CCan entropy save bacteria?
Suckjoon Jun ∗ FAS Center for Systems Biology, Harvard University, 7 Divinity Avenue, Cambridge, MA 02138, USA (Dated: October 8, 2018)
This article presents a physical biology approach to understanding organization and segregationof bacterial chromosomes. The author uses a “piston” analogy for bacterial chromosomes ina cell, which leads to a phase diagram for the organization of two athermal chains confinedin a closed geometry characterized by two length scales (length and width). When applied torod-shaped bacteria such as
Escherichia coli , this phase diagram predicts that, despite strongconfinement, duplicated chromosomes will demix, ı.e., there exists a primordial physical drivingforce for chromosome segregation. The author discusses segregation of duplicating chromosomesusing the concentric-shell model, which predicts that newly synthesized DNA will be found inthe periphery of the chromosome during replication. In contrast to chromosomes, these resultssuggest that most plasmids will be randomly distributed inside the cell because of their small sizes.An active partitioning system is therefore required for accurate segregation of low-copy numberplasmids. Implications of these results are also sketched, e.g., on the role of proteins, segregationmechanisms for bacteria of diverse shapes, cell cycle of an artificial cell, and evolution.
Contents
I. Introduction II. Physical biology approach to bacterialchromosomes in a cell.
III. Discussion and frequently asked questions
IV. Final remarks Acknowledgments A. More technical description of the phase diagram inFig. 4. B. Self-consistent mapping between the Pincus andthe close-packed chains. References ∗ Electronic address: [email protected]
I. INTRODUCTION
Chromosomes are a cornerstone of fundamental pro-cesses of any cell, and harmony between their physi-cal properties and biological functions is an evolution-ary consequence. In bacteria, two defining processes ofthe cell cycle, DNA replication and chromosome segre-gation, progress hand-in-hand. The model system
Es-cherichia coli contains a single circular chromosome in arod-shaped cell. DNA replication starts at a unique ori-gin of replication ( ori ), creating a replication bubble thatgrows bidirectionally, and the two replication forks meetat the terminus ( ter ) located at approximately the op-posite clock position of ori on the circular chromosome(Fig. 1). In slowly growing cells, there is a one-to-onecorrespondence between each complete round of replica-tion and the cell cycle. In fast growing cells, on the otherhand, the cell divides more frequently than the progres-sion of the forks from ori to ter and, thus, new replica-tion has to initiate before the completion of the previousround of duplication, leading to “multifork” replication.In either case, replication and segregation are coordi-nated with the growth of the cell, and recent visualizationexperiments have begun to reveal how replicating chro-mosomes move and segregate during the bacterial cellcycle. Although details vary from organism to organism,the observation common to all bacteria of rod-shapedcells studied so far shows directed movement of duplicat-ing chromosomes along the long axis of the cell [as seenin E. coli (Bates and Kleckner, 2005; Elmore et al. , 2005;Nielsen et al. , 2006; Wang et al. , 2006),
Caulobacter cres-centus (Viollier et al. , 2004),
Bacillus subtilis (Berkmenand Grossman, 2006) and
Vibrio cholerae (Fiebig et al. ,2006; Fogel and Waldor, 2006)]. Experimental data alsosuggest that, once duplicated, there is a fair degree ofcorrelation between the relative clock position of circularchromosome and their average positions along the longaxis of the cell, which often (but not always) shows the a r X i v : . [ q - b i o . S C ] A ug Pre-replication stagePost-replication stage ori ( ) ter ( ) FIG. 1 Schematic illustration of the bacterial cell cycle.Replication begins at the 12’ position ( ori ) on the circularchromosome. Replication forks grow bidirectionally and meetat the opposite clock position ( ter ). During the bacterial cellcycle, replication and segregation progress hand-in-hand, cou-pled with the growth of the cell. principal linear ordering of chromosome (Teleman et al. ,1998; Viollier et al. , 2004; Wang et al. , 2006).Perhaps the most influential model on chromosomesegregation so far was proposed by Jacob, Brenner, andCuzin (1963) in their seminal paper on the replicon modelof
E. coli : if replicating chromosomes (e.g., duplicated ori ’s) are attached to the elongating cell-wall membrane,they can be segregated passively by insertion of mem-brane material between the attachment points duringgrowth. The model is intuitive and elegant – but wrong.(See Sec. II.F).Since the work of Jacob and colleagues, two classesof models have been proposed to explain the observeddirected drift of duplicating chromosomes: (a) biologicalmechanisms such as DNA replication (Lemon and Gross-man, 2001), co-transcriptional translation of membraneproteins (Woldringh, 2002), RNA transcription (Dworkinand Losick, 2002); and (b) physical/mechanical driv-ing forces such as mechanical pushing between chro-mosomes (Bates and Kleckner, 2005; Kleckner et al. ,2004) or conformational entropy of duplicating chromo-somes (Danchin et al. , 2000; Fan et al. , 2007; Jun andMulder, 2006). These two classes of models are not mu-tually exclusive.The second class of model (b) is important in that itprovides a basic physical framework to examine the ne-cessity and the roles of proteins or machinery (if any)involved in the cellular process and, if so, what condi-tions these putative biological players need to satisfy inorder to successfully carry out their biological functions.For instance, when the cell is lysed, the bacterial chromo-some expands to several times the size of the encapsulat-ing cell (Cunha et al. , 2001) (Fig. 3A). This observationis puzzling since duplicated chromosomes must stay seg-regated without mixing until cell division for the viabilityof the cell, and yet the natural size of each chromosome is larger than the confining volume of the cell itself; whydo the chromosomes not mix when they should occupyas much volume as possible? Thus, if the basic physi-cal principles predict that the duplicated chromosomesshould mix inside bacterial cell, this would explain whyputative proteins must be actively involved in segrega-tion of replicating chromosomes. However, if the physi-cal analysis predicts otherwise, this would imply the ex-istence of a primordial physical driving force underlyingchromosome segregation in bacteria and that active bi-ological mechanisms may be present for other reasons(e.g., when the cell cycle is perturbed by external stress).The purpose of this paper is to present a compre-hensive and rigorous physical model of bacterial chro-mosomes that makes experimentally testable predictionsbased on measurable physical parameters of the bacte-rial cell and its chromosomes. We have developed ourmodel with the idea that physical properties of the chro-mosomes must interplay with the shape and size of thecell, and bacteria might have evolved to learn how toregulate these parameters to ensure proper segregationof chromosomes.This article is organized as follows: First, we beginwith the concept of entropy, providing specific examplesto explain how order can emerge from disorder drivenby entropy. Second, we explain, using a piston analogyof bacterial chromosomes confined in a cell, how thegeometric conditions of the cell interplay with thephysical state of the confined chains. Based on thisanalogy, we present a phase diagram for the organizationof self-avoiding chains in a box whose dimensions aredefined by two lengthscales (width and length). Toour knowledge, this is the first direct application ofthe de Gennes-Pincus blob theory (de Gennes, 1979;Pincus, 1976) to closed geometries with more than onelength scale (as opposed to, e.g., open channels, slits, orspheres, which can be characterized by a single lengthscale). Then, we critically examine the model system
E. coli and predict that duplicated chromosomes ofthis organism should segregate spontaneously (with theordering of individual chromosomes being principallylinear unless interfered by tethering to the wall). Wealso predict the opposite behavior for plasmids (namely,random distribution) because of their small sizes.We discuss various implications concerning entropy-driven organization and segregation of chromosomesin bacteria of diverse shapes as well as in an artificial cell. For physical approach to eukaryotic chromosomes, the reader isencouraged read Marko and Siggia (1997). In his celebrated book,
Scaling Concepts in Polymer Physics ,de Gennes noted “This analysis can be extended to chains thatare squeezed in slits and to other geometries provided that theconfining object is characterized by a single length D .” [see p.51 in de Gennes (1979)]. (a) chain connectivity(b) mixing of particlessegregation of chains FIG. 2 Subtle aspect of entropy. (Molecular dynamics sim-ulation.) (a) Two species of particles ( N = N = 200),initially separated by a wall, mix as the wall is removed. (b)If chain connectivity is introduced to this system by linearlyconnecting the particles of the same species, the two chainssegregate. ( N = N = 512) II. PHYSICAL BIOLOGY APPROACH TO BACTERIALCHROMOSOMES IN A CELL.A. Entropy measures the degrees of freedom of the system
In his influential book,
What is life? , Schr¨odinger(1944) asserted that the only thermodynamically equi-librium state of a living being is death. Defining entropyas a measure of ‘disorder’, in the last chapter ‘Is LifeBased on the Laws of Physics?’, he struggled to explainhow order (life) can be achieved from disorder. This as-sociation of life with minimal entropy, however, can bemisleading.In Fig. 2(a), we show snapshots from molecular dy-namics simulations, where two species of N and N par-ticles in equilibrium (blue and red, respectively) in a longrectangular box, initially separated by a wall. As we re-move the wall from the system, the two species start tomix. The driving force of this process is the well-known“entropy of mixing,” which can be estimated as∆ S A = k B (ln Ω (cid:48) A − ln Ω A , ) (1) ≈ − k B (cid:20) N ln (cid:18) N N (cid:19) + N ln (cid:18) N N (cid:19)(cid:21) > , (2)where k B is the Boltzman constant, N = N + N , andΩ (cid:48) A = N ! /N ! N ! (Ω A , = 1) denotes the total number ofconfigurations of the system after (before) mixing.Entropy, however, is more subtle than a simple mea-sure of disorder. To see this, let us start with the mixedstate of Fig. 2(a) and connect the particles of the samespecies, creating two long linear chains, one painted withblue and the other with red [Fig. 2(b), left]. Anotherimportant condition is the excluded-volume interactionbetween the particles. The reader is encouraged to per-form this simple computer simulation, and she or he willsee that the two chains de-mix, ı.e., that “order” emergesout of disorder (for the exact condition of segregation, see Sec. II.C).There are also other examples in soft condensed mat-ter physics where entropy leads to ordering. For instance,Onsager’s hard-rod model (1949) of the nematic-isotropictransition of liquid crystals is based on a similar physicalinsight of the trade-off between loss of orientational en-tropy and gain in positional entropy when hard rods areoriented parallel to one another. Crystallization of hardspheres (Alder and Wainwright, 1957; Wood and Jacob-son, 1957) is another example where ordering allows alarger room for fluctuations.The above examples clarify what entropy really mea-sures, namely, the degrees of freedom, or the size of thephase space, of the system. For the main theme of this ar-ticle, the specific case in Fig. 2 points out the importanceof chain connectivity, a keyword in polymer physics. Inthis example [Fig. 2(b)], chain connectivity defines a con-formational space Ω B within the configuration space Ω A ,and within Ω B the typical conformations of the chainsare those de-mixed. This emergence of order from disor-der due to entropy is our starting point of understandingchromosome segregation in bacteria. B. Piston analogy of bacterial cells and chromosomes.
