Microscopic mechanisms of cooperative communications within single nanocatalysts.

Catalysis is a method of accelerating chemical reactions that is critically important for fundamental research as well as for industrial applications. It has been recently discovered that catalytic reactions on metal nanoparticles exhibit cooperative effects. The mechanism of these observations, however, remains not well understood. In this work, we present a theoretical investigation on possible microscopic origin of cooperative communications in nanocatalysts. In our approach, the main role is played by positively charged holes on metal surfaces. A corresponding discrete-state stochastic model for the dynamics of holes is developed and explicitly solved. It is shown that the observed spatial correlation lengths are given by the average distances migrated by the holes before they disappear, while the temporal memory is determined by their lifetimes. Our theoretical approach is able to explain the universality of cooperative communications as well as the effect of external electric fields. Theoretical predictions are in agreement with experimental observations. The proposed theoretical framework quantitatively clarifies some important aspects of the microscopic mechanisms of heterogeneous catalysis.

Stochastic dynamics of hairballs in single-polymer growth.

Real-time monitoring of the single-chain growth of synthetic polymers shows that their end-to-end extension during polymerization in living conditions does not increase continuously. Instead, it remains in a non-equilibrium state, exhibiting stochastic wait-and-jump events when one end of the polymer is subjected to a constant force and the other end is clamped. This wait-and-jump observation was attributed to the stochastic formation and unwinding of conformational entanglements, referred to as hairballs, which result from intrachain and non-bonded interactions within the polymer. In this work, we propose a new theoretical approach to investigate the microscopic dynamics of a single hairball formation and unravelling process during single-chain polymerisation. A discrete state stochastic approach is adopted to analyse the respective wait-and-jump events, which provides fully analytical solutions for all dynamic properties under non-equilibrium conditions. Our theory suggests that dynamic conformation fluctuations of the hairball may be responsible for the experimentally observed complex non-exponential behaviour in the waiting times. Excellent quantitative agreements with existing experimental data provide strong support for our theory. Further, using a Monte Carlo simulation approach, we analysed the correlations between the waiting time and extension of polymer in a single jump, which indicates the possibility of more complex dynamics of polymer growth.

Coupling Effects of Electrostatic Interactions and Salt Concentration Gradient in Polymer Translocation through a Nanopore: A Coarse-Grained Molecular Dynamics Simulations Study.

We study the influence of polymer pore interactions and focus on the role played by the concentration gradient of salt in the translocation of polyelectrolytes (PE) through nanopores explicitly using coarse-grained Langevin dynamics simulations. The mean translocation time is calculated by varying the applied voltage, the pH, and the salt concentration gradient. Changing the pH can alter the electrostatic interaction between the protein pore and the polyelectrolyte chain. The polymer pore interaction is weakened by the increase in the strength of the externally applied electric field that drives translocation. Additionally, the screening effect of the salt can reduce the strong charge-charge repulsion between the PE beads which can make translocation faster. The simulation results show there can be antagonistic or synergistic coupling between the salt concentration-induced screening effect and the drift force originating from the salt concentration gradient thereby affecting the translocation time. Our simulation results are explained qualitatively with free energy calculations.

Dynamics of chemical reactions on single nanocatalysts with heterogeneous active sites.

Modern chemical science and industries critically depend on the application of various catalytic methods. However, the underlying molecular mechanisms of these processes still remain not fully understood. Recent experimental advances that produced highly-efficient nanoparticle catalysts allowed researchers to obtain more quantitative descriptions, opening the way to clarify the microscopic picture of catalysis. Stimulated by these developments, we present a minimal theoretical model that investigates the effect of heterogeneity in catalytic processes at the single-particle level. Using a discrete-state stochastic framework that accounts for the most relevant chemical transitions, we explicitly evaluated the dynamics of chemical reactions on single heterogeneous nanocatalysts with different types of active sites. It is found that the degree of stochastic noise in nanoparticle catalytic systems depends on several factors that include the heterogeneity of catalytic efficiencies of active sites and distinctions between chemical mechanisms on different active sites. The proposed theoretical approach provides a single-molecule view of heterogeneous catalysis and also suggests possible quantitative routes to clarify some important molecular details of nanocatalysts.

