Investigation of Bacteriophage MS2 Viral Dynamics Using Model DiscriminationAnalysis and the Implications for Phage Therapy

Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava*

Department of Chemical, Materials and Biomolecular Engineering, University of Connecticut, Storrs, Connecticut 06269

Lytic phages infect their bacterial hosts, use the host machinery to replicate, and finally lyseand kill their hosts, releasing progeny phages. Various mathematical models have been developedthat describe these phage-host viral dynamics. The aim of this study was to determine whichof these models best describes the viral dynamics of lytic RNA phage MS2 and its hostEscherichia coliC-3000. Experimental data consisted of uninfected and infected bacterial celldensities, free phage density, and substrate concentration. Parameters of various models wereeither determined directly through other experimental techniques or estimated using regressionanalysis of the experimental data. The models were evaluated using a Bayesian-based modeldiscrimination technique. Through model discrimination it was shown that phage-resistant cellsinhibited the growth of phage population. It was also shown that the uninfected bacterialpopulation was a quasispecies consisting of phage-sensitive and phage-resistant bacterial cells.When there was a phage attack the phage-sensitive cells died out and the phage-resistant cellswere selected for and became the dominant strain of the bacterial population.

Introduction

Lytic phages have been applied in therapeutic treatment ofbacterial infections since the 1930s (1). Today, phage therapyis experiencing renewed interest due to its potential applicationagainst antibiotic-resistant bacteria (2-8). The main character-istic of lytic bacteriophages that make them viable in treatmentof bacterial infections is the death of their hosts at the end ofthe infection cycle instead of the integration of the viral genomewith the host genome, as is the case of lysogenic phages. Theinfection process starts with the adsorption of the phage on thehost receptor and the introduction of the genome into thebacteria. There follows a period where the phage genome isreplicated and the capsule components are being made usingthe host machinery. Finally, the genome is encapsulated andlysis of the bacteria releases the progeny phage that infect newhosts (9).

Mathematical models describing the phage-host viral dynam-ics have been instrumental in understanding the theoretical basisof phage therapy (10, 11). Various mathematical models havebeen developed at both intracellular and intercellular levels.Intracellular models include those of bacteriophagesλ (12), T7(13, 14), T4 (15), and Qâ (16). At the intercellular level, whichis the focus of this study, the basic model describing phage-bacterial viral dynamics was presented by Levin et al. in 1977.Thereafter, many studies have modified Levin’s model andapplied it to different phage-host systems (10, 11, 17-20).Some of these models have also incorporated the phage resist-ance aspect (18, 19, 21). Analogous to the observance ofpenicillin-resistant mutants soon after the discovery of penicillin(22), phage-resistant bacterial mutants were also observed asearly as in the 1930s (23). Since then, phage resistance has beenstudied extensively (19, 24) and is predominantly due to muta-

tions in the host, resulting in the alteration or loss of receptorsto which the bacteriophages adsorb. Although there are otherproblems associated with phage therapy, such as the hostimmune response to the phages themselves (25, 26) and deliveryissues of the phages to the area of infection (26), the focus ofthis study has primarily been bacterial resistance to phages.

Although all of the above-mentioned intercellular models aredifferent, the basic mechanisms involved are the same. The rateat which the phage-sensitive bacterial population declines, theresistant bacterial population increases, and the phage populationincreases depends primarily on five parameters and fivevariables. The parameters are the adsorption rate of the phageto the phage-sensitive bacteria, the burst size or the number ofprogeny that are produced from a single cell, the latent periodwhich is the time between adsorption and burst, and themaximum rate of bacterial growth of the sensitive and resistantpopulations. The variables are the density of the sensitive, theinfected and the resistant bacteria, as well as the phage densityand the concentration of substrate available for bacterial growth.

As can be seen in this study, predictions defining thedynamics of a phage-host system of interest can be obtainedby the application of any of the aforementioned models. Mostof the models do not give a precise prediction of all variablesinvolved. For example, some models will accurately predictphage density but not cell densities, whereas other models willaccurately predict cell densities but not phage density. To findout which model among these models best captures the biologyof the phage-host system, one may use a model discriminationtechnique. Such a technique was developed by Stewart andSorenson (27) based on Bayes Theorem. The method has beendescribed and utilized in several previous works since then (27-31). It should be noted that the model discrimination techniqueis a model comparison tool rather than a model formulatingtool. This method points out the most probable model among agroup of models that best describes the experimental data.

* To whom correspondence should be addressed. Tel:+1-860-486-2802.Fax: +1-860-486-2959. E-mail: srivasta@engr.uconn.edu.

1650 Biotechnol. Prog. 2006, 22, 1650−1658

10.1021/bp060161s CCC: $33.50 © 2006 American Chemical Society and American Institute of Chemical EngineersPublished on Web 11/10/2006

In the present work, we have studied the viral dynamics ofthe bacteriophage MS2 and its hostEscherichia coliC-3000 inminimal media. MS2 is a lytic RNA phage that was among thefirst RNA phages to be discovered (32). It belongs to theLeViViridae family and infects F+ E. coli cells. Experimentswere carried out in both batch and continuous culture settingsas necessary. Uninfected and infected bacterial densities, freephage density, and substrate concentrations were measured atdifferent time points. Throughout this work, free phage densityrefers to the phages that had not yet infected their hosts; it doesnot include the phages that had already infected their hosts. Avariety of different models, each of which represented a differenthypothesis, were applied to the viral dynamics of MS2/C-3000.Some of the parameters, such as the burst size and latent period,were determined directly from experiments. Other parameterswere estimated using regression analysis of the experimentaldata. The models were then evaluated using the model dis-crimination method to find out which model was the mostprobable for describing MS2/C-3000 viral dynamics.

Materials and Methods

Bacterial Strain, Phage, and Culture Condition.Escheri-chia coli C-3000 (ATCC 15597) was grown in minimal M9medium supplemented with 2 mM CaCl2 and 10 µg/mLthiamine overnight at 37°C in shake flasks in a C24 IncubatorShaker (New Brunswick Scientific, Edison, NJ) at 250 rpm.Next, 57 mL of fresh media was inoculated with 3 mL of theovernight culture in a shake flask and grown at 37°C in theincubator/shaker at 250 rpm. This culture was allowed to reachan OD600 of 0.8 and then was infected with MS2 phage (ATCC15597-B1) at an MOI of 0.5. OD600 measurements were carriedout using a Biomate 3 spectrophotometer (Thermo Spectronic,Rochester, NY). The combined total of uninfected and infectedcells densities were quantified by the OD600 readings. Glucoseconcentration was measured using a YSI 2700 Select Biochemi-cal Analyzer (YSI, Yellow Springs, OH). Two types of plaqueassays were carried out: one to measure the combined sum ofinfected cells and free phage densities and the other to measurefree phage density alone. All data points obtained were theaverage of at least three replicates.

Quantification of Combined Total of Infected Cells andFree Phage Densities.A sample of the culture was diluted sothat its plaque titer was between 300 and 3000 PFU/mL. Next,100 µL of this dilution was mixed with 3 mL of E medium(also called EM and having a final concentration of 10 g/L oftryptone, 1 g/L of yeast extract and 8 g/L of NaCl) soft agarand 100µL of the plating culture (an overnight culture ofE.coli C-3000). This mixture was poured on prewarmed EM platesand incubated overnight at 37°C. Plaques were counted thenext day.

Quantification of Free Phage Density.The method used toquantify free phage was similar to the technique used forquantifying infected cells and free phage together. The onlydifferences were the use ofE. coli C-3000+pBAD102 (ampi-cillin-resistant) as the plating culture and the addition of 100µg/mL ampicillin to EM soft agar and EM plates. The presenceof ampicillin did not allow the infected cells to grow as theywere ampicillin-sensitive. As a result, the plaques that wereformed were due to only the free phage.

Transformation of C-3000 with pBAD102. The poly-ethylene glycol (PEG)-mediated colony transformation protocolwas used to transformE. coli C-3000 with pBAD102 (Invitro-gen) making the cells ampicillin resistant. In brief, cells weresuspended in a 0.1 M CaCl2 and 10 mM PEG 8000 (pH 8.0)

mixture, which was then placed on ice for 5 min. DNA wasadded to the cell suspension, and the mixture was again placedon ice for 8 min. The mixture was heat shocked at 42°C for30 s and placed back on ice for an additional 2 min. SOC wasadded, and the mixture was plated and incubated at 37°C for20 h.

Mathematical Modeling of MS2/C-3000 Viral Dynamicsin Minimal M9 Medium. Predictions of several models werecompared with the experimental data of MS2/C-3000 viraldynamics. The key features of the models are shown in Table1. It should be noted in all models that parameters representingthe same phenomena as in Model 1 were given identical nota-tions. This method gave a better frame of reference to comparedifferent models. For example,k1 in all models represented theadsorption constant, although the actual value ofk1 differed frommodel to model as a result of the influence of other properties.

Model 1 was an adaptation of the original model suggestedby Levin et al. (21). This model segregated the bacterial cellsinto the uninfected type and the infected type, where the infectedbacterial cells did not grow. Increase in bacterial density wasdue to the growth of uninfected cells only.X represented theuninfected bacterial density,Y the infected bacterial density,Pthe free phage density, andS the substrate concentration. Anumber of assumptions were required for Model 1. First, it wasassumed that all uninfected bacterial cells were sensitive tophage infection. Next, it was assumed that the infected bacterialcells did not grow and their consumption of substrate wasnegligible. Finally, the loss of infected cells was assumed to bedue to only phage maturation initiated lysis and not due tonormal death of bacterial cells. Monod’s equation (33) relatedthe substrate concentration and the growth rate of the bacterialpopulation. The model was as follows:

where µX was the growth rate of the uninfected cells,k1

represented the rate of infection,k2 was the death rate ofuninfected cells,k3 was the rate of lysis of infected cells,k4

was the rate at which the progeny phage was produced,k5 wasthe rate at which the free phage particles were degraded,YX/S

was the yield factor relating production of uninfected bacterialcells to substrate consumed,µmax,X was the maximal growth

Table 1. Key Features of Models Describing the Viral Dynamics ofBacteriophage MS2 and Its HostEscherichiacoli C-3000 in MinimalM9 Medium

Model key features

1 only uninfected cells grow2 uninfected and infected cells grow at the same rate3 uninfected and infected cells grow at different rates4 only uninfected cells grow; phage resistant cells are present5 phage resistant cells are present; only uninfected cells

grow; no infected cells lyse in the first 60 min

dXdt

) µXX - k1PX - k2X (1)

dYdt

) k1PX - k3Y (2)

dPdt

) k4Y - k1PX - k5P (3)

dSdt

)-µXX

YX/S(4)

µX )µmax,XS

KS,X + S(5)

Biotechnol. Prog., 2006, Vol. 22, No. 6 1651

rate of the uninfected cells, andKS,X was the Monod constantfor uninfected cells.