Imagine a cylinder containing two long linear moleculeswith excluded-volume, closed by two pistons which areinitially far apart and do not perturb the chains, as shownin Fig. 3(b) top. In this dilute, confined solution of poly-mers, the chain conformations are well-described by theblob model; each chain is a series of blobs, stretched alongthe long axis of the cylinder without large-scale backfold-ing (de Gennes, 1979; Pincus, 1976). The origin of this linear ordering of confined chain is entropic because over-lapping blobs costs free energy (Grosberg et al. , 1982; Jun et al. , 2007). As we shall discuss below, it can explainthe principal linear chromosome organization observed inrod-shaped bacteria.Here, the notion of “blob” is the central concept ofour approach to bacterial chromosomes. It can formallybe defined as an imaginary sphere within which the localchain segment densities are correlated (Doi and Edwards,1986). In the above-mentioned case of dilute solution ina cylinder, the blob size ξ is defined by the width of thecylinder D . It is important to note that a blob is also aunit of free energy of the chain and each blob contributes ∼ k B T to the free energy, regardless of its size (Jun et al. , 2007). As we shall discuss in Sec. III, for bacterialchromosomes, the closest concept to the polymer blobsin the literature is macrodomains (Esp´eli and Boccard,2006) or “topological domains” (Deng et al. , 2005). For Note that polymer physics was still in its infant stage duringSchr¨odinger’s time. For instance, Flory’s magnum opus,
Prin-ciples of Polymer Chemistry (1953), appeared almost 10 yearsafter
What is Life? (1944). linear ordering of chain conformation pushingmixing c o m p r e ss i on random ordering of chain conformation “blob” ξ (a) (b) FIG. 3 Piston analogy of bacterial chromosomes in confine-ment. (a) When the cell is lysed, the isolated chromosomesexpands to several times the size of the cell. For our analy-sis, we used the published values of the nucleoid size beforeand after lysis, 0.24 µ m (Woldringh and Odijk, 1999) and3.3 µ m (Romantsov et al. , 2007), respectively, for E. coli
B/rstrain. (b) Two athermal chains (ı.e., with excluded-volumein good solvent) are confined in a cylinder whose pore widthmuch smaller than the size of an unperturbed chain. Theorganization of the chain depends on the aspect ratio of thecylinder and the density of the monomers. instance, the latter denotes the largest independent chro-mosomal subunits, whose topological changes (such ascutting) do not affect other neighboring domains of thechromosome (for example, one may imagine loops thatare independently anchored to one another). Indeed, itis also our view that chromosomes are best understood asa series of blobs, where each blob acts as an independentstructural unit [see, also, Romantsov et al. (2007)].If we compress the pistons further, the two chains willtouch one another and then partly overlap. This, how-ever, is not a favorable situation because overlappingblobs costs conformational entropy and, thus, the twochains repel each other (Bates and Kleckner, 2005; Junand Mulder, 2006; Kleckner et al. , 2004) and retract tooccupy each half of the cylinder. Since the total acces-sible volume within the cylinder has decreased by themoving pistons, the total internal energy has to increase.In other words, the blob size will decreases accordingly(see above) [Fig. 3(b) upper box].We emphasize that the principal organization ofthe retracted chains is still linear, and this averagelinear ordering is maintained as long as the chains stay This chain demixing can be understood intuitively. If we freezethe red chain, its inner envelope volume can be considered as asystem of fixed obstacles. Thus, the blue chain moves away fromthe red chain, because the blue chain’s gain in conformationalentropy exceeds its entropic loss by occupying smaller volumefree of the obstacles. This phenomenon is due to the chain con-nectivity. segregated. If the piston continues to move and increasethe compression of the chains, the blobs also continueto become smaller to reflect the increased free energy.Once the blobs reach their critical size, the chains gainmore conformational entropy by mixing with each other,and the chains lose their principal linear ordering andinstead become a random walk of the connected blobs(Fig. 3(b) lower box).
C. Phase diagram for mixing and de-mixing of polymersolution in a closed geometry.
The piston analogy illustrates how the physical prop-erties of the chromosomes interplay with the confininggeometry and its volume. We have obtained by rigorousanalysis the phase diagram in Fig. 4, which describes theorganizations of two chains in a rectangular box charac-terized by two length scales, width D and length L of thebox.The dimensionless quantity x = R F /D is the ratio be-tween the size of the unperturbed chain ( R F ) and thewidth of the cylinder; thus, the x -axis represents howstrong the confinement is. The y -axis indicates how densethe polymer solution is in the box, because the correla-tion length (blob size), ξ bulk , decreases monotonically asa function of the monomer density of the polymer in so-lution (see Appendix A).Our phase diagram is divided into two regions, onewhere chains segregate (blue), the other where they mix(red) (Fig. 4). The diagram is further divided into sixregimes. The boundary between segregation and mixingis given by the following geometric condition k = LD = Dξ bulk , (3)ı.e., this is where the aspect ratio of the box, k , equalsthe ratio between the width of the box and the size ofthe blobs. The rest of the phase diagram can be bestunderstood as we move along the horizontal dotted linein Fig. 4. Since ξ bulk depends only on the chain density,for chains with fixed contour length, this line representsthe change of the aspect ratio of the confining box ( k )from filamentous ( k >
1) to slab-like ( k <
1) throughcubic/spherical ( k = 1) shapes, keeping constant the vol-ume of the box. Note that the longer the box is the betterthe chains segregate, and that the chains can readily mixin a sphere ( k = 1) (Jun et al. , 2007).We leave a more complete description and technicaldetails of the phase diagram to Appendix A. D. Application to bacterial chromosomes.
