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The rapid worldwide spread of severe viral infections, often involving novel mutations of viruses, poses major challenges to our health-care systems. This means that tools that can efficiently and specifically diagnose viruses are much needed. To be relevant for broad applications in local health-care centers, such tools should be relatively cheap and easy to use. In this paper, we discuss the biophysical potential for the macroscopic detection of viruses based on the induction of a mechanical stress in a bundle of prestretched DNA molecules upon binding of viruses to the DNA. We show that the affinity of the DNA to the charged virus surface induces a local melting of the double helix into two single-stranded DNA. This process effects a mechanical stress along the DNA chains leading to an overall contraction of the DNA. Our results suggest that when such DNA bundles are incorporated in a supporting matrix such as a responsive hydrogel, the presence of viruses may indeed lead to a significant, macroscopic mechanical deformation of the matrix. We discuss the biophysical basis for this effect and characterize the physical properties of the associated DNA melting transition. In particular, we reveal several scaling relations between the relevant physical parameters of the system. We promote this DNA-based assay as a possible tool for efficient and specific virus screening.
Aging, the dependence of the dynamics of a physical process on the time t(a) since its original preparation, is observed in systems ranging from the motion of charge carriers in amorphous semiconductors over the blinking dynamics of quantum dots to the tracer dispersion in living biological cells. Here we study the effects of aging on one of the most fundamental properties of a stochastic process, the first-passage dynamics. We find that for an aging continuous time random walk process, the scaling exponent of the density of first-passage times changes twice as the aging progresses and reveals an intermediate scaling regime. The first-passage dynamics depends on t(a) differently for intermediate and strong aging. Similar crossovers are obtained for the first-passage dynamics for a confined and driven particle. Comparison to the motion of an aged particle in the quenched trap model with a bias shows excellent agreement with our analytical findings. Our results demonstrate how first-passage measurements can be used to unravel the age t(a) of a physical system.
We study the parameter sensitivity of hetero-polymeric DNA within the purview of DNA breathing dynamics. The degree of correlation between the mean bubble size and the model parameters is estimated for this purpose for three different DNA sequences. The analysis leads us to a better understanding of the sequence dependent nature of the breathing dynamics of hetero-polymeric DNA. Out of the 14 model parameters for DNA stability in the statistical Poland-Scheraga approach, the hydrogen bond interaction epsilon(hb)(AT) for an AT base pair and the ring factor. turn out to be the most sensitive parameters. In addition, the stacking interaction epsilon(st)(TA-TA) for an TA-TA nearest neighbor pair of base-pairs is found to be the most sensitive one among all stacking interactions. Moreover, we also establish that the nature of stacking interaction has a deciding effect on the DNA breathing dynamics, not the number of times a particular stacking interaction appears in a sequence. We show that the sensitivity analysis can be used as an effective measure to guide a stochastic optimization technique to find the kinetic rate constants related to the dynamics as opposed to the case where the rate constants are measured using the conventional unbiased way of optimization.
We study the elastic deformations in a cross-linked polymer network film triggered by the binding of submicron particles with a sticky surface, mimicking the interactions of viral pathogens with thin films of stimulus-responsive polymeric materials such as hydrogels. From extensive Langevin Dynamics simulations we quantify how far the network deformations propagate depending on the elasticity parameters of the network and the adhesion strength of the particles. We examine the dynamics of the collective area shrinkage of the network and obtain some simple relations for the associated characteristic decay lengths. A detailed analysis elucidates how the elastic energy of the network is distributed between stretching and compression modes in response to the particle binding. We also examine the force-distance curves of the repulsion or attraction interactions for a pair of sticky particles in the polymer network film as a function of the particle-particle separation. The results of this computational study provide new insight into collective phenomena in soft polymer network films and may, in particular, be applied to applications for visual detection of pathogens such as viruses via a macroscopic response of thin films of cross-linked hydrogels. (C) 2014 AIP Publishing LLC.
Molecular motors pulling cargos in the viscoelastic cytosol: how power strokes beat subdiffusion
(2014)
The discovery of anomalous diffusion of larger biopolymers and submicron tracers such as endogenous granules, organelles, or virus capsids in living cells, attributed to the viscoelastic nature of the cytoplasm, provokes the question whether this complex environment equally impacts the active intracellular transport of submicron cargos by molecular motors such as kinesins: does the passive anomalous diffusion of free cargo always imply its anomalously slow active transport by motors, the mean transport distance along microtubule growing sublinearly rather than linearly in time? Here we analyze this question within the widely used two-state Brownian ratchet model of kinesin motors based on the continuous-state diffusion along microtubules driven by a flashing binding potential, where the cargo particle is elastically attached to the motor. Depending on the cargo size, the loading force, the amplitude of the binding potential, the turnover frequency of the molecular motor enzyme, and the linker stiffness we demonstrate that the motor transport may turn out either normal or anomalous, as indeed measured experimentally. We show how a highly efficient normal active transport mediated by motors may emerge despite the passive anomalous diffusion of the cargo, and study the intricate effects of the elastic linker. Under different, well specified conditions the microtubule-based motor transport becomes anomalously slow and thus significantly less efficient.
We study the thermal Markovian diffusion of tracer particles in a 2D medium with spatially varying diffusivity D(r), mimicking recently measured, heterogeneous maps of the apparent diffusion coefficient in biological cells. For this heterogeneous diffusion process (HDP) we analyse the mean squared displacement (MSD) of the tracer particles, the time averaged MSD, the spatial probability density function, and the first passage time dynamics from the cell boundary to the nucleus. Moreover we examine the non-ergodic properties of this process which are important for the correct physical interpretation of time averages of observables obtained from single particle tracking experiments. From extensive computer simulations of the 2D stochastic Langevin equation we present an in-depth study of this HDP. In particular, we find that the MSDs along the radial and azimuthal directions in a circular domain obey anomalous and Brownian scaling, respectively. We demonstrate that the time averaged MSD stays linear as a function of the lag time and the system thus reveals a weak ergodicity breaking. Our results will enable one to rationalise the diffusive motion of larger tracer particles such as viruses or submicron beads in biological cells.
