A biological interpretation of transient anomalous subdiffusion. I. Qualitative model

Anomalous subdiffusion has been reported for two-dimensional diffusion in the plasma membrane and three-dimensional diffusion in the nucleus and cytoplasm. If a particle diffuses in a suitable infinite hierarchy of binding sites, diffusion is well known to be anomalous at all times. But if the hierarchy is finite, diffusion is anomalous at short times and normal at long times. For a prescribed set of binding sites, Monte Carlo calculations yield the anomalous diffusion exponent and the average time over which diffusion is anomalous. If even a single binding site is present, there is a very short, almost artifactual, period of anomalous subdiffusion, but a hierarchy of binding sites extends the anomalous regime considerably. As is well known, an essential requirement for anomalous subdiffusion due to binding is that the diffusing particle cannot be in thermal equilibrium with the binding sites; an equilibrated particle diffuses normally at all times. Anomalous subdiffusion due to barriers, however, still occurs at thermal equilibrium, and anomalous subdiffusion due to a combination of binding sites and barriers is reduced but not eliminated on equilibration. This physical model is translated directly into a plausible biological model testable by single-particle tracking.     

Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions

We introduce fractional Nernst-Planck equations and derive fractional cable equations as macroscopic models for electrodiffusion of ions in nerve cells when molecular diffusion is anomalous subdiffusion due to binding, crowding or trapping. The anomalous subdiffusion is modelled by replacing diffusion constants with time dependent operators parameterized by fractional order exponents. Solutions are obtained as functions of the scaling parameters for infinite cables and semi-infinite cables with instantaneous current injections. Voltage attenuation along dendrites in response to alpha function synaptic inputs is computed. Action potential firing rates are also derived based on simple integrate and fire versions of the models. Our results show that electrotonic properties and firing rates of nerve cells are altered by anomalous subdiffusion in these models. We have suggested electrophysiological experiments to calibrate and validate the models.      

Anomalous subdiffusion in fluorescence photobleaching recovery: a Monte Carlo study

Anomalous subdiffusion is hindered diffusion in which the mean-square displacement of a diffusing particle is proportional to some power of time less than one. Anomalous subdiffusion has been observed for a variety of lipids and proteins in the plasma membranes of a variety of cells. Fluorescence photobleaching recovery experiments with anomalous subdiffusion are simulated to see how to analyze the data. It is useful to fit the recovery curve with both the usual recovery equation and the anomalous one, and to judge the goodness of fit on log-log plots. The simulations show that the simplest approximate treatment of anomalous subdiffusion usually gives good results. Three models of anomalous subdiffusion are considered: obstruction, fractional Brownian motion, and the continuous-time random walk. The models differ significantly in their behavior at short times and in their noise level. For obstructed diffusion the approach to the percolation threshold is marked by a large increase in noise, a broadening of the distribution of diffusion coefficients and anomalous subdiffusion exponents, and the expected abrupt decrease in the mobile fraction. The extreme fluctuations in the recovery curves at and near the percolation threshold result from extreme fluctuations in the geometry of the percolation cluster.     

Analysis of Reaction-Diffusion Systems with Anomalous Subdiffusion

Reaction-diffusion equations are the cornerstone of modeling biochemical systems with spatial gradients, which are relevant to biological processes such as signal transduction. Implicit in the formulation of these equations is the assumption of Fick's law, which states that the local diffusive flux of species i is proportional to its concentration gradient; however, in the context of complex fluids such as cytoplasm and cell membranes, the use of Fick's law is based on empiricism, whereas evidence has been mounting that such media foster anomalous subdiffusion (with mean-squared displacement increasing less than linearly with time) over certain length scales. Particularly when modeling diffusion-controlled reactions and other systems where the spatial domain is considered semi-infinite, assuming Fickian diffusion might not be appropriate. In this article, two simple, conceptually extreme models of anomalous subdiffusion are used in the framework of Green's functions to demonstrate the solution of four reaction-diffusion problems that are well known in the biophysical context of signal transduction: fluorescence recovery after photobleaching, the Smolochowski limit for diffusion-controlled reactions in solution, the spatial range of a diffusing molecule with finite lifetime, and the collision coupling mechanism of diffusion-controlled reactions in two dimensions. In each case, there are only subtle differences between the two subdiffusion models, suggesting how measurements of mean-squared displacement versus time might generally inform models of reactive systems with partial diffusion control.     

Probing Position-Dependent Diffusion in Folding Reactions Using Single-Molecule Force Spectroscopy

Folding of proteins and nucleic acids involves a diffusive search over a multidimensional conformational energy landscape for the minimal-energy structure. When examining the projection of conformational motions onto a one-dimensional reaction coordinate, as done in most experiments, the diffusion coefficient D is generally position dependent. However, it has proven challenging to measure such position-dependence experimentally. We investigated the position-dependence of D in the folding of DNA hairpins as a simple model system in two ways: first, by analyzing the round-trip time to return to a given extension in constant-force extension trajectories measured by force spectroscopy, and second, by analyzing the fall time required to reach a given extension in force jump measurements. These methods yielded conflicting results: the fall time implied a fairly constant D, but the round-trip time implied variations of over an order of magnitude. Comparison of experiments with computational simulations revealed that both methods were strongly affected by experimental artifacts inherent to force spectroscopy measurements, which obscured the intrinsic position-dependence of D. Lastly, we applied Kramers’s theory to the kinetics of hairpins with energy barriers located at different positions along the hairpin stem, as a crude probe of D at different stem positions, and we found that D did not vary much as the barrier was moved along the reaction coordinate. This work underlines the difficulties faced when trying to deduce position-dependent diffusion coefficients from experimental folding trajectories.     

Mechanism of translocation of uracil-DNA glycosylase from Escherichia coli between distributed lesions

Uracil–DNA glycosylase (Ung) is a DNA repair enzyme that excises uracil bases from DNA, where they appear through deamination of cytosine or incorporation from a cellular dUTP pool. DNA repair enzymes often use one-dimensional diffusion along DNA to accelerate target search; however, this mechanism remains poorly investigated mechanistically. We used oligonucleotide substrates containing two uracil residues in defined positions to characterize one-dimensional search of DNA by Escherichia coli Ung. Mg²⁺ ions suppressed the search in double-stranded DNA to a higher extent than K⁺ likely due to tight binding of Mg²⁺ to DNA phosphates. Ung was able to efficiently overcome short single-stranded gaps within double-stranded DNA. Varying the distance between the lesions and fitting the data to a theoretical model of DNA random walk, we estimated the characteristic one-dimensional search distance of ∼100 nucleotides and translocation rate constant of ∼2 × 10⁶ s⁻¹.     

Quantitative Analysis of Single Particle Trajectories: Mean Maximal Excursion Method

An increasing number of experimental studies employ single particle tracking to probe the physical environment in complex systems. We here propose and discuss what we believe are new methods to analyze the time series of the particle traces, in particular, for subdiffusion phenomena. We discuss the statistical properties of mean maximal excursions (MMEs), i.e., the maximal distance covered by a test particle up to time t. