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Free Diffusivity of Icosahedral and Tailed Bacteriophages: Experiments, Modeling, and Implications for Virus Behavior in Media Filtration and Flocculation Ruth E. Baltus,† Appala Raju Badireddy,‡ Armin Delavari,† and Shankararaman Chellam*,§,∥ †

Department Department § Department ∥ Department ‡

of of of of

Chemical and Biomolecular Engineering, Clarkson University, Potsdam, New York 13699-5705, United States Civil and Environmental Engineering, University of Vermont, Burlington, Vermont 05405, United States Civil Engineering, Texas A&M University, College Station, Texas 77843-3136, United States, Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122, United States

S Supporting Information *

ABSTRACT: The aqueous bulk diffusivities of several near-spherical (icosahedral) and nonspherical (tailed) bacterial viruses were experimentally determined by measuring their flux across large pore membranes and using dynamic light scattering, with excellent agreement between values measured using the two techniques. For the icosahedral viruses, good agreement was also found between measured diffusivity values and values predicted with the Stokes−Einstein equation. However, when the tailed viruses were approximated as spheres, poor agreement was found between measured values and Stokes−Einstein predictions. The shape of the tailed organisms was incorporated into two modeling approaches used to predict diffusivity. Model predictions were found to be in good agreement with measured values, demonstrating the importance of the tail in the diffusive transport of these viruses. Our calculations also show that inaccurate estimates of virus diffusion can lead to significant errors when predicting diffusive contributions to flocculation and to single collector efficiency in media filtration.

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of this approach.21,32 Importantly, to our knowledge, very few diffusivity measurements of nonspherical viruses have been undertaken. Exceptions include the tailed phage T4, the rodlike tobacco mosaic virus, and the filamentous virus fNEL whose diffusivities have been reported.24,29,33−35 In this paper, we report the free aqueous diffusion of several icosahedral (near spherical) and tailed viruses measured with two independent techniques: dynamic light scattering and using a diffusion cell with large pore membranes. We use bacteriophages since they are commonly employed as environmental tracers36 and (imperfect) surrogates for pathogenic enteric viruses.37,38 Phages are also important in their own right since they are ubiquitous in the environment with an estimated population of 1030-1032 on earth.39 Additionally, they are thought to play an important role in the ecology of bacterial populations and are vectors for the transport of genetic information, potentially playing a role in horizontal evolution.

INTRODUCTION The transport and fate of pathogenic viruses in the environment1−3 and their removal from drinking water supplies4 and wastewater5 are important aspects of public health protection. Viruses are known to persist in the subsurface for long durations and can be transmitted over long distances in groundwater.2,3,6 They can be physically removed from contaminated water supplies by coagulation,7−11 or membrane filtration.12−16 The diffusion coefficient is a necessary parameter to quantitatively analyze virus behavior in these processes and to better understand their aggregation and interactions with surfaces.17−20 Virus diffusion also plays a role in host−virus interactions and virus inactivation,21 in perfusion chromatography22 and is important in developing novel strategies to control biofilm formation and growth.23−26 We have recently reported hindered convection of tailed and spherical viruses through membranes with pores of size comparable to their characteristic dimensions.15,17,27 However, paradoxically, the number of direct measurements of the free diffusivity of viruses have been limited.23,28−31 Instead, their diffusive transport has sometimes been simplistically estimated by assuming a spherical shape and the Stokes−Einstein model without sufficient experimental evidence supporting the validity © 2016 American Chemical Society

Received: Revised: Accepted: Published: 1433

October 21, 2016 December 29, 2016 December 30, 2016 December 30, 2016 DOI: 10.1021/acs.est.6b05323 Environ. Sci. Technol. 2017, 51, 1433−1440

