D iscrete charge patterns,C oulom b correlations and interactions in protein solutions
E.A llahyarov1,H .Lowen1,A .A .Louis2,J.P.H ansen2
arXiv:cond-mat/0109427v1 [cond-mat.soft] 23 Sep 2001
1
Institut fur T heoretische Physik II,H einrich-H eine-U niversitat D usseldorf,D -40225 D usseldorf,G erm any
2
D epartm ent ofC hem istry,Lens eld Rd,C am bridge C B 2 1EW ,U K
T he e ective C oulom b interaction betw een globular proteins is calculated as a function of m onovalent salt concentration cs ,by explicit M olecular D ynam ics sim ulations ofpairs ofm odelproteins
in the presence ofm icroscopic co and counterions. For discrete charge patterns ofm onovalent sites
on the surface,the resulting osm otic virialcoe cient B 2 is found to be a strikingly non-m onotonic
function of cs . T he non-m onotonicity follow s from a subtle C oulom b correlation e ect w hich is
com pletely m issed by conventional non-linear Poisson-B oltzm ann theory and explains various experim ental ndings.
PA C S:82.70.D d,61.20.Q g,87.15.A a
tween globular proteins m ediated by the m icroscopic co
and counterions, and on the resulting B 2. T he conventionalD erjaguin-Landau-Verwey-O verbeek approach
[11],borrowed from colloid science,leads one to expect
thatB 2 w illm onotonically decrease asthe concentration
ofsalt increases,since higher salt concentrations lead to
enhanced screening (i.e. reduction of D ),and hence to
a decrease of the e ective protein diam eter. T his behavior rests on the standard \coarse-grained" m odelof
uniform ly charged colloids and sm oothed localdensities
of the m icroions. W e show that the discrete nature of
the protein surfacechargedistribution,togetherw ith the
C oulom b correlations between allcharges involved,lead
to a striking non-m onotonicvariation ofB 2 w ith saltconcentration cs. T he occurrence ofa m inim um ofB 2 as a
function ofcs hasrecently been reported in lysozym e solutionsforcs = 0:3 M [12]and in A poferritin solutionsfor
cs = 0:15 M [13].R elated experim ental ndingsare nonm onotonic variations ofother quantities w hich strongly
correlate w ith B 2 [12,14]such as the interaction param eter [15,16],the cloud point tem perature [17,18],and the
solubility [19]. A llthese trends can be qualitatively understood by our calculation.
A m ore fundam ental understanding of the interactions between nano-sized biom olecules is criticalto the
long-term advance of m odern biom edical research [1].
T he best strategy for a predictive calculation is to
study sim ple coarse-grained m odels w here e ects can be
clearly separated and approxim ations can be system atically tested. W hile for m icron-sized colloidalparticles
such coarse-grained m odelshaveled to a quantitativeunderstanding ofthe e ective interactions[2],the challenging question ishow farthisconceptcan be transferred to
nano-particles.
A particularissueistheaggregation and crystallization
ofglobular proteins in solution,driven by their m utual
interactions,including steric repulsion,van derW aalsattraction, C oulom bic interactions, hydration forces, hydrophobic attraction and depletion forces [2]. M ost of
these are e ective interactions w hich depend sensitively
on solution conditions. In particular C oulom bic forces
are functions of pH (w hich determ ines the totalcharge
ofthe proteins) and of electrolyte concentration,w hich
controlstheD ebyescreening length D ,and hencetheeffectiverangeofC oulom bicinteractions.T hisdependence
on solution conditionsisexploited in \salting-out" experim entsw herelargesaltconcentrationsareused to trigger
protein crystallization,a crucialstep towards the determ ination oftheirstructureby X -ray di raction [3].W hile
the forces acting between m icro-sized colloidalparticles
are dom inated by generic interactions,and are directly
m easurable by opticalm eans [4{6],the interactions between globularproteinsarehighly speci c atshortrange,
and are lessdirectly accessible.O ne possible indirectdeterm ination ofthe totalforce between two proteins m ay
be achieved via m easurem entsofthe osm otic equation of
state by static light scattering,w hich in the low protein
concentration regim e yields the value of the second osm otic virialcoe cientB 2 [7,8].T he variation ofB 2 w ith
solution conditionsyieldsvaluableinform ation on theunderlying e ective protein pair interactions. M oreover,it
has been show n em pirically that there is a strong correlation between the m easured values ofB 2 and the range
ofsolution conditionsunderw hich protein crystallization
is achieved [7{10].
