Interaction of charged patchy protein models with like-charged polyelectrolyte brushes

2 MB·

by  Cemil Yigit

Interaction of charged patchy protein models with like-charged polyelectrolyte
brushes
Cemil Yigit,1,2,3 Matej Kanduč,2 Matthias Ballauff,1,2,3 and Joachim Dzubiella1,2,3
1Institut für Physik, Humboldt-Universität zu Berlin, 12489 Berlin, Germany
2Institut für Weiche Materie und Funktionale Materialien,
Helmholtz-Zentrum Berlin, 14109 Berlin, Germany
3Helmholtz Virtual Institute, Multifunctional Biomaterials for Medicine, 14513 Teltow, Germany
We study the adsorption of charged patchy particle models (CPPMs) on a thin film of a like-
charged and dense polyelectrolyte (PE) brush (of 50 monomers per chain) by means of implicit-
7
solvent, explicit-salt Langevin dynamics computer simulations. Our previously introduced set of
1
0 CPPMsembraceswell-definedone-,andtwo-patchedsphericalglobules,eachofthesamenetcharge
2 and (nanometer) size, with mono- and multipole moments comparable to those of small globular
proteins. We focus on electrostatic effects on the adsorption far away from the isoelectric point of
n
typical proteins, i.e., where charge regulation plays no role. Despite the same net charge of the
a
J brush and globule we observe large binding affinities up to tens of the thermal energy, kBT, which
are enhanced by decreasing salt concentration and increasing charge of the patch(es). Our analysis
0
ofthedistance-resolvedpotentialsofmeanforcetogetherwithaphenomenologicaldescriptionofall
1
leadinginteractioncontributionsshowsthattheattractionisstrongestatthebrushsurface,driven

] bymultipolar,Born(self-energy),andcounterion-releasecontributions,dominatinglocallyoverthe
ft monopolar and steric repulsions.
o
s
t. I. INTRODUCTION In principle, the brush and the protein should repel
a
each other because of two reasons, namely i) elec-
m
trostaticrepulsion,andii)theelectrostericrepulsion
d- Polyelectrolyte brushes consist of polyelectrolyte betweentheproteinandthebrushlayer: Insertinga
n chains grafted to a planar or curved surface [1–3]. protein into a polyelectrolyte brush will lead to un-
o The overall structure of such a system is mainly favorable steric interactions with tethered chains of
c determined by the osmotic pressure of the coun- the brush as well as to a raise of the osmotic pres-
[
terions. In the osmotic limit, that is, at low salt sure of the confined counterions. Hence, there must

1 concentrations, the chains are strongly stretched beastrongattractiveforcewhichiscapableofover-
v whereas in the salted brush generated by high coming these strongly repulsive forces. Three major
7
salt concentrations the electrostatic interaction is causeshavebeensuggestedtoexplainthestrongat-
5
4 stronglyscreenedandtheresultingspatialstructure traction observed for a broad variety of systems:
of the brush layer resembles the case of uncharged
2
0 brushes [4–11]. In this way polyelectrolyte brushes • First,chargeinversionoftheproteinimmersed
1. present adaptive systems that have been discussed in a polyelectrolyte brush [13, 15]. The pH-
0 forabroadvarietyofapplications[3]. Some10years dependent protein concentration within the
7 ago it has been found that polyelectrolyte brushes brush layer my be lower than outside in the
1 strongly adsorb proteins with like net charge in the bulk and below the isoelectric point. Hence,
:
v osmotic limits, while virtually no interaction took the net charge of the protein changes its sign.
i place in the limit of the salted brush [12]. Thus, a
X
spherical polyelectrolyte brush consisting of a solid • As a second driving force counterion re-
r
a core of approximately 100 nm diameter carrying lease was invoked [14, 22, 26]. Proteins in
long grafted chains of poly(acrylic acid) (PAA) ad- general carry patches of positive charge on
sorbshighamountsofbovineserumalbumine(BSA) their surface even above the isoelectric point.
above its isoelectric point where the overall charge These patches can interact with the nega-
of the protein is negative as well. This discovery of tively charged polyelectrolyte chains, thereby
theadsorptionofproteinsatthe“wrongside” ofthe releasingacertainnumberofpreviouslybound
isoelectric point has led to a number of experimen- counterions into the bulk phase. Since the
tal and theoretical studies on planar and spherical osmotic pressure within the brush layer at
systems [13–20]. low salt concentration is quite high, it has
1
been argued that the effect of counterion re- fixed (pH-independent) local charge distributions.
lease should be strong and virtually indepen- The general goal of this study is a fully quantita-
dent of the proton concentration within the tive assessment of the local electrostatic effects that
brush [12, 21]. drive like-charged attraction with a sharpened fo-
cus on the details of dipolar and counterion-release
• As a third cause for attraction, a heteroge- effects in the limit of highly charged PE brushes.
neously charged protein globule may be at- Our simulations allow us to accurately average out
tracted into the brush because the polyelec- all conformational effects and explore in detail the
trolyte brush interacts asymmetrically with ionic mechanisms taking place at the PE-patch in-
thedipoleoftheproteincausedbylargecharge terface. Based on this, the various contribution to
patches of opposite sign [18, 23–25]. protein uptake by the brush can be properly char-
acterized and quantitatively compared to analytical
Thecombinedproblemofpatchinessandcounterion models.