Article

Environmental Science & Technology Two modeling approaches were used to quantitatively examine the influence of virus tails on their diffusive transport. A public domain program, HYDRO++, was used to predict diffusivity for multibead structures assembled to model each tailed virus. COMSOL Multiphysics was also used to calculate the hydrodynamic drag on rigid tailed model structures, from which diffusivity was calculated. In this paper, we compare measured D∞ values to Stokes−Einstein predictions for the icosahedral and tailed viruses and to HYDRO++ and COMSOL model predictions for the tailed viruses. The principal objectives of this research were to (i) rigorously model free diffusion coefficients of tailed and icosahedral viruses and compare them with laboratory measurements and (ii) examine potential errors introduced when predicting virus transport during granular media filtration and flocculation using spherical models to describe the structure of the tailed virus.

Table 1. Images, Dimensions, and Measured Diffusivity Values for the Near Spherical (Icosahedral) Viruses

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EXPERIMENTAL DETAILS Bacteriophage Stock and Enumeration. Near-spherical phages MS2, Qβ, and ϕX174 as well as tailed phages T4, P1, Mu1, and P22 were purchased from American Type Culture Collection (ATCC, 15597-B1, 23631-B1, 13706-B1, 11303-B4, 25404-B1, 23724-B9, and 19585-B1 respectively). They were grown on lawns of their respective bacterial hosts Escherichia coli (ATCC 15597, 23631, 13706, 11303, 25404, 23724), and Salmonella typhimurium LT2 (19585) by the agar-overlay method. See Supporting Information for more details. PRD1 was obtained from David Metge, (United States Geological Survey, Boulder, CO) and cultured using Salmonella typhimurium LT2 (ATCC 19585). Electron microscopy images of MS2, Qβ, ϕX174 and PRD1 are shown in Table 1 and of T4, P1, Mu1, and P22 are shown in Table 2. Diffusion Cell Measurements. A plexiglass cell having two rectangular chambers separated by an 8 μm track-etched polycarbonate membrane (Osmonics, Minnetonka, MN) was specially constructed for this work. The membrane pore size is approximately 1−2 orders of magnitude larger than the size of the viruses employed, thereby minimizing steric hindrances to their diffusion within its pores. Agitators, driven by high torque motors, vigorously stirred the solutions in both chambers in order to reduce boundary layer resistances and to negate any spatial gradients within each cell. After calibrating the cell with KCl and NaCl, diffusivities of CaCl2 and sucrose were measured to be within 6% of literature values40,41 and that of nearly monodispersed 45 nm spherical colloidal silica (Snowtex OL, Nissan Chemicals) was within 4% of the Stokes−Einstein prediction. This demonstrates the capability of the cell to accurately measure diffusivities of dissolved substances and particles. Before performing experiments with bacteriophages, the entire apparatus was washed with bleach, rinsed five times with nanopure water, and then thoroughly air-dried. Then, both cells were filled with PBS and the solutions were stirred overnight. The top cell was then spiked to an initial concentration of ∼108 PFU/mL and both the upper and bottom cells were sampled over a ∼2-day period. Viruses were enumerated in triplicate using the plaque assay technique. Importantly, the concentrations employed were sufficiently dilute to avoid virus−virus interactions allowing the calculation of the bulk diffusion coefficients of individual bacteriophages. Separate experiments demonstrated that virus viability was maintained over the time frame of our measurements.

a

Images for MS2 and PRD1 are from our laboratories. The Qβ image is used by permission from Martin Bachmann. The image of ϕX174 is from https://en.wikipedia.org/wiki/Phi_X_174. bMembrane flux measurements represent the average and standard deviation of 3 to 5 separate experiments performed using different phage stocks on different dates. cDynamic light scattering measurements represent the average and standard deviation of 2 separate experiments performed using different phage stocks on different dates. Note that the instrument software reports the average of three measurements each time.

