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Modeling bacteriophage-induced inactivation of Escherichia coli utilizing particle aggregation kinetics
Water Research 171 (2020) 115438
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Modeling bacteriophage-induced inactivation of Escherichia coli utilizing particle aggregation kinetics Ethan Hicks a, b, Mark R. Wiesner a, b, *, Claudia K. Gunsch a a b
Department of Civil & Environmental Engineering, Duke University, Durham, NC, USA Center for the Environmental Implications of Nanotechnology (CEINT), Duke University, Durham, NC, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 August 2019 Received in revised form 5 December 2019 Accepted 22 December 2019 Available online 23 December 2019
Targeted inactivation of bacteria using bacteriophages has been proposed in applications ranging from bioengineering and biofuel production to medical treatments. The ability to differentiate between desirable and undesirable organisms, such as in targeting filamentous bacteria in activated sludge, is a potential advantage over conventional disinfectants. Like conventional disinfectants, bacteriophages exhibit non-linear concentration-time (Ct) dynamics in achieving bacterial inactivation. However, there is currently no workable model for predicting these observed non-linear inactivation rates. This work considers an approach to predicting bacteriophage-induced inactivation rates by utilizing classical particle aggregation theory. Bacteriophage-bacteria interactions are represented as a two-step process of transport by Brownian motion, differential settling, and shear, followed by attachment. Modifying classical expressions for particle-particle aggregation to include bacterial growth, death, and bacteriophage reproduction, the model was calibrated and validated using literature data. The calibrated model captures much of the observed non-linearity in inactivation rates and reasonably predicts the final host concentration. This model was shown to be most useful in systems more likely to reflect an industrial setting, where the initial multiplicity of infection, or MOI (the ratio of bacteriophage to host organisms), was 1 or greater. For systems of an initial MOI of less than 1 the model showed increased sensitivity to changes in input parameters and a less pronounced ability to reasonably predict inactivation rates. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Aggregation Bacteriophage Disinfection Water treatment Phage therapy
1. Introduction A contemporary need of water quality engineers is the development of targeted disinfection technology. One example of where this technology would be beneficial is the activated sludge process. The generation of filamentous bacteria often results in sludgebulking thus reducing process efficiency. The inactivation of these specific organisms while allowing other necessary and beneficial organisms to remain viable would improve process optimization. Several disinfection technologies currently exist for water treatment, such as chlorination and UV-radiation, however these methods cannot be used selectively to target microorganisms (Martins et al., 2004). One possible means of organism-specific disinfection involves the use of bacteriophages, or phages. Phages are obligate parasites
* Corresponding author. Box 90287, 120 Hudson Hall, Durham, NC, 27708-0287, USA. E-mail address: [email protected] (M.R. Wiesner). https://doi.org/10.1016/j.watres.2019.115438 0043-1354/© 2019 Elsevier Ltd. All rights reserved.
that infect specific bacterial hosts. Phages are ubiquitous and abundant in the environment with a population estimated to be on the order of 1031 (Hendrix, 2002). Phage population is selfgenerated through host-specific infection and lysis. Taken together, these features qualify phages as an autocatalytic means of inactivating target organisms. Upon contact with a bacterium’s surface a phage will either form an irreversible attachment mediated by an adequate surface binding protein (one which could lead to infection) or will only temporarily dwell upon the host surface due to a lack of the necessary binding receptors, thus not resulting in infection. If infection is successful, the host will begin producing replicates of the infecting phage until after some time when the host lyses to release the phage within it e thus killing the bacterial host while producing newly formed phages and allowing the process to continue (Abedon, 2009). There are still some challenges which need to be overcome before phages receive a wider acceptance as tools for disinfection and biocontrol for many applications, most of which have to do with a lack of a more comprehensive understanding of phage ecology and their impact on environmental microbiomes
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(Goodridge, 2004). Most concerns revolve around questions of both the development of phage-resistant hosts and whether or not phages could be over-specific in their bacterial inactivation range to be effective disinfectants (Choi et al., 2011). However, when it comes to being able to predict the rates of bacterial inactivation by bacteriophages there is still need for a quantitative model for phage-bacteria interactions allowing for the possible management of these interactions in an engineered setting to achieve a desired result. A review of different models proposed for particular phagehost systems was previously summarized by Storms and Sauvageau (2015). While each of these models have particular strengths, there is still a need to predictively model the rate of phage-induced disinfection for a broad range of phage/host systems where the relative initial concentration of bacteriophage to the concentration of viable hosts, referred to as the initial multiplicity of infection, or MOI, may vary (Hyman and Abedon, 2009). The current study builds on data produced in one such previous effort, conducted by Worley-Morse and Gunsch (2015), that illustrated some of the challenges in the development of a more predictive model based upon theory rather than statistical or curve fitting relationships. The work presented here continues to advance predictive phage-induced bacterial inactivation models by incorporating host infection using classical particle aggregation kinetics and theory as articulated principally by Smoluchowski (1917). Particle aggregation theory has been used broadly to describe phenomena ranging from the evolution of aerosols (Friedlander, 2000) to the growth of flocs in wastewater treatment (O’Melia, 1980) as well as to model phage-host attachment. Schlesinger (1932), Delbrück (1940), and Schwartz (1976) considered Smoluchowski’s formulation of particle diffusion and used it to estimate the maximum rate of phage attachment for the purpose of interpreting the dynamics of phagehost interactions. Worley-Morse and Gunsch (2015) implemented Smoluchowski’s particle-particle aggregation terms into population balances to model phage-mediated disinfection but considered limited particle transport mechanisms and did not integrate the rate kernels into their population balances. The current work approaches the modeling of phage-induced bacterial inactivation through the use of population balances and considers three transport mechanisms likely to influence disinfection rates under the hypothesis that particle-particle interactions, as described by aggregation theory, may capture and predict the observed nonlinearity in concentration-time (Ct) curves. This is to be done with minimal data fitting and relying upon experimentally and theoretically derived parameter values.
2. Theory The present model represents phage-induced disinfection as two consecutive, non-reversible reactions: k1
k2
PþB/I/P
1
where P, B, and I represent the concentrations of phage, uninfected bacteria and infected bacteria, respectively. In this model, the formation of an infected host will be treated as a non-reversible attachment between a phage and a healthy bacterium. First, a second-order reaction describes phage-bacteria “aggregation” where the rate constant k1 , encompasses transport of phage, P, to the bacteria, B. followed by attachment, to yield an infected bacterium, I. The choice to describe the first step as a second-order reaction is to reflect that the formation of an infected host is the product of a two-body interaction. The next reaction describes the generation of new phage(s). This is assumed to occur as a first order reaction rate with constant, k2 , following the lysis of infected
bacteria, I, since the generation of new phages is assumed to be solely dependent upon the rate of host lysis.
2.1. Particle aggregation kinetics Particle aggregation can be considered to occur as a sequence of physical transport of the particles to a near-field region where then chemical factors dominate and determine attachment between the particles. Following Smoluchowski’s work, the transport parameter is known as the collision frequency (b) with units of a 2nd order rate constant and the latter as attachment efficiency (a). From the perspective of phage-host interactions a and b were applied to quantify the formation of a phage-host complex leading to an infected host so that one may write:
k1 ¼ ab ½ ¼
L3 M ,T
2
where the product of a and b provides a numerical representation of the effective collisions (in this context the number of phage-host collisions resulting in infection) as a second-order rate constant. Mathematical expressions representing collision frequency are rooted in the envisioned movement of fluid around particles as they move closer towards one another. In one model, particle collisions are not affected by the compression of streamlines as they approach one another. This so-called rectilinear model was first utilized by Smoluchowski (1917). A more realistic model for two approaching solid spheres includes this hydrodynamic effect, which reduces the number of collisions in comparison with the rectilinear model and is more important for approaching particles of greater difference in size (Han and Lawler, 1992). In this work, the rectilinear model of the collision rate kernel, b, was used since analytical solutions for non-spherical particles are not available. The effect of using the rectilinear model is to combine the hydrodynamic retardation of particle collision with the chemical factors affecting attachment in the parameter a. Particles may come into contact through three transport mechanisms; Brownian motion (BR), shear motion (S), and differential settling (DS). These three particle collision mechanisms are assumed to be additive. The components of the rectilinear collision rate kernel can be described by equations (3)e(6) (Han and Lawler, 1992; O’Melia, 1980; Smoluchowski, 1917).
2k T di þ dj bBR ði; jÞ ¼ B 3m di dj
bS ði; jÞ ¼
2
3 1 d þ dj G 6 i
bDS ði; jÞ ¼
pg rp rl di þ dj 3 di dj 72m
bði; jÞ ¼ bBR ði; jÞ þ bS ði; jÞ þ bDS ði; jÞ
3
4
5 6
Here, kB represents the Boltzmann’s constant, T is absolute temperature, d is the diameter of size-differentiable spherical particles i and j, m is the absolute fluid viscosity, G is the velocity gradient, g is acceleration due to gravity, and rp and rl are the particle and liquid densities, respectively. The model represents phages as transported to the surface of a bacterium as described by the collision rate kernel. Once a phage is in the near field of a bacterium, the probability of phage attachment to its host is a function of factors including dwell time, high-affinity receptor surface concentration, and phage-tail orientation amongst
E. Hicks et al. / Water Research 171 (2020) 115438
others (Chatterjee and Rothenberg, 2012; Schwartz, 1976). This probability, accounting for all of these intricate surface interactions of surface attachment, as well as near-field hydrodynamic factors, is captured by the parameter a, the attachment efficiency. The attachment efficiency, a, can be viewed as the ratio of the number of surface contacts leading to attachment to the total number of surface contacts (Smoluchowski, 1917). In addition to the factors cited above, particle attachment efficiency is affected by electrostatic repulsive forces, van der Waals attractive forces, as well as hydrophobic and steric interactions (Derjaguin and Landau, 1993; Verwey and Overbeek, 1948). Therefore, a is a global correction parameter converting the value of b, the total number of collisions, to total number of collisions leading to attachment assumed to yield infection. Experimental methods exist for determining values of a but were considered to be beyond the scope of the current work. Without experimentally derived values for a we assume a value of unity. This assumption could prove poor for various phagehost systems such as those of low surface receptor concentration but may also be a reasonable assumption given the specificity of viral attachment interactions. Treatment of a as an additional fitting coefficient would certainly lead to improved fits to data. However, the goal of the current study was to determine the extent to which particle aggregation kinetics can represent the nonlinearities in the bacterial inactivation process with a minimum of adjustable variables. Once a phage has infected a susceptible host, a phage uses the host’s replicating faculties to generate new phages within the host itself, ultimately leading to cell lysis and the release of the newly formed phage. Escherichia coli has been observed to lyse approximately 45 min after infection with l phage (Ryan and Rutenberg, 2007), corresponding to a lysis rate (kL) of 1.33 h1 in this example. We assume here that the phage generation rate (k2 in equation (1)) is well approximated by the host lyses rate following infection, kL , such that:
k2 ¼ kL
7
2.2. Incorporation into population balances In a system of both bacteria and free phages one would expect to find at any given time uninfected bacteria, infected (but not yet lysed) bacteria, unbound (or free) phage, and phage bound to bacteria. As a result, population balances should account for the growth of bacteria, the loss of healthy bacteria and free phage through attachment, and phage production through lysis of infected bacteria:
dB ¼ mnet B abBP dt
8
dI ¼ abBP kL I dt
9
dP ¼ kL IY kI P abðB þ IÞP dt
10
Eq. (8) describes the change in the uninfected host concentration (B, in #CFU /mL) as a function of time (t, in hours) by accounting for the net growth rate of the host (m, per hour) and the loss of uninfected hosts due to phage attachment. It is here that the product of a and b (mL hour1 #PFU or CFU 1) act together as a second-order rate constant for phage attachment. Upon phage attachment the host is considered infected (I, in #CFU /mL). Therefore, the rate of phage attachment is assumed to be equal to the rate
3
of infection (Equation (9)). Here the replication of an infected host is not considered and this model does not account for the formation of phage-insensitive bacteria. Eq. (10) described changes in the concentration of free phages, P ( #PFU /mL), that are generated through the lysis of infected hosts according to the burst size (Y; #PFU =#CFU Þ or lost either through inactivation over time according to a first-order reaction rate, kinact (per hour) or through attachment to an uninfected host or an already-infected host. These balances are similar to those used by Worley-Morse and Gunsch (2015), differing only in 1) the inclusion of a phage inactivation rate and 2) the allowance of free phage to attach to both uninfected and previously infected hosts. Both the uninfected and infected host populations were included in the bacteriophage population balance to allow for the possibility that more than one phage could possibly attach to a still viable yet infected host. 3. Methods The model was calibrated using literature values and theoretical calculations for each of the rate constants with the exception of the growth rate and burst size of the phage-host system. Experimental