Molecular motors pulling cargos in the viscoelastic cytosol: how power strokes beat subdiffusion
(2014)
The discovery of anomalous diffusion of larger biopolymers and submicron tracers such as endogenous granules, organelles, or virus capsids in living cells, attributed to the viscoelastic nature of the cytoplasm, provokes the question whether this complex environment equally impacts the active intracellular transport of submicron cargos by molecular motors such as kinesins: does the passive anomalous diffusion of free cargo always imply its anomalously slow active transport by motors, the mean transport distance along microtubule growing sublinearly rather than linearly in time? Here we analyze this question within the widely used two-state Brownian ratchet model of kinesin motors based on the continuous-state diffusion along microtubules driven by a flashing binding potential, where the cargo particle is elastically attached to the motor. Depending on the cargo size, the loading force, the amplitude of the binding potential, the turnover frequency of the molecular motor enzyme, and the linker stiffness we demonstrate that the motor transport may turn out either normal or anomalous, as indeed measured experimentally. We show how a highly efficient normal active transport mediated by motors may emerge despite the passive anomalous diffusion of the cargo, and study the intricate effects of the elastic linker. Under different, well specified conditions the microtubule-based motor transport becomes anomalously slow and thus significantly less efficient.
Diffusion of finite-size particles in two-dimensional channels with random wall configurations
(2014)
Diffusion of chemicals or tracer molecules through complex systems containing irregularly shaped channels is important in many applications. Most theoretical studies based on the famed Fick–Jacobs equation focus on the idealised case of infinitely small particles and reflecting boundaries. In this study we use numerical simulations to consider the transport of finite-size particles through asymmetrical two-dimensional channels. Additionally, we examine transient binding of the molecules to the channel walls by applying sticky boundary conditions. We consider an ensemble of particles diffusing in independent channels, which are characterised by common structural parameters. We compare our results for the long-time effective diffusion coefficient with a recent theoretical formula obtained by Dagdug and Pineda [J. Chem. Phys., 2012, 137, 024107].
We study generalized fractional Langevin equations in the presence of a harmonic potential. General expressions for the mean velocity and particle displacement, the mean squared displacement, position and velocity correlation functions, as well as normalized displacement correlation function are derived. We report exact results for the cases of internal and external friction, that is, when the driving noise is either internal and thus the fluctuation-dissipation relation is fulfilled or when the noise is external. The asymptotic behavior of the generalized stochastic oscillator is investigated, and the case of high viscous damping (overdamped limit) is considered. Additional behaviors of the normalized displacement correlation functions different from those for the regular damped harmonic oscillator are observed. In addition, the cases of a constant external force and the force free case are obtained. The validity of the generalized Einstein relation for this process is discussed. The considered fractional generalized Langevin equation may be used to model anomalous diffusive processes including single file-type diffusion.
Recent experiments show that transcription factors (TFs) indeed use the facilitated diffusion mechanism to locate their target sequences on DNA in living bacteria cells: TFs alternate between sliding motion along DNA and relocation events through the cytoplasm. From simulations and theoretical analysis we study the TF-sliding motion for a large section of the DNA-sequence of a common E. coli strain, based on the two-state TF-model with a fast-sliding search state and a recognition state enabling target detection. For the probability to detect the target before dissociating from DNA the TF-search times self-consistently depend heavily on whether or not an auxiliary operator (an accessible sequence similar to the main operator) is present in the genome section. Importantly, within our model the extent to which the interconversion rates between search and recognition states depend on the underlying nucleotide sequence is varied. A moderate dependence maximises the capability to distinguish between the main operator and similar sequences. Moreover, these auxiliary operators serve as starting points for DNA looping with the main operator, yielding a spectrum of target detection times spanning several orders of magnitude. Auxiliary operators are shown to act as funnels facilitating target detection by TFs.
We study generalized anomalous diffusion processes whose diffusion coefficient D(x, t) similar to D-0x(alpha)t(beta) depends on both the position x of the test particle and the process time t. This process thus combines the features of scaled Brownian motion and heterogeneous diffusion parent processes. We compute the ensemble and time averaged mean squared displacements of this generalized diffusion process. The scaling exponent of the ensemble averaged mean squared displacement is shown to be the product of the critical exponents of the parent processes, and describes both subdiffusive and superdiffusive systems. We quantify the amplitude fluctuations of the time averaged mean squared displacement as function of the length of the time series and the lag time. In particular, we observe a weak ergodicity breaking of this generalized diffusion process: even in the long time limit the ensemble and time averaged mean squared displacements are strictly disparate. When we start to observe this process some time after its initiation we observe distinct features of ageing. We derive a universal ageing factor for the time averaged mean squared displacement containing all information on the ageing time and the measurement time. External confinement is shown to alter the magnitudes and statistics of the ensemble and time averaged mean squared displacements.
We study noisy heterogeneous diffusion processes with a position dependent diffusivity of the form D(x) similar to D-0 vertical bar x vertical bar (alpha 0) in the presence of annealed and quenched disorder of the environment, corresponding to an effective variation of the exponent a in time and space. In the case of annealed disorder, for which effectively alpha(0) = alpha(0)(t), we show how the long time scaling of the ensemble mean squared displacement (MSD) and the amplitude variation of individual realizations of the time averaged MSD are affected by the disorder strength. For the case of quenched disorder, the long time behavior becomes effectively Brownian after a number of jumps between the domains of a stratified medium. In the latter situation, the averages are taken over both an ensemble of particles and different realizations of the disorder. As physical observables, we analyze in detail the ensemble and time averaged MSDs, the ergodicity breaking parameter, and higher order moments of the time averages. (C) 2015 AIP Publishing LLC.