Diffusion Coefficient Calculations. Consider purely Fickian diffusion across two cells of unequal volumes V1 and V2 separated by a membrane of effective surface area A. Combining solute balances written on each cell yields a relationship between concentrations in the upper (cu) and bottom (cb) cells, time (t) and the solute diffusion coefficient D∞: −1⎤ ⎡ (c u − cb)t = 0 ⎤ ⎡⎛ 1 k2 ⎞ ⎥ 1 ⎞⎛ k 1 ⎢ ⎜ ⎟ + ln⎢ ⎥ = ⎜ + ⎟⎜ ⎟ At V2 ⎠⎝ D∞ ⎣ (c u − cb)t ⎦ ⎢⎣⎝ V1 D∞2/3 ⎠ ⎥⎦

(1)

where k1 and k2 are empirical cell constants. The cell constants capture the total mass transfer resistance and depend on the membrane properties (e.g., porosity, pore size, and distribution), the cell geometry, and stirring speed, but not on the nature of the solutes.42 To determine D∞, experimental data were analyzed by first transforming the measured virus concentrations as a function of time by plotting ln [(cu − cb)t=0/(cu − cb)t] versus t. Dynamic Light Scattering (DLS). Viruses (∼108 PFU/ mL) were dispersed in PBS at room temperature and the decay of the scattered light intensity arising from Brownian motion was monitored at a fixed 173° angle (Zetasizer Nano ZS, Malvern) using a He−Ne laser (633 nm). The refractive index was set as 1.332, the instrument was equilibrated for 60s, the measurement duration was set in automatic mode and three measurements made on each sample. Reported values are the average of two different measurements made on separate days. 1434

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Table 2. Images, Dimensions, Model Structures, and Measured Diffusivity Values for the Tailed (Non-Spherical) Viruses

a

T4 image is from our lab. P1 image is reprinted from Virology, vol 417, Liu, J., Chen, C.-Y., Shiomi, D., Niki, H., Margolin, W., Visualization of bacteriophage P1 infection by cryo-electron tomography of tiny Escherichia coli, pp. 304−311 Copyright (2011), with permission from Elsevier.60 P22 image is reprinted from Structure, 14, Chang, J., Weigele, P., King, J., Chiu, W. and Jiang, W., Cryo-EM Asymmetric Reconstruction of Bacteriophage P22 Reveals Organization of its DNA Packaging and Infecting Machinery, 1073−1082, Copyright (2006), with permission from Elsevier.56 Mu1 image is reprinted from Virology, 134, Grundy, F.J., Howe, M.M., Involvement of the invertible G segment in bacteriophage Mu tail fiber biosynthesis, 296-317, Copyright (1984) with permission from Elsevier.57 bDetermination of the average and standard deviation of free diffusion coefficients are described in Table 1.

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tail fibers on T4, P1, and Mu1 were not included in our model structures. Bead Models and the HYDRO++ Program. The hydrodynamic properties of nonspherical macromolecules or particles was initially examined by Kirkwood and Riseman who developed procedures for calculating the properties of models comprised of collections of spherical beads.61,62 Since this early work, there have been a number of reports from efforts to revise the Kirkwood-Riseman approach, both to improve its accuracy as well as its computational efficiency.63−66 Calculations involve inversion of a 3N × 3N matrix, where N is the number of beads in the model. This matrix accounts for the hydrodynamic interactions between the beads in the structure. Conveniently, Garcia de la Torre and co-workers have provided a public domain program HYDRO++ in which the inversion is implemented.65,66 One provides the position and size of each bead in the model as well as temperature and solvent viscosity and the program calculates the translational diffusion coefficient. (Note that the program can also calculate other properties; some of these require the molecular weight and the buoyancy factor of the structure. These properties were not of interest to us in this study.) To use HYDRO++ to calculate the bulk phase diffusivity of the nonspherical viruses, a bead model was constructed to represent the structure, consistent with the dimensions listed in Table 2. The HYDRO++ bead models are shown in Table 2. With the size and relative position of each bead specified, HYDRO++ was used to calculate the diffusivity.

