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Polysaccharides Induced Crystallization of Tobacco Mosaic Virus Particles Masayuki Imai* Department of Physics, Faculty of Science, Ochanomizu University, Otsuka, Bunkyo, Tokyo 112-0012, Japan
Naohito Urakami Department of Physics, Biology and Informatics, Yamaguchi University, Yoshida, Yamaguchi 753-8512, Japan
Atsumi Nakamura, Ran Takada, and Reiko Oikawa Department of Physics, Faculty of Science, Ochanomizu University, Otsuka, Bunkyo, Tokyo 112-0012, Japan
Yoh Sano Faculty of Pharmaceutical Sciences, Setsunan University, Hirakata, Osaka 573-0101, Japan Received May 20, 2002. In Final Form: September 3, 2002 The effect of added polysaccharides on the phase behavior of (tobacco mosaic virus) TMV suspension has been investigated from experimental and computer simulation points of view. The addition of chondroitin sulfate (Chs) brings two interesting phenomena: (1) Chs induces aggregations of TMV particles at very dilute TMV concentration (enhancement of depletion interaction) and (2) the inter-rod distance in the TMV aggregates depends on the Chs concentration. The chain rigidity and electrostatic interaction between colloidal rod particles are responsible for the observed phenomena.
1. Introduction Soft matter shows various ordered meso structures as a result of competitions between attractive and repulsive interactions. To understand such structure formation, it is very useful to investigate the colloid particle systems, because the shape of colloid particles is well described by simple geometrical forms (sphere, rod, etc.) and interactions between the colloidal particles can be expressed by the well-characterized potentials (hard core, van der Waals, electrostatic, etc.). In this context, there are extensive studies on the structure formation of colloid particles, including colloidal particle mixtures. Recently, Adams et al.1 found the unique ordered meso structures such as columnar, lamellar, and filament structures in colloidal rod-sphere mixtures. These structures are stabilized by the entropic attractive force between colloidal particles. Similar structures are reported for rodpolymer mixtures, where a polymer chain is regarded as a spherical coil.2 The attractive force can be explained by simple entropic interaction, so-called depletion interaction.3,4 To maximize free volume of the spherical colloids, the excluded volume region (depletion zone) surrounding each rod particle should be overlapped despite lowering of the mixing entropy. This overlapping of the depletion zones brings effective attractive force between rod particles. * To whom correspondence may be addressed. E-mail: imai@ phys.ocha.ac.jp (1) Adams, M.; Dogic, Z.; Keller, S. L.; Fraden, S. Nature 1998, 393, 349-352. (2) Adams, M.; Fraden, S. Biophys. J. 1998, 74, 669. (3) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255. (4) Vrij, A. Pure Appl. Chem. 1976, 48, 471.
On the basis of the depletion interaction concept, Warren5 derived the semigrand potential of the rodlike colloidal particle-polymer mixtures system, and Matsuyama et al.6 improved the potential taking into account the rod end contribution. The phase diagram presented by Matsuyama et al. consists of the isotropic phase, the nematic phase, and the coexistence phase having a leaning-chimney shape. The coexistence curve is shifted to lower rod concentration with increasing the polymer concentration, which well described observed phase behavior of the rod tobacco mosaic virus (TMV)-polymer poly(ethylene oxide) (PEO) mixture system.2 On the other hand, Flory7 presented phase diagrams of rodlike particle and random coiled polymer chain mixtures using the lattice model. This theory shows that addition of flexible polymer chain induces phase separation of isotropic and nematic phases at fairly high rod particle concentration and agrees with the Matsuyama’s theory, qualitatively. The TMV has strong infection ability to tobacco leaves and is a rodlike particle having length of 300 nm and diameter of 18 nm. Recently, Sano8,9 found that chondroitin sulfate (Chs), one of the polysaccharides, shows a high inhibitory activity against TMV infection. Electron microscopy revealed that the TMV suspension show monodisperse distribution without Chs, whereas, in the presence of Chs, TMV particles form raftlike aggregates. The large aggregates prevent the penetration of TMV into the cell membrane, which causes the antivirus activity. In (5) Warren, P. B. J. Phys. I 1994, 4, 237. (6) Matsuyama, A.; Kato, T. Eur. Phys. J. E 2001, 6, 15. (7) Flory, P. J.; Macromolecules 1978, 11, 1138. (8) Sano, Y. Macromol. Symp. 1995, 99, 239. (9) Sano, Y. Carbohydr. Polym. 1997, 33, 125.