release has been re-considered by de Vos et al. in a
We systematically start by studying the interaction
field-theoreticalstudy[25]. Theseauthorsconcluded
of uncharged spheres with the brush layer to obtain
that the effect of counterion release is operative but
the simplest steric interaction. In the second step,
of minor importance. However, a recent study by
the interaction of a CPPM without patches but a
He et al. [27] using molecular dynamics simulations
netchargewiththelike-chargedbrushlayerisdeter-
clearlyunderscoredtheimportantroleofcounterion
minedtostudyexclusivelytherepulsiveelectrostatic
release when considering the interaction of proteins
interactionsbasedonmonopoleinteractionsandthe
withpolyelectrolytebrushes. Thus,theroleofcoun-
contributions by the osmotic pressure of the coun-
terionreleaseanditsmagnitudeisstillnotfullyelu-
terions. Finally, we study CPPMs with additional
cidated.
charge patches (with opposite sign of the charge
In order to re-consider this problem, we first de- than the brush) that indeed can display a strong,
veloped a model for proteins with patchy surface like-charge attraction. We qualitatively discuss the
charge (charged patchy protein model, CPPM) [28]. balancebetweenallleading-orderinteractionmecha-
The interaction of these model proteins with single nisms at hand of a phenomenological brush–CPPM
polyelectrolyte chains of like charge was studied ex- binding model. As in our previous paper,[29] we
tensively by implicit-solvent/explicit-salt Langevin findthatthecounterion-releaseeffect(involving2–3
computer simulations [28, 29]. The results obtained ions) is one of the dominant driving force for pro-
inthisstudyconfirmedthegeneralconceptofcoun- tein adsorption. However, this interaction can be
terion release for highly charged polyelectrolytes. supportedsignificantlybythedipolarattractionand
Moreover, a simple theoretical model was shown to Born (self-energy) terms. We demonstrate that the
capture the salient features of the simulations. In combination of all terms lead to a stable adsorption
particular, the binding affinity derived from these of the protein at the surface of the brush layer.
simulations using the CPPM-model was shown to
scale dominantly with the logarithm of the salt con-
centration in the bulk multiplied by the number of
II. METHODS
released counterions [29]. This relation had been
predicted by Record and Lohman [30] a long time
ago and found to be valid in a number of experi- A. Charged patchy protein models
mental studies [31, 32]. It demonstrates the direct (CPPMs). As introduced earlier, [28, 29]
relationbetweenthebindingaffinityandthetransla- we employ a set of spherical patchy protein
tional entropy of the released counterions. It can be models with well-defined charge patchiness and
deriveddirectlyfortheproblemunderconsideration multipolarity to mimic the electrostatic features
here as shown by Henzler et al.[21]. of nanometer-sized globular proteins or similar
Here we extend our previous studies [28, 29] to ex- nanoparticles. Briefly, the CPPMs are constructed
plore the interaction of patchy proteins with a thin by distributing 642 atom-sized ((cid:39)0.3 nm) beads on
planar film of a dense like-charged polyelectrolyte a spherical surface with radius R = 2 nm. The
P
brush. In order to keep the problem as simple as latter is typical for small globular proteins such as
possible, we do not consider the effect of charge re- lysozyme or lactoglobulin [33].
versal and focuson the electrostatic mechanisms for To build a charge patch, one bead on the surface is
2
TABLEI:Asummaryofourchargedpatchyparticlemodels(CPPMs)denotedbyPm. Theindexmstandsforthe
s
number of patches, while s denotes the number of positive charges on each patch. In the images of the CPPMs in
the top row, the pink beads depict the negatively charged atoms, while turquoise beads depict the positively ones.
Yellowandwhiteatomsdepictthesameneutralatomsandareonlydistinguishedheretobetterillustratethepatch
region,whichroughlyhasanareaof3nm2. AllCPPMshavearadiusofR =2nmandanetchargeofQ =−8e.
P P
Thepatchyglobulescarryindividualdipolemomentsasalsosummarizedinthetable. Thecorrespondingquadrupole
(tensorial) moments are provided in previous work [29].
Label P0 P1 P1 P1 P2 P2
0 8 12 16 8 12
Radius R [nm] 2 2 2 2 2 2
P
Patch area A [nm2] 0 3 3 3 3 (x2) 3 (x2)
P
Total charge Q [e] -8 -8 -8 -8 -8 -8
P
Dipole moment µ [D] 159 896 1329 1633 206 151
P
randomlychosenandsubsequentlytheclosestneigh- CPPM as coordinate origin for the charge distribu-
boring beads are selected until a roughly circular tion,areintherangeof159Debyeto1633Debye,cf.
patch area A (cid:39) 3 nm2 is achieved. This area is TableI,alsocomparabletoproteinsofthissize. The
P
of the same order as the size of some naturally oc- smalllactoglobulin,forinstance,has730Debye[36].
curring larger clusters of charged amino acids of the
same sign, based on the inspection of crystal struc-
tures of small globular proteins [34]. Afterwards,
B. PEbrush model. ThesimulatedPEbrush
s positive charges are placed on randomly chosen
is composed of 16 flexible PE chains fixed by har-
beads on the patch. We construct patchy globules
monicconstraintsatoneendonaneutralandplanar
with one (m = 1) or two (m = 2) patches. In
surface in equidistant spacings on a square lattice.
