Calculating tracer currents through narrow ion channels: beyond the independent particle model.

Discrete state models of single-file ion permeation through a narrow ion channel pore are employed to analyze the ratio of forward to backward tracer current. Conditions under which the well-known Ussing formula for this ratio hold are explored in systems where ions do not move independently through the channel. Building detailed balance into the rate constants for the model in such a way that under equilibrium conditions (equal rate of forward versus backward permeation events) the Nernst equation is satisfied, it is found that in a model where only one ion can occupy the channel at a time, the Ussing formula is always obeyed for any number of binding sites, reservoir concentrations of the ions and electric potential difference across the membrane which the ion channel spans, independent of the internal details of the permeation pathway. However, numerical analysis demonstrates that when multiple ions can occupy the channel at once, the nonequilibrium forward/backward tracer flux ratio deviates from the prediction of the Ussing model. Assuming an appropriate effective potential experienced by ions in the channel, we provide explicit formulae for the rate constants in these models.

Effects of cross-linking on partitioning of nanoparticles into a polymer brush: Coarse-grained simulations test simple approximate theories.

The effect of cohesive contacts or, equivalently, dynamical cross-linking on the equilibrium morphology of a polymer brush infiltrated by nanoparticles that are attracted to the polymer strands is studied for plane-grafted brushes using coarse-grained molecular dynamics and approximate statistical mechanical models. In particular, the Alexander-de Gennes (AdG) and Strong Stretching Theory (SST) mean-field theory (MFT) models are considered. It is found that for values of the MFT cross-link strength interaction parameter beyond a certain threshold, both AdG and SST models predict that the polymer brush will be in a compact state of nearly uniform density packed next to the grafting surface over a wide range of solution phase nanoparticle concentrations. Coarse grained molecular dynamics simulations confirm this prediction, for both small nanoparticles (nanoparticle volume = monomer volume) and large nanoparticles (nanoparticle volume = 27 × monomer volume). Simulation results for these cross-linked systems are compared with analogous results for systems with no cross-linking. At the same solution phase nanoparticle concentration, strong cross-linking results in additional compression of the brush relative to the non-crosslinked analog and, at all but the lowest concentrations, to a lesser degree of infiltration by nanoparticles. For large nanoparticles, the monomer density profiles show clear oscillations moving outwards from the grafting surface, corresponding to a degree of layering of the absorbed nanoparticles in the brush as they pack against the grafting surface.

Free Energy of Nanoparticle Binding to Multivalent Polymeric Substrates.

Characterization of the interactions between nanosize ligands and polymeric substrates is important for predictive design of nanomaterials and in biophysical applications. The multivalent nature of the polymer-nanoparticle interaction and the dynamics of multiple internal conformations of the polymer chains makes it difficult to infer microscopic interactions from macroscopic binding assays. Using coarse-grained simulations, we estimate the free energy of binding between a nanoparticle and a surface-grafted polymeric substrate as a function of pertinent parameters such as polymer chain length, nanoparticle size, and microscopic polymer-nanoparticle attraction. We also investigate how the presence of the nanoparticle affects the internal configurations of the polymeric substrate, and estimate the entropic cost of binding. The results have important implications for the understanding of complex macromolecular assemblies.

Simple biophysics underpins collective conformations of the intrinsically disordered proteins of the Nuclear Pore Complex.

Nuclear Pore Complexes (NPCs) are key cellular transporter that control nucleocytoplasmic transport in eukaryotic cells, but its transport mechanism is still not understood. The centerpiece of NPC transport is the assembly of intrinsically disordered polypeptides, known as FG nucleoporins, lining its passageway. Their conformations and collective dynamics during transport are difficult to assess in vivo. In vitro investigations provide partially conflicting results, lending support to different models of transport, which invoke various conformational transitions of the FG nucleoporins induced by the cargo-carrying transport proteins. We show that the spatial organization of FG nucleoporin assemblies with the transport proteins can be understood within a first principles biophysical model with a minimal number of key physical variables, such as the average protein interaction strengths and spatial densities. These results address some of the outstanding controversies and suggest how molecularly divergent NPCs in different species can perform essentially the same function.

Protein viscoelastic dynamics: a model system.

A model system inspired by recent experiments on the dynamics of a folded protein under the influence of a sinusoidal force is investigated and found to replicate many of the response characteristics of such a system. The essence of the model is a strongly overdamped oscillator described by a harmonic restoring force for small displacements that reversibly yields to stress under sufficiently large displacement. This simple dynamical system also reveals unexpectedly rich behavior-exhibiting a series of dynamical transitions and analogies with equilibrium thermodynamic phase transitions. The effects of noise and of inertia are briefly considered and described.