10.1021/la0204747 CCC: $22.00 © 2002 American Chemical Society Published on Web 11/12/2002
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Figure 1. Chemical structure of Chs.
the case of TMV-Chs mixtures, the aggregation of TMV particles occurs in the very dilute region at FTMV (TMV concentration) ) 0.08 mg/mL and FChs (Chs concentration) ) 1 mg/mL in 1 mM phosphate buffer solution. Thus, the TMV concentration where TMV starts to aggregate for Chs addition is about 1/1000 compared with that for PEO addition or the theoretical prediction presented by Matsuyama et al. Furthermore, this phenomenon closely related to the protein crystallization induced by the addition of nonadsorbing polymer.10-12 The purpose of this study is to elucidate the Chs induced aggregation of TMV particles phenomenon from experimental and computer simulation points of view. 2. Experiments TMV, Japanese common OM, was separated from systematically infected leaves of Nicotina tabacum L. var. Bright Yellow and purified by poly(ethylene glycol) precipitation and differential centrifugation. The TMV suspensions in 1 mM phosphate buffer solution were dialyzed about 1 week at 4 °C and then were stocked at 4 °C. Just after the dialysis, the TMV particles were almost monodisperse, judging from electronmicroscopy and ultracentrifugation data. TMV suspensions (80 µg/mL) were prepared by diluting a small volume of the stock suspension (64 mg/mL). The concentration of TMV suspension was determined by measuring UV adsorption. All samples were prepared in 1 mM phosphate buffer solution. It is reported that the polydispersity of the TMV particles increases with age, but the data reported here show good reproducibility during a series of experiments. In this experiment, Chs C-type having molecular weight of 68 000 was used as a polymer sample. The chemical structure of Chs C-type is shown in Figure 1. The Chs sample was purchased from Seikagaku Co. Ltd. (Tokyo, Japan) and further purified by Sephadex G-200 gel chromatography. The Chs sample showed a single sharp peak on the gel chromatography, indicating small polydispersity. A confocal scanning laser microscope (CSLM, Lasertec 1LM15) was used to observe the aggregations in TMV suspensions. The sample was contained in a hole slideglass and sealed with a coverglass. A water-immersed objective was used to reduce reflections from air and coverglass interface. The CSLM image was acquired on a computer through an image grabber board, and brightness and contrast were adjusted by image processing software (National Institutes of Health Image). The aggregation behavior is followed by light scattering and small angle neutron scattering measurements. The elastic light scattering measurements were performed with a light scattering spectrophotometer (Otsuka DLS7000). The incident light source is He-Ne laser (6328 Å, 10 mW) and the covering q range (q ) 4π sin θ/λ: magnitude of scattering vector) is 0.0025-0.0006 Å-1. For the elastic light scattering measurements, the scattering intensity was calibrated to toluene. During the measurements, the temperature was kept at 25.0 ( 0.05 °C. The small angle neutron scattering (SANS) measurements were performed using the SANS-U instrument of Institute of (10) George, A.; Wilson, W. Acta Crystallogr., Sect. D: Biol. Crystallogr. 1994, 50, 361. (11) Rosenbaum, D.; Zamora, P. C.; Zukoski, C. F. Phys. Rev. Lett. 1996, 76, 150. (12) Poon, W. C. K. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1997, 55, 3762.
Figure 2. Phase diagram for TMV + Chs mixture and TMV + PEO mixture.2 The buffer solution was phosphate buffer with an ionic strength of 1 mM for TMV + Chs system and sodium borate buffer with the ionic strength of 7 mM for TMV + PEO system. Inset photograph is raftlike aggregates observed in the TMV + Chs mixture. A bar indicates 0.3 µm. Solid State Physics, University of Tokyo, at JRR-3M reactor of Japan Atomic Energy Research Institute.13 In the SANS measurements, the covering q range was from 0.01 to 0.3 Å-1, and the obtained scattering profiles were normalized to a constant monitor count and then corrected for background scattering.
3. Results and Discussions Addition of polymer to the rodlike particle suspension brings aggregation of the rodlike particles. Figure 2 shows phase behavior of TMV-Chs mixtures obtained by CSLM observations and that of TMV-PEO (molecular weight ∼100 000 and radius of gyration ∼10 nm) mixtures obtained by Adams and Fraden.2 The pure TMV suspension shows the isotropic phase below 75 mg/mL, the isotropic-nematic coexistence phase between 75 and 200 mg/mL, and the nematic phase above 200 mg/mL,2,14 although these values depend on the ionic strength of the buffer solution. For both cases, the addition of polymer lowers the aggregation concentration of the TMV particles. In the case of PEO, the phase boundary curve agrees with Matsuyama’s prediction. On the other hand, it is obvious that the Chs promotes the aggregation of the TMV remarkably, and the phase boundary curve is almost independent of FTMV in the observed concentration range. We examined the reversibility of the isotropic-aggregation transition by addition of buffer solution. It is noteworthy that the same behavior was observed for other polysaccharides, such as hyaluronic acid and alginic acid.15 Thus, this effect is generally observed for the polysaccharides having polyelectrolyte nature. An inset photograph in Figure 2 shows the raftlike aggregates of TMV particles observed in the TMV-Chs mixture, where the particles are aligned in side-by-side manner. According to the microscope observations, morphologies of aggregates in the TMV + Chs system are very close to those of the TMV + PEO system, although we have not confirmed the side(13) Ito, Y.; Imai, M.; Takahashi, S. Physica B (Amsterdam) 1995, 213&214, 889. (14) Fraden, S.; Maret, G.; Caspar, D. L. D. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1993, 8, 2816. (15) Imai, M.; et al. In preparation.
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Figure 4. SANS profiles for the TMV and Chs mixture as a function of FChs. Figure 3. Elastic light scattering profiles for the TMV and Chs mixture as a function of FChs.
by-side aggregates in the TMV + PEO system. The raftlike aggregates transform to the fibril-like aggregates with increasing of Chs concentration. Thus, at some point, the aggregate stops the lateral growth and grows to the longitudinal direction.15 The similar transformation to the fibril-like structure is observed in the TMV + PEO system.2 We consider that the TMV + polymer systems show the general phase behavior, although the transition concentration is shifted owing to the nature of the polymer chain. Here, we followed the aggregation process induced by Chs, using the light scattering and SANS techniques. The elastic light scattering profiles for the TMV (FTMV ) 80 µg/mL) and Chs mixture as a function of Chs concentration FChs is shown in Figure 3. For the pure TMV sample and TMV + Chs (0.5 mg/mL) sample (region 1: FChs < 0.7 mg/mL), the scattering functions show monotonic profiles with low slope. Judging from the dimension of TMV particles (diameter of 18 nm and length of 300 nm), the observed q range (8 × 10-4 < q < 3 × 10-3 Å-1) of the light scattering measurements mainly corresponds to the so-called Guinier region, where we can estimate overall size of the isolated scattering particles. Actually, Guinier analysis of the observed profiles gives the radius of gyration (Rg) of about 900 Å, which agrees with the Rg of the TMV particle (Rg ) 870 Å). Thus, the observed scattering profiles well describe the dilute TMV suspensions. For the intermediate Chs concentration range (region 2, 0.7 < FChs e 1.0 mg/mL), the scattering profiles show a crossover from q-R (R ∼ 2 in high q range) to q-β (β ∼ 4 in low q range). According to the microscope observation, in the intermediate Chs concentration range, the TMV starts to aggregate at FChs ) 0.8 mg/mL and forms the raftlike aggregates at FChs ) 1.0 mg/mL. Thus, we consider that the q-R profile in high q range reflects aggregation manner of TMV particles inside of the aggregate, and the q-β profile in low q range reflects shape of the aggregate and interaggregates structure, which gives steep exponents larger than 3. Here we discuss the origin of the exponent R ∼ 2. For FChs ) 1.0 mg/mL sample, the two-dimensional raftlike aggregates are responsible for the exponent of 2. On the other hand, for FChs ) 0.8 mg/mL sample, the TMV particles do not form the definite raftlike aggregates. Then there are two candidates to
