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The Role of Electrostatics in ProteinProtein Interactions of aMonoclonal AntibodyD. Roberts, R. Keeling, M. Tracka, C. F. van der Walle, S. Uddin, J. Warwicker, and R. Curtis*,

School of Chemical Engineering and Analytical Science, The University of Manchester, Sackville Street, Manchester M13 9PL, U.K.Formulation Sciences, MedImmune Ltd., Aaron Klug Building, Granta Park, Cambridge CB21 6GH, U.K.

*S Supporting Information

ABSTRACT: Understanding how proteinprotein interactions depend on thechoice of buffer, salt, ionic strength, and pH is needed to have better control overprotein solution behavior. Here, we have characterized the pH and ionic strengthdependence of proteinprotein interactions in terms of an interaction parameterkD obtained from dynamic light scattering and the osmotic second virialcoefficient B22measured by static light scattering. A simplied proteinprotein

interaction model based on a Baxter adhesive potential and an electric doublelayer force is used to separate out the contributions of longer-ranged electrostaticinteractions from short-ranged attractive forces. The ionic strength dependenceof proteinprotein interactions for solutions at pH 6.5 and below can beaccurately captured using a DeryaguinLandauVerweyOverbeek (DLVO)potential to describe the double layer forces. In solutions at pH 9, attractiveelectrostatics occur over the ionic strength range of 5275 mM. At intermediatepH values (7.25 to 8.5), there is a crossover effect characterized by a nonmonotonic ionic strength dependence of proteinprotein interactions, which can be rationalized by the competing effects of long-ranged repulsive double layer forces at low ionicstrength and a shorter ranged electrostatic attraction, which dominates above a critical ionic strength. The change of interactionsfrom repulsive to attractive indicates a concomitant change in the angular dependence of proteinprotein interaction fromisotropic to anisotropic. In the second part of the paper, we show how the Baxter adhesive potential can be used to predict valuesofkDfrom tting to B22measurements, thus providing a molecular basis for the linear correlation between the two protein protein interaction parameters.

KEYWORDS: proteinprotein interactions, osmotic second virial coefficients, diffusion interaction parameters, monoclonal antibody,electrostatics

INTRODUCTION

Recent studies of nonspecic proteinprotein interactions havebeen driven by the need for developing stable liquidformulations of antibodies or antibody derived products,which have rapidly become a large market of biotherapeutics.Antibody self-interactions are linked to problems such asprotein aggregation,13 liquidliquid phase separation andopalescence,410 and high viscosities.1118 These problems areoften exagerated at high protein concentrations, which are oftenneeded in liquid formulations to meet patient dose require-ments. Problems can be minimized by manipulating thesolution conditions, such as choices of buffers, pH, ionicstrength, and the inclusion of other small molecule additivessuch as sugars, amino acids, and other excipients. A betterunderstanding of proteinprotein interactions and how theydepend on solution conditions can aid in formulationdevelopment by reducing the solvent space to be screenedfor protein stability.

Weak proteinprotein interactions are most commonlycharacterized in terms of the osmotic second virial coefficient,B22, which provides a direct link to a solvent-mediatedinteraction between a pair of proteins averaged over their

separation and relative orientations. Much of what is knownabout proteinprotein interactions was learned from studiesmotivated by the discovery of a link between B22 andcrystallizability; in order for protein crystallization to bepossible, the value ofB22must fall in a window of interactionstrength. More recently, studies have shown that in low ionicstrength solutions, the value of B22 is correlated with theprotein aggregation behavior.2,3,1922 Even for solutions whereproteins are more likely to form partially folded aggregation

prone states, lowered aggregation propensities occur when theprotein has a high net charge. In this case, repulsive double-layer forces increase the colloidal stability of the protein.Conversely, in solutions near to the isoelectric pH (pI) of theprotein, attractive electrostatic interactions between proteinshave been linked to increased aggregation propensities.12 Otherstudies have correlated increased viscosities of concentratedprotein solutions with attractive proteinprotein interactions

Received: March 28, 2014Revised: May 15, 2014

Accepted: June 3, 2014

Article

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characterizedindilute protein solutions under the same solventconditions.1218

These studies highlight the utility of proteinproteininteraction measurements in dilute solutions, but an improvedunderstanding of protein solution behavior is needed forpredicting concentrated solution properties from dilutemeausurements of proteinprotein interactions.23,24 As astarting point, there is a need for being able to separate out

the different energetic contributions to the molecularinteractions. In the context of protein formulations, studieshave suggested the presence of an adhesive proteinproteininteraction when itis masked due to the presence of long-rangedouble layer forces.12,25,26 In solutions at low ionic strength, thenet proteinprotein interaction measured in dilute solution isrepulsive, however, a short-ranged attraction has still beeninferred indirectly from rheological characterization in moreconcentrated protein solutions, where the average separation ofproteins is signicantly reduced such that the short-rangedattraction is entropically favored. In addition, electrostaticrepulsion can prevent aggregation at low protein concentration,but not at high concentrations where the short-rangedattraction becomes signicant.27

Because most measurements such as B22 only provide anaveraged proteinprotein interaction, deconvoluting theinteraction into individual contributions requires matchingthe measurements to interaction models. Often the molecular-level interpretation gained from the studies depends on thelevel of coarse-graining and the force eld used to describe theprotein and the solvent. Detailed calculations using all-atomisticdescriptions of proteins and an implicit water model indicatethat proteinprotein interactions are more likely anisotropic ina manner similar to specic proteinprotein interationsbecause the interaction is dominated by a small number ofhighly attractive congurations determinedby shape comple-mentarity and van der Waals forces.2830 In this case, the

energetically favored orientations are on the order of tens ofkBT.30 The latter view is consistent with the intuition that

interactions should be anisotropic in order to trap the proteinin an orientationally constrained conguration needed forcrystallization. For globular proteins, the crystal structure isrepresentative of the solution protein conformation, in whichcase protein surfaces buried in crystal contacts should reectthe proteinprotein interactions formed in solution. However,simulations using an explicit model for water indicate that thecalculated strength of orientationally restricted interactionscorrespondingto lysozymelysozyme crystal contacts is only acouple kBT.

31 Interactions of this magnitude are not largeenough to constrain protein orientational interactions insolution. The latter is supported by the nding that a mutation

to hemoglobin signicantly alters the protein crystal solubility,although the liquidliquid phase separation remains unaffectedindicating that anisotropic interactions formed in the crystal arenot the same as those sampled in solution.32 A mutation studyof an antibody suggested that proteinprotein interactionsdepended on the properties of a charged patch, suggesting thatelectrostatic interactions, rather than shape complementarity,control theproteinprotein sampling in solution at low ionicstrength.33 These large discrepancies in estimating theinteraction energetics reect the difficulty in deconvolutingthe proteinprotein interaction into individual contributions.

The aim of this work is to carry out a detailed proteinprotein interaction study as a function of pH and ionic strengthto provide more insight into the molecular origin of protein

protein interactions for monoclonal antibodies. The measure-ments are matched to a simplied interaction model, based on aBaxter adhesive potential and a double layer force taken fromDeryaguinLandauVerweyOverbeek (DLVO) theory. Wealso carry out -potential measurements and calculations of theprotein electrostatic properties to provide more insight into theorigin of proteinprotein interactions. The model is not meantto provide an exact description of the proteinprotein

interaction but instead is a tool that allows us to separate outthe effects of electrostatics from excluded volume forces andother short-ranged interactions. Deviations from the modelprovide insights into whether interactions are anisotropic andthe key stumbling blocks to building a more realistic, predictivemodel for proteinprotein interactions.

Recent investigations of proteinprotein interactions havecharacterized them in terms of an interaction parameter kD,which is given by the slope of the mutual diffusion coefficientversus protein concentration. Experimentally, studies haveshown a linear relationship between kD and B22 indicatingthat the parameter can be used as a surrogate forB22.