DIFFUSION MODELS

The hydrodynamic properties of biological macromolecules, viruses and other biological particles depend upon the particle’s size and shape. Analytical expressions for the Stokes mobilities (from which bulk diffusivity can be calculated) of simple model structures (i.e., spheres, ellipsoids, rods, or disks) to particle dimensions are available;43 however, no such expressions are available for tailed viruses. While spherical models (e.g., Stokes−Einstein) can be expected to reasonably represent the structure of MS2, Qβ, ϕX174 and PRD1 (see Table 1, which also lists their diameter obtained from image analysis and from the literature15,44−49), more complex model structures are needed to describe the properties of the tailed viruses T4, P1, P22, and Mu1, whose dimensions are listed in Table 2. The T4 capsid has been reported to be an elongated icosahedron 120 nm in length and 86 nm in width, which is attached to a tail tube with a contractile sheath and six long tail fibers.50,51 P1 is similar to T4 in that it also has an icosahedral head that is attached to a contractile tail with six kinked tail fibers.52,53 The genome of these viruses consists of double stranded DNA. The P22 virus has a complex multitail structure, as detailed in several literature reports from crystallography and electron cryo-microscopy.54−56 The dimensions listed for P22 in Table 2 were compiled from these reports. The phage Mu1 has an icosahedral head, a contractile tail with six tail fibers and has been extensively studied by Howe and co-workers57,58 from whom the dimensions and image in Table 2 were obtained. The 1435

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Environmental Science & Technology To confirm the validity of the bead models, an ellipsoid shaped structure comprised of 44 beads (similar to the head of the T4 model) was assembled. The diffusivity predicted for this model structure using HYDRO++ was in excellent agreement with the value predicted from an analytical equation for the diffusivity of a prolate ellipsoid with the same dimensions.67 COMSOL Modeling. Using Einstein’s original thermodynamic argument, the translational diffusion coefficient of a particle is defined by68 D∞ =

kBT f̅

(2)

where kB is Boltzmann’s constant, T is the absolute temperature and f ̅ is the mean friction coefficient. To calculate f ̅, a particle is placed at a fixed position and orientation in an unbounded flowing fluid and the drag force on the particle is calculated when flow is in the x, y and z directions. The particle orientation is selected so that the necessary symmetry is preserved. Details are presented in the Supporting Information.

Figure 2. Comparison of the Stokes−Einstein diffusivity with measured values for near spherical viruses. Stokes−Einstein diffusivities were determined from eq 3 using virion particle diameters listed in Table 1. Closed symbols are from this work; open symbols are values reported in the literature.28,69−73 Measured values are the average of D∞ from light scattering and membrane flux measurements.

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RESULTS AND DISCUSSION Experimental Results. The measured diffusivity values are listed in Tables 1 and 2. A comparison of diffusivity values determined from light scattering measurements to values determined from membrane flux measurements is shown in Figure 1. For all viruses, excellent agreement between results

determined from imaging is representative of the characteristics of the viruses in solution and that their icosahedral capsids can be modeled as spheres. The Stokes−Einstein diffusivities for Qβ, ϕX174, and MS2 are also compared in Figure 2 to values reported in the literature.28,69−73 This comparison shows reasonable agreement between values determined in this study and values determined using other techniques. It is important to note that values for MS2 cover a relatively broad range (13.1 × 10−12 to 19.9 × 10−12 m2/s), demonstrating the inherent variability associated with diffusivity measurements by various research groups using different techniques. To examine the impact of the complex morphology on the diffusive transport of the tailed viruses, the measured diffusivity of each phage was compared to Stokes−Einstein diffusivity predictions with the capsid diameter and with the total virus length as the important particle length scales. Results are shown in Figure 3 and in Table S1 in the Supporting Information. Because T4 has a nonspherical capsid, the diffusivity of a 120

Figure 1. Comparison of virus diffusion coefficients measured from light scattering and membrane flux.

