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APS/123-QED Knot-controlled ejection of a polymer from a virus capsid Richard Matthews, A.A. Louis, and J.M. Yeomans Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford 0X1 3NP, England (Dated: January 15, 2009) Wepresent anumericalstudyof theeffect ofknottingon theejection of flexibleand semiflexible polymers from a spherical, virus-like capsid. The polymer ejection rate is primarily controlled by the knot, which moves to the hole in the capsid and then acts as a ratchet. Polymers with more complex knots eject more slowly and, for large knots, the knot type, and not the flexibility of the polymer, determines the rate of ejection. We discuss the relation of our results to the ejection of DNAfromviralcapsidsandconjecturethatthisprocesshasthebiological advantageofunknotting theDNAbefore it entersa cell. 9 0 PACSnumbers: 87.15.-v,82.35.Lr,02.10.Kn 0 2 n Icosahedralbacteriophagesarevirusesthatinfect bac- DNA packing [12], the fact that DNA is highly knotted a teria. They typically consist of an almost spherical cap- is remarkable. Knots can prevent the transcription of J sid head with dimensions of several tens of nanometers, DNA by RNA polymerase, and cells have developed a 5 on the order of the persistence length of DNA, and a number of active ways to control these entanglements, 1 narrow cylindrical tail with an internal diameter of only for example through the use of molecular machines like a few nanometers, through which the phage injects its topoisomerases [1]. It seems unlikely that viruses would ] M DNAintobacteria[1]. RemarkablylongstrandsofDNA use topoisomerases to unknot their DNA, so instead a can be packed to almost crystalline densities inside the different mechanism must be at work. B rigidcapsidheads. Forexample,λphagehasagenomeof Motivated by these experiments, our aimin this letter o. length16micronssqueezedintoasphericalcapsidwitha is to study the effect of knots on the ejection of a viral i diameterofjust58nm. Internalpressurescanthusbeon genome from a phage capsid. The head-tail connector b theordertensofatmospheres. Anumberofviruses,such hasa channelofdiameter 2.3nm inλ [15]and1.7nm in - q asasλandφ29phages,exploitthispressuretoforcetheir φ29 [16], comparedwith an interstrandspacing of about [ DNA through their tail, and into their bacterial hosts. 2.5 nm for packed DNA [17, 18]. Considering that for a knot to pass multiple strands must go through simul- 1 Experiments, using fluorescent staining [2] and light v scattering [3, 4], have recently investigated DNA ejec- taneously, even simple knots in DNA are expected to be 3 tion from viral capsids. These show that the ejection too large to fit. We assume that instead the DNA must 9 beextrudedbyreptatingthroughtheknots. Abiological rate can be affected by temperature [3, 4], the presence 2 advantage would be that the viral DNA would enter the ofbindingproteins[4],genomelength[4]andtheconcen- 2 bacteria unknotted. tration of salt or other ions [3]. Further work has shown . 1 that ejection can be suppressed, so that only a fraction Buildingonthesuccessofpreviousmodeling[9,10,19] 0 that reproduced generic effects seen in experiments, the of the genome is emitted, by adding polyethylene glycol 9 approach we take is to represent the DNA by a simple to change the osmotic pressure of the solution surround- 0 bead-spring polymer. The polymer is initially confined ing the capsid [5]. In addition, pauses, which may be : v correlated with position along the chain, are seen in the to a spherical capsid and is coupled to a coarse-grained i solvent. It is allowed to eject through a small hole, and X ejection of certain phages [2]. A number of the generic thedrivingforceisthepressureofthepackedDNAinthe features of these experiments can be explained by treat- r a ing the DNA as a simple model polymer which is driven capsid. We find that the knots controlthe rate at which the polymer can leave the capsid, with slower ejection from the capsid by the energetic and entropic penalty of rates observed for more complex knots. The knot acts close confinement [6, 7, 8, 9, 10]. as a ratchet, with the polymer being ejected as the knot In anintriguingset ofexperiments, Arsuagaet al. [11, reptates along it. 12], directly extracted DNA from tailless bacteriophage The coarse-grained polymer we consider comprises mutants and showed that it was highly knotted, sug- beads connected linearly by springs. The beads interact gesting that this may also be the case inside the cap- through the potential sid. Moreover, recent simulations have provided further emveidr:enMceichfoerletthtieeptraelv.a[l1e3n]csehoofwkendotthsaotnthaecpornofibnaebdiliptoylyo-f V = kR02 ln 1− |~ri−~ri−1 | 2 knotting for a polymer contained in a sphere increases 2 i " (cid:18) R0 (cid:19) # X with the polymer length and the degree of confinement, σ 12 in agreement with earlier work [14]. Although the spec- + 4ǫ |~r −~r | trum of knots seen in experiments was not exactly the j>i i (cid:20) i j (cid:21) XX same as those from simulations of a random polymer, a + κ (~ri+1−~ri)·(~ri−~ri−1). (1) fact which presumably reflects the chiral nature of the i X 2 1 0.9 0.8 g nin0.7 ai m e0.6 n r FIG. 1: Schematic of initial configurations: Left, a spooled e fractio00..45 chain, right, a random configuration is achieved by packing g a the chain through a hole at the bottom of the capsid, which Aver0.3 is closed before equilibration. 0.2 0.1 The first term in Eq. (1) is the FENE spring potential, 00 2000 4000 6000 8000 10000 12000 Ejection time (simulation units) the second a repulsive Lennard-Jones term representing an excluded volume interaction between the beads and the final term is a bending potential which can be used FIG. 2: Fraction of beads remaining in the capsid as a func- to control the persistence length of the polymer. We tion of time for unknotted flexible (•) and semiflexible (◦) choose ǫ = σ = 1, k = 30 and R0 = 1.5. We consid- polymersandforflexiblepolymerswith31 (N),41 ((cid:4))and61 ((cid:7)) knots. The knots have a clear effect on the shape of the ered both flexible chains, κ=0, and semiflexible chains, ejectioncurve,andtherateofejectionisprimarilydetermined κ = 10, which corresponds to a persistence length of 10 byknot type. beads, on the order of the diameter of the capsids, as is the case for real bacteriophages. The dynamics of the beadswassimulatedbyusingavelocityVerletmolecular left in the capsid as a function of time from the release dynamics algorithm. The viral capsid was modeled as of the polymer. For this figure we used a small capsid a hard spherical shell by applying a force of magnitude k T/(σf4) to beads when their positions satisfied the and a random initial configuration but very similar re- B inequality | f |≤ 0.2, where f = 1−(x2+y2+z2)/R2. sults are found for other packings, initial conditions and capsid sizes. We compare results for unknotted chains In addition, we added a single hole small enough that onlyonebeadatatimecouldpassthrough. Twosizesof to those with the knots 31, 41 and 61. (Here we use the notation C , where C gives the minimal number of capsid were considered: one with a radius of R = 3.02σ k crossings in a projection of the knot onto a plane and k and a second with R = 4.36σ, which leads to a volume is a standard way to distinguish between knots with the three times larger. The polymer was coupled to a sol- same number of crossings [21].) Fig. 2 shows that there vent modeled using a stochastic rotation dynamics algo- is a clear slowing of the rate of ejection when a knot is rithm [20]. This provides a thermostat, which conserves present. Moreover, the rate of ejection depends on the momentum, and hence means that hydrodynamic inter- type of knot: the more complex the knot, the slower the actions between polymer beads are included in the sim- rate of ejection. ulation. The capsid was permeable to the solvent, the physical case for phage capsids. Atveryearlytimestheratesofejectionaresimilarand Two types of initial configuration, shown in Fig. 1, relativelyhighforbothknottedandunknottedpolymers. were considered. In both cases the knot was put in by Thiscorrespondstotheknotbeingtightenedandpushed hand in a tight configurationnear the exit, its type con- tothecapsidentranceasanyfreebeadsbetweentheknot firmed using the Alexander polynomial. In the first case andthe exitareejected. The knotis toolargeto escape, theremainingbeadswerepositionedtoformaspool[17]. so once it has moved to the hole it is held there by the Inthesecondtheywereinitializedinarandomconfigura- excesspressureinsidethecapsid. Nowthepolymerhasto tionoutsidethecapsidandthepolymerwasthenpacked reptatethroughtheknotbeforeanymonomersarefreeto byamotorpullingitthroughasecondholeinthecapsid, allowfurther ejection. Essentiallythe knot is acting as a opposite to the first. After the packing was completed, ratchet. Ifitdiffusesasmalldistanceintothecapsid,this the second hole was closed. In both cases the polymer freesalengthofpolymerbetweentheknotandthecapsid was equilibrated (within its local minimum), with the entrance. The knot is then quickly pushed back towards first bead held in positionjust outside the capsid, before the entrance by the driving force and the free section of the start of the ejection. polymer is ejected. Near the end of the ejection, when Polymers of length 100 beads were used with the only about ∼ 30−40 beads remain in the capsid, the smaller capsid,and polymers of length300 and230were ejection speeds up and there is a shoulder in the curves usedinthelargercapsidsfortherandompackingandthe inFig.2. Thiscorrespondstotheknotbecomingundone. spooledpackingrespectively. All reportedsimulationre- To confirm the effect of the knot on the dynamics, sults are averagedover at least 50 independent runs. Fig. 3 compares similar ejection curves for a semiflexi- Fig. 2 shows typical results for the fraction of beads ble polymer for length N =100 for three different initial 3 1 0.04 0.9 e) 0.035 m ning00..78 er unit ti 0.03 Average fraction remai0000....3456 e ejection rate (beads p0000..00..00121255 0.2 ag er 0.1 Av0.005 0 0 0 5000 10000 15000 0 2 4 6 8 10 12 Ejection time (simulation units) Number of knot crossings FIG. 3: Fraction of beads remaining in the capsid as a func- FIG.4: AverageejectionratesforC1knots: flexiblepolymers tionoftimeforsemiflexiblepolymers: (i)unknottedpolymer inthesmallercapsid(•);semiflexiblepolymersinthesmaller (◦) (ii) polymer with a 61 knot initially at the mid-point of capsid (△); flexible polymers in the larger capsid ((cid:4)); semi- the polymer chain (△) and (iii) a polymer with a 61 knot flexible polymers in the larger capsid (♦). For more complex initially near thecapsid entrance((cid:3)). knots,theratesforflexibleandsemiflexiblepolymersarevery similar. conditions (i) unknotted, (ii) with a knot initially at the the polymer mid-point (iii) with a knot initially at cap- polymers and capsids, but for more complex knots, the sid exit. For case (ii) the polymer initially ejects with rates converge and seem to be almost completely deter- the same speed as the unknotted polymer (i), but, once mined by the knot type. This suggests that, for tight, the knot has reached the capsid exit, the rate becomes confined knots, the rate of reptation of a knot may be similar to case (iii). independent of its flexibility. A similar independence of Simulations for longer polymers in a capsid of radius knot properties - the minimum length-to-radius ratio re- 4.36σ showed the same generic behavior. Some small quired for a tube forming a knot [22] and the diffusion differences were that the final unknotting occurred with constant [23] - from flexibility has been reported. a somewhat larger number of beads left in the capsid Finally, note that the ejection rates show a weak os- because there was more free space, and that the rates cillation as a function of knot size. The plausible expla- sometimesshowedaslightdecreasewithtime,mostlikely nation for this is that the knots we consider here with because the pressure decrease has an effect on knot dy- even and odd crossings belong to families with different namics. For the N = 300 randomly packed polymer we topologies. Knots31,51and71aretorusknotsand41,61 sometimesalsoobservedinitialjamming,withlargevari- and81 areeventwistknots. Itisreasonablethatthereis ations in ejection time between runs. This occurred be- a weak dependence of the reptation rate along the chain cause the knot was initially trapped in a very tight con- on the knot topology. We have observed a similar os- figuration. However once ejection did start, the rate was cillation of the polymer diffusion coefficients of knots on independent of the initial jamming time. unconfined polymers under tension [24]. A quantitative comparison of ejection rates is shown Our modelof a bacteriophageis highly simplified. For in Fig. 4. Results are presented for both flexible and example, we ignored the tail. We checked that this does semiflexiblepolymersforknotswithupto6crossingsfor not have an important effect on the qualitative behav- N =100chainsinthesmallcapsid,andforknotswithup ior by simulating a capsid with a tail attached. Per- to 12 crossing for N = 230 chains in large capsids. The haps more importantly, to make simulation feasible, our resultsplottedarefortheknotsoftypeC1. Otherknots, model of DNA inevitably ignores much chemical detail suchas52and62,wereconsidered. Theresultswerevery and our polymer diameter to persistence length ratio is similar to the C1 knots with the same C. Average rates much larger than in real DNA. Moreover, viral DNA is were obtained by measuring the time taken for between typicallymuchlongerthanthe polymerswestudy. How- 75% to 35% of the beads to be expelled. This protocol ever, we argue that the generic effects we observe: that waschosentoavoidanyearlytime jammingandthe late the ejection rate is dominated by the rate of knot repta- time unravelling of the knot. Small changes in the cut- tion, and that the knot is unravelled as it emerges from offs gave no qualitative changes in the results. thecapsid,arebasicpropertiesofknottedpolymersmov- Fig. 4 showsthe trend ofdecreasingejection rate with ing through a narrow exit hole, and should therefore be increasing knot complexity. For the unknotted case, robust to the inclusion of these details. Knots in phages there are clear differences between the different kinds of may initially be loose but we expect that they will be 4 of a largercapsid. Fig. 5 comparesthe averagedejection 0.8 curve to that for a single 121 knot, showing that the knots stack up at the capsid entrance and behave like 0.7 a single prime knot with a similar number of crossings. g0.6 In the longer DNA in a viral capsid one might expect n maini0.5 to see multiple knots which would sequentially collect at n re the exit. Thusthe knotlengthwouldincreasewithtime, actio0.4 which is expected to slow the ejection rate. It will be ge fr0.3 interesting to see if any sign of this effect can be seen era experimentally. Av0.2 To summarize, we have used simulations to determine 0.1 theeffectofknottingontheejectionofflexibleandsemi- flexible polymers from a spherical capsid. We find that 0 0 1 2 3 4 5 6 7 8 9 the ejection rate is controlled primarily by the knot, not Ejection time (simulation units) x 104 the pressureof confinement. The knot movesto the hole in the capsid and then acts as a ratchet. Polymers with FIG. 5: Fraction of beads remaining in the capsid as a func- more complex knots eject more slowly. For tightly con- tion of time for semiflexible polymers with three knots ran- fined knots the flexibility of the polymer is not key in domly selected from 31, 41, 51, 52 and 61 (△) and the prime knot 121 (◦). determining the rate of ejection. Repeatingpackingexperiments[25]withknottedDNA would shed light on the dynamics of knots on polymer tightened as the DNA is forced through the capsid en- chains, and it is also of interest to ask whether knots trance. can affect the motion of biomolecules as they traverse Experimentssuggestthattherecouldbemultipleknots nanopores [26]. in the DNA in a viral capsid [11, 12]. 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