Polymer-Mediated Compaction and Internal ... - Science Direct

Journal of Structural Biology 136, 53– 66 (2001) doi:10.1006/jsbi.2001.4420, available online at http://www.idealibrary.com on

Polymer-Mediated Compaction and Internal Dynamics of Isolated Escherichia coli Nucleoids So´nia Cunha,*,† Conrad L. Woldringh,* and Theo Odijk†,1 *Swammerdam Institute for Life Sciences, BioCentrum Amsterdam, University of Amsterdam, Kruislaan 316, 1098 SM Amsterdam, The Netherlands; and †Section Theory of Complex Fluids, Kluyver Institute for Biotechnology, Delft University of Technology, P.O. Box 5057, 2600 GB Delft, The Netherlands Received August 20, 2001, and in revised form November 7, 2001

a small volume (0.1– 0.3 ␮m3), i.e., about 15%, of the intracellular space (Woldringh and Nanninga, 1985). Previous studies (Stonington and Pettijohn, 1971; Worcel and Burgi, 1972; Sloof et al., 1983; see for a review Pettijohn and Sinden, 1985) have shown that isolated nucleoids can be analyzed by sucrose density gradients. Because the isolated nucleoids remain in a rather compact state (Hecht et al., 1975; Cunha et al., 2001), these folded chromosomes provide a convenient system for studying the mechanisms involved in DNA packaging in vivo. Three major DNA-compacting factors that need not be independent have been proposed: (i) superhelicity, (ii) binding of histone-like proteins and polyamines, and (iii) macromolecular crowding. Although bacterial DNA is negatively supercoiled (Worcel and Burgi, 1972; Pettijohn and Hecht, 1973; Sloof et al., 1983), its superhelicity leads only to a limited degree of compaction (Boles et al., 1990; for a new assessment see the Appendix). Histone-like proteins may bend and crosslink the DNA, forming more compact forms of DNA structures (Spassky et al., 1984; Dame et al., 2000). The degree of electrostatic binding of polyamines, such as spermidine, at intracellular ionic strengths is not precisely known at present and could be limited (Cayley et al., 1991; Pelta et al., 1996). Thus, the first two factors seem too weak to explain the compact state of the DNA within bacteria. We here focus on the third factor, macromolecular crowding or depletion. As the nucleoid coexists in the cell with a large number of proteins and other macromolecules (300 – 400 mg/ml of cytoplasm) (Zimmerman and Trach, 1991), macromolecular crowding has been suggested to promote phase separation within the cell (Zimmerman and Murphy, 1996; Odijk, 1998; Woldringh and Odijk, 1999). Its import has already been demonstrated by the in vitro condensation of linear DNA in the presence of cytoplasmic extracts and other macromole-

Nucleoids of Escherichia coli were isolated by osmotic shock under conditions of low salt in the absence of added polyamines or Mg2ⴙ. As determined by fluorescence microscopy, the isolated nucleoids in 0.2 M NaCl are expanded structures with an estimated volume of about 27 ␮m3 according to a procedure based on a 50% threshold for the fluorescence intensity. The nucleoid volume is measured as a function of the concentration of added polyethylene glycol. The collapse is a continuous process, so that a coil– globule transition is not witnessed. The Helmholtz free energy of the nucleoids is determined via the depletion interaction between the DNA helix and the polyethylene glycol chains. The resulting compaction relation is discussed in terms of the current theory of branched DNA supercoils and it is concluded that the in vitro nucleoid is crosslinked in a physical sense. Despite the congested and crosslinked state of the nucleoid, the relaxation rate of its superhelical segments, as monitored by dynamic light scattering, turns out to be purely diffusional. At small scales, the nucleoid behaves as a fluid. © 2001 Elsevier Science (USA) Key Words: dynamic light scattering; free energy; light microscopy; nucleoids; osmotic stress; osmotic shock; polyethylene glycol; supercoiling.

INTRODUCTION

Despite the absence of both nucleosome-like structures and a nuclear membrane, the chromosomal DNA of bacteria is organized as a compact structure called the nucleoid (Mason and Powelson, 1956; Kellenberger et al., 1958). The existence of the nucleoid implies an extreme compaction of the circular DNA double helix (⬃1.6 mm long) as it is confined to 1 To whom correspondence should be addressed at P.O. Box 11036, 2301 EA Leiden, The Netherlands. Fax: ⫹31 71 5145346. E-mail: [email protected].

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1047-8477/01 $35.00 © 2001 Elsevier Science (USA) All rights reserved.

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CUNHA, WOLDRINGH, AND ODIJK

cules (Lerman, 1971; Murphy and Zimmerman, 1994, 1995, 1997). Nevertheless, the magnitude and origin of the forces responsible for this compaction are only now becoming clear (Odijk, 1998). A theoretical analysis of DNA compaction in Escherichia coli based on excluded volume estimations predicts that the cross-interaction between the DNA and the soluble proteins present in the cytoplasm gives rise to so-called depletion forces that may be high enough to cause the formation of the nucleoid of the appropriate size (Odijk, 1998). In the study reported here, we used a nucleoid isolation procedure based on osmotic shock (Sloof et al., 1983), and simulated the depletion forces by submitting the isolated nucleoids to the crowding action of the neutral polymer polyethylene glycol (PEG) of molar mass 20 kg mol⫺1. The dimensions of the nucleoid are monitored by fluorescence microscopy with a resolution of about 300 nm. We are able to determine a measure of the Helmholtz free energy of the nucleoids by using a recent expression for the DNA–PEG interaction (de Vries, 2001) and by solving the coexistence equations for the osmotic pressure and PEG chemical potential. We also present the first investigation of the internal hydrodynamic properties of the isolated nucleoids via dynamic light scattering (DLS), a technique commonly used for studying the structure and internal modes of motion of flexible macromolecules. Microscopy monitors only a global state of the nucleoid; DLS is important to gauge the state of the genome at small scales. Our results are discussed in the light of the current theory of branched DNA supercoils (reviewed in the Appendix). MATERIALS AND METHODS