Using the phase diagram presented in Fig. 4, wecan predict whether a specific organism can segregate
FIG. 4 Phase diagram of a 2-chain polymer solution confined in a closed geometry of dimensions D × D × L . The x-axis representsthe shape of the geometry. The y-axis represents the concentration of the polymer solution. Regimes with a red backgroundrepresent mixed states, while those with a blue background represent segregated states. The dotted horizontal line showsthe effect of aspect ratio of confinement on two chains of constant length N . Shown along this line are a few representativechain conformations (b)-(g). For cubic confinement with side D = R F (a), two chains mix at the free-energy cost of order k B T (Grosberg et al. , 1982; Jun et al. , 2007). (h) Same as (d), but the two chains are in partial overlap of penetration depth σ .Due to compression, the correlation length ξ bulk of the confined chains is smaller than the width of the rectangular box D (seethe stacking blobs). [ R is the equilibrium end-to-end distance of an unperturbed chain in a long cylinder, and k the aspectratio of the box.] See Appendix A for more technical details. its chromosomes using only conformational entropy –with no active control mechanism. To this end, weexamined the model system E. coli
B/r (H266), oneof the most well-studied and well-documented bacterialstrains (Woldringh and Odijk, 1999). When in steadystate, slowly growing (doubling time of 150 minutes)
E.coli of this strain have a rod-shaped cell with hemispher-ical caps, with average length is 2.5 µ m and the width0.5 µ m. However, the actual dimension in which chro-mosomes are confined inside a cell, namely the nucleoid,is smaller because of the well-known phenomena of nu-cleoid compaction (Woldringh and Odijk, 1999). As aresult, these values are estimated to be D = 0 . µ m and L = 1 . µ m for new-born cells containing a single chro-mosome .On the other hand, the correlation length ξ bulk is amore difficult quantity to measure experimentally, be-cause a number of factors contribute to chromosome or- The population average is ¯ L = 1 . µ m (Woldringh and Odijk,1999) with L ≈ . L (Kubitschek, 1981), and we assumed alinear relationship between cell size and nucleoid length. ganization in bacteria (e.g., molecular crowding, super-coiling and DNA-protein interactions) (Stavans and Op-penheim, 2006). Nevertheless, we can safely assume thatthe lower bound of ξ bulk should be the persistence lengthof dsDNA, (cid:96) p = 50nm. Note that recent experimen-tal studies have shown that there are as many as (cid:39)
E. coli chromosome, correspond-ing (cid:39) et al. , 2005). If we assumethat each domain occupies d of a nucleoid volume, weobtain d (cid:39) (cid:96) p of bare DNA.Very recently, Krichevsky and colleagues have mea-sured ξ bulk of nucleoid isolated from the same E. coli strain [B/r (H266)] using a more direct method offluorescence correlation spectroscopy (FCS) (Romantsov et al. , 2007). Their careful analysis of the amplitude ofthe FCS correlation functions of randomly fluorescent-labeled nucleoids revealed ≈ d (cid:39) in vivo [based on the nucleoid volume ≈ µm (Woldringh For supercoiled DNA, the persistence length of the plectoneme isestimated to be twice that of dsDNA (Marko and Siggia, 1995). x = R F D I II IIIIVVVI y = R F ξ bulk E. coli (chromosome)plasmid mixed segregated
FIG. 5 Segregation of DNA in
E. coli . E. coli is in thecompressed-cigar regime III (the bars represent 50% rangeof the average values x = 13 . y = 37 .
5, which we addedto show the robustness of our estimate). Thus, the duplicatedchromosomes have principal linear organization in the cell andsegregate spontaneously to maximize their conformational en-tropy. On the other hand, plasmids are typically orders ofmagnitude smaller than the chromosomes, ı.e., the point forchromosome should move on the phase diagram parallel tothe y = x line and enters the mixing regime (e.g., V). Thus,during the course of evolution, plasmids may have acquired anadditional, more active segregation mechanism other than en-tropic repulsion to ensure each daughter cell receives at leastone copy. and Odijk, 1999)]. They also calculated the physicalsize of the unit based on less direct measurement of thediffusion constant of isolated nucleoids. Their estimatedvalue was d = 70 ± d = 87nm above. Indeed, we have used theseparameters including the average size of the isolatednucleoids R F = 3 . µ m (Romantsov et al. , 2007) (fromthe average spherical volume of natively supercoilednucleoid, 18 µ m ) and obtained x = 13 . y = 37 .
5– the
E. coli
B/r strains are in the segregation regimeII (Fig. 5). Thus, the replicating chromosomes couldsegregate purely driven by conformational entropy, andthey will remain demixed (Bates and Kleckner, 2005;Jun and Mulder, 2006).
E. Application to plasmid segregation – evolutionarynecessity for segregation strategies.
Another important biological question concerns plas-mid segregation in bacteria. Importantly, most low-copynumber plasmids, such as R1, are believed to rely on morebiological, protein-based active segregation mechanisms[e.g., par system (Garner et al. , 2007; Schumacher et al. ,2007)]. Our phase diagram can explain the need for ded-icated segregation mechanisms as follows: Since the sizeof typical plasmids is ∼ ∼ kb). At the insertion oftwo such plasmids in bacteria, the total amount of DNA(and, thus, ξ bulk ) remains practically constant, althoughthe total number of chains increases to n = 4. In ourphase diagram, this corresponds to changing both x and y of chromosome by a factor of ∼ (since R F ∼ N / ),ı.e., the plasmids enter the mixing regime V and theywill distribute randomly inside the cell. Indeed, recentexperimental results on the mobility and distribution ofsynthetic minimalized RK2 plasmid lacking the partition-ing system are fully consistent with our prediction of ran-dom distributions (Derman et al. , 2008).The quantitative reasoning above has broader im-plications for segregation strategies and copy-numbercontrol of plamids (Nordstr¨om and Gerdes, 2003): (i) An active segregation mechanism, especially, for low-copynumber plasmids (Adachi et al. , 2006; Derman et al. ,2008; Gerdes et al. , 2004), or (ii) “Random” segregation,ı.e., without a protein-based segregation mechanism,plasmids may produce multiple copies so that, despitetheir small sizes, their chance of survival increasesupon cell division. In random segregation, copy-numbercontrol is an interesting and important issue [See, forexample, Brenner and Tomizawa (1991) and referencestherein]. F. Segregation of replicating nucleoid.
So far, we have discussed the physical conditions inwhich duplicated chromosomes will stay segregate, ı.e.,the conditions for a primordial driving force for chromo-some segregation. How then can we explain observedtrajectories?Figures. 6 and 7 show experimental data of the aver-age positions of the chromosome loci in
E. coli (Bates andKleckner, 2005) and
C. crescentus (Viollier et al. , 2004).In
E. coli , for instance, ori first moves towards the mid-cell position and, then, replication starts. The duplicated ori ’s split and move, on average , towards the 1/4 and3/4 positions. In the mean time, ter drifts slowly fromthe cell pole towards the midcell, crossing one of the ori trajectories. Note that, in principle, there are four determining parameters ofour phase diagram, where the fourth parameter is the number ofchains n . If we release the constraint imposed on n [e.g., n = 2(chromosomes) and n = 4 (chromosomes + plasmids)], whilekeeping constant D and ξ bulk , the two cells are connected by adiagonal line in the phase diagram because, then, R F is the onlyadditional variable (Fig. 5). There is an ongoing debate about how long the duplicated ori and other loci stay together before splitting during chromosomesegregation (“cohesion”). See, for example, Bates and Kleckner(2005); Nanninga et al. (2002); Nielsen et al. (2006); Sunako et al. (2001).
Snapshot Simulation SimulationExperiment with replication factory without replication factory ori ori ter nucleoid pole F r a c t i on -r ep li c a t ed f f = 00.20.40.60.80.99 replisome oriter FIG. 6 Chromosome segregation in
E. coli : simulation vs. experiment. (Left) A series of typical conformations of a replicatingcircular chain from 0% to 99% replication (un-replicated strand in gray, replicated strands in red and blue). We also presenttwo sets of segregation pathways ( ori-ter trajectories during replication) with and without “replication factory”; the third set ofsimulation where ori starts at the midcell can be found in Supporting Information in Jun et al. (2007). We simulated replicationfactories by enforcing physical proximity of the two replisomes during replication, but we did not fix their position within thecell. The dotted lines show the results of 10 individual simulations, and the thick solid lines show the average trajectoriesof ori (red and blue) and ter (black). (Center-to-Right) We juxtapose the simulations with the published data in Bates andKleckner (2005) in an attempt to capture the main features of the experimental observations. For comparison, we used thefraction-replicated f as our “universal clock” and scaled the height of the simulation trajectories to match 0 < f < In C. crescentus , loci trajectories seem even more strik-ing (Viollier et al. , 2004): one ori stays at the cellpole and the duplicated ori moves much faster than thegrowth rate of the cell (Fig. 7). Indeed, this large dif-ference between the cell growth rate and the speed of ori is the evidence against Jacob and colleagues’ modelmentioned earlier, at least, in
C. crescentus . The rest ofthe loci follow similar trajectories, creating a mirror-likeimage of the organization of duplicated circular chromo-somes.