data collected by Worley-Morse and Gunsch (2015) were used to validate the model output for changes in bacteria and phage concentration over time. In brief, the work performed in their studies explored the extent of phage-induced bacterial inactivation as a function of multiplicity of infection across a range of phage/host concentrations for a system of Escherichia coli K-12 MG1655 and coliphage Ec2 (Worley-Morse et al., 2014). Based upon a training sub-set of the data, a value of 0.098 h1 and 4.5 #PFU #CFU 1 was used for the growth rate and burst size of this phage-host system, respectively. The velocity gradient, G, was assumed to be zero as the experimental design did not involve continuous shear mixing. As a result, the shear component of the contact frequency did not contribute to the overall contact frequency calculation in this model validation. The impact of this decision will be an increased dependency upon the ability of differential settling and Brownian motion to describe the inactivation trends and a decreased impact on the use of the rectilinear assumption. Since E. coli is a rod-shaped bacterium, it was assigned a representative spherical diameter of 1 mm. This estimation was based upon calculating the corresponding diameter of a sphere based upon the volume of E. coli, taken as 0.58 mm3 (Kubitschek, 1990) while the diameter of coliphage Ec2 was assigned a representative diameter of 0.1 mm, using coliphage T4 size as a model of phage size (Yap and Rossmann, 2014). Particle density for both phage and bacteria was taken to be 1.105 g/mL, as reported by Martínez-Salas et al. (1981), while the fluid density was treated as the density of water, taken to be 1000 kg m3. The lysis rate was represented by the lysis time of E. coli after infection by l phage, as presented earlier in this article (section 2.2). Considering that the experiments were only conducted over a 4-h period and did not utilize the phage inactivation rate parameter, this parameter was disregarded. Using the assumed values for rate constants, the system of differential equations (Eqs. (8)e(10)) was solved numerically using Euler integration. Calculations were carried out using MATLAB, R2019a 64-bit academic use license, on a 2018 MacBook Pro laptop. A model MATLAB script can be found in the supplemental material of this article. 4. Results and discussion Looking first at Ct curves, Fig. 1 compares the model outcomes with the experimentally observed data points reported by WorleyMorse and Gunsch (2015) where phages were introduced to the system at time 0. The only input parameter values which were
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Fig. 1. Comparing disinfection rates across MOI and phage/bacteria concentration. Moving from left to right in each row, the initial bacteria concentration was held constant while initial phage concentration was increased by one order of magnitude in each column. Values of population balance parameters were as follows: a ¼ 1 (assumed), mnet ¼ 0.098 h1, kinact ¼ 0 h1, kL ¼ 1.33 h1, Y ¼ 4.5 #PFU #CFU 1, dP ¼ 0.1 mm, dB ¼ 1 mm, G ¼ 0 h1, rp ¼ 1105 kg m3, g ¼ 1.27 1014 mm h2, T ¼ 21o C, rl ¼ 1000 kg m3 and m was calculated as a function of temperature according to the Vogel equation. N represents the sum of uninfected and infected bacteria (B þ I), since an infected host which has not yet lysed is still a living bacterium which could be insensitive to infection while N0 is the initial uninfected bacteria concentration (assuming no infected bacteria exist at time 0). Error bars are the standard deviation between trials.
changed between runs was the initial phage and host concentrations. Bacterial inactivation by phages varies as both a function of time and the relative initial concentrations of phage and bacteria, or MOI. For example, at a MOI of 1, and a phage concentration of 108 plaque forming units per mL, approximately 2 h are required to achieve one log inactivation of bacteria. Were a linear Ct applicable, this would suggest that at a concentration of 107 (#PFU /mL), some 20 h would be required to achieve the same log inactivation, while, in fact this is observed to occur after approximately 2.5 h. This not only demonstrates how poorly a linear Ct curve would represent some of these systems but also allows one to comment on the kinetics of the interactions. At the highest phage concentration of 108, inactivation curves were more similar than those produced at 107 suggesting that the kinetics approaches more of a first order behavior while at the lowest tested phage concentrations of 106, inactivation curves were more suggestive of strong second-order kinetics, except at a MOI of 1. For example, according to plot G, where MOI is 0.01, the Ct curve predicts one log inactivation after about 3 h though no inactivation was observed experimentally. Taken together these trends would indicate that as MOI diverges further from 1 the model begins to show strong bias either towards a pseudo-first-order or second-order reaction rate which would tend to significantly over estimate the rate of disinfection at lower relative phage concentrations. When the MOI is closer to 1 the model is able to better capture the non-linear behavior seen in experimental trials. However, even at a MOI greater than 1, the
pseudo-first-order kinetics better represents the real system than the second-order kinetics represents the real system when MOI is less than 1. This is a beneficial characteristic of the model since most applications in which disinfection is desired an initial MOI of at least 10 is used in order to ensure inactivation occurs (Abedon, 2009). This demonstrates the model’s value as a tool for several practical and industrial applications. This model’s ability to capture both first-order and second-order kinetic behavior, especially at MOIs approaching 1, demonstrates the utility of Smoluchowski (1917) particle aggregation theory to predict rates of phageinduced disinfection. The non-linearity of these Ct curves is not trivial to predict. Particle aggregation theory appears to capture most of the observed non-linearity in disinfection rates as observed during early contact times. This is an improvement over other previous modeling attempts offered by Worley-Morse and Gunsch (2015). The most significant improvement being the ability to generate non-linear Ct curves as a result of the choice to use aggregation theory, rather than data fitting, to approximate the rate of phage-host attachment. The most significant challenge to this model is the difficulty in predicting the disinfection rate of systems of low relative phage concentration (MOI <1), as previously discussed. For its ability to predict disinfection rates, this model is best suited for systems of high phage and host concentrations (at least 106 CFUs or PFUs per mL) at a MOI of 1 or greater. Looking more closely at the changes in the phage and host concentrations, not just the disinfection rate, one can see more
E. Hicks et al. / Water Research 171 (2020) 115438
5
Fig. 2. Plots of the change in phage and host concentration as a function of time under the conditions described in Fig. 1. A) 108/106 #PFU =#CFU , B) 107/108 #PFU = #CFU , and C) 106/ 107 #PFU =#CFU. Note that the E. coli concentration in figures A thru C are the sum of both infected and uninfected hosts (B þ I).
agreement between the model predictions and experimental outcomes. Fig. 2A shows strong agreement between the observed and predicted changes in both phage and host concentrations, corresponding to the conditions in Fig. 1C. For the two systems where MOI is less than 1 (plots B and C) characteristics of the population balances become more evident. For example, Fig. 2B shows a sharp inflection in the phage concentration after about 0.5 h before a sharp increase. This is indicative of the loss of free phage as a result of host attachment where the population only begins to increase again once infected hosts begin to lyse and release more free phages. This inflection is not as dramatically seen in Fig. 2C, likely resulting from the lower initial free phage concentration. Fig. 2A does not show any significant inflection as the initial phage concentration was higher than the initial host concentration. All three models reasonably predict the final host and phage concentrations. This is an improvement over the previous modeling work done by Worley-Morse and Gunsch (2015) especially with regards to the changes in host concentration under a range of initial conditions. This is seen most clearly in the predicted final value of the host concentration. This was done by providing a predictive approach to representing changes in host and phage concentrations from both theory and experimentation while also avoiding data fitting. Compared to other models this effort highlights the ability of particle aggregation theory to reasonably predict the rate of phagemediated disinfection utilizing only a few assumptions and without the employment of fitting parameters. This could have significant design implications on water treatment, especially regarding the potential of phage-based biocontrol efforts in the activated sludge process.
For different combinations of phage and host systems one would expect the physical parameters likely to change most would be that of burst size and lysis rate. To examine model sensitivity, the simulated disinfection rates were plotted against the experimentally observed rates to determine an R2 value. These R2 values were collected for several different values for different model parameters. The model appeared to be most sensitive to changes in burst size, Y. Fig. 3 shows the changes in R2 as a function of different values of Y, ranging from 0 (no bursting) to 100 #PFU #1 CFU. The conditions of this figure are the same as those provided in Fig. 1 while allowing only the burst size to vary. Interestingly, the model’s sensitivity to changes in burst size was largely confined to the lowest MOI condition. As MOI increased the R2 values remained more consistent. Model sensitivity toward changes in lysis rate, kL, was examined for values ranging from 0 to 4 h1. It was similarly observed that for the MOI ¼ 0.1 condition that the R2 varied most dramatically, with values ranging from 0.24 to 0.73, peaking slightly at a kL of ~1 h1. However, for the other MOI conditions the R2 stabilized at ~0.95 (except when kL ¼ 0 h1, no lysis). Phage inactivation rate had the least effect on model stability, where again only for the MOI ¼ 0.1 condition was the R2 seen to vary (between 0.68 and 0.81 for values of 0e2 h1). This analysis suggests that the model’s sensitivity is a function of the ratio of the phage and its host. This makes sense given that at higher and higher relative phage concentrations factors such as increased burst size and lysis rates have less and less effect on the uncertainty of the rate of inactivation. Therefore, this analysis corroborates the previous observations that this modeling approach is best suited under conditions of initial MOI of 1 or greater at the tested
Fig. 3. Change in R2 as a function of burst size for three different MOIs, with an initial host concentration of 107 #CFU mL1.
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E. Hicks et al. / Water Research 171 (2020) 115438
concentrations and has the ability to predict the inactivation rate with minimal data fitting. 5. Conclusions Classical Smoluchowski (1917) particle aggregation theory has shown itself to be a useful tool in the construction of phageinduced disinfection models. In summary, this approach is most beneficial under conditions where the initial system MOI is 1 or greater, granted that both phage and host concentrations are no less than 106 colony or plaque forming units per milliliter. This is mostly due to the model’s ability to capture both a pseudo-firstorder and second-order kinetic behavior depending upon the relative phage and host concentrations using minimal fitting factors. While future work will be needed to test this model’s applicability to other phage-host systems which may behave differently under similar conditions, the results presented here suggest that particle aggregation kinetics could provide a stronger foundation for future modeling work. More significantly, this effort also demonstrates the ability of some complex biological interactions to be reimagined within the context of particle-particle interactions which directly influence the health of environmental systems. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The authors would like to thank Dr. Thomas O. Worley-Morse, for contributing his data to this work. This work was also partially funded through the Center for the Environmental Implications of Nanotechnology (CEINT) under NSF Cooperative Agreement Number EF-0830093. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.watres.2019.115438. References Abedon, S.T., 2009. Kinetics of phage-mediated biocontrol of bacteria. Foodborne Pathogens and Disease 6 (7), 807e815. https://doi.org/10.1089/fpd.2008.0242.
