Chromosome segregation in Escherichia coli division A free energy
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Computational Biology and Chemistry 31 (2007) 257–264 Chromosome segregation in Escherichia coli division: A free energy-driven string model J. Fan, K. Tuncay, P.J. Ortoleva ∗ Center for Cell and Virus Theory, Indiana University, Bloomington, IN 47405, United States Received 27 April 2007; accepted 6 May 2007 Abstract Although the mechanisms of eukaryotic chromosome segregation and cell division have been elucidated to a certain extent, those for bacteria remain largely unknown. Here we present a computational string model for simulating the dynamics of Escherichia coli chromosome segregation. A novel thermal-average force ﬁeld accounting for stretching, bending, volume exclusion, friction and random ﬂuctuation is introduced. A Langevin equation is used to simulate the chromosome structural changes. The mechanism of chromosome segregation is thereby postulated as a result of free energy-driven structural optimization with replication introduced chromosomal mass increase. Predictions of the model agree well with observations of ﬂuorescence labeled chromosome loci movement in living cells. The results demonstrate the possibility of a mechanism of chromosome segregation that does not involve cytoskeletal guidance or advanced apparatus in an E. coli cell. The model also shows that DNA condensation of locally compacted domains is a requirement for successful chromosome segregation. Simulations also imply that the shape-determining protein MreB may play a role in the segregation via modiﬁcation of the membrane pressure. Published by Elsevier Ltd Keywords: Prokaryotic cell division; Chromosome segregation; DNA compaction; Escherichia coli; Self-organization 1. Introduction tion of E. coli division. Many mathematical models have been proposed to explain experimental observations (Howard et al., Bacteria are simple organisms which maintain precise repli- 2001; Meinhardt and de Boer, 2001; Kruse, 2002; Huang et al., cation, segregation and division. Although much has been 2003; Drew et al., 2005; Kerr et al., 2006; Pavin et al., 2006). understood for eukaryotic cells, how replication, segregation and To better understand the dynamics of chromosome replication, division are coordinated in prokaryotic cells remains elusive. segregation and division, mathematical models and quantitative In eukaryotic cells, chromosomes are wrapped into nucleo- analysis of the mechanism is of help. somes around highly positively charged histone proteins, further There are several hypotheses on how chromosomes segre- compacted by condensing and tied by cohesions. After replica- gate in bacteria. In 1963 Jacob et al. proposed a model in which tion, chromosomes are separated by a dedicated cytoskeletal bacterial chromosomes are attached to the membrane and they apparatus (Kline-Smith and Walczak, 2004). Highly conserved, are separated as a result of the elongation of the membrane. As histones and cytoskeletal apparatus appeared later in evolu- new membrane material is continuously inserted between the tion and are not present in bacteria. Instead, bacteria appear two attachments, the chromosomes are dragged apart (Jacob and to deploy simpler mechanisms to orchestrate precise replica- Brenner, 1963; van Helvoort and Woldringh, 1994). Later ﬁnd- tion, segregation and division. For example, in Escherichia coli, ings on the speed of chromosome movement and cell elongation placement of the division plane is determined in combination have shown that this model cannot fully explain experimen- of the MinCDE and the nucleoid occlusion systems (Norris et tal observations (Teleman et al., 1998; Daniel and Errington, al., 2004; Margolin, 2006). The MinCDE system coordinates 2003). In Bacillus subtilis, the average movement of chromo- an active oscillation of Min proteins that determines the loca- some is 0.17 m min−1 , while the speed of cell elongation is 0.011–0.025 m min−1 (Webb et al., 1998). Chromosomes move much faster than the cell elongation rate. In 2001, Lemon ∗ Corresponding author. Tel.: +1 812 855 2717; fax: +1 812 855 8300. and Grossman proposed an extrusion capture model. A station- E-mail address: firstname.lastname@example.org (P.J. Ortoleva). ary replisome stays at the middle of the cell and it constantly pulls 1476-9271/$ – see front matter. Published by