Modeling Self-Assembly Processes Driven by Nonbonded Interactions

Jul 18, 2008 - Nicolas D. Winter is currently a postdoctoral research fellow at North- ... George C. Schatz is a Charles E. and Emma H. Morrison Profe...
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CENTENNIAL FEATURE ARTICLE Modeling Self-Assembly Processes Driven by Nonbonded Interactions in Soft Materials† Martin McCullagh, Tatiana Prytkova, Stefano Tonzani, Nicolas D. Winter, and George C. Schatz* Department of Chemistry, Northwestern UniVersity, EVanston, Illinois 60208-3113 ReceiVed: April 13, 2008; ReVised Manuscript ReceiVed: May 16, 2008

This Centennial Feature Article provides an overview of research in the general area of self-assembly modeling, with particular emphasis on the self-assembly of molecules into soft nanoscale structures where the driving force for assembly is provided by nonbonded interactions (hydrogen bonds and electrostatics). The models have been developed at many different levels of theory, going all the way from simple analytical models of packing effects to atomistic descriptions using molecular dynamics methods. In between these limits are mean-field and coarse-grained models, including models for DNA, peptides, and lipids that can be used to describe the assembly of hybrid (amphiphilic) materials. Several recent applications to specific systems are discussed, including the description of peptide amphiphile assembly to make cylindrical micelles, the assembly and melting of DNA hairpins, the use of DNA tethers to assemble nanoparticles into aggregates and crystalline structures, and the use of coarse-grained lipid models to make lamellar and high-curvature phases. These examples demonstrate the difficulties associated with brute force atomistic methods, and they also show the opportunities (but uncertainties and ambiguities) associated with simpler models such as coarse-grained models. The examples also demonstrate the usefulness of successful modeling methods in the design of new materials, including an understanding of the relationship between structure and function. I. Introduction Describing and understanding the self-assembly of complex materials is a significant challenge in 21st century science. Selfassembled structures are ubiquitous in biology, and indeed, this is how nature creates complex structures. However, humans have struggled to learn how to mimic self-assembly, and often, we are stuck at a basic level where it is not possible to predict what structures will arise from the aggregation of a large number of identical molecules in a homogeneous system. There are important reasons for learning how to make such predictions as there are already many examples of self-assembled structures which have provided new capabilities for multifunctional materials, including applications in biomimetics, 1 medical diagnostics,2 therapeutics,3 molecular motors,4 photonic materials,5 and many others. The modeling of self-assembly processes is, in many respects, a young field, but at the same time, many of the commonly used ideas and techniques go back to the beginning of theoretical and computational chemistry and physics. Thus, the modeling of reaction-diffusion equations which lead to the formation of complex spatial structures (at macroscopic length scales) in a system that is well away from equilibrium has been of interest for many decades.6–9 In addition, continuum elastic models of † This year marks the Centennial of the American Chemical Society’s Division of Physical Chemistry. To celebrate and to highlight the field of physical chemistry from both historical and future perspectives, The Journal of Physical Chemistry is publishing a special series of Centennial Feature Articles. These articles are invited contributions from current and former officers and members of the Physical Chemistry Division Executive Committee and from J. Phys. Chem. Senior Editors. * To whom correspondence should be addressed.

membranes10,11 and simple considerations of packing effects in micelles and membranes12 are now described in standard texts.12 However, in the last 10 years, there have been important advances in self-assembly experiments leading to nanoscale structures that require a more sophisticated analysis, and this has stimulated new modeling activities, including several approaches that provide different capabilities than have been available in the past. However, there are still many intractable problems that limit what can be modeled; therefore, many of the synthetic strategies still have no modeling component, and progress is largely based on guesswork. I.1. Themes of Self Assembly. Self-assembly is an enormous field.13–16 As mentioned above, self-assembly is a key component of biology, and indeed, there is an incredible diversity of self-assembly processes that nature uses. However, there are some basic themes involved which can be organized according to the nature of the physical interactions between molecules that are involved. CoValent bond formation is involved in the synthesis of the key building blocks in nature, such as peptides and single-stranded DNA, from the basic monomer units, amino acids, and nucleotides. Hydrogen bonds, hydrophobic interactions, and electrostatics bind peptides into proteins, duplex DNA from oligonucleotides, and they also bind lipids into membranes. These nonbonded interactions, as well as packing effects, are responsible for the formation of protein aggregates from individual proteins (such as amyloid fibril self-assembly3 and Actin filaments interacting with lysozyme proteins17). Examples of these bonded and nonbonded interactions are easily found in synthetic processes that are used to generate soft nanomaterials. Many supramolecular structures are generated by a series of reactions which involve covalent bond

10.1021/jp803192u CCC: $40.75  2008 American Chemical Society Published on Web 07/18/2008