Multiple loop formation in polymer macromolecules is an important feature of the chromatin organization and DNA compactification in the nuclei. We analyse the size and shape characteristics of complex polymer structures, containing in general f(1) loops (petals) and f(2) linear chains (branches). Within the frames of continuous model of Gaussian macromolecule, we apply the path integration method and obtain the estimates for gyration radius R-g and asphericity (A) over cap of typical conformation as functions of parameters f(1), f(2). In particular, our results qualitatively reveal the extent of anisotropy of star-like topologies as compared to the rosette structures of the same total molecular weight.
Acanthamoebae are free-living protists and human pathogens, whose cellular functions and pathogenicity strongly depend on the transport of intracellular vesicles and granules through the cytosol. Using high-speed live cell imaging in combination with single-particle tracking analysis, we show here that the motion of endogenous intracellular particles in the size range from a few hundred nanometers to several micrometers in Acanthamoeba castellanii is strongly superdiffusive and influenced by cell locomotion, cytoskeletal elements, and myosin II. We demonstrate that cell locomotion significantly contributes to intracellular particle motion, but is clearly not the only origin of superdiffusivity. By analyzing the contribution of microtubules, actin, and myosin II motors we show that myosin II is a major driving force of intracellular motion in A. castellanii. The cytoplasm of A. castellanii is supercrowded with intracellular vesicles and granules, such that significant intracellular motion can only be achieved by actively driven motion, while purely thermally driven diffusion is negligible.
We define and study in detail utraslow scaled Brownian motion (USBM) characterized by a time dependent diffusion coefficient of the form D(t) similar or equal to 1/t. For unconfined motion the mean squared displacement (MSD) of USBM exhibits an ultraslow, logarithmic growth as function of time, in contrast to the conventional scaled Brownian motion. In a harmonic potential the MSD of USBM does not saturate but asymptotically decays inverse-proportionally to time, reflecting the highly non-stationary character of the process. We show that the process is weakly non-ergodic in the sense that the time averaged MSD does not converge to the regular MSD even at long times, and for unconfined motion combines a linear lag time dependence with a logarithmic term. The weakly non-ergodic behaviour is quantified in terms of the ergodicity breaking parameter. The USBM process is also shown to be ageing: observables of the system depend on the time gap between initiation of the test particle and start of the measurement of its motion. Our analytical results are shown to agree excellently with extensive computer simulations.
Optimization and universality of Brownian search in a basic model of quenched heterogeneous media
(2015)
The kinetics of a variety of transport-controlled processes can be reduced to the problem of determining the mean time needed to arrive at a given location for the first time, the so-called mean first-passage time ( MFPT) problem. The occurrence of occasional large jumps or intermittent patterns combining various types of motion are known to outperform the standard random walk with respect to the MFPT, by reducing oversampling of space. Here we show that a regular but spatially heterogeneous random walk can significantly and universally enhance the search in any spatial dimension. In a generic minimal model we consider a spherically symmetric system comprising two concentric regions with piecewise constant diffusivity. The MFPT is analyzed under the constraint of conserved average dynamics, that is, the spatially averaged diffusivity is kept constant. Our analytical calculations and extensive numerical simulations demonstrate the existence of an optimal heterogeneity minimizing the MFPT to the target. We prove that the MFPT for a random walk is completely dominated by what we term direct trajectories towards the target and reveal a remarkable universality of the spatially heterogeneous search with respect to target size and system dimensionality. In contrast to intermittent strategies, which are most profitable in low spatial dimensions, the spatially inhomogeneous search performs best in higher dimensions. Discussing our results alongside recent experiments on single-particle tracking in living cells, we argue that the observed spatial heterogeneity may be beneficial for cellular signaling processes.
We study the dynamics of polymer chains in a bath of self-propelled particles (SPP) by extensive Langevin dynamics simulations in a two-dimensional model system. Specifically, we analyse the polymer looping properties versus the SPP activity and investigate how the presence of the active particles alters the chain conformational statistics. We find that SPPs tend to extend flexible polymer chains, while they rather compactify stiffer semiflexible polymers, in agreement with previous results. Here we show that higher activities of SPPs yield a higher effective temperature of the bath and thus facilitate the looping kinetics of a passive polymer chain. We explicitly compute the looping probability and looping time in a wide range of the model parameters. We also analyse the motion of a monomeric tracer particle and the polymer's centre of mass in the presence of the active particles in terms of the time averaged mean squared displacement, revealing a giant diffusivity enhancement for the polymer chain via SPP pooling. Our results are applicable to rationalising the dimensions and looping kinetics of biopolymers at constantly fluctuating and often actively driven conditions inside biological cells or in suspensions of active colloidal particles or bacteria cells.
Ageing first passage time density in continuous time random walks and quenched energy landscapes
(2015)
We study the first passage dynamics of an ageing stochastic process in the continuous time random walk (CTRW) framework. In such CTRW processes the test particle performs a random walk, in which successive steps are separated by random waiting times distributed in terms of the waiting time probability density function Psi (t) similar or equal to t(-1-alpha) (0 <= alpha <= 2). An ageing stochastic process is defined by the explicit dependence of its dynamic quantities on the ageing time t(a), the time elapsed between its preparation and the start of the observation. Subdiffusive ageing CTRWs with 0 < alpha < 1 describe systems such as charge carriers in amorphous semiconducters, tracer dispersion in geological and biological systems, or the dynamics of blinking quantum dots. We derive the exact forms of the first passage time density for an ageing subdiffusive CTRW in the semi-infinite, confined, and biased case, finding different scaling regimes for weakly, intermediately, and strongly aged systems: these regimes, with different scaling laws, are also found when the scaling exponent is in the range 1 < alpha < 2, for sufficiently long ta. We compare our results with the ageing motion of a test particle in a quenched energy landscape. We test our theoretical results in the quenched landscape against simulations: only when the bias is strong enough, the correlations from returning to previously visited sites become insignificant and the results approach the ageing CTRW results. With small bias or without bias, the ageing effects disappear and a change in the exponent compared to the case of a completely annealed landscape can be found, reflecting the build-up of correlations in the quenched landscape.