A Polymer-Brush-Based Nanovalve Controlled by Nanoparticle Additives: Design Principles.

Polymer-grafted surfaces and channels are increasingly used for the design of responsive materials and sensors due to robust performance and ease of use. Various strategies for the control of the nanoscale morphologies of such materials and devices are being tested. Entropic repulsion between the polymer chains in a grafted brush of sufficient density causes the chains to extend in the direction perpendicular to the grafting surface in comparison to the position of unattached polymers. When nanoparticles having attractive interactions with the polymers are introduced into the solvent, these nanoparticles tend to infiltrate into the brush and reduce its extension. Under certain conditions, a sharp reduction in brush height extension can occur over a narrow range of nanoparticle concentrations in solution. We describe a way of controlling transport through polymer-functionalized nanochannels with nanoparticle additives, relying on the physics of nanoparticles and polymer brushes under confinement, and we suggest a blueprint for the creation of a tunable nanovalve. The nanovalve is modeled as a cylinder with a polymer brush grafted on its inside surface. Brush properties such as the chain length and the grafting density are chosen so that the brush chains extend into the center of the cylinder in the absence of nanoparticles, occluding the flux of analyte molecules through the pore. When nanoparticles that are attracted to the polymers are introduced into solution, they infiltrate into the brush and partially collapse it against the cylindrical grafting surface, opening space in the center of the cylinder through which analyte molecules can flow. The operation of such a nanovalve is analyzed via self-consistent field theory calculations in the strong-stretching approximation. Self-consistent field analysis is supported by Langevin dynamics simulations of the underlying coarse-grained model of the polymer-nanoparticle system.

Morphology of polymer brushes infiltrated by attractive nanoinclusions of various sizes.

Addition of nanoparticles can control the morphologies of grafted polymer layers that are important in a variety of natural and artificial systems. We study the morphologies of grafted polymer layers interacting attractively with nanoparticle inclusions, as a function of particle size and the interaction strength, using self-consistent field theory and Langevin dynamics simulations. We find that the addition of nanoparticles causes distinctive changes in the layer morphology. For sufficiently strong interaction/binding, increasing the concentration of nanoparticles causes a compression of the polymer layer into a compact, low height state, followed by a subsequent rebound and swelling at sufficiently high concentrations. For nanoparticles of small size, the compression of the layer is sharp and occurs over a narrow range of nanoparticle concentrations. The transition region widens as the nanoparticle size increases. The transition is initiated via a dense layer of tightly bound monomers and nanoparticles near the grafting surface, with a low density region above it. For nanoparticles much larger than the characteristic graft spacing in the brush, the behavior is reversed: the nanoparticles penetrate only the dilute region near the top of the polymer layer without causing the layer to collapse.

Morphological control of grafted polymer films via attraction to small nanoparticle inclusions.

Control of the morphologies of polymer films and layers by addition of nanosize particles is a novel technique for design of nanomaterials and is also at the core of some important biological processes. In order to facilitate the analysis of experimental data and enable predictive engineering of such systems, solid theoretical understanding is necessary. We study theoretically and computationally the behavior of plane-grafted polymer layers (brushes) in athermal solvent, decorated with small nanoparticle inclusions, using mean field theory and coarse-grained simulations. We show that the morphology of such layers is very sensitive to the interaction between the polymers and the nanoparticles and to the nanoparticle density. In particular, the mean field model shows that for a certain range of parameters, the nanoparticles induce a sharp transition in the layer height, accompanied by a sharp increase in the number of adsorbed nanoparticles. At other parameter values, the layer height depends smoothly on the nanoparticle concentration. Predictions of the theoretical model are verified by Langevin dynamics simulations. The results of the paper are in qualitative agreement with experiments on in vitro models of biological transport and suggest strategies for morphological control of nanocomposite materials.

Event-driven simulations of a plastic, spiking neural network.

We consider a fully connected network of leaky integrate-and-fire neurons with spike-timing-dependent plasticity. The plasticity is controlled by a parameter representing the expected weight of a synapse between neurons that are firing randomly with the same mean frequency. For low values of the plasticity parameter, the activities of the system are dominated by noise, while large values of the plasticity parameter lead to self-sustaining activity in the network. We perform event-driven simulations on finite-size networks with up to 128 neurons to find the stationary synaptic weight conformations for different values of the plasticity parameter. In both the low- and high-activity regimes, the synaptic weights are narrowly distributed around the plasticity parameter value consistent with the predictions of mean-field theory. However, the distribution broadens in the transition region between the two regimes, representing emergent network structures. Using a pseudophysical approach for visualization, we show that the emergent structures are of "path" or "hub" type, observed at different values of the plasticity parameter in the transition region.