Centennial Feature Article Martin McCullagh was born in London, England, in 1983. He received his Bachelors of Science in chemistry from Emory University in 2005. During his undergraduate research under Keiji Morokuma, he investigated excited-state molecular quenching using electronic structure theories. He is currently working on his Ph.D. at Northwestern University under George Schatz. His research includes the dynamical properties of DNA and lipids including self-assembly of both systems. Tatiana Prytkova is a postdoctoral research fellow in the group of Professor Schatz. Her research involves developing methodology to study the thermodynamic and structural properties of self-assembled structures made from DNA. She focuses on modeling the sharp melting, hybridization and phase transition phenomena in these materials. She obtained her Ph.D. in theoretical chemistry at Duke University with David Beratan. Her Ph.D. work involved performing simulations of electron-transfer reactions on proteins to predict electron-transfer rates by using ab initio level quantum chemistry, studying dynamical and solvation effects in protein electrontransfer reactions, modeling the structural basis for molecular recognition of androgen receptor ligands and coactivator proteins, and studying the mechanism of the repair of the UV-damaged DNA in DNA photolyase. Stefano Tonzani was born in Perugia, Italy, in 1977. He received his B.S. And M.S. in chemistry in 2001 from University of Perugia where he worked with V. Aquilanti. He then moved to the group of Prof. C. H. Greene at the University of Colorado, Boulder, where he received a Ph.D. in chemical physics in 2006 while working on electron-molecule interactions. He is currently a research associate at Northwestern University in the group of Prof. G. C. Schatz, and his research involves charge transfer in biomolecules and self-assembly. Nicolas D. Winter is currently a postdoctoral research fellow at Northwestern University. He completed his Ph.D. in chemistry in 2006 at the University of California, Santa Cruz, with Ilan Benjamin, where he studied chemical reactions of small molecules at liquid interfaces using semiclassical molecular dynamics computer simulations. In particular, he examined photodissociation of ICN at the water/vapor interface using a surface hopping model as well as dissociation of tert-butyl chloride at aqueous/organic liquid interfaces using an empirical valence bond model. His current research interests include using coarse-grained molecular dynamics simulations to understand the self-assembly, structure, and phase behavior of lipid mixtures and liposomes. George C. Schatz is a Charles E. and Emma H. Morrison Professor of Chemistry at Northwestern University. Degrees include a B.S. (1971) at Clarkson University and Ph.D. (1976) at Caltech, both in chemistry. He was a postdoc at MIT and has been at Northwestern since 1976. Schatz is Editor-in-Chief of the Journal of Physical Chemistry. Schatz’s research is concerned with theory and computational modeling in a variety of nanoscience topics as well as in related biophysics and materials areas. His nanoscience work has specialized in studies of the optical properties of noble metal nanoparticles, nanoholes in films, and other nanostructured materials of relevance to chemical and biological sensing applications. He has contributed to theories of DNA melting of nanoparticle aggregates, and he has studied a variety of problems in self-assembly modeling, DNA and protein structures, transport in ion channels, the deposition and patterning of molecules on surfaces, and the formation of water droplets on nanoscale structures. In addition, he has worked actively in the modeling of the mechanical properties of hard materials, including diamond films and carbon nanotubes.

formation to sequentially link together chemically similar units.18,19 Often, this is done through a series of irreversible steps, such as via “click” chemistry,20–22 as this enables the formation of complex molecules despite entropic penalties. Hydrogen bond formation, hydrophobic interactions, and electrostatics are involved in many self-assembly processes as these allow for reversible steps which, in some cases, can produce complex organized structures involving very large numbers of monomer units. A good example of this is the self-assembly of amphiphilic molecules (molecules which have both hydrophobic and hydrophilic components),1,13,23 where the interactions use structural elements such as beta-sheet formation for selected peptides and hydrophobic interactions among alkane chains. Hydrogen bonds are also involved in the assembly of DNA tiles and other supramolecular structures in which the programmable properties of DNA enable the synthesis of complex shapes.4 In

J. Phys. Chem. B, Vol. 112, No. 34, 2008 10389 addition, hydrogen bonds and hydrophobic interactions are important in many kinds of organized polymer aggregates and gels.24 Additional complexity arises from the self-assembly of two or more kinds of molecules (or nanostructures) that have distinct chemical and structural properties. One example of this is the use of one kind of molecule as a tether to link together monomers of another kind of molecule. DNA-linked gold particles25 and DNA-linked polymers26,27 are two examples of this theme, and both lead to the production of gel-like aggregates. Since hydrogen bonds are responsible for linking the DNAs, these aggregates are capable of transforming back to isolated monomers as a result of DNA dehybridization (melting) either when the temperature is raised or the salt concentration is reduced. Self-assembly can also lead to complex phase behavior in materials whose underlying components are relatively simple, as in liquid crystals formed by DNA duplexes.28 Templated self-assembly is another important theme, in which a molecule or nanostructure controls the formation of aggregates of other molecules. An example of this is the self-assembly of molecules on surfaces, where the surface serves as a template for the formation of complex organized structures.29,30 Selfassembled monolayers (SAMs) on surfaces and layers in a Langmuir trough are two early examples of this theme, and a great deal of modeling has been devoted to them.31–34 More recent applications have involved structures that are fabricated using dip-pen nanolithography and other scanning probe lithography approaches.35–37 In addition, templated self-assembled structures have been generated in homogeneous solution by combining a protein scaffold with lipid monomers to produce a disk-shaped lipid raft.38 The self-assembly processes described so far have involved monomers with nanometer sizes or less. However, self-assembly processes can take place with larger subunits, also driven by electrostatics and hydrogen bonding. The assembly of colloidal crystals is a good example of electrostatically driven assembly.39 There are many other self-assembly processes of this type, including magnetic self-assembly, mechanical self-assembly, and assembly that arises from nonlinear diffusion processes and sedimentation. However, the modeling of these larger structures is not the focus of this work. I.2. Self-Assembly Modeling. Self-assembly modeling can be done at many different levels of complexity. At the simplest level, simple analytical models can be used to show how molecules can pack together12,40 or what low-energy structures arise from the consideration of electrostatic interactions between the monomers.41 Lattice models are also useful for describing complex materials,35,36,42 and thermodynamic models are excellent for problems where the interactions can be broken down into discrete binding events (such as in DNA hybridization43). The brute force approach to self-assembly modeling is to simply represent all of the atoms in the system and let molecular dynamics (MD) or Monte Carlo (MC) calculations (either with empirical or electronic structure potentials) determine the lowestenergy (or free-energy) structures. Unfortunately, this is only useful for very simple problems due to the enormous number of structures that must be sampled to determine which ones have low energies for Monte Carlo calculations or the long times needed to explore structures in molecular dynamics calculations. However, this brute force approach is incredibly useful when it works, and there has been a lot of work on developing more efficient methods for sampling structures (such as configurationbiased Monte Carlo,44 rejection-free Monte Carlo,45 on-the-fly kinetic Monte Carlo46) or for accelerating molecular dynamics