CPPMs with two patches the patches are antipo- With a surface area of 100 nm2, the grafting den-
dally directed, that is, on the exact opposite sides. sity thus corresponds to τ = 0.16 molecules/nm2.
B
In order to assign a net charge Q to the CPPMs,
P A single flexible PE is modeled in a coarse-grained
we fixed the number of negatively and positively
fashion as a sequence of N = 50 freely jointed
mon
charged beads to be N = 37 and N = 29 in
n p beads. Each bead represents a monomer with a di-
all CPPMs. Thus, the resulting net charge of the
ameter σ and an electric charge of one negative
LJ
patchyglobuleisQ =−8eforallCPPMs, compa-
P elementary charge −e. The PE monomers are con-
rable to absolute net charges of proteins of similar
nected by a harmonic bond potential with an equi-
size at physiological conditions [34]. The N neg-
n librium bond length b =0.4 nm and a force con-
mon
ative charges are homogeneously distributed on the stant K = 4100 kJ/(mol nm2). A harmonic an-
mon
surface around the positive patch(es). The remain-
gle potential is applied in which the angle between
ingNp−m·spositivechargesaredistributedinsuch a triplet of monomers is γ = 120◦ and the force
a way that charged beads (positive or negative) are constant is K = 418 kJ/(mol rad2). An illustrat-
γ
not immediately adjacent. Our CPPMs are denoted
ing snapshot of the PE brush (including salt) and a
byPm wheremspecifiesthenumberofpatchesand
s CPPM is shown in Fig. 1.
s the number of positively charged beads per patch.
Illustrative snapshots and a summary of the CPPM
features, in particular, the dipole moments, are
C. Simulation details. The dynamics of each
listedinTableI.Inourmodelsweconsiderm=1,2
ofthebeads(creatingtheCPPMandthebrush)and
ands=8,12,16resultinginmeanpatchchargeden-
explicit ions is governed by Langevin’s equation of
sities of around 1 to 2 e/nm2 corresponding to a lo-
motion
cal assembly of a few amino acids separated from
each other by a few angstroms [34, 35]. The dipole d2r dr
moments, calculated using the center-of-mass of the mi dt2i =−miξi dti +∇iU +Ri(t) (1)
3
where the force R (t) is a Gaussian noise pro-
i
cess with zero mean (cid:104)R (t)(cid:105) = 0 and satisfies the
i
fluctuation–dissipation theorem
(cid:104)R (t)·R (t(cid:48))(cid:105)=2m ξ k Tδ(t−t(cid:48))δ . (2)
i j i i B ij
The m and ξ are the mass and friction constant of
i i
theithbead, respectively. U isthesystempotential
energy and includes intra- and intermolecular inter-
actions and position constraints. All interatomic in-
teractions are composed of the Lennard-Jones (LJ)
potential between the beads, ions, and the grafting
surface (modeled by the 9–3 potential), as well as
the Coulomb interaction between all charged beads
andions. LJinteractionsbetweenneighboringpoly-
mer beads are excluded. The solvent is modeled as
a continuous medium with a static dielectric con-
stant (cid:15) = 78.44. All beads and ions have mass
m = 1 amu, a LJ diameter σ = 0.3 nm, energy
i LJ
well (cid:15) =0.1 k T, and integer charges q = 0, +e,
LJ B i
or −e.
The simulations are performed using the GRO-
FIG.1: AnillustrationoftheuptakeofaCPPM(yellow
MACS4.5.4softwarepackage[37]. Aleap-frogalgo-
patchy globule) by a planar PE brush (magenta colored
rithmwithatimestepof2fsisusedtointegratethe
connected beads) in the presence of co- and counterions
equationsofmotion. TheLangevinthermostatwith (small magenta and green beads, respectively). (a) The
ξi = 0.5 ps−1 keeps the temperature at T = 298 K CPPM is situated in the bulk region. (b) for the PMF
and generates a canonical ensemble (NVT). Cen- calculation the CPPM is moved with a constant pulling
ter of mass translation of the system is removed ev- ratevp towardsthebrushand(c)furtherpulledthrough
ery 10 steps. The rectangular simulation box with the brush layer until it finally reaches the grafting sur-
L =L =10 nm, and L =30 nm is periodic only face.
x y z
inthex,y-directions. Thewalls,placedatz =σand
z =30 nm, are represented by the 9–3 LJ potential
to ensure electroneutrality of the system. Addition-
with parameters σ = 0.3 nm, (cid:15) = 0.1 k T, and pa-
B
ally, monovalent salt is added to the system lead-
rameter ’wall-density’ = 0.5 nm−3. One end of the
ingto(bulk)concentrationsc =15,32,78,137,and
PE chains is position-restrained in the immediate s
259mM.Thesystemisrelaxedfor100pstoremove
vicinity of the wall at z = 0 nm by a harmonic po-
tential with a force constant of 4100 kJ/(mol nm2). local contacts and afterwards equilibrated for 30 ns.
