New Conceptual Approaches to Modeling and Simulation of Complex Systems A Funding Initiative of the Volkswagen Foundation First call for proposals on Computer Simulation of Molecular and Cellular Biosystems as well as Complex Soft Matter New Algorithms in Charged Soft and Biological Matter R. Everaers Max-Planck-Institut f¨ ur Physik komplexer Systeme N¨ othnitzerstr. 38 01187 Dresden Germany. A.C. Maggs Laboratoire de Physico-Chime Th´ eorique, ESPCI-CNRS, 10 rue Vauquelin, 75231 Paris Cedex 05, France. Proposed duration: 3 years starting in January 2005
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New Conceptual Approaches to Modeling and Simulation of
Complex Systems
A Funding Initiative of the Volkswagen Foundation
First call for proposals on
Computer Simulation of Molecular and Cellular Biosystems as well as Complex
Soft Matter
New Algorithms in Charged Soft and Biological Matter
of polyelectrolytes with hydrophobic backbones. The chains are adsorbed on Mica surfaces and
observed by AFM (from Ref. (Kiriy et al., 2002))
in such a way as to render the effective interactions between any two charges short-ranged),
micro phase separation or the formation of pearl-necklace conformations in polymers (see
Fig. 1) due to a mechanism analogous to the Rayleigh instability of charged droplets. How-
ever, extremely complex behavior arises as a consequence of the interdependence of these
effects (Borue and Erukhimovich, 1988, 1990; Dobrynin and Rubinstein, 2000, 2001; Do-
brynin et al., 1996; Joanny and Leibler, 1990; NYRKOVA et al., 1993; Schiessel, 1999;
Schiessel and Pincus, 1998) (see, for example, Fig. 2).
Many of the theoretical predictions have not been tested via experiments or simulations.
Experimentally, the characteristic length scales are partially accessible through scattering
experiments (e.g. Ref. (Baigl et al., 2003)). However, due to the multitude of effects that
need to be accounted for, the interpretation of such experiments may require the introduction
of such a large number of adjustable parameters as to render them inconclusive. In contrast,
an underlying simplicity may be apparent from a single glance at the microscopic structure
(Fig. 1).
Computer simulations of polyelectrolyte systems are playing an increasingly important
role. Compared to neutral polymers polyelectrolyte simulations of many-chain systems are
still in their infancy (Chang and Yethiraj, 2003; Limbach et al., 2002; Micka and Kremer,
2000). Even single-chain simulations (Everaers et al., 2002; Limbach et al., 2002; Lyulin
et al., 1999; Micka and Kremer, 1997; Nguyen and Shklovskii, 2002; Yamakov et al., 2000)
6
)
Figure 3. Phase diagram of a semidilute solution of macro-ions without extra salt as a function of υ and lB
-1= εT/e2. See
also Table 1.
Counterion Condensation on Polyelectrolytes 5677
Figure 7. Diagram of state of a hydrophobically modifiedpolyelectrolyte. Shaded area corresponds regimes with coun-terion condensation. Regime I corresponds to conformationsof polyelectrolyte chains unperturbed by hydrophobic interac-tions. In regimes II, IIa, III, IIIa, IV, IVa, V, and Va, chainsform necklaces. Regime VI corresponds to cylindrical micelles.Logarithmic scales.
120
80
40
0
I, a.
u.
0.80.60.40.2q, nm-1
f = 39% f = 56% f = 71% f = 91%
Fig. 1 – SAXS intensity (arbitrary units) as a function of the wave vector q for PSS at Cp = 0.1 mol/L,N = 410 monomers and various chemical charge fractions f .
FIG. 2 Scaling predictions from Refs. (Schiessel, 1999) and (Dobrynin and Rubinstein, 2000).
Experimentally, the length scales are partially accessible from small-angle neutron scattering, e.g.
from variations of the position of the peak of the structure factor with parameters such as solvent
quality, salt concentration, charge fraction etc. (from Ref. (Baigl et al., 2003)).
can be computationally extremely demanding, but have made important contributions to
the emerging quantitative understanding of polyelectrolyte behavior.
