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 fur Physik komplexer Systeme Nothnitzerstr. 38 01187 Dresden Germany.

A.C. Maggs

Laboratoire de Physico-Chime Theorique, ESPCI-CNRS, 10 rue Vauquelin, 75231 Paris Cedex05, France.

Proposed duration: 3 years starting in January 2005

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I. SUMMARY

Many biologically relevant macromolecules carry ionizable side-groups and contain hy-

drophobic as well as hydrophilic parts. Since the non-polar, hydrophobic elements typically

have a very low dielectric constant compared to water, electrostatic (solvation) self-energies

and charge-charge interactions have to be evaluated in dielectrically inhomogeneous media.

Due to technical reasons, these effects are neglected in all current (bio-)molecular simulations

using implicit solvent models. We propose applying a new family of electrostatic algorithms

to study the conformational properties of charged soft and biological matter. The algorithms

allow direct incorporation of arbitrary dielectric effects in a manner which is both easy to

program and numerically efficient.

II. ZUSAMMENFASSUNG

Viele biologisch relevante Makromolekule tragen ionisierbare Seitengruppen und enthal-

ten sowohl hydrophile als auch hydrophobe Teile. Letztere sind durch, im Vergleich zu

Wasser, kleine dielektrische Konstanten charakterisiert. Dies macht die Auswertung elek-

trostatischer Selbstenergien und von Wechselwirkungen zwischen Ladungen wesentlich kom-

plizierter, da sie in dielektrisch inhomogenen Medien zu erfolgen hat. Aufgrund tech-

nischer Probleme vernachlassigen die derzeitigen molekularen Simulationen mit impliziten

Losungsmitteln diesen Effekt. Wir schlagen vor, geladene weiche und biologische Materie

mit Hilfe einer neue Familie von Algorithmen zur Auswertung elektrostatischer Wechsel-

wirkungen zu untersuchen. Die Algorithmen erlauben die Berucksichtigung dielektrischer

Effekte und sind sowohl numerisch effizient als auch einfach zu programmieren.

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III. INTRODUCTION

In aqueous solution polymers with ionizable side-groups dissociate into charged macroions

and small counter ions. Macromolecules of this type are commonly referred to as polyelec-

trolytes, a class which comprises proteins and nucleic acids as well as synthetic polymers

such as sulfonated polystyrene and polyacrylic acid. The prominence of polyelectrolytes

among biologically relevant macromolecules is no accident: life as we know it depends on

the interplay of carbon-based macromolecules in water. As a general rule, organic molecules

are non-polar and do not mix with water, while polyelectrolytes become soluble due to

the gain in translational entropy of dissociated counter-ions. Nature uses a combination

of ionic, hydrophilic and hydrophobic interactions on all levels of the complex intracellular

self-organization from the folding of proteins to the formation of the cytoskeleton and the

cell membrane. Since the non-polar, hydrophobic parts of such systems typically have a

very low dielectric constant compared to water, electrostatic self-energies and charge-charge

interactions have to be evaluated in dielectrically inhomogeneous media. These effects are

neglected in all current implicit solvent simulations, but can be included in an algorithm

recently introduced by one of us. Here we propose (i) to adapt this algorithm to the needs of

macromolecular simulations and (ii) to employ it for the first systematic investigation of the

interplay between the ion distribution, the macromolecular structure and the electrostatic

field.

IV. STATE OF THE ART

Modeling Polyelectrolytes

The theoretical description of electrostatic interactions in (biological) soft matter can set

in on different levels of coarse-graining of the involved polarizabilities, dipole moments and

charges. For an improved physical understanding as well as for practical purposes one is

interested in integrating out “uninteresting” solvent and small ion degrees of freedom. So far,

the choice is often between detailed descriptions whose excessive demands on computational

resources preclude proper equilibration and control of finite size effects and the investigation

of models based on (sometimes uncontrolled) simplifying assumptions. Typical levels of

description are:

• the inclusion of explicit salt and counter-ions and of (dipolar) solvent molecules

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• the inclusion of explicit salt and counter-ions combined with an implicit description of

the solvent via a dielectric constant and effective interactions between the constituents

of the macromolecules representing, for example, poor/theta/good solvent conditions

• implicit solvent and salt degrees of freedom (in the form of a non-linear Poisson-

Boltzmann theory or its linearized form, Debye-Huckel theory)

Understanding Polyelectrolytes

Much insight has been gain from atomistic simulations of proteins, nucleic acids and lipid

membranes (Leach, 2002). However, atomistic simulations start out from (plausible) guesses

for the atomistic structure and follow the dynamics of time scales in the nanosecond range.

Slow conformational rearrangements and the actual folding of proteins are still out of reach

in such studies.

Similar to simplified descriptions of the solvent, one is also interested in integrating out

uninteresting macromolecular degrees of freedom. Coarse-graining to bead-spring or lattice

models is justified in cases where one is either interested in generic properties and/or in

behavior on scales which are much larger than the monomer size. Our comparatively good

understanding of solutions of neutral polymers or amphiphiles owes much to this approach.

