Regular Articles

2012

Coarse-Grained and Atomistic Simulations of the Salt-Stable Cowpea Chlorotic Mottle Virus (SS-CCMV) Subunit 26–49: β-Barrel Stability of the Hexamer and Pentamer Geometries

Tristan Bereau, Christoph Globisch, Markus Deserno, and Christine Peter

J. Chem. Theory Comput., Article ASAP

A combination of coarse-grained (CG) and atomistic simulations provides a suitable computational framework to study unstructured regions of proteins, for which experimental data are often lacking or limited. In this work, we combine CG and atomistic simulations with clustering algorithms and free energy reweighting methods to explore the conformational equilibrium of certain regions of the salt-stable cowpea chlorotic mottle virus (SS-CCMV). In particular, we focus on the geometry of converging strands (residues 26–49) from contacting subunits at the 3-fold (hexamer) and 5-fold (pentamer) symmetry points of the capsid. We show the following: (i) The simulations reproduce the experimentally observed β-barrel for the hexamer. (ii) The pentamer geometry is unable to stabilize a β-barrel conformation; it assumes various states instead, again in accordance with the experimental results which do not indicate a well-defined structure for the pentameric interface. (iii) Atomistic simulations of the backmapped CG structures remain relatively stable, indicative of plausible CG conformations and slow kinetics on the atomistic level.

Determining the Gaussian curvature modulus in simulations

Mingyang Hu, John J. Briguglio, and Markus Deserno

Biophys. J. 102, 1403–1410 (2012).

The Gaussian curvature modulus κ' of lipid bilayers likely contributes more than 100 kcal/mol to every cellular fission or fusion event. This huge impact on membrane remodeling energetics might be a factor that co-determines the complex lipid composition of biomembranes through tuning of κ'. Yet, its value has been measured only for a handful of simple lipids, and no simulation has so far determined it better than a factor of two, rendering a systematic investigation of such enticing speculations impossible. Here we propose a highly accurate method to determine κ' in computer simulations. It relies on the interplay between curvature stress and edge tension of partially curved axisymmetric membrane discs and requires determining their closing probability. For a simplified lipid model we obtain κ' and its relation to the normal bending modulus κ for membranes differing both in stiffness and spontaneous lipid curvature. The elastic ratio κ'/κ can be determined with a few percent statistical accuracy. Its value agrees with the scarce experimental data, and its change with spontaneous lipid curvature is compatible with theoretical expectations, thereby granting additional information on monolayer properties. We also show that an alternative determination of these elastic parameters based on moments of the lateral stress profile gives markedly different and unphysical values.

Effective field theory approach to fluctuation-induced forces between colloids at an interface

Cem Yolcu, Ira Z. Rothstein, and Markus Deserno

Phys. Rev. E 85, 011140 (2012).

We discuss an effective field theory (EFT) approach to the computation of fluctuation-induced interactions between particles bound to a thermally fluctuating fluid surface controlled by surface tension. By describing particles as points, EFT avoids computing functional integrals subject to difficult constraints. Still, all information pertaining to particle size and shape is systematically restored by amending the surface Hamiltonian with a derivative expansion. The free energy is obtained as a cumulant expansion, for which straightforward techniques exist. We derive a completedescription for rigid axisymmetric objects, which allows us to develop a full asymptotic expansion—in powers of the inverse distance—for the pair interaction. We also demonstrate by a few examples the efficiency with which multibody interactions can be computed. Moreover, although the main advantage of the EFT approach lies in explicit computation, we discuss how one can infer certain features of cases involving flexible or anisotropic objects. The EFT description also permits a systematic computation of ground-state surface-mediated interactions, which we illustrate with a few examples.

2011

Effective field theory approach to Casimir interactions on soft matter surfaces

Cem Yolcu, Ira Z. Rothstein, and Markus Deserno

Europhys. Lett. 96 20003 (2011).

A shorter version can also be found here: arXiv:1007.4760v2 [cond-mat.soft]

We utilize an effective field theory approach to calculate Casimir interactions between objects bound to thermally fluctuating fluid surfaces or interfaces. This approach circumvents the complicated constraints imposed by such objects on the functional integration measure by reverting to a point particle representation. To capture the finite-size effects, we augment the Hamiltonian by a term ΔH that encapsulates the particles' response to external fields. ΔH is systematically expanded in a series of terms, each of which scales homogeneously in the two power counting parameters: λ = R/r, the ratio of the typical object size (R) to the typical distance between them (r), and δ = k_BT/k, where k is the modulus characterizing the surface energy. The coefficients of the terms in ΔH correspond to generalized polarizabilities and thus the formalism applies to rigid as well as deformable objects. We first illustrate and verify our approach by re-deriving known pair forces between circular objects bound to films or membranes. To demonstrate its efficiency and versatility, we then derive a number of new results, among them the triplet interactions present in these systems, a higher-order correction to the film interaction, and general scaling laws for the leading-order interaction valid for objects of arbitrary shape and internal flexibility.

Membrane mediated interactions between circular particles in the strongly curved regime

Benedict J. Reynwar and Markus Deserno

Soft Matter 7, 8567–8575 (2011).

Particles binding to a fluid lipid membrane can induce bilayer deformations, for instance when these particles are curved. Since the energy of two overlapping warp fields depends on the mutual distance between the two particles creating them, they will experience forces mediated by the curvature of the membrane. If the deformations are sufficiently weak, the associated differential equations for the membrane shape are linear, and the resulting interactions are understood very well; but for stronger curvature imprint very little is known, owing to the highly nonlinear nature of the problem. Here we numerically calculate the magnitude of such membrane-mediated interactions in the case of two axisymmetric particles over a wide range of curvature imprints, deep into the nonlinear regime. We show that over an intermediate distance range the sign of the force reverses beyond a sufficiently strong deformation. These findings are quantitatively confirmed by a simple analytical close-distance expansion. The sign flip can be traced to a change in magnitude between the two principal curvatures midway between the two particles, which can only occur at sufficient particle tilt, a condition which is by construction ruled out in the linearized description. We also show these large perturbation results to agree with coarse-grained molecular dynamics simulations and suggest that a favorable comparison is indeed more likely to hold in the strongly deformed regime.

Structural basis of Folding Cooperativity in Model Proteins:
Insights from a Microcanonical Perspective

Tristan Bereau, Markus Deserno, and Michael Bachmann

Biophys. J. 100, 2764–2772 (2011).