Model 2 was a modification of Model 1. The suggestion byPayne et al. (11) that infected bacterial cells were growing andthe rate of growth was the same as that of the uninfectedbacterial cells was incorporated into Model 2. A growth termwas added in eq 2 that reflected the growth of infected cells,and a loss term was added in eq 4 showing the consumption ofsubstrate by the infected cells:

An obvious alteration to Model 2 would be a different growthrate of infected cells as compared to that of the uninfected cells.This modification was incorporated in Model 3 by changingthe growth rate of the growth term in eq 7. As was the casewith the uninfected cells, the utilization of substrate by theinfected cells was also modeled using Monod’s equation:

whereµY was the growth rate of infected cells,µmax,Y was themaximal growth rate of infected cells,KS,Y was the Monodconstant for infected cells, andYY/S was the yield factor relatingproduction of infected bacterial cells to substrate consumed.

Lenski’s suggestion (19) of the presence of phage-resistantbacterial cells was incorporated into Model 4. As in all previousmodels,X still represented the uninfected bacterial populationdensity. Unlike the previous models, the uninfected bacterialpopulation now comprised two subpopulations, the phage-sensitive bacterial population and the phage-resistant bacterialpopulation. The phage-sensitive bacterial population density wasrepresented byZ, and the phage-resistant bacterial populationdensity was represented byR. Both populations had differentgrowth rates that were associated with the substrate concentra-tion by Monod’s equation. In addition, the new differential

equation forR only had a single source term due to cellulargrowth and one loss term due to the death of the cells. As phage-resistant bacteria cannot be infected, no additional loss termswere necessary.

wherek6 was the death rate of resistant cells and the resistantcell growth parameters wereµR, the growth rate,µmax,R, themaximal growth rate, andKS,R, the Monod constant. Thepercentage of phage-sensitive cells initially present in theuninfected bacterial population was also considered as aparameter to be determined.

Model 5 was a variation of Model 4 that included an alterationof the lysis rate to deal with the incorporation of the latentperiod. The lysis rate was multiplied by a unit step functionthat had a value of zero for the first 60 min, corresponding tothe latent period of the cells that were initially infected.Thereafter it had a value of 1. As a result, the loss term in theinfected cells equation and the source term in the free phageequation were effectively zero for the first 60 min:

dXdt

) µXX - k1PX - k2X (6)

dYdt

) µXY + k1PX - k3Y (7)

dPdt

) k4Y - k1PX - k5P (8)

dSdt

)-µX(X + Y)

YX/S(9)

µX )µmax,XS

KS,X + S(10)

dXdt

) µXX - k1PX - k2X (11)

dYdt

) µYY + k1PX - k3Y (12)

dPdt

) k4Y - k1PX - k5P (13)

dSdt

)-µXX

YX/S-

µYY

YY/S(14)

µX )µmax,XS

KS,XS(15)

µY )µmax,YS

KS,Y + S(16)

dZdt

) µZZ - k1PZ - k2Z (17)

dRdt

) µRR - k6R (18)

dYdt

) k1PZ - k3Y (19)

dPdt

) k4Y - k1PZ - k5P (20)

dSdt

)-µZZ

YZ/S-

µRR

YR/S(21)

µZ )µmax,ZS

KS,Z + S(22)

µR )µmax,RS

KS,R+ S(23)

X ) Z + R (24)

dZdt

) µZZ - k1PZ - k2Z (25)

dRdt

) µRR - k6R (26)

dYdt

) k1PZ - k3YU (27)

dPdt

) k4YU - k1PZ - k5P (28)

U ) 0 for t e latent period,) 1 for t > latent period (29)

dSdt

)-µZZ

YZ/S-

µRR

YR/S(30)

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Parameter Estimation. Parameters such as the burst size,latent period,KS,X, µmax,X, YX/S, andk5 were determined directlyfrom experiments. The remaining parameters were determinedvia regression of experimental data.

Determination of Burst Size and Latent Period.One-stepgrowth experiments (34) were conducted to determine the burstsize and the latent period. Exponentially growingE. coli C-3000cells were infected with MS2 phage. The phages were allowedto attach to the bacterial hosts for 5 min. This culture was thendiluted one million fold to avoid secondary infections later inthe experiment. Every 10 min, 100µL of the culture wascollected and mixed with 3 mL of EM soft agar and 100µL ofthe plating culture (an overnight culture ofE. coli C-3000),which was then poured over prewarmed EM plates. These plateswere then incubated overnight at 37°C, and the plaques werecounted the next day. A plot of the log of titer value versustime revealed the burst size and the latent period.

Determination of Monod’s Constant for Uninfected Cells(KS,X). To determineKS,X, E. coli C-3000 in minimal M9 mediawas grown in a chemostat. The chemostat was an airtight flaskwith provisions for incoming filtered air and an inlet and outletfor the flow of minimal M9 media. The reactor was immersedin a water bath to maintain the contents of the chemostat at37 °C. Preheated (to 37°C) sterile minimal M9 medium waspumped into the chemostat from a reservoir using a 323S/MC4peristaltic pump (Watson Marlow Bredel Pumps, Wilmington,MA). The outlet stream flow was due to the hydrostatic pressurehead created by the difference in the positions of the inlet andthe outlet. The position of the outlet was changed according tothe incoming flow rate so that they were both the same and thevolume of the culture in the chemostat remained constant at250 mL. Glucose concentration of the output stream wasmeasured at steady state for each of the flow rates. The growthrate was determined on the basis of the fact that the growthrate is equal to the flow rate at steady state (33). KS,X was cal-culated from a plot of the growth rate versus glucose concentra-tion that followed Monod growth kinetics (35). The flow ratewas varied from 0.6 to 1.4 mL/min in steps of 0.1 mL/min.Washout was observed at 1.2 mL/min.