Recent experiments show that transcription factors (TFs) indeed use the facilitated diffusion mechanism to locate their target sequences on DNA in living bacteria cells: TFs alternate between sliding motion along DNA and relocation events through the cytoplasm. From simulations and theoretical analysis we study the TF-sliding motion for a large section of the DNA-sequence of a common E. coli strain, based on the two-state TF-model with a fast-sliding search state and a recognition state enabling target detection. For the probability to detect the target before dissociating from DNA the TF-search times self-consistently depend heavily on whether or not an auxiliary operator (an accessible sequence similar to the main operator) is present in the genome section. Importantly, within our model the extent to which the interconversion rates between search and recognition states depend on the underlying nucleotide sequence is varied. A moderate dependence maximises the capability to distinguish between the main operator and similar sequences. Moreover, these auxiliary operators serve as starting points for DNA looping with the main operator, yielding a spectrum of target detection times spanning several orders of magnitude. Auxiliary operators are shown to act as funnels facilitating target detection by TFs.
We examine by extensive computer simulations the self-diffusion of anisotropic star-like particles in crowded two-dimensional solutions. We investigate the implications of the area coverage fraction phi of the crowders and the crowder-crowder adhesion properties on the regime of transient anomalous diffusion. We systematically compute the mean squared displacement (MSD) of the particles, their time averaged MSD, and the effective diffusion coefficient. The diffusion is ergodic in the limit of long traces, such that the mean time averaged MSD converges towards the ensemble averaged MSD, and features a small residual amplitude spread of the time averaged MSD from individual trajectories. At intermediate time scales, we quantify the anomalous diffusion in the system. Also, we show that the translational-but not rotational-diffusivity of the particles Dis a nonmonotonic function of the attraction strength between them. Both diffusion coefficients decrease as the power law D(phi) similar to (1 - phi/phi*)(2 ... 2.4) with the area fraction phi occupied by the crowders and the critical value phi*. Our results might be applicable to rationalising the experimental observations of non-Brownian diffusion for a number of standard macromolecular crowders used in vitro to mimic the cytoplasmic conditions of living cells.
Many chemical reactions in biological cells occur at very low concentrations of constituent molecules. Thus, transcriptional gene-regulation is often controlled by poorly expressed transcription-factors, such as E. coli lac repressor with few tens of copies. Here we study the effects of inherent concentration fluctuations of substrate-molecules on the seminal Michaelis-Menten scheme of biochemical reactions. We present a universal correction to the Michaelis-Menten equation for the reaction-rates. The relevance and validity of this correction for enzymatic reactions and intracellular gene-regulation is demonstrated. Our analytical theory and simulation results confirm that the proposed variance-corrected Michaelis-Menten equation predicts the rate of reactions with remarkable accuracy even in the presence of large non-equilibrium concentration fluctuations. The major advantage of our approach is that it involves only the mean and variance of the substrate-molecule concentration. Our theory is therefore accessible to experiments and not specific to the exact source of the concentration fluctuations.
We define and study in detail utraslow scaled Brownian motion (USBM) characterized by a time dependent diffusion coefficient of the form . For unconfined motion the mean squared displacement (MSD) of USBM exhibits an ultraslow, logarithmic growth as function of time, in contrast to the conventional scaled Brownian motion. In a harmonic potential the MSD of USBM does not saturate but asymptotically decays inverse-proportionally to time, reflecting the highly non-stationary character of the process. We show that the process is weakly non-ergodic in the sense that the time averaged MSD does not converge to the regular MSD even at long times, and for unconfined motion combines a linear lag time dependence with a logarithmic term. The weakly non-ergodic behaviour is quantified in terms of the ergodicity breaking parameter. The USBM process is also shown to be ageing: observables of the system depend on the time gap between initiation of the test particle and start of the measurement of its motion. Our analytical results are shown to agree excellently with extensive computer simulations.
We investigate the ergodic properties of a random walker performing (anomalous) diffusion on a random fractal geometry. Extensive Monte Carlo simulations of the motion of tracer particles on an ensemble of realisations of percolation clusters are performed for a wide range of percolation densities. Single trajectories of the tracer motion are analysed to quantify the time averaged mean squared displacement (MSD) and to compare this with the ensemble averaged MSD of the particle motion. Other complementary physical observables associated with ergodicity are studied, as well. It turns out that the time averaged MSD of individual realisations exhibits non-vanishing fluctuations even in the limit of very long observation times as the percolation density approaches the critical value. This apparent non-ergodic behaviour concurs with the ergodic behaviour on the ensemble averaged level. We demonstrate how the non-vanishing fluctuations in single particle trajectories are analytically expressed in terms of the fractal dimension and the cluster size distribution of the random geometry, thus being of purely geometrical origin. Moreover, we reveal that the convergence scaling law to ergodicity, which is known to be inversely proportional to the observation time T for ergodic diffusion processes, follows a power-law similar to T-h with h < 1 due to the fractal structure of the accessible space. These results provide useful measures for differentiating the subdiffusion on random fractals from an otherwise closely related process, namely, fractional Brownian motion. Implications of our results on the analysis of single particle tracking experiments are provided.
The looping of polymers such as DNA is a fundamental process in the molecular biology of living cells, whose interior is characterised by a high degree of molecular crowding. We here investigate in detail the looping dynamics of flexible polymer chains in the presence of different degrees of crowding. From the analysis of the looping-unlooping rates and the looping probabilities of the chain ends we show that the presence of small crowders typically slows down the chain dynamics but larger crowders may in fact facilitate the looping. We rationalise these non-trivial and often counterintuitive effects of the crowder size on the looping kinetics in terms of an effective solution viscosity and standard excluded volume. It is shown that for small crowders the effect of an increased viscosity dominates, while for big crowders we argue that confinement effects (caging) prevail. The tradeoff between both trends can thus result in the impediment or facilitation of polymer looping, depending on the crowder size. We also examine how the crowding volume fraction, chain length, and the attraction strength of the contact groups of the polymer chain affect the looping kinetics and hairpin formation dynamics. Our results are relevant for DNA looping in the absence and presence of protein mediation, DNA hairpin formation, RNA folding, and the folding of polypeptide chains under biologically relevant high-crowding conditions.