Chatterjee, S., Rothenberg, E., 2012. Interaction of bacteriophage l with its E. coli receptor, LamB. Viruses 4 (11). https://doi.org/10.3390/v4113162. Choi, J., Kotay, S.M., Goel, R., 2011. Bacteriophage-based biocontrol of biological sludge bulking in wastewater. Bioengineered Bugs 2 (4), 214e217. https:// doi.org/10.4161/bbug.2.4.16211. Delbrück, M., 1940. Adsorption OF bacteriophage under various physiological conditions OF the host. J. Gen. Physiol. 23 (5), 631e642. https://doi.org/10.1085/ jgp.23.5.631. Derjaguin, B., Landau, L., 1993. Theory of the stability of strongly charged lyophobic sols and of the adhesion of strongly charged particles in solutions of electrolytes. Prog. Surf. Sci. 43 (1), 30e59. https://doi.org/10.1016/0079-6816(93) 90013-L. Friedlander, S.K., 2000. Smoke, Dust, and Haze, second ed. Oxford University Press, New York & Oxford. Goodridge, L.D., 2004. Bacteriophage biocontrol of plant pathogens: fact or fiction? Trends Biotechnol. 22 (8), 384e385. https://doi.org/10.1016/ j.tibtech.2004.05.007. Han, M., Lawler, D.F., 1992. The (relative) insignificance of G in flocculation. Journal (American Water Works Association) 84 (10), 79e91. Hendrix, R.W., 2002. Bacteriophages: evolution of the majority. Theor. Popul. Biol. 61 (4), 471e480. https://doi.org/10.1006/tpbi.2002.1590. Hyman, P., Abedon, S.T., 2009. Practical methods for determining phage growth parameters. In: Clokie, M.R.J., Kropinski, A.M. (Eds.), Bacteriophages: Methods and Protocols, Volume 1: Isolation, Characterization, and Interactions. Humana Press, Totowa, NJ, pp. 175e202. Kubitschek, H.E., 1990. Cell volume increase in Escherichia coli after shifts to richer media. J. Bacteriol. 172 (1), 94. https://doi.org/10.1128/jb.172.1.94-101.1990. Martínez-Salas, E., Martín, J.A., Vicente, M., 1981. Relationship of Escherichia coli density to growth rate and cell age. J. Bacteriol. 147 (1), 97e100. Martins, A.M.P., Pagilla, K., Heijnen, J.J., van Loosdrecht, M.C.M., 2004. Filamentous bulking sludgeda critical review. Water Res. 38 (4), 793e817. https://doi.org/ 10.1016/j.watres.2003.11.005. O’Melia, C.R., 1980. ES Features: aquasols: the behavior of small particles in aquatic systems. Environ. Sci. Technol. 14 (9), 1052e1060. https://doi.org/10.1021/ es60169a601. Ryan, G.L., Rutenberg, A.D., 2007. Clocking out: modeling phage-induced lysis of Escherichia coli. J. Bacteriol. 189 (13), 4749. https://doi.org/10.1128/JB.00392-07. Schlesinger, M., 1932. Über die Bindung des Bakteriophagen an homologe Bakterien. Zeitschrift für Hygiene und Infektionskrankheiten 114 (1), 136e148. https:// doi.org/10.1007/BF02176514. Schwartz, M., 1976. The adsorption of coliphage lambda to its host: effect of variations in the surface density of receptor and in phage-receptor affinity. J. Mol. Biol. 103 (3), 521e536. https://doi.org/10.1016/0022-2836(76)90215-1. Smoluchowski, M.v., 1917. Versuch einer mathematischen Theorie der Koagula€sungen. Zeitschrift fuer physikalische Chemie 92, tionskinetik kolloider Lo 129e168. Storms, Z.J., Sauvageau, D., 2015. Modeling tailed bacteriophage adsorption: insight into mechanisms. Virology 485, 355e362. https://doi.org/10.1016/ j.virol.2015.08.007. Verwey, E.J.W., Overbeek, J.T.G., 1948. Theory of the Stability of Lyophobic Colloids. Elsevier, Amsterdam, p. 218p. Worley-Morse, T.O., Gunsch, C.K., 2015. Modeling phage induced bacterial disinfection rates and the resulting design implications. Water Res. 68, 627e636. https://doi.org/10.1016/j.watres.2014.10.025. Worley-Morse, T.O., Zhang, L., Gunsch, C.K., 2014. The long-term effects of phage concentration on the inhibition of planktonic bacterial cultures. Environ. Sci.: Processes & Impacts 16 (1), 81e87. https://doi.org/10.1039/C3EM00427A. Yap, M.L., Rossmann, M.G., 2014. Structure and function of bacteriophage T4. Future Microbiol. 9 (12), 1319e1327. https://doi.org/10.2217/fmb.14.91.
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Modeling bacteriophage-induced inactivation of Escherichia coli utilizing particle aggregation kinetics Ethan Hicks a, b, Mark R. Wiesner a, b, *, Claudia K. Gunsch a a b
Department of Civil & Environmental Engineering, Duke University, Durham, NC, USA Center for the Environmental Implications of Nanotechnology (CEINT), Duke University, Durham, NC, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 August 2019 Received in revised form 5 December 2019 Accepted 22 December 2019 Available online 23 December 2019
Targeted inactivation of bacteria using bacteriophages has been proposed in applications ranging from bioengineering and biofuel production to medical treatments. The ability to differentiate between desirable and undesirable organisms, such as in targeting filamentous bacteria in activated sludge, is a potential advantage over conventional disinfectants. Like conventional disinfectants, bacteriophages exhibit non-linear concentration-time (Ct) dynamics in achieving bacterial inactivation. However, there is currently no workable model for predicting these observed non-linear inactivation rates. This work considers an approach to predicting bacteriophage-induced inactivation rates by utilizing classical particle aggregation theory. Bacteriophage-bacteria interactions are represented as a two-step process of transport by Brownian motion, differential settling, and shear, followed by attachment. Modifying classical expressions for particle-particle aggregation to include bacterial growth, death, and bacteriophage reproduction, the model was calibrated and validated using literature data. The calibrated model captures much of the observed non-linearity in inactivation rates and reasonably predicts the final host concentration. This model was shown to be most useful in systems more likely to reflect an industrial setting, where the initial multiplicity of infection, or MOI (the ratio of bacteriophage to host organisms), was 1 or greater. For systems of an initial MOI of less than 1 the model showed increased sensitivity to changes in input parameters and a less pronounced ability to reasonably predict inactivation rates. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Aggregation Bacteriophage Disinfection Water treatment Phage therapy
1. Introduction A contemporary need of water quality engineers is the development of targeted disinfection technology. One example of where this technology would be beneficial is the activated sludge process. The generation of filamentous bacteria often results in sludgebulking thus reducing process efficiency. The inactivation of these specific organisms while allowing other necessary and beneficial organisms to remain viable would improve process optimization. Several disinfection technologies currently exist for water treatment, such as chlorination and UV-radiation, however these methods cannot be used selectively to target microorganisms (Martins et al., 2004). One possible means of organism-specific disinfection involves the use of bacteriophages, or phages. Phages are obligate parasites
* Corresponding author. Box 90287, 120 Hudson Hall, Durham, NC, 27708-0287, USA. E-mail address: [email protected] (M.R. Wiesner). https://doi.org/10.1016/j.watres.2019.115438 0043-1354/© 2019 Elsevier Ltd. All rights reserved.
