Self-consistent field theory of chiral nematic worm-like chains.

Many macromolecules of biological and technological interest are both chiral and semi-flexible. DNA and collagen are good examples. Such molecules often form chiral nematic (or cholesteric) phases, as is well-documented in collagen and chitin. This work presents a method for studying cholesteric phases in the highly successful self-consistent field theory of worm-like chains, offering a new way of studying many biologically relevant molecules. The method involves an effective Hamiltonian with a chiral term inspired by the Oseen-Frank (OF) model of liquid crystals. This method is then used to examine the formation of cholesteric phases in chiral-nematic worm-like chains as a function of polymer flexibility, as well as the optimal cholesteric pitch and distribution of polymer segment orientations. Our approach not only allows for the determination of the isotropic-cholesteric transition and segment distributions, beyond what the OF model promises, but also explicitly incorporates polymer flexibility into the study of the cholesteric phase, offering a more complete understanding of the behavior of semiflexible chiral-nematic polymers.

Collapse transition of a heterogeneous polymer in a crowded medium.

Long chain molecules can be entropically compacted in a crowded medium. We study the compaction transition of a heterogeneous polymer with ring topology by crowding effects in a free or confined space. For this, we use molecular dynamics simulations in which the effects of crowders are taken into account through effective interactions between chain segments. Our parameter choices are inspired by the Escherichia coli chromosome. The polymer consists of small and big monomers; the big monomers dispersed along the backbone are to mimic the binding of RNA polymerases. Our results show that the compaction transition is a two-step process: initial compaction induced by the association (clustering) of big monomers followed by a gradual overall compaction. They also indicate that cylindrical confinement makes the initial transition more effective; for representative parameter choices, the initial compaction accounts for about 60% reduction in the chain size. Our simulation results support the view that crowding promotes clustering of active transcription units into transcription factories.

Characteristic time for the end monomers of a spherically confined polymer to find a nano-pore.

Translocation of a polymer through a nano-pore is relevant in a variety of contexts such as passage of RNAs through a nuclear pore and transportation of proteins across a membrane. An essential step in polymer translocation is for the end monomers to search the pore. This process requires a characteristic time, referred to as the "attempt time" in this work. Here, we study the attempt time τ of a confined polymer inside a spherical surface by combining a scaling approach and Langevin dynamics simulations. For a moderately to strongly confined polymer, our results suggest that τ ∼ R3.67 for R > P and τ ∼ R2.67 for R < P, where R is the radius of the spherical surface and P is the persistence length of the polymer. All simulation data obtained for an intermediate range of the volume fraction of monomers ϕ(≲ 0.2) tend to collapse onto each other. This implies that τ does not explicitly depend on ϕ, in agreement with the theoretical predictions. These results will be useful for interpreting translocation as a two-step process: the initial attempt to find the pore and eventual pore crossing.

Nematic ordering of worm-like polymers near an interface.

The phase behavior of semi-flexible polymers is integral to various contexts, from materials science to biophysics, many of which utilize or require specific confinement geometries as well as the orientational behavior of the polymers. Inspired by collagen assembly, we study the orientational ordering of semi-flexible polymers, modeled as Maier-Saupe worm-like chains, using self-consistent field theory. We first examine the bulk behavior of these polymers, locating the isotropic-nematic transition and delineating the limit of stability of the isotropic and nematic phases. This effort explains how nematic ordering emerges from the isotropic phase and offers insight into how different (e.g., mono-domain vs multi-domain) nematic phases form. We then clarify the influence of planar confinement on the nematic ordering of the polymers. We find that while the presence of a single confining wall does not shift the location of nematic transition, it aligns the polymers in parallel to the wall; above the onset of nematic transition, this preference tends to propagate into the bulk phase. Introducing a second, perpendicular, wall leads to a nematic phase that is parallel to both walls, allowing the ordering direction to be uniquely set by the geometry of the experimental setup. The advantage of wall-confinement is that it can be used to propagate mono-domain nematic phases into the bulk phase.

Toward building a physical model for membrane selectivity of antimicrobial peptides: making a quantitative sense of the selectivity.