We study the diffusion of a tracer particle, which moves in continuum space between a lattice of excluded volume, immobile non-inert obstacles. In particular, we analyse how the strength of the tracer-obstacle interactions and the volume occupancy of the crowders alter the diffusive motion of the tracer. From the details of partitioning of the tracer diffusion modes between trapping states when bound to obstacles and bulk diffusion, we examine the degree of localisation of the tracer in the lattice of crowders. We study the properties of the tracer diffusion in terms of the ensemble and time averaged mean squared displacements, the trapping time distributions, the amplitude variation of the time averaged mean squared displacements, and the non-Gaussianity parameter of the diffusing tracer. We conclude that tracer-obstacle adsorption and binding triggers a transient anomalous diffusion. From a very narrow spread of recorded individual time averaged trajectories we exclude continuous type random walk processes as the underlying physical model of the tracer diffusion in our system. For moderate tracer-crowder attraction the motion is found to be fully ergodic, while at stronger attraction strength a transient disparity between ensemble and time averaged mean squared displacements occurs. We also put our results into perspective with findings from experimental single-particle tracking and simulations of the diffusion of tagged tracers in dense crowded suspensions. Our results have implications for the diffusion, transport, and spreading of chemical components in highly crowded environments inside living cells and other structured liquids.
Polymer looping is controlled by macromolecular crowding, spatial confinement, and chain stiffness
(2015)
We study by extensive computer simulations the looping characteristics of linear polymers with varying persistence length inside a spherical cavity in the presence of macromolecular crowding. For stiff chains, the looping probability and looping time reveal wildly oscillating patterns as functions of the chain length. The effects of crowding differ dramatically for flexible versus stiff polymers. While for flexible chains the looping kinetics is slowed down by the crowders, for stiffer chains the kinetics turns out to be either decreased or facilitated, depending on the polymer length. For severe confinement, the looping kinetics may become strongly facilitated by crowding. Our findings are of broad impact for DNA looping in the crowded and compartmentalized interior of living biological cells.
What are the fundamental laws for the adsorption of charged polymers onto oppositely charged surfaces, for convex, planar, and concave geometries? This question is at the heart of surface coating applications, various complex formation phenomena, as well as in the context of cellular and viral biophysics. It has been a long-standing challenge in theoretical polymer physics; for realistic systems the quantitative understanding is however often achievable only by computer simulations. In this study, we present the findings of such extensive Monte-Carlo in silico experiments for polymer-surface adsorption in confined domains. We study the inverted critical adsorption of finite-length polyelectrolytes in three fundamental geometries: planar slit, cylindrical pore, and spherical cavity. The scaling relations extracted from simulations for the critical surface charge density sigma(c)-defining the adsorption-desorption transition-are in excellent agreement with our analytical calculations based on the ground-state analysis of the Edwards equation. In particular, we confirm the magnitude and scaling of sigma(c) for the concave interfaces versus the Debye screening length 1/kappa and the extent of confinement a for these three interfaces for small kappa a values. For large kappa a the critical adsorption condition approaches the known planar limit. The transition between the two regimes takes place when the radius of surface curvature or half of the slit thickness a is of the order of 1/kappa. We also rationalize how sigma(c)(kappa) dependence gets modified for semi-flexible versus flexible chains under external confinement. We examine the implications of the chain length for critical adsorption-the effect often hard to tackle theoretically-putting an emphasis on polymers inside attractive spherical cavities. The applications of our findings to some biological systems are discussed, for instance the adsorption of nucleic acids onto the inner surfaces of cylindrical and spherical viral capsids.
Brownian motion is ergodic in the Boltzmann-Khinchin sense that long time averages of physical observables such as the mean squared displacement provide the same information as the corresponding ensemble average, even at out-of-equilibrium conditions. This property is the fundamental prerequisite for single particle tracking and its analysis in simple liquids. We study analytically and by event-driven molecular dynamics simulations the dynamics of force-free cooling granular gases and reveal a violation of ergodicity in this Boltzmann-Khinchin sense as well as distinct ageing of the system. Such granular gases comprise materials such as dilute gases of stones, sand, various types of powders, or large molecules, and their mixtures are ubiquitous in Nature and technology, in particular in Space. We treat-depending on the physical-chemical properties of the inter-particle interaction upon their pair collisions-both a constant and a velocity-dependent (viscoelastic) restitution coefficient epsilon. Moreover we compare the granular gas dynamics with an effective single particle stochastic model based on an underdamped Langevin equation with time dependent diffusivity. We find that both models share the same behaviour of the ensemble mean squared displacement (MSD) and the velocity correlations in the limit of weak dissipation. Qualitatively, the reported non-ergodic behaviour is generic for granular gases with any realistic dependence of epsilon on the impact velocity of particles.
Many chemical reactions in biological cells occur at very low concentrations of constituent molecules. Thus, transcriptional gene-regulation is often controlled by poorly expressed transcription-factors, such as E.coli lac repressor with few tens of copies. Here we study the effects of inherent concentration fluctuations of substrate-molecules on the seminal Michaelis-Menten scheme of biochemical reactions. We present a universal correction to the Michaelis-Menten equation for the reaction-rates. The relevance and validity of this correction for enzymatic reactions and intracellular gene-regulation is demonstrated. Our analytical theory and simulation results confirm that the proposed variance-corrected Michaelis-Menten equation predicts the rate of reactions with remarkable accuracy even in the presence of large non-equilibrium concentration fluctuations. The major advantage of our approach is that it involves only the mean and variance of the substrate-molecule concentration. Our theory is therefore accessible to experiments and not specific to the exact source of the concentration fluctuations.