Current numerical methods for simulating charged polymers
Simulating charged condensed matter systems is demanding due to the long range nature
of the Coulomb interaction. The direct evaluation of the Coulomb sum for N particles,
Uc =∑
i<j eiej/4πε0rij, requires computation of the separations rij between all pairs of
particles, which implies O(N 2) operations are needed per sweep or time step. Conventional
fast algorithms including the classic Ewald sum and fast multipole algorithms suffer from
either poor scaling with system size, high coding complexity or inefficiency in multiprocessor
environments (Schlick et al., 1998).
The preferred method at the present is to perform simulations using molecular dynamics
while the electrostatic problem is solved using the fast Fourier transform after interpolating
charges to a grid. Even more serious that the limitations in system size may be those in
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the time scales which are accessible in simulations with explicit solvents. In fact, unless a
simulation last for at least nanoseconds (corresponding to 106 elementary integration steps)
it is not even possible to correctly establish the correct distribution of ions in a Debye layer
about a charged molecule, resulting in considerable artefacts in the electrostatic energy and
the configurational entropy. Equilibration of more strongly bound charges requires even
longer simulation times. There is a slowly growing awareness in the field that these effects
may be of considerable importance.
Electrostatic interactions in systems with spatially varying dielectric constant
The largest conceptual and technical problem concerns simulations of systems where the
dielectric properties are variable leading to a generalized Poisson equation
div (ε(r)grad φ) = −ρ (1)
Examples include systems with interfaces and all implicit solvents in bio-molecular simula-
tion. If one manages to solve the Poisson equation (1) the energy of a a set of charges in a
dielectric background is given by1
2
∫ρφ (2)
This integral describes two very different effects: Firstly the long ranged pair interaction
between two different charges as modified by the dielectric constant. Secondly the self
energy of each charge interacting with itself. In a region of uniform dielectric properties
the self energy can be evaluated as U = e2/(8πεa) where a is the Born radius of the ion.
The theory of the Born energy shows that this energy is dominated by short length scales:
those comparable to the radius a. Modifications of the polarizability at the scale of nearest
neighbours about an ion can lead to enormous variations in this energy. A simple estimate
for a monovalent ion gives an energy of U = 2kBT in water and U = 30KBT in a nonpolar
background.
We see that the correct treatment of the Poisson equation, and the associated energy are
crucial if we want to describe correctly the conformations of a polymer within an implicit
solvent model. As a charge is enclosed by a collapsing polymer the energy increases at first
only gradually. However at the moment of complete collapse there is a strong co-operative
increase in the Born energy which at least qualitatively behaves like a short range, multibody
potential. In fact, the additinal electrostatic self-energy for a single embedded charge is of
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the same order as folding free energy entire proteins in their native state!
In practice solving the Poisson equation (1) is so time consuming that one rarely uses
the correct expression eq(2). Instead approximate, but largely ad-hoc approximations for
the Born energy have to be used (Lee et al., 2002). In fact (Schutz and Warshel, 2001),
the results obtained by the approximated implicit solvent models are very sensitive to the
value used for the dielectric constant, which turns out not to be a universal constant but
simply a parameter that depends on the model used. Extreme examples quoted by Warshel
and co-workers include the fact that εp comes out positive when fitting the Born energy of
a single charge, but negative when calculating the mutual influence of two charges.
V. PREVIOUS WORK AND SPECIFIC STRENGTHS OF THE TWO GROUPS
Over the last couple of years, one of us (ACM) has developed new approaches to the
simulation of electrostatic interactions. The methods are simpler to implement than those
previously known and have the great advantage of including the full configuration dependent
Born energy of charges. The algorithms allow one to avoid the difficult problem of solving
the generalized Poisson equation eq. (1). It is as easy to simulate the problem with arbitrary
dielectric properties as a system with uniform dielectric constant. We believe that this is a
major technical advance over earlier simulation techniques.
The other laboratory (RE) has concentrated on combining computer simulations with
analytical and scaling approaches, among other things to study well-defined polyelectrolyte
model systems. Here we propose to adapt these new simulation technique to the specific
needs of macromolecular systems and to apply it actual (bio-)physical problems.
