics (MD) algorithm with a stochastic thermostat, which models the thermal fluctuations and viscosity of an implicit solvent. The system’s temperature is kept constant at a value K. Each particle (DNA beads, proteins and piston) obeys a Langevin equation
where is the position of the centre of mass of the particle with mass , is the friction due to the solvent (typically ) and is a vector representing random uncorrelated noise, such that
where and indicate Cartesian components, and denote Kronecker’s delta, and denotes Dirac’s delta function. Eq. (8) is integrated with a constant time step , where is the simulation time unit, for a total of time steps or more.
ii.2 Mapping between simulation and physical units
In our simulations we use energy units of (where K), length units of nm (the size of a DNA bead that corresponds to the hydrated thickness of B-DNA), and mass units of , where the mass of a DNA bead . With this choice there is a natural simulation time unit . Therefore, in our coarse-grained model, each DNA monomer represents bp of B-DNA (for which the distance between consecutive base pairs is nm).
In our physical system, there are two further timescales, beyond . First, there is an inertial timescale, , where is the mass of a DNA or protein bead, and is its friction. This quantity gives the time after which the velocity of the beads becomes uncorrelated. The second is the Brownian time, , which gives the (order of magnitude of the) time taken for a bead to diffuse across its own diameter, . The diffusion coefficient is set by the friction through Einstein’s formula . We use , which leads to a Brownian timescale (simulation units). Since we are interested in long time behaviours, i.e., of the order of several ms, and not in resolving fine details of the inertial collisions (which play no role in our overdamped system), in order to map time from simulation to physical units, we therefore match the Brownian timescale. As a sphere of diameter nm in an aqueous fluid (viscosity cP) has ns, it follows that one simulation time unit corresponds to ns Brackley et al. (2015). Note that our choice of timescales means that we do not correctly describe the inertial dynamics before , but this is not an issue for our purposes, as we are interested in timescales much exceeding this. At the same time, our choice of ensures that the dynamics in our simulations is overdamped, which is the physically relevant case for our system.
By using the mapping given above for energy, length and time we can map all of the other physical quantities we measure. For instance for the force, one simulation force unit is , which corresponds to pN.
As detailed above, the system studied here is composed of three types of beads: DNA monomers, non-DNA-binding proteins (crowders) and DNA-binding proteins. The number of beads is chosen so that the ratio of the number of proteins to the number of DNA beads is that found in vivo: The chromosome of E. coli consists of bp, which in our model corresponds to a polymer with beads. In bacterial cells, the estimated total number of proteins is , from which are DNA-binding. This corresponds to a protein-DNA bead ratio of . Since the computational cost associated with a bead polymer is extremely high, we instead consider a smaller system with beads. We then explore the influence of proteins in the system, by varying the protein number up to , which represents the true protein-DNA ratio. For simulations where both crowders and DNA-binding proteins are present, the number of DNA-binding proteins is .
Iii Summary of Scaling Theory for Cylindrically Confined Polymers
iii.1 A Review of the Entropic Spring Theory
According to the well known blob scaling concept in polymer physics, a self-avoiding polymer confined in a cylindrical pore can be seen as a linear chain of blobs Brochard and de Gennes (1977). In [Jun, Thirumalai, and Ha, 2008], this blob-scaling approach is used to derive an expression for the force required to compress a polymer confined in a cylindrical nanopore, assuming it behaves like an entropic spring. The “renormalized” free energy for a confined polymer of beads is given by
where is the extension of the polymer along the axis of the cylindrical pore (see Fig. 1), , A and B are constants, and g is the number of monomers inside a blob of diameter D. From this, a universal scaling relation for the (external) force-extension relation can be derived,
where can be seen as the confining cylinder’s width. Here is the extension of the relaxed (zero force) confined polymer. In the regime of weak deformations Pelletier et al. (2012), this reduces to
where is the dimensionless polymer spring constant.