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Figure 1. Self-assembly of amphiphile block copolymers: a) structure of the copolymer unit, b) schematic representation of a PA (top) and of a patch of the micelle surface (bottom), and c) six bead model and assembly of the seven bead model into a cylindrical micelle structure.

calculations46 (hyperdynamics, parallel replica dynamics, temperature-accelerated dynamics). Much of the recent work in self-assembly has been concerned with developing methods that are somewhere between simple models and brute force. Mean-field models47 use simple expressions for the free-energy function of a system composed of many monomers that describe interactions between molecules by averaging over many degrees of freedom. Because of this averaging process, detailed structural information is usually not provided; however, other information, such as the average structure and thermodynamic properties of the material, are easily obtained. Coarse-grained molecular dynamics (CGMD) methods48–50 group atoms into “beads” in which bead-bead interactions average over atomic-level detail. Then molecular dynamics (or Monte Carlo) methods are used to generate the properties of interest for the coarse-grained system. In this approach, more detailed structural information is available than in typical mean-field models; however, atomistic detail is still missing, and thermodynamic properties require significant computational effort. Of course, there are ways to combine coarse-grained and mean-field approaches. Another direction for self-assembly modeling is to use MD or MC methods to simulate the disassembly of a self-assembled structure rather than its formation from individual components. An example of this involves the use of atomistic molecular dynamics to describe protein denaturation (“unfolding”), such as has been studied experimentally using ion mobility measurements.51 Such studies can determine microscopic pathways in both directions (i.e., folding as well as unfolding) but taking advantage of the fact that the time scale for unfolding is often many orders of magnitude faster than that for folding due to the entropic driving force for unfolding. In addition, unfolding

can be simulated under nonequilibrium conditions (high temperatures), which shortens time scales. I.3. Specific Applications. The remainder of this manuscript describes three examples of self-assembly modeling that have been studied in our group. In the first, we describe the selfassembly of peptide amphiphiles into cylindrical micelle structures based on experiments done by Stupp and co-workers.1 For this problem many different levels of theory have been used, ranging from simple models to atomistic, and we illustrate one of these applications using a coarse-grained molecular dynamics model. In the second, we describe a variety of assembly and disassembly processes involving DNA, DNA hairpins, and DNA-linked nanoparticle aggregates. This presentation will again reveal that different levels of theory are important for different kinds of problems, even when all involve the same molecule. In the third, we study the structure of lipid phases using coarse-grained molecular dynamics. II. Self-Assembly of Peptide Amphiphiles II. One Overview of Past Work. In recent years, a new class of self-assembled aggregate structures has been synthesized by Stupp and co-workers,1,52–56 built from amphiphilic molecules composed of a peptide head linked to an alkane tail (Figure 1a). These peptide amphiphile (PA) aggregates often form cylindrical micelle structures52,56,57 that have found use in various applications including tissue regeneration54 and to carry biomolecular signals.55,58 The amino acids used in these peptide amphiphiles include charged residues at the head of the molecule and beta-sheet formers (a tract of four cysteine residues) in the center. This results in the formation of long cylindrical fibers when the pH is low enough that the amphiphiles are neutral.