We examine the non-ergodic properties of scaled Brownian motion (SBM), a non-stationary stochastic process with a time dependent diffusivity of the form D(t) similar or equal to t(alpha-1). We compute the ergodicity breaking parameter EB in the entire range of scaling exponents a, both analytically and via extensive computer simulations of the stochastic Langevin equation. We demonstrate that in the limit of long trajectory lengths T and short lag times Delta the EB parameter as function of the scaling exponent a has no divergence at alpha - 1/2 and present the asymptotes for EB in different limits. We generalize the analytical and simulations results for the time averaged and ergodic properties of SBM in the presence of ageing, that is, when the observation of the system starts only a finite time span after its initiation. The approach developed here for the calculation of the higher time averaged moments of the particle displacement can be applied to derive the ergodic properties of other stochastic processes such as fractional Brownian motion.
Aging scaled Brownian motion
(2015)
Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of passive tracers in complex and biological systems. It is a highly nonstationary process governed by the Langevin equation for Brownian motion, however, with a power-law time dependence of the noise strength. Here we study the aging properties of SBM for both unconfined and confined motion. Specifically, we derive the ensemble and time averaged mean squared displacements and analyze their behavior in the regimes of weak, intermediate, and strong aging. A very rich behavior is revealed for confined aging SBM depending on different aging times and whether the process is sub- or superdiffusive. We demonstrate that the information on the aging factorizes with respect to the lag time and exhibits a functional form that is identical to the aging behavior of scale-free continuous time random walk processes. While SBM exhibits a disparity between ensemble and time averaged observables and is thus weakly nonergodic, strong aging is shown to effect a convergence of the ensemble and time averaged mean squared displacement. Finally, we derive the density of first passage times in the semi-infinite domain that features a crossover defined by the aging time.
We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.
We consider anomalous stochastic processes based on the renewal continuous time random walk model with different forms for the probability density of waiting times between individual jumps. In the corresponding continuum limit we derive the generalized diffusion and Fokker-Planck-Smoluchowski equations with the corresponding memory kernels. We calculate the qth order moments in the unbiased and biased cases, and demonstrate that the generalized Einstein relation for the considered dynamics remains valid. The relaxation of modes in the case of an external harmonic potential and the convergence of the mean squared displacement to the thermal plateau are analyzed.
What are the physical laws of the mutual interactions of objects bound to cell membranes, such as various membrane proteins or elongated virus particles? To rationalise this, we here investigate by extensive computer simulations mutual interactions of rod-like particles adsorbed on the surface of responsive elastic two-dimensional sheets. Specifically, we quantify sheet deformations as a response to adhesion of such filamentous particles. We demonstrate that tip-to-tip contacts of rods are favoured for relatively soft sheets, while side-by-side contacts are preferred for stiffer elastic substrates. These attractive orientation-dependent substrate-mediated interactions between the rod-like particles on responsive sheets can drive their aggregation and self-assembly. The optimal orientation of the membrane-bound rods is established via responding to the elastic energy profiles created around the particles. We unveil the phase diagramme of attractive–repulsive rod–rod interactions in the plane of their separation and mutual orientation. Applications of our results to other systems featuring membrane-associated particles are also discussed.
We investigate the ensemble and time averaged mean squared displacements for particle diffusion in a simple model for disordered media by assuming that the local diffusivity is both fluctuating in time and has a deterministic average growth or decay in time. In this study we compare computer simulations of the stochastic Langevin equation for this random diffusion process with analytical results. We explore the regimes of normal Brownian motion as well as anomalous diffusion in the sub- and superdiffusive regimes. We also consider effects of the inertial term on the particle motion. The investigation of the resulting diffusion is performed for unconfined and confined motion.
We study the adsorption–desorption transition of polyelectrolyte chains onto planar, cylindrical and spherical surfaces with arbitrarily high surface charge densities by massive Monte Carlo computer simulations. We examine in detail how the well known scaling relations for the threshold transition—demarcating the adsorbed and desorbed domains of a polyelectrolyte near weakly charged surfaces—are altered for highly charged interfaces. In virtue of high surface potentials and large surface charge densities, the Debye–Hückel approximation is often not feasible and the nonlinear Poisson–Boltzmann approach should be implemented. At low salt conditions, for instance, the electrostatic potential from the nonlinear Poisson–Boltzmann equation is smaller than the Debye–Hückel result, such that the required critical surface charge density for polyelectrolyte adsorption σc increases. The nonlinear relation between the surface charge density and electrostatic potential leads to a sharply increasing critical surface charge density with growing ionic strength, imposing an additional limit to the critical salt concentration above which no polyelectrolyte adsorption occurs at all. We contrast our simulations findings with the known scaling results for weak critical polyelectrolyte adsorption onto oppositely charged surfaces for the three standard geometries. Finally, we discuss some applications of our results for some physical–chemical and biophysical systems.
Biomembranes are exceptionally crowded with proteins with typical protein-to-lipid ratios being around 1:50 - 1:100. Protein crowding has a decisive role in lateral membrane dynamics as shown by recent experimental and computational studies that have reported anomalous lateral diffusion of phospholipids and membrane proteins in crowded lipid membranes. Based on extensive simulations and stochastic modeling of the simulated trajectories, we here investigate in detail how increasing crowding by membrane proteins reshapes the stochastic characteristics of the anomalous lateral diffusion in lipid membranes. We observe that correlated Gaussian processes of the fractional Langevin equation type, identified as the stochastic mechanism behind lipid motion in noncrowded bilayer, no longer adequately describe the lipid and protein motion in crowded but otherwise identical membranes. It turns out that protein crowding gives rise to a multifractal, non-Gaussian, and spatiotemporally heterogeneous anomalous lateral diffusion on time scales from nanoseconds to, at least, tens of microseconds. Our investigation strongly suggests that the macromolecular complexity and spatiotemporal membrane heterogeneity in cellular membranes play critical roles in determining the stochastic nature of the lateral diffusion and, consequently, the associated dynamic phenomena within membranes. Clarifying the exact stochastic mechanism for various kinds of biological membranes is an important step towards a quantitative understanding of numerous intramembrane dynamic phenomena.