explain the exponent of R. One is that the aggregate is faint and isotropic. The electron density distribution of the faint aggregate has the Gaussian form, which gives a scattering profile with Orstein-Zernike form. The other is a precursor of the two-dimensional raftlike aggregates. At present, we cannot conclude which state is responsible for the observed scattering profile. For the highest Chs concentration samples (region 3: 1.0 mg/mL < FChs < 5.0 mg/mL), TMV particles form definite raftlike aggregates having about 0.5 × 0.5 µm2 dimension, although the size distribution is fairly broad. Thus, the observed light scattering profile gives information of aggregate shape and interaggregates structure. The scattering function of this sample shows q-3.5 law. We attributed the origin of the power law to the sharp boundary of the aggregates or the interaggregate structure, including the size polydispersity. The SANS profiles (FTMV ) 80 µg/mL) as a function of FChs is shown in Figure 4. In regions 1 and 2, the scattering profiles show monotonic curves, indicating that no regular ordered structure appears in the suspension. At FChs ) 1.0 mg/mL, a Bragg peak suddenly appears at q ) 0.014 Å-1, indicating formation of the raftlike aggregates. The repeat distance of TMV particles in the raftlike aggregates is 45 nm, which is about 2.5 times larger than the bare diameter of the TMV particle. The peak position is shifted to the higher q side as increasing FChs and obeys a power law of FChs1/4 as shown in Figure 5. Thus, the inter-rod distance is determined by the balance of the attractive and repulsive inter-rod interactions. From these experiments we obtained two important results: (1) Chs chains strongly enhanced the depletion interaction effect and (2) the inter-rod distance in the aggregates depends on the Chs concentration. In this study we focus our attention on these two issues. 3.1. Enhancement of Depletion Interaction Effect. The most significant difference between Chs and PEO is the chain conformation in the solution. In the Matsuyama model, polymer chains are treated as spherical coils with the radius of gyration R. This assumption is valid for PEO, but Chs chains have fairly large persistence length in the 1 mM phosphate buffer solution due to the rigid backbone structure and the polyelectrolyte nature. Actually, SANS profiles of the Chs phosphate buffer solution show broad but definite peaks at a certain scattering vector qm whose values depend on the Chs concentration. The peak
TMV Particle Crystallization
Figure 5. Relationship between Bragg peak position and FChs observed in TMV and Chs mixture. Solid line indicates theoretical prediction based on eqs 1-6.
positions qm can be scaled as FChs1/4. A similar behavior is generally observed for polysaccharides having a polyelectrolyte nature16 and interpreted as a hexagonal packing of locally stretched segments. Thus, the Chs molecules should be treated as semirigid chains having large persistence length.17 In this case, the depletion zone around the TMV particle is fairly large compared with that for flexible Gaussian chains. This may be responsible for the enhancement effect. To confirm this model, we performed Monte Carlo simulations for rod particle and semirigid chain mixture system.18 In this simulation, the TMV particle was regarded as a hard spherocylinder with the diameter of d and the length of 20d, and the Chs chain was modeled as a semirigid chain consisting of nine segments with the segment length of 6d, where d denotes the unit length in this simulation. The interactions between the rod particle and the chain were simple hard core potential. Using this simulation model, first we examined the isotropic-nematic transition of rod particles without polymer. Our simulation model gives the isotropicnematic transition at the volume fraction ΦIN rod ) 0.14, which agrees with the experimental results or theoretical predictions.14,19 Next, we introduced a hard sphere with radius of R ) 0.8d into the rod system and confirmed that the hard spheres induce the isotropic-nematic transition. However, this depletion interaction effect is fairly limited because the radius of the spherical colloid particle is close to the radius of the rod particle. The addition of hard spheres shifts the isotropic-nematic transition volume IN fraction from ΦIN rod ) 0.14 (without sphere) to Φrod ) 0.135 at ΦSphere ∼ 0.006, or corresponding to FPEO ∼ 0.24 mg/mL in Figure 1, which agrees with Matsuyama’s prediction. Then, we investigated the rod and semirigid chain mixture system using the above model. (16) Borsali, R.; Rinaudo, M.; Noirez, L. Macromolecules 1995, 28, 1085. (17) Berth, G.; Dautzenberg, H.; Christensen, B. E.; Harding, S. E.; G. Rother Smidsrød, O. Macromolecules 1996, 29, 3491. (18) Urakami, N.; Imai, M.; Sano, Y.; Takasu, M. J. Chem. Phys. 1999, 111, 2322. (19) Graf, H.; Lo¨wen, H. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1999, 59, 1932.
