Short-Time Infrequent Metadynamics for Improved Kinetics Inference

Infrequent Metadynamics is a popular method to obtain the rates of long time-scale processes from accelerated simulations. The inference procedure is based on rescaling the first-passage times of the Metadynamics trajectories using a bias-dependent acceleration factor. While useful in many cases, it is limited to Poisson kinetics, and a reliable estimation of the unbiased rate requires slow bias deposition and prior knowledge of efficient collective variables. Here, we propose an improved inference scheme, which is based on two key observations: (1) the time-independent rate of Poisson processes can be estimated using short trajectories only. (2) Short trajectories experience minimal bias, and their rescaled first-passage times follow the unbiased distribution even for relatively high deposition rates and suboptimal collective variables. Therefore, by basing the inference procedure on short time scales, we obtain an improved trade-off between speedup and accuracy at no additional computational cost, especially when employing suboptimal collective variables. We demonstrate the improved inference scheme for a model system and two molecular systems.

Enzymatic Activity Profiling Using an Ultrasensitive Array of Chemiluminescent Probes for Bacterial Classification and Characterization

Identification and characterization of bacterial species in clinical and industrial settings necessitate the use of diverse, labor-intensive, and time-consuming protocols as well as the utilization of expensive and high-maintenance equipment. Furthermore, while cutting-edge identification technologies such as mass spectrometry and PCR are highly effective in identifying bacterial pathogens, they fall short in providing additional information for identifying bacteria not present in the databases upon which these methods rely. In response to these challenges, we present a robust and general approach to bacterial identification based on their unique enzymatic activity profiles. This method delivers results within 90 min, utilizing an array of highly sensitive and enzyme-selective chemiluminescent probes. Leveraging our recently developed technology of chemiluminescent luminophores, which emit light under physiological conditions, we have crafted an array of probes designed to rapidly detect various bacterial enzymatic activities. The array includes probes for detecting resistance to the important and large class of β-lactam antibiotics. The analysis of chemiluminescent fingerprints from a diverse range of prominent bacterial pathogens unveiled distinct enzymatic activity profiles for each strain. The reported universally applicable identification procedure offers a highly sensitive and expeditious means to delineate bacterial enzymatic activity fingerprints. This opens new avenues for characterizing and identifying pathogens in research, clinical, and industrial applications.

Combining stochastic resetting with Metadynamics to speed-up molecular dynamics simulations

Metadynamics is a powerful method to accelerate molecular dynamics simulations, but its efficiency critically depends on the identification of collective variables that capture the slow modes of the process. Unfortunately, collective variables are usually not known a priori and finding them can be very challenging. We recently presented a collective variables-free approach to enhanced sampling using stochastic resetting. Here, we combine the two methods, showing that it can lead to greater acceleration than either of them separately. We also demonstrate that resetting Metadynamics simulations performed with suboptimal collective variables can lead to speedups comparable with those obtained with optimal collective variables. Therefore, applying stochastic resetting can be an alternative to the challenging task of improving suboptimal collective variables, at almost no additional computational cost. Finally, we propose a method to extract unbiased mean first-passage times from Metadynamics simulations with resetting, resulting in an improved tradeoff between speedup and accuracy. This work enables combining stochastic resetting with other enhanced sampling methods to accelerate a broad range of molecular simulations.

Microscopic theory of adsorption kinetics

Adsorption is the accumulation of a solute at an interface that is formed between a solution and an additional gas, liquid, or solid phase. The macroscopic theory of adsorption dates back more than a century and is now well-established. Yet, despite recent advancements, a detailed and self-contained theory of single-particle adsorption is still lacking. Here, we bridge this gap by developing a microscopic theory of adsorption kinetics, from which the macroscopic properties follow directly. One of our central achievements is the derivation of the microscopic version of the seminal Ward-Tordai relation, which connects the surface and subsurface adsorbate concentrations via a universal equation that holds for arbitrary adsorption dynamics. Furthermore, we present a microscopic interpretation of the Ward-Tordai relation that, in turn, allows us to generalize it to arbitrary dimension, geometry, and initial conditions. The power of our approach is showcased on a set of hitherto unsolved adsorption problems to which we present exact analytical solutions. The framework developed herein sheds fresh light on the fundamentals of adsorption kinetics, which opens new research avenues in surface science with applications to artificial and biological sensing and to the design of nano-scale devices.