Elsevier Ltd doi:10.1016/j.compbiolchem.2007.05.003 258 J. Fan et al. / Computational Biology and Chemistry 31 (2007) 257–264 the mother chromosome while pushes the new daughter chro- mosomes away from the center (Lemon and Grossman, 2001). However, in Caulobacter crescentus, the replisome was found to be moving during replication (Jensen et al., 2001). In 2002, Dworkin and Losick proposed that chromosomes are repelled by RNA polymerase extending between two duplicated replica- tion origins (ori) (Dworkin and Losick, 2002). However, RNA Fig. 2. The E. coli geometry is approximated by a cylinder of length H and two polymerase is not stationary or partially immobilized but resides hemispheres of radius R at either end. The origin of the coordinate system (x, y, everywhere around the nucleoid (Lewis et al., 2000). Woldringh z) is placed at the center of the bacterium. presented a coupled transcription–translation–insertion (transer- tion) model in which daughter chromosomes compete for 2. Methods membrane attachment space which leads to bidirectional segre- gation (Woldringh, 2002). However, the model requires daughter 2.1. A free energy-driven string model chromosomes tethered via a different set of proteins for different space (Rocha et al., 2003) and the mechanism is too complicated To analyze the dynamics of spatial organization of chro- to be stable enough to achieve the rapid and precise separation mosomes during replication and segregation, we developed a of DNA segments (Ronen and Sigal, 2006). computational model to simulate the dynamics of chromosome Jun and Mulder presented an entropy-driven spontaneous structures over the cell cycle. The dynamics is driven by gra- segregation model and they applied the model to E. coli and dients of the free energy. This theme has been used to develop C. crescentus (Jun and Mulder, 2006). They considered DNA theories of nanoparticles, viruses and macromolecules (Miao to be polymer chains and used entropy maximization to guide and Ortoleva, 2006a,b). each bead to less crowded positions. This model successfully The E. coli geometry is approximated by a cylinder of length describes the replication-segregation process of a cell cycle. H and two hemispheres of radius R at either end (Fig. 2). Chro- However, they introduced an inner tube (a rod shaped enve- mosome movement in E. coli is restricted in this boundary. lope) to restrain the mother chromosome’s movement and only We construct the free energy of the chromosomal system allowed daughter chromosomes to occupy both the inner tube as follows. Two neighboring domains are assigned a stretching and outer tube space. When replication starts, daughter chro- energy mosomes occupy the empty space outside the inner tube and are pushed toward the poles. Since the mother and daughter Us (d) = ks (d − d0 )2 (stretching), (1) chromosomes consist of the same DNA material, the spatial pref- where ks is the stretching rigidity, while d and d0 are the erence should be implied by the mathematical physico-chemical actual/equilibrium distances of the two linked domains. The model, not imposed during simulations arbitrarily. Furthermore, stretching force on domain i is computed via whether such a differential space restriction for the mother and daughter chromosomes exists awaits additional experimental Fis = −2ks [(di−1 − d0 )ui−1 − (di − d0 )ui ] (stretching), (2) evidence. In this study we present a free energy-driven string model where di is the distance between domains i and i + 1, and ui is that neither introduces differential spatial preference nor with the unit vector pointing from i to i + 1. advanced mitotic cytoskeletal guidance. Chromosomes are orga- Each domain possesses a harmonic bending potential due to nized into a string of distinct topological domains (Staczek and its interaction with left and right neighbors Higgins, 1998; Postow et al., 2004) (Fig. 1). The migration of Ub (θ) = kb (θ − θ0 )2 (bending), (3) individual domains is predicted mathematically with our model and their locations can be compared to experimental obser- where kb is the bending rigidity, while θ and θ 0 are the vations of loci movement. Here we present the mathematical actual/equilibrium angles between two links. The corresponding development of the proposed free energy-driven string model bending force on domain i is given by and simulation results. Fib = −2kb (Ai − Bi + Bi−1 − Ai+1 ) (bending) (4) with ui + ui−1 cos θi Ai = (θi − θ0 ) , di−1 sin θi ui−1 + ui cos θi Bi = (θi − θ0 ) . (5) di sin θi Since distances between domains are much larger than the van Fig. 1. Bacterial DNA organized into locally compacted–connected domains. A circular chromosome in Escherichia coli is thus represented as a string of der Waals interaction range or the electrostatic Debye length of interlinked domains. the intracellular medium, we approximate these interactions by a J. Fan et al. / Computational Biology and Chemistry 31 (2007) 257–264 259 short-range excluded volume interaction potential as proposed to 2.2. Simulation parameters and model details predict chromosome structures (Munkel and Langowski, 1998): To test the free energy-driven string model, parameter values r 4 − 2rc r 2 2 were chosen based on available experimental data. In a culture Ue (r) = ke 1+ 4 (excluded volume), (6) rc medium of ﬁxed nutrition level, E. coli elongates with ﬁxed width (Marr et al., 1966). In our numerical simulations, we used where ke is the excluded volume coefﬁcient, while r and rc are the R = 0.5 m as the cell radius and elongated H from 2.0 m to actual and cut-off distances between two domains. All domains 4.0 m in one cell cycle. The elongation rate of cell membrane with a distance r to domain i contribute a force on domain i: was kept constant at 0.025 m min−1 . A slow growth condition (rji − rc )rji 2 2 was used to simplify the model (i.e. only one initiation of repli- Fie = 4ke 4 (excluded volume), (7) cation occurs per cell cycle). The cell doubling time was taken rc j=i to be 80 min: 20 min before initiation of chromosome replica- where rji is the distance between j and i (rji ≤ rc ). tion, 40 min between initiation and completion of chromosome To keep bacterial DNA inside the cell and simulate the effect replication, and 20 min for post preparation of segregation and of membrane pressure, a boundary force is introduced: division. The E. coli chromosome consists of 4.6 million base pairs nmi and is compacted into a circular ring of topological domains. Fim = km (membrane pressure), (8) 2 rmi Recent studies of ﬂorescent tracking of loci movement show that loci ∼200 kbp apart can occupy separate positions in the cell where km is the membrane pressure coefﬁcient, rmi the nearest and segregate independently (Viollier et al., 2004; Fekete and distance from the membrane to domain i, and nmi is the unit Chattoraj, 2005). Thus we used 22 locally compacted domains vector pointing from the nearest membrane point to domain i. in our simulations to represent a complete chromosome. The To account for energy exchange between a given domain and average size of a domain is 210 kbp. the solution or the other domains, a friction force is introduced: The stretching rigidity of DNA has been actively studied over the past decade (Bustamante et al., 2003). The physical Fif = −kf vi (friction force), (9) properties of a DNA strand can be approximated using a worm- where kf is the friction coefﬁcient and vi is the velocity of domain like chain (WLC) model. However, the detailed conﬁguration of i. DNA that links two domains is still unresolved. We assume the A small random force is introduced to account for ﬂuctuations stretching rigidity could be approximated as kB T/A, where kB in the system and to avoid being locked in states of local energy is Boltzmann’s constant, T is absolute temperature and A is the minima. The random force (Fir ) is generated in all directions at ﬂexural persistence length (Bustamante et al., 2003). We used a every time step via stretching rigidity of ks = 60 kB T m−2 (Munkel and Langowski, 1998) and an equilibrium link distance of d0 = 0.1 m in our Fir = kr ε (ﬂuctuation force), (10) simulation as a rough estimate (van den Engh et al., 1992). In the unwounding state, the stretching rigidity is weakened to where ε is a vector, each of whose three components is chosen 0.06 kB T m−2 . We used an angle of 70◦ as the equilibrium har- independently and at random subject to the interval (−1,1). monic bending angle since protein FIS in E. coli bends DNA Assuming inertial forces are small compared to frictional with an angle between 50◦ and 90◦ upon binding (Altuvia et forces, the position and velocity of each domain is calculated al., 1994; Stavans and Oppenheim, 2006). The bending rigidity over time t via numerical simulation of Langevin equations: was approximated as kb = 0.75 kB T. The cut-off distance of the volume exclusive effect was taken to be rc = 0.4 m and the vol- dr = vi , (11) ume exclusive coefﬁcient is ke = 1.5 kB T. We used a membrane dt pressure force coefﬁcient of km = 0.05 kB T m and a friction kf vi = Fi , (12) coefﬁcient of kf = 35 kB T m−2 s. The amplitude of the random force was taken to be kr = 0.001 kB T m−1 . where Fi is the sum of all forces (except friction) on i. With the above six force ﬁelds (stretching, bending, volume With the above Langevin dynamics, along with the elonga- exclusion, membrane pressure, friction and ﬂuctuation), we sim- tion of cell (i.e. H increases over time) and events of replication, ulated E. coli self-organized chromosomal structural segregation we simulated the cycle of E. coli elongation and chromosome over one cell cycle. The forces are all essential as any subset of replication and segregation. For domains undergoing replica- them alone cannot reproduce the segregation correctly. With- tion, they are unwound by replisomes before replication and out the stretching force, chromosome domains are disconnected then refolded after replication is over. Thus bending force of and become scattered across the cell. Without the bending force, replicating domains is missing temporally and the stretching structural details of angles between nearby domains are missed force is weakened in the unwounding state. After replication of and the segregated chromosomes cannot maintain correct loci each domain, daughter domains are assigned to the same posi- conﬁgurations (i.e. a mother chromosome with the ori region in tion as of their mother domain. In subsequent time steps, they its center may produce daughters with the ori regions close to are positioned appropriately by the force ﬁeld. the poles). Without the volume exclusion force, chromosomes 260 J. Fan et al. / Computational Biology and Chemistry 31 (2007) 257–264 become so entangled that they cannot segregate. Membrane pressure keeps chromosomes inside the cell and helps segre- gate daughter chromosomes further apart and tends to retain structural loci conﬁguration. Without the friction force, domains oscillate in large amplitude around their equilibrium positions and the system is highly unstable. Without the random force, the system may be locked up in unstable states or local minima. 3. Results and discussion The simulation was started at the equilibrium conﬁgurations and such that the ori and ter domains are close to the cell center, while the left and right arms of the chromosome resided in the two cell halves (Nielsen et al., 2006a,b). Simulated chromoso- mal structures at various stages of the cell cycle are shown in Fig. 3. During the ﬁrst 20 min there was only one chromosome. At the 20th min the ori domain was unwound and replication was initiated. Once the ori domain was replicated, two daughter ori domains were formed by DNA recompaction near the posi- tion of the mother domain was located. Over the next 40 min, domains in the mother chromosome were replicated sequentially and bidirectionally until all domains were replicated. In the last 20 min, the daughter chromosomes were physically unconnected and separated. Before the ﬁnal septum formed at the cell equa- tor, the two chromosomes created a low concentration area of nucleoid in the mid-cell which is noted as one of the indepen- dent forces to position the cell division plane (Sun and Margolin, 2001). To track the positions of the ori domains during replication- segregation, we colored the ori domains in green in Fig. 3. Before initiation of replication, there was only one ori domain close to the middle of the cell. Then the mother ori domain was replicated into two daughter ori domains. As replication continued, the two daughter ori domains moved apart toward opposite poles. After completion of replication, the two ori domains relocated to the middle of the two daughter chromosomes. The ter domain was replicated after the ori domains had been fully apart. The loca- tions of the ori and ter domains during the cell cycle are shown in Fig. 4. The daughter chromosomes migrated to the left and right cell halves as they reorganized into their ﬁnal conﬁgurations. At the end of replication-segregation, topology of chromosomal Fig. 3. Simulated conﬁguration of E. coli chromosome structures. Green beads structures was faithfully inherited from the