2,17,34 Theadvantage is that kD can be measured using a multiwell platedynamic light scattering plate reader, which leads to much more

rapid measurements and minimal sample consumptioncompared with approaches for assessing the osmotic secondvirial coefficient using static light scattering. Here, we quantifyproteinprotein interactions from kD measurements and thenbenchmark them againstB22meaurements for selected solutionconditions. The observed correlation betweenB22and kDis notsurprising in that kDis related to proteinprotein interactionsthrough well-established relationships.3538 Using the latterapproaches combined with the simplied protein model, weshow how both interaction parameters provide equivalentinformation and thus provide a molecular basis for thecorrelation.

EXPERIMENTAL SECTION

Sample Preparation. mAb1 is an IgG1 with sequencemolecular weight equal to 144.8 kDa. The mAb1 was obtainedas a pure liquid formulation at a concentration of 41 g/L in anaqueous solutions containing 25 mM histidine buffer and 7%w/v sucrose at pH 6.0. The extinction coefficient of mAb1 is1.42 mL/(g cm).

A size exclusion high performance liquid chromatographyrun with an online Heleos multiangle light scattering detector(from Wyatt Technology) was used to check for the presenceof aggregates in the formulation. An injection of 25 L of themAb1 at 10 g/L was used. The step was carried out using aTSKgel HPLC column (G3000SWXL, 7.8 mm ID 30 cm, 5m particle size) at a ow rate of 1 mL/min with a mobile

phase composed of 100 mM sodium phosphate buffercontaining 100 mM disodium sulfate at pH 6.8.

Before each experiment, 4 L of the buffer solution wasprepared using a buffer concentration that xed the ionicstrength at either 5, 10, or 25 mM as calculated according to theHendersonHasselbach equation. All buffer solutions wereltered through a vacuum ltration unit using a 0.2 m PESmembrane (Stericup, Thermo Scientic Ltd., UK). The mAb1formulation was concentrated to approximately 150 g/L usingan Amicon Ultracentrifuge lter of 30K MWCO (Merck-Millipore Ltd., Ireland) and then placed in a Slide-A-Lyserdialysis cassette with a maximum volume of 3 mL and a 10KMWCO (Thermo Scientic Ltd., U.K.). The dialysis cassettewas placed in 2 L of a dialysis buffer and stirred for 4 h. The

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dialysis buffer was then exchanged, and the dialysis wascontinued overnight. The dialyzed sample and the dialyzatewere then saved and used for all subsequent sample dilutions.

Static Light Scattering.Static light scattering is used herefor determining the osmotic second virial coefficient,B22. For asystem containing protein, water, and salt, the light scatteringequation is given by3941

=

Kc n c

R Mk T ( / ) 1T

T

2

, ,

B , ,

w s

w s (1)

where is the osmotic pressure of the protein solution overthat of the salt solution, subscripts w and s correspond to waterand salt, respectively, R is the measured excess Rayleighscattering ratio of the protein solution over the solvent, M isprotein molecular weight, is the protein number concen-tration, which is related to protein masss concentration cby=NAc/Mwith Avogadros number represented by NA. K is thelight scattering constant given by 22n0

2/(NA4), where n0 is

the refractive index of the solvent, and is the wavelength ofthe light. (n/c)T,w,sis the refractive index increment of the

protein solution measured at constant chemical potential of thewater and salt,sand w. In solutions at low salt concentration

n

c

n

cT T p m, , , ,w s s (2)

The refractive index increment at constant pressure pand saltmolalitymshas been measured extensively for protein solutionsand is equal to approximately 0.185 L/g.

The osmotic compressibility can be calculated from thederivative of the virial expansion

= + +

k Tb O

11 2 ( )

TB , ,

222

w s (3)

where b22 is the osmotic second virial coefficient, whichprovides the link to the molecular level proteinproteininteractions. The terms omitted from the sum on the right sideof the equation are related to higher order virial coefficients.Thus, the second virial coefficient is strictly dened as the slopeof osmotic compressibility plot at zero protein concentration.In practice, the virial coefficient should only be calculated fromthe linear part of the plot at low protein concentrations.Substituting the result into eq1

= +

Kc

R M

b N

Mc

12 22 A

2(4)

Often, measured values ofb22are reported in units of mLmol/g2, which corresponds to the denition

=Bb N

M22

22 A2 (5)

In a light scattering experiment, Ris measured for a series ofsamples with varying protein concentrations. A plot of the leftside of eq4versuscyields a slope equal to 2B22and the inverseof the y-intercept equal to M.

A Wyatt miniDAWN TREOS 3 angle detector was used forthe static light scattering experiments. The instrument uses a 60mW GaAs diode laser with vertically polarized light at awavelength of 658 nm. There is no angle dependent scatteringwhen the particle size is less than 1/20th the wavelength of

light. In these instances, the best data quality is obtained usingthe 90 detector to minimize the effect of dust particulates.Measurements were made using a ow cell. Samples weredelivered using the Calypso, an automated syringe pumpmanufactured by Wyatt Technology. The Calypso containsthree programmable syringe pumps used to generate veryprecise step gradients in concentrations. The ow from eachsyringe pump goes through an in-line 0.1 m pore size

membrane. A static mixer is used to mix the streams after theltration step before the sample enters the owcell. In a typicallight scattering experiment, the protein solution dissolved inonly the dialysis buffer at a bulk concentration of 16 g/L isplaced in syringe pump 1. The dialysis buffer is placed insyringe pump 2, and syringe pump 3 contains a salt solutionprepared in the same dialysis buffer. The Calypso was thenprogrammed to generate a series of experiments. Each virialcoefficient determination was based on two experiments, (1)using step gradients of increasing protein concentration and (2)with gradients of decreasing protein concentration. In eachexperiment, protein concentration is varied in equal stepchanges between concentrations of 2 and 8 g/L at xed saltconcentration. Protein concentrations were determined from

dilution factors calculated from the relative ow rates of thepumps. The concentrations were veried from absorbancereadings at 280 nm measured with a Waters 2487 absorbancedetector connected in series to the outlet of the light scatteringowcell. We used a variable path length owcell in order toaccurately determine the protein concentration. The pathlength of the owcell was set to 0.05 cm such that theabsorbance of the protein solutions falls within the range overwhich the BeerLambert law is valid.

Dynamic Light Scattering.In a dynamic light scatteringexperiment, macromolecular diffusion coefficients are quanti-ed from tting to the measured intensity autocorrelationfunction over time. The function C(t) is given byC(t) = Ag(t)2

+B, whereA is an instrument-dependent optical constant andBis a background term, determined for each sample in the limitof large delay times.42 For a monodisperse solution where thewavelength of light is much larger than the size of the scatteringparticle, the rst order autocorrelation functiong(t) is related to

= t( ) e Dq t2

(6)

where q is the magnitude of the scattering vector (q= 4n0sin(/2)/); is the scattering angle). The main quantitymeasured in dynamic light scattering experiment is thetranslational diffusion coefficient of the particle D, whichcontrols the decay time for correlations in the uctations ofscattered light. For monodisperse solutions, D can bedetermined from tting g(t) to the measured autocorrelationfunction. However, protein solutions always contain a smallamount of impurities, either irreversibly formed aggregates, asmall fraction of other protein impurities, or dust particles.Accounting for these requires tting the correlation function toa population distribution. For polydispersed solutions, g(t) isgiven by

=

tM

M( )

ei i i

Dq t

i i i

2

2

i2

(7)

which corresponds to a z-weighted average over all sizes ofparticles at molecular wightMiand number density i. Relatingthe measured correlation function to the diffusion coefficient is


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an ill-conditioned problem because different distributions canyield the same measured g(t). A general approach toovercoming this problem for samples with mononodaldistributions is to use the cumulant expansion. In this case,the natural logarithm of the correlation function is t to asecond order polynomial in the delay time

= +g t q D tq t

Dln ( )2

( )z z

24 2

2

(8)

where the subscript z denotes the z-average of the property.The rst order coefficient in the expansion corresponds to thez-average of the diffusion coefficient. The second moment ofthe distribution is equal to the second order coefficient, whichis the z-average in the uctuations of the diffusion coefficient;larger ucuations are related to broader size distributions. Ameasure of the polydispersity in the sample is often dened asP= (D)z

2/Dz2, which is equal to zero for monodisperse samples.