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The chain concentration dependence of the rod particle configuration was simulated by decreasing the number of chains at a fixed box size 80d and fixed the number of rod particles Nrod ) 45, corresponding to the TMV concentration of about 1.0 mg/mL or TMV volume fraction of Φrod ) 0.0014. We show the snapshots of the simulation for (Nchain, Nrod) ) (12, 45), (60, 45), and (120, 45) in Figure 6. In the low chain concentration region, the rod particles are in the isotropic state and are well mixed with chains (Figure 6a). With an increase in the chain concentration, the rod particles start to separate from the solution, but the rod particles in the aggregate are still in the isotropic state as shown in Figure 6b. The chain concentration / cChain where chains start to separate is around 60 chains per box, which corresponds to FChs ∼ 2.4 µg/mL. Further increase of the chain concentration brings the isotropicnematic transition of rod particles in the aggregation as IN where shown in Figure 6c. The chain concentration cChain chains start to align is around 100 chains per box, which corresponds to FChs ∼ 3.9 µg/mL. Thus, the semirigid chains remarkably enhance the depletion interaction effect and induce the nematic transition of rod particles through the isotropic-isotropic phase separation state. The large difference of the chain concentration between simulations and experiments is probably due to the model of Chs chain. In this simulation, a Chs chain is composed of 9 segments having length of 6d, corresponding to the persistence length of about 100 nm, which is nearly equal to the contour length of the Chs chain used in this study. Thus, the chain model used in this simulation does not correspond to the real Chs chain, but due to the simulation hardware limitation, we adopted it for this model. According to our experimental results, in region 2, the light scattering profile indicates the formation of TMV aggregates, but the SANS profile shows no short range order of TMV particles. This may correspond to the isotropic-isotropic phase separation state. However, we need more detailed examinations to confirm the isotropicisotropic transition state. 3.2. Chs Concentration Dependence of Inter-TMV Distance in Raftlike Aggregate. As shown in Figures 4 and 5, the inter-TMV distance in the raftlike aggregate depends on FChs, and the peak position qm obeys the power law of qm ∼ FChs1/4. To explain this behavior, we assume that the total inter-TMV interaction is expressed by two contributions
utot ) uele + udep
(1)
The electrostatic interaction uele between TMV particles originates from their polyelectrolyte nature. It is well known that in aqueous solution, protons dissociate from the surface proteins leading to a linear charge density of 10-20 e/nm.14,19 Here, we adopt Derjaguin-LandauVerway-Overbeek (DLVO) theory20 to describe rodlike macro-ions where the charge is distributed on segmentlike beads along the rods and the charged rods are assumed to be aligned in a raftlike manner, without shift as shown in Figure 7. In this case, the pair electrostatic interaction between rods is expressed by where r is a center-of-mass
{
∞ 2 uele(r) ) (Ze/Ns)
rd
(2)
r
(20) Derjaguin, B. V.; Landau, L. Acta Physicochim. URSS 1941, 14, 633.
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Figure 7. Schematic representation of structure model for the raftlike TMV aggregate with depletion zone and colloidal polymer. TMV particle (length lTMV and diameter d) is surrounded by excluded volume shell of thickness R corresponding to the radius of gyration of colloidal polymer. The overlap volume is indicated by the dark region.
difference vector between rods, Z is a total charge number, is the dielectric constant of the solvent, the rod is consisting of Ns beads, and the summation is taken over Ns beads labeled R ) β, for simplicity. The bead segments are arranged in such a spacing (dseg) that the quadrupole moments of the uniformly charged rod and the segment rod are identical, which reads to
dseg )
lTMV ((Ns - 1)(Ns + 1))1/2
(3)
where lTMV is the rod length. The inverse Debye screening length
κ)
(
)
4πe2 (ZF + 2ns) kBT
1/2
(4)
is composed of two contributions, the density of the counterions stemming from the rodlike macroions (ZF) and the salt concentration (ns). In eq 4, kBT is the thermal energy, and F ) N/V is the rod density. This electrostatic interaction is repulsive and dominates for a short distance. A simple expression for the depletion interaction udep between two rods induced by hard-core sphere of radius R is expressed by3,21-23
udep ) -kBTnpolyVov
Figure 6. The typical configurations in Monte Carlo simulation at (Nchain, Nrod) ) (12, 45) (a), (60, 45) (b), and (120, 45) (c).