Antimicrobial peptides (AMPs) are naturally-occurring peptide antibiotics. AMPs are typically cationic and utilize their electrostatic interactions with the bacterial membrane to selectively attack bacteria. The way they work has inspired a vigorous search for optimized peptides for fighting resistant bacteria. Here, we present a physical model of membrane selectivity of AMPs. The challenge for theoretical modeling of membrane-peptide systems arises from the simultaneous presence of several competing effects, including lipid demixing and peptide-peptide interactions on the membrane surface. We first examine critically a number of models of peptide-membrane interactions and map out one, which incorporates adequately these competing effects as well as the geometry of various regions in membranes, occupied by bound peptides, anionic lipids within the interaction range of each peptide, and those outside this range. This effort leads to a systematically-improved model for peptide selectivity. Using the model, we relate peptide's intrinsic (Ccell-independent) selectivity to an apparent, Ccell-dependent one, and clarify the relative roles of peptide parameters and cell densities in determining their selectivity. This relationship suggests that the selectivity is more sensitive to peptide parameters at low cell densities; as a result, the optimal peptide charge, at which the selectivity is maximized, increases with the cell density in such a manner that this notion becomes less meaningful at high cell densities.

Effects of molecular crowding and confinement on the spatial organization of a biopolymer.

A chain molecule can be entropically collapsed in a crowded medium in a free or confined space. Here, we present a unified view of how molecular crowding collapses a flexible polymer in three distinct spaces: free, cylindrical, and (two-dimensional) slit-like. Despite their seeming disparities, a few general features characterize all these cases, even though the ϕc-dependence of chain compaction differs between the two cases: a > ac and a < ac, where ϕc is the volume fraction of crowders, a is the monomer size, and ac is the crowder size. For a > ac (applicable to a coarse-grained model of bacterial chromosomes), chain size depends on the ratio aϕc/ac, and "full" compaction occurs universally at aϕc/ac ≈ 1; for ac > a (relevant for protein folding), it is controlled by ϕc alone and crowding has a modest effect on chain size in a cellular environment (ϕc ≈ 0.3). Also for a typical parameter range of biological relevance, molecular crowding can be viewed as effectively reducing the solvent quality, independent of confinement.

How are molecular crowding and the spatial organization of a biopolymer interrelated.

In a crowded cellular interior, dissolved biomolecules or crowders exert excluded volume effects on other biomolecules, which in turn control various processes including protein aggregation and chromosome organization. As a result of these effects, a long chain molecule can be phase-separated into a condensed state, redistributing the surrounding crowders. Using computer simulations and a theoretical approach, we study the interrelationship between molecular crowding and chain organization. In a parameter space of biological relevance, the distributions of monomers and crowders follow a simple relationship: the sum of their volume fractions rescaled by their size remains constant. Beyond a physical picture of molecular crowding it offers, this finding explains a few key features of what has been known about chromosome organization in an E. coli cell.

How Cell Concentrations Are Implicated in Cell Selectivity of Antimicrobial Peptides.

Antimicrobial peptides (AMPs) are known to selectively bind to and kill microbes over host cells. Contrary to a conventional view, there is now evidence that AMP's cell selectivity varies with cell densities and is not uniquely determined. Using a coarse-grained model, we study how the cell selectivity of membrane-lytic AMPs, defined as the ratio between their minimum hemolytic (MHC) and minimum inhibitory concentrations (MIC), depends on cell densities or on the way it is measured. A general picture emerging from our study is that the selectivity better captures peptide's intrinsic properties at low cell densities. The selectivity, however, decreases and becomes less intrinsic as the cell density increases, as long as it is chosen to be the same for both types of cells. Importantly, our results show that the selectivity can be excessively overestimated if higher host cell concentrations are used; in contrast, it becomes mistakenly small if measured for a mixture of both types of cells, even with similar choices of cell densities (i.e., higher host cell densities). Our approach can be used as a fitting model for relating the intrinsic selectivity to the apparent (cell-density-dependent) one.

Polymers under confinement: single polymers, how they interact, and as model chromosomes.

How confinement or a physical constraint modifies polymer chains is not only a classical problem in polymer physics but also relevant in a variety of contexts such as single-molecule manipulations, nanofabrication in narrow pores, and modelling of chromosome organization. Here, we review recent progress in our understanding of polymers in a confined (and crowded) space. To this end, we highlight converging views of these systems from computational, experimental, and theoretical approaches, and then clarify what remains to be clarified. In particular, we focus on exploring how cylindrical confinement reshapes individual chains and induces segregation forces between them - by pointing to the relationships between intra-chain organization and chain segregation. In the presence of crowders, chain molecules can be entropically phase-separated into a condensed state. We include a kernel of discussions on the nature of chain compaction by crowders, especially in a confined space. Finally, we discuss the relevance of confined polymers for the nucleoid, an intracellular space in which the bacterial chromosome is tightly packed, in part by cytoplasmic crowders.