Stochastic Resetting for Enhanced Sampling

We present a method for enhanced sampling of molecular dynamics simulations using stochastic resetting. Various phenomena, ranging from crystal nucleation to protein folding, occur on time scales that are unreachable in standard simulations. They are often characterized by broad transition time distributions, in which extremely slow events have a non-negligible probability. Stochastic resetting, i.e., restarting simulations at random times, was recently shown to significantly expedite processes that follow such distributions. Here, we employ resetting for enhanced sampling of molecular simulations for the first time. We show that it accelerates long time scale processes by up to an order of magnitude in examples ranging from simple models to a molecular system. Most importantly, we recover the mean transition time without resetting, which is typically too long to be sampled directly, from accelerated simulations at a single restart rate. Stochastic resetting can be used as a standalone method or combined with other sampling algorithms to further accelerate simulations.

Diffusion with partial resetting

Inspired by many examples in nature, stochastic resetting of random processes has been studied extensively in the past decade. In particular, various models of stochastic particle motion were considered where, upon resetting, the particle is returned to its initial position. Here we generalize the model of diffusion with resetting to account for situations where a particle is returned only a fraction of its distance to the origin, e.g., half way. We show that this model always attains a steady-state distribution which can be written as an infinite sum of independent, but not identical, Laplace random variables. As a result, we find that the steady-state transitions from the known Laplace form which is obtained in the limit of full resetting to a Gaussian form, which is obtained close to the limit of no resetting. A similar transition is shown to be displayed by drift diffusion whose steady state can also be expressed as an infinite sum of independent random variables. Finally, we extend our analysis to capture the temporal evolution of drift diffusion with partial resetting, providing a bottom-up probabilistic construction that yields a closed-form solution for the time-dependent distribution of this process in Fourier-Laplace space. Possible extensions and applications of diffusion with partial resetting are discussed.

Mitigating long queues and waiting times with service resetting

What determines the average length of a queue, which stretches in front of a service station? The answer to this question clearly depends on the average rate at which jobs arrive at the queue and on the average rate of service. Somewhat less obvious is the fact that stochastic fluctuations in service and arrival times are also important, and that these are a major source of backlogs and delays. Strategies that could mitigate fluctuations-induced delays are, thus in high demand as queue structures appear in various natural and man-made systems. Here, we demonstrate that a simple service resetting mechanism can reverse the deleterious effects of large fluctuations in service times, thus turning a marked drawback into a favorable advantage. This happens when stochastic fluctuations are intrinsic to the server, and we show that service resetting can then dramatically cut down average queue lengths and waiting times. Remarkably, this strategy is also useful in extreme situations where the variance, and possibly even mean, of the service time diverge-as resetting can then prevent queues from "blowing up." We illustrate these results on the M/G/1 queue in which service times are general and arrivals are assumed to be Markovian. However, the main results and conclusions coming from our analysis are not specific to this particular model system. Thus, the results presented herein can be carried over to other queueing systems: in telecommunications, via computing, and all the way to molecular queues that emerge in enzymatic and metabolic cycles of living organisms.

Gated reactions in discrete time and space

How much time does it take for two molecules to react? If a reaction occurs upon contact, the answer to this question boils down to the classic first-passage time problem: find the time it takes for the two molecules to meet. However, this is not always the case as molecules switch stochastically between reactive and non-reactive states. The reaction is then said to be "gated" by the internal states of the molecules involved, which could have a dramatic influence on kinetics. A unified, continuous-time, approach to gated reactions on networks was presented in a recent paper [Scher and Reuveni, Phys. Rev. Lett. 127, 018301 (2021)]. Here, we build on this recent advancement and develop an analogous discrete-time version of the theory. Similar to continuous-time, we employ a renewal approach to show that the gated reaction time can always be expressed in terms of the corresponding ungated first-passage and return times, which yields formulas for the generating function of the gated reaction-time distribution and its corresponding mean and variance. In cases where the mean reaction time diverges, we show that the long-time asymptotics of the gated problem is inherited from its ungated counterpart. However, when molecules spend most of their time non-reactive, an interim regime of slower power-law decay emerges prior to the terminal asymptotics. The discretization of time also gives rise to resonances and anti-resonances, which were absent from the continuous-time picture. These features are illustrated using two case studies that also demonstrate how the general approach presented herein greatly simplifies the analysis of gated reactions.

Unified Approach to Gated Reactions on Networks

For two molecules to react they first have to meet. Yet, reaction times are rarely on par with the first-passage times that govern such molecular encounters. A prime reason for this discrepancy is stochastic transitions between reactive and nonreactive molecular states, which results in effective gating of product formation and altered reaction kinetics. To better understand this phenomenon we develop a unifying approach to gated reactions on networks. We first show that the mean and distribution of the gated reaction time can always be expressed in terms of ungated first-passage and return times. This relation between gated and ungated kinetics is then explored to reveal universal features of gated reactions. The latter are exemplified using a diverse set of case studies which are also used to expose the exotic kinetics that arises due to molecular gating.

Resetting transition is governed by an interplay between thermal and potential energy

A dynamical process that takes a random time to complete, e.g., a chemical reaction, may either be accelerated or hindered due to resetting. Tuning system parameters, such as temperature, viscosity, or concentration, can invert the effect of resetting on the mean completion time of the process, which leads to a resetting transition. Although the resetting transition has been recently studied for diffusion in a handful of model potentials, it is yet unknown whether the results follow any universality in terms of well-defined physical parameters. To bridge this gap, we propose a general framework that reveals that the resetting transition is governed by an interplay between the thermal and potential energy. This result is illustrated for different classes of potentials that are used to model a wide variety of stochastic processes with numerous applications.

Experimental Realization of Diffusion with Stochastic Resetting

Stochastic resetting is prevalent in natural and man-made systems, giving rise to a long series of nonequilibrium phenomena. Diffusion with stochastic resetting serves as a paradigmatic model to study these phenomena, but the lack of a well-controlled platform by which this process can be studied experimentally has been a major impediment to research in the field. Here, we report the experimental realization of colloidal particle diffusion and resetting via holographic optical tweezers. We provide the first experimental corroboration of central theoretical results and go on to measure the energetic cost of resetting in steady-state and first-passage scenarios. In both cases, we show that this cost cannot be made arbitrarily small because of fundamental constraints on realistic resetting protocols. The methods developed herein open the door to future experimental study of resetting phenomena beyond diffusion.

Ribosome Composition Maximizes Cellular Growth Rates in E. coli

Bacterial ribosomes are composed of one-third protein and two-thirds RNA by mass. The predominance of RNA is often attributed to a primordial RNA world, but why exactly two-thirds remains a long-standing mystery. Here we present a quantitative analysis, based on the kinetics of ribosome self-replication, demonstrating that the 1∶2 protein-to-RNA mass ratio uniquely maximizes cellular growth rates in E. coli. A previously unrecognized growth law, and an invariant of bacterial growth, also follow from our analysis. The growth law reveals that the ratio between the number of ribosomes and the number of polymerases making ribosomal RNA is proportional to the cellular doubling time. The invariant is conserved across growth conditions and specifies how key microscopic parameters in the cell, such as transcription and translation rates, are coupled to cellular physiology. Quantitative predictions from the growth law and invariant are shown to be in excellent agreement with E. coli data despite having no fitting parameters. Our analysis can be readily extended to other bacteria once data become available.

