author's personal copy · author's personal copy near a charged interface, either...

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Modelling electrostatic interactions in complex soft systems

D.A. Pink a, C.B. Hanna b,*, B.E. Quinn a, V. Levadny c, G.L. Ryan a,1,L. Filion a,2, A.T. Paulson d

a Department of Physics, St. Francis Xavier University, Antigonish, NS, Canada B2G 2W5b Department of Physics, Boise State University, Boise, Idaho 83725-1570, USA

c Departamento de Ciencias Experimentales, Universidad Jaume I, 12080 Castellon, Spaind Department of Process Engineering and Applied Science, Dalhousie University, Halifax, NS, Canada B3J 2X4

Received 23 January 2006; accepted 24 August 2006

Abstract

Electrostatic interactions are important in food systems, and modeling these systems correctly can lead to insights into their structureand dynamics when electrostatic interactions are correctly included. Here we give a brief review of electrostatics relevant to food systems,and discuss ways of approximating the electric fields and potentials. We emphasize the importance of correctly representing the boundaryconditions. We also stress the importance of recognizing the various length- and time-scales associated with electrically charged systems,as well as their interactions. We present four illustrative applications: (i) a simulation illustrating the validity of the linearized Poisson–Boltzmann approximation, by comparing to the exact solution of the electrostatic problem of many charges at a finite temperature; (ii)the effect of temperature on gellan aggregation by divalent cations; (iii) a comparison between explicitly modeled monovalent ions versustheir representation as a continuous charge density, using a monolayer of rough-mutant lipopolysaccharides as the example; (iv) the abil-ity of the cationic antimicrobial peptides (CAPs) protamine and lactoferricin B, and the antibiotic gentamicin, to penetrate an O-side-chain brush at the surface of a Gram-negative bacteria. We show that (i) the linearization approximation is remarkably good except closeto a charge; (ii) that we may represent the effect of monovalent ions as a suitable combination of explicitly represented ions and an aver-age continuous distribution yielding a Debye screening length; and (iii) that protamine, lactoferricin B, and gentamicin all penetrate anO-sidechain brush composed of uncharged sugars and reach the membrane surface, and (iv) that the two CAPs become trapped in theouter segments of a charged O-sidechain brush; but that gentamicin penetrates the charged brush.� 2006 Published by Elsevier Ltd.

Keywords: Modelling food systems; Electrostatic interactions; Linearized Poisson–Boltzmann equation; Gellan; Divalent cations; Monte Carlo calcul-ations; Lipopolysaccharides; Cationic antimicrobial peptides; Protamine; Lactoferricin; Gentamicin; Gram-negative bacteria

1. Introduction

Mathematical modeling has come to be recognized as animportant technique in understanding the behaviour of sys-tems, especially when supplemented by computer simula-

tion of the models. In order to model complex softmatter such as food systems, it is necessary that the inter-actions between the various components be correctly takeninto account. Many food systems are aqueous dispersionscontaining inorganic ions and aggregates of fats, polysac-charides, and proteins, ranging in size from nanometersto microns, along with larger-scale phases (Walstra, 1996,Chapter 3). Aggregates arise via molecular interactionsthat are due to Coulomb forces, which lead to covalent-bond formation/breaking (‘‘chemical’’ interactions involv-ing energies �5 · 10�19 J and greater) and lower energy‘‘physical’’ interactions, whose energies are one to two

0963-9969/$ - see front matter � 2006 Published by Elsevier Ltd.doi:10.1016/j.foodres.2006.08.001

* Corresponding author. Address: Department of Physics, St. FrancisXavier University, Antigonish, NS, Canada B2G 2W5.

E-mail address: [email protected] (C.B. Hanna).1 Present address: Department of Physics, Dalhousie University, Hal-

ifax, NS, Canada B3H 3J5.2 Present address: Department of Physics, McMaster University, Ham-

ilton, ON, Canada L8S 4M1.

www.elsevier.com/locate/foodres

Food Research International 39 (2006) 1031–1045

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orders of magnitude smaller. Broadly speaking, physicalinteractions may be taken as comprising the ubiquitous(charge-fluctuation induced) London–van der Waals typeinteractions (Israelachvili, 1991; Landau & Lifshitz, 1968;Leckband & Israelachvili, 2001; Lifshitz, 1956; Parsegian,2006), and ‘‘electrostatic’’ interactions – interactionsbetween those moieties that carry electric charges on suffi-ciently large (long) spatial (time) scales (Cevc, 1990; Cevc &Marsh, 1987; McLaughlin, 1989). The last-named encom-pass electric multipole (e.g., dipole–dipole) interactions.Hydrogen bonding is a third, weaker, Coulombic effect,which is specific to certain proton-containing moietiesand is short-range (<0.3 nm) and directional (Israelachvili,1991). Electrical charges are ubiquitous in food systems,and this leads to the importance of electrostatic interac-tions that are isotropic and long-range.

From the point of view of their physical attributes, threeproperties characterize food systems: (i) many of theminvolve aqueous solutions; (ii) they are composed of salts,proteins, fats, and polysaccharides, some of which are intheir ionic form; (iii) they can be in the form of emulsionsor other ordered structures, or simply heterogeneous mix-tures. Accordingly, their composition is such that theirstructure can involve separation into many coexistingphases – possibly microphases – separated by interfaces ofdifferent characteristics and complexities. In order to incor-porate, for example, an antimicrobial compound into afood system and have it remain available and bioactive, itis important to understand how it interacts with food com-ponents, what the effects of those interactions might be, andhow it might attack bacteria in its environment. Apart fromspecific interactions, it is likely that in many systems, it iselectrostatics that plays the most important role in theenergy component of the free-energy function. For example,the importance of electrostatic interactions in determiningthe minimum inhibitory concentration (MIC) of cationicantimicrobial peptides (CAPs) against bacteria was investi-gated for protamine acting against Pseudomonas aeruginosaPAO1 (Pink et al., 2003a) and against 21 food-related bac-teria (Potter, Truelstrup Hansen, & Gill, 2005). Anotherstudy showed that protamine, with a total charge of +20e,destabilized certain emulsions (Pink, Quinn, Paulson, Rous-seau, & Speers, 2003b) because of electrostatic interactions.

Electric fields are associated with long-range forces sincethe static field at a distance r from a point charge is propor-tional to 1/r2. Even if the effects of such a charge is screenedby the presence of monovalent ions (see below), the rangeof the field, using a linearized approximation to the Pois-son–Boltzmann equation, is proportional to e�jr/r2, andthis can have an effect beyond the range of hydrogen-bondor van der Waals interactions. It should also be noted that,although the attractive part of the van der Waals interac-tion between atomic-scale moieties is proportional to1/r6, the resulting mesoscale (summed van der Waals) inter-action between two hard surfaces that are an average dis-tance D apart is proportional to 1/D2. It has recentlybeen shown that if the molecular motion leading to the

resulting averaging of the two soft surfaces is much fasterthan the rate at which this average distance D betweenthe surfaces is changing, then the attractive van der Waalsinteraction between these two mesoscale soft interfaces isproportional to ‘n(1/D) (Hanna, Pink, & Quinn, in press).