T his report focuses on the e ective interactions be-
FIG .1. Snapshotofa typicalM D -generated m icroion conguration around tw o proteins,separated by r = 1:7 p . T he
proteins carry 15 discrete charges e; m onovalent salt m olarity is cs = 0:206M ol=l. T he globular protein m olecules are
show n as tw o large gray spheres. T he em bedded sm alldark
spheres on their surface m im ic the discrete protein charges
in the D C M m odel. T he sm allgray spheres are counterions,
w hile the black spheres are coions.
1
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ter is c = 0:267nm . N ote that the SC M always im plies
vanishing m ultipolem om ents,w hereasw ithin thepresent
D C M , the only charge pattern w ith a non-vanishing
dipolem om entisthatforZ = 15.A snapshotofa typical
equilibrium m icroion con guration around two proteins
is show n in Fig.1,for the case Z = 15. N ote that the dim ensionless C oulom b coupling param eter for a proteincounterion contact,nam ely = e2=[ kB T ( + c=2)]for
the D C M ,and = 2Z e2=[ kB T ( p + c)]for the SC M ,
are com parable and ofthe order of = 3 at room tem perature. T he total force F~1 = F~2 depends only on
the centre-to-centre distance r for the SC M ,but is also
a function ofthe orientations ofthe two charge patterns
ofthe D C M ,em bodied by two unit vectors ~
!1 and ~
!2;
F~1 = F~1 (r;~
!1;~
!2). T he anisotropy ofthe force turnsout
to be relatively weak. T he e ective radialpair interactions between proteins,V (r),follow from integration of
the radialprojection ofan orientationally averaged force
F~1 along the centre{to{centre vector~
r,according to:
Z1
~
r ~ 0
V (r)=
dr0h
(1)
F (r ;~
!1;~
!2)i! 1 ! 2 :
j~
rj
r
W econsidertwo sphericalproteinsofdiam eter p ,each
carrying a totalchargeZ e (w hereZ dependson pH ),surrounded by m onovalent co and counterions,assum ed to
have identicaldiam eters c. T he solvent (water) is assum ed to be a dielectric continuum ofperm ittivity ;this
sim pli cation,w hich ignoresthe m oleculargranularity of
the solvent,am ounts to the standard \prim itive" m odel
ofionic solutions [20].
In the case of highly charged colloidal particles, the
totalcharge Z e is usually assum ed to be uniform ly distributed on the surface,a situation w hich w illbe referred
to as the \sm eared charge m odel" (SC M ).T his sim plication is m uch less justi ed for the sm aller, weakly
charged proteins (w here Z ’ 10). W e have hence
adopted a second, discrete charge m odel(D C M ) w here
Z m onovalentdiscrete pointchargesare distributed over
the surface of a sphere of diam eter p 2 = 0:96 p
(i.e.slightly inside the surface ofthe protein),in such a
way as to m inim ize the electrostatic energy of the distribution; the resulting pattern does not correspond to
the realcharge distribution on any speci c protein,but
doesprovidea well-de ned discretem odelforcom parison
w ith the SC M ,and between di erentvaluesofZ .A tthis
stage the two m odels (SC M and D C M ) involve only excluded volum e and bare C oulom bic interactions(reduced
by a factor 1= to account for the solvent) between all
particles (proteins and m icroions). H owever,in view of
the large size asym m etry,the e ective force between the
proteins, w hich ultim ately determ ines the second virial
coe cient, involves a statistical average over m icroion
con gurations in the eld oftwo xed proteins [21]. For
distancesr > p between the centresofproteins1 and 2,
the totalforce F~1 = F~2 acting on each ofthe proteinsis
(1)
(2)
(3)
the sum ofthree contributions,F~1 = F~1 + F~1 + F~1 ,
(1)
w here F~1 is the direct C oulom b repulsion between the
(2)
chargeson the two proteins,F~1 isthe m icroion induced
(3)
electrostatic force and F~1 is the depletion force due to
the im balance ofthe m icroion osm otic pressure acting on
(2)
(3)
opposite sides ofthe proteins [21]. B oth F~1 and F~1
are averages over m icroion con gurations; according to
(3)
