Assessment of the Nonlinear Electrophoretic Migration of Nanoparticles and Bacteriophages

Bacteriophage therapy presents a promising avenue for combating antibiotic-resistant bacterial infections. Yet, challenges exist, particularly, the lack of a straightforward purification pipeline suitable for widespread application to many phage types, as some phages are known to undergo significant titer loss when purified via current techniques. Electrokinetic methods offer a potential solution to this hurdle, with nonlinear electrophoresis emerging as a particularly appealing approach due to its ability to discern both the size and shape of the target phage particles. Presented herein is the electrokinetic characterization of the mobility of nonlinear electrophoresis for two phages (SPN3US and ϕKZ) and three types of polystyrene nanoparticles. The latter served as controls and were selected based on their sizes and surface charge magnitude. Particle tracking velocimetry experiments were conducted to characterize the mobility of all five particles included in this study. The results indicated that the selected nanoparticles effectively replicate the migration behavior of the two phages under electric fields. Further, it was found that there is a significant difference in the nonlinear electrophoretic response of phages and that of host cells, as first characterized in a previous report, illustrating that electrokinetic-based separations are feasible. The findings from this work are the first characterization of the behavior of phages under nonlinear electrophoresis effects and illustrate the potential for the development of electrokinetic-based phage purification techniques that could aid the advancement of bacteriophage therapy.


Introduction
Antibiotics are widely regarded as one of the most important medical developments of the 20th century [1].The discovery of penicillin in 1928 marked the beginning of the antibiotic revolution, which transformed the way infectious diseases were treated worldwide [2].In less than a hundred years since the discovery of penicillin, mankind is in grave danger of being faced yet again with a lack of effective treatment options for bacterial infections.This threat can be attributed to the emergence of antibiotic-resistant bacteria (ARB) as a consequence of the extensive misuse and overuse of existing antibiotics and the limited discovery of novel ones [2][3][4].On top of having a terrible impact on human health, ARB-related infections have a US $20 billion societal annual cost, and if left alone this health crisis could lead to a 2.0-3.5% reduction from the gross domestic product annually by the year 2050 [5].The threat of antibiotic resistance is real and, as such, there is an immediate need for the development of alternative and novel treatment strategies to combat bacterial infections.One such potential alternative treatment option, that predates the discovery and widespread availability of antibiotics, is the use of bacterial viruses, known as bacteriophages or phages, for the treatment of bacterial infections [6,7].The earliest documented report of phages being used as a treatment for a bacteria-caused disease was in 1919 when the French physician Felix d'Herelle used phages to treat dysentery by Although knowledge of EP NL effects is not new, with the earliest mentions in the early 1970s by Dukhin and collaborators [39,40], there are a surprisingly low number of experimental studies on the characterization of EP NL [41].This study presents the first experimental characterization of the migration of phages under the influence of EP NL effects.A series of particle tracking velocimetry (PTV) experiments were performed employing the two distinct phages (SPN3US and ϕKZ) and three types of polystyrene nanoparticles that had size and surface charge characteristics similar to those of the phages.An additional objective of this work was assessing the potential of employing polystyrene nanoparticles as proxies for phages, to be used when testing new separation systems.The results illustrated that the selected nanoparticles behave similarly to the phages under EK forces, and thus, can be employed as proxies for the phages.The PTV experiments allowed for the characterization of the migration of all five particles (two phages and three nanoparticles) in this work under the effects of linear and nonlinear EP effects, which in turn resulted in the quantification of the mobilities of EP L and EP NL velocities.The dependence of the mobility of the EP NL velocity, µ EP,NL , on particle size and shape was studied, where the size and shape of the phages were quantified employing the hydrodynamic diameter and the parameter of the sphericity [42], respectively.The results were consistent with previous reports, in which the magnitude of µ EP,NL increases with particle size [35] and deviations from spherical shape [42,43], as indicated by lower sphericity values.Although the observed trend was subtle, it could be attributed to the limited size range (99-200 nm range) of the studied particles.A major outcome of this work is the identification of the significant difference between the values of µ EP,NL of phages and those of bacterial host cells, which can enable future studies on the continuous separation between phages and host cells by exploiting these differences in µ EP,NL values.These findings can contribute towards the development of novel phage purification protocols based on nonlinear EK phenomena.

Theory
Electrokinetic phenomena are classified according to their dependence on the electric field (E) as either linear or nonlinear.Linear EK phenomena, also called first-kind, have a linear dependence on E and are governed by the permanent surface charge.The linear EK phenomena considered here are EO and linear EP (EP L ).Following the Helmholtz-Smoluchowski Equation, the EO velocity is defined as [44]: where µ EO is the EO mobility, ε m and η represent the media permittivity and viscosity, and ζ W is the zeta potential of the channel wall.Regarding EP L , due to the small size of the nanoparticles and phages studied hereby, Henry's formula [45] will be employed to estimate the mobility of EP L (µ EP,L ).A discussion on the selection of this expression is included in the Supplementary Materials.The expression of the EP L velocity is as follows: where ζ P is the zeta potential of the particle, κ is the inverse of the Debye length (κ = λ −1 D ), and α is the radius of the particle.Further information on these expressions describing EP L is included in the Supplementary Materials.
Nonlinear EK phenomena, also called the second kind, have a nonlinear dependency with E and are a function of the bulk charge.This study considers nonlinear EP (EP NL ), which in contrast to EP L , the mobility of EP NL (µ EP,NL ) does depend on the magnitude of E. Several models describe EP NL ; these models are described by the dimensionless parameters of Peclet (Pe) and Dukhin (Du) numbers as well as the dimensionless field strength β.A description of these three dimensionless parameters is included in the Supplementary Materials.Analytical expressions have been derived to describe the migration of particles under EP NL in the limiting cases of small Pe (Pe ≪ 1) and large Pe (Pe ≫ 1); no analytical expressions exist for intermediate Pe values [46].The expressions for these two limiting cases, illustrating the dependencies with E, for the velocity of EP NL are [46] as follows: Due to the small size of both the polystyrene nanoparticles and phages used in this work, Pe values above 1 were not reached with the current experimental conditions.Thus, for describing EP NL effects, only the µ EP,NL values, which correspond to the moderate field regime (E 3 ), will be reported.Details on the values of the Pe and Du numbers are included in Tables S1 and S2 in the Supplementary Materials for both the weak (E 1 ) and moderate (E 3 ) electric regimes.With this, the overall velocity of a particle under the influence of an electric field in a post-less microchannel, such as the one shown in Figure 1, becomes the following: It is important to note that there are no dielectrophoretic effects present in this system since the electric field has a uniform distribution.The electrokinetic equilibrium condition (E EEC , i.e., the electric field at which v P = 0) was also determined for each particle in this study.This parameter, proposed by Cardenas-Benitez et al. [33], can be used as an additional approach to estimate µ EP,NL values for particles that are under the moderate regime when they reach v P = 0.
Since the phage capsids are non-spherical, and particle shape influences particle migration under EP NL [42,43], it is necessary to quantify particle shape.Following a previous study from our group [42], the shape parameters of sphericity (ψ) was used to assess phage shape.The following is the expression for estimating ψ [47]: A P (7) where V P is the volume of the particle and A P is the surface area of the particle, sphericity varies from 0 to 1, where 1 means a perfect sphere.
E. Several models describe EPNL; these models are described by the dimensionless parameters of Peclet (Pe) and Dukhin (Du) numbers as well as the dimensionless field strength β.A description of these three dimensionless parameters is included in the Supplementary Materials.Analytical expressions have been derived to describe the migration of particles under EPNL in the limiting cases of small Pe (Pe ≪ 1) and large Pe (Pe ≫ 1); no analytical expressions exist for intermediate Pe values [46].The expressions for these two limiting cases, illustrating the dependencies with E, for the velocity of EPNL are [46] as follows: Due to the small size of both the polystyrene nanoparticles and phages used in this work, Pe values above 1 were not reached with the current experimental conditions.Thus, for describing EPNL effects, only the  , ( ) values, which correspond to the moderate field regime ( ), will be reported.Details on the values of the Pe and Du numbers are included in Tables S1 and S2 in the Supplementary Materials for both the weak ( ) and moderate ( ) electric regimes.With this, the overall velocity of a particle under the influence of an electric field in a post-less microchannel, such as the one shown in Figure 1, becomes the following: It is important to note that there are no dielectrophoretic effects present in this system since the electric field has a uniform distribution.The electrokinetic equilibrium condition (EEEC, i.e., the electric field at which  = 0) was also determined for each particle in this study.This parameter, proposed by Cardenas-Benitez et al. [33], can be used as an additional approach to estimate  , ( ) values for particles that are under the moderate regime when they reach  = 0.
Since the phage capsids are non-spherical, and particle shape influences particle migration under EPNL [42,43], it is necessary to quantify particle shape.Following a previous study from our group [42], the shape parameters of sphericity ( ) was used to assess phage shape.The following is the expression for estimating  [47]: where   is the volume of the particle and   is the surface area of the particle, sphericity varies from 0 to 1, where 1 means a perfect sphere.