from the two measurements was observed, providing validity to both approaches and to the reproducibility of our laboratory protocols. An average of the diffusivities measured using the two techniques is used in further analyses in this paper when comparing experimental observations to model predictions. The Stokes−Einstein equation relates the translational diffusivity to the hydrodynamic diameter of a diffusing sphere:

D∞ =

kBT 3πμd p

(3)

where μ is the solvent viscosity, and dp is the sphere diameter. A comparison of the measured diffusivities to the Stokes−Einstein diffusivity for the near spherical viruses is shown in Figure 2. The Stokes−Einstein diffusivity was determined using the virion diameter obtained from imaging (listed in Table 1). The good agreement between the two values indicates that the size

Figure 3. Predicted versus measured diffusivity for nonspherical viruses. For T4, the Stokes−Einstein diffusivity based on the capsid diameter is the prediction from an analytical equation relating D∞ to axes lengths for a prolate ellipsoid.67 Measured values are the average of values measured with light scattering and membrane flux. 1436

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Environmental Science & Technology nm × 86 nm prolate ellipsoid was calculated from an analytical expression available in the literature for this shape67 rather than using the capsid diameter (see equation S1 in the Supporting Information). Comparing the measured diffusivity to Stokes−Einstein predictions with the capsid diameter shows that the measured diffusivities for P1, P22, and Mu1 are significantly smaller than predictions, indicating that the effective hydrodynamic size of these viruses is significantly larger than the capsid diameter. This is not unexpected for P1 and Mu1 because these viruses have relatively long, thin tails which increase their hydrodynamic size, thereby decreasing their diffusion rate compared to values expected for only the capsid. The tail structure on P22 is relatively small yet complex54−56 (six protein tail spikes and a center tail hub), which apparently also increases the frictional resistance experienced by this phage as it moves in solution. The agreement between measured D∞ and the value predicted for the ellipsoidal capsid for T4 was unexpected. Its tail is shorter and larger in diameter when compared to the tails of P1 and Mu1 and is expected to add to the hydrodynamic resistance experienced by this virus. For T4, Mu1, and P1, the measured diffusivity values are significantly larger than the Stokes−Einstein prediction based on the total virus length, as expected. This comparison for P22 shows reasonable agreement between these values, indicating that the small tail structure apparently contributes a relatively large hydrodynamic resistance for this virus, consistent with our comparison of measured D∞ to the Stokes−Einstein prediction using the capsid size only. The comparisons presented in Figure 3 show that, in general, diffusivities are overestimated when one considers the capsid size as the governing particle length scale and are underestimated when one assumes that the important length scale is the virus length. These results demonstrate the importance of considering the complex morphology when predicting the transport properties of these tailed viruses. Modeling Results. The tails on T4, P1, P22, and Mu1 were included in the model structures examined using HYDRO++ and COMSOL, shown in Table 2. A comparison of model predictions from HYDRO++ and COMSOL to the measured diffusivity values is shown in Figure 4; values are listed in Table