In the regime of strong compression (), the free energy in the blob-scaling formalism takes another form and the force-extension relation becomes Jun, Thirumalai, and Ha (2008)
iii.2 A Scaling Theory for the Polymer Expansion dynamics
Concerning polymer dynamics, there are theoretical models that give an estimate of the evolution of the polymer extension in time, during polymer expansion under confinement. Taking to be the confinement-deformation free energy, the polymer deformation force is given by . The equation of motion for the polymer extension becomes Jung et al. (2013)
where is the polymer friction coefficient. This equation can be solved given the initial condition – the extension of the compressed polymer just before it starts expanding. An explicit form for depends on the physical problem at hand. Following the approach in [Jun, Thirumalai, and Ha, 2008], we use the force-extension relations in Eqs. (11-13). Taking Eq. (12) for the weak compression regime and substituting it into Eq. (14), one gets
By defining and , this becomes
Taking into account that during the polymer expansion , the integral of this equation leads to the solution
where and are constants that are defined by the initial condition : .
According to the solution , there is an initial offset corresponding to the extension of the polymer after being compressed by the piston. For intermediate times (), one can expand the solution about zero to get , which reveals the scaling . For late times the polymer relaxes completely reaching a saturation regime where the extension corresponds to the value at equilibrium in the absence of a compression force: .
In the strong compression regime [Eq. (13)], the extension-time scaling relation for intermediate times becomes .
These scaling relations are derived for a polymer confined in a cylindrical pore in the absence of other particles. However, the bacterial cellular environment is rich in proteins and so one can ask how the scaling would change in the presence of proteins either non-DNA-binding (crowders) or DNA-binding. Crowders have been shown to increase the viscosity of the medium Toan et al. (2006), which slows the polymer diffusion and hence the polymer expansion dynamics. Therefore one might expect a scaling relation for intermediate times where the exponent is smaller than that derived above in the absence of crowders. DNA-binding proteins with more than one binding site (such as the ones considered in the present study) bridge together DNA segments, reducing the polymer gyration radius and the extension of the confined polymer. This bridging effect increases the local polymer stiffness, reducing the effective polymer elasticity. Therefore, one may also expect a slow-down effect of the polymer expansion in the presence of such proteins and a further decrease of the exponent .
iv.1 Set-up of the simulation
First, we discuss our simulation geometry and set-up (see Fig. 1). We considered two cases: DNA-only simulations (Fig. 1), and a DNA molecule interacting with proteins (both DNA-binding and non-DNA-binding, Fig. 2(a) and Fig. 2(b) respectively). Note that the two top pictures in Figure 1 correspond to transient configurations during compression of the DNA molecule, depicting the time evolution of the simulation. The bottom picture corresponds to the fully equilibrated system.
To simulate DNA, we considered a model linear DNA molecule of beads, within cylinders of two different diameters ( and , the latter case is shown in Fig. 1(a)). Results for the two diameters are very similar, hence we show below only those for . For each configuration we measure the DNA extension along the longitudinal direction of the cylinder, defined as the maximum distance between any two beads in the chain – we denote this quantity as . We then take the average of over 10 configurations. The radius of gyration of the unconfined DNA (in the absence of the cylinder) was estimated to be , while its extension within the cylinder in the absence of the piston was measured as . Therefore, we work in the semi-dilute regime which is realistic for the bacterial chromosome. Although in E. coli the DNA is circular, the scaling theory and previous work suggests that the main contribution are given by polymer confinement (entropy and free energy loss). Hence we expect the polymer topology to be not too important – the simulations with circular DNA described in Section IV.6 further support this view.
For the DNA-protein simulations, we considered two situations. In the first (Fig. 2(a)), we only included non-DNA-binding proteins (green spheres in Fig. 2(a)), which are the majority in vivo: for instance, in E. coli there are about proteins, of which only are estimated to be DNA-binding. As the bacterial cell volume of E. coli is about m, and the average diameter of a bacterial protein is nm Phillips et al. (2013), the volume fraction occupied by all bacterial proteins is a few %, which is larger than the volume occupied by DNA. In our simulations, we typically considered proteins, modelled as spheres (matching the ratio found in vivo between number of proteins and number of “DNA beads”). In the second situation, we included a fraction of DNA-binding proteins (magenta spheres in Fig. 2(b)); we simulated non-specific binding, as is the case, to a first approximation, for the histone-like bacterial H-NS protein Dame, Noom, and Wuite (2006), or for the DPS protein, which is activated following bacterial starvation Wolf et al. (1999). While we include no direct protein-protein attractive interaction, DNA-binding proteins naturally cluster, through the so-called “bridging-induced attraction” described previously in Refs. [Brackley et al., 2013, 2016]. This attraction is associated with a simple positive feedback loop: proteins bind to the DNA in multiple places forming bridges, this increases the local concentration of DNA, which in turn recruits further proteins, etc., ultimately resulting, for the protein size considered here, in the formation of elongated protein clusters associated with a local tubular folding of DNA where the protein-associated segments are parallel to each other. Ref. [Brackley et al., 2013] shows that larger proteins lead to DNA wrapping around them. However this is not realistic for H-NS which is thought to form linear clusters Brackley et al. (2013); Dame, Noom, and Wuite (2006), as in Figure 2.