Centennial Feature Article Understanding why the PAs assemble into certain shapes as opposed to others is a complex problem, and this leaves a large domain for theory to play a role. Because these aggregates are very large structures, brute force simulations of the fiber formation with atomistic molecular dynamics force fields can only play a limited role. Our group has been involved40,41,59–61 in understanding the mechanisms by which these amphiphilic building blocks assemble into cylindrical structures using a variety of theories that range from simple models to restricted atomistic calculations. Atomistic simulation of small clusters of the PA molecules (a 4 × 4 cluster is schematically displayed in Figure 1b) shows that the charged head groups assume an arrangement such as to minimize the dipolar and charge interactions, while the hydrophobic tails interact with each other. The presence of voids in the hydrophobic tail region as a result of the interplay between these forces drives the cluster to form a cylindrical (rather than spherical) portion of the micelle surface.41 A simple coarse-grained level of theory has involved representing each molecule using a string of six beads, with sizes chosen to mimic the cone shape of the atomistic structure (Figure 1c) and including electrostatic interactions (a dipole) in the head bead and hydrophobic interactions (represented by an effective potential) near the tail. Monte Carlo calculations (starting from a random distribution of PAs but with a small seed aggregate) show that the cone-shaped structures of the bead model form spherical micelles when the electrostatic interactions are turned off but cylindrical micelles (Figure 1c) when they are on.60 Beta-sheet-forming residues also induce the formation of cylindrical structures, as has been shown experimentally, and in, fact there is some question as to whether it is β-sheet or dipolar interaction effects which play a bigger role in determining which structures are formed. Recently, Olvera de la Cruz and collaborators have used an electrostatic model combined with Monte Carlo simulations, which are designed to describe β-sheet formation but not dipolar interactions, to study micelle formation.62 Their model, starting from random molecules in solution, leads to the formation of beta-sheets and subsequently to their aggregation into helical structures that represent portions of the micelle. However, computational limitations of the model did not allow for a more complete description. II.2. Coarse-Grained Modeling. The limitations of the models described above are clear: the six-bead coarse-grained model has too little detail concerning the structure of residues to resolve the beta-sheet formation question, while atomistic force fields not only cannot assemble a fiber from molecules in solution but cannot even describe a sizable chunk of the fiber since the simulations would take far too long. An alternative is a finer level of coarse-grained molecular dynamics than the bead model described above, such as has become popular in recent years for describing self-assembly in lipids,48 nanodiscs,38 and block copolymers.63 Coarse-graining in this case consists of fusing several (usually four) heavy atoms plus associated hydrogens into a single interaction center or “bead”.48 This carries two advantages: it reduces the number of particles in the system, and it also allows the use of longer time steps since the fastest motions (such as C-H stretching) have been included in an effective interaction that varies much more slowly with time. Of course, this can lead to artifacts, such as a difference between CG time scales and atomistic time scales; therefore, validation of each force field and the properties that can realistically be obtained from it is important. We will discuss aspects of this validation throughout this paper. Using as a starting point the model of Marrink et al.48 for the alkane chain and that of Schulten et al.38 for the peptide

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Figure 2. Coarse-grained molecular dynamics description of a small disk-shaped section of a cylindrical amphiphile fiber. The coarse-grained monomer and its atomistic representation are pictured at the bottom. The top panel is the view perpendicular to the cylinder axis, while the bottom is along the cylinder axis. The pictures on the left represent a low-pH situation, after 200 ns, where the fiber maintains its cylindrical structure. The middle panel shows what happens when the cysteins are methylated (250 ns) and a spherical “micelle” forms, while the right panel is a depiction of a high pH-system, (450 ns) in which the amphiphiles are charged and assume more disordered conformations.

residues, we developed a force field that is capable of describing a chunk of the cylindrical fiber immersed in solvent for time scales on the order of 100 ns such that stability of the resulting structures can be determined. The problem of formation of the fiber from molecules randomly dispersed in solution requires much longer time scales than we can simulate (more than a µs); therefore, this is a limitation of this model compared to the simpler coarse-grained models described above. This also stands in contrast to what happens for lipids (described later in this paper), for which hydrophobicity provides a very strong force that leads to fast ( 8), where the CYS residues (pKa ) 8) are deprotonated, the amphiphiles are again not able to form betasheets. At this point, the cylindrical structure morphs into a more disordered structure (right column of Figure 2), again in agreement with experimental data in which the fibers disassemble at this pH.52 Note that the dipoles at the end of the peptide are largely invariant to this pH change; therefore, apparently, the dipoles alone are not adequate to preserve the micelle structure. The simulations that we present are between 200 and 450 ns in length. We are able to reach such long time scales by ignoring the dihedral potential of the peptides. We have verified that by using a dihedral potential similar to that of Shih et al.38 but adapted for beta-sheets instead of alpha-helices, the cylindrical structures are still preserved for the entire length of the simulation (30 ns). The simulations with dihedral potentials are constrained to a 5 fs time step, while the others can use a much longer time step (25 fs). As this is still work in progress, it has not been possible for us to do a more quantitative comparison of properties with experiment. In addition, comparisons between theory and experiment are only likely to be qualitative in nature as high-resolution structural information (such as from X-ray measurements) about the micelles is not available. III. DNA, DNA Hairpins, DNA-Linked Oligonucleotides III.1. Simulation of DNA Melting. DNA plays an important role in developing self-assembly models, and a number of thermodynamic, bead-spring and site models have been developed.43,65–73 These models provide important capabilities for determining melting/dehybridization temperatures for various base pair (BP) compositions, and there are also models which describe mechanical properties. One of the still unsolved challenges with DNA is describing thermal dehybridization at an atomistic level. A similar problem arises in modeling DNA mechanical dehybridization.74 A standard method to model many DNA properties is with atomistic force fields and molecular dynamics simulations. Figure 3 (lower left) shows the structure that is generated with this approach, based on the Amber force field, for a 10 BP duplex. Since DNA is a polyelectrolyte, the structure also includes counterions (typically Na+) that are not shown, as well as explicit waters. A bottleneck with MD simulations on structures like that in Figure 3 is that computations are not feasible, in general, for the >µs time scale that is needed to describe melting. The total number of atoms in the atomistic simulation in Figure 3 is ∼9500 if water is included; therefore, even 100 ns simulations are quite difficult. Fewer base pairs can be considered, but duplexes with fewer than six BP are not sufficiently stable at room temperature to enable experimental measurements of melting temperatures. To circumvent this limitation, there has been much interest recently in coarse-grained approaches, wherein several atoms in the molecule are represented by one coarse-grained bead. Although there were attempts to do this prior to 2000,75 the first model which allowed for unconstrained