Determination of Maximal Growth Rate (µmax,X) and YieldFactor (YX/S) of Uninfected Cells.A 71 mL portion of freshminimal M9 medium was inoculated with 4 mL of an overnightculture ofE. coli C-3000. OD600 and glucose concentration weremeasured every 30 min. A plot of ln(OD600) versus time wasprepared, and the slope of the line in the log phase was used asthe effectiveµmax. Also, a correlation of OD600 to the numberof cells was made using a Petroff-Hausser counting chamber.Dividing the increase in the number of cells by the loss inglucose concentration every 30 min gave a value ofYX/S forthat time period. The average value ofYX/S over the entireexperimental time period was taken as the effectiveYX/S.

Determination of Rate of Degradation of Free PhageParticles (k5). A 57 mL portion of fresh minimal M9 mediawas mixed with MS2 phage at a final concentration of 200 PFU/mL. Plaque assays, as described above, were done every 30 min

to keep track of the phage concentration. The value ofk5 wasequal to the rate at which the phage concentration decreasedover time.

Parameter Estimation by Nonlinear Regression Analysis. Theremainder of the parameters were estimated by fitting the modelsto the experimental data and using nonlinear regression tech-niques available with the Berkeley Madonna software developedby Robert Macey and George Oster of the University ofCalifornia at Berkeley (http://www.berkeleymadonna.com).

Model Discrimination. Model discrimination was carried outusing Stewart’s method (31). All models were assigned an equalprior probability. The posterior probability of each model wascalculated on the basis of the experimental data. Calculation ofthe posterior probability of thejth model was based on theproportionality equation developed by Stewart et al. (30, 31):

whereMj was modelj, p(Mj) was the prior probability ofMj, Ywas the matrix of weighted experimental data,pj was the numberof estimated parameters by regression analysis in modelj, andVe was the number of degrees of freedom. The term 2-pj /2 is afunction that naturally arises in the derivation of eq 34 (31). Iteffectively penalizes models having too many parameters andover-fitting the data.

Vj was the matrix of the products of the deviation of the datafrom the predicted value for modelj, evaluated at the maximumlikelihood of the parameter vectorθj. The ikth element ofVj

was calculated by

whereF was the weighted value predicted by the model, whichwas a function of the vectors of independent variables,êu, andparameters,θj. Weighting in this study was done by multiplyingthe observed value by the inverse of the standard deviation ofthe observed value (31). The subscriptsi and k indicated avariable, andu represented the event or time point at which thedata was collected.

Each posterior probability was normalized to form theposterior probability share,π:

The model with the largestπ was selected as the most probable.The model discrimination method was programmed in

Mathematica(v 5.2).

Results

Experimental Results. Figure 1A shows the changes inuninfected and the infected cell densities with time. Theuninfected cell density increased for the first 110 min, decreased,and then increased again. The infected bacterial cell densityincreased and diminished to undetectable levels quickly, withina time interval of 110 min. Figure 1B shows the change in freephage density that declined by 84% in the first 30 min, thenincreased, and finally leveled out. Small increase in free phagedensity could still be seen toward the end of the experimentaltime period. This could be due to the experimental limitations

µZ )µmax,ZS

KS,Z+ S(31)

µR )µmax,RS

KS,R+ S(32)

X ) Z + R (33)

p(Mj|Y) ∝ p(Mj)2-pj /2|Vj|-Ve/2 (34)

Vik(θj) ) ∑u)1

n

[Yiu - Fji(êu, θj)] [Yku - Fjk(êu, θj)] (35)

π(Mj|Y) )p(Mj|Y)

∑k

p(Mk|Y)(36)

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of measuring low cell densities and free phage density. Figure1C shows the substrate concentration that decreased continu-ously.

OD600 values were used to quantify the total number of cellsthat included both the uninfected and the infected cells. Plaqueassays were used to quantify the infected cells and the freephages. It should be noted that the use of the plaque assay tech-nique only quantifies the viable phage particles and does notinclude the noninfective phage particles. The number of infectedcells subtracted from the total number of cells gave the number

of uninfected cells. Glucose was the only carbon source availablefor bacterial growth in minimal M9 medium.

Parameter Estimation Results. Parameters determined experi-mentally are shown in Table 2. Based on the growth curve ofC-3000 (data not shown) in minimal M9 medium and in theabsence of MS2 phage, the uninfected bacterial population didnot enter the death phase during the experimental time interval.Therefore, the value ofk2 was assumed to be negligible.k3, therate of lysis was the reciprocal of the latent period (11, 17),which was found to be 60( 2 min (n ) 5). k4, the rate atwhich the progeny phage is produced, was equal to the burstsize multiplied by the lysis rate (11, 17). The burst size wasdetermined to be 10( 1 PFU/cell (n ) 5). Also,k5, the rate atwhich the free phage particles are degraded during the experi-mental time interval, was found to be zero (data not shown).The death rate of the resistant cells during the experimental timeperiod was also assumed to be zero.YX/S was found to be 2.4× 1013 cells/g of glucose. Maximal growth rate of uninfectedbacterial cells was found to be 0.0113 min-1 and that of phage-resistant cells was 0.0080 min-1. KS,X, Monod’s constant foruninfected cells was found to be 0.0069 mM glucose.

