Theoretical Investigation of Bacteria Polarizability under Direct Current

Article pubs.acs.org/Langmuir

Theoretical Investigation of Bacteria Polarizability under Direct Current Electric Fields Naga Neehar Dingari and Cullen R. Buie* Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States ABSTRACT: We present a theoretical model to investigate the influence of soft polyelectrolyte layers on bacteria polarizability. We resolve soft-layer electrokinetics by considering the pH-dependent dissociation of ionogenic groups and specific interactions of ionogenic groups with the bulk electrolyte to go beyond approximating soft-layer electrokinetics as surface conduction. We model the electrokinetics around a soft particle by modified Poisson−Nernst−Planck equations (PNP) to account for the effects of ion transport in the soft layer and electric double layer. Fluid flow is modeled by modified Stokes equations accounting for soft-layer permeability. Two test cases are presented to demonstrate our model: fibrillated and unfibrillated Streptococcus salivarius bacteria. We show that electrolytic and pH conditions significantly influence the extent of soft-particle polarizability in dc fields. Comparison with an approximate analytical model based on Dukhin−Shilov theory for soft particles shows the importance of resolving soft-layer electrokinetics. Insights from this study can be useful in understanding the parameters that influence soft-particle dielectrophoresis in lab-on-a-chip devices.

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the fixed charge density and Donnan potential, in addition to the bulk electrolyte concentration. Hence, the electrophoretic mobility can be used as a parameter to assess the surface properties of cells. The ease of measurement and the ability to integrate into lab-on-a-chip devices make electrokinetics a promising cell envelope phenotyping tool. With recent advances in electrokinetics and the ability to create strong electric field gradients in microchips, dielectrophoresis (DEP) is receiving widespread attention in electrokinetics. Discovered by Pohl16 60 years ago, DEP refers to the force that arises on a polarizable particle in a nonuniform electric field. Recent advances in insulator-based dielectrophoresis (iDEP) devices have eliminated the need for electrodes to create electric field gradients.17,18 In iDEP, geometric alterations to insulating channels are used to create gradients in an otherwise uniform electric field, usually at dc.19 Under dc fields, particles or cells experience a combination of electrophoretic, dielectrophoretic, and viscous drag forces and a suitable combination of these forces can lead to particle trapping at channel constrictions. This technique has been applied to separate live bacteria from dead bacteria,17 to determine the cell concentration,20 and to enhance the kinetics of DNA hybridization.21 In an attempt to reduce Joule heating, various researchers have come up with iDEP channel designs employing 2D or 3D geometric features,19,22,23 expanding the range of applicability for iDEP-based devices. The determination of the induced dipole moment on a particle under the influence of an externally applied electric field is the key to predicting the dielectrophoretic force. When

INTRODUCTION The importance of the outer cell envelope of single-celled microorganisms and its interactions with the surrounding environment has been a topic of considerable interest. Various processes such as adhesion, aggregation, and biofilm formation depend on the surface properties of cells.1−3 As a result, there has been growing interest in understanding the cell envelope properties and their connection to pathogenicity. Auger et al.4 demonstrated a relation between the ability of Bacillus cereus to form a biofilm and its pathogenicity. Others demonstrated that the ability to form biofilms strongly depends on the electrostatic and hydrophobic properties of Staphylococcus aureus and Staphylococcus epidermidis.5 It has also been shown that there is a relation between the surface hydrophobicity and pathogenicity of Xanthomonas campestris.6 Because of these correlations between cell envelope properties and pathogenicity, there have been numerous attempts to study the cell envelope on the microscopic level. Atomic force microscopy (AFM) is a powerful technique to image the morphology of cell surfaces.7−9 For example, Pelling et al.7 studied the nanomechanical motion of a cell wall using AFM. Microelectrophoresis is another tool to probe the cell envelope, which has been receiving attention in recent years because of extensive quantitative insights on the electrokinetics of soft particles by Ohshima and co-workers10−13 and others.14 For instance, electrophoresis was used to detect the difference in surface properties of Escherichia coli and S. aureus15 by measuring the mobilities experimentally and then analyzing the results via the formula for electrophoretic mobility. The electrophoretic mobility of colloidal particles is a function of their zeta potential and the bulk electrolyte concentration. Conversely, the mobility of soft biological particles is a strong function of the electrokinetic properties of the soft layer such as © 2014 American Chemical Society

Received: January 21, 2014 Revised: March 20, 2014 Published: March 25, 2014 4375