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Figure 3. Models of duplex DNA structures: (bottom left) atomistic representation; (bottom right) coarse-grained model from DePablo et al.; (top) snapshots of melting process from the Drukker et al. coarsegrained model.

bead motion in three dimensions was developed by Drukker and Schatz in 2000.76 In this model (see Figure 3, top), every DNA base pair is represented by two beads, and the sugar/ phosphate backbone is an additional bead. A simulation time of µs was attainable with this resolution, and this allowed the direct simulation of DNA melting (Figure 3, top) and to distinguish the relative stability of A-T and G-C duplexes as well as mismatches. However, this DNA model omitted electrostatic interactions, which means that the force field parameters have to implicitly account for counterion effects. This means that the only way to describe salt effects on melting (an important effect) is to develop different parameters for different salt concentrations. A more recent coarse-grained DNA model was proposed by de Pablo and co-workers. 77 In this model (Figure 3, bottom right), each DNA nucleotide is represented by three beads that correspond to base, sugar, and phosphate. Charged phosphates are included in this model, and van der Waals and hydrogen bond interactions are included by using Goj-type potentials.78 As a result, the variation of melting temperature with salt concentration can be described. III.2. Melting of DNA Hairpins. Another approach to simpler models of DNA structures and melting involves considering hairpin structures which have a very small number of base pairs such that a fully atomistic description is possible. In other words, here, we simplify the molecule being studied rather than the theoretical model being used. This issue is especially important for DNA as duplexes shorter than six BP are not stable at room temperature; therefore, there are no experimental studies of duplexes that are of most interest to atomistic-level theory. However, hairpins with as few as one BP can easily be studied. Such hairpin structures have recently been synthesized by Lewis and co-workers,79 and a number of studies of hairpin structures and spectroscopy have been

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Figure 4. Structures of the loop regions for triethylene glycol (EG3), hexaethylene glycol (EG6), and stilbenedicarboxamide (SA) A-tract hairpins.

Figure 5. Equilibrated trimer hairpin structures. (a) EG3-3H, (b) EG6-6H, and (c) SA-6H.

presented.79–84 Some of these studies included the use of thermodynamic models;83 however, the most promising for atomistic studies is a recently designed hairpin in which a short linker causes some base pairs to be deliberately disrupted.85 In that work, the effect of capping the DNA with the following three linkers was investigated: triethylene glycol (EG3), hexaethylene glycol (EG6), and stilbenedicarboxamide (SA). The loop regions are depicted in Figure 4. The base pair regime can be varied over a wide range, but here, we limit our considerations to poly(A)-poly(T) (A-tracts) with three BPs. Previous studies have shown that SA significantly stabilizes the DNA duplex due to its rigidity and π-stacking ability.79 The EG linkers, which are floppier than SA and do not provide π stacking, also stabilize DNA but less significantly than SA.85 The shorter EG3 linker provides less stability than the longer EG6, resulting in an experimental temperature difference equivalent to losing a BP. To simulate these structures with atomistic molecular dynamics, the three species were created using ideal B-form DNA. The EG and SA linkers were added and minimized while restraining the BPs. The system was then neutralized, solvated, and run using Amber 8 with the parm99 force field.86 This force field has been parametrized using the building up approach for bond, angle, dihedral, and van der Waals parameters.87 This implies that all parameters involving the same atoms in the same order and with the same hybridization will have the same value. The partial charges are more dependent on local environment

and so are calculated for each residue separately using the restrained electrostatic potential (RESP) method.88 This parametrization was shown to correctly predict B-form DNA in an aqueous environment. On top of that, the building up approach makes this force field particularly adaptable to perturbed DNA structures such as the ones discussed here. In the present calculations, periodic boundary conditions were employed with a particle mesh Ewald treatment of the long-range electrostatic potential. Each run consisted of slow heating followed by a 1 ns equilibration and finally a 6 ns production run. Three temperatures were investigated, 300, 350, and 400 K. The equilibrated structures at 300 K for all species are shown in Figure 5. The EG3 species in Figure 5a has a partially stacked thymine adjacent to the EG3 residue, while the EG6 and SA species do not display this behavior. The SA equilibrated species retains the most B-DNA type BP regime out of all three species. This might be correlated with the π-stacking ability of SA compared to the nonchromophoric EG6. Figure 6 shows all three structures at the end of simulations at 400 K. Significant deviation from Watson-Crick (WC) base pairing can be seen in all species. Again, the SA species seems to have the least deviation from B-DNA, while EG3 has the most. The EG6 species is almost completely unrecognizable as DNA, with all WC type hydrogen bonds broken. As compared to the EG3 species, the EG6 species seems to have more interstrand π stacking taking place.