Materials and Methods
Creation of microdevices.The microchannels used for the PTV experiments were made from polydimethylsiloxane (PDMS, Dow Corning, MI, USA) employing a standard cast-molding technique [48].All microchannels, which featured no insulating posts, had the same dimensions: 10.16 mm in length, 0.88 mm in width, and 40 µm in depth.A schematic of the device employed for PTV experiments is shown in Figure 1.
Suspending medium.The suspending medium employed was 0.2 mM K 2 HPO 4 solution with the addition of 0.05% (v/v) of Tween 20 to avoid particle adhesion to the device surface.This medium had a conductivity of 40.7 ± 4.0 µS/cm and a pH of 7.3 ± 0.2, which resulted in a wall zeta potential (ζ W ) of −60.1 ± 3.7 mV and a µ EO of 4.7 ± 0.3 × 10 −8 m 2 V −1 s −1 in the PDMS channels, as measured with current monitoring experiments [49].In summary, the current monitoring experiment consisted of filling three parallel channel systems with the suspending medium.Platinum wire electrodes were placed at the reservoir tops, and 1000 V of DC voltage was applied to the channels.Initial stable current signals were recorded before changing the solutions in the reservoirs, applying the same potential, and recording the time response until the electric current reached a second plateau.A detailed description of this experimental procedure is included in the Supplementary Materials and in the original publication of this methodology [49].
Nanoparticles.Three distinct types of fluorescent polystyrene nanoparticles (Magsphere Pasadena, CA, USA and ThermoFisher Scientific, Waltham, MA USA) of varying sizes and electrical charges were studied.Their properties are listed in Table 1.Nanoparticle samples were created by diluting the concentrated stock with the suspending medium.The concentration of each particle employed varied depending on size for optimum visualization and varied from 2.8 × 10 9 -9.0 × 10 10 particles/mL as reported in Table S1.Viral samples.High titer stocks (10 10 -10 12 pfu/mL) of two phages were employed in this study: SPN3US infective for Salmonella enterica Typhimurium LT2, and ϕKZ infective for Pseudomonas aeruginosa.To eliminate bacterial debris, all phage stocks underwent a low-speed centrifugation at approximately 8000 rpm for 10 min at 4 • C. The SPN3US samples were then fluorescently labeled using the following procedure: a 1 mL aliquot of phage stock was centrifuged at 13,000 rpm for 10 min.After discarding the supernatant, the resulting pellet was resuspended in 0.5 mL of distilled water.Next, 2 µL of SYTO 11 dye (Invitrogen, Carlsbad, CA, USA) was added to the sample and incubated for 20 min.Excess dye was then removed, and the sample was resuspended in 0.5 mL of the suspending medium.No dye was used for the ϕKZ samples.The properties of the characterized phages are listed in Table 2.The titers for each phage sample employed ranged from 8 × 10 11 -8 × 10 12 pfu/mL as reported in Table S2.Additional characteristics of the phages, including estimates of virion dimensions are shown in Table S3.A discussion on how the hydrodynamic diameter (D H ) of both phages was estimated, along with the equations used to do so, is included in the Supplementary Materials.
Equipment and software.The LabSmith Sequencer software (V1.167) was used to control a high-voltage power supply (Model HVS6000D, LabSmith, Livermore, CA, USA) that applied constant DC voltage sequences to the microchannels using platinum-soldered electrodes.An inverted microscope was used to record the experimental runs: a Leica DMi8 (Wetzlar, Germany) microscope.Experimental procedure.To ensure a reproducible EO flow, the microchannel was conditioned with the suspending media for 8-12 h before experiments, and the liquid levels at both reservoirs were balanced to mitigate the effect that pressure-driven flow may have had on the system.A volume of 5-10 µL of the nanoparticle or phage sample, was introduced to the inlet reservoir, after which platinum wire electrodes were placed and fixed in both reservoirs.Nanoparticle and phage migration were observed and recorded at a range of applied voltages to observe both linear and nonlinear EK effects.Low-voltage PTV experiments in which the Pe value was below 1 were conducted to obtain ζ P and µ EP,L under conditions of the weak field regime (Table 1, Table 2 and Table S1).High-voltage PTV experiments were subsequently conducted to acquire µ EP,NL under conditions of the moderate field regime (Table 1, Table 2 and Table S2).For the estimation of µ  S5.As expected, these results are similar to those in Tables 1 and 2. All experiments were conducted in triplicate, and both the ImageJ and Tracker software (Version 5.1.5)were employed to determine particle velocity.A detailed description of the PTV experimental procedure and data analysis is included in the Supplementary Materials.