The cut-off radius is set to 1.2 nm to calculate the
real-space interactions, while Particle-Mesh-Ewald
(PME) is implemented to account for long-range D. PMF calculations. For calculating the
electrostatics [38]. The reciprocal summation of PMF between the brush and CPPMs the pull code
the PME method is computed on a 3D FFT grid of the software package GROMACS was used with
but with spacings of 0.32 nm in x,y-directions and the umbrella method [37]. Here, the center-of-mass
0.23 nm in z-direction using a fourth-order interpo- (COM) of the patchy particle is restrained in space
lation. Because of the periodicity, a correction term by an external time-dependent force. This force is
to the Ewald summation in the z-direction is added applied by a harmonic potential that is moved with
toproduceapseudo-2Dsummation[39,40]toavoid a constant pulling velocity v to steer the CPPM in
p
artefacts of the system’s instantaneous net dipole the prescribed direction [41]. The reaction coordi-
moment.. nate is z, the distance of the COM of the CPPM to
The CPPM is initially placed at z ≈ 27 nm. After thegraftingsurfaceatz =0. Afterseveraltestruns,
the simulation box with brush and CPPM is set up, andcomparisontostandardumbrellasampling,[29]
the corresponding number of counterions is added the pulling rate v = 0.1 nm/ns and a harmonic
p
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force constant K = 2500 kJ mol−1 nm−2 were cho- difficult to develop, we will explicitly calculate the
sen to yield the best performance with respect to PMF between a neutral CPPM interacting with a
accuracy versus computational effort. The simula- charged brush in the simulation. However, qualita-
tion time of ∼250 ns is required to pull the CPPMs tively the contributions to w (z) can be dis-
excl+vdW
deep into the brush (z (cid:39) 2 nm) from a separated cussed as follows.
state (z (cid:39)27 nm). The standard deviation was cal-
In our simulations, the vdW interactions are mod-
culated by block averages to specify the statistical
eled by the attractive part of the LJ interaction.
error.
This interaction is small because of our choice of
After a completion of a run, the constant friction
the (cid:15) -parameter of only 0.1 k T. The excluded
force F = −mξv was subtracted from the total LJ B
p volume part originates on one hand from the con-
force and averaged within a specific interval of dis-
figurational response of the PEs to the CPPM ex-
crete spacing ∆z to obtain the mean force. Accord-
cluded volume and on the other hand from the os-
ing to our simulation setup, the mean force was in-
motic pressure of confined counterions, which also
tegrated backwards to get the PMF. We emphasize
penalizes the intrusion of the globule. Based on
that in all our simulations the patchy particles were
classical laws of the osmotic pressure of semi-dilute
able to rotate freely and, thus, all our results are
polymer solutions [42] the former is expected to be
orientation-averaged with the appropriate and cor-
repulsive on the order of several k T due to the
rect Boltzmann weight. B
polymer configurational entropy loss. We will cal-
culate it by simulating a neutral CPPM interact-
ing with a completely neutral brush. Note, how-
E. CPPM orientation. The patch vector p
ever,thatchargingthepolymersstretchesthebrush
points from the particle center to the patch center
and may modify this contribution, likely decrease it
and provides also a very good approximation of the
slightly as the brush effectively swells upon charg-
dipole direction µ(cid:126) of the P1 models. In our analy-
s ing. The second excluded volume contribution, let
sis we computed the distance-resolved cosine of the
us write it as w (z), arises from the volume work
angle θ(z) by osm
of the penetrating CPPM with volume V against
P
(cid:28)p·e (cid:29) the osmotic pressure of the confined counterions.
cos[θ(z)]= z . (3)
Assuming ideal gas behavior, we can expect it to
|p|
z scale roughly linearly with counterion density as
The distance-dependent angular correlation of the w (z) = k T[c (z) − c ] · V , where c (z)
osm B free s P free
patch vectors is calculated via the second Legendre denotes the local number density of free (not con-
polynomialP (cos[θ])withP (x)=(3x2−1)/2. For densed)counterionswithinthePEbrush,whilec is
2 2 s
CPPMs with two patches only one patch is used to the bulk number density of salt ions. For our mod-
calculate the orientation since the patches are an- els,V (cid:39)33.5nm3,andforabrushchargedensityof