Constant gradient FEXSY: A time-efficient method for measuring exchange

Filter-Exchange NMR Spectroscopy (FEXSY) is a method for measurement of apparent transmembranal water exchange rates. The experiment is comprised of two co-linear sequential pulsed-field gradient (PFG) blocks, separated by a mixing period in which exchange takes place. The first block remains constant and serves as a diffusion-based filter that removes signal coming from fast-diffusing water. The mixing time and the gradient area (q-value) of the second block are varied on repeated iterations to produce a 2D data set that is analyzed using a bi-compartmental model which assumes that intra- and extra-cellular water are slow and fast diffusing, respectively. Here we suggest a variant of the FEXSY method in which measurements for different mixing times are taken at a constant gradient. This Constant Gradient FEXSY (CG-FEXSY) allows for the determination of the exchange rate by using a smaller 1D data set, so that the same information can be gathered during a considerably shorter scan time. Furthermore, in the limit of high diffusion weighting, such that the extra-cellular water signal is removed while the intra-cellular signal is retained, CG-FEXSY also allows for determination of the intra-cellular mean residence time (MRT). The theoretical results are validated on a living yeast cells sample and on a fixed porcine optic nerve, where the values obtained from the two methods are shown to be in agreement.

Time-dependent density of diffusion with stochastic resetting is invariant to return speed

The canonical Evans-Majumdar model for diffusion with stochastic resetting to the origin assumes that resetting takes zero time: upon resetting the diffusing particle is teleported back to the origin to start its motion anew. However, in reality getting from one place to another takes a finite amount of time which must be accounted for as diffusion with resetting already serves as a model for a myriad of processes in physics and beyond. Here we consider a situation where upon resetting the diffusing particle returns to the origin at a finite (rather than infinite) speed. This creates a coupling between the particle's random position at the moment of resetting and its return time, and further gives rise to a nontrivial cross-talk between two separate phases of motion: the diffusive phase and the return phase. We show that each of these phases relaxes to the steady state in a unique manner; and while this could have also rendered the total relaxation dynamics extremely nontrivial, our analysis surprisingly reveals otherwise. Indeed, the time-dependent distribution describing the particle's position in our model is completely invariant to the speed of return. Thus, whether returns are slow or fast, we always recover the result originally obtained for diffusion with instantaneous returns to the origin.

Gumbel central limit theorem for max-min and min-max

The max-min and min-max of matrices arise prevalently in science and engineering. However, in many real-world situations the computation of the max-min and min-max is challenging as matrices are large and full information about their entries is lacking. Here we take a statistical-physics approach and establish limit laws-akin to the central limit theorem-for the max-min and min-max of large random matrices. The limit laws intertwine random-matrix theory and extreme-value theory, couple the matrix dimensions geometrically, and assert that Gumbel statistics emerge irrespective of the matrix entries' distribution. Due to their generality and universality, as well as their practicality, these results are expected to have a host of applications in the physical sciences and beyond.

Poisson-process limit laws yield Gumbel max-min and min-max

"A chain is only as strong as its weakest link" says the proverb. But what about a collection of statistically identical chains: How long till all chains fail? The answer to this question is given by the max-min of a matrix whose (i,j) entry is the failure time of link j of chain i: take the minimum of each row, and then the maximum of the rows' minima. The corresponding min-max is obtained by taking the maximum of each column, and then the minimum of the columns' maxima. The min-max applies to the storage of critical data. Indeed, consider multiple backup copies of a set of critical data items, and consider the (i,j) matrix entry to be the time at which item j on copy i is lost; then, the min-max is the time at which the first critical data item is lost. In this paper we address random matrices whose entries are independent and identically distributed random variables. We establish Poisson-process limit laws for the row's minima and for the columns' maxima. Then, we further establish Gumbel limit laws for the max-min and for the min-max. The limit laws hold whenever the entries' distribution has a density, and yield highly applicable approximation tools and design tools for the max-min and min-max of large random matrices. A brief of the results presented herein is given in: Gumbel central limit theorem for max-min and min-max [Eliazar, Metzler, and Reuveni, Phys. Rev. E 100, 020104 (2019)10.1103/PhysRevE.100.020104].