Mean-field theory of a plastic network of integrate-and-fire neurons.

We consider a noise-driven network of integrate-and-fire neurons. The network evolves as result of the activities of the neurons following spike-timing-dependent plasticity rules. We apply a self-consistent mean-field theory to the system to obtain the mean activity level for the system as a function of the mean synaptic weight, which predicts a first-order transition and hysteresis between a noise-dominated regime and a regime of persistent neural activity. Assuming Poisson firing statistics for the neurons, the plasticity dynamics of a synapse under the influence of the mean-field environment can be mapped to the dynamics of an asymmetric random walk in synaptic-weight space. Using a master equation for small steps, we predict a narrow distribution of synaptic weights that scales with the square root of the plasticity rate for the stationary state of the system given plausible physiological parameter values describing neural transmission and plasticity. The dependence of the distribution on the synaptic weight of the mean-field environment allows us to determine the mean synaptic weight self-consistently. The effect of fluctuations in the total synaptic conductance and plasticity step sizes are also considered. Such fluctuations result in a smoothing of the first-order transition for low number of afferent synapses per neuron and a broadening of the synaptic-weight distribution, respectively.

The "weighted ensemble" path sampling method is statistically exact for a broad class of stochastic processes and binning procedures.

The "weighted ensemble" method, introduced by Huber and Kim [Biophys. J. 70, 97 (1996)], is one of a handful of rigorous approaches to path sampling of rare events. Expanding earlier discussions, we show that the technique is statistically exact for a wide class of Markovian and non-Markovian dynamics. The derivation is based on standard path-integral (path probability) ideas, but recasts the weighted-ensemble approach as simple "resampling" in path space. Similar reasoning indicates that arbitrary nonstatic binning procedures, which merely guide the resampling process, are also valid. Numerical examples confirm the claims, including the use of bins which can adaptively find the target state in a simple model.

Efficient and verified simulation of a path ensemble for conformational change in a united-residue model of calmodulin.

The computational sampling of rare, large-scale, conformational transitions in proteins is a well appreciated challenge-for which a number of potentially efficient path-sampling methodologies have been proposed. Here, we study a large-scale transition in a united-residue model of calmodulin using the "weighted ensemble" (WE) approach of Huber and Kim. Because of the model's relative simplicity, we are able to compare our results with brute-force simulations. The comparison indicates that the WE approach quantitatively reproduces the brute-force results, as assessed by considering (i) the reaction rate, (ii) the distribution of event durations, and (iii) structural distributions describing the heterogeneity of the paths. Importantly, the WE method is readily applied to more chemically accurate models, and by studying a series of lower temperatures, our results suggest that the WE method can increase efficiency by orders of magnitude in more challenging systems.

Designing synthetic vesicles that engulf nanoscopic particles.

We examine the interaction of a lipid bilayer membrane with a spherical particle in solution using dissipative particle dynamics, with the aim of controlling the passage of foreign objects into and out of vesicles. Parameters are chosen such that there is a favorable adhesive interaction between the membrane and the particle. Under these conditions, the membrane wraps the particle in a process resembling phagocytosis in biological cells. We find that, for a homogeneous membrane with a uniform attraction to the particle, the membrane is unable to fully wrap the particle when the adhesion strength is below a certain value. This is observed even in the limit of zero membrane tension. When the adhesion strength is increased above the threshold value, the membrane fully wraps the particle. However, the wrapped particle remains tethered to the larger membrane. We next consider an adhesive domain, or raft, in an otherwise nonadhesive membrane. We find that, when the particle is wrapped by the raft, the line tension at the raft interface promotes fission, allowing the wrapped particle to detach from the larger membrane. This mechanism could be used to allow particles to cross a vesicle membrane.

Transition-event durations in one-dimensional activated processes.