Two-state cooperativity is an important characteristic in protein folding. It is defined by a depletion of states that lie energetically between folded and unfolded conformations. There are different ways to test for two-state cooperativity; however, most of these approaches probe indirect proxies of this depletion. Generalized-ensemble computer simulations allow us to unambiguously identify this transition by a microcanonical analysis on the basis of the density of states. Here, we present a detailed characterization of several helical peptides obtained by coarse-grained simulations. The level of resolution of the coarse-grained model allowed to study realistic structures ranging from small α–helices to a de novo three-helix bundle without biasing the force field toward the native state of the protein. By linking thermodynamic and structural features, we are able to show that whereas short α–helices exhibit two-state cooperativity, the type of transition changes for longer chain lengths because the chain forms multiple helix nucleation sites, stabilizing a significant population of intermediate states. The helix bundle exhibits signs of two-state cooperativity owing to favorable helix-helix interactions, as predicted from theoretical models. A detailed analysis of secondary and tertiary structure formation fits well into the framework of several folding mechanisms and confirms features that up to now have been observed only in lattice models.

2010

Systematic implicit solvent coarse-graining of bilayer membranes:
lipid and phase transferability of the force field

Zun-Jing Wang and Markus Deserno

New Journal of Physics 12, 095004 (2010).

We study lipid and phase transferability of our recently developed systematically coarse-grained solvent-free membrane model. The force field was explicitly parametrized to describe a fluid 1-palmitoyl-2-oleoyl-phosphatidylcholine (POPC) bilayer at 310 K with correct structure and area per lipid, while gaining at least three orders of magnitude in computational effciency (see Ref. 1). Here we show that exchanging CG tails, without any subsequent re-parameterization, creates reliable models of 1,2-dioleoylphosphatidylcholine (DOPC) and 1,2-dipalmitoylphosphatidylcholine (DPPC) lipids in terms of structure and area per lipid. Furthermore, all CG lipids undergo a liquid-gel transition upon cooling, with characteristics as observed in experiments and all-atom simulations during phase transformation. These studies suggest a promising transferability of our force field parameters to different lipid species and thermodynamic state-points, properties that are prerequisite for even more complex systems, such as mixtures.

Interplay between secondary and tertiary structure formation in protein folding cooperativity

Tristan Bereau, Michael Bachmann, and Markus Deserno

J. Am. Chem. Soc. 132, 13129–13131 (2010).

Protein folding cooperativity is defined by the nature of the finite-size thermodynamic transition exhibited upon folding: two-state transitions show a free energy barrier between the folded and unfolded ensembles, while downhill folding is barrierless. A microcanonical analysis, where the energy is the natural variable, has shown better suited to unambiguously characterize the nature of the transition compared to its canonical counterpart. Replica exchange molecular dynamics simulations of a high resolution coarse-grained model allow for the accurate evaluation of the density of states, in order to extract precise thermodynamic information, and measure its impact on structural features. The method is applied to three helical peptides: a short helix shows sharp features of a two-state folder, while a longer helix and a three-helix bundle exhibit downhill and two-state transitions, respectively. Extending the results of lattice simulations and theoretical models, we find that it is the interplay between secondary structure and the loss of non-native tertiary contacts which determines the nature of the transition.

Cell Model Approach to Membrane Mediated Protein Interactions

Martin M. Müller and Markus Deserno

Progr. Theor. Phys. Suppl. 184, 351–363 (2010).

Membrane-deforming proteins can interact through the curvature fields they create. In the case of many such proteins a cell model approach can be used to calculate the energy per protein and predict, whether it would lead to phase segregation or bud-formation. Using covariant differential geometry exact results are derived for the lateral pressure in terms of geometric properties at the cell boundary. Numerical solutions of the exact shape equations in the highly nonlinear regime are found and it is seen that both phase segregation and bud formation can occur.

A systematically coarse-grained solvent-free model for quantitative phospholipid bilayer simulations

Zun-Jing Wang and Markus Deserno

J. Phys. Chem. B 114, 11207 (2010).

We presented an implicit solvent coarse-grained (CG) model for quantitative simulations of
1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) bilayers. The use of implicit solvent enables membrane simulations on large length- and time-scales at moderate computational expense. Despite an improved computational efficiency, the model preserves chemical specificity and quantitative accuracy. The bonded and nonbonded interactions together with the effective cohesion mimicking the hydrophobic effect were systematically tuned by matching structural and mechanical properties from experiments and all-atom bilayer simulations, such as saturated area per lipid, radial distribution functions, density and pressure profiles across the bilayer, P₂ order, etc. The CG lipid model is shown to self-assemble into a bilayer starting from a random dispersion. Its elastic properties, such as bending and stretching modulus, are semi-quantitatively consistent with experiments. The effects of (i) reduced molecular friction and (ii) more efficient integration combine to an overall speedup of three to four orders of magnitude compared to all-atom bilayer simulations. Our CG lipid model is especially useful for studies of large-scale phenomena in membranes which nevertheless require a fair description of chemical specificity, e.g. membrane patches interacting with movable and transformable membrane proteins and peptides.

In-plane homogeneity and lipid dynamics in tethered Bilayer Lipid Membranes (tBLMs)

Siddharth Shenoy, Radu Moldovan, James Fitzpatrick, David J. Vanderah, Markus Deserno, and Mathias Lösche

Soft Matter 6, 1263 (2010).

Tethered bilayer lipid membranes (tBLMs) were prepared by the self-assembly of thiolated lipidic anchor molecules on gold, followed by phospholipid precipitation via rapid solvent exchange. They were characterized in their in-plane structure, dynamics and dielectric properties. We find that the in-plane homogeneity and resistivity of the tBLMs depends critically on a well-controlled sample environment during the rapid solvent exchange procedure. The in-plane dynamics of the systems, assessed by fluorescence correlation spectroscopy (FCS) as the diffusivity of free, labeled phospholipid dissolved in the membrane, depends on the density of the lipidic anchors in the bilayer leaflet proximal to the substrate as well as on details of the molecular structure of the anchor lipid. In DOPC tBLMs in which tethers are laterally dilute (sparsely-tethered bilayer lipid membranes, stBLMs), measured diffusivities, D ~ 4 μm²/s, are only slightly greater than those reported in physisorbed bilayers. However, when we distinguish label diffusion in the proximal and in the distal bilayer leaflets, we observe distinct diffusivities, D ~ 2 μm²/s and ~ 7 μm²/s, respectively. The value observed in the distal leaflet is identical to that in free membranes. stBLMs completed with phytanoyl lipids (DPhyPC) show consistently lower label diffusivity than those completed with unsaturated chains (DOPC). As the flexibility of the tether chain increases, a reduction in the apparent diffusivityis observed, which we interpret as an increased propensity of the proximal bilayer leaflet to host free lipid. Finally, in laterally heterogeneous bilayers, the label diffusivity varies only by a factor of ~ 2× – 4×, indicating that the distinct regions in the bilayers do not correspond to distinct phases such as a fluid phase coexisting with a gel phase.