mother cell into the are domains that contain the ori region. Purple beads are domains undergoing new generation. replication. Red beads are domains of mother chromosome. Cyan and blue beads Hansen and colleagues used a ﬁne-tuned GFP-parB/parS sys- are domains of the two daughter chromosomes. tem to label the loci and recorded their positions over a cell cycle (Nielsen et al., 2006a). Our simulation reproduced many of their apart to their respective daughter chromosomes. The separation observations. For example, they reported that the ori region seg- speed of ter domains was faster than that of ori domains. Posi- regation displays a noticeable delay after replication. We predict tions of selected domains containing the loci studied by Hansen a similar phenomenon wherein the ori domains tend to be co- and colleagues are plotted in Fig. 5. The overall behavior of our localized longer than the other domains after they are replicated simulated loci migration is similar to their observations. The (Fig. 4). This is due to the closeness of the left and right neigh- ori domains separate near the cell center and they move slowly bors of the ori domain when the ori domain was being replicated. outward until they reach the cell quarter positions. Intermediate Since both of the replicated ori regions are connected to the same loci are segregated from positions that are away from the cen- neighbor domains transiently, they are tied to a position close to ter. However, the segregation of some early replicated domains each other for a small period. However, when the ter domain was seems faster in Hansen’s chart than those in our simulated results. replicated, the left and right domains were on different daugh- We believe that it is because foci close to each other may not ters. Hence the two newly replicated ter domains moved quickly be identiﬁable as distinct spots in experimental measurements. J. Fan et al. / Computational Biology and Chemistry 31 (2007) 257–264 261 Fig. 4. Location of the ori and ter regions over an E. coli cell cycle. The two regions are located near the middle of the cell before initiation of replication. Fig. 6. Migration of loci in the left and right chromosome arms and the center. After replication, the two regions move toward the two opposite poles and ﬁnally The single E. coli chromosome is organized with the left and right chro- localize near the quarter-cell positions. mosome arms in separate cell halves. After replication and segregation, the relative position of individual chromosome loci is preserved. The cell exhibits a “left-right-left-right” pattern before cell division which is in agreement with Only foci that are apart far enough were reported as separated observations. spots in Hansen’s chart. Also since cells may not initiate repli- cation exactly at the same cell length, the mapping from cell ric segregation in E. coli. Furthermore, in ﬁlamental cells that length to cell cycle time in the experimental chart introduced contain four or eight nucleoids, the majority of them had the errors (Nielsen et al., 2006a). During the structural reorganiza- asymmetric pattern polarized throughout the ﬁlament. In a cell tion process, locations of domains ﬂuctuated in the space and just prior to division, the arrangement of left and right arms of may have fast movement. The speed of loci movement usually chromosome is “left-right-septum-left-right” and in a ﬁlamen- maximizes right after their replication. The maximum speed of tal cell is “left-right-left-right-left-right-left-right . . .”. We ran a domain movement in our simulation is about 0.6 m min−1 . As simulation with initial conﬁguration of the left and right chro- maximum chromosome foci movement has been observed to be mosome arms in the left and right cell halves as observed in 0.5 m in a matter of seconds (Nielsen et al., 2006a) and many the experiments (Nielsen et al., 2006b). The simulated migra- loci can move 1 m apart in 2 min right after their replication tion paths of 5 individual loci along the chromosome, 2 left arm (Viollier et al., 2004), our speed of loci movement is reasonable. loci (64.1 and 74.1 ), 2 right arm loci (92.5 and 3.8 ) and the Sherratt and coworkers monitored the position of the ter mid-cell ori loci (84.3 ), are shown in Fig. 6. At the end of the region in a population of cells that had duplicated the ter region replication-segregation, the spatial conﬁguration of chromoso- and were approaching cell division (Wang et al., 2005). They mal loci is the same as the mother chromosome. The E. coli reported that 97% of sister nucleoids pairs exhibited asymmet- cell before division exhibited a “left-right-left-right” polarized asymmetric pattern which is in agreement with experimental observations. Earlier studies of chromosomal loci positions proposed that the ori and ter regions are located near the two opposite poles of E. coli in newly divided cells. Before the initiation of replica- tion, the ori and ter regions move to mid-cell. After replication, the ori and ter regions move back to the poles (Niki and Hiraga, 1998; Niki et al., 2000; Bates and Kleckner, 2005). However, in recent studies it is proposed that the ori and ter regions remain near the center of the cell before replication and move to quar- ter positions after replication (Wang et al., 2005; Nielsen et al., 2006a,b). We adopted the most recent ﬁndings so that the ori and ter domains were located near cell center while the left and right arms were placed in the two cell halves before the initiation of replication. We also tested the initial conﬁguration with the ori and ter domains close to cell poles. The simulated results with polar positioned ori and ter domains showed that Fig. 5. Predicted migration of loci in a cell cycle. The loci picked are from the work of Hansen and colleagues (Nielsen et al., 2006a). Loci positions are the chromosomes can segregate but the loci migration path does calculated from the distance to the nearest pole as Hansen et al. prepared their not agree with experimental E. coli observations (Nielsen et al., experimental chart. 2006a). After the ori domain has been replicated at the pole, 262 J. Fan et al. / Computational Biology and Chemistry 31 (2007) 257–264 Fig. 7. Simulated E. coli chromosome structure with double number of domains representing a chromosome (each domain size is half of the normal). The chro- mosomes are entangled and therefore cannot segregate readily. Shown is the conﬁguration just before division. Here green beads are domains containing ori region. The two daughter chromosomes are colored in cyan and blue. Fig. 8. Simulated results of E. coli chromosome structure with decreased mem- brane pressure coefﬁcient (10-fold smaller and 100-fold smaller). Chromosomes become wider in the radial direction of the cell and both cells cannot segregate one newly replicated ori domain moved toward cell center and in agreement with observations. Here green beads are domains that contains ori stayed there after replication has ﬁnished. The other ori domain region. The rest of the domains of two daughter chromosomes are colored in remained at the original pole during the whole process. When cyan and blue, respectively. the ter domain was replicated at its original pole, one newly replicated ter domain moved quickly to the cell center while Recent studies show that cell shape-determining protein the other remained at the pole. The orientation of chromosomal MreB plays an important role in chromosome segregation (Kruse structures was thus preserved after replication. In C. crescen- et al., 2003, 2005; Figge et al., 2004). The E. coli cell membrane tus, the ori and ter domains are at two poles before replication imposes high pressure on chromosomes inside the cell (Odijk, initiation. But after the ori domain is replicated, one newly repli- 2000). If the membrane is lysed, DNA can “explode” out of the cated ori domain moves rapidly to the opposite pole and stays cell (Odijk, 2000) with a liberated volume of 100-fold (Cunha et there (Jensen et al., 2001; Viollier et al., 2004; Thanbichler et al., 2005). The MreB protein of E. coli forms helical ﬁlaments al., 2005). Just before cell division, the chromosomes exhibit a beneath the cell surface and maintains the cell in a rod-shape “ori-ter-septum-ter-ori” conﬁguration. It is proposed that some (Kruse et al., 2003). Depletion of MreB results in spherical or polar localizing proteins may function in C. crescentus to tether enlarged cells and unsuccessful segregations (Kruse et al., 2003). the ori regions to both poles (Mohl and Gober, 1997; Jacobs et Thus we hypothesize that a decrease in cell membrane pressure al., 1999). might cause the segregation defects in MreB depleted cells. We Sufﬁcient compaction of chromosome plays an important simulated a 10-fold smaller membrane pressure coefﬁcient case role in chromosome segregation. The E. coli DNA is compacted with the same cell size in our model. The simulated chromosome by supercoiling (DNA gyrase), small nucleoid-associated pro- structures before cell division are shown in Fig. 8. As membrane teins (H-NS, HU, etc.) and SMC-like complexes (MukBEF). The pressure decreases, the chromosomes become wider in the radial HU and H-NS are histone-like proteins that assemble DNA into direction of the cell. The spatial conﬁguration of each chromo- nucleoprotein complexes. Certain mutations in the genes encod- some tends to be circular and the chromosomes movement space ing HU and H-NS cause nucleoids partition defects (Huisman is enlarged. These results indicate a random localization of ori et al., 1989; Dri et al., 1991; Kaidow et al., 1995). Mutations regions is more likely in the MreB depleted daughter cells than in mukB of E. coli result in chromosome decompaction and in the wild-type cells. The simulated results also indicate the irregular distribution of nucleoids along an elongated cell (Niki two daughter chromosomes cannot segregate as observed exper- et al., 1991, 1992). To test whether chromosome compaction imentally. When the membrane pressure coefﬁcient was further is a requirement for segregation, we constructed a model with decreased to 100-fold, the two daughter chromosomes stayed twice as many domains representing a chromosome (the size side by side in parallel (Fig. 8). The pair up of daughter chro- of each distinct domain is half of the normal). As shown in mosomes is observed in rich medium of MreB depleted cells Fig. 7, our result shows the two daughter chromosomes remain (Kruse et al., 2003). We also simulated a situation of 10-fold entangled together after replication. There is no nucleoid-empty larger membrane pressure coefﬁcient in our model. The sim- area at mid-cell. Hence, the E. coli chromosomes cannot form ulation results showed that domains became clustered toward a preferred cell division plane at mid-cell. This will generate the cell center but segregation was successfully accomplished; a nucleoid cells or elongated (ﬁlamental) cells without divi- in agreement with observations that over expression of wild- sion. The prediction is consistent with observations on cells type MreB does not perturb chromosome segregation (Kruse et with mutant compaction genes (Huisman et al., 1989; Dri et al., 2003). However, with a 100-fold larger membrane pressure al., 1991; Niki et al., 1991, 1992; Kaidow et al., 1995). How- coefﬁcient, chromosomes failed to segregate as the pressure is ever, by compacting DNA into fewer domains (increase domain too large. size while keeping all the other parameters same), our model shows that chromosomes can segregate successfully. It can leave 4. Conclusion excessive nucleoid empty areas around the mid-cell and in the two poles. Furthermore, the two daughter chromosomes may A free energy-driven chromosome structural optimization not precisely be located at the quarter-cell positions after seg- approach that simulats the E. coli replication-segregation cell regation (one is close to the middle while the other is in the cycle is presented. The model uses a string of topologically quarter). compacted domains to represent the chromosome structure. The J. Fan et al. / Computational Biology and Chemistry 31 (2007) 257–264 263 movement of individual domains is guided by a force ﬁeld Jun, S., Mulder, B., 2006. Entropy-driven spatial organization of highly con- accounting for stretching, bending, volume exclusion, mem- ﬁned polymers: lessons for the bacterial chromosome. Proc. Natl. Acad. Sci. U.S.A. 103, 12388–12393. brane pressure, friction, and random ﬂuctuations. The elongation Kaidow, A., Wachi, M., Nakamura, J., Magae, J., Nagai, K., 1995. Anucleate of cell membrane and sequential replication of domains are cell production by Escherichia coli delta hns mutant lacking a histone-like accounted for in the model. Migration of individual loci sim- protein H-NS. J. Bacteriol. 177, 3589–3592. ulated in our model agrees well with experimental observations. Kerr, R.A., Levine, H., Sejnowski, T.J., Rappel, W.J., 2006. Division accuracy in The simulated results indicate E. coli chromosomes may segre- a stochastic model of min oscillations in Escherichia coli. Proc. Natl. Acad. Sci. 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