sozyme buffer (3.5 mM sodium phosphate (NaPi) pH 7.2, 0.3 g liter⫺1 lysozyme (Sigma Aldrich), 10 mM EDTA pH 7.2, 60 mM NaCl, 0.8 M sucrose (27% w/v)). In general within 20 min of incubation ⬃90% of the cells became spheroplasts. A sample (10 ␮l) of the spheroplast suspension was added to 990 ␮l of a 10 mM NaCl solution. Lysis was achieved by carefully mixing this suspension in a round-bottomed tube and confirmed by phase-contrast and fluorescence microscopy. Nonviscous lysates where the nucleoids were stable at room temperature for several days were obtained. Polymer-mediated compaction. Polyethylene glycol (molar mass 20 kg mol⫺1) was obtained from Sigma Chemical Company. Aliquots of 100 ␮l of the crude lysates were diluted with 300 ␮l of an aqueous solution containing DAPI, NaCl, and PEG. The final ionic strength and DAPI concentration of each sample were 0.2 M and 0.5 ␮M, respectively. After 30 min incubation at room temperature, samples were kept on ice until use. Samples of freshly prepared nucleoids (10 ␮l) with and without PEG were applied to object slides, covered with a coverslip (18 ⫻ 18 mm2), sealed with nail polish, and left for 10 min at room temperature. Microscopy and image analysis. Microscopic imaging was performed using a Princeton RTE 1317-k/1 cooled CCD camera (Princeton instruments, Groenekan, The Netherlands) mounted on an Olympus fluorescence microscope (BX60 Olympus, Tokyo, Japan) equipped with a 100-W mercury lamp. Images were made using the public domain program OBJECTIMAGE 1.62n (Vischer et al., 1994). In our measurements we assumed the nucleoids to have ellipsoidal shapes with the major axis lying horizontal. The images were thresholded such that the fluorescence inside the boundary was 50% of total fluorescence. As a first approximation only the 50% axes were used to calculate an index volume. The ellipsoidal volume was calculated by V ⫽ major axis ⫻ minor axis2 ⫻ ␲/6. Absolute values can be higher by a factor of at most 3. Nucleoids were first photographed with a UV filter (U-MWU/ narrow wide band cube; illuminated at 330 –385 nm and with an emission filter at 420 nm) and then in phase-contrast mode to check whether or not the nucleoids were attached to empty cell ghosts. In some experiments, the fluorochrome FM4-64 (Molecular Probe No. T-3166) was added (5 ␮M) to visualize ghosts or envelope remnants. A green filter (U-MNG/narrow band cube; illuminated at 530 –550 nm and with an emission at 590 nm) was then used. Care was taken to photograph free-floating nucleoids (not attached to cell remnants or to the glass surface).

Bacterial strain and growth conditions. E. coli cells of strain PJ 4271 (Jensen et al., 1999; Collection No. LMC1236) were grown at 37°C in batch cultures in glucose minimal medium containing, per liter, 6.33 g K2HPO4 䡠 3H2O, 2.95 g KH2PO4, 1.05 g (NH4)2SO4, 0.10 g MgSO4 䡠 7H2O, 0.28 mg FeSO4 䡠 7H2O, 7.1 mg Ca(NO3)2 䡠 4H2O, 5 g glucose, 50 mg lysine, 40 mg leucine, 40 mg isoleucine, 40 mg valine, 4 mg vitamin B1 and supplemented with 200 ␮g ampicillin. NaCl was added to adjust the osmolarity of the medium to 300 mosM (Micro-Osmometer, Advanced instruments). The doubling time (Td) was about 60 min. Cell growth was monitored by measuring the optical density (OD) at 450 nm with a Gilford microsample spectrophotometer and cell number with an electronic particle counter. Exponential growth was maintained by periodic dilution of the culture. Cells were harvested in steady state of growth, defined by the constancy of average cell mass (OD/cell number). DAPI (4⬘,6-diamidino-2-phenylindole, dihydrochloride; Molecular Probe No. D-1306) was added to the cell culture to a final concentration of 0.5 ␮M at least 15 min before harvesting.

Dynamic light scattering. All the solutions used during the isolation procedure, from growth medium until the lysis buffer, were first filtered through a 0.22-␮m polycarbonate membrane filter (Millipore S. A.). Cultures were started by adding a thick unfiltered bacterial suspension of 500 ␮l to 50 ml of filtered medium. A second dilution was made by adding 2 ␮l of this cell suspension to 50 ml of filtered medium. No DAPI was added to the cell culture. The final samples consisted of a 5-ml nucleoid suspension (10 mM NaCl). DLS measurements were performed with a Krypton ion laser (Series 2020 –11, Spectra-Physics) operating at a wavelength of 647.1 nm. The scattered light was collected by a photomultiplier (H7155-01 Photon Counting Head, Hamamatsu) and analyzed with a Malvem Multi 8 7032 CE 128-point correlator. A cylindrical glass scattering cell of 1-cm diameter was used and samples were thermostatted at 298 K. For data analysis, we used the program ORIGIN to fit the autocorrelation functions Y for the scattered intensity to double exponentials. Here the baseline is Y0 and A1 and A2 are amplitudes:

Preparation of nucleoids. Samples of 10 ml cell culture were harvested at an OD450 of 0.2 (107 cells ml⫺1) and centrifuged (5 min, 20,000g, room temperature). The pelleted cells were resuspended at room temperature in 200 ␮l of freshly prepared ly-

Y ⫽ Y 0 ⫹ A 1e ⫺t/␶1 ⫹ A 2e ⫺t/␶2.

(1)

The relaxation times obtained, ␶1 and ␶2, were converted into effective diffusion coefficients using the relation Deff ⫽ (2k2␶)⫺1

COMPACTION AND DYNAMICS OF Escherichia coli NUCLEOIDS where k is the scattering vector (k ⫽ 4␲n0 sin (␪/2)/␭) in which n0 is the refractive index of the solution, ␭ the wavelength of the incoming coherent light, and ␪ the scattering angle.

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II). To gain insight into the diffusional nature of the fast mode, the fast relaxation rates (1/␶2) are plotted as a function of k2 (Fig. 5).