Characterization of the Velocity Behavior of Polystyrene Nanoparticles and Bacteriophages
A series of PTV experiments were conducted by varying the magnitude of the electric fields (25-2400 V/cm) to determine the overall particle velocity of the three types of nanoparticles and the two phages studied in this work.The results are shown summarized in Table 1 and Figure 2. The electrophoretic velocity depicted in Figure S1, which considers the linear and nonlinear components, was obtained by subtracting the electroosmotic component from the overall particle migration.Figure S1 illustrates that at high electric fields, the electrophoretic migration is no longer linear with the electric field, further supporting the presence of EP NL .The three nanoparticles were selected with two criteria in mind: (1) the nanoparticles must possess a diameter akin to the hydrodynamic diameter (D H ) of the employed phages, and (2) the nanoparticles must possess a charge similar to that of the phages.Given the intermediary nature of the phages' hydrodynamic and capsid diameters included in this study, these being ~150 nm, both 100 nm and 200 nm nanoparticles were studied.It was decided to utilize nanoparticles with an aminated surface functionalization since, from previous results in our laboratory [36], aminated particles possess electrical charges similar to those of microorganisms, which have a lower magnitude than the ζ W , allowing the particles to move forward, as represented in the inset in Figure 1.The selection of appropriate proxies for the phages is confirmed in Table 1.The ζ P of the nanoparticles are similar to those of the phages and all have a magnitude below the ζ W value of −60.1 mV.From Figure 2a, which depicts particle velocity vs. electric field, it is seen that all three particles follow the expected behavior: a linear increase of their velocity at low electric fields, a maximum immediately followed by a decrease in velocity as the electric field increases, reaching negative velocity values.The two larger nanoparticles (Particles 2 and 3) crossed the zero-velocity threshold at a lower magnitude electric field, which is also an expected result since the effects of EPNL, the phenomena attributed to cause the decrease in velocity magnitude, increases at high electric fields and increases with particle size.The EEEC values of the nanoparticles range between 1564.6 and 1710.5 V/cm.
Regarding the phages, their migration behavior is illustrated in Figure 2b, confirming that the selected nanoparticles and the phages behave in a similar manner, even though the phages have slightly larger magnitudes in their  values.Furthermore, the phage ϕKZ crosses the zero-velocity line roughly 200 V/cm before SPN3US, whose genome length is smaller than that of ϕKZ (Table S5).The EEEC values of the phages are 1640.6 and 1431.0 for the SPN3US and ϕKZ, respectively.It is noteworthy that the EEEC values for both From Figure 2a, which depicts particle velocity vs. electric field, it is seen that all three particles follow the expected behavior: a linear increase of their velocity at low electric fields, a maximum immediately followed by a decrease in velocity as the electric field increases, reaching negative velocity values.The two larger nanoparticles (Particles 2 and 3) crossed the zero-velocity threshold at a lower magnitude electric field, which is also an expected result since the effects of EP NL , the phenomena attributed to cause the decrease in velocity magnitude, increases at high electric fields and increases with particle size.The E EEC values of the nanoparticles range between 1564.6 and 1710.5 V/cm.
Regarding the phages, their migration behavior is illustrated in Figure 2b, confirming that the selected nanoparticles and the phages behave in a similar manner, even though the phages have slightly larger magnitudes in their ζ P values.Furthermore, the phage ϕKZ crosses the zero-velocity line roughly 200 V/cm before SPN3US, whose genome length is smaller than that of ϕKZ (Table S5).The E EEC values of the phages are 1640.6 and 1431.0 for the SPN3US and ϕKZ, respectively.It is noteworthy that the E EEC values for both nanopar-ticles and phages fall within a relatively narrow range of 1550 ± 150 V/cm.It is important to highlight that while both size and shape influence the EP NL behavior of particles, the size and shape (in terms of sphericity values) of all nanoparticles and phages examined in this study are highly similar.In terms of their hydrodynamic diameter all particles range between 99 to 200 nm; while the sphericity values estimated for both phages are almost identical.Consequently, it is expected that all the nanoparticles and phages studied would exhibit very similar velocity behaviors, as evidenced in Figure 2.