Occupancy correlations in the asymmetric simple inclusion process

The asymmetric simple inclusion process (ASIP)-a lattice-gas model for unidirectional transport with irreversible aggregation-has been proposed as an inclusion counterpart of the asymmetric simple exclusion process and as a batch service counterpart of the tandem Jackson network. To date, the analytical tractability of the model has been limited: while the average particle density in the model is easy to compute, very little is known about the joint occupancy distribution. To partially bridge this gap, we study occupancy correlations in the ASIP. We take an analytical approach to this problem and derive an exact formula for the covariance matrix of the steady-state occupancy vector. We verify the validity of this formula numerically in small ASIP systems, where Monte Carlo simulations can provide reliable estimates for correlations in reasonable time, and further use it to draw a comprehensive picture of spatial occupancy correlations in ASIP systems of arbitrary size.

Light-Controlled Selective Collection-and-Release of Biomolecules by an On-Chip Nanostructured Device

The analysis of biosamples, e.g., blood, is a ubiquitous task of proteomics, genomics, and biosensing fields; yet, it still faces multiple challenges, one of the greatest being the selective separation and detection of target proteins from these complex biosamples. Here, we demonstrate the development of an on-chip light-triggered reusable nanostructured selective and quantitative protein separation and preconcentration platform for the direct analysis of complex biosamples. The on-chip selective separation of required protein analytes from raw biosamples is performed using antibody-photoacid-modified Si nanopillars vertical arrays (SiNPs) of ultralarge binding surface area and enormously high binding affinity, followed by the light-controlled rapid release of the tightly bound target proteins in a controlled liquid media. Two important experimental observations are presented: (1) the first demonstration on the control of biological reaction binding affinity by the nanostructuring of the capturing surface, leading to highly efficient protein collection capabilities, and (2) the light-triggered switching of the highly sticky binding surfaces into highly reflective nonbinding surfaces, leading to the rapid and quantitative release of the originally tightly bound protein species. Both of these two novel behaviors were theoretically and experimentally investigated. Importantly, this is the first demonstration of a three-dimensional (3D) SiNPs on-chip filter with ultralarge binding surface area and reversible light-controlled quantitative release of adsorbed biomolecules for direct purification of blood samples, able to selectively collect and separate specific low abundant proteins, while easily removing unwanted blood components (proteins, cells) and achieving desalting results, without the requirement of time-consuming centrifugation steps, the use of desalting membranes, or affinity columns.

First Passage under Restart with Branching

First passage under restart with branching is proposed as a generalization of first passage under restart. Strong motivation to study this generalization comes from the observation that restart with branching can expedite the completion of processes that cannot be expedited with simple restart; yet a sharp and quantitative formulation of this statement is still lacking. We develop a comprehensive theory of first passage under restart with branching. This reveals that two widely applied measures of statistical dispersion-the coefficient of variation and the Gini index-come together to determine how restart with branching affects the mean completion time of an arbitrary stochastic process. The universality of this result is demonstrated and its connection to extreme value theory is also pointed out and explored.

Single-molecule theory of enzymatic inhibition

The classical theory of enzymatic inhibition takes a deterministic, bulk based approach to quantitatively describe how inhibitors affect the progression of enzymatic reactions. Catalysis at the single-enzyme level is, however, inherently stochastic which could lead to strong deviations from classical predictions. To explore this, we take the single-enzyme perspective and rebuild the theory of enzymatic inhibition from the bottom up. We find that accounting for multi-conformational enzyme structure and intrinsic randomness should strongly change our view on the uncompetitive and mixed modes of inhibition. There, stochastic fluctuations at the single-enzyme level could make inhibitors act as activators; and we state—in terms of experimentally measurable quantities—a mathematical condition for the emergence of this surprising phenomenon. Our findings could explain why certain molecules that inhibit enzymatic activity when substrate concentrations are high, elicit a non-monotonic dose response when substrate concentrations are low.