2009

Generic coarse-grained model for protein folding and aggregation

Tristan Bereau and Markus Deserno

J. Chem. Phys. 130, 235106 (2009).

A generic coarse-grained (CG) protein model is presented. The intermediate level of resolution (four beads per amino acid, implicit solvent) allows for accurate sampling of local conformations. It relies on simple interactions that emphasize structure, such as hydrogen bonds and hydrophobicity. Realistic alpha/beta content is achieved by including an effective nearest-neighbor dipolar interaction. Parameters are tuned to reproduce both local conformations and tertiary structures. The thermodynamics and kinetics of a three-helix bundle are studied. We check that the CG model is able to fold proteins with tertiary structures and amino acid sequences different from the one used for parameter tuning. By studying both helical and extended conformations we make sure the force field is not biased toward any particular secondary structure. The accuracy involved in folding not only the test protein but also other ones show strong evidence for amino acid cooperativity embedded in the model. Without any further adjustments or bias a realistic oligopeptide aggregation scenario is observed.

Multiscale modeling of emergent materials: biological and soft matter

Teemu Murtola, Alex Bunker, Ilpo Vattulainen, Markus Deserno, Mikko Karttunen

Phys. Chem. Chem. Phys. 11, 1869–1892 (2009).

In this review, we focus on four current related issues in multiscale modeling of soft and biological matter. First, we discuss how to use structural information from detailed models (or experiments) to construct coarse-grained ones in a hierarchical and systematic way. This is discussed in the context of the so-called Henderson theorem and the inverse Monte Carlo method of Lyubartsev and Laaksonen. In the second part, we take a different look at coarse graining by analyzing conformations of molecules. This is done by the application of self-organizing maps, i.e., a neural network type approach. Such an approach can be used to guide the selection of the relevant degrees of freedom. Then, we discuss technical issues related to the popular dissipative particle dynamics (DPD) method. Importantly, the potentials derived using the inverse Monte Carlo method can be used together with the DPD thermostat. In the final part we focus on solvent-free modeling which offers a different route to coarse graining by integrating out the degrees of freedom associated with solvent.

Mesoscopic Membrane Physics: Concepts, Simulations, and Selected Applications

Markus Deserno

Macromol. Rapid. Comm. 30, 752–771 (2009).

The window of a few tens to a few hundred nanometers in length scale is a booming field in lipid membrane research, owing largely to two reasons. First, many exciting biophysical and cell biological processes take place within it. Second, experimental techniques manage to zoom in on this sub-optical scale, while computer simulations zoom out to system sizes previously unattainable, and both will be meeting soon. This paper reviews a selection of questions and concepts in this field and demonstrates that they can often be favorably addressed with highly simplified simulation models. Among the topics discussed are membrane adhesion to substrates, mixed lipid bilayers, lipid curvature coupling, pore formation by antimicrobial peptides, composition-driven protein aggregation, and curvature driven vesiculation.

2008

Membrane composition-mediated protein-protein interactions

Benedict J. Reynwar, Markus Deserno

Biointerphases 3, FA117 (2008).

The authors investigate membrane composition-mediated interactions between proteins adsorbed onto a two-component lipid bilayer close to critical demixing using coarse-grained molecular dynamics simulations and a phenomenological Ginzburg-Landau theory. The simulations consist of three-bead lipids and platelike proteins, which adsorb onto the membrane by binding preferentially to one of the two lipid species. The composition profile around one protein and the pair correlation function between two proteins are measured and compared to the analytical predictions. The theoretical framework is applicable to any scalar field embedded in the membrane, and although in this work the authors treat flat membranes, the methodology extends readily to curved geometries. Neglecting fluctuations, both lipid composition profile and induced protein pair potential are predicted to follow a zeroth order modified Bessel function of the second kind with the same characteristic decay length. These predictions are consistent with our molecular dynamics simulations, except that the interaction range is found to be larger than the single profile correlation length.

Coarse-grained modeling of interactions of lipid bilayers with supports

Matthew I. Hoopes, Markus Deserno, Margie L. Longo, Roland Faller

J. Chem. Phys. 129, 175102 (2008).

We characterize the differences between supported and unsupported lipid bilayer membranes using a mesoscopic simulation model and a simple particle-based realization for a flat support on to which the lipids are adsorbed. We show that the nanometer roughness of the support affects membrane binding strength very little. We then compare the lipid distributions and pressure profiles of free and supported membranes. The surface localization of the proximal leaflet breaks the symmetry seen in a free bilayer, and we quantify the entropic penalty for binding and the increased lateral compression modulus.

Adhesion promotes phase separation in mixed-lipid membranes

Vernita D. Gordon, Markus Deserno, Carolyn M. J. Andrew, Stefan U. Egelhaaf, Wilson C. K. Poon

Europhys. Lett. 84, 48003 (2008).

We investigate the interplay of domain formation and adhesion in mixed-lipid membranes. Giant unilamellar vesicles consisting of two- and three-component lipid mixtures are studied using confocal fluorescence microscopy. Upon driving the system towards the demixing transition, phase separation is invariably found to occur first in regions where membranes adhere to one another, despite identical lipid headgroups and negligible curvature effects. We propose a simple generic mechanism based on the suppression of thermal shape fluctuations to explain these observations. Our findings suggest novel possibilities by which biomembranes can create and utilize lateral lipid heterogeneities.

Coarse-Grained Simulation Studies of Peptide-Induced Pore Formation

Gregoria Illya, Markus Deserno

Biophys. J. 95, 4163–4173 (2008).

We investigate the interactions between lipid bilayers and amphiphilic peptides using a solvent-free coarse-grained simulation technique. In our model, each lipid is represented by one hydrophilic and three hydrophobic beads. The amphiphilic peptide is modeled as a hydrophobic-hydrophilic cylinder with hydrophilic caps. We find that with increasing peptide-lipid attraction the preferred state of the peptide changes from desorbed, to adsorbed, to inserted. A single peptide with weak attraction binds on the bilayer surface, while one with strong attraction spontaneously inserts into the bilayer. We show how several peptides, which individually bind only to the bilayer surface, cooperatively insert. Furthermore, hydrophilic strips along the peptide cylinder induce the formation of multipeptide pores, whose size and morphology depend on the peptides' overall hydrophilicity, the distribution of hydrophilic residues, and the peptide-peptide interactions. Strongly hydrophilic peptides insert less readily, but prove to be more destructive to bilayer integrity.