RESULTS

Polymer-mediated compaction of isolated nucleoids. The appearance of DAPI-stained nucleoids in an intact living cell or spheroplast and after osmotic shock is illustrated in Fig. 1. The response of isolated nucleoids to different PEG concentrations is shown in Fig. 2A. The average nucleoid volumes, summarized in Fig. 2B and Table I, confirm that a gradual (never discontinuous) compaction of the nucleoids occurs with increasing PEG concentration. Note that the volume distributions are rather broad (Fig. 3) and show a fairly constant coefficient of variation (CV). The polydispersity of the distribution, using the detergent method, was already studied in the past via sedimentation through sucrose gradients at 1 M NaCl (Worcel and Burgi, 1972). This property was partly attributed to the variation in DNA content of the nucleoids at different stages of replication. However, it is difficult to compare their qualitative analysis with our results because we lack a quantitative theory of nucleoid sedimentation. (One of us, T. Odijk, is currently developing such a theory (manuscript in preparation).) Also, the respective ionic conditions differ. Based on our microscopic observations, the breadth in the distribution is, in part, related to the fact that the nucleoids exhibit a variety of shapes. For instance, in the same lysate rod-shaped and globular nucleoids can be seen. In addition, during the excitation at low PEG concentrations the isolated nucleoids are very unstable and expand rapidly, becoming more diffuse structures. Accordingly, we attempt to monitor initial shapes before the onset of instability. The same phenomenon was observed in previous microscopic studies (Hecht et al., 1975), where ethidium bromide was used as a fluorescence probe. In our experiments, addition of PEG appeared to contribute to a stabilization of the nucleoids in this respect (This could simply imply that PEG compactifies the DNA even if it is nicked or broken. We have not investigated this problem.) Because the expansion on excitation is more severe in the absence of PEG, the sizes we present at low PEG concentrations should be seen as only estimates of the real sizes. For this reason we exclude data below 0.01 g ml⫺1 PEG in our quantitative analysis. Dynamic light scattering. The time dependence of the scattered light intensity is interpreted in terms of two exponential decays. In Fig. 4 we can see that the biexponential fit is excellent for isolated nucleoids in 10 mM NaCl. Two well-separated decay times were obtained at all scattering angles (Table

DISCUSSION

Because the depletion interaction between linear DNA and PEG has recently been established (de Vries, 2001), it is possible to derive the free energy Fn of the nucleoids. It has become fairly routine to determine the equation of state of compacted linear DNA (Podgornik et al., 1989; Rau and Parsegian, 1992; Strey et al., 1999). In these osmotic stress experiments, the DNA gel is separated from the polymeric compacting agent by a semipermeable membrane. Compaction of linear DNA by PEG in the same solution is well known (Lerman, 1971; Laemmli, 1975) (or by polyethylene oxide (PEO), which is identical to PEG except possibly for the end groups (Bailey and Koleske, 1976)). Here the nucleoids are bound to retain proteins that are associated with the original supercoiled genome in a fashion that is not fully understood. The problem is then whether PEG may be considered inert, since there have been recent intimations that PEG or PEO may exert attractive forces on certain proteins, at least at high polymer concentrations (Sheth and Leckband, 1997; Israelachvili, 1997; Halperin and Leckband, 2000; Kulkarni et al., 2000). Nevertheless, in many other investigations, there is little or no evidence for any kind of forces besides depletion. (Depletion is the entropic interaction of a flexible polymer chain with a hard particle, caused by the loss of polymeric configurational degrees of freedom. The density of polymer segments must be zero at the particle surface (de Gennes, 1979).) Wills et al. (1995) determined the interaction between PEG and a whole host of proteins by gel chromatography. This interaction is well described by the depletion theory of Jansons and Phillips (1990). Partitioning experiments between PEO and a variety of small proteins were interpreted in terms of depletion although fairly weak attractive interactions had to be invoked if the explanation was to be quantitative (Abbott et al., 1992a– c). A range of proteins retarded by PEG in gel capillary electrophoresis show no anomalies that could be attributed to attractive forces (Radko and Chrambach, 1997). Indeed, for small proteins the retardation agrees quantitatively with a depletion theory of protein transport (Odijk, 2000a). On the whole, we conclude that PEG may be considered inert in general, but we have to remain alert to possible anomalies. We next assume that PEG interacts with the DNA of the nucleoid only. The fraction of proteinaceous material attached to the DNA superhelix is probably

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COMPACTION AND DYNAMICS OF Escherichia coli NUCLEOIDS

TABLE I Image Cytometric Determination of the Average Index Volumes (Vn) of the Isolated E. coli Nucleoids at Various PEG Concentrationsa PEG (g/ml)

␲os (atm) Vnb (␮m3) CV (%)

0

0.005

0 27 53

0.0087 32 49

0.01

0.015

0.02

0.025

0.03

0.04

0.06

0.08

0.10

0.021 28 57

0.036 21 61

0.056 18 45

0.079 10.0 47

0.107 8.3 39

0.175 6.4 48

0.372 2.8 52

0.661 0.52 78

1.057 0.37 60

a The osmotic pressure ␲os of the pure PEG solution is also displayed (calculated with the help of the empirical expression of Parsegian et al., 1986). b At each PEG concentration 40 – 60 nucleoids were measured.

small so the neglect of the PEG–protein depletion interaction should be acceptable (See the Appendix for a discussion of this issue. Note that at the level of approximation adopted here, we may neglect cosolvent effects as discussed by Timasheff (1998). We assume the system is incompressible so our starting point will be the Helmholtz free energy.) The PEG– DNA interaction (per nm of the DNA helical contour) introduced by de Vries (2001), f dep ⫽ E1w ⫹ E2w9/4 , kBT

(2)

has the correct form at low and high PEG or PEO weight fractions (w in g ml⫺1). The free energy is scaled by kBT, where kB is Boltzmann’s constant and T the temperature; the constant E1 is 4 and E2 is 50 so that the fdep is given in kBT/nm when w is in g ml⫺1. Equation (2) allows for a quantitative description of DNA condensation in PEO solutions (Frisch and Fesciyan, 1979; Vasilevskaya et al., 1995; de Vries, 2001). We now express the Helmholtz free energy of the nucleoid in a suspension of PEG coils as

冉冊

m F tot ⫺ m ⫹ k2mw ⫽ m ln kBT V

where the weight fraction w ⫽ mM/VNav with m the number of PEG coils in volume V (M ⫽ 20 kg mol⫺1, the molar mass of PEG, Nav ⫽ Avogadro’s number). The free energy of the nucleoid of contour length L is Fn. It is presupposed that the protein content of the nucleoid remains constant as we vary w. In Eq. (3) we distinguish an ideal free energy of the polymer chains (the first two terms) and two virial terms containing the coefficients k2 and k3. The latter are determined by evaluating the osmotic pressure ␲os ⫽ ⫺⭸Ftot/⭸V of the PEG solution without the nucleoid. This must yield the empirical osmotic pressure, which we here approximate as a virial series for convenience:

␲ os ⫽ Aw ⫹ Bw2 ⫹ Cw3. 共w ⱕ 0.1兲

The coefficients A ⫽ 1.237 ⫻ 105, B ⫽ 7.02 ⫻ 106, and C ⫽ 2.46 ⫻ 107 are chosen so that we have the correct dilute limit for PEG as w 3 0 and Eq. (4) conforms to well-established pressure data within a few percent (Parsegian et al., 1986: ␲os in N m⫺2 if w is in g ml⫺1; k2 ⬅ B/A; k3 ⬅ C/2A). We next assess the possible states of thermodynamic equilibrium resulting from Eq. (3). The osmotic pressure balance is given by w o ⫹ k2 w2o ⫹ 2k3 w3o ⫽ wi ⫹ k2 w2i ⫹ 2k3 w3i

Lfdep Fn ⫹ k3mw ⫹ ⫹ , kBT kBT 2

(3)

(4)

0.349wi 9.85w9/4 3.34 ⫻ 10⫺8 G i ⫹ ⫹ ⫹ Vn Vn Vn

(5)

FIG. 1. Phase-contrast and fluorescence images of cells and nucleoids at different stages of isolation: (A) cells, (B) spheroplasts, (C) isolated nucleoids in the presence of 10 mM NaCl. Nucleoids are stained with 0.5 ␮M DAPI and envelopes with 5 ␮M FM4-64. Bar ⫽ 2.5 ␮m. These images were made using a CoolSNAP color CCD camera (Roper Scientific). FIG. 2. PEG-Mediated compaction of isolated E. coli nucleoids. (A) DAPI-stained free-floating nucleoids at several representative PEG concentrations (indicated in g ml⫺1). The contrast of the images was inverted. Bar ⫽ 2.5 ␮m. (B) Average index volume of the isolated nucleoids as a function of PEG concentration in 0.2 M NaCl. Volumes were measured using a macro of Object-Image written by N. Vischer (BioCentrum, University of Amsterdam, manuscript in preparation).

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FIG. 3. Examples of index volume distributions of isolated nucleoids at two PEG concentrations: (A) 0.02 and (B) 0.1 g ml⫺1. Average values and coefficients of variation are given in Table I. Note the different scales.

for a nucleoid coexisting with an outside reservoir (i ⫽ inside the nucleoid, o ⫽ outside). Here we have introduced an empirical measure of the free energy Fn of the nucleoid that we wish to determine: G⬅ ⫺

V n ⭸Fn . kBT ⭸Vn

(6)

In Eq. (5) the two depletion terms inversely proportional to the nucleoid volume Vn are computed with the help of Eq. (2) using the fact that the contour of the DNA helix in the average nucleoid is 2.6 mm. The average amount of DNA per nucleoid estimated by image cytometry (Huls et al., 1999) has been found for the present cells to be 1.64 chromosome equivalents (unpublished results); the contour

length of a nonreplicating chromosome is 1.6 mm (note that in Eq. (5), Vn is in ␮m3 and w is in g ml⫺1). The weight fraction of PEG inside the nucleoid, wi, is slightly less than the weight fraction outside, wo. The difference is caused by the depletion interaction (Eq. (2)) and is computed via the balance of the polymer chemical potentials (⫽⭸Ftot/⭸m) inside and outside the nucleoid: ln wo ⫹ 2k2 wo ⫹ 3k3 w2o 0.349 9.85w5/4 i ⫽ ln wi ⫹ 2k2 wi ⫹ 3k3 w ⫹ ⫹ . Vn Vn

It is now straightforward to obtain empirical values for the quantity G from Eqs. (5) and (7) (see Table III). In general, for a macromolecular chain one expects power laws for thermodynamic quantities, especially in the asymptotic limit of long chains as we have here (de Gennes, 1979). This is indeed the case for ln G versus ln Vn is linear within the experimental margin of error (see Fig. 6). It is useful to rewrite the scaling relation obtained in terms of the Helmholtz free energy:

冉冊

Fn ⫽ g

Vo V

1.34

kBT

共 g ⫽ 362兲.

(8)

TABLE II Decay Times Obtained from DLS Measurements for the Isolated Nucleoids in the Absence of PEG FIG. 4. The autocorrelation function in arbitrary units at a scattering angle of 50° (equivalent to a magnitude of the scattering vector 1.092E ⫹ 7 m⫺1), defined in Eq. (17) (Appendix), is plotted against t (s). Also shown is a biexponential fit (see Eq. (1)) given by I2 ⫽ 1.05E ⫹ 14 (⫾8.79E ⫹ 10) ⫹ 6.43E ⫹ 13e⫺t/(0.02187) ⫹ 4.16E ⫹ 13 e⫺t/(0.00425).

(7)

2 i

k2 (m⫺2)

␶1 (s)

␶2 (s)

1.19E ⫹ 14 2.47E ⫹ 14 3.92E ⫹ 14 5.26E ⫹ 14

0.02187 0.01139 0.00692 0.00471

0.00425 0.00195 0.00114 0.00076


FIG. 5. Fast relaxation rates (1/␶2) as deduced from a biexponential fit are plotted as a function of k2 (refractive index of water n ⫽ 1.331). A least-squares linear plot is given by 1/␶2 ⫽ ⫺114.5 (⫾61.5) ⫹ 2.6E⫺12(⫾1.7E ⫺ 13)k2.

Here we have introduced the volume at zero compression (Vo ⬇ 27 ␮m3). The slight deviations from the double-logarithmic curve (Fig. 6) give us confidence in the entire procedure (isolation and identification of the nucleoids, calibration, use of Eq. (2)). Let us discuss the implications of Eq. (8). In polymer theory, the general scaling structure of a compressed coil is (de Gennes, 1979) F coil ⬇

冉冊

Vo d kBT. V

(9)

This is independent of the nature of the interactions or chain structure. In physical terms, this expression implies that only a few of degrees of freedom are involved when the relative deformation is of order unity and hence is only global. From this point of view, the large coefficient g in Eq. (8) is anoma-