Determination of the Mobility of Nonlinear Electrophoresis of Polystyrene Nanoparticles and Bacteriophages
The determination of the mobility of EP NL is necessary for the design of EK-based separations.As was mentioned in the theory section, due to the minute dimensions of both the nanoparticles and phages, only the moderate electric field regime is reached, in which the EP NL velocities have a cubic dependence on the electric field (Equation ( 4)).Previous studies have demonstrated that the magnitude of µ (3) EP,NL increases with increasing particle size [35,40] and increases with increasing deviations from spherical shape (decreasing values of sphericity, ψ) [42,43].This was considered in Figure 3, which illustrates the µ EP,NL with D H /ψ could be the small overall size of all particles employed here, which diminishes the effect of particle size under this limited particle size range, since particle diameters only varied from 99-200 nm.Further, the overall differences in the estimated shape between the two phages is negligible, as illustrated by their almost identical sphericity values.nanoparticles and phages fall within a relatively narrow range of 1550 ± 150 V/cm.It is important to highlight that while both size and shape influence the EPNL behavior of particles, the size and shape (in terms of sphericity values) of all nanoparticles and phages examined in this study are highly similar.In terms of their hydrodynamic diameter all particles range between 99 to 200 nm; while the sphericity values estimated for both phages are almost identical.Consequently, it is expected that all the nanoparticles and phages studied would exhibit very similar velocity behaviors, as evidenced in Figure 2.