Ori is found at the old cell pole and ter at thenew cell pole; importantly, there is a one-to-one corre-spondence between the clock position of the chromoso-mal loci and their average physical locations along thelong axis of the cell. In
E.coli , recent data from severallabs [e.g., (Nielsen et al. , 2006; Wang et al. , 2006)] alsosuggest similar principal linear organization, except thattheir clock positions are rotated by 90 degrees with thetwo chromosome arms between ori and ter occupyingeach cell half.To explain these experimental data, we have previouslyproposed the concentric-shell model (Jun and Mulder,2006), which was inspired by the observation of nucleoidcompaction in
E. coli (see, also, Fig. 8). Here, our pre-diction was that the newly synthesized DNA will move much faster in the periphery of the nucleoid near the cell-wall membrane than inside the nucleoid body (which is ameshwork of chromosomal DNA), faster than the typicaltimescale of the cell cycle of
E. coli ( >
20 min.) or
C.crescentus [ ≈
240 min. in Viollier et al. (2004)]. Indeed,this minimal model and assumption reproduced most ofthe major features of the experimental data in both or-ganisms. For
E. coli , we note that Fan et al. (2007)have also used a free energy-driven string model to ex-plain the experimental data. The most important resultin their study is that the size of the chromosomal domainis important (the larger the domain is, the better chro-mosomes segregation), which we have explained aboveusing our phase diagram.We emphasize that the concentric-shell model is con-sistent with other models, as long as they also implythe preferential occupation of the nucleoid periphery vol-ume by newly synthesized DNA in the early stage ofcell cycle. These models include the transertion model In our previous study (Jun and Mulder, 2006), this gap was evensmaller than the width of the chain in our simulations.
Fraction-replicated f Time (minutes) F r a c . o f nu c l eo i d l eng t h D i s t. f r o m c e ll po l e ( µ m ) Simulation Experiment orioriterterori ori f =0.5) FIG. 7 Chromosome segregation of
C. crescentus , comparingsimulation (Left) vs. experiment Viollier et al. (2004) (Right).The simulated trajectories are the average of 26 individualsimulation runs (or cells); we show the trajectories of ninerepresentative loci (including ori and ter ) on the right-arc ofa circular chromosome for the entire duration of replication(up to 99.9%), whereas experimental data are only availablefor trajectories up to 50% of replication. For clarity, we showthe trajectories only from the onset of replication of each lo-cus. A full trajectory of ter is shown, however, to emphasizeits slow drift from the cell pole to the cell center during repli-cation, in contrast to the fast, directed diffusion of ori2 inthe nucleoid periphery. A main difference from the
E.coli simulation is the additional assumption that we kept ori1 inthe volume near the stalked pole until 10%-20% of the chainhas been replicated. The concentric-shell model used in thesimulation is formal and is not inconsistent with other addi-tional mechanisms that may act on the directed movement of ori2 , although in our simulation we did not need any such as-sumptions. [Reprinted from Fig. 6 of Jun and Mulder (2006).Copyright (2006) National Academy of Sciences, U.S.A.] by Woldringh (2002), which assumes an interaction be-tween the nucleoid and the inner-cell wall membrane viaco-transcriptional translation and protein translocation,or even the hypothetical role of
Par proteins on ori trans-portation in the cytoplasmic space, namely, outside thenucleoid volume (Viollier et al. , 2004). We believe time-lapse experiments with higher spatiotemporal resolutionwill reveal the nature of directed motion between diffu-sive, biased random walk vs. transportation by polymer-ization or motor proteins.
III. DISCUSSION AND FREQUENTLY ASKEDQUESTIONS
Based on the piston model of the bacterial cell, we havepresented the above physical biology model of bacterialchromosomes. Importantly, our model makes an exper-imentally testable predictions whether duplicated chro-mosomes will mix or segregate, and we have criticallyexamined our predictions against the published data of
E. coli
B/r strain.We emphasize the importance of the concept of corre-lation length ξ , a central parameter of our chromosomemodel. To recapitulate, this is the size of the chromoso- stretching and twistingstabilization by proteinsclose-packing in a cell Pincus blobTopologically indepen-dent structural unit
FIG. 8 Illustration of chromosome as a close packing of astring of Pincus blobs (see, also, Appendix. B). The blobsschematically represent correlation-length structural units ofthe chromosome, which can be formed by stretching (andtwisting) of the chain. (Although not essential to our the-ory presented in this article, independence of the topologi-cal states of individual blobs may be achieved by nucleoid-associated proteins.) To maintain an average linear organi-zation, we envision the blobs forming a helical coil that fillsthe nucleoid volume. If the helicity reverses, nucleoid enve-lope may develop constricted, bi-lobed or multi-lobed shape.Molecular crowding and possible interactions between the in-ner cell-wall membrane and dsDNA protruding from the nu-cleoid is not shown for clarity. The gap between the nucleoidand the inner cell-wall membrane is not drawn to scale andmay vary from organism to organism. mal structural unit within which the density fluctuations(and, loosely, the motions) of the chain segments are cor-related. The major assumption underlying our model isthat, although a real bacterial chromosome is a very com-plex entity because of supercoiling and the presence ofvarious nucleoid-associated proteins (whose roles are stilllargely unknown), there is a single length scale ξ thatcharacterizes the nucleoid. Then, the chromosome canbe interpreted as an entity consisting of these structuralunits, and chromosome organization in bacteria can beunderstood as a close packing of a string of Pincus blobs.We illustrate this view in Fig. 8 and leave the scalinganalysis for equivalence between the Pincus chain (Pin-cus, 1976) and the compressed chain in Appendix. B.Based on our results discussed in the previous sections,there are two general principles in understanding chromo-some organization and segregation in bacteria.I. The larger the correlation length is, the betterchromosomes segregate.II. The larger the aspect ratio is (ı.e., the longerand/or the narrower the cell is), the morechromosomes tend to demix.Below, we discuss various questions concerning ourchromosome model. A. Role of proteins
Proteins directly and indirectly change the physicalproperties of chromosomes and the cell morphology and,thus, they are important for chromosome organizationand segregation in the following three contexts.
Correlation length of chromosome . Various SMC ornucleoid-associated proteins such as
MukBEF, HU, H-NS, IHF and gyrase may change the size of the chromo-somal structural unit and, thus, can help ensure success-ful chromosome segregation (Stavans and Oppenheim,2006). From our point of view, the best evidence has beengiven by Sawitzke and Austin (2000) [see, also, Holmesand Cozzarelli (2000) and Dasgupta et al. (2000)]. Theseauthors have demonstrated that severity of disruptionof chromosome segregation of muk − E.coli can be con-trolled by the level of supercoiling in the cell. Thus,neither class of proteins is the dedicated segregation ma-chinery, but their role can be understood as, within thetheoretical framework presented here, to increase the ef-fective correlation length of the chromosome above thecritical size (Eq. 3), which is sufficient for proper segre-gation (see I above).We also note that, because of supercoiling, a more re-alistic coarse-grained topology of a bacterial chromosomeis that of the branched polymer rather than a purelylinear chain. This branched structure will enhance thetendency of chromosome pushing even more (Jun andMulder, 2006; Trun and Marko, 1998; Vilgis, 2000).
Cell shape,
MreB , and bacterial “cytoskeletal” proteins . MreB and other proteins (Gerdes et al. , 2004) areimportant to maintain the high aspect ratio of thecell (see II above). It is our contention that, if theseproteins contribute to chromosome segregation, it isabout changes in cell shape as our model suggests.
Cell growth and division.
Since the cell size (relative tothe correlation length) is also important, proteins thatregulate cell growth and division are also important toensure that the cell reaches the appropriate size or massfor proper segregation (Weart et al. , 2007). In this con-text, the length-measuring “devices” such as
MinCDE are also important, and proteins involved in cell divisionsuch as
FtsZ are also relevant [see (Lutkenhaus, 2007)and references therein], if cell constriction should helpresolve partially overlapping nucleoids (Huls et al. , 1999).