The measured diffusion coefficient can be expanded in aseries about protein concentration

= +D D k c[1 ]0 D (9)

D0 corresponds to the innite dilution value of the diffusioncoefficient, which is often reported in terms of the hydro-dynamic radius, Rh, using the StokesEinstein relationship

=R

k T

D6h

B

0 (10)

Here Rh is the radius of a sphere that has the same diffusioncoefficient as the protein and is the solvent viscosity. TheparameterkDreects the interactions between proteins, whichcontrols their collective motion. Experimental studies havealready shown a monotonic correlation between kD and B22indicating that the parameter provides a direct measure ofproteinprotein interactions. A discussion on the theoretical

connection between kDand the protein

protein pair potentialof mean force, W, is given later in the text when we introducethe proteinprotein interaction model.

Diffusion coefficients were measured using a DynaProPlateReader (Wyatt, Santa Barbara, CA) at a laser wavelengthof 838.88 nm with 30 L samples in a 384 well plate. Eachmeasurement included ten 5 s acquistions. For each solventcondition, six samples were measured with equally spacedprotein concentrations and a maximum protein concentrationof 25 g/L. For each protein concentration, measurementswere carried out in triplicate and then averaged. Cumulantsanalysis was done using the Wyatt Technology DynamicsSoftware. Values of polydispersity, P, greater than 0.1 indicaterelatively broad size distributions and are not considered when

analyzing the data. All measurements were performed at 25 C.Zeta () Potential Measurements. -Potential was

measured by massively-parallel phase analysis light scattering(MP-PALS) using a Mobius mobility instrument manufacturedby Wyatt Technology. The velocity of charged particles underan applied electric eld leads to a frequency (or Doppler) shiftin the scattered light, which is used to determine theelectrophoretic mobility by MP-PALS. The instrument containsa dynamic light scattering detector, which allows forsimultaneous determination of electrophoretic mobility andparticle diffusion coefficient. A 70 mW solid state laser at awavelength of 532 nm provides light source and 30 detectorchannels are used to improve the sensitivity and speed of themeasurement. Assuming the protein behaves as a uniformly

charged spherical particle, the Henry equation can be used torelate the electrophoretic mobilityeto the potential

=

f a3

2 ( )0e

(11)

wherea is the particle radius, 0is the vacuum permittivity, isthe dielectric constant of the medium, and is the inverseDebyeHuckel screening length, DH

= = e N I1/ 2 /( )DH2

A 0 (12)

I is the ionic strength of the electrolyte solution, inversetemperature is given by = 1/(kBT), and e is the electroniccharge.f(a) is the Henrys function. The Henrys function canbe approximated by

= + ++

f aa a

( ) 1 1

21

2.5

[1 2 exp( )]

3

(13)

Henrys equation is valid for values of < 50 mV, whichcorreponds to when the polarization of the diffusedouble layeris small and the surface conductivity is negligible.43

THEORY

Electrostatic Calculations. The electrostatic charge andpotential distribution were calculatedfor a homology model ofmAb1 using an in-house program.44 We report percentage ofsolid angle covered by the positive and negative surfacepotentials corresponding to magnitudes of the potential greaterthan 25 mV or less than 25 mV, respectively. In addition,contours of the surface potential at 25 mV and 25 mV wereused to determine the sizes of the largest positively charged andnegatively charged patches. The patch size corresponds to thepercentage of solid angle coverage. A comparative model wasmade based on PDB id 1hzh, with subsequent adjustment tomake the two Fab-Fc orientations approximately symmetric.

ProteinProtein Interaction Model. The main motiva-tion for carrying out the complementaryB22studies is that theparameter provides a direct link to proteinprotein inter-actions, whereas the link to kD measurements is less wellestablished especially for particles with nonspherical shapes.Thus, we initially focus on intrepreting theB22studies in termsof a simplied proteinprotein interaction model and thenexamine the transferability to describing the kDmeasurements.The exact relation between the osmotic second virial coefficientand the proteinprotein interactions, as characterized in termsof the protein pair potential of mean force, is given by

=

b g r r r

1

16[1 ( , )] d d22 2

0

2

(14)

where r is the center-to-center separation between a pair ofproteins and g(r,) is the pair distribution function given by

= r W r( , ) exp[ ( , )] (15)

The protein pair potential of mean force W(r,) i s a ninteraction free energy averaged over all the solvent andcosolvent degrees of freedom. Equation14contains a multipleintegral over the set of angles dening the relative orientationsbetween a pair of proteins .

For rigid molecules, the potential of mean force can befurther broken up into the sum of an excluded volumecontribution,Wex(r,), and the sum of all other longer-rangedinteractions denoted byWsoft(r,). The excluded volume term


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is equal to innity for the set of separations and relativeorientations in which the pair of proteins overlap with eachother and zero otherwise. The integral over the excludedvolume term in eq14 yields

= +

b b W r r

r

1

16[1 exp( ( , ))]

d d

d22 22

ex2

( )

soft 2

c

c

(16)

wherecdenotes the set of orientations in which the proteinsdo not overlap with each other, and dc is the center-to-centerseparation between proteins at contact. b22

ex is the excludedvolume contribution to the osmotic second virial coefficient,which corresponds to the volume that is made inaccessible tothe center of a protein due to the presence of the interactingpartner averaged over all their relative orientations. Thedifference b22

soft = b22 b22ex often referred to as the energetic

part of the proteinprotein interaction accounts for allinteractions except the entropic excluded-volume contribu-tion.45 If the difference is less than zero the net energeticinteractionWsoft is attractive, whereas a difference greater thanzero reects a net repulsive interaction. A recent study carriedout the detailed integration given by eq14 to determine whatlevel of coarse graining was needed in order to capture theexcluded volume calculated from all-atomistic description of theprotein based on its crystal structure.46 The keynding is thatthe contribution can be approximated by the excluded volumeof a sphere with a radius given by the experimentally measuredhydrodynamic radius of the protein, in which case

=b R16

322ex

h3

(17)

The approximation holds true for a range of proteins withdegrees of complexity ranging from single domain proteinsthrough to a monoclonal antibody.

The energetic contributions to proteinprotein interactionscan be further broken down according to the range of the force.The longer-ranged interaction arises from electrical doublelayer forces, whereas short-ranged forces can be composed ofall types including dispersion forces, solvation forces such ashydrophobic or hydration, or other forms of electrostaticinteractions such as salt bridges. The molecular origin of short-ranged interactions between proteins are not well-known ashighlighted in theIntroductionof this article. Here, we do notattempt to relate the short-ranged proteinprotein interactionsto a physical model but instead use an adjustable parameter toreect the adhesive forces between proteins or congurationswhere the separation between surfaces of proteins are closeenough (within a solvent layer) to be considered in contact. Weuse the Baxter adhesive potential wsr, which is the limiting caseof a square well potential when the well depth goes to innityas the width goes to zero, chosen in such a way that thecontribution of the interaction toB22remains constant. As such,the advantage of using this model is that only one parameter isneeded to describe the interaction. The potential given by

= < < + r r( ) for (1 )sr (18)

where corresponds to an effective hard sphere diameter, istaken in the limit of 0 where is dened by

= +

1ln 1

1

12 (19)

where corresponds to the strength of the adhesive forcebetween the particles. In the Baxter limit, the second virialcoefficient becomes

= B

B

422sr 22

ex

(20)

Previous studies have shown the Baxter model can capture thegeneric phase behavior and osmotic compressibility for globularproteins solutions in moderately concentrated salt solu-tions.47,48 In addition, small-angle X-ray scattering studieshave indicated that the proteinprotein interaction has a short-range on the order of one-tenth the diameter of the protein,49

and the static structure factor curve can be accurately t usingthe Baxter potential.50

The energetic part of the proteinprotein interaction is givenby the sum of a short ranged term and a long-ranged electricaldouble layer force. In this case,

= +B B B22soft

22sr

22el

(21)