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Figure 6. Hairpin structures after a 7 ns trajectory at 400 K. (a) EG3-3H, (b) EG6-6H, and (c) SA-6H.

To quantify the degree of melting, the length of the interstrand hydrogen bonds was monitored. To allow for the possibility that significant deviation from WC base pairing can occur at high temperatures, seven possible hydrogen bonds between each adenine and thymine pair were monitored. The six hydrogen bonds with the shortest distances were then summed, and this value was plotted as a function of time. The plots for each species are shown in Figure 7. The EG3 plot (Figure 7a) shows that while at 300 and 350 K the sum of hydrogen bonds fluctuates around the WC value of ∼12 Å for the entire duration of the trajectory, at 400 K, we get significant deviation from this as a result of the strands separating. The temperature trend is different for EG6 and SA, as seen in Figure 7b and c. Here, we see strand separation for these linkers even at 350 K, but the strand separation is overall smaller than that in Figure 7a. Of course, these results refer to individual trajectories; however, what might be inferred from this is that stability at 300 K does not necessarily dictate what will happen at higher temperature. These results will provide important benchmarks for studies with coarse-grained models. III.3 DNA-Linked Molecules, Polymers, and Nanoparticles. Mirkin and co-workers have developed a method for fabrication of DNA-linked gold nanoparticle aggregates that has been widely used for DNA and protein sensing.25,89,90 These structures have melting (thermal dehybridization) transitions which are much sharper (∼1 °C rather than 10 °C) than those for the same duplexes in solution.2 This property can be used in sensing applications because sharp melting can efficiently be used to discriminate even single BP mismatches; therefore, there has been much interest in the mechanism of this process.10,91–100 In related work, Nguyen and co-workers have synthesized DNA-linked polymer structures that also have sharp melting transitions.26 Both statistical mechanical models and mean-field models have been developed to describe this system.101,102 Common to the models of the DNA-linked gold particles, or DNA-linked polymers, is a primitive description of the structures of the underlying components. For example, in the mean-field model of Kudlay et al.,102 the free energy of the DNA-linked polymer system was described, assuming that the polymers are monodisperse, with a fixed number of DNAs per monomer. The DNAs have no defined structure; instead, they serve as attachment sites whereby the polymers can stick together. The overall free energy is thus the sum of a kinetic energy term associated

with the polymers, a steric term associated with the free volume that is available after polymer and DNA volumes are removed, and enthalpic and entropic terms associated with DNA hybridization. After minimization of the free energy for a given temperature, we obtain limited structural information about the system such as the polymer density and the degree of hybridization. This is useful as it leads to a description of the sharp melting curves resulting from a phase transition from gel to dilute states. However, some quantitative details of the properties underlying the phase transition are incorrect, and a recent experimental test of this mechanism27 has demonstrated that it is missing cooperative interactions between the DNA chains that enhance the sharpness of the melting when dehybridization does not change gel density. Of course, the mean-field model can be (and has now been) modified to include this effect;102 however, this demonstrates the pitfalls associated with simple free-energy functions. The original DNA-linked gold nanoparticle structures were found to be disordered, with either amorphous three-dimensional or fractal structures.103,104 Recently, it has been found that when the DNA linker structure is suitably chosen and the aggregates are formed with an appropriate choice of temperature control, DNA programmable crystal structures can be generated. 105 Figure 8 shows a schematic representation of one structure that was obtained, where we have omitted the large density of singlestranded DNA around each nanoparticle that otherwise makes these particles into fuzzy balls. The structure depicted is a bodycentered cubic structure, which results from the hybridization of a DNA/gold particle system for which each particle has single-stranded DNA that can only hybridize to a complementary partner on another nanoparticle. This “binary” topology is easily produced by using two batches of particles that have cDNAs. This leads to a nonclosest-packed structure, as the thermodynamicminimumforthebinarytopologyisacesium-chloride lattice with eight-fold coordination. The key to obtaining thermodynamically determined structures for the DNA-linked gold particle aggregates is to use DNA linkers that have only a few (4-6) hybridizable linkers. Earlier work was done with 24 or more linkers, and this led to kinetic structures due to the strong driving force for hybridization when the complimentary DNAs were mixed. Indeed, this demonstrates one of the pitfalls of self-assembly modeling, which is that both theory and experiment need to generate thermodynamic minimum-freeenergy structures in order for a meaningful comparison to be

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Figure 8. Schematic of the body-centered-cubic DNA-linked gold nanoparticle crystal structure.