the contact theorem F~1 is directly related to the integralofthe m icroion contact density over the surface of
the protein [22,23]. T he statistical averages leading to
(2)
(3)
F~1 and F~1 were com puted using M olecularD ynam ics
(M D ) sim ulations. T he two proteins were placed sym m etrically w ith respect to the centre along the body diagonalofa cubicsim ulation celloflength L = 4 p ,w hich
also contained m onovalentco and counterionsin num bers
determ ined by theirbulk concentrations;periodicboundary conditions were adopted. T he choice ofL was m ade
to ensure that the box length is m uch larger than the
range ofthe total(e ective) protein-protein interaction,
so that the results would be independent ofL. For our
m odelto be a rough representation oflysozym e,wechose
p = 4nm ,and Z = 6;10 and 15,corresponding to three
di erent values ofthe solution pH .T he m icroion diam e-
H ere h:::i! 1 ;! 2 refersto a statisticalaverage overm utual
orientations ofthe two proteins [24]. T he second virial
coe cient in units ofits value 2 p3 =3 for hard spheres
(H S )
ofdiam eter p ,B 2 = B 2=B 2
,can then be proven to
be given by:
Z
3 1
drr2 [1 exp f V (r)=k B T g]; (2)
B2 = 1+ 3
p
p
a result form ally identical to that valid for spherically
sym m etric forces. R esults for B 2 as a function of salt
concentration are show n in Fig.2 for the SC M and D C M
m odels,w ith three values ofthe totalprotein charge.In
order to obtain values of B 2 com parable to m easured
virialcoe cients,we have taken short-range attractions
between proteinsinto account,by adding to the e ective
C oulom b potentialin Eq.(2) a \sticky" hard-sphere potentialofthe B axterform [25],w ith potentialparam eters
= 0:02 p and = 0:12,w hich are know n to yield reasonable osm otic data forlysozym e solutions[9,26]in the
high salt concentration regim e,w here C oulom bic interactions are essentially screened out.
T he key result,illustrated in Fig.2,liesin the considerable qualitative di erence between the predictions ofthe
SC M and the D C M m odels for the variation ofB 2 w ith
m onovalentsaltconcentration cs,irrespectiveofthetotal
protein charge Z e. W hile the SC M (dashed curves) predicts a m onotonic decay ofB 2 w ith cs,the D C M leads
to a m arkedly non-m onotonic variation,involving an initialdecay towards a m inim um followed by a subsequent
increase to a m axim um and a nal decrease towards a
high cs value sim ilar to that predicted by the SC M .T he
location of the m inim um and of the m axim um shift to
higher values ofcs for largerprotein chargesZ .
T he origin ofthe non-m onotonic variation ofB 2 w ith
cs can be traced back to the dependence ofthe e ective
2
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m icroion density pro le (r) = + (r)+
(r) around a
single isolated protein is show n in Fig.3,for Z = 10,and
two salt concentrations (the pro les are orientationally
averaged in the case ofthe D C M ).A tthe lowersaltconcentration (cs = 0:206M =l) the SC M and D C M m odels
both yield an accum ulation ofthe m icroion density near
contact,in sem i-quantitative agreem entw ith the prediction ofstandard Poisson-B oltzm ann (PB )theory.A tthe
highersaltconcentration,however,there isa m arked depletion in the m icroion density,signaled by a m inim um
of (r) wellbelow the asym ptotic bulk value. T his correlation e ect is ofcourse absent in the (non-linear) PB
theory,w hich alwayspredictsa m onotonically decreasing
density pro le (r). N ote howevera signi cantdi erence
between the SC M and D C M pro les. W hile the latter
predicts a contact value c(r = ( p + c)=2) larger than
the bulk value,SC M predicts a m uch stronger m icroion
depletion nearcontact.T his nding illustratesthe sensitivity ofcorrelation e ectsto the assum ed chargepattern
at the surface ofa protein: taking into account the discreteness ofthe surface charges leads to a signi cant reduction ofm icroion depletion atcontact,com pared to the
sim pli ed picture ofa uniform ly sm eared charge (SC M ).