In these systems the range of interactions between monomers is much smaller than the

scale determining the physical properties of the solution, i.e. the size of the polymer coil or

micelles or the various correlation lengths characterizing the solutions.

The monomer description is usually reduced to the effective interaction with the solvent

(i.e. hydrophilic (P) versus hydrophobic (H) ). Such models can be used to describe struc-

ture formation in amphiphilic systems (including the formation of lipid bilayers). Similarly

(lattice) H–P–models of proteins and protein-like co-polymers have contributed much to our

current understanding of generic features of the foldable sequences and of the free energy

landscape for protein folding (DILL et al., 1995).

From the point of view of Statistical Mechanics, the inclusion of electrostatic interactions

even into the simplest polymer models remains a challenge. Polyelectrolyte solutions (Barrat

and Joanny, 1996) are controlled by an intricate interplay of short- and long-range inter-

actions. Interestingly, it is often possible to discuss the behavior on different hierarchical

levels in terms of simple, intuitive concepts such stretching, stiffening or swelling of polymers,

over-charging and charge renormalization due to counter ion condensation, screening of elec-

trostatic interactions (i.e. the tendency of oppositely charged objects to spatially arrange

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FIG. 1 Lhs: Snapshot from a Monte Carlo simulation of long polyampholyte chains in the necklace

regime (from Ref. (Yamakov et al., 2000)). Rhs: Experimentally observed necklace conformations

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

12

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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Borue, V. Y., and I. Y. Erukhimovich, 1988, Macromolecules 21, 3240.

Borue, V. Y., and I. Y. Erukhimovich, 1990, Macromolecules 23, 3625.

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//link.aip.org/link/?JCP/118/6634/1.

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Everaers, R., A. Johner, and J.-F. Joanny, 1997a, Europhys. Lett. 37, 275.

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Everaers, R., A. Johner, and J.-F. Joanny, 1997b, Macromolecules 30, 8478.

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10606, URL http://link.aip.org/link/?JCP/116/10606/1.

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http://link.aps.org/abstract/PRL/v88/e196402.

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Letters 85, 4305.

15

VIII. COSTS

A. Dresden

We apply for one BAT II/a position as well as for travel money allowing the members

of the Dresden group to spend per year 4 times one weeks in Paris (Travel 500 Euros plus

120 Euros per day). In addition we calculate 1000 Euros for participation in international

conferences. We have also included the approximate costs for a 16-processor Linux compute

server which we would like to buy in the course of the first year, when we get ready to start

larger scale data production.

Personnel scientific personnel 154800 Euros

other personnel —

scholarships —

Recurring costs travel costs 19080 Euros

other —

Non-recurring costs equipment 40000 Euros

other —

B. Paris

We apply for one position with a salary corresponding to the EU Marie-Curie incoming

international fellow level in France (total cost 3880 Euros per month) as well as for travel

money allowing the members of the Paris group to spend per year 4 times one weeks in

Desden (Travel 500 Euros plus 100 Euros per day). In addition we calculate 1000 Euros

for participation in international conferences. We have also included the approximate costs

for two 2-processor Linux workstations needed for code development and small scale data

production.

Personnel scientific personnel 139680 Euros

other personnel —

scholarships —

Recurring costs travel costs 17400 Euros

other —

Non-recurring costs equipment 10000 Euros

other —

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IX. RECIPIENT INSTITUTIONS

Max-Planck-Institut fur Physik komplexer Systeme

Nothnitzerstr. 38

01187 Dresden

Germany

Laboratoire de Physico-Chime Theorique

ESPCI-CNRS

10 rue Vauquelin

75231 Paris Cedex 05

France

X. STATEMENT

We have neither submitted this nor a similar proposal to other funding agencies. We will

immediately inform the Volkswagenstiftung of future applications for financial support for

related activities.

XI. CURRENT OR RECENTLY FINISHED RELATED PROJETS

The recent work of RE on charged polymer discussed in Section V was supported by the

Deutsche Forschungsgemeinschaft in the framework of a Emmy-Noether-grant for ”Modeling

the structure and dynamics of DNA beyond atomistic time and length scales” (November

1999 to October 2002).

XII. SIGNATURES

Dresden and Paris, June 15th 2004

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XIII. ENCLOSURES

We enclose CVs and lists of relevant recent publications for the applicants as well as for

two post-doctoral candidates who have expressed interest in working with us on this project.

I. Pasichnyk will defend his thesis on implementation of a fully parallel Molecular Dynamics

of the algorithm presented in Sec. V on June 16th in Mainz. T.T. Nguyen has worked with

B. Shklovskii and A. Grosberg in Minnesota and T. Witten in Chicago on screening and

overcharging phenomena in polyelectrolytes.