that infect specific bacterial hosts. Phages are ubiquitous and abundant in the environment with a population estimated to be on the order of 1031 (Hendrix, 2002). Phage population is selfgenerated through host-specific infection and lysis. Taken together, these features qualify phages as an autocatalytic means of inactivating target organisms. Upon contact with a bacterium’s surface a phage will either form an irreversible attachment mediated by an adequate surface binding protein (one which could lead to infection) or will only temporarily dwell upon the host surface due to a lack of the necessary binding receptors, thus not resulting in infection. If infection is successful, the host will begin producing replicates of the infecting phage until after some time when the host lyses to release the phage within it e thus killing the bacterial host while producing newly formed phages and allowing the process to continue (Abedon, 2009). There are still some challenges which need to be overcome before phages receive a wider acceptance as tools for disinfection and biocontrol for many applications, most of which have to do with a lack of a more comprehensive understanding of phage ecology and their impact on environmental microbiomes
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(Goodridge, 2004). Most concerns revolve around questions of both the development of phage-resistant hosts and whether or not phages could be over-specific in their bacterial inactivation range to be effective disinfectants (Choi et al., 2011). However, when it comes to being able to predict the rates of bacterial inactivation by bacteriophages there is still need for a quantitative model for phage-bacteria interactions allowing for the possible management of these interactions in an engineered setting to achieve a desired result. A review of different models proposed for particular phagehost systems was previously summarized by Storms and Sauvageau (2015). While each of these models have particular strengths, there is still a need to predictively model the rate of phage-induced disinfection for a broad range of phage/host systems where the relative initial concentration of bacteriophage to the concentration of viable hosts, referred to as the initial multiplicity of infection, or MOI, may vary (Hyman and Abedon, 2009). The current study builds on data produced in one such previous effort, conducted by Worley-Morse and Gunsch (2015), that illustrated some of the challenges in the development of a more predictive model based upon theory rather than statistical or curve fitting relationships. The work presented here continues to advance predictive phage-induced bacterial inactivation models by incorporating host infection using classical particle aggregation kinetics and theory as articulated principally by Smoluchowski (1917). Particle aggregation theory has been used broadly to describe phenomena ranging from the evolution of aerosols (Friedlander, 2000) to the growth of flocs in wastewater treatment (O’Melia, 1980) as well as to model phage-host attachment. Schlesinger (1932), Delbrück (1940), and Schwartz (1976) considered Smoluchowski’s formulation of particle diffusion and used it to estimate the maximum rate of phage attachment for the purpose of interpreting the dynamics of phagehost interactions. Worley-Morse and Gunsch (2015) implemented Smoluchowski’s particle-particle aggregation terms into population balances to model phage-mediated disinfection but considered limited particle transport mechanisms and did not integrate the rate kernels into their population balances. The current work approaches the modeling of phage-induced bacterial inactivation through the use of population balances and considers three transport mechanisms likely to influence disinfection rates under the hypothesis that particle-particle interactions, as described by aggregation theory, may capture and predict the observed nonlinearity in concentration-time (Ct) curves. This is to be done with minimal data fitting and relying upon experimentally and theoretically derived parameter values.
2. Theory The present model represents phage-induced disinfection as two consecutive, non-reversible reactions: k1
k2
PþB/I/P
1
where P, B, and I represent the concentrations of phage, uninfected bacteria and infected bacteria, respectively. In this model, the formation of an infected host will be treated as a non-reversible attachment between a phage and a healthy bacterium. First, a second-order reaction describes phage-bacteria “aggregation” where the rate constant k1 , encompasses transport of phage, P, to the bacteria, B. followed by attachment, to yield an infected bacterium, I. The choice to describe the first step as a second-order reaction is to reflect that the formation of an infected host is the product of a two-body interaction. The next reaction describes the generation of new phage(s). This is assumed to occur as a first order reaction rate with constant, k2 , following the lysis of infected
bacteria, I, since the generation of new phages is assumed to be solely dependent upon the rate of host lysis.
2.1. Particle aggregation kinetics Particle aggregation can be considered to occur as a sequence of physical transport of the particles to a near-field region where then chemical factors dominate and determine attachment between the particles. Following Smoluchowski’s work, the transport parameter is known as the collision frequency (b) with units of a 2nd order rate constant and the latter as attachment efficiency (a). From the perspective of phage-host interactions a and b were applied to quantify the formation of a phage-host complex leading to an infected host so that one may write:
k1 ¼ ab ½ ¼
L3 M ,T
2
where the product of a and b provides a numerical representation of the effective collisions (in this context the number of phage-host collisions resulting in infection) as a second-order rate constant. Mathematical expressions representing collision frequency are rooted in the envisioned movement of fluid around particles as they move closer towards one another. In one model, particle collisions are not affected by the compression of streamlines as they approach one another. This so-called rectilinear model was first utilized by Smoluchowski (1917). A more realistic model for two approaching solid spheres includes this hydrodynamic effect, which reduces the number of collisions in comparison with the rectilinear model and is more important for approaching particles of greater difference in size (Han and Lawler, 1992). In this work, the rectilinear model of the collision rate kernel, b, was used since analytical solutions for non-spherical particles are not available. The effect of using the rectilinear model is to combine the hydrodynamic retardation of particle collision with the chemical factors affecting attachment in the parameter a. Particles may come into contact through three transport mechanisms; Brownian motion (BR), shear motion (S), and differential settling (DS). These three particle collision mechanisms are assumed to be additive. The components of the rectilinear collision rate kernel can be described by equations (3)e(6) (Han and Lawler, 1992; O’Melia, 1980; Smoluchowski, 1917).
2k T di þ dj bBR ði; jÞ ¼ B 3m di dj
bS ði; jÞ ¼
2
3 1 d þ dj G 6 i
bDS ði; jÞ ¼
pg rp rl di þ dj 3 di dj 72m
bði; jÞ ¼ bBR ði; jÞ þ bS ði; jÞ þ bDS ði; jÞ
3
4
5 6
Here, kB represents the Boltzmann’s constant, T is absolute temperature, d is the diameter of size-differentiable spherical particles i and j, m is the absolute fluid viscosity, G is the velocity gradient, g is acceleration due to gravity, and rp and rl are the particle and liquid densities, respectively. The model represents phages as transported to the surface of a bacterium as described by the collision rate kernel. Once a phage is in the near field of a bacterium, the probability of phage attachment to its host is a function of factors including dwell time, high-affinity receptor surface concentration, and phage-tail orientation amongst