Figure 7. Sum of the six shortest hydrogen bonds between bases for (a) the EG3 trimer hairpin, (b) the EG6 trimer hairpin, and (c) the SA trimer hairpin, as function of time for 300, 350, and 400 K.

expected. The appearance of crystalline structures for DNAlinked gold nanoparticles provides more attractive tests for selfassembly modeling than has been available previously; therefore, we expect it to stimulate significant new work in this field. IV. Lipid Models Lipids provide one of the best examples of self-assembling amphiphiles, typically forming flat bilayer structures as a result

of a combination of hydrophobic interactions between lipid tails and hydrophilic interactions of the head groups with solvent (water) molecules. Atomistic representations of bilayer structures are often generated,106–113 and these have proven useful for understanding the structural properties of lipids, the diffusion of molecules through lipid membranes, and the interaction of lipids with proteins, but the self-assembly of randomly structured lipids into a bilayer structure cannot be simulated due to the large system size needed and relatively long time scales involved. Recently, there has been much interest in modeling the selfassembly of lipid structures using coarse-grained molecular dynamics.48,114–124 For example, Marrink and co-workers48 have used a 4:1 coarse-graining to describe many typical lipids, and they find that it is possible to use molecular dynamics to simulate lipid formation from random starting conditions, with time scales on the order of 100 ns for simulations with a few thousand lipids and comparable coarse-grained waters. The most recent version of their force field, denoted MARTINI,125 includes more interaction energy levels and particle types than the original CG force field48 and was parametrized in a more quantitative way through the analysis of the free energy of hydration, the free energy of vaporization, and the partitioning free energies between water and various organic phases for different CG particles, making comparisons with the corresponding experimental values. The CGMD results have also been compared to all-atom (AA) models for validation. For example, the spontaneous curvature of a lipid bilayer can be evaluated via the stress profile ∑(z) ) 〈Pz - P||(z)〉 across the bilayer. The MARTINI force field picks up the main features of the stress profile quite nicely when compared to the analogous AA result. Marrink and co-workers also calculated the potential of mean force for extracting a lipid from and traversing a lipid through a bilayer using both the AA and CG models and found excellent agreement. The MARTINI force field also gives reasonably good agreement with experiment for properties such as the area per head group, phosphate-phosphate distance, interfacial tension, spontaneous curvature, and the conditions for a lamellar to micellar transition. The interpretation of time scales in CGMD is not straightforward. Compared to AA models, the dynamics observed with CG models are faster due to the smoother energy landscape and hence lower friction. By examining diffusion constants of

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Figure 9. Mapping of the coarse-grained model onto the atomistic representation of C18:0/C10-EPC, C18:1/C10-EPC, DOPG, and water.

both the CG and AA models, the effective time sampled using the MARTINI force field appears to be between 2- and 10-fold larger. To a first approximation, the time axis can be scaled by a factor of 4. For example, water permeation rates across a DPPC membrane and lipid lateral diffusion rates are observed to be in good agreement with experimental measurements after scaling the rates by a factor of 4.23. As an example of the CGMD approach, we consider the phase behavior of mixtures of cationic and anionic phospholipids. The overall goal is to determine which lipid structures are most suitable for preparing lipoplexes (complexes of DNA with lipids) that are effective for delivery of viral DNA to bacterial cells, that is, have a high transfection efficiency. This process is a model for the delivery of DNA as a drug, for example, in gene therapy, where the lipid is used as a protective encapsulant for the DNA until it is delivered to the cell. Experimental results have suggested that lipids which are close to phase transitions from flat membrane (lamellar) phases to high-curvature phases (cubic or hexagonal phases) produce the highest transfection efficiency for lipoplexes that are prepared using these lipids. Presumably, this is because the lipoplexes under these circumstances are relatively unstable and, therefore, exhibit maximum leakiness. However, the correlation of lipid-phase behavior with lipoplex transfection ability is only an empirical observation. As an example of this behavior, Koynova and co-workers126 have found that mixtures of the cationic phospholipid oleoyldecanoyl-ethylphosphatidylcholine (C18:1/C10-EPC) with the anionic phospholipid dioleoylphosphatidylglycerol (DOPG) serve as a remarkably effective transfection agent under physiological conditions. This behavior was not observed in the analogous mixture with the saturated cationic phospholipid stearoyldecanoyl-ethylphosphatidylcholine (C18:0/C10-EPC) which, as apparent from Figure 9, has a more linear tail structure. Using small-angle X-ray diffraction, a 1:1 mixture of C18:1/ C10-EPC and DOPG was observed to undergo a lamellar to nonlamellar phase transition near 333 K, whereas the analogous C18:0/C10-EPC mixture did not. Thus, this provides an example where DNA release is correlated with the proximity to a phase transition. To see if theory supports this effect, we have performed CGMD studies of lipid structures for these lipid mixtures at different temperatures. Adapting from the model of Marrink,125 we have mapped a coarse-grained topology for C18:0/C10-EPC and C18:1/C10-EPC, as depicted in Figure 9. From initial random distributions, we observe formation of a lamellar phase after ∼100 ns in CGMD simulations of both (C18:0/C10-EPC)/ DOPG and (C18:1/C10-EPC)/DOPG mixtures at 298 K. An example from this work in progress is shown in the top panel of Figure 10. However, at higher temperatures, the (C18:1/C10EPC)/DOPG mixture has a tendency to form higher-curvature