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lous. One resolution suggests itself in the context of polymers gels: g represents the number of crosslinks (de Gennes, 1979). At V ⫽ Vo the dimensionless free energy is a measure of the number of degrees of freedom frozen in. However, at the same time, the exponent in Eq. (8) is substantially larger than what one would expect for a crosslinked random coil (a similar expression with g Ⰷ 1, but the exponent d is then 2/3 (de Gennes, 1979)). We can adduce further evidence for possible crosslinks of the nucleoid by estimating its dimensions if it were not crosslinked. The genome is viewed as a branched plectonemic supercoil perturbed by the excluded-volume effect. We here simply assume we may use equilibrium statistical mechanics (Odijk, 1998; Woldringh and Odijk, 1999). We set the specific linking difference of the supercoil equal to ␴eff ⬇ ⫺0.025 (Bliska and Cozzarelli, 1987; Bates and Maxwell, 1993). The associated proteins are hypothesized to cause a homogeneous decrease in the degree of supercoiling, in the absence of any evidence to the contrary. We give a detailed analysis of the size of a branched E. coli genome at an ionic strength of 0.2 M in the Appendix. We combine results from Monte-Carlo simulations on annealed branched chains, the current theory of plectonemes, and simulations yielding the number of supercoiled branches. If the supercoil segments did not interact at all (i.e., if the coil were phantom-like), the volume Veq of the nucleoid containing 1.64 chromosome equivalents of DNA would be 3.2 ␮m3, which is less than Vo. The segments, however, repel each other by volume exclusion so that the supercoil expands to a size of Veq ⫽ 166 ␮m3, which is much greater than

TABLE III Measure of the Free Energy G of the Isolated Nucleoids at Various PEG Concentrations wo (g/ml)

wi (g/ml)a

Vn (␮m3)

Ga

0.015 0.02 0.025 0.03 0.04 0.06 0.08 0.10

0.014899 0.019866 0.024736 0.029656 0.039498 0.05864 0.07200 0.08772

21 18 10.0 8.3 6.4 2.8 0.52 0.37

600 850 1760 2420 3930 1.29 ⫻ 104 8.61 ⫻ 104 1.52 ⫻ 105

a Variables derived from the coexistence equations (5) and (7). See Discussion for details.

FIG. 6. Relation between G and the index volumes Vn for isolated nucleoids at various PEG concentrations. The data in Table III are plotted in a double-logarithmic format. A linear least-squares fit gives ln G ⫽ 10.6(⫾0.07) ⫺ 1.34(⫾0.034) In Vn.

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Vo ⫽ 27 ␮m3. We are quite certain of this inequality. Though the theories quoted in the Appendix are not free from approximation, the value Veq ⫽ 166 ␮m3 for an uncrosslinked genome is very likely a lower bound (e.g., the Kuhn length could be greater owing to electrostatic interactions within the supercoil; the branching density could saturate to a value higher than the one at finite sizes estimated in the Appendix). The small size of the nucleoid is then a second indication that it could be crosslinked. A further peculiarity of Eq. (8) is that the exponent is larger then unity, which is difficult to understand in the context of theories for uncrosslinked supercoils as discussed in the Appendix. The lead excluded-volume term in Eq. (16) scales as V⫺1 n . If one solves the coexistence equations for an uncrosslinked genome within the bacterial cytoplasm (Odijk, 1998), the transition is first-order (in the sense defined in Lifshitz et al., 1978). Intriguingly enough, if we use the free energy of Eq. (8) instead of the lead term of Eq. (16) to solve these equations, we get a continuously decreasing compaction as we increase the protein concentration. This is caused by the V⫺1.34 dependence in Eq. (8). The compression n displayed in Table I is also continuous for the very same reason (by contrast if we use Eq. (16) in the coexistence Eqs. (5) and (7) for an uncrosslinked supercoil compacted by PEG, we get a first-order transition or jump for the predicted supercoil volume at a certain PEG concentration). A second issue is whether Eq. (8) is able to explain the nucleoid volume (0.080 ␮m3) measured by Valkenburg and Woldringh (1984). Within the context of the theory of Odijk (1998), the coefficient g would need to be two and a half times lower if our nucleoid free energy is to be compatible with the known dimensions of the E. coli nucleoid in vivo (assuming it contains one chromosome equivalent of DNA). The slight curvature in Fig. 6 and the neglect of protein virials in the phase separation scheme (Odijk, 1998) could well account for such a factor. We conclude that Eq. (8) is consistent with the dimensions of the genome within E. coli (Valkenburg and Woldringh, 1984; Odijk, 1998). These considerations aside, it is gratifying to note that the free energy (⫽2.0 ⫻ 105 kBT) given by the empirical Eq. (8) when suitably scaled down for a nucleoid of one chromosome equivalent is close to that (⫽1.3 ⫻ 105kBT) predicted by Eq. (16). The crosslinking that we have introduced must be interpreted in terms of the equilibrium statistical thermodynamics employed. Crosslinking could arise because dynamic entanglements are frozen in as the genome escapes rapidly from the envelope after osmotic shock. It is conceivable that tight knots could arise lasting longer than the time scale of the experiments.

It is also tempting to associate g with the “crosslinks” connected with the independent domains within the nucleoid hypothesized a long time ago (Worcel and Burgi, 1972; Sinden and Pettijohn, 1981; Pettijohn and Hecht, 1973; Pettijohn, 1990). Currently, these domains are often reinterpreted as stochastic and dynamic in nature rather than static, on the basis of genetic experiments (Miller and Simons, 1993; Pavitt and Higgins, 1993; Higgins et al., 1996; Schreirer and Higgins, 2001). These crosslinks are viewed as impediments to supercoil diffusion. Whether our statistical crosslinking number g is indeed related to the domain number (ᏻ (100), for our nucleoids) is an issue that could be verified by strand nicking experiments (like those of Sinden and Pettijohn, 1981) on nucleoids compacted by PEG. As we discussed above, the dependence of Eq. (8) on nucleoid size is a bit more marked than one would expect on the basis of excluded-volume arguments (Odijk, 1998) (Eq. (16) in the Appendix). It can be shown that this discrepancy cannot be explained by incorporating higher-order virials in the theories outlined in the Appendix. The exponent ⫺1.34 could, however, arise by indirect crowding (Zimmerman and Harrison, 1987; Zimmerman and Trach, 1991). Even under in vitro conditions, there ought to be proteins associated with the DNA helix; this interaction would be strengthened by an increasing PEG concentration. The specific linking difference would tend to decrease and so the supercoil diameter would be enhanced (see Appendix). On the whole the compression of the nucleoids would be more difficult than expected because the excluded-volume effect is not constant but rather increases on addition of PEG. The continuity of nucleoid compression or expansion could be tested by systematically changing the concentration of the cytoplasm. Recent leakage experiments have been transient in nature (Murphy and Zimmerman, 2001). It would be interesting to carry out more gradual experiments to see whether the nucleoid expansion exhibits a first-order transition or not. The discussion above has dealt with the average or global behavior of the nucleoids. An important issue is whether the crosslinking we have found has any bearing on the dynamics of the superhelical segments of the nucleoid at small scales. Recently, it was argued that this could be fast even in the case of appreciable crosslinking (Odijk, 2000b). In light scattering experiments, one monitors a region of size k⫺1 at a wavevector k associated with the scattering angle ␪. Accordingly, the variable kRg is a measure of the degree one looks inside the supercoil (Rg is the root-mean-square radius of gyration; for a discussion of Rg and the dynamic light scattering by a