Figures 1 and 2 show examples of compression simulations, where different forces were applied to the piston, and the resulting DNA conformations were analysed. We have also performed further simulations, as follows. First, we considered compression simulations with modified protein density. Second, we simulated the dynamics of entropic expansion, by first compressing the DNA with a large force, allowing the system to reach equilibrium, and subsequently removing the piston. Third, we have varied the strength of the interaction between DNA-binding proteins and DNA. Finally, we have also considered the compression of supercoiled DNA loops (as opposed to the linear DNA shown in Figs. 1 and 2).
iv.2 Excluded volume effects lead to quantitative deviations from the predicted entropic spring response in DNA-only simulations
We begin by studying the entropic compression in DNA-only simulations (Fig. 3(a)). We find two regimes. First, for the entropic spring theory revised in Section III.1 (which is also the theory used in Ref. [Pelletier et al., 2012]) works well, and we observe a very sensitive dependence of the extension on compression force: the extension reduces to 10% of its free value, , for a force of just simulation units (corresponding to about 5 pN – see mapping in Sec. II.2). Second, for , we find a sharp deviation from the theory: our numerical estimate of the entropic force is more than an order of magnitude larger than predicted. The origin of this deviation is in the assumption in the scaling theory of an infinitesimally thin polymer: clearly, this is not applicable to the tightly confined regime which we reach by the end of our simulation, where the DNA segments are forced into close contact, so that steric interactions dominate. Figure 3(b) further suggests that close packing and many body interactions are the main contributions to this strong steric repulsion in the tightly confined regime, as it can be seen from the fact that the volume fraction reaches a large value () for the largest compression force.
This result suggests that steric repulsion is an important factor to consider when estimating the entropic force resisting compression. At the same time, we note that the effect found here (a 10-fold increase) is an overestimate of the correction needed for a real bacterial chromosome. This is due to the scaled down dimensions of the DNA we use: the same value of corresponds to a much denser volume fraction of DNA in our simulations with respect to the experiments in Ref. [Pelletier et al., 2012].
iv.3 Non-DNA-binding proteins greatly increase the entropic force and pressure exerted on the piston
Next, we analyse the effect of bacterial proteins on the entropic elasticity of the system. The osmotic pressure of proteins in the cytosol can be estimated as , where is the number of proteins, and is the confinement volume; for the case of E. coli this is 0.01 atm, which is larger than, or at least of the same order of, the pressures recorded in the experiment in Ref. [Pelletier et al., 2012].
We show in Figure 4 how the presence of (non-DNA-binding) proteins (modelled as spheres which interact both with each other and with the DNA solely via excluded volume) affects the force-extension curves. We find that proteins make an important contribution, especially for moderate compression (relatively large values of ): in this regime, the force required for a given extension is orders of magnitude larger than in the case where the proteins are not included. More specifically, the force is approximately linear in the number of proteins , as expected for an osmotic contribution (see Fig. 4, inset 1). Interestingly, for intermediate forces, we find that the extension recorded at a given force also depends linearly on (Fig. 4, inset 2).