Determination of the Mobility of Nonlinear Electrophoresis of Polystyrene Nanoparticles and Bacteriophages
The determination of the mobility of EPNL is necessary for the design of EK-based separations.As was mentioned in the theory section, due to the minute dimensions of both the nanoparticles and phages, only the moderate electric field regime is reached, in which the EPNL velocities have a cubic dependence on the electric field (Equation ( 4)).Previous studies have demonstrated that the magnitude of  , ( ) increases with increasing particle size [35,40] and increases with increasing deviations from spherical shape (decreasing values of sphericity, ) [42,43].This was considered in Figure 3 with DH/ could be the small overall size of all particles employed here, which diminishes the effect of particle size under this limited particle size range, since particle diameters only varied from 99-200 nm.Further, the overall differences in the estimated shape between the two phages is negligible, as illustrated by their almost identical sphericity values.As observed in Figure 3, the  , ( ) magnitude of the ϕKZ is around ~30% higher than that of SPN3US, which could be considered unexpected as both phages have similar characteristics in size, shape, and zeta potential.There are two potential causes for the higher magnitude in  , ( ) for ϕKZ: (i) the longer genome length (~40 kb longer, Table  As observed in Figure 3, the µ EP,NL magnitude of the ϕKZ is around ~30% higher than that of SPN3US, which could be considered unexpected as both phages have similar characteristics in size, shape, and zeta potential.There are two potential causes for the

Figure 1 .
Figure 1.Schematic representation of the flat, post-less microchannel used for PTV experiments, including dimensions.The inset shows the considered EK forces along with their respective directions for a negatively charged particle and a channel with a negatively charged surface.

Figure 1 .
Figure 1.Schematic representation of the flat, post-less microchannel used for PTV experiments, including dimensions.The inset shows the considered EK forces along with their respective directions for a negatively charged particle and a channel with a negatively charged surface.
values, velocity data obtained at electric field values that were the closest to the E EEC value were employed.Additional estimations of µ (3) EP,NL values obtained by interpolating the E EEC from velocity data are included in Table

Figure 2 .
Figure 2. Overall velocity ( ) as a function of the electric field (E) for (a) nanoparticles and (b) phages.Markers indicate experimental data, and the dashed lines are included for ease of visualization.Error bars denote standard deviation.

Figure 2 .
Figure 2. Overall velocity (v P ) as a function of the electric field (E) for (a) nanoparticles and (b) phages.Markers indicate experimental data, and the dashed lines are included for ease of visualization.Error bars denote standard deviation.

( 3 )
EP,NL magnitude as a function of the ratio of D H /ψ. According to previous experimental studies, the magnitude of µ (3) EP,NL should increase as a function of D H /ψ. This trend is only weakly observed.A potential cause of this slight variation of µ (3) , which illustrates the  , ( ) magnitude as a function of the ratio of DH/.According to previous experimental studies, the magnitude of  , ( ) should increase as a function of DH/.This trend is only weakly observed.A potential cause of this slight variation of  , ( )

Figure 3 .
Figure 3.The absolute value of the mobility of nonlinear electrophoresis  , ( ) as a function of particle diameter divided by the particle sphericity.Absolute values were plotted for  , to aid visualization as these values are negative.For the phages, the hydrodynamic diameter was used.The sphericity of the spherical polystyrene nanoparticles was set to 1. Markers indicate experimental data, and the dashed line is included to denote the data trend, although the phage ϕKZ is outside the trend.Error bars denote standard deviation.

Figure 3 .
Figure 3.The absolute value of the mobility of nonlinear electrophoresis µ

( 3 )
EP,NL as a function of particle diameter divided by the particle sphericity.Absolute values were plotted for µ EP,NL to aid visualization as these values are negative.For the phages, the hydrodynamic diameter was used.The sphericity of the spherical polystyrene nanoparticles was set to 1. Markers indicate experimental data, and the dashed line is included to denote the data trend, although the phage ϕKZ is outside the trend.Error bars denote standard deviation.

Table 1 .
Characteristics of the particles employed in this study.

Table 2 .
Characteristics of the phages employed in this study.The hydrodynamic diameter (D H ) of the phages was estimated considering the entire volume of the phage (capsid and tail).Equation (S7) and the data in TableS5from the Supplementary Materials were employed for the estimation of the D H values. ** Additional estimations of µ(3)EP,NL were performed at the E EEC condition.These results, which are similar to the values in this table, are included in TableS6in the Supplementary Materials. *