Parameters estimated by the nonlinear regression techniqueare shown in Table 3. In addition, the percentage of phage-sensitive cells initially present in the uninfected bacterialpopulation in Model 4 was found to be 61( 5% and in Model5 was 65( 5%. An important point to be noted here is that forModels 1, 2, and 3,X was fitted to the experimental data ofuninfected cells, whereas for Model 4 and 5, (Z + R ) X) wasfitted to the experimental data of uninfected cells. Also, inModels 4 and 5, the maximal growth rate of the uninfected cells(0.0113 min-1) was the weighted average of the maximal growthrates of the phage-sensitive and phage-resistant bacterial cells.

The model predictions in comparison with the experimentaldata are shown in Figure 2. Visual inspection showed thatModels 4 and 5 gave a good fit to the uninfected cells data, butit was difficult to ascertain which model best fitted the infectedcells and the free phage data. In fact, all models gave poorpredictions of the infected cell density. Also, in the case ofsubstrate concentration, all models gave good predictions incomparison to the experimental data. Therefore, it was difficultto conclude which model among these five models best repre-sented the experimental data. It was here that model discrimina-tion was utilized to compare the models and determine whichwas the most accurate among all of the models.

Evaluation of Models by Model Discrimination Method.All models were evaluated by the model discrimination methodto determine which model best described the experimental data.The posterior probability share of each model was calculatedand is shown in Table 4. Models 4 and 5 were highly favoredover other models. Model 5, having a posterior probability shareof 58%, fared better than Model 4, which had a posterior proba-bility share of 42%.

Figure 1. Experimental data for MS2/C-3000 viral dynamics inminimal M9 medium. (A) Uninfected bacterial cell density (X)increases, decreases, and then increases again. Infected bacterial celldensity (Y) increases and diminishes quickly within a short time interval.(B) Free phage density (P) decreases initially, then increases, and finallylevels out. (C) Glucose concentration (S) decreases continuously. Errorbars represent standard errors of at least three replicates.

Table 2. Parameters Determined Directly from Experimentsa

parameter parameter value

k2 0 min-1

k3 0.0167( 0.0006 min-1

k4 0.1667( 0.0098 PFU/cell‚mink5 0 min-1

k6 0 min-1

µmax,X 0.0113( 0.0003 min-1

µmax,R 0.0080( 0.0003 min-1

KS,X 0.0069 mM glucoseYX/S 2.4× 1013 ( 2.3× 1012cells/g of glucose

a Errors represent standard errors of at least three replicates.

1654 Biotechnol. Prog., 2006, Vol. 22, No. 6

Therefore, Model 5 was selected as the most probable modelamong the five models for describing MS2/C-3000 viraldynamics. Two major conclusions from this result were thatthe infected cells did not grow and the phage-resistant cellsplayed a very significant role in MS2/C-3000 dynamics. Model5 predicted that the bacterial population, even prior to a phageattack, was a heterogeneous population or a quasispeciesconsisting of phage-sensitive and phage-resistant cells. Whenthis quasispecies was infected by phages, the resistant cells were

selected for and dominated the surviving uninfected bacterialcell population (Figure 3).

Several other modifications to the above models were con-sidered in this study. Growth of infected cells was included inModels 4 and 5. Fitting these models to the experimental datagave a negligible value to the growth rate parameter of theinfected cells. Also, mutational changes leading to the conver-sion of phage-sensitive cells to resistant cells was included inModels 4 and 5. Fitting these models to the experimental dataas well gave a very small value to the mutational change param-eter. As these modifications did not result in significant changesin the model predictions, they were not considered for the finalanalysis.

Discussion

Phage Resistance.Bacteria become resistant to phages eitherby producing defective pili, pili at lower densities, or no pili at

Figure 2. Model predictions in comparison with the experimental data for (A) uninfected cell density, (B) infected cell density, (C) free phagedensity, and (D) substrate concentration (glucose concentration). Error bars represent standard errors of at least 3 replicates. Key: (- - -) Model 1,(- -) Model 2, (- - -, black) Model 3, (- - -) Model 4, (- - -, gray) Model 5, (b) expt.

Table 3. Parameters Estimated by a Nonlinear Regression Techniquea

parameter Model 1 Model 2 Model 3 Model 4 Model 5

-k1 (min × phageparticles/mL)-1

2.95× 10-13

( 2.02× 10-142.34× 10-12

( 5.51× 10-135.30× 10-13

( 1.43× 10-133.89× 10-13

( 2.00× 10-146.10× 10-13

( 1.14× 10-13

µmax,Y (min-1) 0.021( 0.003KS,Y (mM) 1.83( 0.71YY/S (cells/g of glucose) 1.01× 1014

( 2.75× 1013

KS,R (mM) 0.45( 0.05 0.009( 0.002YR/S (cells/g of glucose) 6.54× 1013

( 1.67× 10137.59× 1013

( 2.76× 1013

a Errors represent the standard deviations.

Table 4. Posterior Probability Share of Each Model

Model posterior probability share (%)

1 0.00002 0.00003 0.00014 41.97335 58.0266

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all (19, 36, 37). These effects are due to mutations in the hostgenome. Bacterial mutants resistant to phage infection eithercan arise when there is a phage attack or can be present priorto the phage attack (38, 39) and then be selected for when thereis one. We have modeled both cases, the latter being Model 4or 5. The former case, that of generation of mutants after phageattack, was the same as Model 4 or 5 with additions of a lossterm (k7Z) in the equation of the phage-sensitive uninfected cellsand a source term in the equation of the resistant cells, wherek7 was the mutational rate.