As a starting point to describe the double layer force betweenproteins, we use DLVO theory. In this approach, the protein is

treated as a uniformly charged sphere immersed in a solvent,where the ions are treated as point charges and the water is acontinuum characterized by the relative dielectric permittivityof water equal to 78.4. The proteinprotein electrostaticrepulsion occurs due to an osmotic repulsive force when theelectrical double layers overlap, because the local concentrationof ions in the double layer is greater than the ion concentrationin the bulk. As such, the range of the force is related to thethickness of the double layer, which is characterized by theinverse screening length,1. Analytical solutions to the doublelayer force between two charged spheres only exist in certainlimits. Of relevance to this study is an approximate formderived using the DebyeHuckel approximation and in thelimit ofa< 551

=

+

w r

Z

a

r a

r( )

(1 )

exp[ ( 2 )]el2

B2

(22)

Here, Zcorresponds to the protein valency or the effectivecharge on the protein per electronic charge,e, and includes theions bound in the Stern layer surrounding the protein; a isequal to the distance from the protein center to the outerHelmholtz plane (OHP), which separates the diffuse layer fromthe Stern layer. The Bjerrum length, B = e

2/(40) is theionion separation when the Coulomb energy is equal tothermal energy, kBT, equal to 0.7 nm for water at roomtemperature. The model potential has provided accurate ts tothe pH and ionic strength trends ofB22for protein solutions at

low ionic strength (below 200 mM).5254

Deviations to DLVO theory are greatest at pH values near tothe pI, where the electrostatic repulsion is weakened due tolowering the net charge. The anisotropic distribution of chargebecomes signicant and is often linked to the presence ofattractive electrostatic interactions, which have been observedfor chymotrypsinogen28 and for antibodies in slightly basicsolutions close to their pI.13,21,25,33 Nevertheless, DLVO theoryprovides a good starting point to interpret the interactions insolutions at pH removed from the pI. Here, we show thatdeviations from DLVO theory provide strong evidence for thepresence of anisotropic electrostatic interactions.

Relating ProteinProtein Interactions tokD.There is aless direct correspondence between the potential of mean force


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and the interaction parameter kD. All measurements wereperformed in the long-time scale regime, where the character-istic time scale probed by the DLS experiment is much greaterthan the velocity correlation time for the protein. In this limit,uctuations in local particle concentration can be described bythe diffusion equation

= t

D r t( , )2

(23)Particle diffusion is controlled by a chemical potential gradient,the drag force exerted on a particle from the solvent, andinterparticle hydrodynamic forces as mediated by the solvent.Calculating hydrodynamic forces requires solving the Navier Stokes equations for incompressible ow about the particles.Various appoximations have been used to solve these equationsand then solve for kDby matching the solution to eq23. Herewe use the result obtained by Felderhof,35 in which case, thevalue ofkDis given by

= + + + +k k k k k kD V O A S FD (24)

where the virial term kV is the thermodynamic contribution

driving diff

usion. kV is related to the osmotic second virialcoefficient bykV= 2B22M

=

k B M g x x x6 [ ( ) 1] dV 22ex

0

2

(25)

The effects of hydrodynamic interactions are also related tothe potential of mean force according to

=

k B M g x x x3 [ ( ) 1] dO 22ex

0 (26)

which is the Oseen term,

=

k B M

x xg x x x6

9

8

1

(2 )

5

4

1

(2 )( ) dA 22

ex

0 6 4

2

(27)

and

=

k B Mg x

xx

75

8

( )

(2 )dS 22

ex

0 5

(28)

are short-range hydrodynamic interactions, and

=k 1FD (29)

is the force dipole part. In the above integrals, the substitutionof x = r/(2a) has been made to make the representationssimplistic. There have been several studies that have t theinteraction parameters to DLVO type potentials for lysozymesolutions and found realistic values for the net charge and

Hamaker constant.36,55

In other studies, the contribution of thesoft part of the interaction has only been included in the Oseenand virial terms, which follows on from a study demonstratingthat the short-range hydrodynamicinteractions arising from theYukawa potential are negligible.56 With the latter approachapplied to solutions of a monoclonal antibody, DLVOparameters t to the value of kD agreed closely to the tsobtained fromthe correponding B22values measured by staticlight scattering.52

When we include the short-range Baxter adhesive potential inthe model to account for short-range forces, the interactionparameter is given by the sum

= + +k k k kD Dex

Del

Dsr

(30)

The excluded volume terms is givenby the sum over all virialand hydrodynamic terms to give35,56

=k B M0.39Dex

22ex

(31)

The double layer term only contains the Oseen and virialcontributions

= +k k kDel

Oel

Vel

(32)

which are evaluated using a lower limit of integration given bythe effective hard sphere diameter, 2a. The short-range term is

= + + +k k k k kDsr

Osr

Vsr

Asr

Ssr

(33)

The model described above only accounts for repulsive doublelayer forces, and no term is included for a long-rangedelectrostatic attraction.

RESULTS AND DISCUSSION

Electrostatic Properties.We rst report the results of the-potential measurements, which are used to probe the chargedproperties of the protein. The -potential corresponds to theelectricalpotential at the slip plane between the protein and the

solvent.43 The effective charge of the protein can be related tothepotential using DebyeHuckel theory assuming that theOHP and the slip plane are located at the same position

=

+Z

a

e

4 (1 )0 d(34)

wheredis the electrical potential evaluated at the beginning ofthe diffuse layer (or at the OHP). Within this model Zis thexed charge arising from the ionized groups on the protein andthe ions bound in the Stern layer.

We measured the-potential for 1 g/L mAb1 solutions at anionic strength of 25 mM containing sodium actetate buffer atpH 5 and 6.5 and containing tris-chloride buffer and sodium

chloride at pH 8 and 9. In addition, measurements were carriedout as a function of pH for a solution containing 1 g/L mAb1and 10 mM NaCl. In the latter measurement, the samples wereobtained by titrating a solution of mAb1 at an initial pH of 4.5with dilute NaOH dissolved in 10 mM NaCl. All the measuredpotentials were less than 20 mV and correspond to conditionswhere Henrys equation (eq 11) is valid. The results arereported in terms ofZshown in Figure1as a function of pH.The error bars correspond to the standard error taken overmeasurements of three independent runs. The values reportedhere forZat pH 6.5 are in the same range as those reported inother -potential studies of antibodies at pH 6 in histidine-hydrochloride buffer at ionic strength between 15 and 20mM.15,34 There is an expected decrease in Zdue to reducing

the net positive charge arising from the protein ionizablegroups. In acidic solutions, the values of Z are lower inmagnitude than the theoretical calculations of the proteincharge shown in the inset toFigure 1, which is consistent withthe ndings of other studies.15,34,57,58 The xed charge on theprotein surface can be either calculated from theoreticallyderived pKavalues or measured using a potentiometric titration.In either case, the value does not account for the charges arisingfrom ions bound in the Stern layer, which contribute to themeasured -potential. Thus, the lower value of Zrelative tocharge estimated from the titration curve indicates preferentialbinding of chloride ion to the positively charged protein groups.This result is consistent with other -potential studies; forinstance, in sodium chloride solutions of BSA, the excess


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number of bound chloride ions over bound sodium rangedfrom20 at pH 4 to 10 ions at pH 5.3, the isoelectric point ofBSA.58 Because similar trends in ion specic binding areobserved across many different proteins,5962 a similar amountof preferential anion binding is expected to occur with themAb1 investigated here. The measured values ofZat xed pHare similar for the solutions at 25 and 10 mM ionic strength.Thus, there are no discernible effects on the protein net chargewhen sodium is replaced by tris in alkaline solutions or whenchloride is replaced by acetate in acidic solutions.