Single molecule approaches demonstrated that enzymatic catalysis is stochastic which could lead to deviations from classical predictions. Here authors rebuild the theory of enzymatic inhibition to show that stochastic fluctuations on the single enzyme level could make inhibitors act as activators.

Ribosomes are optimized for autocatalytic production

Many fine-scale features of ribosomes have been explained in terms of function, revealing a molecular machine that is optimized for error-correction, speed and control. Here we demonstrate mathematically that many less well understood, larger-scale features of ribosomes-such as why a few ribosomal RNA molecules dominate the mass and why the ribosomal protein content is divided into 55-80 small, similarly sized segments-speed up their autocatalytic production.

First Passage under Restart

First passage under restart has recently emerged as a conceptual framework suitable for the description of a wide range of phenomena, but the endless variety of ways in which restart mechanisms and first passage processes mix and match hindered the identification of unifying principles and general truths. Hope that these exist came from a recently discovered universality displayed by processes under optimal, constant rate, restart-but extensions and generalizations proved challenging as they marry arbitrarily complex processes and restart mechanisms. To address this challenge, we develop a generic approach to first passage under restart. Key features of diffusion under restart-the ultimate poster boy for this wide and diverse class of problems-are then shown to be completely universal.

Optimal Stochastic Restart Renders Fluctuations in First Passage Times Universal

Stochastic restart may drastically reduce the expected run time of a computer algorithm, expedite the completion of a complex search process, or increase the turnover rate of an enzymatic reaction. These diverse first-passage-time (FPT) processes seem to have very little in common but it is actually quite the other way around. Here we show that the relative standard deviation associated with the FPT of an optimally restarted process, i.e., one that is restarted at a constant (nonzero) rate which brings the mean FPT to a minimum, is always unity. We interpret, further generalize, and discuss this finding and the implications arising from it.

Michaelis-Menten reaction scheme as a unified approach towards the optimal restart problem

We study the effect of restart, and retry, on the mean completion time of a generic process. The need to do so arises in various branches of the sciences and we show that it can naturally be addressed by taking advantage of the classical reaction scheme of Michaelis and Menten. Stopping a process in its midst-only to start it all over again-may prolong, leave unchanged, or even shorten the time taken for its completion. Here we are interested in the optimal restart problem, i.e., in finding a restart rate which brings the mean completion time of a process to a minimum. We derive the governing equation for this problem and show that it is exactly solvable in cases of particular interest. We then continue to discover regimes at which solutions to the problem take on universal, details independent forms which further give rise to optimal scaling laws. The formalism we develop, and the results obtained, can be utilized when optimizing stochastic search processes and randomized computer algorithms. An immediate connection with kinetic proofreading is also noted and discussed.

Occupation probabilities and fluctuations in the asymmetric simple inclusion process

The asymmetric simple inclusion process (ASIP), a lattice-gas model of unidirectional transport and aggregation, was recently proposed as an "inclusion" counterpart of the asymmetric simple exclusion process. In this paper we present an exact closed-form expression for the probability that a given number of particles occupies a given set of consecutive lattice sites. Our results are expressed in terms of the entries of Catalan's trapezoids-number arrays which generalize Catalan's numbers and Catalan's triangle. We further prove that the ASIP is asymptotically governed by the following: (i) an inverse square-root law of occupation, (ii) a square-root law of fluctuation, and (iii) a Rayleigh law for the distribution of interexit times. The universality of these results is discussed.