Parameter estimation gave a very small value to the muta-tional rate in corroboration with previous studies (19). This resultfavors the latter case, i.e., the presence of a quasispeciesconsisting of a phage-resistant bacterial strain and a phage-sensitive bacterial strain. The mutant strain resistant to phageinfection is selected for and becomes the dominant strain inthe face of a phage attack. In the absence of phages, the resistantcells would compete with the sensitive cells for substrate. Theresistant cells having a growth rate (0.008 min-1) lower thanthat of the sensitive cells (0.0134 min-1) would be out-competed,and therefore the sensitive cells form the larger portion of thebacterial population. This is in agreement with the modelprediction of phage-sensitive cells making up 65% of the totalbacterial population before the phage attack.

In the event of a phage attack, the resistant population wouldbe selected for and would become the dominant strain in theuninfected bacterial population (Figure 3). This would suggestthat the phage population would eventually die out, but naturehas its ways of getting around this problem. Though not seenin the current study, the phages could survive by mutatingthemselves and adapting to the resistant bacterial population(19, 40, 41).

Adsorption of MS2 to Its Host E. coli C-3000.The infectionprocess of a host bacterial cell starts with the adsorption of thephage to the bacterial surface. The phage adsorption organellebinds to a highly specific receptor site on the bacterial surface.In the case of MS2 phage, it adsorbs specifically onto F-pili.F-pili are extracellular filaments expressed by F+ strains ofE. coli. They are required in early stages of conjugal DNAtransfer to establish specific and secure cell-cell contacts (42).In comparison with the other kinds of filaments, which arepresent at high multiplicities, only one or a few F-pili are presenton an individual cell (19, 43). Also, the occurrence of F-pili onthe bacterial surface ofE. coli changes under different growth

conditions (43-45). The concentration of F-pili in minimalmedium is lower than in rich medium (43). All of the abovestated reasons could explain the very low value of the adsorptionconstant in this study as compared to other phage-host systems(21, 46). This value is in concordance with Lerner’s experi-mental study of MS2 phage (47).

Treatment of Latent Period. The latent period is the periodbetween the instant of infection and that of lysis, during whichthe phage “reproduces” inside the bacterial cell. There is norelease of progeny during this period. Various studies haveincorporated this effect into their models in different ways. Levinet al. have ignored the latent period (48), whereas other studies(11, 17) and this study (Models 1-4) include this idea byassigning the value of the lysis rate to be the reciprocal of thelatent period.

Experimental observations of MS2/C-3000 viral dynamicsshowed that free phage concentration decreased by 84% in thefirst 30 min and after that there was a continuous increase thatfinally leveled out (Figure 1B). A possible explanation for theseobservations could be that when phages are introduced in abacterial host population, some but not all of the phages willinfect their hosts at time zero. From then on, a continuous, slowinfection process will take place, as corroborated by the factthat the adsorption rate (10-13 min phage particles/mL) was verylow. The cells infected at time zero would be the “first” cellsto lyse, after which there would be a continuous lysis of hosts.As the latent period has been found to be 60 min, the cells thatwere infected at time zero would lyse at the 60th minute. Inother words, there would be no lysis of infected cells between0 and 60 min. Also, between 0 and 60 min, the free phageswould only be consumed as part of the infection process. Nonew free phages would be produced in this period as there wouldbe no cell lysis.

In contrast to the experimental observations, Models 1-4predicted a continuous increase of free phage density in the0-60 min time interval. This was due to the use of differentialequations. Differential equations generally describe continuousfunctions. As a result, the loss term in the infected cell equationand the source term in the free phage equation were active evenin the 0-60 min time interval. Consequently, lysis of infectedcells was continuously taking place in the model, leading to acontinuous increase in the free phage density.

The continuous burst problem was overcome in Model 5 bymaking a simple modification to the lysis rate. The lysis rate(reciprocal of the latent period) was multiplied by the unit stepfunction. This function had a value of zero for the first 60 min,and thereafter it had a value of 1. As a result, the loss term inthe infected cells equation and the source term in the free phageequation were zero in the 0-60 min time interval.

Other hypotheses have been proposed in favor of a continuousburst of infected cells. One such proposal is the presence ofphages that have shorter latent periods as compared to theaverage phage population behavior (49). However, one shouldnote that the key word here is “average.” The latent period itselfis an “average” property, a property of a typical cell in thebacterial population. In fact, deterministic modeling as suchdescribes an average population behavior.

Model Discrimination. The model discrimination methodis a model comparison tool rather than a model formulatingtool. This method points out the most probable model amongan assortment of models in describing the experimental data.The most probable model does not necessarily mean that it isthe best model. The selected model would just be the best amongthe group of models that it is competing with. As can be seen

Figure 3. Uninfected bacterial cell densities predicted by Model 5.Z represents the phage-sensitive cells,R the phage-resistant cells, andX the total uninfected cell population (X ) Z + R). After the phage-sensitive cells die out, the phage-resistant cells are selected for andbecome the dominant strain.

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in this study, Model 5 was chosen as the most probable model,but one could imagine better results for capturing infected celldynamics.