A Local MC scheme for the implementation of electrostatic interactions
All current large scale simulation codes incorporating electrostatic interactions are based
on molecular dynamics rather than Monte-Carlo dynamics. Since the solution of the Poisson
equation is unique, even the motion of a single charge requires the global calculation of the
electrostatic potential. Thus Monte-Carlo methods with local moves must (apparently) lead
to hopelessly inefficient algorithms. While molecular dynamics are indeed required in detail
studies of dynamic processes, Monte-Carlo methods do have a number of major advantages
when only thermodynamic information is required. Among the advantages of Monte-Carlo
are
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• Absolute numerical stability
• No artifacts in the generated configurations due to finite step size
• Existence of ”smart” non-local algorithms: cluster/pocket algorithms in fluids, pivot
methods in polymer simulation.
• Monte-Carlo code can be much easier to write and maintain since only the energy and
not the force is calculated.
It is surprising that the Coulomb interaction poses such tremendous difficulty; after all the
underlying Maxwell equations are local. The above methods, however, do not solve Maxwell’s
equations, but rather search for the electrostatic potential φp(r), from which the electric field
E(r) is deduced. Over the last few years in ESPCI, Paris we have explored exactly what
parts of Maxwell’s equations are required in order to generate Coulombic interactions and
have generated a number of algorithms and codes that allow different approaches in the
simulation of charged systems. These codes include both molecular dynamics and efficient
O(N) Monte-Carlo implementations.
We have found that in classical electromagnetism there are just two requirements in order
to generate Coulomb’s law in a numerically efficient and purely local manner (Maggs and
Rossetto, 2002; Rottler and Maggs, 2004): A local expression for the field energy
U =
∫D2
2ε(r)d3r
where D is the electric displacement field, and the imposition of Gauss’ law
div D− ρ = 0
This leads to the following partition function for the electromagnetic field (Maggs, 2002):
Z({ri}) =
∫DD
∏
r
δ(div D− ρ({ri}))e−U/kBT . (3)
where the charge density, ρ(r) =∑
i eiδ(r− ri); the charge of the i’th particle is ei. It turns
out that codes based on sampling the constrained partition function eq. (3) can be far more
efficient than those based on solutions of the Poisson equation (1): especially when ε has
non trivial spacial structure (Maggs, 2004).
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Combining large scale computer simulations and scaling theories
Combining (scaling) theories and computer simulation can lead to important synergetic
effects: Theoretical insight helps to design simulations and to rationalize the abundantly
available microscopic information. On the other hand, simulations provide the necessary
independent evidence for the justification of phenomenological models or the relevance of
certain length or time scales. One of the present authors (RE) was involved in two such
investigations of single chain properties of charged polymers. Both studies depended heavily
on the combination of the numerical approach with scaling theories for the choice of simu-
lation parameters, observables and data analysis. In the first study (Yamakov et al., 2000),
we settled a controversy on the applicability of the necklace model to quenched random
polyampholytes at infinite dilution, thereby establishing the basis of earlier considerations
on the solution behavior at finite concentrations (Everaers et al., 1997a,b). The second
study (Everaers et al., 2002) concerned the interdependence of stretching, stiffening and
swelling effects on various length scales in intrinsically flexible polyelectrolytes with Debye–
Huckel interactions and confirmed the existence of an electrostatic persistence length in
these systems.
VI. SPECIFIC SCIENTIFIC QUESTIONS AND PROJECTS
The proposed project involves two major parts
• Adapting the local electrostatic algorithm to a form suitable for simulating polymeric
systems.
• Using the codes so developed to further the understanding of the fundamental physics.
These projects, can rather naturally be split between the two sites in Paris and Dresden.
At each point in the project extensive interaction will be needed between the groups: In the
implementation of a new algorithm there are always many cross-checks which are needed
to ensure that the method is useful in solving real physical problems. New and challenging
scientific questions are also an excellent motivation for finding new simulation methods.
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Year one
Dresden:
In the first year the Dresden side of the collaboration will be mainly responsible for
the first numerical studies of dielectric effects in simple model (lattice) polymers. Bond
fluctuation models are easy to implement and are well understood in neutral polymers. The
Paris group already has codes which permit the simulation of a lattice electrolyte in the
presence of arbitrary dielectric background.