The first passage is a generic concept for quantifying when a random quantity such as the position of a diffusing molecule or the value of a stock crosses a preset threshold (target) for the first time. The last decade saw an enlightening series of new results focusing mostly on the so-called mean and global first passage time (MFPT and GFPT, respectively) of such processes. Here we push the understanding of first passage processes one step further. For a simple heterogeneous system we derive rigorously the complete distribution of first passage times (FPTs). Our results demonstrate that the typical FPT significantly differs from the MFPT, which corresponds to the long time behaviour of the FPT distribution. Conversely, the short time behaviour is shown to correspond to trajectories connecting directly from the initial value to the target. Remarkably, we reveal a previously overlooked third characteristic time scale of the first passage dynamics mirroring brief excursion away from the target.
What are the physical laws of the mutual interactions of objects bound to cell membranes, such as various membrane proteins or elongated virus particles? To rationalise this, we here investigate by extensive computer simulations mutual interactions of rod-like particles adsorbed on the surface of responsive elastic two-dimensional sheets. Specifically, we quantify sheet deformations as a response to adhesion of such filamentous particles. We demonstrate that tip-to-tip contacts of rods are favoured for relatively soft sheets, while side-by-side contacts are preferred for stiffer elastic substrates. These attractive orientation-dependent substrate-mediated interactions between the rod-like particles on responsive sheets can drive their aggregation and self-assembly. The optimal orientation of the membrane-bound rods is established via responding to the elastic energy profiles created around the particles. We unveil the phase diagramme of attractive-repulsive rod-rod interactions in the plane of their separation and mutual orientation. Applications of our results to other systems featuring membrane-associated particles are also discussed.
We study the adsorption-desorption transition of polyelectrolyte chains onto planar, cylindrical and spherical surfaces with arbitrarily high surface charge densities by massive Monte Carlo computer simulations. We examine in detail how the well known scaling relations for the threshold transition demarcating the adsorbed and desorbed domains of a polyelectrolyte near weakly charged surfaces-are altered for highly charged interfaces. In virtue of high surface potentials and large surface charge densities, the Debye-Huckel approximation is often not feasible and the nonlinear Poisson-Boltzmann approach should be implemented. At low salt conditions, for instance, the electrostatic potential from the nonlinear Poisson-Boltzmann equation is smaller than the Debye-Huckel result, such that the required critical surface charge density for polyelectrolyte adsorption sigma(c) increases. The nonlinear relation between the surface charge density and electrostatic potential leads to a sharply increasing critical surface charge density with growing ionic strength, imposing an additional limit to the critical salt concentration above which no polyelectrolyte adsorption occurs at all. We contrast our simulations findings with the known scaling results for weak critical polyelectrolyte adsorption onto oppositely charged surfaces for the three standard geometries. Finally, we discuss some applications of our results for some physical-chemical and biophysical systems.
A single predator charging a herd of prey: effects of self volume and predator-prey decision-making
(2016)
We study the degree of success of a single predator hunting a herd of prey on a two-dimensional square lattice landscape. We explicitly consider the self volume of the prey restraining their dynamics on the lattice. The movement of both predator and prey is chosen to include an intelligent, decision making step based on their respective sighting ranges, the radius in which they can detect the other species (prey cannot recognise each other besides the self volume interaction): after spotting each other the motion of prey and predator turns from a nearest neighbour random walk into directed escape or chase, respectively. We consider a large range of prey densities and sighting ranges and compute the mean first passage time for a predator to catch a prey as well as characterise the effective dynamics of the hunted prey. We find that the prey's sighting range dominates their life expectancy and the predator profits more from a bad eyesight of the prey than from his own good eye sight. We characterise the dynamics in terms of the mean distance between the predator and the nearest prey. It turns out that effectively the dynamics of this distance coordinate can be captured in terms of a simple Ornstein–Uhlenbeck picture. Reducing the many-body problem to a simple two-body problem by imagining predator and nearest prey to be connected by an effective Hookean bond, all features of the model such as prey density and sighting ranges merge into the effective binding constant.
When does a diffusing particle reach its target for the first time? This first-passage time (FPT) problem is central to the kinetics of molecular reactions in chemistry and molecular biology. Here, we explain the behavior of smooth FPT densities, for which all moments are finite, and demonstrate universal yet generally non-Poissonian long-time asymptotics for a broad variety of transport processes. While Poisson-like asymptotics arise generically in the presence of an effective repulsion in the immediate vicinity of the target, a time-scale separation between direct and reflected indirect trajectories gives rise to a universal proximity effect: Direct paths, heading more or less straight from the point of release to the target, become typical and focused, with a narrow spread of the corresponding first-passage times. Conversely, statistically dominant indirect paths exploring the entire system tend to be massively dissimilar. The initial distance to the target particularly impacts gene regulatory or competitive stochastic processes, for which few binding events often determine the regulatory outcome. The proximity effect is independent of details of the transport, highlighting the robust character of the FPT features uncovered here.
We study the properties of ageing Scher-Montroll transport in terms of a biased subdiffusive continuous time random walk in which the waiting times between consecutive jumps of the charge carriers are distributed according to the power law probability with . As we show, the dynamical properties of the Scher-Montroll transport depend on the ageing time span between the initial preparation of the system and the start of the observation. The Scher-Montroll transport theory was originally shown to describe the photocurrent in amorphous solids in the presence of an external electric field, but it has since been used in many other fields of physical sciences, in particular also in the geophysical context for the description of the transport of tracer particles in subsurface aquifers. In the absence of ageing () the photocurrent of the classical Scher-Montroll model or the breakthrough curves in the groundwater context exhibit a crossover between two power law regimes in time with the scaling exponents and . In the presence of ageing a new power law regime and an initial plateau regime of the current emerge. We derive the different power law regimes and crossover times of the ageing Scher-Montroll transport and show excellent agreement with simulations of the process. Experimental data of ageing Scher-Montroll transport in polymeric semiconductors are shown to agree well with the predictions of our theory.
We investigate the ensemble and time averaged mean squared displacements for particle diffusion in a simple model for disordered media by assuming that the local diffusivity is both fluctuating in time and has a deterministic average growth or decay in time. In this study we compare computer simulations of the stochastic Langevin equation for this random diffusion process with analytical results. We explore the regimes of normal Brownian motion as well as anomalous diffusion in the sub- and superdiffusive regimes. We also consider effects of the inertial term on the particle motion. The investigation of the resulting diffusion is performed for unconfined and confined motion.