We next address the role of DNA-binding proteins. These tend to compact the DNA, and should lead to a change in the extensional elasticity. Mobile cross-links, or slip-links, have previously been shown to strongly modify the force-extension curves of polymers in stretching experiments Metzler, Kantor, and Kardar (2002); for this reason, DNA-binding proteins were proposed in Ref. [Pelletier et al., 2012] as a possible explanation for the quantitative significant discrepancy between the experiments and the entropic spring theory in compression experiments. Therefore we ask to what extent the presence of DNA-associating proteins affect our results. Fig. 5 shows that these effects are minor in the simulations. This is consistent with our previous finding that osmotic forces from non-DNA-binding proteins are more important quantitatively than DNA entropic forces: DNA-binding proteins only affect the polymer response, and hence do not have much bearing on the overall curve. We should stress here that the DNA-binding proteins we consider form clusters, as could be the case for H-NS Brackley et al. (2013); Dame, Noom, and Wuite (2006), or DPS Wolf et al. (1999); while the dynamic crosslinks invoked in Ref. [Pelletier et al., 2012] will in practice interact entropically (see [Metzler et al., 2002]), their collective behaviour may be different.
In summary, we find that non-DNA-binding proteins exert a significant osmotic pressure on the piston, and this is much larger than the force exerted by the DNA. Furthermore, any reduction in the force exerted by the DNA due to DNA-binding proteins is dwarfed by the contribution of the non-DNA-binding proteins. Our estimates suggest that even in the experimental situation the presence of non-DNA-binding proteins could significantly affect the force measured via the set-up used in Ref. [Pelletier et al., 2012]; therefore it would be of interest to compare in more detail those experiments with in vitro compression experiments with different size DNA and different protein environments.
iv.4 The expansion dynamics of DNA depends on macromolecular crowding and DNA-protein interactions
We next examine the dynamical behaviour of a DNA molecule (with or without proteins), where the DNA is first compressed under a strong force, and then let free to expand after the piston is removed. Figure 6 shows how the extension increases in time following the piston removal. We note that the curves for a single realisation are very noisy, so in Figure 6 each curve (or point) corresponds to an average over 10 independent runs.
First, for the DNA-only case, one may expect a scaling behaviour for with (for self-avoiding chains, see Sec. III.2 and Ref. [Jung et al., 2013]). Our results typically show a similar, but on average slightly smaller, exponent: the values are also consistent with those found numerically in Ref. [Jung et al., 2013]. Our simulations show a different apparent exponent for different compression force (Fig. 6(a)): this may either point to some dependence on the initial condition, or to a wide variability of the exponent (which is apparent from Fig. 6(d) and further discussed below).
Second, we consider the case with proteins. The expansion dynamics is much slower (i.e., the apparent exponent is smaller, Fig. 6(b)) in the presence of non-DNA-binding proteins: this is because the proteins create a crowded environment which hampers DNA unfolding. At the same time, the DNA extension is larger at , for the same initial force, because of the osmotic pressure of the proteins which opposes the compression force from the piston; as a result the value of is always larger when proteins are present in our simulations. For late times, the proteins diffuse away from the DNA and become dilute, so the effect of crowding diminishes. Hence for large the apparent exponents for our curves with and without proteins become approximately equal.
DNA-binding proteins lead to even slower progress in the DNA expansion (Fig. 6(c)), especially at early times. Therefore, DNA-binding proteins have a much more detectable signature in the expansion dynamics, with respect to the minor contribution noted in the force-extension curves in Figure 5 where DNA-binding proteins were absent. To characterise this effect more in detail, we performed simulations with only DNA-binding proteins (Fig. 6(d)): the exponents found in this case should then be compared with those for DNA-only simulations (Fig. 6(a)). These results confirm that when proteins bind to the DNA they may reduce the apparent exponent; however the effect is subtle, and there is a large stochastic element in the dynamical curves (as each of the values in Fig. 6(d) is computed by averaging 10 different expansion runs).
iv.5 “Popping-off” dynamics with DNA-binding proteins
In Figure 6 (bottom two panels), we analysed the effect of DNA-binding proteins on the dynamics of DNA expansion. By examining the trajectories of the system as a function of DNA-protein interaction strength, we observed some further interesting phenomena in the kinetics of the system, which we now describe.
If the DNA-protein interaction is weak (), proteins only transiently bind to DNA. Since increased concentration favours the bound state (the entropy loss upon binding is smaller), more proteins bind upon compression; however they detach immediately after the piston is released (see Fig. 7, left column, and Fig. 7, blue line in bottom panel). As a result, the DNA responds elastically as a protein-free polymer – this is consistent with our finding that the apparent exponent measured for a low DNA-protein interaction strength , in the simulations with only DNA-binding proteins, is comparable with the exponent measured for the DNA-only simulations (see Fig. 6).