Better predictions could be obtained by improving mathe-matical modeling techniques, such as the use of delay differentialequations (50, 51). Delay differential equations incorporate atime-dependent segregation of infected cells. However, accurateparameter estimation for models using delay differential equa-tions required experimental data at more frequent time points.The physical constraints of the present experimental techniquesrestricted us from sampling at more frequent time points. Devel-opment of experimental techniques to overcome the frequentsampling constraints would further enhance the predictions ofmathematical models. Furthermore, improving experimentaltechniques that would be capable of detecting lower numbersof cells and phages would be an added refinement.

Through model discrimination, we have shown that phage-resistant cells play a very important role in MS2/C-3000 dynam-ics. These resistant cells are present in substantial numbers evenin the absence of phages and are part of the quasispecies of thebacterial population. Not surprisingly, when a phage infectiondoes take place, these phage-resistant cells are selected for anddominate the bacterial population. The implications for phagetherapy are profound. If such therapy is ever to be successful,establishment of phage-host dynamics where the host is aquasispecies will be vital. Only then will it be possible toconsider the development and optimization of efficacioustreatment strategies for bacterial pathogens.

Acknowledgment

The authors would like to acknowledge Pfizer and theConnecticut Business and Industry Association for supportingthis research.

References and Notes

(1) Summers, W. C. Bacteriophage Research: early history. InBacteriophages: Biology and Applications; Kutter, E., Sulakvelidze,A., Eds.; CRC Press: Boca Raton, 2005.

(2) Jado, I.; Lopez, R.; Garcia, E.; Fenoll, A.; Casal, J.; Garcia, P.,Phage lytic enzymes as therapy for antibiotic-resistant Streptococcuspneumoniae infection in a murine sepsis model.J. Antimicrob.Chemother.2003,52, (6), 967-973.

(3) Cheng, Q.; Nelson, D.; Zhu, S.; Fischetti, V. A., Removal of groupB streptococci colonizing the vagina and oropharynx of mice witha bacteriophage lytic enzyme.Antimicrob. Agents Chemother.2005,49 (1), 111-117.

(4) Loeffler, J. M.; Nelson, D.; Fischetti, V. A., Rapid killing ofStreptococcus pneumoniae with a bacteriophage cell wall hydrolase.Science2001, 294 (5549), 2170-2172.

(5) Yoong, P.; Schuch, R.; Nelson, D.; Fischetti, V. A., PlyPH, abacteriolytic enzyme with a broad pH range of activity and lyticaction againstBacillus anthracis. J. Bacteriol.2006,188(7), 2711-2714.

(6) Sulakvelidze, A.; Barrow, P. Phage therapy in animals andagribusiness. InBacteriophage: Biology and Applications; Kutter,E., Sulakvelidze, A., Eds.; CRC Press: Boca Raton, 2005.

(7) Sulakvelidze, A.; Kutter, E. Bacteriophage therapy in humans. InBacteriophages: Biology and Applications; Kutter, E., Sulakvelidze,A., Eds.; CRC Press: Boca Raton, 2005.

(8) Schuch, R.; Nelson, D.; Fischetti, V. A. A bacteriolytic agent thatdetects and kills Bacillus anthracis.Nature2002, 418(6900), 884-889.

(9) Ellis, E. L.; Delbruck, M. The growth of Bacteriophage.J. Gen.Physiol.1939, 22 (3), 365-384.

(10) Weld, R. J.; Butts, C.; Heinemann, J. A. Models of phage growthand their applicability to phage therapy.J. Theor. Biol.2004, 227(1), 1-11.

(11) Payne, R. J.; Jansen, V. A. Understanding bacteriophage therapyas a density-dependent kinetic process.J. Theor. Biol.2001, 208(1), 37-48.

(12) Arkin, A.; Ross, J.; McAdams, H. H. Stochastic kinetic analysisof developmental pathway bifurcation in phage lambda-infectedEscherichia colicells.Genetics1998, 149 (4), 1633-1648.

(13) Endy, D.; Kong, D.; Yin, J. Intracellular kinetics of a growingvirus: a genetically structured simulation for bacteriophage T7.Biotechnol. Bioeng.1997, 55 (2), 375-389.

(14) You, L.; Suthers, P. F.; Yin, J. Effects ofEscherichia coliphysiology on growth of phage T7 in vivo and in silico.J. Bacteriol.2002, 184 (7), 1888-1894.

(15) Rabinovitch, A.; Hadas, H.; Einav, M.; Melamed, Z.; Zaritsky,A. Model for bacteriophage T4 development in Escherichia coli.J.Bacteriol.1999, 181 (5), 1677-1683.

(16) Kim, H.; Yin, J. Energy-efficient growth of phage Q Beta inEscherichia coli.Biotechnol. Bioeng.2004, 88 (2), 148-156.

(17) Beretta, E.; Kuang, Y. Modeling and analysis of a marinebacteriophage infection.Math. Biosci.1998, 149 (1), 57-76.

(18) Burroughs, N. J.; Marsh, P.; Wellington, E. M. Mathematicalanalysis of growth and interaction dynamics of streptomycetes anda bacteriophage in soil.Appl. EnViron. Microbiol. 2000, 66 (9),3868-3877.

(19) Lenski, R. E. Dynamics of interactions between bacteria andvirulent bacteriophage.AdV. Microb. Ecol.1988, 10, 1-44.

(20) Weitz, J. S.; Hartman, H.; Levin, S. A. Coevolutionary arms racesbetween bacteria and bacteriophage.Proc. Natl. Acad. Sci. U.S.A.2005, 102 (27), 9535-9540.

(21) Levin, B. R.; Stewart, F. M.; Chao, L. Resource-limited growth,competition and predation: A model and experimental studies withbacteria and bacteriophage.Am. Nat.1977, 111 (977), 3-24.