For a monovalent salt solution, DebyeHuckel theorycorresponds to the limit ed < 1, in which case thePoissonBoltzmann equation can be linearized when solvingfor the mobile ion distribution about the protein. For thesolutions at pH 5, the measured value of ed 1. As aconsequence, we have checked the validity of eq34by using asemiempirical equation that provides an accurate approxima-tion of the nonlinear PoissonBoltzmann equation whencalculating the ion distribution about a charged sphere51

= +

Z a

a e

a

e( ) 2 sinh

2

4tanh

4B

d d

(35)

The calculations ofZusing eq35are shown by the dashed line

in Figure 1. The largest deviations occur at pH 5, but therelative difference is only 3% indicating that the DebyeHuckeltheory provides a reasonable estimate within the experimentalerror of the measurement.

pH and Ionic Strength Dependence of ProteinProtein Interactions for mAb1.Next we characterized theionic strength and pH dependence of the osmotic second virialcoefficient, which is related to the protein pair potential ofmean force by eq14. Positive values of the parameter reectrepulsive proteinprotein interactions, whereas negative valuescorrespond to attractive forces. In Figure 2 are shownexperimental data obtained for an experiment using a bufferof sodium acetate with an ionic strength contribution of 25 mMat pH 5.75. A B22 measurement is obtained at xed salt

concentration by collecting the light scattering data from aseries of samples with a decreasing gradient in proteinconcentration (given by the closed symbols in Figure 2),followed by a series of samples with increasing concentration(open symbols in Figure 2). A linear regression of thecombined data set was used to calculate the slope from the plot(see eq4) equal to 2B22; reported error bars correspond to thestandard error in the slope estimation. The molecular weight ofmAb1 is equal to the inverse of the regressed intercept. Themeasured values of the molecular weight from all experimentsfell in the range of 190 to 200 kDa, which is greater than thesequence molecular weight estimated to be 148 kDa. Ameasured monomer molecular weight equal to 145 kDaA wasobtained for the monomer fraction from an SEC-MALLS stepindicating the refractive index increment of 0.185 mL/g isaccurate. Only a small fraction of trimer (less than 0.1% w/v)

eluted before the monomer peak during the SEC-MALLS step.The larger weight-average molecular weight observed in thebatch light scattering experiment is thus due to a smallpopulation of irreversibly formed aggregates, which are lteredout by the guard column and SEC step. The aggregates werenot resolved in the dynamic light scattering experimentindicating that the aggregate hydrodynamic size cannot begreater than approximatelyve times the monomer value. Thisis consistent with the nding that aggregates are removed bythe guard column of a SEC step, but not byltering through asyringe top lter with a 0.02 m pore size. The effect ofirreversibly formed aggregates on the measured B22 value isexpected to be negligible since the aggregates occur at a smallrelative mass fraction.54

The value ofkDis obtained from the slope of a plot of thenormalized mutual diffusion coefficient D/D0 versus proteinconcentration. Sample data are shown in Figure 3 forexperiments using a 25 mM sodium acetate buffer at pH 5.75with different NaCl concentrations. The innite dilutioncoefficient D0 was rst obtained from the intercept of thebest t line to a plot of D versus protein concentration. Thehydrodynamic radius Rh was calculated from the StokesEinstein relation using a viscosity equal to 0.89 cP, whichcorresponds to water at 25 C. The measured values ofRhfromall the experiments were in the range between 5.0 and 5.3 nm,which is similar to other DLS studies of IgG1 antibodies.2,18

The error bars reported on the values ofkDcorrespond to thestandard error in the slope estimate from plottingD/D0versus

Figure 1. Plot of charge estimated from -potential measurements.Open circles correspond to solutions of mAb1 containing approx-imately 10 mM NaCl with no buffer, where the protein charge Zhas

been calculated using the DebyeHuckel approximation, and thedashed line corresponds to the full solution of the PoissonBoltzmannequation. Closed symbols correspond to solutions containing sodiumacetate buffer at an ionic strength contribution of 25 mM at pH 5 andat pH 6.5. Solutions at pH 8 and 9 are at an ionic strength of 25 mMcontaining sodium chloride and a tris chloride buffer at an ionicstrength contribution of 10 mM and 5 mM, respectively. The inset

contains the theoretical calculation of charge Zarising from proteinioinizable groups.

Figure 2.Experimental static light scattering data obtained in solutionsat pH 5.75 containing sodium acetate buffer at an ionic strength of 25mM for solutions with varying NaCl concentration. Closed symbolsand open symbols correspond to data obtained with decreasing andincreasing gradients of protein concentration, respectively.


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protein concentration. In some instances, error bars are notvisible when they have the same size or are smaller than thecorresponding symbols used in the plots. Polydispersities lessthan 0.05 were obtained for most DLS measurements

indicating a very small amount of large molecular weightimpurities. A regularization analysis indicated that the data is tby a single mononodal distribution, which is consistent with thelow measured polydispersity.

In Figure4is shown measured values ofB22for mAb1 undermildly acidic conditions as a function of ionic strength. In each

case, the ionic strength contribution of the sodium acetatebuffer is 25 mM. Proteinprotein repulsion is reduced with

increasing ionic strength at xed pH or by increasing pH atxed ionic strength. The curves shown in the gure are the tsto the proteinprotein interaction model, which includescontributions from the excluded volume, a short rangedadhesive force, and a double-layer repulsion. For each pHvalue, the model contains only two t parameters, the proteinnet charge given byZ, which is shown in the gure legend, andthe sum B22

ex + B22sr . The purpose of using the model is to

discriminate between electrostatic interactions and short-ranged forces. The tting is sensitive to the data over theionic strength range of 25100 mM where there is a sharpdecrease in B22 values due to screening of the electrostaticinteractions. The physical meaning of the t parameters arediscussed later in the text. The good agreement obtained for

the tting indicates that the double layer potential provides anadequate model for capturing the ionic strength dependence ofB22. Treating the antibody as a uniformly charged sphere seemsto be an oversimplication of the protein shape and surfaceproperties. The reason such an approach can be used has beendiscussed previously when describing the ionic strengthdependence of lysozymelysozyme interactions.53 B22 valuesmuch greater than the excluded volume contribution indicate

that the proteinprotein interactions are long-ranged andrepulsive. Under these conditions, there will be no bias to therelative orientations sampled by a pair of proteins, so that theaveraging process will correspond to the averaged proteinsurface properties, rather than the distribution of surfacecharge. This approach is supportedby a nding that proteinsundergo rapid rotational motion,63 and other studies, whereDLVO theory has been applied to obtain accurate ts inprotein solutions at low ionic strength and at pH valuesremoved from the isoelectric point.36,5255

The strength of the short-ranged attraction is reected by thevalues for the t parameters,B22

sr . At pH 5 and 5.75, the value ofB22

sr is equal to 6.1 105 mLmol/g2 for pH 5 and 5.75,whereas at pH 6.5, the value is equal to 6.5 105 mLmol/g2. In obtaining these values, we have used the calculated valueofB22

ex equal to 6.7 105 mLmol/g2 for anRhequal to 5.2 nm(see eq17). These estimates of the short-ranged interactionsare calculated with a model including the electric double layerpotential. However, at an ionic strength of 275 mM, we expectthat the electrostatic interactions are weak and a similar rangeto the adhesion force as the screening length is on the order ofa solvent layer. This is also supported by the small change inB22values when ionic strength increases from 125 to 275 mM.Because the forces act over a similar range, it is not possible toseparate the contributions to the potential of mean force.Furthermore, the DLVO potential is not expected to be arealistic representation of the electrostatic interactions at

moderate ionic strength, where the soft part of the pr otein

protein interaction is attractive and likely anisotropic.28,29 Amore realistic quantication of the short ranged forces is givenby the experimentally measured value of B22 minus theexcluded volume term. With either approach, however, wend that there is a signicant short-ranged attraction atmoderate ionic strength. When electrostatic interactions are notincluded, the short-ranged interaction is independent ofchanging pH from 5 to 6.5, whereas, a small pH dependenceis observed when electrostatics is included in the model.

In Figure5is shown the measured and tkDvalues obtainedin solutions of mAb1 with sodium acetate buffer at pH 5, 5.75,and 6.5 and for solutions at pH 7.25 using a tris chloride bufferwith a contribution to the ionic strength equal to 10 mM.

Similar trends to those found in the B22 studies are observed,where at xed pH, the effect of increasing ionic strength is toscreen the proteinprotein repulsion, and the effect ofincreasing pH is to lower the double layer repulsion fromreducing the protein net positive charge. As with the B22studies, for each pH value, two parameters, Zand the sum ofthe contributions from the short-ranged adhesion and excludedvolume terms, kD

sr + kDex, are determined by tting the ionic

strength dependence ofkDto eq30. The t curves agree wellwith the experimental values for solutions at pH 6.5 and below.In solutions at pH 7.25, the kD values at moderate ionicstrength are overestimated by the tting, perhaps providingevidence for the presence of attractive proteinproteininteractions, which are not captured by DLVO theory. In

Figure 3.Plots of mutual diffusion coefficient as a function of proteinconcentration obtained at pH 5.75 with sodium acetate buffer at anionic strength contribution of 25 mM for solutions with varying NaClconcentration.