P
tipodally directed. about 1 nm−3, then tens of the thermal energy k T
B
repulsionareeasilypossible. Wewillindeedseethat
thiscontributiondominatesthenon-electrostaticin-
F. Phenomenological separation of the teraction.
PMF. Inordertodescribethedominantphysical
If the CPPM is charged, explicit electrostatic in-
contributionstothePMFbetweenthesingleCPPM
teractions come into play, incorporated in w (z)
and the brush, we employ a phenomenological ap- elec
in eq. (4). In principle one could employ numer-
proach where we divide up the total free energy in
ical solutions of mean-field theories [18]. Here we
three major contributions, via
proceed on a simpler but analytical and thus more
w (z)=w (z)+w (z)+w (z). (4) transparent way: based on analytical solutions of
tot excl+vdW elec cr
thelinearizedPoisson–Boltzmannequationinacell-
With the first term w (z) we represent all model,[43]theleadingorderelectrostaticinteraction
excl+vdW
contributions for a neutral globule, that is, the energy for a strictly monopolar globule within a ho-
excluded-volume (including steric and osmotic) and mogeneousPEmatrixisgivenbytwoterms: first,a
van der Waals (vdW) interactions between a totally monopole term that simply describes charge repul-
neutralCPPM(nobeadischarged)andthecharged sion by the electrostatic (Donnan) potential. Sec-
PEbrush. Asafullanalyticaltheoryforthispartis ondly, a Born term that describes the change of the
5
self-energy of electrostatic charging the globule in bytransferringaCPPMfrombulkintothePEbrush
the bulk solvent versus the brush environment. The reads in the monopole–dipole approximation
latter has significantly different screening properties
∆w (z)=w (κ (z))−w (κ ), (7)
than the bulk and the self-energy of charging can Born Born brush Born bulk
change considerably. Here, we tie up on these ideas √
where κ = 8πλ c represents the inverse De-
and include the next leading term in a multipolar bulk B s
bye screening length in the bulk and κ (z) =
expansion of the problem, that is, the dipolar con- (cid:112) (cid:80) brush
4πλ c (z)isthelocalinversescreeninglength
tribution to both the electrostatic interaction [44] B i i
within the PE brush. Here, the summation i =
and the Born terms. We thus arrive at
+,−,m runs over the densities of cations, anions,
and PE monomers. With the latter, we have as-
w (z) = Q ·Φ(z) (5)
elec P
(cid:20) (cid:21) sumed that the PE monomers are mobile and thus
sinh[βµE(z)]
− k T ln +∆w (z), fully contribute to the local screening. Since the
B βµE(z) Born
PE are flexible and far from close packing, this
should be a better approximation than assuming a
where β = (k T)−1, and the first two terms on the
B fixed, non-screening background of the PE matrix.
right hand side characterize the (point) monopole
In our comparison to the fully simulated PMFs we
and (point) dipole contribution to the direct inter-
will take the salt density profiles c (z) directly from
action of the CPPM with the electrostatic poten- i
reference simulations without CPPM. The constant
tial Φ(z) and field E(z) = −dΦ/dz, respectively.
λ = e2/(4π(cid:15)(cid:15) k T) is the Bjerrum length and has
The latter are determined consistently in our work B 0 B
the value of 0.71 nm for our systems, i.e., aqueous
from the charge density profiles through integration
solvent at normal conditions.
of Poisson’s equation. Z < 0 and µ > 0 are the
P P Finally, for ourhighlychargedPE brushinteracting
netchargeanddipolemomentoftheCPPM,respec-
with oppositely charged patches, we expect explicit
tively. Note that for our like-charged systems, the
counterion-release effects to play a role, which we
monopole-term is repulsive, while the dipole term is
treatseparatelytotheelectrostaticinteractionsjust
attractive.
introducedabove. Inourpreviousworkontheinter-
For the leading order of the Born energy up to the
actionbetweenCPPMsandsinglePEs,[29]wehave
dipole level we derive (see Appendix A)
argued that counterion-release, within the frame
Q2 1 of the Onsager-Manning-Oosawa treatment,[46–48]
wBorn(κ) = 8π(cid:15)(cid:15)PR (1+κR ) (6) only happens for ions condensed at the PEs, not
0 p P
those accumulated at the protein patch. After com-
3µ2 (1+κR )[2+2κR +(κR )2]
+ P P P P . plexation,thepositiveCPPMpatchchargesbecome
8πεε R3 [3+3κR +(κR )2]2
0 P P P neutralized by one or more PE chains in the brush
and consequently a corresponding amount of con-
The first term is the classical result of a monopolar
densed ions will be liberated. The release entropy
sphere with valence Z and radius R in a salty
P P
per ion can be of considerable magnitude, that is,
environment with inverse screening length κ in the
of the order of several k T. Inspired by the classi-
Debye-Hückel approximation [45]. The second term B
calworkofRecordandLohman[30]andmorerecent
is the equivalent expression for a point dipole with
work,[21]wehaveshownthatthetotalgainintrans-
moment µ centered in the sphere.
P
lational entropy of released ions during CPPM-PE
Since we are dealing with an inhomogeneous sys-
complexation can be well expressed by [28, 29]
tem, i.e., locally varying ion densities perpendicular
to the grafting wall, we apply a local field approxi- (cid:20) (cid:21)
c
mation, that is, we assume we can apply the Born wcr(z)=−kBTN+(z)ln ccond , (8)
s
theory developed for a homogeneous system also in
the case of la ocal (z-dependent) salt distribution. where N is the number of released ions from the
+
This should be a good approximation as long as the PEs upon complexation. As before, c is the bulk
s
length scales of the inhomogeneities are larger than salt concentration and c (typically c (cid:29) c )
cond cond s
the globule size. This is not always the case in our specifies the concentration of condensed counteri-
workbutthesimpletheorywillstillserveasagood, ons in the solvation shell around the PE monomers.
