How the excluded volume architecture influences ion-mediated forces between proteins
V. Dahirel, M. Jardat, J.-F. Dufrêche, and P. Turq
Laboratoire Liquides Ioniques et Interfaces Chargées, UMR CNRS 7612, Université Pierre et Marie Curie–Paris 6,
case courrier 51, 4 place Jussieu F-75252, Paris Cedex 05, France
Received 28 June 2007; published 4 October 2007
The effective interactions between model proteins of various shapes are computed by means of Monte Carlo
simulations. In particular, we determine how the modification of the excluded volume architecture influences
both entropic and purely electrostatic ion-mediated forces between proteins. We find that interprotein interac-
tions are strongly affected by protein shape, which results in a high decrease of electrostatic screening for
typical active site geometries. Effective interactions are then closer to the direct Coulombic interactions, and
both affinity and selectivity are enhanced by several orders of magnitude.
DOI: 10.1103/PhysRevE.76.040902 PACS numbers: 87.15.Aa, 31.15.Qg, 82.70.Dd
Protein recognition and binding lie at the heart of many
biological phenomena, but also result in many human dis-
eases. Deciphering the role of protein architecture in the as-
sociation process is crucial to design better pharmalogical
inhibitors 1,2. The specificity of molecular recognition is
often explained by a “lock and key” mechanism, i.e., by
shape and charge complementarities of the molecular part-
ners. Such a rigid-body model lacks the plasticity required to
describe the molecular events during the binding process, but
it is sufficient to compare the energy of different complex
structures 3,4. Although the influence of shape complemen-
tarity on the interaction between proteins in vacuum can be
easily understood and estimated, it becomes less intuitive as
far as water-mediated and salt-mediated interactions are con-
cerned. In this Rapid Communication, we focus on the role
of salt ions, which screen the direct Coulombic interactions
between charged proteins. So far, shape has never been re-
garded as a factor of screening optimization. Most theoretical
works have been devoted to suspensions of spherical or rod-
like particles 5,6, showing that ion-induced interactions in-
fluence both their static 7,8 and dynamical properties
9–11. Nonspecific interprotein potentials deduced from
small-angle neutron scattering experiments are well de-
scribed by screened Coulombic potentials 12, which indi-
cates the relevance of considering electrostatic screening as a
key phenomenon in protein solutions.
In this work, we use explicit-ion, continuum-dielectric
Monte Carlo simulations to derive exact ion-averaged poten-
tials of mean force PMFs between proteins, whose level of
modeling is chosen to unravel the role of the protein shape.
Although calculations at the Poisson-Boltzmann level are
frequently used, explicit-ion simulations are required for na-
nometric particles, since correlations between ions are not
negligible 13. The primitive model that describes ions as
charged spheres is used for its ability to explain the finest
trends of ion-mediated electrostatic interactions, such as the
attraction between like-charged particles 14,15. We extend
this model to nonspherical particles. For proteins that differ
only by their shape, we quantify the intensity of ion-
mediated forces for like-charged +10 and +10 and oppo-
sitely charged proteins +10 and -10. The charges of the
model proteins are located at a single point. The impact of
charge distribution within the protein has been considered
elsewhere see Ref. 16. We stress that, in all cases, the
direct force between the proteins in vacuum is the same, so
that we can isolate ion-induced effects from any other forces.
We first present the results for two spherical nanoparticles.
This enables us to justify the choice of the protein models
designed to investigate the effect of shape on the ion-
mediated PMFs. We find that ion-averaged forces dramati-
cally affect both the affinity between oppositely charged
partners and the repulsion between like-charged ones when
the shape changes: the architecture of the excluded volume
can be optimized to amplify the magnitude of Coulombic
forces. This is due to an important variation of the electro-
static effective force, which is not compensated by an en-
tropic counterpart.
The species interact through the pair potential V
ij
r
ij
= Z
i
Z
j
e
2
/4
0
r
1/ r
ij
+ V
H
r
ij
, where
0
is the permittivity
of the vacuum,
r
is the relative permittivity of water taken
equal to 78.25, e is the elementary charge, Z
i
is the charge
of particle i, and r
ij
is the distance between particles i and j .
The hard potential V
H
r
ij
is infinite in the particles and zero
outside. The salt ions are hard spheres of radius a
m
= 0.15 nm. For this model of interactions, the mean force
includes two contributions in addition to the direct Coulom-
bic interaction between the two proteins: the electrostatic in-
teractions of the proteins with the surrounding ions, plus an
osmotic term or collision force. Both forces are functionals
of ion density: collision forces are due to hard-core repul-
sions, and are thus functionals of surface density; electro-
static forces are due to long-range Coulombic interactions,
and are thus functionals of volume density. More precisely,
we denote as F
dir
1
the direct electrostatic force exerted by the
protein 2 P
2
on the protein 1 P
1
. F
el
1
is the electrostatic
force exerted by ions on P
1
. It reads
F
el
1
=
Z
P
1
e
2
4
0
r
=1
2
j=1
N
r
1
Z
|r
j
- r
1
|
, 1
where r
1
is the position of P
1
, r
j
is the position of ion j of
type and charge Z
, and N
is the number of ions of type
. The angular brackets denote the canonical ensemble aver-
age. The third contribution F
coll
1
is entropic, i.e., it scales with
the thermal energy k
B
T. This collision force results from the
asymmetry of the ion density at the surface of the proteins. It
may be expressed as 17
PHYSICAL REVIEW E 76, 040902R2007
RAPID COMMUNICATIONS
1539-3755/2007/764/0409024 ©2007 The American Physical Society 040902-1