S1 in the Supporting Information. These results show generally good agreement between diffusivity values predicted using the bead models with HYDRO++ and COMSOL, supporting the validity of the model structures and the computational techniques used to predict D∞. The diffusivity predicted for only the capsid of P1 is ∼50% larger than the measured value (Figure 3). When the tail on this virus is included, the predicted diffusion coefficients agree within 6% of the measured value. Similarly, the diffusivity predicted for only the capsid of Mu1 is also ∼50% larger than the measured value. When the tail is “added” to the model, the predicted diffusivity decreases and agrees within 16% (HYDRO ++) and 30% (COMSOL) with the measured value. These comparisons demonstrate the importance of the entire virion morphology on diffusive transport. The diffusivity predicted for a 120 nm × 86 nm prolate ellipsoid (with no tail) agrees well with the value measured for T4 (Figure 3). Addition of the tail to the model structure decreases the predicted diffusivity, yielding predictions that are ∼30% smaller than the measured value. Included in Figure 4 is D∞ for T4, reported by Barr et al.29 who measured the diffusivity of fluorescently labeled phages by tracking particle motion in a microfluidic device, yielding a value that is in reasonable agreement with our measurements and in very good agreement with our model predictions. Hu et al.23 examined the diffusion of T4 using a two chamber diffusion device, conceptually similar to the membrane system used in this study. The diffusivity through filter paper was measured to be 28 × 10−12 m2/s, over six times larger than the values measured in this study. We have opted to not include this value in Figure 4 because it would require expanding the axes considerably, making it difficult to make other comparisons. The difference between the value reported by Hu and the values measured in this study and reported by Barr is much larger than differences found between various reports for MS2 (Figure 2). It appears unlikely that these differences can be attributed to variations between experimental techniques or between virus samples but nevertheless demonstrate inherent difficulties in making such measurements. The model structures developed in this study for P1, Mu1, and T4 include the tails that are major characteristics of these viruses. However, these structures are simplified representations which neglect a number of other characteristics. Our modeling approaches assume particle surfaces are smooth and experience no slip. The validity of the no slip boundary condition has been shown to depend upon surface roughness, topology, and wettability74 (and the references cited therein). P1, Mu1, and T4 have six fine tail fibers that were not included in our model structures. These fibers may add to the frictional resistance experienced by the viruses, decreasing the diffusivity values. Additionally, the tails on our models are assumed to be rigid whereas they may have some flexibility, leading to an increase in diffusivity. The generally good agreement between our measured and predicted diffusivity values shown in Figure 4 indicates that it is important to consider the overall tailed morphology when estimating the diffusivity of viruses such as T4, P1, and Mu1. While other virus features (i.e., nonsmooth surface, slender tail fibers and tail flexibility) may play a role in transport, our results point to that role being a minor one. An interesting feature in Figure 3 and Figure 4 is that measured diffusivities for the different viruses varied less than the theoretical values. Although we do not have a clear explanation for this behavior,

Figure 4. Comparison of predicted to measured virus diffusivity values for the nonspherical viruses. Measured values are the average of values measured with light scattering and membrane flux. The measured value reported by Barr et al.29 is compared to the average of D∞ predicted using HYDRO++ and COMSOL. 1437

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leading to a 110% overestimation of flocculation rates whereas the diffusivity predicted using the Stokes−Einstein equation with the virus length is underestimated by 65%, leading to an underestimation of flocculation rates by 130%. In addition, the collision frequency function depends on both the relative diffusivities as well as the particle surface areas. Particle surface area will be underestimated if a spherical model is used to describe the tailed viruses; the impact of this underestimation will depend upon the relative size of capsid to tail. A quantitative comparison of the impact of using the various models to estimate virus diffusion coefficients in collector efficiency and flocculation rate predictions for each tailed virus is summarized in the Supporting Information. In this manuscript, we report that measured bulk phase diffusion coefficients of the icosahedral viruses are in good agreement with Stokes−Einstein predictions when virus diameters from microscopy imaging are used. Since both DLS and membrane flux experiments agree closely, we recommend the more facile light scattering technique to measure bulk Brownian diffusivities of viruses. Model predictions for the tailed viruses illustrate the importance of including the tail morphology when estimating virus diffusion. Our results also demonstrate that significant errors can arise if the tail structure is neglected when estimating diffusion contributions to flocculation and media filtration. We are currently extending this work to quantitatively investigate more complex morphologies including filamentous viruses.