The message we wish to convey is that electrostaticsplays a fundamental role in food systems, and that in orderto understand its effects at a molecular level, one must havean appreciation of how to correctly model the electrostaticeffects involved. These effects are not subtle. Thus, (i) ifboundary conditions at surfaces are not taken into accountsufficiently correctly, then the behavior of macromoleculessuch as proteins at such an interface could be modeledincorrectly. (ii) As we mentioned above, van der Waalsinteractions between two soft surfaces can be almostperipheral when compared to electrostatics. (iii) Multiva-lent ions cannot be treated as part of an averaged Debyescreening function: not only do they contribute to screen-ing but, equally importantly, they can take part in bridging.Ion bridging – in particular, divalent ion bridging, as thename suggests – might mistakenly be thought of as staticconfigurations, on the time scales of the system being stud-ied, in which a divalent ion bridges a pair of monovalentmoieties. This view is not tenable. Multivalent ion bridgingcan be a dynamical process in which the bridging ionsmove, on a possible rapid time-scale, between local energyminima and in which their entropy places an importantrole. If multivalent ion bridging were to be modeled as astatic process, it is likely that incorrect phenomena wouldbe predicted. (iv) In the presence of charged multicompo-nent polymers in which the soft interfaces are changingon a time-scale similar to that of the movement of the var-ious charged species present, the correct application of theboundary conditions are essential. (v) Another point con-cerns the use of buffers, which might be thought of as sim-ply a pH-controlling technique. Buffer molecules, however,become ionized and give rise to multivalent molecular ionsthat, in sufficiently high concentrations, can change thescreening in a dielectric. Such possible effects must at leastbe considered when electric fields are calculated.

In this paper we shall briefly review the fundamentals ofelectrostatics in dielectric media, and show how polyelec-trolytes and the effects of multivalent ions in an aqueoussolution can be modeled. We shall begin with the integralform of Gauss’s law and the differential form of it, Pois-son’s equation. We shall then briefly describe the impor-tance of correctly identifying the boundary conditionsand show how these can be accounted for, in ion-free aque-ous solutions that contain locally planar interfaces, byimage charges. We shall then consider aqueous solutionscontaining ions and outline how monovalent ions can bedescribed using the Poisson–Boltzmann equation, whichyields the Debye screening length, and compare them tocases of multivalent ions and certain buffers. Finally, weshall outline applications to three systems: (a) the effectsof monovalent and divalent ions in a solution containinggellan molecules, (b) the representation of monovalent ions

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near a charged interface, either explicitly or by a continu-ous charge distribution, and (c) the interaction of the outercell wall of a Gram-negative bacterium with protamine,lactoferricin, and gentamicin.

We emphasize that this paper is neither about modelsnor computer simulation. It is concerned with the correctapplication of electrostatics to any model, regardless oftechnique. The particular models that we use to illustrateelectrostatic problems involving computer simulation alladdress mesoscale problems that represent molecularmulticomponent systems with dimensions greater than10 nm. For such systems, it is impracticable to useatomic-scale models.

2. Poisson’s equation, uniqueness theorem, boundary

conditions and image charges (Frankl, 1986; Lorrain,

Corson, & Lorrain, 1988)

2.1. Poisson’s equation

The electric field ~Eð~rÞ at any point in space can, in theabsence of time-dependent magnetic fields, be derived fromthe electric potential wð~rÞ via ~Eð~rÞ ¼ � ~rwð~rÞ. The interac-tion of an electrical charge with other charges is propor-tional to the charge itself and to the electric potentialcreated by the other charges. Accordingly, the problem ofcharge–charge interactions reduces to the correct calcula-tion of the electric potential distribution for the system ofinterest. If the charges are all at fixed positions, then thecalculation need only be done once. If the charges are‘‘free’’ (to move), then one has a self-consistent problem:the movement of the charges is determined by the electricalpotential and vice versa. In the latter case, the potential hasto be calculated at each instant of time in order to deter-mine how the accelerations – and hence positions – of thecharges change with time. The integral and differentialforms of Gauss’s lawZ

S

~Eð~rÞ � d~sð~rÞ ¼ Qe; ð1aÞ

r2wð~rÞ ¼ � qð~rÞe

ð1bÞ

can be used to determine the electric potential (distribu-tion). Here Q is the total charge contained inside the vol-ume bounded by the surface S over which the integral ofthe electric field ~Eð~rÞ is carried out, qð~rÞ is the electriccharge density at point ~r, and e = e0er is the permittivity,where e0 is the permittivity of free space and er is the rela-tive permittivity describing the polarizability of the dielec-tric. Eq. (1a) is used to calculate the electric field when thesystem possesses sufficiently high symmetry; for example,when ~Eð~rÞ has a constant magnitude over the surface S.

2.2. Uniqueness theorem

An existence theorem states that the electrical potentialis unique if it satisfies Poisson’s equation and the boundary

conditions. This very powerful theorem means that we mayuse any technique whatsoever to obtain the electric poten-tial, subject only to those two requirements.

2.3. Boundary conditions, polarization, induced charges,

and energy

To obtain a correct solution for the potential from Pois-son’s equation, it is necessary to satisfy the boundary con-ditions. In the case of a charge Q located at the origin in auniform dielectric, these are limr!1wð~rÞ ¼ 0 andlimr!1

owðrÞor ¼ 0. In this case, we obtain as a solution to

Eq. (1b),

wðrÞ ¼ 1

4peQr: ð2Þ

The boundary conditions above are not valid for the caseof a non-uniform space, such as a system possessing aninterface. Examples of this include emulsions, bacterial sur-faces, and micellar delivery systems. Indeed it is unlikelythat any food system does not possess interfaces within it.Interfaces can become electrically polarized, whichamounts to inducing electrical dipoles at the interface,and it is therefore necessary to determine the boundaryconditions at interfaces. In order to apply the uniquenesstheorem, we need to satisfy the boundary conditions, whichare the continuity of the potential, w1ð~rSÞ ¼ w2ð~rSÞ, thecontinuity of the tangential component of the electric fieldE*

,

½E*

2ð~rSÞ � E*

1ð~rSÞ� � nð~rSÞ ¼ 0; or E2T ð~rSÞ ¼ E1T ð~rSÞ;ð3Þ

and the continuity of the normal component of the electricdisplacement D

*

,

½~D2ð~rSÞ�~D1ð~rSÞ� � nð~rSÞ ¼ rFð~rSÞ; or ~D2N ð~rSÞ�~D1N ð~rSÞ ¼ rFð~rSÞ;ð4Þ

where

~Dmð~rÞ ¼ e0~Emð~rÞ þ~P mð~rÞ ¼ em

~Emð~rÞ ð5Þ

is the electric flux density. Here, the vector~rS locates a po-sition on the interface in question, EmT ð~rSÞ is the compo-nent of the electric field in region m that is tangential tothe interface at position~rS, ~Dmð~rÞ and ~P mð~rÞ are the electricdisplacement and polarization vectors in region m at ~r,nð~rSÞ is a unit vector perpendicular to the interface at ~rS

and pointing outward from the interface, ~DmN ð~rSÞ is thecomponent of the electric displacement in region m perpen-dicular to the interface at~rS , rFð~rSÞ is the free-charge den-sity at~rS , and em is the permittivity of region m.

The boundary conditions in Eqs. (3)–(5) can be satisfiedby finding the polarization induced at every interface andinside the bulk of every material making up the system.For a linear isotropic dielectric, the electric polarization~P ð~rSÞ at point~rS at an interface that is induced by an elec-tric field ~Eð~rÞ is given by

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~P ð~rSÞ ¼ ðe� e0Þ~Eð~rSÞ: ð6Þ

The induced ‘‘bound’’ surface charge density rbð~rSÞ can beobtained from

rbð~rSÞ ¼ ~P ð~rSÞ � nð~rSÞ: ð7Þ

The surface charge density is referred to as the ‘‘bound’’surface charge density because it arises from the polariza-tion of atoms or molecules essentially fixed in space. It isthus distinguished from ‘‘free’’ charges that can move inspace.

For a given charge distribution, the electrostatic energyof the system inside a volume v is

U ¼ 1

2

Zv

qfð~r0Þwð~r0Þdvð~r0Þ; ð8Þ

where qfð~r0Þ is the ‘‘free’’ charge density at ~r0, wð~r0Þ is theelectric potential at~r0, dvð~r0Þ is the infinitesimal volume at~r0 and the integration is over v. Eq. (8) is, strictly speaking,formally correct only for continuous charge distributionsqfð~r0Þ. When discrete charges are included, care must be ta-ken to exclude the infinite self-energies that arise from theunphysical interaction of a discrete charge with its ownelectrostatic potential. The potential at any point~r0 is

wð~r0Þ ¼ 1

4pe0

Zv

qfð~r00Þ þ qbð~r00Þj~r0 �~r00j dvð~r00Þ

þ 1

4pe0

ZS

rfð~r00Þ þ rbð~r00Þj~r0 �~r0j dAð~r00Þ; ð9Þ

where qfð~r00Þ and qbð~r00Þ is the ‘‘free’’ and ‘‘bound’’ volumecharge densities in the infinitesimal volume dvð~r00Þ, andrfð~r00Þ and rbð~r00Þ are the ‘‘free’’ and ‘‘bound’’ surfacecharge densities in the infinitesimal surface area dAð~r00Þ.The integration is over the volume v and the surface(s) S.Although Poisson’s equation (1b) with Eqs. (3)–(9) are suf-ficient to calculate the energy of a system of charges – fixed

in position – in the presence of interfaces, one can appreci-ate the difficulties of applying these to electrostatics in com-plex systems. When one appreciates that these equationsenable us to calculate only the instantaneous electrostaticproperties of a dynamic system that changes in timeaccording to Newton’s equations of motion, one can seethe magnitude of the problem.

2.4. Application to a complex interface

Consider a portion of the outer membrane (OM) of aGram-negative bacterium in an aqueous solution contain-ing various concentrations of salts. Let us consider onlythat portion of the OM that is composed of B-band lipo-polysaccharides (LPS). Such LPS molecules possesscharged KDO-Core regions that connect the hydrocarbonchains to the highly charged O-sidechain polymers (Rivi-era, Bryan, Hancock, & McGroarty, 1988). Now, let usintroduce a cationic antimicrobial peptide (CAP) into thevicinity of this outer membrane. Will the CAP go to thesurface of the OM, or will it be repelled? In order to see

how the position of the CAP changes with time, we mustmake a model of the membrane, and the simplest is to con-sider the hydrophobic hydrocarbon chain layer as a homo-geneous continuum in which only a bound surface chargemight be induced. To a first approximation, the water-hydrophobic interface is the only interface present andwe might consider it to be essentially a plane. As timechanges, the position of all the charged moieties changeso that, at each instant of time, we must recalculate the elec-tric potential – which involves recalculating the bound sur-face charge density rbð~r00Þ at the interface that is induced byall the charges in the system. This is not a trivial problem.If we are not modeling an experiment that measures rbð~r00Þ,then knowledge of it is not needed – except to calculate thepotential wð~rÞ that is needed for calculating how the vari-ous components change with time. Accordingly, we seekan approach that gets around having to calculate theinduced bound surface charge rbð~r00Þ. Such an approachis provided by the method of image charges.

2.5. Image charges

The method of image charges involves finding the(unique) electrostatic potential that satisfies Poisson’sequation, together with the appropriate boundary condi-tions, by the artifice of introducing fictitious ‘‘images’’ ofthe free charges. Consider a single free electric charge Q1

located at position r*

1 at a perpendicular distance d1 froma plane interface that separates the space into two half-spaces, regions 1 and 2, which have permittivities e1 ande2, respectively. The charge Q1 is in region 1, and theperpendicular to the interface is taken to be the z-axis.Using the boundary conditions, it is easy to see that thepotential at any point ~r in region 1 can be calculated byplacing a fictitious image charge Q01 at position r

*01 in region

2, at a perpendicular distance d1 from the interface(Fig. 1A), so that

r1* � r

*01 ¼ 2d1z; ð10Þ

and taking Q01 to be

Q01 ¼e1 � e2

e1 þ e2

Q1: ð11Þ

The electric potential at point~r in region 1 is then

w1ð~rÞ ¼1

4pe1

Q1

j r*� r

*1jþ Q01j r*� r

*01j

!: ð12Þ

There are two contributions to the electrostatic potentialw1ð r

*Þ in Eq. (12). The first term is the source potential (seeEq. (2)) for the free charge Q1 located at position r

*1. The

second term is in reality the electric potential producedby the bound surface charge density rbð r

*Þ at the interfacez = 0 that is induced by the free charge Q1. Remarkably, in

region 1, this second term is equal to the potential (see Eq.(2) again) that would be produced by a fictitious imagecharge Q01 located at position r

*01 in region 2.


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pyThe free-charge density for a discrete free charge Q1

located at position r*

1 is given by

qfð~rÞ ¼ Q1dð r*� r

*1Þ; ð13Þ

where dð r*Þ is Dirac’s delta function, which is defined by

dð r*Þ ¼ 0; if j r* j 6¼ 0; and

Zdð r*Þd3r ¼ 1: ð14Þ

Note that only the free charge Q1 contributes to the free-charge density qf, not the fictitious image charge Q01. Itfollows from Eqs. (8)–(14) that the electrostatic energyresulting from introducing the free charge Q is

Uðd1Þ ¼1

2

1

4pe1

Q1Q01j r1* � r

*01j¼ Q2

16pe1d1

e1 � e2

e1 þ e2

� �; ð15Þ

where we have dropped the infinite self-energy due to theunphysical interaction of the free charge Q1 with its ownpotential. Eq. (15) is equal to one-half of Q1 times the po-tential at Q1 due to its image charge Q01.

Note that if e1 < e2, then U < 0, and the charge Q inregion 1 is attracted to region 2. This is the essential physicsbehind, for example, the attraction of dry breadcrumbs toan electrically charged object in dry air, such as a chargedrubber comb. But if e1 > e2, then U > 0, which means thatthe charge Q in region 1 will be repelled by region 2. This

latter situation is relevant, for example, to lipid membranesin aqueous solutions. The permittivity of the hydrophobicregion of a cell membrane is emem � 4e0, while that of anaqueous solution is ewater � 81e0. This means that non-organic ions located in solution are repelled from a mem-brane interface, so that the concentration of the ions closeto a membrane carrying zero net charge is less than that ina bulk solution.

Note that the image charge Q01 required for the calcula-tion of the potential in region 1 must be located in region 2.This is necessary in order that the boundary conditions andPoisson’s equation be satisfied. This suggests that, if wewant to calculate the potential in region 2, then we shouldlocate an image charge in region 1. This is true. If wereplace the original charge Q1 by a charge Q001 with

Q001 ¼2e2

e1 þ e2

Q1; ð16Þ

then the electrical potential at any point ~r in region 2 isgiven by

w2ð~rÞ ¼Q001

4pe2

1

j~r �~r1j: ð17Þ

The electrostatic energy of a system of charges depends onthe nature of the surrounding medium. A simple illustra-tion of this is provided by calculating the electrostatic en-ergy of two free charges (Q1 and Q2 at locations r

*1 and

r*

2 that are embedded in a surrounding space that consistsof two semi-infinite dielectric regions (regions 1 and 2, withpermittivities e1 and e2) that are separated by a plane inter-face located at z = 0. In the special case where the permit-tivities are equal (e1 = e2 = e), then there is only a singledielectric region, and the electrostatic energy is simply

U ¼ Q1Q2

4pej r*1 � r*

2j: ð18Þ

But if the charges are embedded in regions of varyingdielectric permittivities (e.g., heterogeneous food systems),then the electrostatic energy is more complex than Eq.(18). For the simple geometry we are considering (twodielectric half-spaces), we can use the method of imagesto calculate the electrostatic potential induced by the freecharges. For the case in which the two free charges Qj

are both in region 1 at distances dj (where j = 1, 2) fromthe interface, the electrostatic potential in region 1 is givenby applying the principle of superposition to the electro-static potential in Eq. (12) that was obtained from themethod of images:

w1ð~rÞ ¼1

4pe1

Q1

j r*� r

*1jþ Q01j r*� r

*01jþ Q2

j r*� r

*2jþ Q02j r*� r

*02j

!;

ð19Þwhere the position r

*0j of image charge Q0j is (cf. Eq. (10))

r*

j � r*0

j ¼ 2djz; ð20Þ

and Eq. (11) gives

Fig. 1. (A) A charge Q1 located at ~r1 in region 1, at a perpendiculardistance d1 from an interface (horizontal line) that separates region 1 fromregion 2. The permittivities are e1 and e2, respectively. The electricpotential w1ð~rÞ at~r in region 1 can be calculated by introducing an imagecharge Q01, located an equal distance from the interface inside region 2.The unit vector z is perpendicular to the interface. (B) Geometry of twocharges Q1 and Q2 in region 1 near an interface (horizontal line). Thepotential anywhere in region 1 can be calculated by introducing two imagecharges Q01 and Q02 in region 2 as shown.


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Q0j ¼e1 � e2

e1 þ e2

Qj: ð21Þ

Evidently, Eq. (19) has contributions both from sourcecharges (Q1 and Q2) and image charges (Q01 and Q02Þ. Thefree-charge density is given by (cf. Eq. (13))

qfð~rÞ ¼ Q1dð r*� r

*1Þ þ Q2dð r

*� r*

2Þ: ð22Þ

The free-charge density qfð~rÞ has contributions only fromfree charges (Q1 and Q2). It follows from Eqs. (8), (19)and (22) that

U ¼ 1

2

Q1

4pe1

Q01j r*1 � r

*01jþ Q2

j r*1 � r*

2jþ Q02j r*1 � r

*02j

!"

þ Q2

4pe1

Q1

j r*2 � r*

1jþ Q01j r*2 � r

*01jþ Q02j r*2 � r

*02j

!#; ð23Þ

where we have again excluded the infinite self-energies dueto the unphysical interaction of the free charges Qj withtheir own potentials. By using the fact that Q1Q02 ¼ Q2Q01and that j r*1 � r

*02j ¼ j r

*2 � r

*01j, we may simplify Eq. (23)

to obtain

Uð r*1; r*

2Þ¼1

4pe1

Q1Q2

j r*1� r*

2jþ1

2

Q1Q01j r*1� r

*01jþ1

2

Q2Q02j r*2� r

*02jþ Q1Q02j r*1� r

*02j

!:

ð24Þ

There are four contributions to the electrostatic energyUð r*1; r

*2Þ in Eq. (24). The first is just the electrostatic en-

ergy between the free charges Q1 and Q2. The second andthird are the electrostatic energies between each free chargeQj and its corresponding image charge Q0j (cf. Eq. (15)).The last term in Eq. (24) is the sum of the electrostatic ener-gies between each free charge Qj and all the non-corre-sponding image charges Q0k (with k 6¼ j). Note thatEq. (24) does not contain any terms corresponding to inter-actions between image charges. This is because the free-charge density qfð~rÞ has contributions only from freecharges, not from any image charges.

We emphasize that the method of image charges is onlyone way, albeit a very convenient one, of satisfying theboundary conditions at interfaces. Formally, this methodis applicable only to a limited class of boundary conditionsin which the interfaces possess relatively high symmetry.Even so, the method of images can be very useful in mod-eling and understanding the physics underlying electrostat-ics in heterogeneous systems, such as food systems.

3. Treating ions in solution as a continuous charge density:

the Poisson–Boltzmann (P–B) equation

Although electrical charges are localized on atoms ormolecules, we can approximate the charge distribution bya continuous function of position, if the spatial scale ofthe system is such that the distances between the chargesis very much less than the characteristic spatial scale inwhich we are interested. Even if we are interested in spatial

scales comparable to the characteristic distance betweencharges, it is possible to approximate their locations byan average distribution if the time-scale of their movementand redistribution is much shorter than that of the systemfor which we are calculating the electric potential. Anexample of this is the electrical potential due to a relativelystationary charge located on a surface or on a polymer inthe presence of ions (free charges) in the surrounding aque-ous solution. One assumes that the free charges in solutionare moving sufficiently rapidly that the stationary chargesees an average ‘‘smeared-out’’ distribution of free chargescharacteristic of some equilibrium. In many cases, theaqueous solution is in thermodynamic equilibrium, charac-terized by an ambient temperature T, so that the spatialdistribution of the free-charge density is determined bythe Boltzmann factor exp(�bU), where U is the energyand b = 1/kBT. Taking into account the requirement oftotal electroneutrality, this converts the (linear) Poissonequation (1b) into the Poisson–Boltzmann (P–B) equation(Israelachvili, 1991), which is non-linear because w appearsin the exponent:

r2w ¼ eere0

Xi

zin0i exp

ziewkBT

� �: ð25Þ

Here e is the elementary charge, and n0i and zi are the con-

centration and valence of ith type of ion in solution.It is worth noting that most continuum models of aque-

ous solutions do not take into account collective propertiesof water such as dynamical hydrogen bonding involvingwater molecules. This effect has been modeled as non-localinteractions (e.g., Belaya, Levadny, & Pink, 1994 and refer-ences therein).

4. Linearized Poisson–Boltzmann equation: the Debye and

Gouy–Chapman solutions

4.1. The linearized P–B equation and the Debye screening

length

In cases where entropic effects (sufficiently-high temper-ature T) dominate the energetics, i.e., when the potential wis sufficiently small so that w� kBT/e (�25 mV at roomtemperature), then the exponential term in Eq. (25), exp(ziew/kBT), can be expanded to first order in ziew/kBT. Thisresults in approximating the non-linear P–B equation by alinearized differential equation (the linearized P–B equa-tion) for which an analytical solution can be obtained. Inthis case, the P–B equation becomes, using the conditionof total charge neutrality,

Pizin0

i ¼ 0,

r2w ¼ we2

ekBT

Xi

z2i n0

i ¼ j2w; ð26Þ

where

j2 ¼ e2

ekBT

Xi

z2i n0

i ¼ k�2; ð27Þ


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and n0i and zi are the concentration and the valence of the

ith type of ion in the bulk solution. The units of j areclearly inverse length (e.g., nm�1). The distance k can betreated as the effective radius of a perturbed volume in aneutral solution that contains mobile ions and is calledthe ‘‘Debye (screening) length’’ (Israelachvili, 1991). Inthe simplest case of a 1:1 electrolyte, where n0

1 ¼ n02 ¼ n,

Eq. (27) becomes

j2 ¼ k�2 ¼ 2e2n0

ekBT: ð28Þ

The Debye screening length of a 100 mM solution at roomtemperature (300 K) is �1 nm.

4.2. Debye and Gouy–Chapman solutions

There are two classical cases of the application of thelinearized P–B equation: (a) the case of spherical geometry(so that the solutions to Eq. (26) are functions of r ¼ j~rjÞ,where the radius of curvature of the surface is comparableto or smaller than the Debye length k = j�1, and (b) thecase of plane geometry (where the solutions to Eq. (26)are functions only of the perpendicular distance, z, fromthe plane), where the radius of curvature of the surface ismuch larger than the Debye length. The solution of Eq.(26) in the first case is the basis of Debye’s theory of solu-tions and is called the ‘‘Debye equation’’ (Hunter, 2001;Israelachvili, 1991):

w ¼ Q4pe

expð�r=kÞr

: ð29Þ

The solution of Eq. (26) in the second case is the basis ofthe Derjaguin–Landau–Vervey–Overbeek (DLVO) theoryof the stability of solutions of colloidal particles (Derjaguin& Landau, 1941; Hunter, 2001; Verwey & Overbeek, 1948),and is called the ‘‘Gouy–Chapmann equation’’ (Chapman,1913; Gouy, 1910; Hunter, 2001; Israelachvili, 1991),

w ¼ w0 expð�z=kÞ ¼ rke

expð�z=kÞ; ð30Þ

where w0 is the surface potential of the particle, and r is itssurface charge density (Hunter, 2001; Israelachvili, 1991).

4.3. The meaning of the Debye screening length

Consider the electric potential in the neighbourhood(r� k) of a spherical particle with a charge Q at its centretaken to be the origin of coordinates. We can write thepotential of Eq. (29) as

w ¼ 1

4peQ expð�r=kÞ

r� 1

4peQr� Q

k

� �: ð31Þ

This shows that the electric potential at any point in theneighborhood of the charge Q is the sum of two terms.The first one is due to the charge Q at the origin, whilethe second is due to free charges in the solution that are at-tracted to, or repelled by, the source charge Q. Because the

net charge of the latter always has the opposite sign to Q, itis called the ‘‘countercharge’’. This manifestation of coun-tercharge is called ‘‘screening’’ because the counterchargereduces the potential that a test charge would experienceas it approached the charge Q – the countercharge, setup by the free charges in the solution, screens Q in its inter-action with other charges.

Eq. (29) shows that the potential, at a distance r from afixed charge in an aqueous solution containing sufficientlyrapidly moving ions, is determined by the product of 1/rand exp(�r/k). The first is the familiar ‘‘long-range’’ 1/r-dependence of the Coulomb potential while the seconddepends upon the ion concentration in the solution. Sincean exponential of this kind will dominate any inversepower of r, it converts the potential to ‘‘short-range’’.

4.4. Application 1. How good is the linearized P–B

approximation? Calculation of j

We can calculate the accuracy of the truncated expan-sion (Eq. (26)) that linearizes Eq. (25). Here we showMonte Carlo calculations of w, in order to see to whatextent it can be approximated by Eq. (29). Fig. 2 showsthe results of computer simulations that calculated the

Fig. 2. The electric potential w(r) as a function of distance from a +e

charged sphere fixed at the origin, in the presence of 479 +e and 480 –e

charged freely moving spheres of radius rS confined to a spherical volumeof radius R = 102a. (a) Heavy black lines indicate ‘n[(r/a)w(r/a)/w0], (b)heavy gray lines indicate the straight line y = C � jr, and (c) thin blackdashed lines indicate a best-fit straight line (using linear regression) to‘n[(r/a)w(r/a)/w0]. The concentration of spheres is 0.180 M, correspondingto a Debye screening parameter of ja = 0.134 when a = 1 A. In A and B,the interaction between charges is unscreened: Q1Q2/r12, with rS = 1.8a

(A) and rS = 0.5a (B). The gray line is for ja = 0.134, and the thin blacklines have slopes �0.139 (A) and �0.136 (B). In C and D, the interaction isscreened: Q1Q2 exp(�j 0r12)/r12, with j0a = 0.1, rS = 1.8a (C), and rS =0.5a (D). The gray line is for ja = 0.167 and the thin black lines haveslopes �0.170 (C) and �0.167 (D). In both cases, the results of thesimulation are in good agreement with the theoretical values for the Debyescreening parameter j.


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potential w(r) as a function of the distance r/a from thecentre of a spherical volume of radius R = 102a (wherea = 0.1 nm). The spherical volume contains N � 1 freepositive (+e) and N free negative (�e) charged spheres ofradius rS that represent spherical ions, with a singlepositively charged sphere, of the same size, fixed at the ori-gin of the spherical volume. The value N = 480, corre-sponding to an ionic concentration of 0.180 M, givesja = 0.134. The interaction between any pair of ions a dis-tance r apart was taken to be the (unscreened) Coulombinteraction 1

4peQ1Q2

r , where Qj is ±e (j = 1,2). If linearizingthe P–B equation is a valid approximation, then the poten-tial should obey the Debye–Huckel equation (26) for r < R.Eq. (29) gives

‘n½ðr=aÞw1=w0� ¼ C � jr ð32Þfor r < R, where ew0 = 10�20 J is a constant with dimen-sions of energy, and

C ¼ ‘n e2=4peaew0

� �� ‘n 231

81

� �� 1:05: ð33Þ

Fig. 2A and B show plots of ‘n[(r/a)w(r)/w0] versus r/afor charged spheres with rS = 1.8a (Fig. 2A) andrS = 0.50a (Fig. 2B). The slopes are �0.139 and �0.136,respectively, which are close to the theoretical slope ofj � �0.134. We then replaced the unscreened interaction(above) by the solution to the linearized P–B equation,1

4peQ1Q2 expð�0:1r=aÞ

r . If the linearized P–B equation is valid, this

is equivalent to increasing ja from ja = 0.134 to

ja ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið0:134Þ2 þ ð0:1Þ2

q¼ 0:167, since Eq. (27) shows that

j2 is additive. Fig. 2C (rS = 1.8a) and 2D (rS = 0.50a) showplots of ln[(r/a)w(r)/w0] versus r/a, and the slopes of �0.170and �0.167 compare well with the expected value ofja = 0.167 inverse length units. This result shows that,given a set of monovalent ions, we may divide it into twosubsets, representing one by a continuum described by aDebye screening length, and treating the ions in the otherset as spheres, or some other localized objects. This hasgreat utility if we want to model the ions as spheres butdo not want to have to deal with infinite-range,(unscreened) 1/r potentials (see below, Application 3,Fig. 5).

5. The linearized P–B equation in the presence of an interface

Netz (e.g., 1991, 2001a, 2001b) has studied the Pois-son–Boltzmann equation in the presence of interfaces.We summarize below the results for two cases of specialinterest: (1) an ionic solution in contact with a conductor(metallic half-space), and (2) the physically related case ofan ionic solution in contact with a dielectric half-space, inthe limit where the ratio of the permittivity of the ionicsolution to that of the dielectric diverge. We focus onthe case in which there is no surface charge density atthe interfaces.

5.1. Ionic solution in contact with a conductor

Consider a positive charge Q located at a distance d

from an infinite plane surface of a conductor that fills thehalf-space z < 0 (region 2). The interface occupies the x–y

plane at z = 0. The half-space z > 0 (region 1) is filled witha solution of permittivity e1 and contains ‘‘free’’ ions whichdefine an inverse Debye length j (Eq. (27)). The positivecharge is located at z = d in the dielectric. If there are nofree ions in the solution, then Eqs. (11), (12), (16) and(17) give the potentials in regions 1 and 2. The questionnow is: with the free ions present, what is the electric poten-tial at any point ~r in the dielectric? The permittivity of aconductor is e2!1. We also know that, in a conductor,the potential is everywhere a constant, so we choose it tobe zero, w2ð~rÞ 0. The fact that it is identically zero meansthat all its derivatives are zero.

If we let e2/e1!1 in Eq. (11), we get the well-knownresult Q 0 = �Q, and Eq. (12) becomes

w1ð~rÞ ¼Q

4pe1

1

r� 1

r0

� �; ð34Þ

where r and r 0 are the distances from the charge Q andfrom its image charge Q 0, respectively, to the point~r, in re-gion 1, at which the potential is calculated (Fig. 1). Eq. (34)automatically satisfies the boundary conditions. We canmodify Eq. (34) if we have free ions in region 1. All we needdo is find a solution to the linearized P–B equation in re-

gion 1 that satisfies the boundary conditions, and a solutionthat does both is a combination of Eqs. (29) and (34),

w1ð~rÞ ¼Q

4pe1

expð�jrÞr

� expð�jr0Þr0

� �: ð35Þ

5.2. An aqueous solution containing ions in contact with a

dielectric

This is one of the problems that we must address since itis relevant to the cases of, for example, a charged moleculeat a bacterial surface or DNA near a membrane. Under-standing the dynamics of any charged molecule in solutionthat approaches a dielectric surface requires a solution tothis problem – and it is not simple, for while there maybe free ions in region 1, there are no free ions in region2. If we let e1/e2!1 in Eq. (11), we get Q 0 = Q, andEq. (12) becomes

w1ð~rÞ ¼Q

4pe1

1

rþ 1

r0

� �; ð36Þ

where r and r 0 are the distances from the charge Q andfrom its image Q 0, respectively, to the point~r in region 1,where the potential is calculated (Fig. 1). Eq. (36) automat-ically satisfies the boundary conditions. We can modify Eq.(36) if we have free ions in region 1. Again, all we need do isto find a solution to the linearized P–B equation in region 1that satisfies the boundary conditions, and a solution thatdoes both is (cf. Eq. (35))


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w1ð~rÞ ¼Q

4pe1

expð�jrÞr

þ expð�jr0Þr0

� �; ð37Þ

which is the solution given in Netz (1999) for e1 /e2!1.In practice, it is convenient to make the approximation

w1ð~rÞ �Q

4pe1

expð�jrÞr

þ e1 � e2

e1 þ e2

expð�jr0Þr0

� �; ð38Þ

which, although not exact, correctly gives Eq. (37) in thelimit e1/e2!1, and Eqs. (11) and (12) in the limit j! 0.

To put the problem of dielectric screening in the pres-ence of interfaces in perspective, the Debye screeninglength for an aqueous solution containing a physiologicalconcentration of monovalent ions of �0.1 M is �1 nm.The thickness of the KDO-Core region of an LPS for aGram-negative bacterium is typically �2–3 nm, while theend-to-end length of an extended O-sidechain can be 20–40 nm. Hence, if we are studying the electrostatics ofcharged molecules in the O-sidechain region, the crudeapproximations described above are probably satisfactory.It is only if we are concerned with effects deep into theKDO-core layer that we need to think again. Also, if wewant to model molecules entering the region-2 dielectricfrom region 1, we must make better models than the simpleones described here.

6. A comment on the superposition principle

The ‘‘superposition principle’’ is one of the basic princi-ples of electrostatics, and follows from the linearity of Pois-son’s equation. It means that the total electric potential dueto two charges is the sum of the electric potentials due toeach of them. This follows from the fact that, if w1 andw2 are two solutions of a linear equation, then so isw1 + w2. This property is not true of non-linear equations,so that at first glance, the superposition principle does nothold for the non-linear P–B equation (25): i.e., if w1 and w2

are two solutions of Eq. (25), then w1 + w2 is not generallya solution. There has been considerable discussion aboutthis. To resolve this seeming paradox, it is necessary torealize that the superposition principle assumes a constant(fixed) number of charges in the system under consider-ation, and that this requirement is not satisfied by thenon-linear P–B equation. Making use of Boltzmann’s dis-tribution function exp (�bew) means that we are consider-ing a system with an infinite number of charges in thereservoir that maintains the temperature. The two poten-tials w1 and w2 could represent the potentials due to twodifferent charges (e.g., two arginine residues, two polarheadgroups, two calcium ions, etc.). If the superpositionprinciple is valid, these two potentials can be added. How-ever, the superposition principle is not generally valid fornon-linear equations. But if the potentials w1 and w2 areconsidered as approximations to the true potentials –which can be added – then this justifies simply adding w1

and w2 to get the total potential. The non-linear P–B equa-tion does not violate the basic superposition principle of

electrostatics, because it describes a system that does notpossess a constant number of charges. However, we makeuse of the superposition principle because the potentialsare approximations to the true potentials for which super-position is valid. It should be borne in mind that we canalways apply the superposition principle to the solutionsto the linearized P–B equation.

7. Modelling multivalent ions and buffers

The use of the P–B equation is valid when the ions in theaqueous solution change their positions on time scalesmuch faster than all other time scales being considered,and when correlation effects are unimportant. Thus, thepotential due to a relatively immobile charge on a polymerin an aqueous solution containing rapidly moving monova-lent ions can be treated using P–B theory – i.e., the rapidlymoving monovalent ions can be treated as a continuouscharge density (above). If the solution contains multivalentions, however, then positional correlations must be consid-ered. What this means is that individual multivalent ionscan form (dynamical) ‘‘bridges’’ between relatively immo-bile charges of the opposite sign, and so a strong correla-tion can exist between three charges – two relativelyimmobile charges and one multivalent ion – a situationnot often observed in the case of monovalent ions in solu-tion. Thus, we can use P–B theory to describe NaCl in solu-tion but not CaCl2, even though the two Cl� ions might bedescribed by a continuous charge density.

The situation becomes more complicated in the presenceof buffers in the solution, since a buffer can contribute mul-tivalent ions. It might be thought that buffers, which areadded to maintain the pH, can be ignored when modelingbuffered systems. This is not so, especially if the buffer con-centration approaches a significant fraction of the concen-tration of monovalent ions in the system. If the buffersyield only monovalent ions then they must be included ina continuum charge density in the P–B equation. Consideran electrolyte containing a buffer. By how much does theDebye screening length differ if we ignore or include theeffect of the buffer ions? Let n0

þ and z0+, n0� and z0� repre-

sent the number of positive and negative ions and thecharges per ion of the electrolyte, respectively. Let nb

þ andzb+, nb

� and z0b represent the number of positive and nega-tive ions and the charges per ion of the buffer, respectively.Then, if we denote the Debye screening length of the elec-trolyte only by k0 and that of the electrolyte plus buffer byk, we obtain

k ¼ k0 1þnbþz2

bþ þ nb�z2

b�n0þz2

0þ þ n0�z2

0�

� ��1=2

� k0 1� 1

2

� �nbþz2

bþ þ nb�z2

b�n0þz2

0þ þ n0�z2

0�

� �: ð39Þ

Let us estimate k when the electrolyte is 0.1 M NaCl andthe buffer is 10 mM Na3PO4 (but see our comment belowregarding the validity of representing multivalent ions by


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a continuum, as can be done for monovalent ions). Substi-tuting the numbers into Eq. (39), we find k � 0.7k0. Thus,the buffer decreases the Debye screening length by about30%. Bearing in mind that this change will take place inan exponential in the expression for the potential, we seethat the presence of buffers can substantially change theelectrostatic interactions in an aqueous system.

There is, however, a further complication if the buffercontributes multivalent ions. This comment holds for themodeling of any multivalent ions. If the buffer yields multi-valent ions (e.g., HEPES or Na3PO4), then those ions can-not be simply included as part of the monovalent ioncontinuum. PO3�

4 ions can, for example, form dynamical‘‘bridges’’ with relatively immobile monovalent cations justas effectively as Ca2+ does with relatively immobile mono-valent anions, and they must be treated as moving chargedobjects. A failure to take this into account might yieldresults that implied that experimental results depend uponthe type of buffer used.

8. Manning–Oosawa condensation and overcharging

It was noted by Manning (1978) and Oosawa (1968) thata finite-size object (i.e., not a point charge) carrying acharge Q > 0 can attract to it two finite-size objects, eachcarrying a negative charge �Q 0, and thereby create a neg-atively charged composite. This is called ‘‘overcharging’’.This effect is easily understood if we realize that a(Q,�Q 0) pair formed from finite-size objects is a finite-sizecharged electric dipole, and so can attract another posi-tively or negatively charged object. However, there is acondition on Q and Q 0. Consider a spherical object ofdiameter a, carrying a charge Q > 0 at its centre. Considertwo spherical objects, each of diameter b, carrying charges�Q 0 < 0 at their centres. The minimum energy occurs whenthe two latter objects touch the positively charged sphere,and this energy is U = [�4QQ 0 + Q 02]/4pe(a + b). For thissystem to be stable, we must have U < 0, and this is satis-fied if Q 0 < 4Q. In a solution, this effect can lead, for exam-ple, to negatively charged polymers becoming bound tonegatively charged interfaces. One should be aware thatwhat determines such binding at finite temperatures is thefree energy – the electrostatic energy plus the entropic com-ponent that is proportional to the temperature – so that, ata sufficiently high temperature, the 3-sphere system will notbe stable. Manning–Oosawa condensation is an example ofthe sorts of correlation effects that are missed by the Pois-son–Boltzmann equation, which considers only thesmeared-out charge distribution.

9. Applications

Here we apply the physical principles discussed above todescribe the modeling of mesoscopic systems relevant tofood science and microbiology. Because of the complexityof these models, all numerical results have been obtained

by Monte Carlo computer simulation using the Metropolisalgorithm (Binder, 1984).

9.1. Application 2: Condensation of charged semi-flexiblepolymers via divalent cations

We modeled a gellan solution as linear polymers inside asimulation volume with periodic boundary conditions. Inits fully ionized form, each gellan polymer possesses onenegative charge (due to the carboxyl ion on glucoronicacid) per repeat group of four monomers (Chandrasekaran& Radha, 1995). We represented the monomers by hardspheres that were connected, centre-to-centre, via stretch-able bonds (Carmesin & Kremer, 1988). The spheres were1 nm in diameter. To model the semi-flexible gellan system,two successive bonds were constrained to form an internalangle of 150 ± 5�. A negative charge of �e was located atthe centre of every fourth monomer sphere. In the simula-tion volume, we distributed other spheres, each carrying apositive charge of +2e at their centres, to represent divalentcations. We assumed that the entire system was located inan aqueous solution containing 0.1 M of a monovalent saltand we used Eq. (29) with a Debye screening lengthk = 1/j = 1 nm to calculate the energetics of the system.The intent was to show how divalent cations form bridges,and to study the entropic effects of temperature. Fig. 3Ashows a snapshot of the system at a high temperature, with

Fig. 3. Model of gellan and calcium ions. Every fourth sugar of eachpolymer possesses a negatively charged sugar. Three linear polymers cancombine to form a helical segment, represented as a rigid cylinder, inwhich the charged moieties spiral around the surface of the cylinder. (A)Typical equilibrium conformation at high temperatures. (B) Equilibriumconformation at low temperatures. Note how the calcium ions havebrought about a collapse of the structure on the right.


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the divalent ions more or less randomly distributed inspace, although some attachments to the polymers can alsobe seen. Fig. 3B shows the effect of lowering the tempera-ture, so that cations condense and act, in some cases, as

bridges. This is supported by the observation that somepolymers have collapsed on each other in the presence ofthe divalent ions.

9.2. Application 3: Accuracy of replacing finite-size ions by a

continuous charge density near an interface using linearizedP–B equations

We modeled LPS molecules on the outer surface of aGram-negative rough-mutant bacterium spread at an air–water interface. In the case of rough mutants, the O-side-chain is absent. The model is similar to that of Application2, in that sugars are represented by spheres connected bystretchable bonds, as shown in Fig. 4, and ions are repre-sented by spheres.

Fig. 4. Representative model of the hydrophilic segment of a lipopolysa-charide (LPS). The sugars are represented by spheres connected by bondsrepresenting the –O– groups. The hydrocarbon chain region is representedby a sphere which can move (±0.15 nm) out of the plane of the membrane.Two of the spheres are charged, �e and �2e. In the case of a roughmutant, the O-sidechain is absent.

Fig. 5. Distributions of sugar spheres (KDO-Core), Ca2+ spheres, Na+

spheres, and Cl� spheres. All Ca2+ ions were represented by spherescarrying charge +2e. Heavy black curves: 480 Na+ and Cl� spheresrepresenting a �170 mM NaCl solution, together with j = 0.5 nm�1,giving a total Debye screening with j = 1.414 nm�1. Gray curves: Allmonovalent ions are represented by Debye screening with j = 1.414 nm�1.Note that, irrespective of which model we chose for the monovalent ions,the KDO-Core and Ca2+ distributions remained unchanged.

Fig. 6. Ball-and-stick models of protamine (A), lactoferricin B (B), andgentamicin (C). Charges of +e are located at the centre of the argininespheres (Ac). In B, a disulfide bond connects the two cysteines. The modelgentamicin carries the charges at the centres of the spheres (Cb).


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The intent here was to see to what extent one can repre-sent monovalent ions as a continuous charge density near anegatively charged interface in the presence of divalentcations. The point is that – notwithstanding the observa-tions following Eq. (31) – representing monovalent ionsby a charge density and a Debye screening length k = j�1

might not correctly represent the possible increase in con-centration of monovalent ions near the charged interface.If the system also possesses divalent cations – which mustbe modeled as individual spheres and not a continuum –then is there any difference in the divalent cation distribu-tion near the charged interface that depends upon howwe model the monovalent ions?

We represented a portion of the monolayer by 100model rough-mutant LPS molecules in the x–y plane.The air–water interface was taken to define z = 0 with anaqueous solution occupying the space 0 < z < zMAX. Wedid not concern ourselves with the details of the hydrocar-bon chains, but represented them as large spheres thatserved to anchor the remainder of the LPS molecule in

the monolayer. The area projected by these spheres ontothe monolayer plane was characteristic of the cross-sectionarea of 6 LPS hydrocarbon chains in a fluid phase. Thenumber of Ca2+ and Cl� spheres (150 and 300) representeda CaCl2 ionic concentration of �0.1 M.

We ran two Monte Carlo simulations. In the first, wealso included 480 Na+ and Cl� spheres representing a�170 mM NaCl solution, and we chose j = 0.5 nm�1. Thischoice of j corresponds to a �0.025 M monovalent ionsolution. For the second simulation, we eliminated theNa+ and Cl� spheres, but included them in the continuouscharge density by using j = 1.414 nm�1 to represent thetotal �0.2 M NaCl solution. Our question was: are thereany differences in the spatial distributions of the Ca2+

and Cl� spheres? Fig. 5 shows the results, and it can be seenthat the two models give nearly identical Ca2+ and Cl�

sphere distributions. Accordingly, within the limits ofvalidity of the linearized P–B equation, one can, for allpractical purposes, represent all monovalent ions by a con-tinuous charge density.

Fig. 7. Distributions of antimicrobial molecules and LPS moieties in a monolayer of model LPS molecules (cf. Fig. 4, but with the O-sidechain present) asa function of distance z perpendicular to the plane of the monolayer: 5 protamine (A and B), 5 lactoferricin B (C and D), and 50 gentamicin (E and F)molecules. The LPS molecules are about 1.6 nm apart, characteristic of the density of such molecules in the outer leaflet of an outer membrane of a Gram-negative bacterium, Pseudomonas aeruginosa PAO1. Distributions shown are LPS (a), Ca2+ (b), and antimicrobial (c). The Cl� has not been shown, since itwas distributed almost uniformly over most of the simulation volume. (A, C and E) Uncharged O-sidechains. (B, D and F) Charged O-sidechainscomposed of trimers of sugars characterized by the charge sequence +,�,�. CaCl2 is present. Almost all antimicrobials penetrate the uncharged O-sidechains to reach the surface of the membrane. Note, however, that although the charged O-sidechains trap the protamines and lactoferricins far fromthe membrane surface, this is not achieved in the case of gentamicin, which penetrates the O-sidechain brush. This might explain why, although the firsttwo exhibit some antimicrobial activity, gentamicin is used as an antibiotic.


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9.3. Application 4: Interaction of B-band LPS with

protamine, lactoferricin, and gentamicin

Our last application compares the effects of interactionsbetween charged antimicrobial molecules and a monolayerof B-band LPS molecules of P. aeruginosa PAO1. Theintent here is to see how CAPs such as protamine (e.g.,Hancock & Chapple, 1999; Islam, Itakura, & Motohiro,1984) and lactoferricin (e.g., Bellamy et al., 1992) are inhib-ited in penetrating the B-band polymer brush compared tothe antibiotic gentamicin (e.g., Enderlin, Morales, Jacobs,& Cros, 1994). Fig. 4 shows the model LPS that we used,where it is seen that the O-sidechain is a polymer composedof trimer repeat units with charges (+,�,�) at physiologi-cal pH. As above, we represented sugars by spheres con-nected via stretchable bonds, ions by charged spheres,and the hydrophobic region of the LPS by an anchoringsphere. A similar approach was taken in modeling thetwo CAPs and the antibiotic, and they are shown inFig. 6A (protamine), B (lactoferricin) and C (gentamicin).

We carried out Monte Carlo simulations with 5 prota-mines, 5 lactoferricins, and 50 gentamicins using the linear-ized P–B expression, Eq. (29), for the electric potentials,with k = 1 nm. The results are shown in Fig. 7A and B(protamine), C and D (lactoferricin), and E and F (genta-micin). In order to compare them, we utilized approxi-mately equal masses of each molecule. This means thatthere are many more gentamicin molecules than eitherprotamine or lactoferricin. We see that both protamineand lactoferricin are trapped by the B-band O-sidechainsof the LPS in Fig. 7B and D, respectively, but that a frac-tion of the gentamicin gets through to the hydrocarbonchain layer in Fig. 7F. Fig. 7A, C, and E, show the resultsof protamine, lactoferricin and gentamicin, respectively, inthe presence of uncharged A-band O-sidechains. There aretwo conclusions that follow from these results: (a) chargedO-sidechains – possessed by the wild-type of this bacterium– provide a good static defense against positively chargedmolecules, and (b) the larger the molecule, the better thecharged polymer brush is able to trap them. Although gen-tamicin carries a charge of only +3e compared to lacto-ferricin of +8e and protamine of +20e, the increased sizeof the latter offer many opportunities for the semi-flexiblebrush to trap them electrostatically. The small size of a gen-tamicin molecule enables it to avoid being trapped. Theeffects of Ca2+ ions on the efficacy of protamine againstP. aeruginosa PAO1 were studied by Pink et al. (2003a).

Finally, we show instantaneous snapshots of typicalconformations for these three cases and both A- and B-band LPS (left and right-hand pictures, respectively). Theseserve to relate the average value distributions above, to thefluctuations that take place in the O-sidechains. Note therelease of some calcium ions in the cases when moleculespenetrate to the KDO-Core. In Fig. 8A and B (protamine)and C and D (lactoferricin), note the large amplitudes ofmovement of the polymer brush and the trapped protamine(B) and lactoferricin (D) in B-band LPS. In Fig. 8E and F

(gentamicin), note also the ability of gentamicin to pene-trate both uncharged (E) and charged (F) O-sidechainbrushes and the large amount of divalent cations releasedin the presence of A-band LPS (E).

10. Conclusions

We have reviewed the fundamentals of electrostatics asapplied to complex systems, such as food systems, pos-sessing various phases and containing free ions, anddescribed how to treat electrostatics at the interfacesbetween the phases. We investigated the validity of the

Fig. 8. Instantaneous snapshots of the six cases shown in Fig. 7, with thethree-dimensional distributions projected onto the x–z plane. The mono-layer lies in the x–y plane with the z-axis vertical. (A and B) 5 protamines,(C and D) 5 lactoferricins, (E and F) 50 gentamicins. The LPS moleculesare about 1.6 nm apart. (A, C and E) Uncharged O-sidechains. (B, D andF) Charged O-sidechains composed of trimers composed of sugarscharacterized by the charged sequence +,�,�. CaCl2 is present. Solidfilled circles indicate Ca2+ ions and antimicrobial moieties. Smaller graycircles connected by lines indicate LPS components. The Cl� has not beenshown.


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linearized Poisson–Boltzmann (P–B) equation, and foundthat it is remarkably good at describing the charge distri-bution of monovalent ions in solution at temperatures andionic concentrations characteristic of food systems. Wediscussed electrostatic boundary conditions, and describedwhen image charges can be usefully employed. We empha-sized the importance of recognizing the various length-and time-scales associated with electrically charged sys-tems, as well as their interactions: the characteristic scalesand interactions of Ca2+ ions, for example, can be verydifferent from those of charged extracellular polysaccha-rides. We then studied the effects of electrostatic interac-tions in models of three complex systems: (i) a solutioncontaining gellan molecules with monovalent and divalentions, and the effect of temperature upon the gelling pro-cess; (ii) the representation of monovalent ions near acharged interface, either explicitly or by a continuouscharge distribution. In this example, the model interfacewas chosen to be composed of a monolayer of rough-mutant lipopolysaccharides; (iii) the interaction betweenthe LPS monolayer on the outer (cell wall) membrane ofa Gram-negative bacterium, and the cationic antimicro-bial peptides protamine and lactoferricin, and the antibi-otic gentamicin.

Our conclusion is that solutions to the linearized P–Bequation are acceptable for a wide range of distances froma charge, depending upon what deviations from the exactsolution are permitted. The latter are determined by theenergetics of any other relevant interactions, and theambient temperature. Thus, for example, at T � 300 Kand in the presence of �50–100 mM monovalent ions,the linear P–B approximation is adequate to within afew Angstroms of a charged ion. In the presence of dielec-tric surfaces, however, the linearization can be usedapproximately to within a couple of Debye screeninglengths of an interface. However, to model the penetrationby a charged molecule from an aqueous environment intoan oily dielectric (e.g., the hydrocarbon-chain region of amembrane) where there are no free (screening) charges,requires further considerations beyond what has been cov-ered here. Although the simple models outlined in thispaper are valid for the applications treated here, an under-standing of molecular penetration into an oily regionmight require the use of atomic-scale models and molecu-lar dynamics techniques.

Our simulations showed that we could represent theeffect of monovalent ions by a suitable combination ofexplicitly represented ions and an average continuous dis-tribution described by a Debye screening length, j�1. Thishas important practical applications. Although there aretechniques for handling infinite-range ‘‘1/r’’ electrostaticswithout Debye screening (j = 0), it is sometimes conve-nient to have a finite screening length, since this makesthe electrostatics easier to handle. Our result shows thatwe may split up the monovalent ion concentration intothose which we represent by a continuum and which giverise to Debye screening, thus bringing about a faster fall-

off of the electric potentials with distance, and those whichwe treat explicitly and which might be used to reflect mono-valent ion behavior at an interface, different from bulkbehavior. There are two comments that follow from this:the first is that one should be careful not to represent multi-valent ions by a continuum with a Debye screening factor,since we will have disregarded any possibility of ‘‘bridg-ing’’. The second pertains to explicitly taking into accountthe effects of buffers, especially if they contribute multi-valent ions. A failure to do so can result in obtaining buf-fer-dependent results.

Our simulations showed that protamine, lactoferricin B,and gentamicin all penetrate an O-sidechain brush com-posed of uncharged sugars and reach the membrane sur-face. However, when the brush is composed of chargedO-sidechains characteristic of B-band P. aeruginosa

PAO1 with the repeat unit being a trimer carrying charges(+,�,�), then only the two CAPs become trapped in theouter segments of the brush. The gentamicin penetratesthe charged brush and reaches the membrane surface.The reason for this has to do with the fact that gentamicinis small but carries a charge of +3e, giving rise to an elec-trostatic energy that is large enough to attract it to theslightly negatively charged bacterial surface and to over-come the loss of entropy that it experiences at the surface.In the case of protamine and lactoferricin, the electrostaticattraction to the outer layers of the brush is sufficientlylarge to overcome the loss of entropy, since the brush canadapt itself to permit the protamine to retain sufficiententropy while being trapped in the outer layers of thebrush. This might help explain the success of gentamicinas an antibiotic compared to the effects of the two CAPs,which are simply antimicrobial. It might also point theway towards requirements for CAPs to perform as success-ful antibiotics.

Acknowledgements

It is a pleasure to acknowledge many enjoyable discus-sions with Terry Beveridge, who introduced DAP to bacte-rial interfaces. CBH thanks St. Francis Xavier Universityfor the award of a James Chair Professorship in 2005–2006. This work was supported by NSERC through Dis-covery Grants to D.A. Pink and to A.T. Paulson andUSRA awards to L. Filion and G. Ryan, and by the USNational Science Foundation through grant DMR-0206681 to C.B. Hanna.

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