2007

Aggregation and vesiculation of membrane proteins by curvature-mediated interactions

Benedict J. Reynwar, Gregoria Illya, Vagelis A. Harmandaris, Martin M. Müller, Kurt Kremer, and Markus Deserno

Nature 447, 461–464 (2007).

Membrane remodelling plays an important role in cellular tasks such as endocytosis, vesiculation and protein sorting, and in the biogenesis of organelles such as the endoplasmic reticulum or the Golgi apparatus. It is well established that the remodelling process is aided by specialized proteins that can sense as well as create membrane curvature, and trigger tubulation when added to synthetic liposomes. Because the energy needed for such large-scale changes in membrane geometry significantly exceeds the binding energy between individual proteins and between protein and membrane, cooperative action is essential. It has recently been suggested that curvature-mediated attractive interactions could aid cooperation and complement the effects of specific binding events on membrane remodelling. But it is difficult to experimentally isolate curvature-mediated interactions from direct attractions between proteins. Moreover, approximate theories predict repulsion between isotropically curving proteins. Here we use coarse-grained membrane simulations to show that curvature-inducing model proteins adsorbed on lipid bilayer membranes can experience attractive interactions that arise purely as a result of membrane curvature. We find that once a minimal local bending is realized, the effect robustly drives protein cluster formation and subsequent transformation into vesicles with radii that correlate with the local curvature imprint. Owing to its universal nature, curvature-mediated attraction can operate even between proteins lacking any specific interactions, such as newly synthesized and still immature membrane proteins in the endoplasmic reticulum.

Contact lines for fluid surface adhesion

Martin M. Müller, M. Deserno, and J. Guven

Phys. Rev. E 76, 011605 (2007); see also cond-mat/0703019

When a fluid surface adheres to a substrate, the location of the contact line adjusts in order to minimize the overall energy. This implies boundary conditions which depend on the characteristic surface deformation energies. We develop a general geometrical framework within which these conditions can be derived in a completely systematic way. We treat both adhesion to a rigid substrate and adhesion between two fluid surfaces, and illustrate our general results for several important Hamiltonians involving both curvature and curvature gradients. Some of these have previously been studied using very different techniques. With the exception of capillary phenomena, the Hamiltonian will not only be sensitive to boundary translations, but may also respond to changes in slope and even in curvature. The functional form of the additional contributions will follow readily from our treatment. We will show that the boundary conditions describing adhesion between two fluid surfaces express the balance of stresses and torques, as one would expect. At a rigid substrate, however, this simple identification will generally fail. This is because local rotations of the surface normal will be entirely "enslaved" to translations on the substrate. As a consequence, stresses and torques enter a single balance condition and cannot be disentangled.

Balancing torques in membrane-mediated interactions: exact results and numerical illustrations

Martin M. Müller, M. Deserno, and J. Guven

Phys. Rev. E 76, 011921 (2007); see also cond-mat/0702340.

Torques on interfaces can be described by a divergence-free tensor which is fully encoded in the geometry. This tensor consists of two terms, one originating in the couple of the stress, the other capturing an intrinsic contribution due to curvature. In analogy to the description of forces in terms of a stress tensor, the torque on a particle can be expressed as a line integral along a contour surrounding the particle. Interactions between particles mediated by a fluid membrane are studied within this framework. In particular, torque balance places a strong constraint on the shape of the membrane. Symmetric two-particle configurations admit simple analytical expressions which are valid in the fully nonlinear regime; in particular, the problem may be solved exactly in the case of two membrane-bound parallel cylinders. This apparently simple system provides some flavor of the remarkably subtle nonlinear behavior associated with membrane-mediated interactions.

2006

A novel method for measuring the bending rigidity of model lipid membranes by simulating tethers

Vagelis A. Harmandaris and Markus Deserno

J. Chem. Phys. 125, 204905 (2006); see also cond-mat/0611012.

The tensile force along a cylindrical lipid bilayer tube is proportional to the membrane's bending modulus and inversely proportional to the tube radius. We show that this relation, which is experimentally exploited to measure bending rigidities, can be applied with even greater ease in computer simulations. Using a coarse-grained bilayer model we efficiently obtain bending rigidities that compare very well with complementary measurements based on an analysis of thermal undulation modes. We furthermore illustrate that no deviations from simple quadratic continuum theory occur up to a radius of curvature comparable to the bilayer thickness.

How to determine local elastic properties of lipid bilayer membranes from atomic-force-microscope measurements: A theoretical analysis

Davood Norouzi, Martin M. Müller, and Markus Deserno

Phys. Rev. E 74, 061914 (2006); see also cond-mat/0602662.

Measurements with an atomic force microscope (AFM) offer a direct way to probe elastic properties of lipid bilayer membranes locally: provided the underlying stress-strain relation is known, material parameters such as surface tension or bending rigidity may be deduced. In a recent experiment a pore-spanning membrane was poked with an AFM tip, yielding a linear behavior of the force-indentation curves. A theoretical model for this case is presented here which describes these curves in the framework of Helfrich theory. The linear behavior of the measurements is reproduced if one neglects the influence of adhesion between tip and membrane. Including it via an adhesion balance changes the situation significantly: force-distance curves cease to be linear, hysteresis and nonzero detachment forces can show up. The characteristics of this rich scenario are discussed in detail in this paper.

Mechanical properties of pore-spanning lipid bilayers probed by atomic force microscopy

Siegfried Steltenkamp, Martin M. Müller, Markus Deserno, Christian Hennesthal, Claudia Steinem, and Andreas Janshoff

Biophys. J. 91, 217–226 (2006).

We measure the elastic response of a free-standing lipid membrane to a local indentation by using an atomic force microscope (AFM). Starting point is a planar goldcoated alumina substrate with a chemisorbed 3-mercaptopropionic acid monolayer displaying circular pores of very well defined and tunable size, over which N,N,-dimethyl-N,N,- dioctadecylammonium bromide or 1,2-dioleoyl-3-trimethylammonium-propane chloride were spread. Centrally indenting these "nanodrums" with an AFM tip yields force-indentation curves, which we quantitatively analyze by solving the corresponding shape equations of continuum curvature elasticity. Since the measured response depends in a known way on the system geometry (pore size, tip radius) and on material parameters (bending modulus, lateral tension), this opens the possibility to monitor local elastic properties of lipid membranes in a well-controlled setting.