A polymer in a crowded and confined space: effects of crowder size and poly-dispersity.

DNA compaction in a bacterial cell is in part carried out by entropic (depletion) forces induced by "free" proteins or crowding particles in the cytoplasm. Indeed, recent in vitro experiments highlight these effects by showing that they alone can condense the E. coli chromosome to its in vivo size. Using molecular dynamics simulations and a theoretical approach, we study how a flexible chain molecule can be compacted by crowding particles with variable sizes in a (cell-like) cylindrical space. Our results show that with smaller crowding agents the compaction occurs at a lower volume fraction but at a larger concentration such that doubling their size is equivalent to increasing their concentration fourfold. Similarly, the effect of polydispersity can be correctly mimicked by adjusting the size of crowders in a homogeneous system. Under different conditions, however, crowding particles can induce chain adsorption onto the cylinder wall, stretching the chain, which would otherwise remain condensed.

Chromosome-like organization of an asymmetrical ring polymer confined in a cylindrical space.

To what extent does a confined polymer show chromosome-like organization? Using molecular dynamics simulations, we study a model Escherichia coli (E. coli) chromosome: an asymmetrical ring polymer, formed by small monomers on one side and big monomers on the other confined in a concentric-shell or simple cylinder with closed ends. The ring polymer is organized in the way observed for the E. coli chromosome: if the big monomers are assumed to be localized in the inner cylinder, the two "subchains" forming the ring are spontaneously partitioned in a parallel orientation with the "body" (big-monomer) chain linearly organized with a desired precision and the crossing (small-monomer) chain residing preferentially in the peripheral region. Furthermore, we show that the introduction of a "fluctuating boundary" between the two subchains leads to a double-peak distribution of ter-proximate loci, as seen in experiments, which would otherwise remain single-peaked. In a simple cylinder, however, a chromosome-like organization of the ring polymer typically requires an external mechanism such as cell-wall attachment. Finally, our results clarify to what degree the spatial organization of the chromosomes can be accomplished solely by ring asymmetry and anisotropic confinement.

Physical manipulation of the Escherichia coli chromosome reveals its soft nature.

Replicating bacterial chromosomes continuously demix from each other and segregate within a compact volume inside the cell called the nucleoid. Although many proteins involved in this process have been identified, the nature of the global forces that shape and segregate the chromosomes has remained unclear because of limited knowledge of the micromechanical properties of the chromosome. In this work, we demonstrate experimentally the fundamentally soft nature of the bacterial chromosome and the entropic forces that can compact it in a crowded intracellular environment. We developed a unique "micropiston" and measured the force-compression behavior of single Escherichia coli chromosomes in confinement. Our data show that forces on the order of 100 pN and free energies on the order of 10(5) k(B)T are sufficient to compress the chromosome to its in vivo size. For comparison, the pressure required to hold the chromosome at this size is a thousand-fold smaller than the surrounding turgor pressure inside the cell. Furthermore, by manipulation of molecular crowding conditions (entropic forces), we were able to observe in real time fast (approximately 10 s), abrupt, reversible, and repeatable compaction-decompaction cycles of individual chromosomes in confinement. In contrast, we observed much slower dissociation kinetics of a histone-like protein HU from the whole chromosome during its in vivo to in vitro transition. These results for the first time provide quantitative, experimental support for a physical model in which the bacterial chromosome behaves as a loaded entropic spring in vivo.

Surface-lattice model describes electrostatic interactions of ions and polycations with bacterial lipopolysaccharides: ion valence and polycation's excluded area.

The bacterial outer membrane (OM) is compositionally distinct and contains polyanionic lipopolysaccharide (LPS) in the outer layer as a main component. It has long been known that the cation-binding ability of LPS is one of the key determinants of OM permeability. Here we present a two-dimensional lattice model of the outer LPS layer, in which the lattice is decorated with bound ions or polycations; while small ions can occupy single binding sites, polycations, modeled as (charged) rods, compete for binding sites through their area exclusion, a consequence of their multisite binding. Our results suggest that in the parameter space of biological relevance, the effect of area exclusion is well-reflected in the competitive binding of Mg(2+) and polycations onto LPS; by reducing the apparent binding affinity of polycations, it enhances Mg(2+) binding. Despite simplifications, our results are generally consistent with the common view of Mg(2+) as OM-stabilizing and polycations as OM-perturbing agents. They will be useful for understanding how cationic antimicrobials can gain entry into the cytoplasmic membrane. We also outline a few strategies for extending our model toward a more realistic modeling of OM permeability.

Electrostatic modification of the lipopolysaccharide layer: competing effects of divalent cations and polycationic or polyanionic molecules.

The outer membrane (OM) of Gram-negative bacteria is asymmetrical with its outer layer mainly populated with polyanionic lipopolysaccharide (LPS). Much empirical evidence shows how OM permeability can be altered electrostatically: if Mg(2+) or divalent cations are required for the integrity of the OM, antimicrobial peptides (AMPs) or ethylene-diaminetetraacetic acid (EDTA) can permeabilize it. Using a coarse-grained model of the outer LPS layer, in which the layer is viewed as forming discrete binding sites for opposite charges, we study how the LPS layer can be modified electrostatically. In particular, we capture systematically ion-pairing and lateral-charge correlations on the LPS layer. Our results offer a clear picture of (competitive) ion binding onto the LPS layer and its impact on the lateral packing of LPS molecules, similarly to what has been seen in experiments: divalent cations such as Mg(2+) not only neutralize the LPS layer but also make its planar charge distribution heterogeneous, thus tightening the LPS layer; on the other hand, polycationic AMPs or polyanionic EDTA can displace Mg(2+) ions from the LPS layer and counteract the favorable effect of Mg(2+). Our result will be useful for clarifying to what extent OM permeability can be modified electrostatically.

Modeling the compaction of bacterial chromosomes by biomolecular crowding and the cross-linking protein H-NS.

Cells orchestrate the action of various molecules toward organizing their chromosomes. Using a coarse-grained computational model, we study the compaction of bacterial chromosomes by the cross-linking protein H-NS and cellular crowders. In this work, H-NS, modeled as a mobile "binder," can bind to a chromosome-like polymer with a characteristic binding energy. The simulation results reported here clarify the relative role of biomolecular crowding and H-NS in condensing a bacterial chromosome in a quantitative manner. In particular, they shed light on the nature and degree of crowder and H-NS synergetics: while the presence of crowders enhances H-NS binding to a chromosome-like polymer, the presence of H-NS makes crowding effects more efficient, suggesting two-way synergetics in chain compaction. Also, the results show how crowding effects promote clustering of bound H-NS. For a sufficiently large concentration of H-NS, the cluster size increases with the volume fraction of crowders.

Modeling selectivity of antimicrobial peptides: how it depends on the presence of host cells and cell density.

Antimicrobial peptides (AMPs), naturally-occurring peptide antibiotics, are known to attack bacteria selectively over the host cells. The emergence of drug-resistant bacteria has spurred much effort in utilizing optimized (more selective) AMPs as new peptide antibiotics. Cell selectivity of these peptides depends on various factors or parameters such as their binding affinity for cell membranes, peptide trapping in cells, peptide coverages on cell membranes required for membrane rupture, and cell densities. In this work, using a biophysical model of peptide selectivity, we show this dependence quantitatively especially for a mixture of bacteria and host cells. The model suggests a rather nontrivial dependence of the selectivity on the presence of host cells, cell density, and peptide trapping. In a typical biological setting, peptide trapping works in favor of host cells; the selectivity increases with increasing host-cell density but decreases with bacterial cell density. Because of the cell-density dependence of peptide activity, the selectivity can be overestimated by two or three orders of magnitude. The model also clarifies how the cell selectivity of AMPs differs from their membrane selectivity.

Interplay between nematic and cholesteric interactions in self-consistent field theory.