Figure 4.Plot containing experimental data forB22and lines of best tas a function of ionic strength for solutions containing sodium chlorideat pH 5, 5.75, and 6.5 with sodium acetate buffer at an ionic strengthcontribution of 25 mM. The values ofZcorresponding to the best tare shown in the legend.


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Table1below are shown the different estimations of proteincharge. The differences between the values obtained by tting

to thekDor to theB22measurements are small, indicating thateqs2528provide an accurate representation ofkDin terms ofproteinprotein interactions. Previous studies have shown theequivalence of using either B22or kDmeasurements to regressproteinprotein potential of mean force parameters for

lysozyme.36,53,55

However, the error associated with assumingthe protein is spherical is expected to be much smaller for aglobular protein such as lysozyme, compared with that for amultidomain antibody. The ndings presented here along witha similar study provide evidence that such an approach worksequally well for antibodies.52

Small angle X-ray scattering studies of proteins indicate thatthe form of the double layer potential does not match themeasured structure factor and very often leads to unrealisticvalues of the t parameters.64 However, DLVO potentials areable to accurately capture the ionic strength dependence ofkDorB22, indicating that the model still provides a semiempiricalapproach to account for the contribution of the averagedelectrostatic interaction to the net proteinprotein potential of

mean force. It is thus still of value to rationalize the proteincharge estimated from the DLVO potential in terms of themeasured-potential. In theSupporting Information,we havedetermined the effect of approximations used in DLVO theoryon the value ofZneeded to match the measured values ofB22in solutions at pH 5. The t value ofZmust fall in a windowcorresponding to the values obtained when solving the double-layer potential using a constant potential and constant chargeboundary conditions. The minimum value is equal to 26.7,which is much greater than the value determined from the -potential measurements equal to 17.1. Within the approx-imations that the protein is a uniformly charged sphere, wewould expect that the values agreed with each other since the-potential corresponds to the electrical potential at the

beginning of the diffuse layer,D, which is also the determiningpotential in the DLVO model. However, assuming a sphericalshape and a uniform charge distribution for the antibody willintroduce error in the estimation of Z. The uncertaintybecomes apparent when we compare the same electrophoreticproperty determined by different methods. For instance, for aseries of mAbs, the measured isoelectric points by capillaryelectrophoresis were consistently less by 0.5 to 1.0 pH units

than those determined from electrophoretic light scattering.34For this pH difference, the change in protein charge can be onthe order of 10; thus drawing denitive conclusions based onthe absolute magnitude ofZis not possible. Nevertheless, thending that the t value ofZis much greater thanZappears toindicate that the double layer potential underpredicts theproteinprotein repulsion; if the value of Z is used in theDLVO potential, the value ofB22 will be much less than themeasurement. This underestimation cannot be reconciled byincluding charge anisotropy in the potential model, because thiswill always lead to a lower estimate in the proteinproteinrepulsion since the attractive congurations in the B22calculation have a greater weighting. Other possibile sourcesof error include neglecting contributions from Donnanequilibria or neglecting higher order contributions in the virialexpansion, both of which will lead to a larger value ofB22.

65

Within DLVO theory, the main effect of changing ionicstrength is to screen repulsive double layer forces. However,there are many instances, where proteinprotein interactionsbecome more repulsive (or lessattractive) with inceasing ionicstrength from 10 to 100 mM.21,33,6670 Over this range, themain effect of salt is to alter electrostatic interactions, in whichcase, the observed pattern of proteinprotein interactionsreects the presence of attractive electrostatic forces. Thisbehavior was rst studied in detail for chymotrypinsogen at pHvalues between 5.2 and the protein isoelectric pH of about 8.69

The presence of attractive electrostatic interactions indicate

con

gurational biasing of attractive orientations betweenproteins, although the forces that control the sampling aredifficult to identify. The chymotrypsinogen data formed thebasis for detailed computational studies in which the integralforB22was approximated using all-atomisticrepresentations ofthe protein based on crystal structure data.28 The main ndingwas that the proteinprotein interactions are dominated by afew highly attractive orientations correponding to a high degreeof surface complementarity, where electrostatics only contrib-ute to the energy of the interactions. In this case, attractiveelectrostatics results from the charge asymmetry in theinteracting congurations. More recently, attractive electro-statics observed in antibody systems have been rationalized byconsidering the distribution of charge on an antibody implying

that electrostatics control the congurations that are beingsampled by interacting proteins.13,21,25,33 For instance, muta-genesis was used to disrupt a negatively charged patch in the VLchain of a mAb. The change altered the proteinproteininteraction from attractive to repulsive at an ionic strength of15 mM demonstrating that attractive electrostatics originatefrom having oppositely charged patches on the same protein.33

In order to examine the effect of mAb1 charge anisotropy, wehave measured B22 and kD values at alkaline pH values. Theresults are shown in Figures6 and7. A tris chloride buffer hasbeen used at an ionic strength contribution of 10 mM at pH 8and an ionic strength of 5 mM for solutions at pH 8.5 and 9.For these plots, the lines are only drawn as guides for the eyesince it is not possible to t the data to a DLVO potential. At

Figure 5.Plot containing experimental data for kDand lines of best tas a function of ionic strength for solutions containing sodium chlorideat pH 5, 5.75, and 6.5 with sodium acetate buffer at an ionic strengthof 25 mM and at pH 7.25 with a tris chloride buffer at an ionicstrength of 10 mM.

Table 1. Tabulated Values of Protein ChargepH Z Z

a Zb

5.0 17.1 34.5 36.8

5.75 13.4 26.3 28.3

6.5 10.6 18.9 22.8aValues ofZ t to the B22data.

bValues ofZ t to the kD data.


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pH 9, there is a substantial increase in the values ofB22or kDwith increasing ionic strength. The trend reects what isexpected for the screening of attractive electrostatic inter-actions. Because B22values are less than the excluded volumecontribution, adding salt is reducing proteinprotein attractionrather than enhancing proteinprotein repulsion. Also, we nda novel nonmonotonic dependence of the proteinproteininteraction with increasing ionic strength from 10 to 275 mM atpH values of 8 and 8.5. This behavior occurs at a pHintermediate of where repulsive and attractive electrostaticshave been observed. Thus, for solutions at pH 8 and 8.5, it islikely that the initial increase of ionic strength screens repulsivedouble layer interactions, leading to attractive electrostatics,

which are then screened by further increased ionic strength. Forthe solutions at pH 7.25, there is also a very shallow minimumin kDvalues at an ionic strength of 125 mM. While difficult todetect from the experimental data, it is more clear whencomparing the t of the potential of mean force model to theexperimental data. The t curve overestimates the experimentalkDvalues (see Figure5) indicating the presence of an additionalionic strength dependent electrostatic attraction. Within ourknowledge, this maximum in proteinprotein attractionoccurring at relatively low ionic strength (less than 200 mM)has only been observed directly from B22 studies of -lactoglobulin,70 which correlated well with the large dipolemoment of -lactglobulin at pH 4 indicating that thenonmonotonic dependence arises from the charge anisotropy.

The observed nonmonotonic dependence of proteinprotein interactions implies that repulsive double layer forcesare greatest at low salt concentration, whereas attractiveelectrostatics dominate at an intermediate ionic strength. Thisexplanation follows directly from considering electrostaticscreening functions, where the screening of a monopolemonopole interaction is greater than that of a monopoledipole interaction, which in turn is greater than that of a

dipoledipole interaction.71,72 A numerical study to determinethe potential of mean force between charged-dipolar spheresindicated that the nonmonotonic dependence occurred over asmall window of charge for a sphere with dipole moment equalto 380 D.71 With decreasing charge from 5 to 3, the maximumbecomes more pronounced and shifts toward lower ionicstrength. Further lowering charge makes the maximum moreshallow until it disappears at a charge equal to 1. The samequalitative behavior is observed here, where at pH 7.25 there isa shallow maximum at an ionic strength of 125 mM, whichshifts to 50 mM at pH 8 and becomes more pronounced at pH8.5, where the maximum occurs at 25 mM. The nonmonotonicbehavior disappears in solutions at pH 9, conditions wheremAb1 still carries a net positive charge as is evident from the -potential measurements. The signicant difference with respectto the charged-dipole model is that the peak in attractionoccurs at a higher ionic strength for the protein solutions. Ashift to higher ionic strengths is expected if a short-rangednonelectrostatic attraction is included in the charged-dipolemodel. Thus, it is likely that the minimum in the electrostaticattraction is controlled both by electrostatics and by thepresence of a short-ranged proteinprotein attraction, whichwill make the nonmonotonic behavior much more pronounced.This cooperativity has been observed in a simulation studyexamining the electrostatic steering between lysozyme andseveral neutral protein fragments, in which case, the presence ofvan der Waals forces signicantly enhanced the free energy of

protein heteroassociation.

73

The nonmonotonic behavior of proteinprotein interactionshas been observed indirectly in the context of protein cloudpoint temperatures, which correspond to the temperature onsetof a liquidliquid phase separation. The trends in temperaturecloud points at xed protein concentration reect changes toproteinprotein interactions, in which case, an increase in acloud point temperature reects an increase in proteinproteinattraction. Studies of lysozyme have found a maximum in thecloudpoint temperature with increasing ionic strength for somesalts.74,75 The proteinprotein interaction pattern inferredfrom these studies differs qualitatively from our observations inthat the maximum occurs at a higher ionic strength and there isa continuous decrease in the cloud point temperature above the

maximum. Both these phenomena provide strong indicationsthat the behavior is not due to screening electrostaticinteractions. However, a more recent cloud point temperaturestudyof a mAb has found a similar trend to what we observedhere.5 In that study, with increasing pH from 5.3 to 6.6 forpotassium chloride solutions, the maximum in the cloud pointtemperature shifted from an ionic strength of 120 to 50 mMand became more pronounced reecting a much strongerproteinprotein attraction. In addition, the maximum dis-appeared in solutions at pH 7.1 indicating the presence of onlyattractive electrostatic interactions. While this type of behaviorhas not been observed for other protein solutions, a similarphenomena has been observed in the context of proteinpolyelectrolyte interactions.76 A maximum in proteinpolyion

Figure 6. B22 values measured as a function of ionic strength forsolutions containing sodium chlroide at pH 5, 5.75, and 6.5 withsodium acetate buffer at an ionic strength of 25 mM and at pH 8 and 9

with tris chloride buffer at an ionic strength of 10 and 5 mM,respectively.

Figure 7. kD values obtained as a function of ionic strength forsolutions with tris chloride at an ionic strength of 5 mM for pH 8.5and 9 and at an ionic strength of 10 mM for pH 7.25 and 8. NaCl wasused to adjust the total ionic strength of all solutions.


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binding affinity is observed in the range of ionic strengthsbetween 5 and 30 mM. The maximum in binding occurs at apH where the protein and the polyion have the same netcharge. As such, the behavior has been rationalized by thecompetition between repulsive electrostatics at low ionicstrength and attractive interactions between the polyion andan oppositely charged patch on the protein surface. Becausethis type of behavior has been oberved for a range of protein

polyion pairs, we similarly expect that the nonmonotonicbehavior in proteinprotein interactions should occur for otherproteins, especially in solutions near to the protein isoelecticpoint.

In order to quantify the anisotropic charge distribution, inFigure 8 we have plotted the percentage of the surface

containing positive potential greater than 25 mV and thatcontaining a negative potential less than 25 mV. These

potentials are chosen because the electrostatic energy of a pointcharge at this potential is approximately 1kBT. The contours ofthe potential cut-offs have been used to dene the largestpositive and largest negative patch for each pH value. The mostdramatic effect on mAb1 electrostatic properties is obervedbetween pH 5 and pH 6.5 as the fraction of the positivelycharged surface decreases by a factor of 2 and a negative patchappears at pH 6.5. When the pH is further increased from 6.5to 8, the size of the negatively charged patch remains relativelyconstant whereas the largest positive patch coverage decreasesfrom 20% to 10% of the protein surface. Denitive conclusionsabout the origin of attractive electrostatics cannot be drawnfrom considering the how the patch sizes change with pH.

More details can be gained from visualizing the electrical

potentials as shown in Figure9. At pH 5, the entire surface iscovered by the positive potential, which extends much furtherinto the solvent than any of the other pH values. When the pHis increased to 6.5, a patch of negative charge becomes apparentin the hinge region of the protein, and the location of thepositive patch contour is much closer to the protein surface.The presence of the negative patch might be linked to theelectrostatic attraction observed in the experimental studies.However, the largest change in the electrotatic potential mapoccurs when pH is changed from 5 to 6.5, whereas muchsmaller differences occur between pH 6.5 and 8. The lattercorresponds to the pH range over which the measuredelectrostatic interactions change from repulsive to attractive.The most notable change in the potential map is a reduction in

the positive patch to a size similar to the negative patch at pH 8.This observation might indicate that at pH 6.5, an interactionbetween the opposite ly charged patch is energet ical lyunfavorable because the outside perimeter of the positivepatch is repelled by the positive potential surrounding the

negative patch. At pH 8, this unfavorable interaction could bemitigated since the positive region neighboring the negativepatch and the size of the positive patch are both signicantlyreduced, so that there would be no repulsive barrier to formingthe attractive interactions. These observations are purelysubjective at this point; understanding the origin of theelectrostatic interactions will require detailed energeticsampling of the relative orientations between proteins, whichwill be made more complicated by the interdomain exibility ofthe antibody.

Relationship betweenkDand B22.In order to explain thecorrelation between kDand B22, we rst need to express kDinterms of the adhesive parameter, by carrrying out the integralsin eqs2528using the denition of the Baxter potential to give

= k B M0.50Vsr 1

22ex

(36)

= k B M0.25Osr 1

22ex

(37)

= k B M0.03Asr 1

22ex

(38)

and

= k B M0.023Ssr 1

22ex

(39)

The equations are written in terms ofB22ex to provide a quick

estimate of the contribution of the term relative to the effect ofexcluded volume,kD

ex = 0.39B22exM= 3.8 mL/g for mAb1. Thus,

in the absence of any proteinprotein attraction, the value of

kDwill always be positive. The largest contributions to kDsr

arefrom a negative virial term, which is double the magnitude of apositive Oseen term. As a consequence, the adhesion forceprovides a negative net contribution to kD. The correlationbetweenB22and kDcan be expressed as

= k B M B M 0.39 0.256D 22ex 1

22ex

(40)

or

= k B M B B M 0.39 1.024[ ]D 22ex

22ex

22 (41)

The contribution of excluded volume relative to the short-rangeadhesion attractive term is smaller for kD than for B22; as aconsequence, at moderate ionic strength, the values ofkD arenegative while those of B22 remain positive. Thus, if the

Figure 8. Plots of the percentage coverage of (a) total surface withpositive potential and negative potential and (b) largest positivelycharged patch and largest negatively charged patch. Cut-offs of 25 mVand 25 mV are used for denining positively and negatively chargedpotentials.

Figure 9.Plots of positive and negative potential as a function of pH.The surface is contoured for absolute values of the potential equal to25 mV.


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proteinprotein interaction is composed of only short-rangedattractive interactions, according to eq41, the correlation plotbetweenkDand B22has a slope close to 1.

The Oseen and virial terms also provide a much largercontribution to kD

el than any other terms. An analytical relationcan be gainedfrom considering the limit of weak interactions,in which case56

= +

k ka2 1/( )

Oel V

el

(42)

In the limit of large a, the relation kOel = 0.5kV

el is the same asthat obtained for adhesive interactions (see eqs 36 and 37).This result is not surprising because large corresponds towhen electrostatic interactions are short-ranged due to ionicscreening. Thus, the correlation between B22 and kD isindependent of the form for the potential of mean force aslong as the interaction is short-ranged.