Some interesting dynamics occur if we choose a larger DNA-protein interaction strength, so as to promote more long-lived binding (see Fig. 7, middle column). In this situation, the interaction strength is such that it favours long-lived binding under compression – in this case, essentially all proteins are bound at all times, and they locally compact the DNA into a toroidal structure, resembling that of DNA within bacteriophages 111In a real bacterial DNA, proteins like H-NS would not bind the whole genome so the 3D structure would likely be different upon compression; however, we expect similar dynamics to occur there as well.. When the piston is removed, the translational entropy of the proteins in the unbound state increases dramatically – as they could now occupy any region of the cylindrical domain; however the bound state is still metastable, and it takes a relatively long time for the proteins to detach. Over time, proteins “pop-off”, typically one-by-one, from the collapsed DNA, and the total energy of the polymer-and-protein system (which is approximately proportional to the number of bound proteins) decreases in magnitude linearly with time. Therefore, the “popping-off” time should increase linearly with number of DNA-binding proteins in the system, so could easily be observable in experiments with bacterial DNA (recall there are an estimated DNA-binding proteins in the bacterial nucleoid). Intriguingly, while the protein kinetics are completely different, the popping-off leaves little detectable signature in the apparent exponents recorded in Figure 6. Finally, the popping-off requires tuning of the interaction, because if this becomes too large, proteins bind to the DNA permanently (at least within our simulation time), and the popping-off kinetics can no longer be observed – the energy now does not appreciably depend on time (see Fig. 7, bottom panel, green curve).
iv.6 Compression curves and dynamics for supercoiled DNA
Thus far, as previously highlighted, we have described the results obtained for linear DNA. Within bacteria, DNA is circular and negatively supercoiled (see Sec. I); it is therefore of interest to ask what the effect of supercoiling is on our results.
In order to address this issue we consider the model introduced in Ref. [Brackley, Morozov, and Marenduzzo, 2014] and described in Section II.1. We consider here the same bending persistence length used previously in this work – nm, appropriate for B-DNA – and a twisting rigidity of nm, which is in line with experimental values Moroz and Nelson (1997). In DNA there is a complete twist every base pairs (corresponding to approximately nm). To set these parameters, we chose a bead size nm, , and . We considered two situations. In the first case the dihedral phase is equal to giving a ribbon with no additional twist (). In the second case giving a ribbon that is undertwisted by a full turn every beads i.e. every base pairs. This second case corresponds to supercoiled DNA, with a supercoiling density of (note that in our model there is symmetry between negative and positive supercoiling because DNA denaturation is disallowed). In both cases, and as previously for the linear DNA, we considered a chain with backbone beads (each with its associated patches).
The chain was first relaxed within a cylinder of radius and length , verifying that the linking number of the circular ribbon was constant and approximately equal to 0 and 50 (in absolute value) for the case of torsionally relaxed and supercoiled DNA respectively (data not shown).
In Figure 8(b) we report the time evolution of the longitudinal extension of the DNA, , relative to its value in the relaxed state – the compression was performed in the presence of non-binding proteins, and by applying a constant force (in simulation units). For this value of the DNA-proteins system is already strongly compressed () independently on the degree of twisting. While the final value of is very similar for the torsionally relaxed and the supercoiled DNA, the dynamics is, quite intriguingly, rather different, as the supercoiled DNA shows a softer response to compression. For mild compression (see Fig. 8(a)) the situation is similar although, as expected, the final state is less compressed than the case (). These conclusions also hold in the absence of non-binding proteins (blue curves in Fig. 8(a)), however in that case the final value of is significantly lower (: as found previously with linear DNA, non-binding proteins contribute significantly to the force opposing compression by the piston.
By considering different values of the force and looking at the equilibrium compressed state, we further computed the compression force as a function of the ratio , for the torsionally relaxed and supercoiled DNA with non-binding proteins. The results are reported in Figure 9: both curves are quite similar, and, remarkably, relatively close to the linear case as well.
Supercoiling or, in other words, the local writhing of the DNA polymer has an effect similar to that of DNA-binding proteins (described in Secs. IV.3 and IV.4) – it leads to a change in the extensional elasticity and hence a softer response to compression. Like in the case of DNA-binding proteins, the effect of supercoiling in the force-compression curves is very small, being more detectable in the dynamics curves.