(22) Livermore, D. M. Antibiotic resistance in staphylococci.Int. J.Antimicrob. Agents2000, 16 Suppl 1, S3-10.

(23) Summers, W. C. Bacteriophage discovered. InFelix d’Herelleand the Origins of Molecular Biology; Yale University Press: NewHaven, 1999; pp 47-59.

(24) Levin, B. R.; Bull, J. J. Population and evolutionary dynamics ofphage therapy.Nat. ReV. Microbiol. 2004, 2 (2), 166-173.

(25) Clark, J. R.; March, J. B. Bacteriophages and biotechnology:vaccines, gene therapy and antibacterials.Trends Biotechnol.2006,24 (5), 212-218.

(26) Sulakvelidze, A.; Alavidze, Z.; Morris, J. G., Jr. Bacteriophagetherapy.Antimicrob. Agents Chemother.2001, 45 (3), 649-659.

(27) Stewart, W.; Sorenson, J. Bayesian estimation of common param-eters from multiresponse data with missing observations.Techno-metrics1981, 23 (2), 131-141.

(28) Knorr, A. L.; Srivastava, R. Evaluation of HIV-1 kinetic modelsusing quantitative discrimination analysis.Bioinformatics2005, 21(8), 1668-1677.

(29) Stewart, W.; Caracatsios, M.; Sorenson, J. Parameter estimationfrom multiresponse data.AIChE J.1992, 38 (5), 641-650.

(30) Stewart, W.; Henson, T.; Box, G. Model discrimination andcriticism with single-response data.AIChE J.1996, 42 (11), 3055-3062.

(31) Stewart, W.; Shon, Y.; Box, G. Discrimination and goodness offit of multiresponse mechanistic models.AIChE J. 1998, 44 (6),1404-1412.

(32) Davis, J. E.; Jr, J. H. S.; Sinsheimer, R. L. InBacteriophageMS2: Another RNA Phage; National Academy of Sciences: La Jolla,1961.

(33) Shuler, M. L.; Kargi, F.Bioprocess Engineering, 2 ed.; PrenticeHall PTR: Upper Saddle River, NJ, 2002.

(34) Eisenstark, A. Bacteriophage techniques. InMethods in Virology;Maramorosch, K., Koprowski, H., Eds.; Academic Press: New York,1967; Vol. 1, pp 492-495.

(35) Bailey, J. E.; Ollis, D. F.Biochemical Engineering Fundamentals,2 ed.; McGraw-Hill: Singapore, 1986.

(36) Manchak, J.; Anthony, K. G.; Frost, L. S. Mutational analysis ofF-pilin reveals domains for pilus assembly, phage infection and DNAtransfer.Mol. Microbiol. 2002, 43 (1), 195-205.

(37) Burke, J. M.; Novotny, C. P.; Fives-Taylor, P. Defective F piliand other characteristics of Flac and HfrEscherichia colimutantsresistant to bacteriophage R17.J. Bacteriol.1979, 140 (2), 525-531.

Biotechnol. Prog., 2006, Vol. 22, No. 6 1657

(38) Luria, S.; Delbruck, M. Mutations of bacteria from virus sensitivityto virus resistance.Genetics1943, 28 (6), 491-511.

(39) Lederberg, J.; Lederberg, E. M. Replica plating and indirectselection of bacterial mutants.J. Bacteriol.1952, 63 (3), 399-406.

(40) Luria, S. Mutations of bacterial viruses affecting their host range.Genetics1945, 30 (1), 84-99.

(41) Hershey, A. Mutation of bacteriophage with respect to type ofplaque.Genetics1946, 31 (6), 620-640.

(42) Silverman, P. M. Towards a structural biology of bacterialconjugation.Molecular Microbiol.1997, 23 (3), 423-429.

(43) Biebricher, C. K.; Duker, E. M. F and type 1 piliation ofEscherichia coli. J. Gen. Microbiol.1984, 130 (4), 951-957.

(44) Stallions, D. R.; Curtiss, R., 3rd. Bacterial conjugation underanaerobic conditions.J. Bacteriol.1972, 111 (1), 294-295.

(45) Novotny, C. P.; Fives-Taylor, P. Effects of high temperature onEscherichia coliF pili. J. Bacteriol.1978, 133 (2), 459-464.

(46) Lenski, R. E.; Levin, B. R. Constraints on the coevolution ofbacteria and virulent phage: A model, some experiments andpredictions for natural communities.Am. Nat.1985, 125 (4), 585-602.

(47) Lerner, F. L. The Population Biology of Male-Specific Bacterio-phages: Existence Conditions. University of Massachusetts, Amherst,1984.

(48) Levin, B. R.; Bull, J. J. Phage therapy revisited: the populationbiology of a bacterial infection and its treatment with bacteriophageand antibiotics.Am. Nat.1996, 147 (6), 881-898.

(49) Adams, M. H.; Wassermann, F. E. Frequency distribution of phagerelease in the one-step growth experiment.Virology 1956, 2 (1),96-108.

(50) Carletti, M. Numerical simulation of a Campbell-like stochasticdelay model for bacteriophage infection.Math. Med. Biol.2006,Epub ahead of print.

(51) Gourley, S. A.; Kuang, Y. A delay reaction-diffusion model ofthe spread of bacteriophage infection.Siam J. Appl. Math.2005, 65(2), 550-566.

Received May 26, 2006. Accepted September 20, 2006.

BP060161S

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