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others (Chatterjee and Rothenberg, 2012; Schwartz, 1976). This probability, accounting for all of these intricate surface interactions of surface attachment, as well as near-field hydrodynamic factors, is captured by the parameter a, the attachment efficiency. The attachment efficiency, a, can be viewed as the ratio of the number of surface contacts leading to attachment to the total number of surface contacts (Smoluchowski, 1917). In addition to the factors cited above, particle attachment efficiency is affected by electrostatic repulsive forces, van der Waals attractive forces, as well as hydrophobic and steric interactions (Derjaguin and Landau, 1993; Verwey and Overbeek, 1948). Therefore, a is a global correction parameter converting the value of b, the total number of collisions, to total number of collisions leading to attachment assumed to yield infection. Experimental methods exist for determining values of a but were considered to be beyond the scope of the current work. Without experimentally derived values for a we assume a value of unity. This assumption could prove poor for various phagehost systems such as those of low surface receptor concentration but may also be a reasonable assumption given the specificity of viral attachment interactions. Treatment of a as an additional fitting coefficient would certainly lead to improved fits to data. However, the goal of the current study was to determine the extent to which particle aggregation kinetics can represent the nonlinearities in the bacterial inactivation process with a minimum of adjustable variables. Once a phage has infected a susceptible host, a phage uses the host’s replicating faculties to generate new phages within the host itself, ultimately leading to cell lysis and the release of the newly formed phage. Escherichia coli has been observed to lyse approximately 45 min after infection with l phage (Ryan and Rutenberg, 2007), corresponding to a lysis rate (kL) of 1.33 h1 in this example. We assume here that the phage generation rate (k2 in equation (1)) is well approximated by the host lyses rate following infection, kL , such that:
k2 ¼ kL
7
2.2. Incorporation into population balances In a system of both bacteria and free phages one would expect to find at any given time uninfected bacteria, infected (but not yet lysed) bacteria, unbound (or free) phage, and phage bound to bacteria. As a result, population balances should account for the growth of bacteria, the loss of healthy bacteria and free phage through attachment, and phage production through lysis of infected bacteria:
dB ¼ mnet B abBP dt
8
dI ¼ abBP kL I dt
9
dP ¼ kL IY kI P abðB þ IÞP dt
10
Eq. (8) describes the change in the uninfected host concentration (B, in #CFU /mL) as a function of time (t, in hours) by accounting for the net growth rate of the host (m, per hour) and the loss of uninfected hosts due to phage attachment. It is here that the product of a and b (mL hour1 #PFU or CFU 1) act together as a second-order rate constant for phage attachment. Upon phage attachment the host is considered infected (I, in #CFU /mL). Therefore, the rate of phage attachment is assumed to be equal to the rate
3
of infection (Equation (9)). Here the replication of an infected host is not considered and this model does not account for the formation of phage-insensitive bacteria. Eq. (10) described changes in the concentration of free phages, P ( #PFU /mL), that are generated through the lysis of infected hosts according to the burst size (Y; #PFU =#CFU Þ or lost either through inactivation over time according to a first-order reaction rate, kinact (per hour) or through attachment to an uninfected host or an already-infected host. These balances are similar to those used by Worley-Morse and Gunsch (2015), differing only in 1) the inclusion of a phage inactivation rate and 2) the allowance of free phage to attach to both uninfected and previously infected hosts. Both the uninfected and infected host populations were included in the bacteriophage population balance to allow for the possibility that more than one phage could possibly attach to a still viable yet infected host. 3. Methods The model was calibrated using literature values and theoretical calculations for each of the rate constants with the exception of the growth rate and burst size of the phage-host system. Experimental data collected by Worley-Morse and Gunsch (2015) were used to validate the model output for changes in bacteria and phage concentration over time. In brief, the work performed in their studies explored the extent of phage-induced bacterial inactivation as a function of multiplicity of infection across a range of phage/host concentrations for a system of Escherichia coli K-12 MG1655 and coliphage Ec2 (Worley-Morse et al., 2014). Based upon a training sub-set of the data, a value of 0.098 h1 and 4.5 #PFU #CFU 1 was used for the growth rate and burst size of this phage-host system, respectively. The velocity gradient, G, was assumed to be zero as the experimental design did not involve continuous shear mixing. As a result, the shear component of the contact frequency did not contribute to the overall contact frequency calculation in this model validation. The impact of this decision will be an increased dependency upon the ability of differential settling and Brownian motion to describe the inactivation trends and a decreased impact on the use of the rectilinear assumption. Since E. coli is a rod-shaped bacterium, it was assigned a representative spherical diameter of 1 mm. This estimation was based upon calculating the corresponding diameter of a sphere based upon the volume of E. coli, taken as 0.58 mm3 (Kubitschek, 1990) while the diameter of coliphage Ec2 was assigned a representative diameter of 0.1 mm, using coliphage T4 size as a model of phage size (Yap and Rossmann, 2014). Particle density for both phage and bacteria was taken to be 1.105 g/mL, as reported by Martínez-Salas et al. (1981), while the fluid density was treated as the density of water, taken to be 1000 kg m3. The lysis rate was represented by the lysis time of E. coli after infection by l phage, as presented earlier in this article (section 2.2). Considering that the experiments were only conducted over a 4-h period and did not utilize the phage inactivation rate parameter, this parameter was disregarded. Using the assumed values for rate constants, the system of differential equations (Eqs. (8)e(10)) was solved numerically using Euler integration. Calculations were carried out using MATLAB, R2019a 64-bit academic use license, on a 2018 MacBook Pro laptop. A model MATLAB script can be found in the supplemental material of this article. 4. Results and discussion Looking first at Ct curves, Fig. 1 compares the model outcomes with the experimentally observed data points reported by WorleyMorse and Gunsch (2015) where phages were introduced to the system at time 0. The only input parameter values which were
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Fig. 1. Comparing disinfection rates across MOI and phage/bacteria concentration. Moving from left to right in each row, the initial bacteria concentration was held constant while initial phage concentration was increased by one order of magnitude in each column. Values of population balance parameters were as follows: a ¼ 1 (assumed), mnet ¼ 0.098 h1, kinact ¼ 0 h1, kL ¼ 1.33 h1, Y ¼ 4.5 #PFU #CFU 1, dP ¼ 0.1 mm, dB ¼ 1 mm, G ¼ 0 h1, rp ¼ 1105 kg m3, g ¼ 1.27 1014 mm h2, T ¼ 21o C, rl ¼ 1000 kg m3 and m was calculated as a function of temperature according to the Vogel equation. N represents the sum of uninfected and infected bacteria (B þ I), since an infected host which has not yet lysed is still a living bacterium which could be insensitive to infection while N0 is the initial uninfected bacteria concentration (assuming no infected bacteria exist at time 0). Error bars are the standard deviation between trials.