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supercoil, see the Appendix). The quantity kRg has not been substantial in previous dynamic light scattering experiments on supercoiled plasmids (Langowski, 1987; Langowski et al., 1994; Seils and Pecora, 1995; Hammermann et al., 1997). By contrast, for our nucleoids in the absence of PEG, the quantity kRg ⫽ ᏻ(10), so we are truly monitoring the internal dynamics of the particles. Figure 5 shows that the rate does not have a k3 dependence characteristic of the internal dynamics of long flexible chains. The k2 dependence implies that the dynamics is pure diffusion (see Appendix). From the slope we get a diffusion coefficient Deff ⫽ 1.3 ⫻ 10⫺12 m2 s⫺1 which should be compared with Dc ⫽ 2.1 ⫻ 10⫺12 m2 s⫺1 at I ⫽ 10 mM estimated with the help of Eq. (20) in the Appendix. The hydrodynamic screening length turns out to be about 79 nm so that the nucleoid is not nondraining but rather porous because it is a compact and congested object. Nevertheless, the internal dynamics is fast for the superhelical segments do not easily entangle (Odijk, 2000b). This is quite reasonable if we take into account that the number of crosslinks (g ⬇ 360) and the number of branches (ns ⬇ 2800) are smaller than the number of superhelical segments (Ns ⬇ 4000) within the nucleoid (see Appendix). CONCLUDING REMARKS

It is important to recognize that the PEG compaction discussed here is only superficially identical to the usual compaction of the E. coli genome by the cytoplasm. Polyethylene glycol of molar mass 20 kg mol⫺1 has a radius of gyration of 6.9 nm (Devanand and Selser, 1991; PEG is known to exhibit a full excluded-volume effect down to 20 kg mol⫺1, Bailey and Koleske, 1976), but is not a compact particle with regard to depletion: the PEG molecule may wrap itself around the DNA helix (de Vries, 2001). The exclusion or depletion radius E is effectively about 6.5 nm (estimated from the first term in Eq. (2)) which is to be compared with E ⫽ 4.7 nm introduced earlier for the cytoplasmic proteins (Odijk, 1998). Furthermore, the PEG coils are not compact particles so at a certain concentration they interpenetrate and the solution becomes semidilute: at higher concentrations, the interpenetrating network interacts with a DNA helix in a qualitatively different way (de Vries, 2001), resulting in the scaling law expressed by the second term in Eq. (2). This effect is of course absent in the case of supercoil compaction by globular proteins (Odijk, 1998). Our analysis of the nucleoid compaction by PEG has led us to conclude that the in vitro nucleoid is crosslinked in a physical sense: (i) In the absence of PEG it is smaller than it should be on the basis of

theoretical arguments as applied to an uncrosslinked supercoil. (ii) The empirical scaling law for compaction (Eq. (8)) does not conform to that expected for uncrosslinked chains; in particular the prefactor g, which should be a measure of the number of crosslinks, is quite large. (iii) The compaction is continuous as a function of the PEG concentration; the nucleoid is difficult to compress so that a coil– globule transition one usually expects (Lifshitz et al., 1978) is preempted. Although we conclude that the E. coli nucleoid is highly congested even in the absence of PEG, the superhelical segments within diffuse quite fast according to our DLS experiments. They do not seem to be dynamically entangled which seems paradoxical but is not, if one recognizes that congested slender rods may reorient quite freely (Odijk, 2000b). APPENDIX: THEORETICAL CONSIDERATIONS

Statistics of branched polymers. We first summarize recent results from the theory of branched flexible polymers and then apply these to a DNA supercoil of genomic size. The theory of branched polymers has a long history dating back to the seminal work of Zimm and Stockmayer (1949) and de Gennes (1968). For a randomly branched chain consisting of a total of N links each of length A and having n3 trifunctional branches (see Fig. 7), the root-mean-square radius of gyration Rg,0 (or radius of gyration for short) is given by R 2g,0 ⬅ 具s2 典0 ⫽

⌳⬅

n3 . N

冉冊

A2 ␲N 4 ⌳

1/ 2

,

(10)

(11)

Here, the index 0 signifies the chain is ideal; i.e., the links are phantom-like and do not interact with each other. For a given configuration of the chain within the entire ensemble of possible states, the vector s is the sum of all vectors from the centerpoints of the links to the center of mass of the chain. The quantity s2 is averaged over all realizations (denoted by 具典), which is equivalent to the actual thermal average. The chain is of course continually buffeted by the thermal motion of the molecules of the surrounding solvent. The small size of the chain (Rg,0 ⬇ N1/4) may be understood qualitatively as follows. We focus on a path spanning the entire chain diameter. It contains m links. The path is a random walk so that we must have Rg,0 ⬃ m1/2 after averaging over all realizations. But we know the number of branches sprouting from this path is pro-

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portional to m. Hence, m2 must be proportional to the total number of links N, which explains the dependence of Eq. (10) on N. We next switch on a repulsive excluded-volume effect between the links. Scaling arguments have been adduced to show that the average size of the chain is now considerably greater than that given by Eq. (10) (Daoud and Joanny, 1981). However, the branching need not be quenched (quenched here means the branches are invariant and independent of external conditions) but may be annealed; in the latter case the degree of branching also depends on the excluded-volume effect (Gutin et al., 1993). These scaling analyses (and other types of theories that we need not quote here) predict that for branched chains Rg ⬇ N␮ with ␮ close to 1/2 (whether the branches are annealed or quenched although the respective exponents are not identical). The only theoretical work that has been numerically precise and thus suitable for our purposes has been via Monte-Carlo simulation (Janse van Rensburg and Madras, 1992; Cui and Chen, 1996). If the excluded volume between two links is ␤, the radius of gyration Rg of an annealed branched chain is expressed in terms of the excluded volume parameter z, which is universal (Cui and Chen, 1996): R 2g ⬇ 1.37z2/5 R2g,0 z⬅

共z Ⰷ 1兲,

冉

16 ␲ 2N ⌳ 2␤ 4 3 共4 ␲ 兲 A ⌳

冊

(12) 5/4

.