Coupling between lipid shape and membrane curvature

Ira R. Cooke and Markus Deserno

Biophys. J. 91, 487–495 (2006).

Using molecular dynamics simulations, we examine the behavior of lipids whose preferred curvature can be systematically varied. This curvature is imposed by controlling the head group size of a coarse grained lipid model recently developed by us. In order to validate this approach we examine self assembly of each individual lipid type and observe the complete range of expected bilayer and micelle phases. We then examine binary systems consisting of lipids with positive and negative preferred curvature and find a definite sorting effect. Lipids with positive preferred curvature are found in greater proportions in outer monolayers with the opposite observed for lipids with negative preferred curvature. We also observe a similar, but slightly stronger effect for lipids in a developing spherical bud formed by adhesion to a colloid (eg a viral capsid). Importantly, the magnitude of this effect in both cases was large only for regions with strong mean curvature (radii of curvature <10nm). Our results suggest that lipid shape must act in concert with other physico-chemical effects such as phase transitions or interactions with proteins to produce strong sorting in cellular pathways.

2005

Solvent free model for self-assembling fluid bilayer membranes: Stabilization of the fluid phase based on broad attractive tail potentials

Ira R. Cooke and Markus Deserno

J. Chem. Phys. 123, 224710 (2005).

We present a simple and highly adaptable method for simulating coarse-grained lipid membranes without explicit solvent. Lipids are represented by one head-bead and two tail-beads, with the interaction between tails being of key importance in stabilizing the fluid phase. Two such tail-tail potentials were tested, with the important feature in both cases being a variable range of attraction. We examined phase diagrams of this range versus temperature for both functional forms of the tail-tail attraction and found that a certain threshold attractive width was required to stabilize the fluid phase. Within the fluid phase region we find that material properties such as area per lipid, orientational order, diffusion constant, inter-leaflet flip-flop rate and bilayer stiffness all depend strongly and monotonically on the attractive width. For three particular values of the potential width we investigate the transition between gel and fluid phases via heating or cooling and find that this transition is discontinuous with considerable hysteresis. We also investigated the stretching of a bilayer to eventually form a pore and found excellent agreement with recent analytic theory.

Interface mediated interactions between particles—a geometrical approach

Martin M. Müller, M. Deserno, and J. Guven

Phys. Rev. E 72, 061407 (2005).

Particles bound to an interface interact because they deform its shape. The stresses that result are fully encoded in the geometry and described by a divergence-free surface stress tensor. This stress tensor can be used to express the force on a particle as a line integral along any conveniently chosen closed contour that surrounds the particle. The resulting expression is exact (i.e., free of any "smallness" assumptions) and independent of the chosen surface parametrization. Additional surface degrees of freedom, such as vector fields describing lipid tilt, are readily included in this formalism. As an illustration, we derive the exact force for several important surface Hamiltonians in various symmetric two-particle configurations in terms of the midplane geometry; its sign is evident in certain interesting limits. Specializing to the linear regime, where the shape can be analytically determined, these general expressions yield force-distance relations, several of which have originally been derived by using an energy based approach.

Tunable generic model for fluid bilayer membranes

Ira R. Cooke, Kurt Kremer, and Markus Deserno

Phys. Rev. E 72, 011506 (2005).

We present a model for the efficient simulation of generic bilayer membranes. Individual lipids are represented by one head bead and two tail-beads. By means of simple pair potentials these robustly self-assemble to a fluid bilayer state over a wide range of parameters, without the need for an explicit solvent. The model shows the expected elastic behavior on large length scales, and its physical properties (e.g. fluidity or bending stiffness) can be widely tuned via a single parameter. In particular, bending rigidities in the experimentally relevant range are obtained, at least within 3–30kT. The model is naturally suited to study many physical topics, including self-assembly, fusion, bilayer melting, lipid mixtures, rafts, and protein-bilayer interactions.

Geometry of surface mediated interactions

Martin M. Müller, M. Deserno, and J. Guven

Europhys. Lett. 69, 482–488 (2005).

Soft interfaces can mediate interactions between particles bound to them. The force transmitted through the surface geometry on a particle may be expressed as a closed line integral of the surface stress tensor around that particle. This contour may be deformed to exploit the symmetries present; for two identical particles, one obtains an exact expression for the force between them in terms of the local surface geometry of their mid-plane; in the case of a fluid membrane the sign of the interaction is often evident. The approach, by construction, is adapted directly to the surface and is independent of its parameterization. Furthermore, it is applicable for arbitrarily large deformations; in particular, it remains valid beyond the linear small-gradient regime.

2004

When do fluid membranes engulf sticky colloids?

Markus Deserno

J. Phys.: Condens. Matter 16, S2061-S2070 (2004).

Binding of a spherical colloid to a fluid membrane, which is an interplay between the energies of adhesion and elastic deformation, is studied within the framework of a Helfrich Hamiltonian. The solution of the full nonlinear shape equations for the membrane profile reveals a continuous binding and a discontinuous envelopment transition, the latter with a tension dependent substantial energy barrier. In the bending dominated regime this scenario is analytically confirmed by a small gradient expansion.

Screening of Spherical Colloids beyond Mean Field—A Local Density Functional Approach

Marcia C. Barbosa, Markus Deserno, Christian Holm, and René Messina

Phys. Rev. E 69, 051401 (2004).

We study the counterion distribution around a spherical macroion and its osmotic pressure in the framework of the recently developed Debye-Hückel-Hole-Cavity (DHHC) theory. This is a local density functional approach which incorporates correlations into Poisson-Boltzmann theory by adding a free energy correction based on the One Component Plasma. We compare the predictions for ion distribution and osmotic pressure obtained by the full theory and by its zero temperature limit with Monte Carlo simulations. They agree excellently for weakly developed correlations and give the correct trend for stronger ones. In all investigated cases the DHHC theory and its computationally simpler zero temperature limit yield better results than the Poisson-Boltzmann theory.

Twist-bend instability for toroidal DNA condensates

Igor M. Kulic, Denis Andrienko, and Markus Deserno

Europhys. Lett. 67, 418–424 (2004).

We propose that semiflexible polymers in poor solvent collapse in two stages. The first stage is the well known formation of a dense toroidal aggregate. However, if the solvent is sufficiently poor, the condensate will undergo a second structural transition to a twisted entangled state, in which individual filaments lower their bending energy by additionally orbiting around the mean path along which they wind. This ``topological ripening'' is consistent with known simulations and experimental results. It connects and rationalizes various experimental observations ranging from strong DNA entanglement in viral capsids to the unusually short pitch of the cholesteric phase of DNA in sperm-heads. We propose that topological ripening of DNA toroids could improve the efficiency and stability of gene delivery.