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Table 1. Some Common Appendages Found in Bacteria and Their Functions33 structure

function(s)

examples

flagella

swimming movement

sex pilus common pili or fimbriae capsules (includes “slime layers” and glycocalyx) inclusions

mediates DNA transfer during conjugation attachment to surfaces; protection against phagotrophic engulfment attachment to surfaces; protection against phagocytic engulfment, occasionally killing, or digestion; reserve of nutrients or protection against desiccation often reserves of nutrients; additional specialized functions

the characteristic length scale of the electric field variation is small compared to the diameter of the particle or cell, Maxwell−Wagner (MW) theory24 can be used to find the * acting on the particle given by F⃗DEP * dielectrophoretic force F⃗DEP 2 = 3/2V*ε*mRe( f)∇E⃗ *RMS , where V* is the particle volume, ε*m is the medium permittivity, E⃗ *RMS is the RMS value of the ac externally imposed electric field, and f is the Clausius−Mosotti factor, which for a spherical particle is given by f = (((εp*) − *))/((2εm *) + (εp*))). εi* is the complex permittivity given by (εm ε*i = ε*i − i((κ*i )/(ω*)), where ε*i is the permittivity of species i, κ*i is the conductivity of species i, ω* is the frequency of the applied electric field, i = p represents the particle, and i = m represents the surrounding medium. When a colloidal particle is suspended in an electrolyte, the polarization of the particle is also due to the migration of ions within and outside the charged layer surrounding the particle called the electric double layer (EDL). The characteristic thickness of the electric double layer known as the Debye length is given by λ*D = ((ε*mkBT*)/(2z2e2c*bulk))1/2, where kB is the Boltzmann constant, T* is the absolute temperature, c*bulk is the bulk electrolyte concentration, z is the valence of ions in a symmetric z/z electrolyte, and e is the elementary electric charge. If the electric field frequency is much larger than the characteristic time scale for ion migration around the particle τD*−1 = ((a*2)/(D*±))−1 (with a* being the radius of the particle and D*± being the diffusivity of positive and negative ions), then the ions do not have sufficient time to respond to the changes in the electric field. Hence, the polarizability can be explained by a modified Maxwell−Wagner theory known as the Maxwell−Wagner−O’Konski25 (MWO) theory. However, this theory is limited to thin electrical double layers where migration and ion convection are approximated by surface conduction. Because of the surface conduction approximation, MWO does not account for the diffusion of ions outside the electric double layer. This is particularly relevant when the frequency of the applied electric field is smaller than τ*D −1. For a typical bacterial cell of diameter 1 μm in an electrolyte with an ion diffusivity of D±* = 2 × 10−9 m2 s−1, then τD*−1 corresponds to 2 kHz. In these cases, MWO is inappropriate because it cannot account for the significant ion migration outside the electric double layer. Alternatively, the Dukhin and Shilov26,27 (DS) model can be used to predict the induced dipole moment coefficient at low frequencies28 by assuming quasi-equilibrium between the thin electric double layer and the bulk neutral electrolyte. Both MWO and DS models are applicable only for thin electric double layers (λ*D < 0.1a*). For higher electric double layer thicknesses, the particle’s electrophoretic motion contributes significantly to the induced dipole moment. To account for the effect of electrophoretic motion on the particle’s polarizability, one needs to solve coupled Poisson− Nernst−Planck (PNP) and momentum equations around the particle. For example, Zhao and Bau29 calculated the polar-

Bacillus cereus, Vibrio cholera, Bacillus brevis Escherichia coli, Streptococcus pyogenes, Pseudomonas aeruginosa. most bacteria Escherichia coli, Pseudomonas, Corynebacterium

izability of a nanoparticle surrounded by a thick electric double layer by adopting a Poisson−Nernst−Planck formalism. In the limit of a small zeta potential, they found an approximate expression for the induced dipole moment coefficient fapprox = −

1 1 (λD + 4PeλD2)ζ 2 + O(ζ 3) + 2 6(1 + iλD2ω) (1)

where λ = ((λ*D)/(a*)) is the dimensionless electric double layer thickness, ω = ((ω*)/(τ*D−1)) is the dimensionless frequency of the applied electric field, ζ = ((ζ*)/(kBT*/e)) is the dimensionless zeta potential, and Pe is the Peclet number defined as Pe = ((ε*mR2T*2)/(μ*D*± F2)), where R is the universal gas constant, T* is the absolute temperature, μ* is the solvent viscosity, D±* is the diffusivity of positive (+) or negative (−) ions, and F is the Faraday constant. Equation 1 shows that as electric double layer thickness increases, the particle’s electrophoretic motion has a significant influence on the induced dipole moment coefficient. Unlike colloidal particles, the surface of soft biological particles is not a well-defined rigid boundary. Many soft particles, such as bacteria, exhibit a range of external appendages, which have significant physiological importance. Table 1 shows some of the kinds of cell appendages found in bacteria. The presence of a charged soft polyelectrolyte layer around bacteria adds an additional layer of complexity to the electrokinetics. Ion transport in the soft layer and in the electric double layer contributes to the induced dipole moment coefficient. The conductivity of the soft layer depends on the availability of mobile ions within the soft layer, which in turn depends on the charge density of fixed ionogenic groups within the soft layer. This is pH-dependent because the fixed charge density is due to dissociation reactions of ionogenic groups within the soft layer. Therefore, the pH is a critical parameter when trying to calculate the induced dipole moment coefficient for soft particles such as bacteria. Uppapalli and Zhao30 studied the polarizability of soft biological particles under the influence of an alternating electric field by solving Poisson−Nernst− Planck (PNP) equations. However, they assumed that the charge density within the soft layer is a constant, independent of other parameters such as the pH, bulk electrolyte concentration, and local electric potential. This is true only if one assumes complete dissociation of ionogenic groups within the soft layer. One of their main results is that the induced dipole moment coefficient decreases with increasing electric double layer thickness at low Donnan potentials. Meanwhile, the opposite is true at high Donnan potentials. This is true only when we assume that one can vary the electric double layer thickness while keeping the Donnan potential constant. This is unlikely because the Donnan potential and electric double layer thickness typically vary simultaneously with the bulk electrolyte concentration and pH, particularly if we account for the 4376

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incomplete dissociation of ionogenic groups within the soft layer. This has been demonstrated by Duval et al.,31 who analyzed the effect of soft-layer properties on the electrophoretic mobility of two bacterial strains, Streptococcus salivarius HB (fibrillated) and Streptococcus salivarius HB-C12 (unfibrillated). Through a combination of titration experiments and theoretical investigations, they measured the soft layer properties for these two strains of bacteria. Electrophoretic mobilities obtained by solving modified Poisson-Nernst− Planck (PNP) equations and Stokes equations for soft particles32 show that electrophoretic mobility is a strong function of the soft-layer properties, especially the charge density and the friction factor of the soft layer. The aim of this study is to elucidate the combined influence of the soft layer and the electric double layer on the soft particle polarizability in various concentration and pH regimes. Note that by the electric double layer we refer to the region of net charge density extending from the membrane surface to the outside of the soft layer. However, the electric double layer within the soft layer has less influence on the polarizability. This is due to the fact that fluid flow experiences a frictional force within the soft layer and the electric double layer thickness within the soft layer is smaller than the electric double layer thickness outside the soft layer (by a factor of (cosh(ψD)))1/2.13 In the present study, we use the soft-layer properties for the two strains of bacteria analyzed by Duval et al.31 to calculate the induced dipole moment coefficient. We consider bulk electrolyte concentrations from 10−5 to 0.1 M to span the range from very thin electric double layers to thicknesses comparable to the soft layer thickness.

Figure 1. Schematic of a soft particle with a rigid core of radius a* surrounded by a (negatively) charged soft layer of thickness δ*.

is the dimensionless electric double layer thickness given by λD = ((λ*D)/(a*)). Equation 2 represents a force balance among the pressure force (∇P), the viscous force (∇2u⃗), the electric body force (1/(2λD2)(C+ − C−)∇ϕ), and the frictional force within the soft layer ( f(r)k0u⃗). The fluid is assumed to be incompressible (3) ∇·u ⃗ = 0 The electric potential ϕ in the medium around the particle satisfies Poisson’s equation given by

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∇2 ϕ = −

MATHEMATICAL MODELING Governing Equations. A uniform electric field E⃗ * is applied far from the particle in the z direction as shown in 1. Assuming a spherically symmetric particle, the coordinate axes in the (r, θ) system are shown with origin at the particle’s center, where er⃗ and eθ⃗ are unit vectors in the r and θ directions, respectively. All variables with superscript * are dimensional whereas variables without * are dimensionless. For dimensionless variables, we use the rigid core radius a* as the pertinent length scale, kBT*/e as the electric potential scale, εm *kB2T*2/ 2 2 (μ*e a*) as the velocity scale, a* /D±* as the time scale, ε*mkB2T*2/(e2a*2) as the pressure scale, ε*mkBT*/(ea*) as the electric charge scale, and c*bulk, the bulk electrolyte concentration, as the concentration scale. Here, kB is Boltzmann’s constant, T* is the absolute temperature, εm * is the medium permittivity, e is the fundamental electric charge, and μ* is the medium viscosity. The electric field drives excess mobile ions within the soft layer and the electric double layer around the particle. These ions drag the fluid along with them, creating a flow around the particle. This relative flow with respect to the particle contributes to the electrophoretic mobility μe. The fluid experiences a resistance due to the soft layer, which can be modeled within the framework of the Debye−Bueche model.34 The fluid flow around the particle is modeled by the Stokes equation −∇p + ∇2 u ⃗ −

1 (C+ − C −)∇ϕ − f (r )k 0u ⃗ = 0 2λD2

(C+ − C − + ρfix f (r )) 2λD2

(4)

where ρfix = ρfix * /(Fcbulk * ) is the dimensionless charge density within the soft layer, explained in the next section, and F is the Faraday constant. Ionic transport around the particle is given by a balance of diffusive (D±∇C± ), electromigrative (D± z±C ±∇ϕ), and convective fluxes (PeC±u⃗) governed by the Nernst−Plank equations ∂C± ∂t

= ∇·(D±∇C± + D±z±C±∇ϕ − PeC±u ⃗)

(5)

where the Peclet number Pe = ((εm *R T* )/(μ*D+*F )) and D± = ((D±*)/(D+*)). For the sake of simplicity, we assume in our analysis that D*+ = D*− . Physical Description of the Soft Layer. Consider a spherical soft particle with a rigid core of radius a* surrounded by a soft polyelectrolyte layer of thickness δ*, nondimensionalized as δ = δ*/a*. Assuming that the ionogenic groups are homogeneously distributed within the soft layer, the dimensionless fixed charge density due to the dissociation of the ionogenic groups ρfix depends on the dimensionless radial position r 2

ρfix = ρ0 f (r )

2

2

(6)

where ρ0 = ρ*0 /(Fc*bulk) is the dimensionless nominal fixed charge density and f(r) denotes the functional form of the radial distribution of ionogenic groups. However, eq 6 is valid only if we assume the complete dissociation of ionogenic groups. Typically, the extent of dissociation depends on the local pH, local electric potential ϕ(r), and pKa of the ionogenic groups as

(2)

where u⃗ is the velocity vector, C± represents the concentrations of positive and negative ions, ϕ is the electric potential, and λD 4377

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ρfix = ρ0 f (r ) g o(r , pH)

by Duval and co-workers.31,32,37,39 Dukhin et al.40 addressed the applicability of the Brinkman equation for soft surface electrokinetics and argued that a revision of the Brinkman equation incorporating the viscous stress discontinuity may not be as important for soft particle electrokinetics as it is for hybrid systems. The experimental investigation of the hydrodynamic penetration length of swollen and collapsed thermoresponsive films by Zimmermann et al.39 supports this proposition. The friction coefficient k depends on the polymeric segments’ distribution within the soft layer. For a high water content polyelectrolyte layer, the Brinkman equation states that the spatial dependence of k can be taken to be the spatial dependence of the volume fraction of polymeric segments within the soft layer f(r). Hence

(7)

where go(r, pH) is an isotherm describing the protolytic properties of the soft layer and can be modeled by Langmuirtype kinetics.35 g o(r , pH) =

10 pH − pK a exp(ϕ(r )) 1 + 10 pH − pK a exp(ϕ(r ))

(8)

The spatial dependence of pH and salt-dependent charges across the soft layer was first introduced by Duval and coworkers,31,32 originally for the purposes of calculating the electrophoretic mobility of soft particles. It was more generally elaborated by Langlet et al.36 for calculating the electrophoretic mobility of multilayered soft particles and by Zimmermann et al.37 for streaming current−potential measurements. The total amount of charge within the soft layer Q*fix can be found by integrating eq 7 over the volume of the soft layer as * = 4πa*3 ρ* Q fix 0 *3 FN0*

∫1

1+δ

∫1

1+δ

k = f (r ) k0

where k0 is the bulk friction coefficient. The bulk softness parameter can thus be defined as

r 2f (r ) g 0(r , pH)dr = 4πa

r 2f (r ) g 0(r , pH)dr

⎛ a* ⎞ 2 k0 = ⎜ ⎟ ⎝ λ 0* ⎠

(9)

where N*0 = ρ*0 /F is the space concentration of ionogenic sites available for dissociation. It has units of mol m−3 and in dimensionless form is N0 = N0*/cbulk * . In addition to dissociation, one may also include specific interactions between the ionogenic groups and the background electrolyte. For example, the binding reaction between a cation X+ and a negative ionogenic group A− given by X+ + A− ⇄ [X+A−] can be modeled as gi(r, cbulk * ) using the Frumkin− Fowler−Guggenheim isotherm35 as follows * )= g i(r , cbulk

* exp( −ϕ(r )) K *cbulk * exp( −ϕ(r )) N 0 1 + K *cbulk

(10)

where Ni = N*i /c*bulk is the dimensionless space concentration of ionogenic sites available for specific interaction with the background electrolyte and K* is the rate constant of the binding reaction. The fixed charge density due to these two mechanisms then becomes

E ⃗ = − χ ∇ϕ

Γ*i =

3FN0*

∫ (1 + δ)3 − δ 3 1 3FNi* 3

(1 + δ) − δ

3

∫1

1+δ

1+δ

Following the approach followed by O’Brien and White, we expand each of the variables as a regular perturbation expansion with χ as a small parameter

(11)

γ = γ (0) + χγ (1) + χ 2 γ (2) + ....

(16)

where γ is any of the variables ϕ,C± , u,⃗ or P. Using symmetry conditions for spherical coordinates,43 we decouple the radial (r) and (θ) dependence of the unknown variables as follows

r 2f (r ) g o(r , pH) dr * ) dr r 2f (r ) g i(r , cbulk

(15) 42

One can now find the volume-averaged fixed charge density within soft layer known as the space charge density Γ*tot = Γ*0 + Γ*i , where Γ*0 and Γ*i are given by Γ*0 =

(14)

where λ*0 is the hydrodynamic penetration length of the fluid within the soft layer.41 Physically, it represents the extent to which fluid can penetrate within the soft layer (i.e., the greater the softness parameter, the lower the friction experienced by the fluid within the soft layer). Using eq 13, we have F⃗fric = k0 f(r)u⃗, which is commonly known as the Darcy−Brinkman equation. Equations 7−13 were originally provided by Duval et al.31 Solution Methodology. We assume that the electric field is weak and only slightly perturbs the equilibrium of the electric double layer. In dimensionless form, the electric field is given by

Ni

* ) + g 0(r , pH)) ρfix (r ) = ρ0 f (r ) (g i(r , cbulk

(13)

(12)

The charged polymer segments of characteristic radius as within the soft layer act as resistance to fluid flow. This frictional force, modeled by the Debye−Beuche model,34 can be represented in the form of Stokes drag that depends on the local fluid velocity (u⃗) as F⃗fric = ku⃗. Here, the dimensionless parameter k is the friction coefficient, which is also positiondependent. This position dependence was first introduced by Hill et al.,38 who studied the electrophoretic mobility of polymer-coated colloidal particles. It was generalized for electrophoresis and streaming current−potential measurements

ϕ(1) = ϕ(̂ r ) cos(θ )

(17)

(1) C±(1) = C±̂ (r ) cos(θ )

(18)

ur = ur̂ (r ) cos(θ )

(19)

uθ = uθ̂ (r ) sin(θ )

(20)

p = p ̂(r ) cos(θ )

(21)

Substituting the decoupled variables (eqs 17−21) into eqs 2−5, we obtain ordinary differential equations in r for each of the variables as follows 4378

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̂ 4uθ̂ (1) 4ur̂ (1) dp(1) 1 d ⎛⎜ 2 d (1) ⎞⎟ − − − r ( u ) ̂ r 2 2 2 ⎝ ⎠ dr dr r dr r r (1) ⎛ (0) ⎞ ̂ (1) (1) dϕ ⎟ 1 ⎜ (0) (0) dϕ ̂ ̂ − − − − ( C C ) ( C C ) + − + − dr dr ⎟⎠ 2λD2 ⎜⎝

dϕ(0) |r = 1 = 0 dr

dϕ ̂ |r = 1 = 0 dr

(29)

Equations 27 and 29 can be combined to give (1)

dC±̂ dr

(22)

=0

(1)

and

(1)

|r = 1 = 0 and

dϕ ̂ |r = 1 = 0 dr

(30)

43

Using symmetry considerations, one can write the dimensionless radial and tangential velocities as ur = − 2/ rh(r)E cos(θ) and uθ = 1/r(d/dr){rh(r)}E sin(θ), respectively. In the reference frame of the particle, we assume a no-slip condition and an impenetrable wall boundary conditions for the fluid flow

̂ 2uθ̂ (1) 2ur̂ (1) p(1) 1 d ⎛⎜ 2 d (1) ⎞⎟ − − + r ( u ) ̂ θ 2 2 2 ⎠ r r dr ⎝ dr r r +

(1) ̂ 1 (0) (0) ϕ ( ) − =0 C C + − 2 r 2λD

(23)

ur̂ |r = 1 = 0 and

2uθ̂ (1) 1 d 2 (1) ( r u ) − =0 ̂ r r r 2 dr

h|r = 1

(24)

(1) (1) (1) ⎞ (1) ⎛ C+̂ − Ĉ − 2ϕ ̂ 1 d ⎜ 2 dϕ ̂ ⎟ r − = − r 2 dr ⎜⎝ dr ⎟⎠ r2 2λD2

+

r

2

+ z±

(0) (1) 2C±̂ ϕ ̂

r

2

+

dC±(0) Peur̂ (1)

(25)

dr

=0

ur̂ |r →∞ = −μe E −

(1)

dĈ − dr

(1) ⎞ ⎛ (0) dϕ ̂ ⎟ (1) dϕ + z −⎜⎜Ĉ − + C −(0) =0 dr dr ⎟⎠ ⎝

2 1 d dh h h=− (rh(r )) ⇒ |r →∞ = |r →∞ r r dr dr r

(32)

(33)

Far away, at r → ∞, the electrolyte is neutral and hence the concentration of positive and negative ions must be the same as cbulk. In dimensionless form, we have C±(0)(r → ∞) = 1 and

C±(1)(r → ∞) = 0

(34)

The perturbed electric potential ϕ must decay to a linear profile in the bulk to match the constant applied electric field. However, the nature of the decay is governed by the induced dipole moment coefficient fdipole as (1)

ϕ(0)(r → ∞) = 0 and (27)

⎛ fdipole ⎞ (1) ϕ ̂ (r → ∞) = ⎜⎜ −r + 2 ⎟⎟ r ⎠ ⎝ (35)

The continuity of the electric displacement field D = εE at the cell surface (by Gauss’ law given that the rigid cell surface does not possess any charge) implies (0)

uθ̂ |r →∞ = μe E , i.e.,

d2h |r →∞ = 0 dr 2

(1) ⎞ ⎛ (0) dϕ ̂ ⎟ (1) dϕ + z+⎜⎜C+̂ + C+(0) =0 dr dr ⎟⎠ ⎝ (1)

and

and

We now use the condition that the total force acting on the particle must be zero. It has been argued by Duval and Ohshima32 that this is equivalent to setting

(26)

Equations 22 and 23 represent the momentum balance equations, equation 24 represents the continuity equation, equation 25 is the Poisson equation, and equation 26 represents the ion-transport equations for positive and negative ions in the steady state. Boundary Conditions. We assume that the rigid cell surface (r = 1) is insulating and impenetrable to ions. Hence, the first-order perturbation expansion of ion flux normal to the cell surface must be zero. This gives us dC+̂ dr

(31)

Far away from the particle, the velocity of the fluid must approach the bulk fluid velocity, −μeEez, where μe is the electrophoretic mobility of the particle. This results in the following conditions,

⎛ ⎛ ⎛ ⎞⎞ (1) (1) ⎛ (0) ⎞⎞ ̂ (1) dϕ ⎟⎟⎟⎟ 1 ⎜ d ⎜ 2⎜ dC±̂ (0) dϕ ⎜ ̂ − z±⎜C± + C± r − dr dr ⎟⎠⎟⎠⎟⎟⎟ r 2 ⎜⎜ dr ⎜ ⎜⎝ dr ⎝ ⎝ ⎠⎠ ⎝ (1) 2C±̂

uθ̂ |r = 1 = 0, i.e., dh = 0 and |r = 1 = 0 dr

Test Cases. To apply our theoretical model, we choose two test casesS. salivarius HB (fibrillated strain) and S. salivarius HB-C12 (unfibrillated strain)whose electrophoretic mobilities were studied theoretically and experimentally by Duval et al.31 The parameters governing specific interactions between ionogenic groups in the cell wall and the background electrolyte are given in Table 3. Using the data mentioned in Tables 2 and 3, one can derive the appropriate forms of go(r, pH) and gi(r, cbulk * ) as done by Duval et al.31

(1)

dϕ dϕ(0) dϕ ̂ εsoft |r = 1 = εcell wall cell wall |r = 1 and εsoft |r = 1 dr dr dr (1) ̂ dϕcell wall = εcell all |r = 1 (28) dr (1) ̂ (1) where ϕ(0) cell wall and ϕcell wall (r, θ) = ϕcell wall(r) cos (θ) are the zeroth- and first-order perturbation expansions of the potential inside the cell wall. Because the soft-layer permittivity εsoft (which is nearly equal to the permittivity of water because of the high water content) is much larger than the permittivity of the cell wall εcell wall, we must have

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RESULTS AND DISCUSSION Figure 2 shows the variation in the space charge density (Γ*tot) of S. salivarius HB (fibrillated strain) and S. salivarius HB-C12 (unfibrillated strain) as a function of bulk electrolyte concentration cbulk * (M) at various pH values. The bulk 4379

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Table 2. Properties of Soft Layers of Fibrillated and Unfibrillated S. salivarius31 property

unfibrillated

fibrillated

cell radius (a*) cell wall thickness (δW *) fibrillar layer thickness (δ*F ) pKa of dissociation groups in the cell wall

577 nm 23 nm

577 nm 23 nm 200 nm pKa1 = 3, pKa2 = 6.1, pKa3 = 9.7

pKa1 = 3, pKa2 = 6.1, pKa3 = 9.7

pKa of dissociation groups in the fibrillar layer hydrodynamic penetration length of the cell wall (λ*0W) hydrodynamic penetration length of the fibrillar layer (λ*0F) space concentration of sites available for dissociation (N0W * ) in the cell wall space concentration of sites available for dissociation N0F * in the fibrillar layer

1.9 nm

pKa1 = 2, pKa2 = 4.8, pKa3 = 9.5, pKa4 = 9.5 1.9 nm 1.3 nm

725 mol m−3

725 mol m−3 23 mol m−3

Table 3. Parameters Governing Specific Interactions with the Background Electrolyte31 pH

space concentration of sites (N*iw/mol M−3)

rate constant of interaction (K*W/M3 mol−1)

5 6 7 8

7 290 638 715

0.2 0.2 0.4 1.3

electrolyte concentration varies from 0.1 to 10−5 M (i.e., from a very thin electric double layer thickness (0.001 times the cell radius) to a relatively thick electric double layer (0.1 times the cell radius)). At high concentrations, because of almost complete dissociation of the ionogenic groups, the space charge density is highest (most negative) and decreases in magnitude as the concentration decreases. The space charge density varies with pH as a result of the pH-dependent dissociation isotherm of ionogenic groups. The space charge density within the soft layer is a strong function of pH and bulk electrolyte concentration, and hence it is important to consider it in conjunction with these parameters while analyzing soft particle electrokinetics. Figure 3 shows the comparison between the electrophoretic mobilities of S. salivarius HB (fibrillated strain) and S. salivarius HB-C12 (unfibrillated strain) as a function of the bulk electrolyte concentration at various pH values. At higher concentrations, the electrophoretic mobility is less dependent upon pH and the surface charge density within the soft layer is almost independent of pH because of nearly complete dissociation of the ionogenic groups. Thus, the physical properties of the soft layer at high concentrations for a given strain are the same across different pH values, giving rise to similar electrophoretic mobilities. At low concentrations, the properties of the soft layer (charge density and friction factor) are a stronger function of pH. The difference in electrophoretic mobilities between S. salivarius HB and S. salivarius HB-C12 can be attributed to the difference in their soft layers. Figure 4 shows a comparison of induced dipole moment coefficients of S. salivarius HB (fibrillated strain) and S. salivarius HB-C12 (unfibrillated strain) as a function of bulk electrolyte concentration at various pH values. At high

Figure 2. Space charge density Γtot * as a function of bulk electrolyte concentration c*bulk(M) at various pH values. (a) An unfibrillated strain. (b) A fibrillated strain. The space charge density is strongly dependent on the pH and bulk electrolyte concentration because of the incomplete dissociation of ionogenic groups.

concentrations, the induced dipole moment coefficient approaches a constant value of −0.5, which is the value for insulating particles consistently used in many dielectrophoresisrelated experiments on bacteria.17,44 At low concentrations, however, the induced dipole moment coefficient departs significantly from this value of −0.5 and is even positive at low concentrations. As in the case of electrophoretic mobility, the pH dependence is higher at low concentrations. However, it is important to note that unlike electrophoretic mobility, in the case of an induced dipole moment coefficient, there is a clear demarcation between the two strains of bacteria. In general, fibrillated strains are more polarizable than unfibrillated strains. Fibrillated strains have a thicker soft layer compared to unfibrillated counterparts, thereby allowing more concentration polarization as a result of ion motion around the particle, thereby increasing their polarizability. It is important to note that the influence of ion motion around the particle upon polarizability is more pronounced in thicker electric double 4380

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Figure 3. Electrophoretic mobility (μe) as a function of bulk * (M) at various pH values. The electrolyte concentration cbulk electrophoretic mobility is strongly dependent on the nature of the soft layer and bulk pH.

Figure 5. Dukhin number (Du) as a function of the bulk electrolyte concentration c*bulk(M) at various pH values. (Top) The unfibrillated strain. (Bottom) The fibrillated strain. Dusoft corresponds to the softlayer contribution to the total Dukhin number (Dutotal). Dutotal approaches zero at high bulk electrolyte concentrations and increases as the bulk electrolyte concentration decreases.

Figure 4. Dipole moment coefficient ( fdipole) as a function of bulk * (M) at various pH values. fdipole deviates electrolyte concentration cbulk significantly from the conventionally assumed value of −0.5 when considering only interfacial polarization.

Du =

layer limits29 and is almost negligible as we approach highconcentration limits. Hence we see that induced dipole moment coefficients are higher at low concentrations, with fibrillated strains being more polarizable than unfibrillated strains. The dimensionless electric double layer thickness (λD) at a low concentration of 10−4 M is roughly on the order of 0.1. This is the regime where the particle electrophoresis contributes significantly to its polarizability45 and results in induced dipole moment coefficients that are significantly higher than −0.5 and even positive. Figure 5 shows the variation of the Dukhin number for S. salivarius HB (fibrillated strain) and S. salivarius HB-C12 (unfibrillated strain) as a function of the bulk electrolyte concentration at various pH values. The Dukhin number is calculated as follows

∞ ∞ −0.5⎡⎣∫ (C+(0) + C −(0) − 2)Eθ r 2 dr + m ∫ (C+(0) − C −(0))uθ r 2 dr ⎤⎦ 1 1

Eθ

(36)

where Eθ is the tangential component of the externally applied electric field. Physically, Du represents the ratio of surface conductivity (the soft layer and the electric double layer) to the bulk electrolyte conductivity {i.e., Du = ((σs)/(Rσb))}. The soft-layer contribution to the Dukhin number (Dusoft) is obtained by integrating eq 36 across the soft layer. The electric double layer contribution (outside the soft layer) is obtained by subtracting Dusoft from the total Dukhin number (Dutotal). From Figure 5 it can be seen that at high concentrations Du approaches zero. Consequently, the surface conductivity is very small and the induced dipole moment coefficient approaches the value of −0.5 (as predicted when assuming Maxwell− 4381

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soft particles, which is valid in regimes δ*/a* ≪ 1 and λ*D/a* ≪ 1. They assume a constant charge density within the soft layer, which does not vary with the electrolyte concentration or pH. Therefore, to utilize their model for the present case, we employ the surface charge density calculated at a given pH and bulk electrolyte concentration. The approximate induced dipole moment coefficient fanalytic is given by

Wagner polarization). As the bulk electrolyte concentration is lowered, surface conductivity increases as a result of an increase in the soft layer conductivity and the electric double layer conductivity. At very low concentrations ( 0.1.45 Thus, as considered in the present theory, one must consider the Poisson−Nernst−Planck-based formalism to calculate the induced dipole moment coefficient. To account for the pH-dependent nature of soft layers, we incorporated the soft layer dissociation isotherms31,35 in conjunction with electrokinetics around the soft particle. Our results show that at low concentrations the convection of ions within the soft layer and electric double layer can have a significant influence on the polarizability. Our comparisons with the Dukhin−Shilov analytical expression for soft particles30 further substantiate this point. A quantitative understanding of soft particle polarizability can result in the more efficient use of dielectrophoresis for cell sorting, separation, and characterization. Coupling soft layer properties and externally measurable quantities such as the induced dipole moment coefficient can promote lab-on-a-chip tools to probe the cell-surface structure. Such tools would be valuable for many applications, including point of care health diagnostics and food safety testing where cell surface properties can be used to assess the bacterial pathogenicity.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the National Science Foundation (NSF award no. 1150615) for funding. This work was also supported by the Institute for Collaborative Biotechnologies through grant W911NF-09-0001 from the U.S. Army Research Office. The content of this information does not necessarily reflect the position or the policy of the government, and no official endorsement should be inferred. We thank Dr. Bruno Figliuzzi for his help in preparing this article.

Figure 7. Comparison of the numerically calculated dipole moment coefficient fdipole with the electric double layer contribution to the induced dipole moment coefficient f EDL calculated using the Poisson− Nernst−Planck (PNP) formalism for (a) unfibrillated and (b) fibrillated strains. At low concentrations, the electric double layer contributes significantly to the induced dipole moment coefficient.

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REFERENCES

(1) Rosan, B. Microbial adhesion to surfaces. Science 1981, 214, 902− 903. (2) Baszkin, A.; Norde, W. Physical Chemistry of Biological Interfaces; CRC Press: Boca Raton, FL, 1999. (3) Van Loosdrecht, M. C.; Lyklema, J.; Norde, W.; Zehnder, A. J. Influence of interfaces on microbial activity. Microbiol. Rev. 1990, 54, 75−87. (4) Auger, S.; Ramarao, N.; Faille, C.; Fouet, A.; Aymerich, S.; Gohar, M. Biofilm formation and cell surface properties among pathogenic and nonpathogenic strains of the Bacillus cereus group. Appl. Environ. Microbiol. 2009, 75, 6616−6618. (5) Otto, M. Staphylococcal infections: mechanisms of biofilm maturation and detachment as critical determinants of pathogenicity. Annu. Rev. Med. 2013, 64, 175−188. (6) Fett, W. F. Relationship of bacterial cell surface hydrophobicity and charge to pathogenicity, physiologic race, and immobilization in attached soyabean leaves. Physiol. Biochem. 1985, 75, 1414−1418. (7) Pelling, A. E.; Sehati, S.; Gralla, E. B.; Valentine, J. S.; Gimzewski, J. K. Local nanomechanical motion of the cell wall of Saccharomyces cerevisiae. Science 2004, 305, 1147−1150. (8) Dufrêne, Y. F. Atomic force microscopy, a powerful tool in microbiology. J. Bacteriol. 2002, 184, 5205−5213.

regime, and hence one cannot use the approximate analytical expression as obtained by Bau and Zhao29 for thick electric double layers.

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CONCLUSIONS At high electrolyte concentrations, we find that the induced dipole moment coefficient approaches a pH-independent value of −0.5, as assumed in many experiments related to the dielectrophoresis of bacteria.23,46 However, at low concentrations (