changed between runs was the initial phage and host concentrations. Bacterial inactivation by phages varies as both a function of time and the relative initial concentrations of phage and bacteria, or MOI. For example, at a MOI of 1, and a phage concentration of 108 plaque forming units per mL, approximately 2 h are required to achieve one log inactivation of bacteria. Were a linear Ct applicable, this would suggest that at a concentration of 107 (#PFU /mL), some 20 h would be required to achieve the same log inactivation, while, in fact this is observed to occur after approximately 2.5 h. This not only demonstrates how poorly a linear Ct curve would represent some of these systems but also allows one to comment on the kinetics of the interactions. At the highest phage concentration of 108, inactivation curves were more similar than those produced at 107 suggesting that the kinetics approaches more of a first order behavior while at the lowest tested phage concentrations of 106, inactivation curves were more suggestive of strong second-order kinetics, except at a MOI of 1. For example, according to plot G, where MOI is 0.01, the Ct curve predicts one log inactivation after about 3 h though no inactivation was observed experimentally. Taken together these trends would indicate that as MOI diverges further from 1 the model begins to show strong bias either towards a pseudo-first-order or second-order reaction rate which would tend to significantly over estimate the rate of disinfection at lower relative phage concentrations. When the MOI is closer to 1 the model is able to better capture the non-linear behavior seen in experimental trials. However, even at a MOI greater than 1, the
pseudo-first-order kinetics better represents the real system than the second-order kinetics represents the real system when MOI is less than 1. This is a beneficial characteristic of the model since most applications in which disinfection is desired an initial MOI of at least 10 is used in order to ensure inactivation occurs (Abedon, 2009). This demonstrates the model’s value as a tool for several practical and industrial applications. This model’s ability to capture both first-order and second-order kinetic behavior, especially at MOIs approaching 1, demonstrates the utility of Smoluchowski (1917) particle aggregation theory to predict rates of phageinduced disinfection. The non-linearity of these Ct curves is not trivial to predict. Particle aggregation theory appears to capture most of the observed non-linearity in disinfection rates as observed during early contact times. This is an improvement over other previous modeling attempts offered by Worley-Morse and Gunsch (2015). The most significant improvement being the ability to generate non-linear Ct curves as a result of the choice to use aggregation theory, rather than data fitting, to approximate the rate of phage-host attachment. The most significant challenge to this model is the difficulty in predicting the disinfection rate of systems of low relative phage concentration (MOI <1), as previously discussed. For its ability to predict disinfection rates, this model is best suited for systems of high phage and host concentrations (at least 106 CFUs or PFUs per mL) at a MOI of 1 or greater. Looking more closely at the changes in the phage and host concentrations, not just the disinfection rate, one can see more
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Fig. 2. Plots of the change in phage and host concentration as a function of time under the conditions described in Fig. 1. A) 108/106 #PFU =#CFU , B) 107/108 #PFU = #CFU , and C) 106/ 107 #PFU =#CFU. Note that the E. coli concentration in figures A thru C are the sum of both infected and uninfected hosts (B þ I).
agreement between the model predictions and experimental outcomes. Fig. 2A shows strong agreement between the observed and predicted changes in both phage and host concentrations, corresponding to the conditions in Fig. 1C. For the two systems where MOI is less than 1 (plots B and C) characteristics of the population balances become more evident. For example, Fig. 2B shows a sharp inflection in the phage concentration after about 0.5 h before a sharp increase. This is indicative of the loss of free phage as a result of host attachment where the population only begins to increase again once infected hosts begin to lyse and release more free phages. This inflection is not as dramatically seen in Fig. 2C, likely resulting from the lower initial free phage concentration. Fig. 2A does not show any significant inflection as the initial phage concentration was higher than the initial host concentration. All three models reasonably predict the final host and phage concentrations. This is an improvement over the previous modeling work done by Worley-Morse and Gunsch (2015) especially with regards to the changes in host concentration under a range of initial conditions. This is seen most clearly in the predicted final value of the host concentration. This was done by providing a predictive approach to representing changes in host and phage concentrations from both theory and experimentation while also avoiding data fitting. Compared to other models this effort highlights the ability of particle aggregation theory to reasonably predict the rate of phagemediated disinfection utilizing only a few assumptions and without the employment of fitting parameters. This could have significant design implications on water treatment, especially regarding the potential of phage-based biocontrol efforts in the activated sludge process.
For different combinations of phage and host systems one would expect the physical parameters likely to change most would be that of burst size and lysis rate. To examine model sensitivity, the simulated disinfection rates were plotted against the experimentally observed rates to determine an R2 value. These R2 values were collected for several different values for different model parameters. The model appeared to be most sensitive to changes in burst size, Y. Fig. 3 shows the changes in R2 as a function of different values of Y, ranging from 0 (no bursting) to 100 #PFU #1 CFU. The conditions of this figure are the same as those provided in Fig. 1 while allowing only the burst size to vary. Interestingly, the model’s sensitivity to changes in burst size was largely confined to the lowest MOI condition. As MOI increased the R2 values remained more consistent. Model sensitivity toward changes in lysis rate, kL, was examined for values ranging from 0 to 4 h1. It was similarly observed that for the MOI ¼ 0.1 condition that the R2 varied most dramatically, with values ranging from 0.24 to 0.73, peaking slightly at a kL of ~1 h1. However, for the other MOI conditions the R2 stabilized at ~0.95 (except when kL ¼ 0 h1, no lysis). Phage inactivation rate had the least effect on model stability, where again only for the MOI ¼ 0.1 condition was the R2 seen to vary (between 0.68 and 0.81 for values of 0e2 h1). This analysis suggests that the model’s sensitivity is a function of the ratio of the phage and its host. This makes sense given that at higher and higher relative phage concentrations factors such as increased burst size and lysis rates have less and less effect on the uncertainty of the rate of inactivation. Therefore, this analysis corroborates the previous observations that this modeling approach is best suited under conditions of initial MOI of 1 or greater at the tested
Fig. 3. Change in R2 as a function of burst size for three different MOIs, with an initial host concentration of 107 #CFU mL1.
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concentrations and has the ability to predict the inactivation rate with minimal data fitting. 5. Conclusions Classical Smoluchowski (1917) particle aggregation theory has shown itself to be a useful tool in the construction of phageinduced disinfection models. In summary, this approach is most beneficial under conditions where the initial system MOI is 1 or greater, granted that both phage and host concentrations are no less than 106 colony or plaque forming units per milliliter. This is mostly due to the model’s ability to capture both a pseudo-firstorder and second-order kinetic behavior depending upon the relative phage and host concentrations using minimal fitting factors. While future work will be needed to test this model’s applicability to other phage-host systems which may behave differently under similar conditions, the results presented here suggest that particle aggregation kinetics could provide a stronger foundation for future modeling work. More significantly, this effort also demonstrates the ability of some complex biological interactions to be reimagined within the context of particle-particle interactions which directly influence the health of environmental systems. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The authors would like to thank Dr. Thomas O. Worley-Morse, for contributing his data to this work. This work was also partially funded through the Center for the Environmental Implications of Nanotechnology (CEINT) under NSF Cooperative Agreement Number EF-0830093. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.watres.2019.115438. References Abedon, S.T., 2009. Kinetics of phage-mediated biocontrol of bacteria. Foodborne Pathogens and Disease 6 (7), 807e815. https://doi.org/10.1089/fpd.2008.0242.
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