Dynamics of the protein search for targets on DNA in quorum-sensing cells.

Quorum sensing is a bacterial cell-cell communication process that regulates gene expression. The search and binding of the autoinducer molecule (AHL)-bound LuxR-type proteins to specific sites on DNA in quorum-sensing cells in Gram-negative bacteria is a complex process and has been theoretically investigated based on a discrete-state stochastic approach. It is shown that several factors such as the rate of formation of the AHL-bound LuxR protein within the cells and its dissociation to freely diffusing AHL, the diffusion of the latter in and out of the cells, positive feedback loops, and the cell population density play important roles in the protein target search and can control the gene regulation processes. Physical-chemical arguments to explain these observations are presented. Our calculations of the dynamic properties are also supplemented by Monte Carlo computer simulations. Our theoretical model provides physical insights into the complex mechanisms of protein target search in quorum-sensing cells.

Theoretical insights into the full description of DNA target search by subdiffusing proteins.

DNA binding proteins (DBPs) diffuse in the cytoplasm to recognise and bind with their respective target sites on the DNA to initiate several biologically important processes. The first passage time distributions (FPTDs) of DBPs are useful in quantifying the timescales of the most-probable search paths in addition to the mean value of the distribution which, strikingly, are decades of order apart in time. However, extremely crowded in vivo conditions or the viscoelasticity of the cellular medium among other factors causes biomolecules to exhibit anomalous diffusion which is usually overlooked in most theoretical studies. We have obtained approximate analytical expressions of a general FPTD and the two characteristic timescales that are valid for any single subdiffusing protein searching for its target in vivo. Our results can be applied to single-particle tracking experiments of target search.

Single-Molecule Kinetics of an Enzyme in the Presence of Multiple Substrates.

Herein, the catalytic activity of a single enzyme in the presence of multiple substrates is studied. Three different mechanisms of bisubstrate binding, namely, ordered sequential, random sequential and ping-pong nonsequential pathway, are broadly discussed. By means of the chemical master equation approach, exact expressions for the waiting-time distributions, the mean turnover time and the randomness parameter as a function of the substrate concentration, such that both concentrations are fixed, but one of them is changed quasi-statically are obtained. The randomness parameter is not equal to unity at intermediate to high substrate concentrations, which indicates the presence of multiple rate-limiting steps in the reaction pathway in all three modes of bisubstrate binding. This arises due to transitions between the free enzyme and the enzyme-substrate complexes that occur on comparable timescales. Such turnover statistics of the single enzyme can also distinguish between the different types of bisubstrate binding mechanisms.

A stochastic theoretical approach to study the size-dependent catalytic activity of a metal nanoparticle at the single molecule level.

The catalytic activity of metal nanoparticles is intrinsically heterogeneous due to the heterogeneous distribution of surface catalytic sites and surface restructuring dynamics. Recent advances in single-molecule fluorescence spectroscopy reveal that the rates of product formation and dissociation exhibit size-dependent activities. Here we present a theoretical method to study the size-dependent catalytic activity of a metal nanoparticle using the stochastic approach based on the superposition of renewal processes. We observe that for a single nanoparticle with fewer surface-active catalytic sites, temporal fluctuations in the reaction rate, a phenomenon commonly known as dynamic disorder, are present in both the product formation and product dissociation events. The increase in the number of surface catalytic sites suppresses the effect of dynamic restructuring of the surface, thereby leading to a decrease in dynamic disorder. The proposed formalism provides a theoretical foundation to understand the size-dependent catalytic activity of metal nanoparticles at the single molecule level.

Statistical properties of fluctuating enzymes with dynamic cooperativity using a first passage time distribution formalism.