(5)
(21) Bolhuis, P. G.; Stroobants, A.; Frenkel, D.; Lekkerkerker, H. N. W. J. Chem. Phys. 1997, 107, 1551. (22) Yaman, K.; Jeppensen, C.; Marques, C. M. Europhys. Lett. 1998, 42, 221. (23) Vliegenthart, G. A.; Lekkerkerker, H. N. W. J. Chem. Phys. 1999, 111, 4153.
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where npoly is the number density of polymer and Vov is the overlap volume of the excluded shells of a pair of rods as shown in Figure 7. In the case of parallel alignment of the cylindrical rods with radius of d/2, Vov is given by
Vov(r) ) lTMV(π((d/2) + R)2 r(((d/2) + R)2 - (r/2)2)1/2 -2((d/2) + R) × 1/2 r/2 (6) tan-1 (((d/2) + R)2 - (r/2)2)
(
)
Thus, the repulsive electrostatic interaction and the attractive depletion interaction give a potential minimum, which determines the inter-rod distance. In the numerical calculation based on the above model, determination of the line charge density of TMV is a fairly difficult problem. Titration experiments of the TMV particle at pH of 7 gives a line charge density of 10-20 e/nm.19 However, some of these charges recondense at the TMV surface. Graf and Lo¨wen proposed that the line charge density of 1-2 e/nm is adequate to reproduce the structure factor of light scattering experiments,19 and Fraden et al. used 4-20 e/nm for TMV buffer suspensions having pH of 7.2-8.0.14 When we calculated the total potential for a pair of parallel aligned rods corresponding to the TMV + Chs system with phosphate buffer solution (pH ∼ 7.2), we adopted the following numerical values: lTMV ) 300 nm, radius of TMV particle dTMV ) 18 nm, Ns ) 17 spheres, Z ) 3000 corresponding to a linear charge density of 10 e/nm, and ) 81. For the rod density, we assumed the hexagonally packed cylinder structure with nearest neighbor distance r, and the salt concentration was evaluated from the elecroconductivity measurements for the Chs phosphate buffer solutions. The effective radius of the Chs chain RChs was assumed to be 30 nm, which is a trial value, because it is hard to estimate the effective radius of the Chs chain. The obtained total potential as a function of inter-rod distance r for FChs ) 1 mg/mL is plotted with the electrostatic and depletion interaction contributions in Figure 8. The total potential curve has a minimum at rm ) 35 nm. From the minimum position, we can calculate corresponding peak position using the simple relation qm ) 2π/rm. The obtained peak position as a function of FChs is shown in Figure 5 with experimental results. The theoretical calculation well describes the FChs dependence of the peak position, although the theoretical values are somewhat shifted to the higher q side. Thus, the observed experimental results are explained by the depletion and electrostatic interactions.
Figure 8. The total inter-rod potential as a function of interrod distance r for FChs ) 1 mg/mL with the electrostatic and depletion interaction contributions.
In this study, we have investigated the effect of added polysaccharides on the phase behavior of TMV suspension from experimental and computer simulation points of view. The addition of Chs brings two interesting phenomena: (1) Chs induces aggregations of TMV particles at very dilute TMV concentration, and (2) the inter-rod distance in the TMV aggregates depends on the Chs concentration. The chain rigidity of the Chs having a polyelectrolyte nature may be responsible for the enhancement of the depletion interaction. Semirigid chains having large persistence lengths cause the large depletion zone around the TMV particle. This model is confirmed by the Monte Carlo simulations for a rod particle and semirigid chain mixture system. The simulation shows that the semirigid chains remarkably enhance the depletion interaction effect and induce the nematic transition of rod particles. The dependence of inter-TMV distance in the raftlike aggregate on FChs can be scaled as qm ∼ FChs1/4. This dependence is explained by the electrostatic interaction between rod particles based on the DLVO model and the depletion interaction. Thus, depletion interaction and electrostatic interaction play an important role in understanding the phase behavior of polysaccharide and TMV mixtures. LA0204747