Combining bond fluctuation dynamics and electrostatic effects should be possible with
very little programming effort; we anticipate that no more than one month of coding is
involved. From this moment on specific physics will be studied in Dresden. From our
experience on running new codes we would expect to spend the next 3 months calibrating
the resulting code: One needs to check that the known phenomenology is reproduced in
simple limits and that there are no equilibration problems due to the combination of bond
fluctuations with electrostatics. The rest of the first year will be then dedicated to studies
of dielectric effects in polymers; in particular we are interested in the qualitative as well as
quantitative modification in conformational properties as a function of the Born self energy
variations.
What is most important in determining the large scale structure of a polymer? Is it the
fact that the range of the electrostatic interaction is long ranged? Or does the Born energy
(which is a function of the whole local environment) act as a strong, short ranged, multibody
potential? We will answer this question by detailed studies of the conformational properties
of polymers as a function of the main physical parameters.
Paris:
In parallel to the preliminary work on simple models in Dresden we will explore further
algorithmic advances. We have tested our algorithm until now in rather uniform situations.
In highly heterogeneous media we expect that the method will become less efficient, due to
the need to introduce large numbers of field degrees of freedom compared with the number
of charged particles.(Note conventional Poisson solvers have exactly the same problem in
similar physical systems). We believe that we have a solution to this problem via more
sophisticated cluster sampling methods (similar to Swendsen-Wang) for the electric field.
Such algorithms have recently been found to be useful in a very different field: simulation
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of quantum spin models which are also defined in terms of a constrained partition function.
Assuming the success of this method we would develop an general implementation of the
algorithm for the both the on-lattice code which is to be used in Dresden the first year and
our off-lattice simulation code for electrolytes in order to test in both situations.
Year Two
Dresden:
In the second year we wish to make a more direct study of the analogies between proteins
and polyelectrolytes as studied by physicists. One possible project is the study foldability
in simple protein models. In models with nearest neighbour interactions it is known that
certain optimal sets of interactions are far easier to fold than a random polymer. What
happens in the presence of a long ranged interaction? Does the Born energy hinder or help
the folding process? Can long ranged interactions act as a guide during the folding process
and lead to more efficient collapse?
Paris:
In the context of application of the constrained partition function to polymer systems it
would be particularly useful to combine the global particle motion, such as that generated by
the pivot algorithm with the cluster algorithm for the field. It is unclear as to the feasibility
of combining the constrained partition function with such global moves. However the great
improvements in efficiency that are possible with such algorithms implies that a effort on the
problem could give major results. This part of the study should however be considered as
more difficult and speculative. We will implement an off-lattice version of the Monte-Carlo
algorithms in a form which will useful for other workers in the field. We will use this code
to check some of the qualitative conclusions of the lattice based approaches.
Year Three
In the third year we wish to move on to more definite comparisons of the coarse grained
models studied earlier in the project and more detailed atomistic models of solvation –
typically at the level of TIP3 models of water. We will be looking at confirming some of the
qualitative features of the work performed in earlier years: By this time we should be able,
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for instance, to simulate the collapse of a structured electrolyte (mimicking a protein with
charges confined to the surface of the folded structure) using firstly simple lattice models,
secondly an off-lattice “bead-spring” model and see how much of the general phenomenology
is stable to the inclusion of a true dynamic solvent.
By this time in the project we would expect that the major part of the algorithmic work
will have been finished and that both the Paris and Dresden groups will be working on
running and interpreting simulations.
VII. FUTURE PERSPECTIVES
We believe that the algorithms and methods that we will be developed during this project
will also be of use in the more fundamental atomic physics simulation community. In par-
ticular, the idea of constrained dynamics has already been implemented in an molecular
dynamics setting. The groups in Paris (Rottler and Maggs, 2003) and in Mainz (Pasichnyk
and Duenweg, 2003) have a working implementation of a constrained molecular dynamics
code for the case of uniform dielectric constant. The Mainz code has already been tested
on a large scale, parallel computer. Further progress with these codes should lead to the
availability of efficient methods of studying biomolecules with implicit solvents and variable
dielectric properties.
References
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Barrat, J.-L., and J.-F. Joanny, 1996, Advances in Chemical Physics 94, 1.
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Borue, V. Y., and I. Y. Erukhimovich, 1990, Macromolecules 23, 3625.
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