We study the temporal fluctuations in catalytic rates for single enzyme reactions undergoing slow transitions between two active states. We use a first passage time distribution formalism to obtain the closed-form analytical expressions of the mean reaction time and the randomness parameter for reaction schemes where conformational fluctuations are present between two free enzyme conformers. Our studies confirm that the sole presence of free enzyme fluctuations yields a non Michaelis-Menten equation and can lead to dynamic cooperativity. The randomness parameter, which is a measure of the dynamic disorder in the system, converges to unity at a high substrate concentration. If slow fluctuations are present between the enzyme-substrate conformers (off-pathway mechanism), dynamic disorder is present at a high substrate concentration. Our results confirm that the dynamic disorder at a high substrate concentration is determined only by the slow fluctuations between the enzyme-substrate conformers and the randomness parameter is greater than unity. Slow conformational fluctuations between free enzymes are responsible for the emergence of dynamic cooperativity in single enzymes. Our theoretical findings are well supported by comparison with experimental data on the single enzyme beta-galactosidase.

Modeling the effect of transcriptional noise on switching in gene networks in a genetic bistable switch.

Gene regulatory networks in cells allow transitions between gene expression states under the influence of both intrinsic and extrinsic noise. Here we introduce a new theoretical method to study the dynamics of switching in a two-state gene expression model with positive feedback by explicitly accounting for the transcriptional noise. Within this theoretical framework, we employ a semi-classical path integral technique to calculate the mean switching time starting from either an active or inactive promoter state. Our analytical predictions are in good agreement with Monte Carlo simulations and experimental observations.

Multivalent Salt-Induced Self-Assembly of Amphiphilic Polyelectrolytes of Different Charge Fractions: A Coarse-Grained Molecular Dynamics Simulation Study.

Amphiphilic polymers with both hydrophobic and hydrophilic blocks are of great interest for their potential applications in drug delivery. Their self-assembly behavior in response to environmental factors like ion charge and multivalent salt concentration has been the subject of recent investigation. Our study utilizes coarse-grained molecular dynamics simulations to investigate the aggregation behavior of amphiphilic copolymers upon introducing tetravalent salt at varying charge fractions. We identify a critical concentration, Cs*, where the aggregation number reaches its maximum for each charge fraction, followed by a subsequent decrease at the excessive salt regime. This study reveals distinct morphological transitions in response to increasing salt concentration and decreasing charged fractions, namely, (i) stable dispersed micelles, (ii) a singular micelle comprising all copolymer chains, and (iii) redispersed micelles, particularly evident at lower charged fractions. Our study highlights the significant influence of tetravalent salt and charge fractions of polyelectrolyte chains on the self-assembly behavior of polyelectrolyte copolymers.

Macromolecular Crowding Facilitates ssDNA Capture within Biological Nanopores: Role of Size Variation and Solution Heterogeneity.

Genetic sequencing is a vital process that requires the transport of charged nucleic acids through transmembrane nanopores. Single-molecule studies show that macromolecular bulk crowding facilitates the capture of these polymers, leading to a high throughput of nanopore sensors. Motivated by these observations, a minimal discrete-state stochastic framework was developed to describe the role of poly(ethylene glycol) (PEG) crowders in varying concentrations in the transport of ssDNA through α-hemolysin nanopores. This theory suggested that the cooperative partitioning of polycationic PEGs controls the capture of ssDNA due to underlying electrostatic interactions. Herein, we investigate the impact of the size variation of PEGs on the capture event. Even though larger crowders attract ssDNA strongly to enhance its capture, our results show that considerable cooperative partitioning of PEGs is also required to achieve high interevent frequency. The exact analytical results are supported by existing single-molecule studies. Since real cellular conditions are heterogeneous, its influence on the ssDNA capture rate is studied by introducing a binary mixture of crowders. Our results indicate that the "polymer-pushing-polymer" concept possibly affects the capture rate depending on the mixture composition. These new findings provide valuable insights into the microscopic mechanism of the capture process, which eventually allows for accurate genome sequencing in crowded solutions.