at least qualitative interpretation of the simulation Fromaveragingtheradialdistributionfunctions(not
results. Hence, the change of the total Born energy shown) in a shell of 0.4 nm thickness we calculate a
6
(a) (b)
1 1
c (z) c (z)
m m
-3m] 00..68 cc+−((zz)) -3m] 00..68 cc+−((zz))
n n
[z) 0.4 [z) 0.4
c(i c(i
0.2 0.2
0 0
(c) 0 (d) 0
nm)] -0.2 e] -1
T / (e -0.4 kT / B -2
[kz)B -0.6 cs = 32 mM Φ [(z) -3 cs = 32 mM
E( cs = 137 mM cs = 137 mM
-0.8 -4
2 6 10 14 18 22 26 2 6 10 14 18 22 26
z [nm] z [nm]
FIG.2: Densityprofilesc (z)forchargedPEmonomers(i=m)andsaltions(i=±)foranisolatedpolyelectrolyte
i
brush at (a) c = 32 mM and (b) c = 137 mM. (c) Electrostatic field and (d) potential for a PE brush at 32 mM
s s
and 137 mM salt concentration. All PE monomer beads carry a charge of −e.
value c (cid:39) 3.5±0.5 M, only slightly depending profilesandelectrostaticpropertiesofanisolatedPE
cond
onsaltconcentration[29]. Werefertopreviouswork brush (without any CPPM) at two salt concentra-
for a detailed analysis of accumulation and release tions. PE monomer density profiles c (z) at ionic
m
ofionsduringtheassociationoftwoCPPMs[28],or strengths of 32 mM and 137 mM are displayed in
a single PE and a CPPM [29] or a better resolved Fig. 2 (a) and (b), respectively. They are a mono-
protein model [32]. In our current work the CPPM tonicallydecreasingfunctionofthedistancefromthe
complexes with many PE chains at the same time grafting surface, quickly converging to zero beyond
in the brush and the number of released ions in the z (cid:38) 12 nm in the bulk region. For the lower salt
simulation is not easily accessible. We circumvent concentration the brush is slightly more stretched,
this problem by counting the local number of PE as expected due to the higher osmotic pressure dif-
monomers, N (z), bound to the patch region and ference of the neutralizing counterions. Similar pro-
m
then assuming that on those their fraction of con- files and trends have been observed in related sim-
densed ions were liberated. The latter is then esti- ulations [6–10] and are predicted by self-consistent
mated using the classical Manning law as 1−Γ−1, fieldtheory[49]. Asexpected,thecounterionprofiles
where Γ = zλ /l is the ’Manning’-parameter for a c (z) closely follow the brush density profiles in the
B +
salt valency z and bond length l (cid:39)b . Hence, we brush layer (apart from a Debye layer at the brush
mon
find interfaceatz ∼12nm),indicatingasignificantelec-
trostatic neutralization of the PE brush, while the
(cid:20) (cid:21)
w (z)=−k T (cid:8)1−Γ−1(cid:9)N (z)ln ccond (9) coions are mostly depleted from the brush. Both
cr B m c
s the counterion and coion profiles converge to their
having used the identity N = (cid:8)1−Γ−1(cid:9)N . For respective concentration values in the bulk region.
+ m
the PEs in our brush we find Γ(cid:39)1.78.
From the charge density profiles we calculate the
electric field E(z) and potential Φ(z) of the sys-
tem via integration of Poisson’s equation as shown
III. RESULTS
in Fig. 2 (c) and (d), respectively. The electrostatic
potentialisnegativethroughthebrushlayerandsat-
A. Polyelectrolyte brush only. We first urates to zero in the bulk region. Near the grafting
present an analysis of the monomer and ion density surface, it reaches roughly −3.1 k T/e at the low
B
7
(a) 6 (b) 30
c = 32 mM
s
4 c = 137 mM
T] T] 20 s
B B
[kw(z) 2 [kw(z) 10
0
-2 0
2 6 10 14 18 22 26 2 6 10 14 18 22 26
z [nm] z [nm]
FIG.3: (a)PMFbetweenafullyneutralglobuleandaneutralPEbrush(solidlineincludingerrorbars)andnosalt
(c =0). Panel (b) shows the PMF for a completely neutral globule and charged PE brush at 32 mM and 137 mM
s
salt concentrations.
(a) 40 (b) 30
Simulation Simulation
30 w (z) w (z)
T] tot T] 20 tot
B B
k 20 k
[w(z) [w(z) 10
10
0 0
2 6 10 14 18 22 26 2 6 10 14 18 22 26
z [nm] z [nm]
FIG. 4: PMF for the charged but patchless CPPM (P0) inserted into a charged brush at a salt concentration of
0
(a) 32 mM and (b) 137 mM. The blue dashed lines are comparisons from eqs.(4)–(7) using the potential and ion
distributions from the reference simulations in Figs. 2 and 3 as input.
ionic strength and about −2 k T/e at the higher brush shrinks. Thereason for thisextremumis that
B
ionic strength. As a rough consistency check we here, in the Debye layer, where electroneutrality is
cancomparethesevalueswiththehypotheticalDon- locallyviolated,thepotentialchangeisextremalbe-
nan potential that builds up for perfect electroneu- fore it deteriorates in the inner part of the brush.
tralization of a charged background connected to This has interesting consequences on the adsorption
a salt reservoir [50, 51]. The ideal Donnan poten- of multipolar particles: if a particle has a strong
(cid:112)
tial in this case reads eβφ = ln(y+ y2+1) with dipolemoment, itsdipolarcouplingtothefieldmay
y = c /(2c ), where c is the charge density of the dominate over other interaction contributions and
B s B
chargematrix(here,thebrush). Ifwetakethemax- the distribution of adsorbed particles may be signif-
imumvalueofthebrushdensityclosetothegrafting icantly different than for simple monopoles.
plane,c (cid:39)1nm−3,wefindfortheelectrostaticpo-
B
tentials -3.4 and -2.1 k T/e at the grafting surface
B
for the lower and higher salt concentration, respec-
B. PMFs in some reference cases. We dis-
tively. These numbers overestimate the simulations
cuss now a few insightful limiting reference cases of
by less than 10%. The small discrepancy may come
thePMFinordertounderstandtheindividualinter-
from steric and electrostatic correlation effects, not
actioncontributionsbetter. InFig.3(a)weplotthe
included in the simple Donnan picture where ions
simplest case where both the brush and the CPPM
are simply Boltzmann distributed.
are completely neutral, i.e., no bead is charged and
The electrostatic field, E(z) = −∂Φ/∂z, plotted in only simple excluded volume and vdW interactions
Fig. 2 (c) shows a distinct minimum at the brush play a role. In the PMF of this neutral system we
surface around z = 10−11 nm, moving to smaller observe a small vdW attraction of about −1 k T at
B
values for the larger salt concentration, when the z ≈ 10 nm, while for smaller distances the PMF is
8
repulsive due to the compression of the brush in a (a) 20
rangeofafewk T asexpectedforsemi-dilutepoly- 10
B
mer solutions [42]. In Fig. 3 (b) we further plot the 0
PMF of a completely neutral CPPM inserted now T]B -10
k
icdnaetsnoesiattyhfeuoflblycruocushhnatriesgreiosdtnrsbe,rtucchfs.hed.FiagInns.dt2hfi(ilsale)doasmnwdiotth(ibc)a.brhTuighshhe [w(z) ---432000 PPP181111 26- --B BB505500
repulsionisconsiderableintherangeoftensofk T, -50
B
(b)
aspredictedfromtheidealgasequation. Hence,the 10
contribution wexcl+vdW(z) to the total free energy, 8 P18 - B50
eq (4), for a neutral globule pushed into a highly P112 - B50
charged brush is dominated by the volume work of z) 6 P116 - B50
(m
the globule against the osmotic pressure of neutral- N 4
izing counterions.
2
As another instructive reference system we plot the
PMF of the rather homogeneously charged (patch- 0
(c)
less) and essentially monopolar (P0) system at c = 0.9
0 s
32 mM and c = 137 mM salt concentrations in 0.6
s
Fig. 4 (a) and (b), respectively. The plots reveal, θ]) 0.3
as intuitively expected, a purely repulsive interac- os[ 0
tion, weaker for the higher ionic strength by about c(2 -0.3 P18 - B50
1ca0sekBmTa.nHyoiwnteevrearc,tiaolnresaadryeipnretsheinstr.elFaitrisvte,lywesimhapvlee P -0.6 PP111126 -- BB5500
-0.9
the osmotic and vdW interactions w (z).
excl+vdW
Secondly, we have the electrostatic term w (z) 2 6 10 14 18 22 26
elec
with the repulsive monopole term but the attrac- z [nm]
tive (monopole) Born term. We can neglect the
influence of the very small dipole of this CPPM FIG. 5: (a) Simulated PMF profiles w(z) for the (P1)
s
(P00) and counterion-release plays no role as no systemwiths=8,12,16atcs =32mM.(b)Numberof
patches are present. The phenomenological predic- accumulatedmonomersNm(z)onthepatchasafunction
tion w (z)+w (z) (cid:39) w (z) is also displayed of z. (c) CPPM orientation with respect to the grafting
osm elec tot
surface.
in the figures. Here, the contribution for w (z)
osm
was taken directly from the previous reference sim-
ulations of the neutral sphere in the charged brush,
−10 k T to −45 k T for c =32 mM.
cf. Fig. 3(b). For the electrostatic contributions, B B s
The released ions are determined from the localiza-
eq. (5)–(7), the values for Φ(z), E(z), and charge
tion of N (z) PE monomers on the positive patch
profiles c (z) were taken from the simulations of an m
i
and are presented in Fig. 5 (b). Once the CPPM
isolated charged brush, cf. Fig. 2. The comparison
comes close to the brush surface, cf. Fig. 1(b), PE
issatisfyinggiventheapproximationsalreadymade.
monomers interact with the positive patch and lead
toaquickriseofthenumberofattachedmonomers.