B. Shouldn’t a hypothetical motor protein be enough tosegregate chromosomes in bacteria?
No. There is increasing experimental evidence that ori and perhaps some other chromosomal loci are local-ized or tethered at specific intracellular positions [see, forexample, Stavans and Oppenheim (2006) and referencestherein]. However, one still must explain how the hy-pothetical transportation of a small fraction (e.g., ori )of millions of basepairs of DNA from one position insidethe cell to another can dictate, if any at all, the directedmovement, segregation and organization of the rest ofthe chromosome. These observations thus require under-standing of more basic physical principles (e.g., Fig. 4).We note that there are special cases where motorproteins are indeed involved in transportation of DNAfrom one position to another inside a bacterial cell. Theexamples include
SpoIIIE for sporulation in
B.subtilis and
FtsK for dimer resolution and other “rescue” tasksin
E.coli [Barre (2007) and references therein]. However,it is important to realize that these proteins translocate
DNA, ı.e., they take advantage of the directionalityprovided by the septum, which is entirely different fromsegregation of replicating chromosomes.
C. What is the timescale?
In the segregation regime, segregation is a driven pro-cess and two intermingled chains drift at typical timescale of τ ∼ N (Arnold and Jun, 2007; Jun and Mul-der, 2006), which is much faster than diffusion (repta-tion) timescale of τ ∼ N in the channel. Timescaleof polymer motions in the presence of confinement in-volves severe finite-size effects (Arnold et al. , 2007), andit is far more difficult to predict the timescale for seg-regation of duplicating chains. For the concentric-shellmodel in our simulations for E.coli and
C. crescentus ,we assumed a separation of timescale that the relaxationtime of the replicating segment of DNA is much smallerthan the typical timescale of cell cycle (Sec. II.F). Suchfast movements of DNA segment have indeed been re-ported in recent experiments, that the 15kb minimalizedplasmid RK2 explores about 50% of the cell volume onlywithin 2 minutes (Derman et al. , 2008).0
D D’ (a) (b)
FIG. 9 Two possible scenarios of the role of molecular crowd-ing on chain interactions. (a) segregation (b) mixing. Eachchain consists of N monomers. Calculations show that crowd-ing, in the first approximation, should not influence mixingvs. de-mixing of the chains (see text). We do feel that the issue of timescale can be addressedadequately only via experiments using novel physicaltechniques.
D. How about topologically catenated chromosomes?
Imagine two topologically concatenated ring polymersconfined in a rectangular box, where our phase diagramin Fig. 4 predicts segregation of two linear chains ofthe same contour length as the rings. These two ringpolymers will still segregate to occupy each half of thebox, ı.e., the topology of the system localizes on averageat the center of the box (unpublished results). It is anintriguing question whether this localization of chaintopology, which is a global property of the system,will influence the action of topoisomerases, which haveaccess to only local information. We speculate thatthe entropy-driven directionality of the movements andrelative orientation of the confined chains will, at leastin part, help topoisomerases decatenate the replicatedchains before cell division.
E. Bacterial cells are very crowded: Will molecularcrowding influence chromosome organization andsegregation?
In the first approximation, no. One may foresee twoopposing arguments: (A) As the chains are compressedby depletion effect, the inner space occupied by the chainshas higher density of polymers, ı.e., less accessible vol-ume by the other chain and, thus, the tendency of de-mixing will increase because of squeezing, or (B) Molec-ular crowding exerts an effective osmotic pressure, whichwill make the chains mix.In fact, a simple calculation suggests that there is abalance between these two effects. To see this, let us con-sider the free energy of the compressed chains in Fig. 9as follows (Cacciuto and Luijten, 2006; Grosberg and Khokhlov, 1994; Jun et al. , 2007; Sakaue and Rapha¨el,2006). β F A = 2 C (cid:18) N ν D (cid:19) ν − (4) β F B = C (cid:20) (2 N ) ν D (cid:48) (cid:21) ν − , (5)where C is the prefactor. These two free energies forthe compressed chains become equal at D (cid:48) = 2 / D ,ı.e., when their total volumes are the same, V A = V B .Thus, compression due to depletion by itself will not in-fluence mixing vs. de-mixing of the chains, even if itcould bring the chains together and change their enve-lope shape (surface tension). Computer simulation re-sults also support the neutral effect of molecular crowding(Axel Arnold, personal communication).On the other hand, crowding can influence the localorganization of chromosome. For instance, looping isone way to achieve compaction of chromosomes, and theentropy gain by depletion attraction between DNA seg-ments can be larger than the entropy loss by DNA loop-ing (Marenduzzo et al. , 2006).
F. Bacteria exist in various cell shapes and composition ofchromosomes. Are there general strategies for successfulchromosome segregation in bacteria?
Since chromosome segregation is one of the definingprocesses of any cell, its basic mechanism must havebeen conserved across branches of life and organisms ofdiverse shapes. Based on our phase diagram (Fig. 4)and the two general rules I & II presented above, wecan speculate about how bacteria may create favorablephysical conditions for partitioning the chromosomes(Fig. 10). Filamentous bacteria.
We propose an entropy-driven,“random” segregation mechanism for filamentousbacteria. As we have shown both analytically and nu-merically, polymers confined in a narrow cylindrical ge-ometry strongly resist overlap and, thus, repel one an-other (Arnold and Jun, 2007; Jun and Mulder, 2006),where the timescale of disentanglement of overlappingpolymers is much shorter than that of pure diffusion.Indeed, recent experimental study on the filamentous The origin of this result is due to the form of the free energyin Eq. 4 (Grosberg et al. , 1982; Jun et al. , 2007; Sakaue andRapha¨el, 2006), which has been constructed to be a functionof only the monomer density. In other words, regardless ofthe shape of the envelope surrounding the chains, the free en-ergies will be the same as long as the volumes are also the same,and vice versa. Additional consideration of the interactions atthe surface will change the envelope shape, but not segrega-tion/mixing. (a) (b) symmetry breaking of cell shape I II IIIIVVVI square bacteria (”2-dimension”)spherical bacteria
FIG. 10 Possible cell-shape-dependent strategies for chromo-some segregation. (a) According to our phase diagram, fil-amentous and rod-shaped cells are most favored by entropy-driven segregation. (b) Thin, square bacteria such as Walsby’ssquare bacterium change their direction of growth after everydivision (typically they are found in composition of 1x1, 2x2,4x4, 8x8 cells and so on). On the other hand, round cellsdivide at alternating perpendicular planes. The arrows in theround cells indicate oscillation of the MinCDE proteins alongthe long axis of the ellipsoidal cells, which help determine themid plane of the cell. cyanobacterium
Anabaena sp. PCC 7120 has revealedGaussian distribution of DNA content in each daugh-ter cells after the septum formation (Hu et al. , 2007).Importantly, the variation of the distribution was muchlarger than a value expected if the two daughter cells wereidentical, suggesting a random event involved in chromo-some segregation. Also,
Anabaena sp. PCC 7120 , likemany other filamentous bacteria, is polyploid and con-tains multicopy of chromosomes per cell, ∼
10 (Hu et al. ,2007), which seems to moderate the effect of random seg-regation. Since chromosomes occupy much larger volumethan plasmids inside the cell, this apparent random seg-regation process is sufficient for the viability of the cellas long as multiple copies of chromosomes are producedbefore division.
Rod-shaped bacteria.
We have critically examined themodel system
E. coli
B/r strain and explained whyduplicated chromosomes will not mix. Thus, otherorganisms of similar cell shape and volume must benefitfrom the entropy-driven driving force of segregation,with or without any organism-specific segregation mech-anisms, as long as the the correlation of chromosome isalso comparable to that of
E. coli ( ≈
100 nm).
Square bacteria.
Perhaps one of the most striking bacte-rial shapes is that of Walsby’s square bacterium (Walsby,2000). This organism has a very thin and square-shapedcell ( ≈ ≈ µ m wide with aspect ratio1), a “2-dimensional” creature resembling a postagestamp. As we have explained above, polymers in highlysymmetric confinement can readily intermingle. In otherwords, entropy-favored condition for segregation can becreated when the cell breaks its symmetry during growthor division. Indeed, microscopy images so far suggest that that square bacteria grow and divide at alternatingperpendicular axes and planes, respectively [Fig. 10(b)].Also, in a thin slab (ı.e., 2D confinement), using theapproach taken in Jun et al. (2007), it is straightforwardto show that polymers repel much more strongly thanin bulk 3D. (Note that the cells are typically observedin 2x2, 4x4, 8x8 stamp-like configurations.) Therefore,we believe that simple symmetry breaking by cellgrowth and the thin 2D geometry suffice to separate thechromosomes entropically. Spherical bacteria.