A Statistical-Thermodynamic Model of Viral Budding

Shelly Tzlil, Markus Deserno, William M. Gelbart, and Avinoam Ben-Shaul

Biophys. J. 86, 2037–2048 (2004).

We present a simple statistical thermodynamic model for budding of viral nucleocapsids at the cell membrane. The membrane is modeled as a flexible lipid bilayer embedding linker (``spike'') proteins, which serve to anchor and thus wrap the membrane around the viral capsids. The free energy of a single bud is expressed as a sum of the bending energy of its membrane coat, the spike-mediated capsid-membrane adhesion energy, and the line energy associated with the bud's rim; all depending on the extent of wrapping (i.e., bud size), and density of spikes in the curved membrane. This ``self-energy'' is incorporated into a simple free energy functional for the many-bud system, allowing for different spike densities, and hence entropy, in the curved (budding) and planar membrane regions, as well as for the configurational entropy of the polydisperse bud population. The equilibrium spike densities in the coexisting, curved and planar, membrane regions are calculated as a function of the membrane bending energy and the spike-mediated adhesion energy, for different spike and nucleocapsid concentrations in the membrane plane, as well as for several values of the bud's rim energy. We show that complete budding (full wrapping of nucleocapsids) can only take place if the adhesion energy exceeds a certain, critical, bending free energy. Whenever budding takes place, the spike density in the mature virions is saturated, i.e., all spike adhesion sites are occupied. The rim energy plays an important role in determining the size distribution of buds. The fraction of fully wrapped buds increases as this energy increases, resulting eventually in an ``all or nothing'' mechanism, whereby nucleocapsids at the plasma membrane are either fully enveloped or completely ``naked'' (just touching the membrane). We also find that at low concentrations all capsids arriving at the membrane get tightly and fully enveloped. Beyond a certain concentration, corresponding approximately to a ``stoichiometric'' spike-to-capsid ratio, newly arriving capsids cannot be fully wrapped, i.e., the budding yield decreases.

Elastic deformation of a fluid membrane upon colloid binding

Markus Deserno

Phys. Rev. E 69, 031903 (2004).

When a colloidal particle adheres to a fluid membrane, it induces elastic deformations in the membrane which oppose its own binding. The structural and energetic aspects of this balance are investigated within the framework of a Helfrich Hamiltonian. Based on the full nonlinear shape equations for the membrane profile, a line of continuous binding transitions and a second line of discontinuous envelopment transitions are found, which meet at an unusual triple point. The regime of low tension is studied analytically using a small gradient expansion, while in the limit of large tension scaling arguments are derived which quantify the asymptotic behavior of phase boundary, degree of wrapping, and energy barrier. The maturation of animal viruses by budding is discussed as a biological example of such colloid-membrane interaction events.

2003

Osmotic Shock and the Strength of Viral Capsids

Amado Cordova, Markus Deserno, William M. Gelbart, and Avinoam Ben-Shaul

Biophys. J. 85, 70–74 (2003).

Osmotic shock is a familiar means for rupturing viral capsids and exposing their genomes intact. The necessary conditions for providing this shock involve incubation in high-concentration salt solutions, and lower permeability of the capsids to salt ions than to water molecules. We discuss here how values of the capsid strength can be inferred from calculations of the osmotic pressure differences associated with measured values of the critical concentration of incubation solution.

Wrapping of a spherical colloid by a fluid membrane

Markus Deserno and Thomas Bickel

Europhys. Lett. 62, 767–773 (2003).

We theoretically study the elastic deformation of a fluid membrane induced by an adhering spherical colloidal particle within the framework of a Helfrich energy. Based on a full optimization of the membrane shape we find a continuous binding and a discontinuous envelopment transition, the latter displaying a potentially substantial energy barrier. A small-gradient approximation permits membrane shape and complex energy to be calculated analytically. While this only leads to a good representation of the complex geometry for very small degrees of wrapping, it still gives the correct phase boundaries in the regime of low tension.

Attraction and ionic correlations between charged stiff polyelectrolytes

Markus Deserno, Axel Arnold, and Christian Holm

Macromolecules 36, 249–259 (2003).

We use molecular dynamics simulations to study attractive interactions and the underlying ionic correlations between parallel like-charged rods in the absence of additional salt. For a generic bulk system of rods we identify a reduction of short-range repulsions as the origin of a negative osmotic coefficient. The counterions show signs of a weak three-dimensional order in the attractive regime only once the rod-imposed charge inhomogeneities are divided out. We also treat the case of attraction between a single pair of rods for a few selected line charge densities and rod radii. Measurements of the individual contributions to the force between close rods are studied as a function of Bjerrum length. We find that even though the total force is always attractive at sufficiently high Bjerrum length, the electrostatic contribution can ultimately become repulsive. We also measure azimuthal and longitudinal correlation functions to answer the question how condensed ions are distributed with respect to each other and to the neighboring rod. For instance, we show that the prevalent image of mutually interlocked ions is qualitatively correct, even though modifications due to thermal fluctuations are usually strong.

2002

Theory and simulations of rigid polyelectrolytes

Markus Deserno and Christian Holm

Mol. Phys. 100, 2941–2956 (2002).

Theoretical and numerical studies are reported on stiff, linear polyelectrolytes within the framework of the cell model, first reviewing analytical results obtained on a mean-field Poisson-Boltzmann level, and then using molecular dynamics simulations to show the circumstances under which these fail quantitatively and qualitatively. For the hexagonally packed nematic phase of the polyelectrolytes the osmotic coefficient is computed as a function of density. In the presence of multivalent counterions it can become negative, leading to effective attractions. This is shown to result from a reduced contribution of the virial part to the pressure. The osmotic coefficient and ionic distribution functions are computed from Poisson-Boltzmann theory with and without a recently proposed correlation correction. Simulation results for the case of poly(p-phenylene) are presented and compared with recently obtained experimental data on this stiff polyelectrolyte. Ion-ion correlations in the strong coupling regime are studied and compared with the predictions of the recently advocated Wigner crystal theories.

The osmotic pressure of charged colloidal suspensions: A unified approach to linearized Poisson-Boltzmann theory

Markus Deserno and Hans-Hennig von Grünberg

Phys. Rev. E 66, 011401 (2002).