(13)

In Eq. (12), the numerical coefficient is estimated from Table I of Cui and Chen (1996). Note that they also estimated the branching density ⌳ (Eq. 11) numerically (a` posteriori). Size of the uncrosslinked E. coli genome. We now use Eqs. (10)–(13) to give a quantitative estimate of Rg for the E. coli genome containing one chromosome equivalent of DNA, if we assume it is a branched supercoil without any crosslinks (a qualitative analysis of the radius of supercoiled DNA was given previously by Marko and Siggia, 1995). We first formulate an equivalent model correct at the mesoscale we focus on (for a discussion of this point, see Woldringh and Odijk, 1999). At an effective specific linking difference ␴ ⬇ ⫺0.025 and an ionic strength of monovalent salt I ⫽ 0.2 M, the supercoil is plectonemic and has a diameter Ds ⫽ 32 nm (Ubbink and Odijk, 1999). The contour length of the DNA helix is L ⫽ 1.6 mm; the contour length of the plectonemic supercoil is then Ls ⫽ 630 ␮m where the superhelical pitch angle ␣ is about 52° (Boles et al.,

FIG. 7. Randomly branched chain consisting of N ⫽ 30 links and n3 ⫽ 7 trifunctional branches.

1990; Ubbink and Odijk, 1999). At high ionic strength, the persistence length of a plectonemic supercoil has been estimated to be Ps ⫽ 2P sin ␣, where P ⫽ 50 nm is the usual persistence length of the DNA helix (see Eq. 8.2 of Odijk, 1996); accordingly the step length or Kuhn length of an effective link is As ⫽ 2Ps ⬇ 158 nm. We next suppose the branches occurring in the genome are purely trifunctional (see Fig. 7). Boles et al. (1990) determined the branching density of DNA plasmids by electron microscopy at a monovalent ionic strength close to 0.1 M. Their result of about 0.45 branch per kilo-base pair is independent of ␴ within the experimental margin of error. The two plasmids used were quite small (3.5 and 7 kb). Branches occurring in longer superhelical DNA (up to 20 kb) were investigated more recently by MonteCarlo simulation, leading to an asymptotic value of 0.6 branch/kb at ␴ ⫽ ⫺0.05 and I ⫽ 0.2 M and at higher contour lengths (Vologodskii and Cozzarelli, 1996). Because of the experimental invariance of branching with respect to the specific linking difference ␴, we here simply use the computationally determined branching density and assume it remains valid at the very high contour length of the E. coli genome. For our equivalent model consisting of Ns Kuhn segments, this yields ⌳s ⫽ n3/Ns ⫽ 0.71. The Kuhn segments interact with each other like hard-core cylinders of length As and diameter Ds. The excluded volume between a pair is ␤ ⫽ ␲A2s Ds/2 (Odijk, 1998; for a didactic exposition, see Woldringh and Odijk, 1999). There are Ns ⫽ Ls/As ⫽ 4000 Kuhn links in the genome. The radius of gyration of the supercoil unperturbed by the excluded-volume effect is Rg,0 ⫽ 0.91 ␮m from Eq. (10). But the effect of segment repulsion is substantial for the excludedvolume parameter is computed to be zs ⬇ 176, yielding a genomic radius of gyration of Rg ⬇ 3.00 ␮m via Eqs. (12) and (13). There is an important consistency check on the latter value. Vologodskii and Cozzarelli (1994) have published one computational set of data

63


of Rg at I ⫽ 0.2 M for a 5.2-kb plasmid. At ␴ ⫽ ⫺0.025, the radius Rg of the supercoil is about 85 nm. Tentatively extrapolating this up to a coil of genomic size by taking into account that Rg ⬃ N1/2 s from Eqs. (12) and (13), we obtain Rg ⬇ 2.56 ␮m which is close to our semianalytical estimate. We shall set Rg ⫽ 3.0 ␮m as our best estimate for the size of a branched E. coli genome without crosslinks but one must remain wary of the theoretical argumentation yielding this number. A further problem is the relation between the radius of gyration Rg and the volume Veq one would measure by microscopy. A branched supercoil without crosslinks and perturbed heavily by the excluded-volume effect has a size similar to that of a linear random coil (Rg ⬃ N1/2 s ; see Eq. (12)). It is then plausible to assume that the segment distributions are also similar. The segment distribution for a long linear random coil turns out to be close to Gaussian with variance equal to the root-mean-square radius of gyration (Yamakawa, 1971). Hence in our case we may express the normalized distribution of Kuhn segments as

P共s兲 ⫽

冉冊 3 2␲R2g

3/ 2

exp ⫺

3s2 . 2R2g

(14)

Here s is the vector distance from the center of mass of the spherically symmetrical distribution. We now also introduce a 50% threshold criterion to ensure consistency with the experimental image analysis. Using tables for the incomplete gamma function (Abramowitz and Stegun, 1970), we readily derive a radius 2.66 ␮m at 50% threshold for an uncrosslinked genomic supercoil with Rg ⫽ 3.00 ␮m. A theoretical estimate of its volume is thus 79 ␮m3. Free energy of compaction. In polymer physics, scaling theories are often powerful in elucidating the qualitative features of the physical behavior of polymer chains (de Gennes, 1979). Marko and Siggia (1995) have used a scaling approach for quenched branched polymers (Daoud and Joanny, 1981) to estimate the size of supercoiled DNA. This turns out to yield the right exponent for annealed chains (see Eq. (12)) fortuitously (for a treatment of annealed polymers, albeit at the mean-field level, see Gutin et al., 1993). For a very large number of segments Ns, we seek relations that are asymptotically valid so it is natural to suppose they are power laws (de Gennes, 1979). The free energy of compacting the uncrosslinked supercoil into a volume V may be written in a mean-field approximation as (Daoud and Joanny, 1981; Marko and Siggia, 1995)

冉冊冉冊

F com V ⬇ 3 kBT Rg,o

2/3

⫹

N ␤s 2V 2 s

共V ⱖ R3g,o兲.