this is manifested as their appearance in a fairly narrow vertical “band” in comparison to the predicted diffusivities. The Stokes−Einstein diffusivity predicted for the P22 capsid only (65 nm diameter sphere) is 6.65 × 10−12 m2/s. The diffusivity predicted for the multibead model (HYDRO++ calculations) of this virus is 6.24 × 10−12 m2/s and the diffusivity predicted from the COMSOL calculations is 6.58 × 10−12 m2/s. The fact that these predictions yield diffusivity values quite close to D∞ for only the capsid is not surprising because the tail structures on this virus are relatively small. However, the measured diffusivity values for this virus are ∼60% of the predicted values. Of the four tailed viruses examined in this study, P22 has the smallest overall dimensions yet exhibited the smallest free diffusivity. It has a complex tail structure that was incorporated in both model structures (see Table 2). The relatively small diffusivity for this virus and the reasonably close agreement between the measured D∞ values and the Stokes−Einstein prediction for a sphere with diameter equal to the virus length (Figure 3) suggests that the multitail structure provides more hydrodynamic resistance than predicted from the tail dimensions alone. More work is necessary to understand these observations. Implications for Virus Filtration by Granular Media and Flocculation. Diffusivity values for P1, P22, and Mu1 predicted using the Stokes−Einstein equation with only the capsid size are 50−75% larger than our measured values; diffusivity values predicted with the Stokes−Einstein equation with the virus length are 30−60% smaller than measured values for T4, P1, and Mu1 (Figure 3). The effect of using only one particle length scale to estimate particle diffusion when designing porous media filtration processes was examined using the correlation equation for single collector efficiency.75 In this equation, diffusive contributions to efficiency scale as Pe−0.715. This contribution to the collector efficiency can therefore be overestimated by 33%−50% if just the capsid size is used to estimate diffusivity, rather than the actual value. Gravitational contributions to collector efficiency scale strongly as Pe−1.11. Hence, its contribution can be overestimated by 60− 85% when diffusivity is overestimated by 50−75%. Using only the capsid size to estimate diffusivity can therefore result in overestimating overall collector efficiency by ∼100%−140%. A similar analysis shows that the contributions of diffusion and gravitation to collector efficiency can be underestimated by 25−50% and 35−65%, respectively, if only the particle length is used to characterize virus diffusion. The net effect is an underestimation of collector efficiency by a factor of ∼2 or more. These comparisons demonstrate that there is considerable uncertainty in collector efficiency predictions if one does not consider the virus morphology when calculating particle diffusivity. Note that, in this analysis, we have neglected contributions to the overall collector efficiency by interception since it scales very weakly with diffusivity; only Pe−0.125. We have similarly examined the effect of inaccurately estimating virus diffusion to predicting collision frequency for perikinetic flocculation of virus particles. The diffusion coefficient characterizing the relative motion of two particles is the sum of the individual diffusivities.76 Therefore, considerable uncertainty arises when the Stokes−Einstein equation with either the capsid diameter or the virus length is used to estimate individual virus diffusion coefficients and the diffusivity for relative particle motion. For example, for P1, the diffusivity predicted using the Stokes−Einstein equation with the capsid diameter is 55% larger than the measured value,

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.est.6b05323. Additional details of bacteriophage enumeration, diffusion cell and phage flux measurements, details for the ellipsoid model for the T4 capsid, COMSOL modeling, comparison of experimental and predicted diffusivities along with their implications for predicting virus fate during media filtration and flocculation are provided (PDF)

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AUTHOR INFORMATION

Corresponding Author

*Phone: (979) 458-5914; e-mail: [email protected]. ORCID

Shankararaman Chellam: 0000-0001-9173-1439 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge the support from the National Science Foundation through grants CBET-0966934, CBET0966939, CBET-1605088 and CBET-1604715.

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REFERENCES

(1) Blanford, W. J.; Brusseau, M. L.; Yeh, T. C. J.; Gerba, C. P.; Harvey, R. Influence of water chemistry and travel distance on bacteriophage PRD-1 transport in a sandy aquifer. Water Res. 2005, 39 (11), 2345−2357. (2) Yates, M. V.; Gerba, C. P.; Kelley, L. M. Virus persistence in groundwater. Appl. Environ. Microbiol. 1985, 49 (4), 778−781.

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DOI: 10.1021/acs.est.6b05323 Environ. Sci. Technol. 2017, 51, 1433−1440

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