We study theoretically the equation of state of a fluid suspension of charged objects (e.g., colloids, polyelectrolytes, clay platelets, etc.) dialyzed against an electrolyte solution using the cell model and linear Poisson-Boltzmann (PB) theory. From the volume derivative of the grand potential functional of linear theory we obtain two expressions for the osmotic pressure in terms of the potential or ion profiles, neither of which coincides with the expression known from nonlinear PB theory, namely, the density of microions at the cell boundary. We show that the range of validity of linearization depends strongly on the linearization point and prove that expansion about the self-consistently determined average potential is optimal in several respects. For instance, screening inside the suspension is automatically described by the actual ionic strength, resulting in the correct asymptotics at high colloid concentration. Together with the analytical solution of the linear PB equation for cell models of arbitrary dimension and electrolyte composition, explicit and very general formulas for the osmotic pressure ensue. A comparison with nonlinear PB theory is provided. Our analysis also shows that whether or not linear theory predicts a phase separation depends crucially on the precise definition of the pressure, showing that depending on the choice, an artificial phase separation in systems as important as DNA in physiological salt solution may result.

Adhesion and Wrapping in Colloid-Vesicle Complexes

Markus Deserno and William M. Gelbart

J. Phys. Chem. B 106, 5543–5552 (2002).

We present a simple theoretical model for adsorption of colloidal particles onto vesicles. The contact energy of adhesion is balanced by the tension and curvature energies of the vesicle membrane under the constraint of fixed volume, with the geometry of the complex determined by a variational calculation. Physical observables, such as the degree of penetration or the membrane tension, are investigated as functions of colloidal size and adhesion, tension, and bending energies. We find various new (and discontinuous) transitions in the geometry of the complex compared to those from a description that neglects the curvature contribution. Particular emphasis is put on the transition from the partly to the fully wrapped state and on unbinding of the complex at weak adhesion energy or small colloidal size. The above model can be thought of as a phenomenological theory of the initial steps involved in biological endocytosis and aims toward an improved physical understanding of this process.

2001

Rayleigh instability of charged droplets in the presence of counterions

Markus Deserno

Eur. Phys. J. E 6, 163–168 (2001).

The classical instability of a charged spherical droplet is reconsidered in the presence of counterions. An ensemble of such droplets is studied within a simplified cell model. Screening of the electric field by the counterions is found to increase the equilibrium droplet size. Furthermore, if the ions can enter the droplet, a first-order phase transition occurs upon increasing Bjerrum length, surface tension or droplet density, leading to a phase separation. Simple scaling properties of the free energy give the shape of the phase boundary and show the system to be scale-invariant there. Pearl-necklace structures of hydrophobic polyelectrolytes are discussed as an application.

Overcharging of DNA in the Presence of Salt: Theory and Simulation

Markus Deserno, Felipe Jiménez-Ángeles, Christian Holm, and Marcelo Lozada-Cassou

J. Phys. Chem. B 105, 10983–10991 (2001).

A study of a model rodlike polyelectrolyte molecule immersed in a monovalent or divalent electrolyte is presented. Results for the local concentration profile, mean electrostatic potential, charge distribution function, and zeta-potential are obtained from hypernetted-chain/mean spherical approximation (HNC/MSA) theory and compared with molecular dynamics (MD) simulations. As a particular case, the parameters of the polyelectrolyte molecule are mapped to those of a DNA molecule. Both HNC/MSA and MD predict the occurrence of overcharging, which is not present in the Poisson-Boltzmann theory. Further an excellent qualitative, and in some cases quantitative, agreement between HNC/MSA and MD is found. Oscillations observed in the mean electrostatic potential, local concentration profiles, and the curvature of the zeta-potential are discussed in terms of the observed overcharging effect. Particularly interesting results are a very nonmonotonic behavior of the zeta-potential, as a function of the rod charge density, and the overcharging by monovalent counterions.

The osmotic coefficient of rod-like polyelectrolytes: Computer simulation, analytical theory, and experiment

Markus Deserno, Christian Holm, Jürgen Blaul, Matthias Ballauff, and Matthias Rehahn

Eur. Phys. J. E 5, 97–103 (2001).

The osmotic coefficient of solutions of rod-like polyelectrolytes is considered by comparing current theoretical treatments and simulations to recent experimental data. The discussion is restricted to the case of monovalent counterions and dilute, salt-free solutions. The classical Poisson-Boltzmann solution of the cell model correctly predicts a strong decrease in the osmotic coefficient, but upon closer look systematically overestimates its value. The contribution of ion-ion-correlations are quantitatively studied by MD simulations and the recently proposed DHHC theory. However, our comparison with experimental data obtained on synthetic, stiff-chain polyelectrolytes shows that correlation effects can only partly explain the discrepancy. A quantitative understanding thus requires theoretical efforts beyond the restricted primitive model of electrolytes.

2000

Ion-ion correlations: an improved one-component plasma correction

Marcia C. Barbosa, Markus Deserno, and Christian Holm

Europhys. Lett. 52, 80–86 (2000).

We propose a new approach for incorporating counterion correlations into Poisson-Boltzmann-like theories, which is based on a local density functional for the system of counterions. Our derivation is inspired by the Debye-Hückel approach to the one-component plasma (OCP). An artificial instability due to the OCP's homogeneous background charge – which is absent in the original counterion system – is overcome by excluding the background from the vicinity of a central ion. The resulting free energy density is convex and thus applicable within a local density approximation. For the simple example of a strongly charged stiff rod surrounded by its counterions we demonstrate that the Poisson-Boltzmann free energy functional augmented by our new correction accounts for the correlations present in this system when compared to molecular dynamics simulations.

A Monte-Carlo approach to Poisson-Boltzmann like free-energy functionals

Markus Deserno

Physica A 278, 405–413 (2000).

A simple technique is proposed for numerically determining equilibrium ion distribution functions belonging to free energies of the Poisson-Boltzmann type. The central idea is to perform a conventional Monte-Carlo simulation using the free energy as the "Hamiltonian" entering the Metropolis criterion and the spatially discretized density as degrees of freedom. This approach is complementary to the possibility of numerically solving the differential equations corresponding to the variational problem, but it is much easier to implement and to generalize. Its utility is demonstrated in two examples: valence mixtures and hard core interactions of ions surrounding a charged rod.

Fraction of condensed counterions around a charged rod: Comparison of Poisson-Boltzmann theory and computer simulations

Markus Deserno, Christian Holm, and Sylvio May

Macromolecules 33, 199–206 (2000).