Microscopic Mechanism of Macromolecular Crowder-Assisted DNA Capture and Translocation through Biological Nanopores.

Biological nanopore sensors are widely used for genetic sequencing as nucleic acids and other molecules translocate through them across membranes. Recent studies have shown that the transport of these polymers through nanopores is strongly influenced by macromolecular bulk crowders. By using poly(ethylene glycol) (PEG) molecules as crowders, experiments have shown an increase in the capture rates and translocation times of polymers through an α-hemolysin (αHL) nanopore, which provides high-throughput signals and accurate sensing. A clear molecular-level understanding of how the presence of PEGs offers such desirable outcomes in nanopore sensing is still missing. In this work, we present a new theoretical approach to probe the effect of PEG crowders on DNA capture and translocation through the αHL nanopore. We develop an exactly solvable discrete-state stochastic model based on the cooperative partitioning of individual polycationic PEGs within the cavity of the αHL nanopore. It is argued that the apparent electrostatic interactions between the DNA and PEGs control all of the dynamic processes. Our analytical predictions find excellent agreements with existing experiments, thereby strongly supporting our theory.

How Heterogeneity Affects Cooperative Communications within Single Nanocatalysts.

Catalysis remains one of the most essential methods in chemical research and industry. Recent experiments have discovered an unusual phenomenon of catalytic cooperativity, when a reaction at one active site can stimulate reactions at neighboring sites within single nanoparticles. While theoretical analysis established that the transport of charged holes is responsible for this phenomenon, it does not account for inhomogeneity in the structural and dynamic properties of single nanocatalysts. Here, we investigate the effect of heterogeneity on catalytic communications by extending a discrete-state stochastic framework to random distributions of the transition rates. Our explicit calculations of spatial and temporal properties of heterogeneous systems in comparison with homogeneous systems predict that the strength of cooperativity increases, while the communication lifetimes and distances decrease. Monte Carlo computer simulations support theoretical calculations, and microscopic arguments to explain these observations are also presented. Our theoretical analysis clarifies some important aspects of molecular mechanisms of catalytic processes.

Influence of Nonspecific Interactions between Proteins and In Vivo Cytoplasmic Crowders in Facilitated Diffusion of Proteins: Theoretical Insights.

The binding of proteins to their respective specific sites on the DNA through facilitated diffusion serves as the initial step of various important biological processes. While this search process has been thoroughly investigated via in vitro studies, the cellular environment is complex and may interfere with the protein's search dynamics. The cytosol is heavily crowded, which can potentially modify the search by nonspecifically interacting with the protein that has been mostly overlooked. In this work, we probe the target search dynamics in the presence of explicit crowding agents that have an affinity toward the protein. We theoretically investigate the role of such protein-crowder associations in the target search process using a discrete-state stochastic framework that allows for the analytical description of dynamic properties. It is found that stronger nonspecific associations between the crowder and proteins can accelerate the facilitated diffusion of proteins in comparison with a purely inert, rather weakly interacting cellular environment. This effect depends on how strong these associations are, the spatial positions of the target with respect to the crowders, and the size of the crowded region. Our theoretical results are also tested with Monte Carlo computer simulations. Our predictions are in qualitative agreement with existing experimental observations and computational studies.

A single-molecule stochastic theory of protein-ligand binding in the presence of multiple unfolding/folding and ligand binding pathways.

Protein folding is a biophysical process by which a protein chain is translated to its native (folded) structure through several intermediate states such that the folded conformation becomes biologically functional. This folded protein can again exist in multiple conformations in its native state and its intrinsic conformational fluctuations are responsible for the protein-ligand recognition and binding to form a specific complex. In this study, we introduce an exactly solvable kinetic model based on a discrete stochastic approach to study the protein-ligand binding by taking into account an arbitrary number of the transient intermediates between the unfolded and the native folded state of the protein. We also examine the conformational fluctuations in the folded state explicitly. The dynamic properties of the system are explicitly evaluated to understand the role of short-lived conformations in the process of protein folding and also conformational fluctuations existing in the folded state of the protein.