Figure 10. Snapshots of a bilayer formed by a 1:1 mixture of C18: 0/C10-EPC and DOPG at 298 K (top) and an inverted hexagonal phase of a 1:1 mixture of C18:1/C10-EPC and DOPG at 333 K (bottom). Both systems include six CG waters per lipid (equivalent to 24 real waters per lipid).

phases such as the inverted hexagonal phase, as shown in the bottom panel of Figure 10. Further work needs to be done to connect this behavior with DNA transfection; however, the CGMD approach shows promise for this class of materials. V. Conclusion The examples provided in this work show that for a fairly diverse set of self-assembly processes, a useful level of coarsegraining is provided by reductions which group approximately four heavy atoms into one coarse-grained bead. Examples where this, or something similar, has been developed include DNA, lipids, and peptides. Hybrid materials in which pieces of each of the components are combined are possible, and this provides significant capabilities for modeling complex systems. This level of description enables the use of relatively conventional force field parametrization, standard molecular dynamics codes, unconstrained molecular motions, and explicit solvent and solvated ions. Time scales of microseconds or more can be described, but of course, this is still not adequate for many self-

Centennial Feature Article assembly processes. Indeed, a major problem even with the coarse-grained molecular dynamics approach is that the driving force for self-assembly is often not strong enough to produce useful structures within feasible simulation times. A number of alternatives to the coarse-grained descriptions were discussed. Atomistic models of self-assembly are extremely useful when they can be used, and there are a number of ways to simplify and accelerate their performance. Monte Carlo methods provide a useful alternative to molecular dynamics, which, if done properly, can completely circumvent the time scale problem. Mean-field models can also circumvent the time scale problem and, in fact, have better capabilities for determining the phase diagrams of complex materials. However, meanfield models capable of describing a broad range of assembly processes with a sophistication similar to what can be done with CGMD, and the simplicity associated with molecular dynamics simulations, have yet to be developed. There are many possible directions for future research in this field. Extending the existing coarse-grained models and meanfield methods to more general classes of molecules and materials will clearly play an important role. Generalizing the coarsegraining process so that finer and coarser levels can be systematically developed is also important; here, we point out the important work of Voth and co-workers in this direction using a force-matching procedure.49 Another direction which has also received some attention49,114,127 is developing multiscale methods that combine coarse-grained and atomic-level descriptions, and from the perspective of self-assembly modeling, a key issue is developing models which systematically remove kinetic barriers, such that the time scale of self-assembly can be greatly reduced. Acknowledgment. This research was supported by the National Science Foundation (Grants CHE-0550497 and CHE0628130) and the Northwestern Center for Cancer Nanobiotechnology Excellence (1 U54 CA119341-01) grant. We also acknowledge a grant of supercomputer time from PNNL and the Network for Computational Nanotechnology. We thank Fred Lewis, Chad Mirkin, Mark Ratner, Samuel Stupp, Julie GibbsDavis, Stefan Tsonchev, Hai Long, Sung Yong Park, Alexander Kudlay, and Hyonseok Hwang for their contributions to parts of the work provided here. References and Notes (1) Stupp, S. I.; Beniash, E.; Hartgerink, J. D.; Sone, E. D. Bio-Implant Interface 2003, 393. (2) Taton, T. A.; Mirkin, C. A.; Letsinger, R. L. Science 2000, 289, 1757. (3) Gazit, E. FEBS J. 2005, 272, 5971. (4) Reif, J. H.; LaBean, T. H.; Sahu, S.; Yan, H.; Yin, P. AdV. Sci. Technol. 2004, 44, 311. (5) Landon, P.; Glosser, R.; Zakhidov, A. Trends Opt. Photonics 2003, 91, 52. (6) Ross, J. Ber. Bunsen-Ges. Phys. Chem. 1985, 89, 605. (7) Ross, J.; Vlad, M. O. Annu. ReV. Phys. Chem. 1999, 50, 51. (8) Epstein, I. R.; Showalter, K. J. Phys. Chem. 1996, 100, 13132. (9) Fialkowski, M.; Bishop, K. J. M.; Klajn, R.; Smoukov, S. K.; Campbell, C. J.; Grzybowski, B. A. J. Phys. Chem. B 2006, 110, 2482. (10) Goodrich, G. P.; Helfrich, M. R.; Overberg, J. J.; Keating, C. D. Langmuir 2004, 20, 10246. (11) Lipowsky, R. Nature 1991, 349, 475. (12) Israelachvili, J. N. Intermolecular and Surface Forces: With Applications to Colloidal and Biological Systems; Academic: New York, 1985. (13) Gompper, G.; Schick, M. Phase Transitions Crit. Phenom. 1994, 16, 1. (14) Drexler, K. E. J. Comput. Theor. Nanosci. 2006, 3, 1. (15) Hamacek, J.; Borkovec, M.; Piguet, C. Dalton Trans. 2006, 1473. (16) Levy, Y.; Onuchic, J. N. Acc. Chem. Res. 2006, 39, 135.

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