The perfect symmetry of the cellshape means that the confined chains do not have anypreferred conformations between mixing and de-mixing,although their global reorganization could readily beachieved (Jun et al. , 2007). Since little data is availableon chromosome organization in spherical bacteria, wecan only speculate about possible contributing factorsto segregation. At the chromosome level, supercoiling-induced branched structures of the chromosome willincrease the tendency of demixing (Jun and Mulder,2006; Trun and Marko, 1998; Vilgis, 2000), perhapshelped by nucleoid-associated proteins to increase thecorrelation length (rule I above). More importantly,at the cell level, symmetry breaking of the cell shapeand invagination of the cell during division could helpresolve partially intermingled chromosomes [Fig. 10(b)].Indeed, real cells are never perfectly spherical, andHuang and Wingreen (2004) have shown numericallythat Min-protein oscillations can be achieved along thelong axis of the nearly round cell, where the equatorialradii differ by as small as 5%. This may explain theobserved division of round cells at alternating perpen-dicular planes (Corbin et al. , 2002). Along with thecase of the filamentous bacteria discussed above, it istempting to speculate that chromosome segregation inspherical bacteria is also a “random” process and, thus,polyploidy can be a strategy to increase the chanceof successful distribution of chromosomes to daughtercells. For example, the endosymbiotic bacteria ofaphids,
Buchnera , which cannot divide by itself outsidethe host eukaryotic cell, contains over 100 copies ofgenome (Komaki and Ishikawa, 1999).
G. Early life and artificial cells
Previously, we proposed that the entropy-driven seg-regation of polymers could be implemented in designingan artificial cell, and that it has implications on earlylife (Jun and Mulder, 2006). How could an artificial cell The free-energy cost of chain overlapping crosses over from β F ∼ n dν d / ( dν d − to ∼ n d/ ( d − , where ν d = 3 / ( d + 2) is the Floryexponent and d the spatial dimension. [Jun et al. (2007); B.-Y.Ha, personal communication] replication and/or polymerizationDivisionPolymer repulsion &elongation of membrane FIG. 11 Entropy-driven segregation of encapsulated polymersand its potential role on the cell cycle of a protocell. Repul-sion between the replicated chains can break the sphericalsymmetry of the vesicle, and, thus, may facilitate its spon-taneous division. Moreover, the daughter vesicles containingthe duplicated polymers may have similar sizes. (The threeminisketches at the top represent input from environment forfeeding the growth.) These scenarios could be tested boththeoretically and experimentally. [Adapted from Figure 3 inSzostak et al. (2001).] achieve its basic cellular processes? While Szostak andcolleagues have proposed a system of a self-replicatingvesicle and replicases as a protocell (Szostak et al. , 2001),cell division mechanism as well as the size regulation ofdaughter cells of a protocell are far from being under-stood.We suggest that two long polymers, formed by replica-tion/ligation of smaller molecules within a spherical vesi-cle repel each other because of entropy. If the polymer-polymer repulsion is strong enough to break the spher-ical symmetry of the vesicle, this process may lead tomembrane fission, preferentially at the mid-cell position,which is defined by the surface of contact between the tworepelling polymers. This would further help regulate thesize distribution of newly formed daughter cell as illus-trated in Fig. 11. Although it is in general a formidabletheoretical problem to predict the shape of the closedmembrane in a given environment, it would be highlydesirable to obtain a phase diagram similar to Fig. 4 forsoft walls, and possibly test our proposed idea here exper-imentally using macromolecules encapsulated in vesiclesor micro drops.
IV. FINAL REMARKS
In recent years, biology has become increasingly inter-disciplinary. In particular, in the new discipline of sys-tems biology, researchers trained in physical and other quantitative sciences are making significant contribu-tions. From a physicist’s point of view, however, it isinteresting and important to notice that many of the con-tributions made by physicists so far have involved bring-ing new tools and ways of thinking to (systems) biology,rather than new understanding about the role that ba-sic physical principles play in governing the fundamentalbiological processes inside the cell. But aren’t biologicalentities also physical objects? The membranes definingthe envelope of the organelles; the polymers constitutingproteins, spindles, and chromosomes – these are physicalobjects and, although biophysical approach has uncov-ered many of their important physical properties, theirbiological implications need to be more fully explored.Thus, in seeking a more direct relationship betweenphysics and biology, with the example of bacterial chro-mosome segregation, I hope to have convinced the readerthat there is a seamless “dialogue” between physical andbiological processes involved in the bacterial cell cycle;and that deeper understanding of the relationship be-tween the physical principles and their biological impli-cations may even shed new light on the major transitionsin evolution (Maynard-Smith and Sz´athmary, 1998). Acknowledgments
This work would not have been possible without mylong-term collaborations with A. Arnold, B.-Y. Ha, N.Kleckner, and C. Woldringh. I thank J. Bechhoefer, M.Brenner, R. D’Ari, D. Frenkel, M. Kardar, O. Krichevsky,P. A. Levin, R. Losick, J. Marko, B. Mulder, A. Murray,D. Nelson, B. Stern, F. Taddei and numerous other col-leagues for helpful suggestions. I am also grateful to N.Kleckner for introducing the notion of “piston” to me,and to R. Hellmiss for the illustration of chromosomemodel in Fig. 8. This article is dedicated to the memoryof J. Raoul-Duval. For instance, in the presence of multiple chromosomes in eu-karyotes, the cell faces an additional challenge in chromosomesegregation because each daughter cell must receive not only thecorrect number but also the correct set of chromosomes. Fromout point of view, this strongly argues why mitosis requires moresophisticated segregation machinery as well as the checkpoints;whereas entropy might be sufficient for chromosome segregationin most bacteria.On the other hand, the hallmark of mitosis is the compactsister chromosomes of well-defined shapes being held together,waiting for the mitotic spindle to separate them. However, thesechromosomes must go through a period of intermingling duringreplication, before mitosis. Thus, there is a process analogous tobacterial chromosome segregation, namely, de-mixing of repli-cated eukaryotic sister chromatids, which we distinguish from segregation by spindle. Although the exact mechanism of thisde-mixing process has remained elusive, we believe that the basicphysics behind is the same as what we presented in this article. APPENDIX A: More technical description of the phasediagram in Fig. 4.