Limit laws for the asymmetric inclusion process

The Asymmetric Inclusion Process (ASIP) is a unidirectional lattice-gas flow model which was recently introduced as an exactly solvable 'Bosonic' counterpart of the 'Fermionic' asymmetric exclusion process. An iterative algorithm that allows the computation of the probability generating function (PGF) of the ASIP's steady state exists but practical considerations limit its applicability to small ASIP lattices. Large lattices, on the other hand, have been studied primarily via Monte Carlo simulations and were shown to display a wide spectrum of intriguing statistical phenomena. In this paper we bypass the need for direct computation of the PGF and explore the ASIP's asymptotic statistical behavior. We consider three different limiting regimes: heavy-traffic regime, large-system regime, and balanced-system regime. In each of these regimes we obtain-analytically and in closed form-stochastic limit laws for five key ASIP observables: traversal time, overall load, busy period, first occupied site, and draining time. The results obtained yield a detailed limit-laws perspective of the ASIP, numerical simulations demonstrate the applicability of these laws as useful approximations.

Asymmetric inclusion process as a showcase of complexity

The asymmetric inclusion process is a lattice-gas model which replaces the "fermionic" exclusion interactions of the asymmetric exclusion process by "bosonic" inclusion interactions. Combining together probabilistic and Monte Carlo analyses, we showcase the model's rich statistical complexity-which ranges from "mild" to "wild" displays of randomness: gaussian load and draining, Rayleigh outflow with linear aging, inverse-gaussian coalescence, intrinsic power-law scalings and power-law fluctuations and condensation.

Dynamic structure factor of vibrating fractals

Motivated by novel experimental work and the lack of an adequate theory, we study the dynamic structure factor S(k,t) of large vibrating fractal networks at large wave numbers k. We show that the decay of S(k,t) is dominated by the spatially averaged mean square displacement of a network node, which evolves subdiffusively in time, ((u[over →](i)(t)-u[over →](i)(0))(2))∼t(ν), where ν depends on the spectral dimension d(s) and fractal dimension d(f). As a result, S(k,t) decays as a stretched exponential S(k,t)≈S(k)e(-(Γ(k)t)(ν)) with Γ(k)∼k(2/ν). Applications to a variety of fractal-like systems are elucidated.

Dynamic structure factor of vibrating fractals: proteins as a case study

We study the dynamic structure factor S(k,t) of proteins at large wave numbers k, kR(g)≫1, where R(g) is the gyration radius. At this regime measurements are sensitive to internal dynamics, and we focus on vibrational dynamics of folded proteins. Exploiting the analogy between proteins and fractals, we perform a general analytic calculation of the displacement two-point correlation functions, <[u(−>)(i)(t)-u(−>)(j)(0)](2)>. We confront the derived expressions with numerical evaluations that are based on protein data bank (PDB) structures and the Gaussian network model (GNM) for a few proteins and for the Sierpinski gasket as a controlled check. We use these calculations to evaluate S(k,t) with arrested rotational and translational degrees of freedom, and show that the decay of S(k,t) is dominated by the spatially averaged mean-square displacement of an amino acid. The latter has been previously shown to evolve subdiffusively in time, <[u(−>)(i)(t)-u(−>)(i)(0)](2)> ~t(ν), where ν is the anomalous diffusion exponent that depends on the spectral dimension d(s) and fractal dimension d(f). As a result, for wave numbers obeying k(2)<u(−>)(2)>≳1, S(k,t) effectively decays as a stretched exponential S(k,t)≃S(k)e(-(Γ(k)t)(β)) with β≃ν, where the relaxation rate is Γ(k)~(k(B)T/mω(o)(2))(1/β)k(2/β), T is the temperature, and mω(o)(2) the GNM effective spring constant describing the interaction between neighboring amino acids. The static structure factor is dominated by the fractal character of the native fold, S(k)~k(-d(f)), with negligible to marginal influence of vibrations. The analytical expressions are first confronted with numerically based calculations on the Sierpinski gasket, and very good agreement is found between simulations and theory. We then perform PDB-GNM-based numerical calculations for a few proteins, and an effective stretched exponential decay of the dynamic structure factor is found, albeit their relatively small size. However, when rotational and translational diffusion are added, we find that their contribution is never negligible due to finite size effects. While we can still attribute an effective stretching exponent β to the relaxation profile, this exponent is significantly larger than the anomalous diffusion exponent ν. We compare our theory with recent neutron spin-echo studies of myoglobin and hemoglobin and conclude that experiments in which the rotational and translational degrees of freedom are arrested, e.g., by anchoring the proteins to a surface, will improve the detection of internal vibrational dynamics.