(screened) C oulom b interaction on salt concentration as
show n in the inset ofFig.2 for Z = 10. W hile the spherically averaged,repulsive e ective potentialV (r) ofthe
D C M is initially strongly reduced as cs is increased,its
am plitude and range increase very signi cantly at interm ediateconcentrations(cs ’ 1M =l),beforeitnearly vanishes at the highest salt concentrations. N ote that V (r)
becom es even slightly attractive at contact (r = p ) for
cs ’ 2M =l. T he enhanced e ective C oulom b repulsion
at interm ediate salt concentrations cannot be rationalized in term s ofsim ple m ean- eld screening argum ents;
it is caused by a subtle correlation e ect w hich leads to
the non-m onotonic behaviorofB 2 w ithin the D C M .T he
protein-m icroion correlationsareofa su ciently di erent
nature in the SC M ,to lead to a m uch m ore conventional,
m onotonic decay ofB 2 w ith cs,sim ilar to that expected
from a linear screening picture.
0.206
0.412
0.824
1.03
1.24
2.061
0.1
V(r)/kBT
0.03
B*2
−0.1
−0.01
−0.3
0.86
1
1.1
r/σp1.2
1.3
0.84
1.4
0.82
ρ(r) [M/l]
0.8
−0.5
−0.7
0
0.5
1
Cs [M/l]
1.5
0.78
0.26
2
0.24
FIG .
2.
N orm alized second virialcoe cient B 2 = B 2 =B 2H S of a protein
solution versus added salt m olarity. R esults are show n for
protein chargesZ = 6 (dashed lines),Z = 10 (solid lines) and
Z = 15 (dot-dashed lines). T he lines w ith (w ithout) sym bols
correspond to the SC M (D C M ) m odel. T he inset show s the
e ective protein-protein interaction V (r) in the D C M m odel
versus separation distance r for Z = 10. Various sym bols in
theinsetrelateto thedi erentadded saltm olarities,indicated
in the legend.
0.22
0.2
0.6
0.7
0.8
r/σp
0.9
1
FIG .3. Totaldensity pro les (r)= + (r)
(r)ofsm all
ions around a single protein, for salt m olarities cs = 0:206
(bottom set ofcurves) and cs = 0:824 (upper set ofcurves).
T he solid and dashed lines are sim ulation results for D C M
and SC M m odels respectively,w hile the dot-dashed lines are
predictions ofnon-linear Poisson-B oltzm ann theory.
Even though the e ective C oulom b potentialbetween
proteins is ofsm allam plitude,only a few percent ofthe
therm alenergy kB T ,the e ecton B 2 isdram atically enhanced by the presence ofthe strong short-range attractive com ponent due to van der W aals and hydrophobic
interactions,w hich we have included in the form ofthe
B axter \sticky" sphere potential. T his potential is independent ofsalt-concentration,and has no in uence on
the qualitative dependence ofB 2 on cs.
In orderto gain furtherinsightinto thephysicalm echanism responsibleforthe unusualvariation ofthe e ective
C oulom b potentialand ofB 2 w ith saltconcentration,we
have investigated in detailthe localm icroion density in
the im m ediate vicinity ofthe protein surface.T he radial
N ext,considerthein uenceofa second near-by protein
on the m icroion distribution nearcontact.W e have com puted the di erence between \inner" and \outer" shell
m icroion contactdensities,asschem atically illustrated in
theinsetto Fig.4.T helocalm icroion density isno longer
spherically sym m etric,due to the interferenceofthe electric double-layers associated w ith the two proteins. T he
di erence = in
out between the m ean num ber of
m icroions w ithin a fraction ofa sphericalshellofradius
R = 0:6 p subtended by opposite 60 cones is plotted in
Fig.4 versussaltconcentration. isalwayspositive,indicating thatm icroions(in factm ostly counterions)tend
to clusterin the region between the proteins,ratherthan
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m issed by non-linear PB theory.