Chirality is a design feature of a number of biomolecules (e.g., collagen). In these molecules, cholesteric (chiral-nematic) behavior emerges from a combination of the tendency for the biopolymers to align (nematic interactions) and for the alignment direction to change with position, rotating around an axis normal to the alignment direction. This paper presents self-consistent field theory (SCFT) of chiral-nematic polymers, which takes into account polymer flexibility and the orientational degrees of freedom of polymer segments. Using the resulting SCFT, we construct a phase diagram showing regions of stability for isotropic, nematic, and cholesteric phases. Furthermore, we find that nematic interactions can stabilize the cholesteric phase, pushing the isotropic-cholesteric phase transition to lower cholesteric interaction strength, until the isotropic-nematic-cholesteric triple point is reached.

Modeling the Protective Role of Bacterial Lipopolysaccharides against Membrane-Rupturing Peptides.

Lipopolysaccharide (LPS) is a key surface component of Gram-negative bacteria, populating the outer layer of their outer membrane. A number of experimental studies highlight its protective role against harmful molecules such as antibiotics and antimicrobial peptides (AMPs). In this work, we present a theoretical model for describing the interaction between LPS and cationic antimicrobial peptides, which combines the following two key features. The polysaccharide part is viewed as forming a polymer brush, exerting an osmotic pressure on inclusions such as antimicrobial peptides. The charged groups on LPS (those in lipid A and the two Kdo groups in the inner core) form electrostatic binding sites for cationic AMPs or cations. Using the resulting model, we offer a quantitative picture of how the brush component enhances the protective role of LPS against magainin-like peptides, in the presence of divalent cations such as Mg2+. The LPS brush tends to diminish the interfacial binding of the peptides, at the lipid headgroup region, by about 30%. In the presence of 5 mM of Mg2+, the interfacial binding does not reach a threshold value for wild-type LPS, beyond which the LPS layer is ruptured, even though it does for LPS Re (the simplest form of LPS, lacking the brush part), as long as [AMP] ≤ 20 µM, where [AMP] is the concentration of AMPs. At a low concentration of Mg2+ (≈1 mM), however, a smaller [AMP] value (≳2 µM) is needed to reach the threshold coverage for wild-type LPS. Our results also suggest that the interfacial binding of peptides is insensitive to their possible weak interaction with the surrounding brush chains.

Modeling Cell Selectivity of Antimicrobial Peptides: How Is the Selectivity Influenced by Intracellular Peptide Uptake and Cell Density.

Antimicrobial peptides (AMPs) are known to attack bacteria selectively over their host cells. Many attempts have been made to use them as a template for designing peptide antibiotics for fighting drug-resistant bacteria. A central concept in this endeavor is "peptide selectivity," which measures the "quality" of peptides. However, the relevance of selectivity measurements has often been obscured by the cell-density dependence of the selectivity. For instance, the selectivity can be overestimated if the cell density is larger for the host cell. Furthermore, recent experimental studies suggest that peptide trapping in target bacteria magnifies the cell-density dependence of peptide activity. Here, we propose a biophysical model for peptide activity and selectivity, which assists with the correct interpretation of selectivity measurements. The resulting model shows how cell density and peptide trapping in cells influence peptide activity and selectivity: while these effects can alter the selectivity by more than an order of magnitude, peptide trapping works in favor of host cells at high host-cell densities. It can be used to correct selectivity overestimates.

Electrostatic bending of lipid membranes: how are lipid and electrostatic properties interrelated?

Electrostatic modification of lipid headgroups and its effect on membrane curvature are not only relevant in a variety of contexts such as cell shape transformation and membrane tubulation but also are intriguingly implicated in membrane functions. For instance, the gating (open vs closed) properties of mechanosensitive channels can be influenced by membrane curvature and ion valence. However, a full theoretical description of membrane electrostatics is still lacking; in the past, membrane bending has often been considered under a few assumptions about how bending modifies lipid arrangements and surface charges. Here, we present a unified theoretical approach to spontaneous membrane curvature, C(0), in which lipid properties (e.g., packing shape) and electrostatic effects are self-consistently integrated. For the description of electrostatic interactions, especially between a lipid charge and a divalent counterion, we implement the Poisson-Boltzmann (PB) approach by incorporation of finite ionic sizes, so as to capture both lateral and transverse charge correlations on the membrane surface. Our results show that C(0) is sensitive to the way lipid rearrangements and divalent counterions are modeled. Interestingly, it can change its sign in the presence of divalent counterions, thus stabilizing reverse hexagonal (H(II)) phases. Our results show how electrostatic modification of headgroups influences the preferred structure of lipid aggregates (inverted micelles vs bilayers).