V Discussion and conclusions
In summary, in this work we have presented Brownian dynamics simulations of the compression and expansion dynamics of a DNA molecule, modelled as a self-avoiding linear or looped polymer, interacting with an ensemble of non-DNA-binding and DNA-binding proteins. Previous work (see, among other, Refs. [de Vries, 2012; Joyeux, 2015; Kim et al., 2015; Shendruk et al., 2015; Brackley et al., 2013]) had clearly demonstrated the importance of proteins on the thermodynamic conformations of bacterial DNA: non-DNA-binding proteins create macromolecular crowding which can promote global collapse of the bacterial chromosomes, while DNA-binding proteins such as H-NS provide local compaction. The novelty of our current work is that, by addressing a set-up which is directly relevant to single molecule experiments probing the entropic elasticity of bacterial DNA, we establish that the the inclusion of proteins further leads to important effects on the force-extension curves recorded upon compression, and also on the expansion dynamics of the polymer.
First, we presented some DNA-only simulations, where there are no proteins as a reference case. We have shown that for weak to intermediate compression our results confirm the entropic spring theory of Ref. [Pelletier et al., 2012], which predicts an abrupt decrease in elongation with compression. For large compression ( in our simulations), however, excluded volume interaction create a strong deviation from the theory.
Second, we have found that the osmotic pressure of non-DNA-binding proteins can dwarf the entropic pressure of the spring-like polymer during compression. In our simulations, we have considered polymer “beads”, and up to non-DNA-binding protein “beads”; although these numbers are comparable, the effect of proteins can be orders of magnitude larger in the compression curves. With respect to these thermodynamic curves, DNA-binding proteins affect the DNA elasticity, hence only have a minor effect overall.
It is interesting to ask whether osmotic pressure from unbound proteins can account for the high force ( pN) observed experimentally in the compression curves in Ref. [Pelletier et al., 2012]. As mentioned by the authors, while the curve overall can be fitted with the entropic spring theory, the numerical value which is expected of a self-avoiding polymer would be significantly smaller than observed: we note that even a small fraction of proteins could lead to an osmotic force which might account for this.
Third, we have explored the impact of proteins (again, non-DNA-binding and DNA-binding), on the expansion dynamics starting from a compressed DNA-and-protein system and following the removal of the force exerted by the piston. We found that the DNA-only simulations lead to a scaling behaviour for the extension as a function of time which is consistent with previous work in the literature Jung et al. (2013). The crowding introduced by non-DNA-binding proteins leads to a much slower dynamics, and to a much decreased apparent exponent. An interesting observation is that, when simulating DNA with DNA-binding-proteins only, tuning of the DNA-protein interaction leads to a popping-off kinetics during DNA expansion, where proteins are metastably bound to the DNA and detach one-by-one after the volume at their disposal increases.
Fourth, we have also performed more realistic simulations for bacterial DNA, where we considered circular polymers, with or without supercoiling. Quite remarkably, the torsionally relaxed and the supercoiled model DNA lead to a very similar force compression curve, which is also not far from the one we found for linear DNA. On the other hand, the dynamic response is significantly different, and markedly dependent on supercoiling.
While the results presented in this work focus on the case of proteins which are not charged, and are the same size as the DNA beads (a choice made for simplicity), we have also performed simulations taking into account the fact that the size of a typical bacterial protein is larger, and about twice the thickness of DNA Phillips et al. (2013) (Appendix A), and that proteins may have a non-negligible charge (Appendix B). The size results are shown in Appendix A, Figures 10, 11 and 12: these correspond to Figures 4, 5 and 6, and were obtained with a protein size of . The charge results are shown in Appendix B, Figures 13, 14, 15: these correspond to Figures 3, 4, 5. In all cases, the trends are qualitatively identical, and confirm our conclusions above. There are quantitative differences for the size simulations when the volume fraction is high: this is expected as, under those situation, it is the volume fraction, rather than number density, which determines the compression pressure and force. It is interesting that the effective exponent for DNA expansion slightly decreases for larger proteins, due to the increased crowding.