Asymmetric inclusion process

We introduce and explore the asymmetric inclusion process (ASIP), an exactly solvable bosonic counterpart of the fermionic asymmetric exclusion process (ASEP). In both processes, random events cause particles to propagate unidirectionally along a one-dimensional lattice of n sites. In the ASEP, particles are subject to exclusion interactions, whereas in the ASIP, particles are subject to inclusion interactions that coalesce them into inseparable clusters. We study the dynamics of the ASIP, derive evolution equations for the mean and probability generating function (PGF) of the sites' occupancy vector, obtain explicit results for the above mean at steady state, and describe an iterative scheme for the computation of the PGF at steady state. We further obtain explicit results for the load distribution in steady state, with the load being the total number of particles present in all lattice sites. Finally, we address the problem of load optimization, and solve it under various criteria. The ASIP model establishes bridges between statistical physics and queueing theory as it represents a tandem array of queueing systems with (unlimited) batch service, and a tandem array of growth-collapse processes.

Genome-scale analysis of translation elongation with a ribosome flow model

We describe the first large scale analysis of gene translation that is based on a model that takes into account the physical and dynamical nature of this process. The Ribosomal Flow Model (RFM) predicts fundamental features of the translation process, including translation rates, protein abundance levels, ribosomal densities and the relation between all these variables, better than alternative ('non-physical') approaches. In addition, we show that the RFM can be used for accurate inference of various other quantities including genes' initiation rates and translation costs. These quantities could not be inferred by previous predictors. We find that increasing the number of available ribosomes (or equivalently the initiation rate) increases the genomic translation rate and the mean ribosome density only up to a certain point, beyond which both saturate. Strikingly, assuming that the translation system is tuned to work at the pre-saturation point maximizes the predictive power of the model with respect to experimental data. This result suggests that in all organisms that were analyzed (from bacteria to Human), the global initiation rate is optimized to attain the pre-saturation point. The fact that similar results were not observed for heterologous genes indicates that this feature is under selection. Remarkably, the gap between the performance of the RFM and alternative predictors is strikingly large in the case of heterologous genes, testifying to the model's promising biotechnological value in predicting the abundance of heterologous proteins before expressing them in the desired host.

General mapping between random walks and thermal vibrations in elastic networks: fractal networks as a case study

We present an approach to mapping between random walks and vibrational dynamics on general networks. Random walk occupation probabilities, first passage time distributions and passage probabilities between nodes are expressed in terms of thermal vibrational correlation functions. Recurrence is demonstrated equivalent to the Landau-Peierls instability. Fractal networks are analyzed as a case study. In particular, we show that the spectral dimension governs whether or not the first passage time distribution is well represented by its mean. We discuss relevance to universal features arising in protein vibrational dynamics.

Vibrational shortcut to the mean-first-passage-time problem

What is the average time a random walker takes to get from A to B on a fractal structure and how does this mean time scale with the size of the system and the distance between source and target? We take a nonprobabilistic approach toward this problem and show how the solution is readily obtained using an analysis of thermal vibrations on fractals. Invariance under scaling and continuity with respect to the spectral dimension are shown to be emergent properties of the solution obtained via vibrational analysis. Our result emphasizes the duality between diffusion and vibrations on fractal structures. Applications to biological systems are discussed.

Proteins: coexistence of stability and flexibility

We introduce an equation for protein native topology based on recent analysis of data from the Protein Data Bank and on a generalization of the Landau-Peierls instability criterion for fractals. The equation relates the protein fractal dimension df, the spectral dimension ds, and the number of amino acids N. Deviations from the equation may render a protein unfolded. The fractal nature of proteins is shown to bridge their seemingly conflicting properties of stability and flexibility. Over 500 proteins have been analyzed (df, ds, and N) and found to obey this equation of state.