(15)

Here the first term is the elastic free energy of dilation and the second signifies the excluded-volume effect of Ns(Ns ⫺ 1)/2 pairs of rodlike Kuhn segments interacting with each other in a volume V. If we minimize Eq. (15) with respect to V, we obtain an expression qualitatively similar to Eq. (12). In the regime V ⱕ R3g,o, we are actually compressing the ideal state (index 0) that we began with. We may now express the free energy of compression as two terms: F com ⫽

冉冊 R3g,o V

2/3

kBT ⫹

冉冊

N s2 ␤s kBT. 2V

(16)

The first term is the work expended in compressing the ideal branched coil within V (Lifshitz et al., 1978). Because N 2s ␤s overwhelms Rg,o3, only the excluded-volume term was used in a theory of the in vivo compaction of the nucleoid by the cytoplasmic proteins within E. coli (Odijk, 1998; Woldringh and Odijk, 1999). Relaxation times of the internal modes. Polymer chains in solution readily scatter light or other types of radiation (e.g., neutrons) without absorption. The incoming beam of laser light is coherent with a characteristic wavelength ␭. The intensity I(t) of the light scattered by the polymer at an angle ␪ at time t can be monitored by a photoelectric cell, and a correlator may be used to determine the intensity– intensity correlation functions under so-called homodyne conditions (Berne and Pecora, 1976): C共t兲 ⬅

具I共0兲I共t兲典 a ⬃ e ⫺共t/␶q兲 . 具I 2共0兲典

(17)

Often, this may be dominated by a stretched exponential form at short times (a ⬍ 1), at least if the radius of the chain is much greater than the inverse magnitude of the scattering vector k (k ⫽ 4␲no sin (␪/2)/␭ with no the index of refraction of the solvent). The relaxation time ␶k is then (Doi and Edwards, 1986)

␶k ⬇

␩0 , kBTk3

(18)

and a ⫽ 2/3 for large linear chains in a solvent of viscosity ␩0 at large scattering angles ␪. This k dependence is nondiffusional and may be understood in physical terms as follows. The wavevector k

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arises because there is interference between various segments of the chain. At a scattering angle ␪, one looks at a region of typical size k⫺1. For a linear polymer chain, the density of segments is sparse so the hydrodynamic interaction between the segments is exerted to the full. We are in the so-called nondraining limit (Yamakawa, 1971; Doi and Edwards, 1986). Thus, a polymeric blob of size k⫺1 acts as one hydrodynamic unit and all its segments move in concert. The diffusion coefficient of the blob is simply given by a Stokes–Einstein law: D(k) ⬇ kBT/ (6␲␩0k⫺1). The time it takes the blob to diffuse across its own length k⫺1 is then ␶k ⬇ k⫺2/D(k), which is Eq. (18). Our PEG-free nucleoids are 15 times denser than the supercoiled genome if it had no branches or crosslinks and no excluded-volume effect. An E. coli nucleoid is thus highly congested so one may argue that so-called hydrodynamic screening is also significant (Odijk, 2000b). In contrast to the case discussed above, the Kuhn segments then act as effectively independent, hydrodynamic units. At high k, one should then see pure diffusion. If the nucleoid is viewed as a collection of interacting Kuhn cylinders, the relaxation time is related to the coefficient of cooperative diffusion Dc (Odijk, 2000b)

␶ k ⬇ 1/共2k2 Dc兲, ␤s NskBT . Dc ⬇ hAs␩0 Veq

(19)

(20)

Note that Dc is proportional to the concentration of segments ⬇Ns/Veq. In Eq. (20) one recognizes the diffusion coefficient of a single Kuhn segment equal to kBT/(hAs␩0). The slowly varying coefficient h depends on the degree of hydrodynamic screening via the so-called hydrodynamic screening length (Odijk, 2000b). PEG–protein interaction within the nucleoid. We wish to compare the depletion interaction of the DNA helix within the nucleoid and the PEG coils with the additional depletion energy between the proteins that are bound and the same PEG coils. Let us focus on the first term of Eq. (2) which is here rewritten as a cross virial in terms of the exclusion radius E ⫽ 6.5 nm of a PEG chain: B c,DNA ⫽ ␲LE2 .

(21)

This may be viewed as the average volume excluded to one PEG coil by the DNA helix of contour length L. The average molar mass of the proteins binding to

the DNA of the E. coli nucleoid is about 15 kg mol⫺1 (Azam et al., 1999). The proteins are quite small (radius a ⫽ 1.7 nm) in comparison with the size of the PEG coils (radius of gyration Rg ⫽ 6.9 nm). The cross virial between such a protein, if it were freely translating, and a PEG chain may be written as B c,protein ⫽ 4␲aR2g .

(22)

This is the leading term of the computation of Jansons and Phillips (1990), which is known to yield an accurate estimate of the depletion energy between proteins and PEG (Wills et al., 1995). If the same protein were to adhere to the DNA helix, the covolume pertaining to the protein–PEG interaction would be less than that given by Eq. (22) because part of the interaction would be cut off by the DNA. Accordingly, if there are b proteins within the nucleoid (b ⬇ 250 000 for proteins of size 15 kg mol⫺1 (Woldringh and Odijk, 1999)), the relative error in our Eq. (2) is at most bBc,protein/Bc,DNA ⬇ 100%. However, only a fraction of the proteins adhere and the covolume is likely to be half that expressed by Eq. (22). At the current level of accuracy of experiment, we deem Eq. (2) to be a reasonable zero-order approximation for the depletion interaction between the nucleoid and the PEG chains. We thank N. Vischer for his help in estimating nucleoid volumes, E. Pas and P. Huls for technical assistance, R. Stuger, R. de Vries, and H. V. Westerhoff for discussions, J. L. Sikorav for correspondence, and B. Kuipers, G. Koenderink, H. Lekkerkerker, and A. Philipse for practical support and use of the DLS equipment. This work is supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM, physics) and the Stichting Aard- en Levenswetenschappen (ALW Project 47-032-P, biology), both subsidiaries of the Dutch organization for Scientific Research NWO. REFERENCES Abbott, N. R., Blankschtein, D., and Hatton, T. A. (1992a) Protein partitioning in two-phase aqueous polymer systems. On the free energy of mixing globular colloids and flexible polymers. Macromolecules 25, 3917–3931. Abbott, N. R., Blankschtein, D., and Hatton, T. A. (1992b) Protein partitioning in two-phase aqueous polymer systems: A neutron scattering investigation of the polymer solution structure and protein–polymer interactions. Macromolecules 25, 3932–3941. Abbott, N. R., Blankschtein, D., and Hatton, T. A. (1992c) Protein partitioning in two-phase aqueous polymer systems: Proteins in solutions of entangled polymers. Macromolecules 25, 5192– 5200. Abramowitz, M., and Stegun, I. A. (1970) Handbook of Mathematical Functions, Dover, New York. Azam, T. A., Iwata, A., Nishimura, A., Ueda, S., and Ishihama, A. (1999) Growth phase-dependent variation in protein composition of the Escherichia coli nucleoid. J. Bacteriol. 181, 6361– 6370.

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