W e chose oursim ple m odelsto help highlightand separatethe e ectsofdiscrete chargepatternsand C oulom b
correlations.Extending ourM D calculationsto the m ore
com plex (pH dependent)charge patternsofrealistic proteins [27]is technically straightforward. W e expect that
the physicalm echanism leading to enhanced protein repulsion atinterm ediate saltconcentration,w hich isillustrated by the m icroion density im balance show n in Fig.4,
w illcarry over.Sincethe second osm oticvirialcoe cient
determ ines m uch of the excess (non-ideal) part of the
chem icalpotentialofsem i-dilute protein solutions,it is
anticipated thatthe non-m onotonicity ofB 2 m ay have a
signi cant in uence on protein crystallization from such
solutions in the course of a \salting-out" process. T he
non-m onotonic behavior also suggests the possibility of
an inverse,\salting-in" e ect,w hereby a reduction ofsalt
concentration m ay bring B 2 into the\crystallization slot"
[8].
T he authors are grateful to R . Piazza, I.L. A lberts,
P.G .B olhuis, G .B ricogne, J.C larke,S. Egelhaaf,J.F.
Joanny,and W .C .K .Poon for usefuldiscussions,and to
Schlum bergerC am bridgeR esearch and theIsaacN ew ton
Trust for nancialsupport.
on the opposite sides, as one m ight expect due to the
enhanced lowering ofthe electrostatic energy forcounterionsshared between the two proteins.H owever,there is
a very signi cantdi erencein thevariation of w ith c s,
between the SC M and the D C M m odels. B oth exhibit
sim ilar behavior for cs . 0:5M =l, w ith a sm all m axim um around 0:2M =l.B eyond 0:5M =l,however,theSC M
predicts a m onotonic decrease of , w hile the D C M
leads to a sharp peak in
for c s ’ 1M =l. T his highly
non-m onotonic behavior clearly correlates w ith the nonm onotonicity ofB 2 evidentfrom Fig.2.T he excessnum ber of m icroions between the two proteins leads to an
im balance in osm otic pressure,w hich is the origin ofthe
increased repulsion between proteins around cs = 1M =l,
asshow n in theinsetofFig.2.ForstrongerC oulom b coupling,asis the case for highly charged colloidalparticles
in the absence of salt, the above depletion m echanism
is inverted,and leads to a depletion attraction between
the particles [21],rather than to the enhanced repulsion
found here in the case ofrelatively weakly charged proteins.
8
σp
7
60
)
6
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r
5
∆ρσp3
o
4
DCM
3
2
SCM
1
0
0
0.5
1
1.5
2
Cs [M/l]
FIG .4. M icroion density im balance
versus salt m olarity for protein charge Z = 10 and separation r = 1:2 p .
T he solid and dashed lines correspond to the D C M and SC M
m odels respectively. T he inset show s the angular range over
w hich
is averaged (see text).
T he m ain nding ofthe present work is that the second osm otic virialcoe cient of protein solutions has a
non-m onotonic dependence on salt concentration if the
charge pattern on the protein surface is discrete (as is
the case forrealproteins)ratherthan uniform ly sm eared
out, as usually assum ed in the related case of chargestabilized colloidal dispersions, involving m uch larger
particles. T he lesson to be learned from this nding
is that one m ust be cautious in attem pting to extend
coarse graining concepts and approxim ations,developed
and routinely used on the colloidal scale, to the nanom etric scale ofproteins. T he discreteness ofthe charge
pattern is crucialto obtain non-m onotonic behavior of
B 2,w hich is a subtle C oulom b correlation e ect,totally
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[27] M .B oyer et al,J.C ryst.G row th 196,185 (1999).
5
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