Below, we provide more technical descriptions of indi-vidual regimes with references so that the reader canreproduce the boundary conditions of the phase diagramin Fig. 4. [See, also, Brochard and de Gennes (1979);Daoud and de Gennes (1977); Teraoka and Wang (2004)]. I . This is the “bulk” dilute regime, where the interchaindistances are much larger than the size of the chain inconfinement. When D = L = R F , the correlation length ξ bulk also equals the size of the chain R F , and the twochains can overlap at the free-energy cost of order k B T [Fig. 4(a)] (Grosberg et al. , 1982; Jun et al. , 2007). II . Imagine dilute solution in a long cylindrical box,where there is a free space between the two chains[Fig. 4(b)]. The size of individual chains scales as R ∼ ( N/g ) D ∼ N D − / (where g is the number of monomersper blob). In case of overlapping, the chains stronglyrepel one another and, thus, their segregation is muchfaster than reptation (typical timescale of τ ∼ N vs. τ ∼ N , respectively) (Arnold and Jun, 2007; Jun andMulder, 2006). Note that ξ bulk > D in this regime,whereas the blob size of the chain due to confinementis D . Thus, the monomer-monomer correlation lengthinside the box is ξ box = D < ξ bulk .If we gradually reduce the aspect ratio of the box,keeping constant the volume of the box (thus, fixing ξ bulk ), the number of blobs of each chain decreases andeventually the two chains make contact at L = 2 R , ı.e., ξ bulk = ξ box = D ( y = x ). This situation is depictedin Fig. 4(c), at which the chains enter the semi-diluteregime inside the elongated box. III . If we continue the above process of reducing the as-pect ratio of the box, the two chains are gradually com-pressed but do not mix as long as their principal con-formations are linear [Fig. 4(d).] As we discussed inSec. II.D, this is the most important regime for rod-shaped bacteria. To see the segregation of overlappingchains, let us employ the following Flory-type free en-ergy of individual chains: β F ( R ) = 12 (cid:20) R ( N/g ) D + D ( N/g ) R (cid:21) , (A1)where β = 1 /k B T (Jun et al. , 2008). On the other hand,it costs ∼ k B T of free energy for two blobs to significantlyoverlap regardless of their size (Grosberg et al. , 1982; Jun et al. , 2007; Jun and Mulder, 2006). Thus, the total freeenergy of the system of two partially overlapping chainsdescribed in Fig. 4(h) can be written as β F tot = (cid:16) σR (cid:17) β F ( R )= k (cid:18) L (cid:48) − R (cid:48) (cid:19)(cid:20) R (cid:48) + 1 R (cid:48) (cid:21) , (A2) where L (cid:48) = L/R ≤ R (cid:48) = R/R ≤ D (cid:48) = D/R .Since ∂β F tot /∂R (cid:48) = 3 k/ > R (cid:48) = L (cid:48) / F tot is a monotonically increasingfunction of R (cid:48) ≥ L (cid:48) / L (cid:48) (cid:39) . IV, V, VI (mixing regimes). As the aspect ratio ap-proaches 1, the two chains lose their linear conformationsand mix with each other. The reason for this transitionis that the bulk correlation length ξ bulk becomes muchsmaller than both D and L , and the chain conformationbecomes a random walk of a string of blobs (blob size ξ bulk ). We described the physics of this regime in detailin Jun et al. (2007).At the boundary between III and IV , transition oc-curs because individual chains are contracted and losetheir linear ordering. The size of individual chain isthat of the random-walk conformation, R (cid:107) ∼ N / Φ − / with the volume fraction of the monomer Φ ∼ N/R (cid:107) D .Based on this, it is easy to show the boundary condition y = x / (Lal et al. , 1997). Since the free energy in thisregime is given by β F IV (cid:39) (cid:18) R F D L (cid:19) ν − (cid:39) D Lξ (A3)(Cacciuto and Luijten, 2006; Grosberg and Khokhlov,1994; Jun et al. , 2007; Sakaue and Rapha¨el, 2006), theabove boundary condition is translated to D = k · ξ bulk ,which is in symmetry with the condition L = k · D . Sim-ilarly, it is straightforward to show that the boundarycondition k = 1 between IV and V is identical to y = x / [Fig. 4(e)].In regime V , the shape of the box resembles a thickslab. As the thickness of the slab reaches the size of theblob, ı.e., ξ bulk = L = k · D [Fig. 4(e)], the two chainsenter the dilute regime in a thin, closed slab [Fig. 4(g)],which is the dual regime of II . Thus, the transition be-tween V and V I occurs when the chains start to feel thesize of the slab, ı.e., at D ∼ R (cid:107) ∼ N / Φ − / , where thevolume fraction is given by Φ ∼ g/ξ , with g ∼ ξ /ν bulk .We thus obtain y = x ( L = k · D ). Since the N /R dependence in Eq. A1 is an well-known overes-timate of Flory-type approach for R (cid:28) R , the L (cid:48) ≈ . R ) is notavailable. APPENDIX B: Self-consistent mapping between the Pincusand the close-packed chains.
In the Pincus picture of a stretched chain, the chainbreaks up into a series of blobs of size ξ (Fig. 8). The freeenergy cost can be estimated by counting the number ofblobs, β F P ∼ Ng ∼ Rξ ∼ N (cid:18) RN (cid:19) − ν , (B1)where g ∼ ξ /ν is the number of monomers per blob and R the end-to-end distance of the chain.Let us now imagine close-packing of the chain consist-ing of blobs of size ξ in a cylinder of diameter D . If thisprocess does not cost any additional free energy, usingEq. B1, we can self-consistently obtain the confinementfree energy accordingly as follows. β F cyl ∼ LD ξ ∼ N (cid:18) RN (cid:19) − ν ∼ (cid:18) LD N ν (cid:19) − ν , (B2)where L is the longitudinal size of the close-packed chainin the cylinder. Eq. B2 has the same scaling form of thefree energy cost for confining a self-avoiding chain in avolume V , β F cyl ∼ (cid:18) VR g (cid:19) − ν (B3)(see Eq. A3), and we thus have a self-consistent picturefor interpreting compressed chain based on the Pincuschain.The essence of our analysis here is that the free energyfor deformation costs ∼ k B T / blob regardless of the na-ture of deformation and, thus, is mappable one another(e.g., between stretching vs. confinement).
References
Adachi, S., K. Hori, and S. Hiraga, 2006, “Subcellular Posi-tioning of F Plasmid Mediated by Dynamic Localization ofSopA and SopB,” J Mol Biol , 850–863.Alder, B. J., and T. E. Wainwright, 1957, “Phase Transitionfor a Hard Sphere System,” J Chem Phys , 12081209.Arnold, A., B. Borzorgui, D. Frenkel, B.-Y. Ha, and S. Jun,2007, “Unexpected relaxation dynamics of a self-avoidingpolymer in cylindrical confinement,” J Chem Phys ,164903.Arnold, A., and S. Jun, 2007, “Timescale of entropic seg-regation of flexible polymers in confinement: Implicationsfor chromosome segregation in filamentous bacteria,” PhysRev E , 031901.Barre, F.-X., 2007, “FtsK and SpoIIIE: the tale of the con-served tails,” Mol Microbiol , 1051–1055.Bates, D., and N. Kleckner, 2005, “Chromosome and Repli-some Dynamics in E. coli : Loss of Sister Cohesion TriggersGlobal Chromosome Movement and Mediates ChromosomeSegregation,” Cell , 899–911. Berkmen, M. B., and A. D. Grossman, 2006, “Spatial andtemporal organization of the
Bacillus subtilis replicationcycle,” Mol Microbiol , 57–71.Brenner, M., and J.-I. Tomizawa, 1991, “Quantitation ofColE1-encoded replication elements,” Proc. Nat. Acad. Sci. , 405–409.Brochard, F., and P. G. de Gennes, 1979, “Conformation ofmolten polymers inside small pores,” J. Phys. (Paris) ,L399–L401.Cacciuto, A., and E. Luijten, 2006, “Self-Avoiding FlexiblePolymers under Spherical Confinement,” Nano Lett , 901–905.Corbin, B. D., X.-C. Yu, and W. Margolin, 2002, “Exploringintracellular space: function of the Min system in round-shaped Escherichia coli ,” EMBO J. , 1998–2008.Cunha, S., T. Odijk, E. S¨uleymanoglua, and C. L.Woldringh, 2001, “Isolation of the Escherichia coli nu-cleoid,” Biochimie , 149–154.Danchin, A., P. Guerdoux-Jamet, I. Moszer, and P. Nitschk´e,2000, “Mapping the bacterial cell architecture into thechromosome,” Phil. Trans. R. Soc, Lond. B , 179–190.Daoud, M., and P. G. de Gennes, 1977, “Statistics of macro-molecular solutions trapped in small pores,” J. Phys.(Paris) , 85–93.Dasgupta, S., S. Maisnier-Patin, and K. Nordstr¨om, 2000,“New genes with old modus operandi : The connection be-tween supercoiling and partitioning of DNA in Escherichiacoli ,” EMBO Reports , 323–327.Deng, S., R. A. Stein, and N. P. Higgins*, 2005, “Organizationof supercoil domains and their reorganization by transcrip-tion,” Mol Microbiol , 1511–1521.Derman, A. I., G. Lim-Fong, and J. Pogliano, 2008, “Intracel-lular mobility of plasmid DNA is limited by the ParA familyof partitioning systems,” Mol Microbiol , 935–946.Doi, M., and S. F. Edwards, 1986, The Theory of PolymerDynamics (Oxford University Press, Oxford, UK).Dworkin, J., and R. Losick, 2002, “Does RNA polymerasehelp drive chromosome segregation in bacteria?,” Proc NatAcad Sci , 14089–14094.Elmore, S., M. M¨uller, N. Vischer, T. Odijk, and C. L.Woldringh, 2005, “Single-particle tracking of oriC -GFPfluorescent spots during chromosome segregation in Es-cherichia coli ,” J Struct Biol , 275–287.Esp´eli, O., and F. Boccard, 2006, “Organization of the Es-cherichia coli chromosome into macrodomains and its pos-sible functional implications,” J Struct Biol , 304–310.Fan, J., K. Tuncay, and P. J. Ortoleva, 2007, “Chromosomesegregation in