For solutions at high ionic strength, it does not make senseto delineate between electrostatic and short-ranged forces;instead, the electrostatic interactions should also be containedin the adhesive parameter. This limit corresponds to the data

taken at 275 mM ionic strength (a= 3.1), which are shown inFigure10. The closed circles correspond to the experimentally

measured kD values, whereas the open circles are thepredictions from the model. The adhesion parameter isdetermined from tting to the experimental B22 data withoutincluding electrostatic interactions in the potential of meanforce model. The t value ofis then used to estimate the valueof kD as represented by the open symbols in Figure 10. Theclose agreement between measurement and calculation indicate

that the model is capturing the generic behavior of a short-ranged attraction between proteins. The approximation that theantibody shape is a sphere with radius equal to thehydrodynamic value is accurate when calculating the value ofB22

ex. However, this approximation works well because theaveraging process used in calculatingB22is similar to averagingthe rotational degrees of freedom during the translationalmotion, which is characterized by the innite dilution value ofthe diffusion coefficient.46 Because the Oseen interactionparameter is calculated by an integral weighted by r insteadofr2 as occurs for the virial term, it does not necessarily followthat the excluded volume contribution to the Oseen term canbe approximated as a sphere with size Rh. More detailedcalculations, which are beyond the scope of this work, would be

required to determine the effect of shape anisotropy on thecontributions to each interaction parameter.

In Figure 11 is shown the experimental and predictedcorrelation ofkDwith B22M. The solid line corresponds to an

interaction model that includes only excluded volume andadhesive interactions, where the values ofrange from 0.17 to10. The dashed line corresponds to the full potential of meanforce model for a value of = 0.29 and protein net charge Zequal to 35. As mentioned previously, when only short-rangedinteractions are included in the model, the correlation line has aslope equal to 1.03. This is similar to the experimental slopeequal to 1.06 when a range of antibodies at low and high ionicstrength are included in the data set.34 When electrostaticinteractions are included in the model there is a slight positivecurvature to the correlation. This curvature would be difficult todetect experimentally and most likely would be reected bylinear correlations with slopes slightly greater than 1.03. Verygood agreement with the calculated correlation is obtained forsolutions where proteinprotein repulsion is screened,especially considering that there are no t parameters and thesimplifying spherical assumptions used in the model.

Recent studies on antibody solutions have rationalized thebehavior ofkDin terms of

77

= k BM v2D (43)

wherev is the practical specic volume of the protein and is

the

rst order coeffi

cient in protein concentration for theexpansion of the sedimentation coefficient or the frictioncoefficient

= +f f c(1 )0 (44)

fis the friction coefficient at protein concentration c and f0 isthe innite dilution value. For hard spheres, eq43is equivalentto eq24, where77

= + + + + k k k k vO A S FD (45)

The concentration dependence of the frictional coefficient isrelated to only hydrodynamic interactions because sedimenta-tion is driven by an external eld. The main contribution ofproteinprotein interactions to the frictional coefficient is

Figure 10.Black circles correspond to experimentalkDvalues obtainedfor solutions at 275 mM ionic strength and open circles are predictions

based on B22 measurements and an interaction model given by thesum of an excluded volume force and the adhesive potential.

Figure 11. Plot containing correlation of kD with B22 for solutionscontaining sodium chlroide at pH 5, 5.75, and 6.5 with sodium acetate

buffer at an ionic strength of 25 mM and at pH 8 and 9 with trischloride buffer at an ionic strength of 10 and 5 mM, respectively. Solid

line corresponds to predicted correlation when the interactioncontains only an excluded volume and an adhesion force. The dashedline is the predicted correlation for a value of equal to 0.29 and Zequal to 35 when ionic strength is varied.


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contained in the Oseen term; for short-ranged interactions kOsr =

0.5kVsr and for electrical double layer forces, kO

el 0.5kV

el.Thus, changing solution conditions will have a similar effect onthe sedimentation and diffusion interaction parameters, becauseboth will be proportional to the change in the Oseeninteraction parameter kO. The predicted behavior is consistentwith a sedimentation velocity study of lysozyme where changesin with increasing ionic strength correlated with a similar

decrease in kD.1

DISCUSSION

The main purpose of breaking up the proteinproteininteraction into a simplied model are twofold. The rst is toseparate out the longer-ranged electrostatic interaction from theshort-ranged adhesive force. The second purpose is to provide alink between thekDand B22measurements. In solutions at lowpH, the validity of using the double layer potential has beendemonstrated by obtaining accurate ts to the ionic strengthdependence of proteinprotein interactions. There arelimitations to the approach because the charge determinedfrom tting the DLVO potential does not match what isexpected from the -potential measurements, but this is notsurprising when a spherical shape and uniform chargedistribution are used to describe an antibody. The approachonly works well at low pH where repulsive electrostaticsdominate. The deciencies are clearly reected by the presenceof attractive electrostatic interactions in solutions at pH 9. Thenonmonotonic ionic strength trend of proteinproteininteractions at intermediate pH values between 7.25 and 8.5can be rationalized by the competition between repulsiveelectrostatics at low ionic strength and attractive electrostaticsabove a critical ionic strength. For proteins carrying a netcharge, attractive electrostatic interactions arise when the netproteinprotein interation is dominated by a small number oflow energy congurations corresponding to surfaces with

charge complementarity. When the protein

protein interac-tions are repulsive, there is less weighting of attractivecongurations in favor of sampling the large ensemble ofrepulsive orientations. Thus, the nonmonotonic behaviorprovides evidence for a transition from a centrosymmetricproteinprotein interaction to an orientationally constrained(anisotropic) interaction at the critical ionic strength. What isnot known is whether electrostatic or nonelectrostatic forcesdetermine the low energy protein congurations sampled abovethe critical ionic strength. The measured pH and ionic strengthpatterns forkDfollow the same trends as those observed from acombined numerical and theoretical study of charged dipolarspheres indicating that the behavior is reconcilable in terms ofelectrostatic effects.71 However, the experimentally observed

maxima in proteinprotein attraction occur at a higher ionicstrength, which might reect an enhancement of the electro-static attraction by other short-ranged forces. The resultspresented here indicate that small changes in either ionicstrength or pH can lead to a switch from repulsive to attractiveelectrostatic interactions.

We have shown that the same correlation betweenB22andkDis obtained if the proteinprotein interaction is only composedof a short-ranged electrostatic repulsion versus a short-rangedattractive interaction as described by the Baxter adhesivepotential. Indeed the correlation is more broadly applicable andis independent of the mathematical form of the pair potential aslong as the force remains short-ranged since slight deviations inthe correlation occur when long-ranged electrostatics were

included in the potential model. Thus, approximating theprotein as a sphere within the Baxter model likely worksbecause the correlation betweenkDand B22is independent ofthe details of the pair potential. That is, when the pair potentialbetween nonspherical sticky particles is angle-averaged, thepotential of mean force with respect to separation will becomemore rugged but remain short-ranged as long as the particlesare not elongated. However, one assumption that has not been

tested here is whether the short-ranged attractive interaction isanisotropic. A similar approach could be taken using a modelspherical particle with a xed number of interacting sites, eachcharacterized by the same stickiness parameter, to test whetheran anisotropic model can also capture the generic behavior ofthe correlation between B22and kD.

ASSOCIATED CONTENT

*S Supporting InformationComparison of the screened Yukawa potential to a numericalsolution obtained using Deryaguins method. This material isavailable free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION

Corresponding Author

*E-mail:[email protected].

Notes

The authors declare no competing nancial interest.

ACKNOWLEDGMENTS

Dorota Roberts was supported by a BRIC/BBSRC GrantNumber BB/I017194/1. Robin Curtis had an ISS grant fromthe Royal Academy of Engineering that supported his part-timeplacement in 2013 at MedImmune. The authors thank WyattTechnology for continued instrument support and helpfuldiscussions, in particular, Kevin Jackson, Dan Some, and Sophia

Kenrick. mAb1 was kindly supplied by MedImmune.

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