Molecular signalling in living cells occurs at low copy numbers and is thereby inherently limited by the noise imposed by thermal diffusion. The precision at which biochemical receptors can count signalling molecules is intimately related to the noise correlation time. In addition to passive thermal diffusion, messenger RNA and vesicle-engulfed signalling molecules can transiently bind to molecular motors and are actively transported across biological cells. Active transport is most beneficial when trafficking occurs over large distances, for instance up to the order of 1 metre in neurons. Here we explain how intermittent active transport allows for faster equilibration upon a change in concentration triggered by biochemical stimuli. Moreover, we show how intermittent active excursions induce qualitative changes in the noise in effectively one-dimensional systems such as dendrites. Thereby they allow for significantly improved signalling precision in the sense of a smaller relative deviation in the concentration read-out by the receptor. On the basis of linear response theory we derive the exact mean field precision limit for counting actively transported molecules. We explain how intermittent active excursions disrupt the recurrence in the molecular motion, thereby facilitating improved signalling accuracy. Our results provide a deeper understanding of how recurrence affects molecular signalling precision in biological cells and novel medical-diagnostic devices.
The dynamics of constituents and the surface response of cellular membranes also in connection to the binding of various particles and macromolecules to the membrane are still a matter of controversy in the membrane biophysics community, particularly with respect to crowded membranes of living biological cells. We here put into perspective recent single particle tracking experiments in the plasma membranes of living cells and supercomputing studies of lipid bilayer model membranes with and without protein crowding. Special emphasis is put on the observation of anomalous, non-Brownian diffusion of both lipid molecules and proteins embedded in the lipid bilayer. While single component, pure lipid bilayers in simulations exhibit only transient anomalous diffusion of lipid molecules on nanosecond time scales, the persistence of anomalous diffusion becomes significantly longer ranged on the addition of disorder through the addition of cholesterol or proteins and on passing of the membrane lipids to the gel phase. Concurrently, experiments demonstrate the anomalous diffusion of membrane embedded proteins up to macroscopic time scales in the minute time range. Particular emphasis will be put on the physical character of the anomalous diffusion, in particular, the occurrence of ageing observed in the experiments the effective diffusivity of the measured particles is a decreasing function of time. Moreover, we present results for the time dependent local scaling exponent of the mean squared displacement of the monitored particles. Recent results finding deviations from the commonly assumed Gaussian diffusion patterns in protein crowded membranes are reported. The properties of the displacement autocorrelation function of the lipid molecules are discussed in the light of their appropriate physical anomalous diffusion models, both for non-crowded and crowded membranes. In the last part of this review we address the upcoming field of membrane distortion by elongated membrane-binding particles. We discuss how membrane compartmentalisation and the particle-membrane binding energy may impact the dynamics and response of lipid membranes. This article is part of a Special Issue entitled: Biosimulations edited by Ilpo Vattulainen and Tomasz Rog. (C) 2016 The Authors. Published by Elsevier B.V.
A combined dynamics consisting of Brownian motion and Levy flights is exhibited by a variety of biological systems performing search processes. Assessing the search reliability of ever locating the target and the search efficiency of doing so economically of such dynamics thus poses an important problem. Here we model this dynamics by a one-dimensional fractional Fokker-Planck equation combining unbiased Brownian motion and Levy flights. By solving this equation both analytically and numerically we show that the superposition of recurrent Brownian motion and Levy flights with stable exponent alpha < 1, by itself implying zero probability of hitting a point on a line, leads to transient motion with finite probability of hitting any point on the line. We present results for the exact dependence of the values of both the search reliability and the search efficiency on the distance between the starting and target positions as well as the choice of the scaling exponent a of the Levy flight component.
A topic of intense current investigation pursues the question of how the highly crowded environment of biological cells affects the dynamic properties of passively diffusing particles. Motivated by recent experiments we report results of extensive simulations of the motion of a finite sized tracer particle in a heterogeneously crowded environment made up of quenched distributions of monodisperse crowders of varying sizes in finite circular two-dimensional domains. For given spatial distributions of monodisperse crowders we demonstrate how anomalous diffusion with strongly non-Gaussian features arises in this model system. We investigate both biologically relevant situations of particles released either at the surface of an inner domain or at the outer boundary, exhibiting distinctly different features of the observed anomalous diffusion for heterogeneous distributions of crowders. Specifically we reveal an asymmetric spreading of tracers even at moderate crowding. In addition to the mean squared displacement (MSD) and local diffusion exponent we investigate the magnitude and the amplitude scatter of the time averaged MSD of individual tracer trajectories, the non-Gaussianity parameter, and the van Hove correlation function. We also quantify how the average tracer diffusivity varies with the position in the domain with a heterogeneous radial distribution of crowders and examine the behaviour of the survival probability and the dynamics of the tracer survival probability. Inter alia, the systems we investigate are related to the passive transport of lipid molecules and proteins in two-dimensional crowded membranes or the motion in colloidal solutions or emulsions in effectively two-dimensional geometries, as well as inside supercrowded, surface adhered cells.
It is quite generally assumed that the overdamped Langevin equation provides a quantitative description of the dynamics of a classical Brownian particle in the long time limit. We establish and investigate a paradigm anomalous diffusion process governed by an underdamped Langevin equation with an explicit time dependence of the system temperature and thus the diffusion and damping coefficients. We show that for this underdamped scaled Brownian motion (UDSBM) the overdamped limit fails to describe the long time behaviour of the system and may practically even not exist at all for a certain range of the parameter values. Thus persistent inertial effects play a non-negligible role even at significantly long times. From this study a general questions on the applicability of the overdamped limit to describe the long time motion of an anomalously diffusing particle arises, with profound consequences for the relevance of overdamped anomalous diffusion models. We elucidate our results in view of analytical and simulations results for the anomalous diffusion of particles in free cooling granular gases.