Escherichia coli division: A free energy-driven string model,” Comp Biol Chem , 257–264.Fiebig, A., K. Keren, and J. A. Theriot, 2006, “Fine-scaletime-lapse analysis of the biphasic, dynamic behaviour ofthe two Vibrio cholerae chromosomes,” Mol Microbiol ,1164–1178.Flory, P., 1953, Principles of Polymer Chemistry (CornellUniversity Press).Fogel, M. A., and M. K. Waldor, 2006, “A dynamic, mitotic-like mechanism for bacterial chromosome segregation,”Genes Dev. , 3269–3282.Garner, E. C., C. S. Campbell, D. B. Weibel, and R. D.Mullins, 2007, “Reconstitution of DNA Segregation Drivenby Assembly of a Prokaryotic Actin Homolog,” Science , 1270–1274.de Gennes, P.-G., 1979, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, NY). Gerdes, K., J. Moller-Jensen, G. Ebersbach, T. Kruse, andK. Nordstr¨om, 2004, “Bacterial Mitotic Machineries,” Cell , 359–366.Grosberg, A. Y., P. G. Khalatur, and A. R. Khokhloo,1982, “Polymeric Coils with Excluded Volume in DiluteSolution: the Invalidity of the Model of ImpenetrableSpheres and the Influence of Excluded Volume on theRates of Diffusion-Controlled Intermacromolecular Reac-tions,” Makromol. Chem., Rapid Commun. , 709–713.Grosberg, A. Y., and A. R. Khokhlov, 1994, StatisticalPhysics of Macromolecules (AIP Press).Holmes, V. F., and N. R. Cozzarelli, 2000, “Closing the ring:Links between SMC proteins and chromosome partitioning,condensation, and supercoiling,” Proc Nat Acad Sci ,13221324.Hu, B., G. Yang, W. Zhao, Y. Zhang, and J. Zhao, 2007,“MreB is important for cell shape but not for chromosomesegregation of the filamentous cyanobacterium Anabaena sp. PCC 7120,” Mol Microbiol , 1640–1652.Huang, K. C., and N. S. Wingreen, 2004, “Min-protein oscil-lations in round bacteria,” Phys. Biol. , 229–235.Huls, P. G., N. O. E. Vischer, and C. L. Woldringh, 1999,“Delayed nucleoid segregation in Escherichia coli ,” Mol Mi-crobiol , 959–970.Jacob, F., S. Brenner, and F. Cuzin, 1963, “On the Regulationof DNA Replication in Bacteria,” J Mol Biol , 329–348.Jun, S., A. Arnold, and B.-Y. Ha, 2007, “Confined Space andEffective Interactions of Multiple Self-Avoiding Chains,”Phys Rev Lett , 128303.Jun, S., and B. Mulder, 2006, “Entropy-driven spatial orga-nization of highly confined polymers: Lessons for the bac-terial chromosome,” Proc Nat Acad Sci , 12388–12393.Jun, S., D. Thirumalai, and B.-Y. Ha, 2008, “Compressionand stretching of a self-avoiding chain under cylindricalconfinement,” Phys Rev Lett (in press) .Kleckner, N., D. Zickler, G. H. Jones, J. Dekker, R. Padmore,J. Henle, and J. Hutchinson, 2004, “A mechanical basis forchromosome function,” Proc Nat Acad Sci , 12592–12597.Komaki, K., and H. Ishikawa, 1999, “Intracellular BacterialSymbionts of Aphids Possess Many Genomic Copies perBacterium,” J Mol Evol , 717–722.Kubitschek, H. E., 1981, “Bilinear Cell Growth of Escherichiacoli ,” J Bacteriol , 730–733.Lal, J., S. K. Sinha, and L. Auvray, 1997, “Structure of Poly-mer Chains Confined in Vycor,” J. Phys. II (France) ,1597–1615.Lemon, K. P., and A. D. Grossman, 2001, “The extrusion-capture model for chromosome partitioning in bacteria,”Genes Dev , 2031–2041.Lutkenhaus, J., 2007, “Assembly Dynamics of the BacterialMinCDE System and Spatial Regulation of the Z Ring,”Annu Rev Biochem , 539–562.Marenduzzo, D., C. Micheletti, and P. R. Cook, 2006,“Entropy-Driven Genome Organization,” Biophys J ,3712–3721.Marko, J. F., and E. D. Siggia, 1995, “Statistical mechanicsof supercoiled DNA,” Phys Rev E , 2912–2938.Marko, J. F., and E. D. Siggia, 1997, “Polymer Models of Mei-otic and Mitotic Chromosomes,” Mol Biol Cell , 22172231.Maynard-Smith, J., and E. Sz´athmary, 1998, The MajorTransitions in Evolution (Oxford University Press, USA).Nanninga, N., M. Roos, and C. L. Woldringh, 2002, “Modelson stickiness of replicated
Escherichia coli oriC ,” Microbi- ology , 3327–3328.Nielsen, H. J., J. R. Ottesen, B. Youngren, S. J. Austin, andF. G. Hansen, 2006, “The
Escherichia coli chromosome isorganized with the left and right chromosome arms in sep-arate cell halves,” Mol Microbiol , 331–338.Nordstr¨om, K., and K. Gerdes, 2003, “Clustering versus ran-dom segregation of plasmids lacking a partitioning func-tion: a plasmid paradox?,” Plasmid , 95–101.Onsager, L., 1949, “The effect of shape on the interaction ofcolloidal particles,” Ann. N.Y. Acad. Sci. , 627659.Pincus, P., 1976, “Excluded Volume Effects and StretchedPolymer Chains,” Macromolecules , 386–388.Romantsov, T., I. Fishov, and O. Krichevsky, 2007, “In-ternal Structure and Dynamics of Isolated Escherichiacoli
Nucleoids Assessed by Fluorescence Correlation Spec-troscopy,” Biophys J , 2875–2884.Sakaue, T., and E. Rapha¨el, 2006, “Polymer Chains inConfined Spaces and Flow-Injection Problems: Some Re-marks,” Macromolecules , 2621–2628.Sawitzke, J. A., and S. Austin, 2000, “Suppression of chromo-some segregation defects of Escherichia coli muk mutantsby mutations in topoisomerase I,” Proc Nat Acad Sci ,1671–1676.Schr¨odinger, E., 1944, What is life? (Cambridge UniversityPress, Cambridge, UK).Schumacher, M. A., T. C. Glover, A. J. Brzoska, S. O. Jensen,T. D. Dunham, R. A. Skurray, and N. Firth, 2007, “Segro-some structure revealed by a complex of ParR with cen-tromere DNA,” Nature , 1268–1271.Stavans, J., and A. Oppenheim, 2006, “DNAprotein interac-tions and bacterial chromosome architecture,” Phys Biol ,R1–R10.Sunako, Y., T. Onogi, and S. Hiraga, 2001, “Sister chro-mosome cohesion of Escherichia coli ,” Mol Microbiol ,1233–1241.Szostak, J. W., D. P. Bartel, and P. L. Luisi, 2001, “Synthe-sizing life,” Nature , 387–390.Teleman, A. A., P. L. Graumann, D. C.-H. Lin, A. D. Gross-man, and R. Losick, 1998, “Chromosome arrangementwithin a bacterium,” Curr Biol , 1102–1109.Teraoka, I., and Y. Wang, 2004, “Computer simulation stud-ies on overlapping polymer chains confined in narrow chan-nels,” Polymer , 3835–3843.Trun, N. J., and J. F. Marko, 1998, “Architecture of a Bac-terial Chromosome,” ASM News , 276–283.Vilgis, T. A., 2000, “POLYMER THEORY: PATH INTE-GRALS AND SCALING,” Phys Rep , 167–254.Viollier, P. H., M. Thanbichle, P. T. McGrath, L. West,M. Meewan, H. H. McAdams, and L. Shapiro, 2004, “Rapidand sequential movement of individual chromosomal loci tospecific subcellular locations during bacterial DNA replica-tion,” Proc Nat Acad Sci , 9257–9262.Walsby, A. E., 2000, “A square bacterium,” Nature , 69–71.Wang, X., X. Liu, C. Possoz, and D. J. Sherratt, 2006, “Thetwo Escherichia coli chromosome arms locate to separatecell halves,” Genes Dev. , 1727–1731.Weart, R. B., A. H. Lee, A.-C. Chien, D. P. Haeusser, N. S.Hill, and P. A. Levin, 2007, “A Metabolic Sensor GoverningCell Size in Bacteria,” Cell , 335–347.Woldringh, C. L., 2002, “The role of co-transcriptional trans-lation and protein translocation (transertion) in bacterialchromosome segregation,” Mol Microbiol , 17–29.Woldringh, C. L., and T. Odijk, 1999, in Organization of the Prokaryotic Genome , edited by R. L. Charlebois (ASMPress, Washington, D.C.), 161–197.Wood, W. W., and J. D. Jacobson, 1957, “Preliminary Re- sults from a Recalculation of the Monte Carlo Equation ofState of Hard Spheres,” J Chem Phys27