C. InteractionbetweenaPEbrushandone– For even closer approach, Nm(z) saturates to a
patchedCPPMs. WenowturntothePMFsbe- constant value in the brush layer. The value de-
tween the PE brush and a like-charged CPPM with pends on the patchiness and is about 3.8 and 9.3
only one patch, i.e., (P1). We particularly inves- (for s = 8 and 16 at z = 2 nm, respectively) PE
s
tigate how the patch charge and ionic strength af- monomers attached to the patch, which then leads
fects the PMF and its individual contributions. In to N+(z) = (cid:8)1−Γ−1(cid:9)Nm(z) (cid:39) 0.44Nm(z) locally
Fig. 5 (a) we present PMFs for patchiness s=8,12, released counterions on average, i.e., about 1.7 and
and 16 ationic strengths of32mM. Aclearly stable 4.1(fors=8and16,respectively)deepinthebrush.
adsorption minimum is found at the brush surface In Fig. 5 (c) the mean angular orientation of the
layer at z ≈ 12 nm. By increasing the patch size patch towards the grafting surfaces is shown. At
from s = 8 to s = 16 the attraction shifts from large distances there is no favorite alignment of
9
30 30 -5
cs = 32 mM PMF Scism =u l1a3ti7o nmM PMF
20 wwtot(z()z) 20 -10 wtot(cs , zmin) wwtot(z()z)
wosm(z) 3.6⋅ln(cs) wosm(z)
elec elec
w (z) w (z)
10 cr 10c)s -15 cr
T] T] (n
[kw(z)Bi 0 [kw(z)Bi 0wβmi -20
-10 20 wwwmdBioopr(nnz(()zz)) -10 -25 20 wwwmdBioopr(nnz(()zz))
-20 0 -20 0
-30
-20 10 -20 100 200 300
2 10 18 26 2 10 18 26
-30 -30 c [mM]
s
2 6 10 14 18 22 26 2 6 10 14 18 22 26
z [nm] z [nm]
FIG. 7: Binding affinity βw (minimum value of the
min
PMFs)asafunctionoftheionicstrengthc forthe(P1)
FIG. 6: Individual free energy contributions to w (z) s 12
tot system. The blue line is a fit according to the function
of the (P112) complex at cs = 32 mM from eq. (4). The βw (c ) = a˜+N˜ln(c ) with a˜ = −36.8 and N˜ = 3.6.
insets shows the monopole (solid line), diopole (dashed min s s
ThegreenlineisadescriptionbyEq.(4),seetextforex-
line), and Born (dotted) contributions to the electro-
planation. Theindividualcontibutiontow areshown
static interaction w (z) in eq. (5). min
elec in Table II.
the CPPM while at intermediate distances around cf. Fig.3(b). Theelectrostaticpartisplottedasthe
z =16nm,closetothebrushsurface,thepatchvec- green curve and further subdivided into monopolar
torisalignedalmostparalleltothegraftingsurface. repulsion and dipole and Born attraction in the in-
Thealignmentisexpectedduetothecouplingofthe set. Here, the monopole repulsion is very large (up
dipoletotheE-field,butinterestinglythemaximum to 23 k T) and partially canceled by the large Born
B
alignment happens for a distance larger than the attraction (up to -18 k T). The dipole part is ex-
B
minimum of the electrostatic field, cf. Fig. 2. The tremal at the brush surface at z (cid:39) 12 nm and re-
reason is probably that single PE chains can reach mains an attractive contribution inside the brush.
out of the brush to touch the patch and thereby For smaller distances z (cid:46) 10 nm the net monopole
strongly orient the CPPM, see also the snapshots in repulsion, however, wins over it. To obtain the pre-
Fig. 1(b) for an exemplary illustration. When the dictionofthetotalPMF,thecounterion-releasecon-
CPPM penetrates into the brush layer, the orienta- tributionhastobefinallyadded. Thelatterishighly
tion correlation weakens likely due to the PEs more attractive, providing up to 10–14 k T favorable en-
B
homogeneously surrounding the CPPM and finally tropy for distances smaller than z (cid:39) 14 nm. In to-
due to the decrease of the electrostatic field within tal the prediction adds up to show a stable adsorp-
the PE brush. tion at the brush surface at z (cid:39) 10−12 nm and
In Fig. 6 we quantitatively compare and discuss an increasing repulsion closer to the grafting sur-
the PMFs within our phenomenological framework face. This description (blue curve) is in qualitative
around eq. (4) for a salt concentration c =32 mM. accordance with the PMF (red curve) directly ex-
s
Here,wecomparethesimulationPMF(redcurve)to tracted from the simulation. Apparently, however,
the total phenomenological prediction w (z) that the phenomenological description is roughly overall
tot
originates from the sum of the vdW and excluded- (cid:39) 10 k T more repulsive. We believe this discrep-
B
volume interactions w (z), the electrostatic ancymightstemfromtheadditionoferrorswehave
excl+vdW
part w , and the counterion-release contribution made in the approximations of the single contribu-
elec
w (z). As discussed before, the excluded volume tions. In particular, our estimation of the Born free
cr
part is highly repulsive and completely dominated energyisprobablytheweakestofallasweperformed
by the osmotic contributions from the counterions only a simple dipolar expansion for the CPPM. The
w . This contribution for w (z) (yellow curve CPPM has large surface charge densities in direct
osm osm
in Fig. 6) was taken directly from the reference sim- contactwiththePEchainsandmobileionsandthus
ulations of the neutral sphere in the charged brush, webelievethattheBorncontributionisinfactmore
10