Theoretical Tools to Quantify Stochastic Fluctuations in Single-Molecule Catalysis by Enzymes and Nanoparticles.

Single-molecule microscopic techniques allow the counting of successive turnover events and the study of the time-dependent fluctuations of the catalytic activities of individual enzymes and different sites on a single heterogeneous nanocatalyst. It is important to establish theoretical methods to obtain the statistical measurements of such stochastic fluctuations that provide insight into the catalytic mechanism. In this review, we discuss a few theoretical frameworks for evaluating the first passage time distribution functions using a self-consistent pathway approach and chemical master equations, to establish a connection with experimental observables. The measurable probability distribution functions and their moments depend on the molecular details of the reaction and provide a way to quantify the molecular mechanisms of the reaction process. The statistical measurements of these fluctuations should provide insight into the enzymatic mechanism.

Simulation Study of the Conformational Properties of Diblock Polyelectrolytes in Salt Solutions.

Coarse-grained molecular dynamics simulations are performed to understand the behavior of diblock polyelectrolytes in solutions of divalent salt by studying the conformations of chains over a wide range of salt concentrations. The polymer molecules are modeled as bead spring chains with different charged fractions and the counterions and salt ions are incorporated explicitly. Upon addition of a divalent salt, the salt cations replace the monovalent counterions, and the condensation of divalent salt cations onto the polyelectrolyte increases, and the chains favor to collapse. The condensation of ions changes with the salt concentration and depends on the charged fraction. Also, the degree of collapse at a given salt concentration changes with the increasing valency of the counterion due to the bridging effect. As a quantitative measure of the distribution of counterions around the polyelectrolyte chain, we study the radial distribution function between monomers on different polyelectrolytes and the counterions inside the counterion worm surrounding a polymer chain at different concentrations of the divalent salt. Our simulation results show a strong dependence of salt concentration on the conformational properties of diblock copolymers and indicate that it can tune the self-assembly behaviors of such charged polyelectrolyte block copolymers.

Effect of Memory and Inertial Contribution on Transition-Time Distributions: Theory and Simulations.

Transition paths refer to the time taken by molecules to cross a barrier separating two molecular conformations. In this work, we study how memory, as well as inertial contribution in the dynamics along a reaction coordinate, can affect the distribution of the transition-path time. We use a simple model of dynamics governed by a generalized Langevin equation with a power-law memory along with the inertial term, which was neglected in previous studies, where memory effects were explored only in the overdamped limit. We derive an approximate expression for the transit-time distribution and discuss our results for the short- and long-time limits and also compare it with known results in the high friction (overdamped) limit as well as in the Markovian limit. We have developed a numerical algorithm to test our theoretical results against extensive numerical simulations.

Understanding the Reaction Dynamics on Heterogeneous Catalysts Using a Simple Stochastic Approach.

Recent experimental advances on investigating nanoparticle catalysts with multiple active sites provided a large amount of quantitative information on catalytic processes. These observations stimulated significant theoretical efforts, but the underlying molecular mechanisms are still not well-understood. We introduce a simple theoretical method to analyze the reaction dynamics on catalysts with multiple active sites based on a discrete-state stochastic description and obtain a comprehensive description of the dynamics of chemical reactions on such catalysts. We explicitly determine how the dynamics of catalyzed chemical reactions depend on the number of active sites, on the number of intermediate chemical transitions, and on the topology of underlying chemical reactions. It is argued that the theory provides quantitative bounds for realistic dynamic properties of catalytic processes that can be directly applied to analyze the experimental observations. In addition, this theoretical approach clarifies several important aspects of the molecular mechanisms of chemical reactions on catalysts.