We investigate the phenomenon of counterion condensation in a solution of highly charged rigid polyelectrolytes within the cell model. A method is proposed which-based on the charge distribution function-identifies both the fraction of condensed ions and the radial extension of the condensed layer. Within salt-free Poisson-Boltzmann (PB) theory it reproduces the well-known fraction 1-1/ξ of condensed ions for a Manning parameter ξ > 1. Furthermore, it predicts a weak salt dependence of this fraction and a breakdown of the concept of counterion condensation in the high-salt limit. We complement our theoretical investigations with molecular dynamics simulations of a cell-like model, which constantly yield a stronger condensation than predicted by PB theory. While the agreement between theory and simulation is excellent in the monovalent, weakly charged case, it deteriorates with increasing electrostatic interaction strength and, in particular, increasing valence. For instance, at a high concentration of divalent salt and large xi our computer simulations predict charge oscillations, which mean-field theory is unable to reproduce.

1998

How to mesh up Ewald sums (II): An accurate error estimate for the P3M algorithm

Markus Deserno and Christian Holm

J. Chem. Phys. 109, 7694–7701 (1998).

We construct an accurate estimate for the root mean square force error of the particle-particle–particle-mesh (P3M) algorithm by extending a single particle pair error measure which has been given by Hockney and Eastwood. We also derive an easy-to-use analytic approximation to the error formula. This allows a straightforward and precise determination of the optimal splitting parameter (as a function of system specifications and P3M parameters) and hence knowledge of the force accuracy prior to the actual simulation. The high quality of the estimate is demonstrated in several examples.

How to mesh up Ewald sums (I): A theoretical and numerical comparison of various particle mesh routines

Markus Deserno and Christian Holm

J. Chem. Phys. 109, 7678–7693 (1998).

Standard Ewald sums, which calculate e.g. the electrostatic energy or the force in periodically closed systems of charged particles, can be efficiently speeded up by the use of the Fast Fourier Transformation (FFT). In this article we investigate three algorithms for the FFT-accelerated Ewald sum, which attracted a widespread attention, namely, the so-called particle-particle–particle-mesh (P3M), particle mesh Ewald (PME) and smooth PME method. We present a unified view of the underlying techniques and the various ingredients which comprise those routines. Additionally, we offer detailed accuracy measurements, which shed some light on the influence of several tuning parameters and also show that the existing methods – although similar in spirit – exhibit remarkable differences in accuracy. We propose combinations of the individual components, mostly relying on the P3M approach, which we regard as most flexible. The issue of estimating the errors connected with particle mesh routines is reserved to part II of this series.

1997

Tricriticality and the Blume-Capel model: A Monte Carlo study within the microcanonical ensemble

Markus Deserno

Phys. Rev. E 56, 5204–5210 (1997).

The microcanonical partition function Ω_α,N(E) of a three-dimensional ferromagnetic simple cubic Blume-Capel model is calculated for several coupling ratios alpha close to and on the first-order side of the tricritical point. To this end a single Monte Carlo simulation of a suitably extended partition function for systems of L×L×L spins with L in {8,10,12,14,18} was performed. A finite system analog of the latent heat is introduced in order to define a tricritical point in finite systems as well. An empirical scaling of its coordinates to the thermodynamic limit yields α_t=2.844 79 +- 0.00030 for the tricritical coupling ratio, and kT_t/J=1.4182 +- 0.0055 for the tricritical transition temperature. The results are compared with values obtained on FCC lattices.

Proceeding, Schools, ...

Membrane Elasticity and Mediated Interactions in Continuum Theory: A Differential Geometric Approach

Markus Deserno

in: Biomembrane Frontiers: Nanostructures, Models, and the Design of Life, Volume 2, edited by Roland Faller, Thomas Jue, Marjorie L. Longo, and Subhash H. Risbud,Humana Press (Springer, New York, 2009).

Objects in contact with membranes can locally deform these membranes, and several such objects can interact by means of the elastic stresses thus created. For fluid surfaces the Hamiltonian is purely geometric, and this permits a very efficient treatment of this scenario using methods of covariant differential geometry. This article provides a pedagogical introduction into the philosophy behind this approach. The relation between geometry and elasticity is first derived in a very intuitive way and then extended to more complex (but very relevant) examples. The relation between stresses and geometry is introduced and illustrated in three very different examples: interactions mediated by (a) composition variations, (b) lipid tilt fields, and (c) membrane curvature.

Solvent-free lipid-bilayer simulations: From Physics to Biology

Markus Deserno

in: Computer Simulation Studies in Condensed Matter Physics XIX, Eds. D.P. Landau, S.P. Lewis, and H.B. Schüttler (Springer, Heidelberg, Berlin, 2006).

Coarse-grained solvent-free simulation models enabling the study of self-assembling fluid lipid bilayers have been the goal of much recent modeling efforts, since their realization appeared to be quite intricate. This contribution reviews some of the challenges faced along the way, presents a surprisingly simple solution, and illustrates its capacity by means of three examples of biological interest.

Cell Model and Poisson-Boltzmann Theory: A brief introduction

Markus Deserno and Christian Holm

in: Electrostatic Effects in Soft Matter and Biophysics, Vol 46 of NATO Advanced Studies Institute, Series II: Mathematics, Physics and Chemistry, ed. by Christian Holm, Patrick Kékicheff, and Rudolf Podgornik (Kluwer, Dordrecht, 2001).

The cell-model and its treatment on the Poisson-Boltzmann level are two important concepts in the theoretical description of charged macromolecules. In this brief contribution we provide an introduction to both ideas and summarize a few important results which can be obtained from them. Our article is organized as follows: Section 1 outlines the sequence of approximations which ultimately lead to the cell-model. Section 2 is devoted to two exact results, namely, an expression for the osmotic pressure and a formula for the ion density at the surface of the macromolecule, known as the contact value theorem. Section 3 provides a derivation of the Poisson-Boltzmann equation from a variational principle and the assumption of a product state. Section 4 applies Poisson-Boltzmann theory to the cell-model of linear polyelectrolytes. In particular, the behavior of the exact solution in the limit of zero density is compared to the concept of Manning condensation. Finally, Section 5 shows how a system described by a cell model can be coupled to a salt reservoir, i.e., how the so-called Donnan equilibrium is established.

Molecular Dynamics Simulation of the Cell Model

Markus Deserno, Christian Holm, and Kurt Kremer

in: Physical Chemistry of Polyelectrolytes, Vol. 99 of the surfactant science series, ed. by Tsetska Radeva (Marcel Dekker, New York, 2001).