Overall, our results provide a generic framework within which to analyse experiments such as those in Ref. [Pelletier et al., 2012]; they also provide further testable predictions for future experiments, e.g., probing the dynamics of bacterial DNA or of supercoiled plasmids in vitro.
Acknowledgements.This work was partially funded by the European Research Council (ERC Consolidator Grant, THREEDCELLPHYSICS, Ref. 648050). MCP acknowlegdes studentship funding from EPSRC under grant no. EP/L015110/1. JL was supported by EU intra-European fellowship623637 DyCoCoS FP7-PEOPLE-2013-IEF.
Appendix A Effect of protein size
Figure 10 shows the compression force versus extension curve: comparison between these results and those of Figure 4 show a very similar trend. The quantitative values are very close together for large enough , and start to deviate significantly below , where the volume fraction is non-negligible, and excluded volume effects are larger for the larger proteins.
The comparison between Figure 11 and Figure 5 leads to the same conclusion – the slightly larger gap between cases with and without DNA-binding proteins is due to the fact that larger DNA-binding proteins lead to multiple binding to the DNA, hence the DNA becomes more compact. Thus the observed effect might not just be due to protein size, as bacterial DNA-binding proteins such as H-NS only have two DNA-binding sites (they are better represented in our model by the smaller proteins considered in the main text).
The effect of the larger protein size on the expansion dynamics is analysed in Figure 12, which shows a slower expansion dynamics. This is because the larger spheres create larger effective friction (as the friction, or viscosity of a hard sphere suspension is proportional to its volume fraction). The effective exponent we find is accordingly smaller, although this should be seen as a measure of the speed of the dynamics rather than a true dynamical exponent, which would require the study of different chain lengths. The decrease in the effective exponent is more marked in the case with DNA-binding proteins: again, this is due to the fact that these proteins can form multiple contacts with DNA.
Appendix B Effect of charge
The electrostatic interaction between particles is modeled by considering the Debye-Hückel potential Dwyer and Bloomfield (1995); Kunze and Netz (2000), in addition to the potentials already described in Section II.1,
where , is the charge of particle , the dimensionless dielectric constant (we consider ), the inverse Debye length, and the radius of the particle.
We consider nm, which is the Debye length in the cell interior. More explicitly, , where is the Bjerrum length in water and mM is the salt concentration inside cells Ando and Skolnick (2010).
Regarding the charge of the particles, DNA is known to carry two negative charges () per base-pair, due to the negatively charged phosphate group. As detailed in Section II.2, DNA bead corresponds to bp in our model. Therefore, there are phosphate groups per DNA bead. However, counterion (or salt) condensation leads to a neutralisation of of the phosphate groups Forrey and Muthukumar (2006). So, in fact, each DNA bead will only carry a charge of . The choice of the value of the proteins’ charge is not as straight forward since it depends on the protein residues. We opted to consider the value of the charge of an average protein, and use that value for negatively and positively charged proteins in our model. In [Runcong and Mitaku, 2001], the average of protein charges in bacteria was measured to lie in the range . For simplification we consider the proteins’ charge to be equal in magnitude to the DNA beads’ charge – – which corresponds to an effective charge due to neutralisation emerging from couterion condensation. Therefore, non-DNA-binding proteins are modeled as beads with charge and DNA-binding proteins with charge .
Figure 13 shows the compression force versus extension in the absence of proteins. Figure 13(a) shows that, like in Figure 3, the entropic spring theory of Ref. [Pelletier et al., 2012] agrees with the model for , but breaks down for . Figure 13(b) shows that, for , the DNA beads’ charge does not play a significant role in the elastic response of the DNA polymer, but that for weak compression forces () the effect of long-range electrostatic interactions is more noticeable: for a given force the DNA extension for the charged polymer is larger than for the neutral polymer.
Figure 14 for charged particles and Figure 4 for neutral particles are remarkably similar. The comparison between inset 2 in both Figures shows, however, that the presence of electrostatic interactions leads to a slightly higher DNA extension for a moderate force, as seen before.
Again, Figure 15 and Figure 5 lead to the same conclusions. In both cases the DNA-binding proteins lead to the formation of DNA clusters, hence compacting the DNA, giving a decrease in the compression force. The striking similarity between the results arising from the charged and neutral models further suggests that the electrostatic interactions do not play a significant role in the overall elastic response of DNA.
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