Despite their importance in activated processes, transition-event durations--which are much shorter than first passage times--have not received a complete theoretical treatment. The authors therefore study the distribution rhob(t) of durations of transition events over a barrier in a one-dimensional system undergoing overdamped Langevin dynamics. The authors show that rhob(t) is determined by a Fokker-Planck equation with absorbing boundary conditions and obtain a number of results, including (i) the analytic form of the asymptotic short-time transient behavior, which is universal and independent of the potential function; (ii) the first nonuniversal correction to the short-time behavior leading to an estimate of a key physical time scale; (iii) following previous work, a recursive formulation for calculating, exactly, all moments of rhob based solely on the potential function-along with approximations for the distribution based on a small number of moments; and (iv) a high-barrier approximation to the long-time (t-->infinity) behavior of rhob(t). The authors also find that the mean event duration does not depend simply on the barrier-top frequency (curvature) but is sensitive to details of the potential. All of the analytic results are confirmed by transition-path-sampling simulations implemented in a novel way. Finally, the authors discuss which aspects of the duration distribution are expected to be general for more complex systems.

Local control of periodic pattern formation in binary fluids within microchannels.

Simulations show that a local perturbation (a "hot spot") at the inlet of a microchannel can be a source of traveling concentration waves in a binary fluid and can be exploited to create periodic, oscillatory patterns within the channel. We isolate two distinctly different types of traveling periodic patterns and estimate the wavelength of the structures. Our findings provide the first example of how a local perturbation can be used to create well-defined periodic patterns in two-phase flow within a confined geometry and can lead to new routes for creating gradient materials.

Newtonian fluid meets an elastic solid: coupling lattice Boltzmann and lattice-spring models.

We integrate the lattice Boltzmann model (LBM) and lattice spring model (LSM) to capture the coupling between a compliant bounding surface and the hydrodynamic response of an enclosed fluid. We focus on an elastic, spherical shell filled with a Newtonian fluid where no-slip boundary conditions induce the interaction. We calculate the "breathing mode" oscillations for this system and find good agreement with analytical solutions. Furthermore, we simulate the impact of the fluid-filled, elastic shell on a hard wall and on an adhesive surface. Understanding the dynamics of fluid-filled shells, especially near adhesive surfaces, can be particularly important in the design of microcapsules for pharmaceutical and other technological applications. Our studies reveal that the binding of these capsules to specific surfaces can be sensitive to the physical properties of both the outer shell and the enclosed fluid. The integrated LBM-LSM methodology opens up the possibility of accurately and efficiently capturing the dynamic coupling between fluid flow and a compliant bounding surface in a broad variety of systems.

Diffusive intertwining of two fluid phases in chemically patterned microchannels.

Via a coarse-grained model, we simulate the flow of a pressure driven binary AB fluid through a three-dimensional microchannel, which is decorated on both top and bottom with distinct A- and B-like patches. The advection is "frustrated" because A-like patches are placed in the path of the B stream and similarly, B-like patches are placed in the path of the A fluid. A competition between two factors, the advection caused by the imposed flow and the interactions between the confined fluids and the patterned substrates, introduces nonlinearity into the system. This nonlinear behavior gives rise to a temporally periodic state, where the A and B fluids are intertwined. In effect, the simple pattern of chemically distinct patches introduces positive feedback, which is responsible for the instability of the interface separating the injected fluids.

Cohesive energy, stability, and structural transitions in polyelectrolyte bundles.

A lattice of uniformly charged, infinitesimally thin rods decorated with an ordered array of counterions exhibits anomalous behavior as the spacing between the rods is varied. In particular, the counterion lattice undergoes a sequence of structural shearing or "tilting," phase transformations as the spacing between the rods decreases. The potential implications of this behavior with respect to the packaging of biologically relevant polyelectrolytic molecules are commented upon.

Periodic droplet formation in chemically patterned microchannels.

Simulations show that, when a phase-separated binary AB fluid is driven to flow past chemically patterned substrates in a microchannel, the fluid exhibits unique morphological instabilities. For the pattern studied, these instabilities give rise to the simultaneous, periodic formation of monodisperse droplets of A in B and B in A. The system bifurcates between time-independent behavior and different types of regular, nondecaying oscillations in the structural characteristics. The surprisingly complex behavior is observed even in the absence of hydrodynamic interactions and arises from the interplay between the fluid flow and patterned substrate, which introduces nonlinearity into the dynamical system.

Self-assembly of a binary mixture of particles and diblock copolymers.

Using theoretical models, we undertake the first investigation into the synergy and rich phase behavior that emerges when binary particle mixtures are blended with microphase-separating copolymers. We isolate an example of spontaneous hierarchical self-assembly in such hybrid materials, where the system exhibits both nanoscopic ordering of the particles and macroscopic phase transformation in the copolymer matrix. Furthermore, the self-assembly is driven by entropic effects involving all the different components. The results reveal that entropy can be exploited to create highly ordered nanocomposites with potentially unique electronic and photonic properties.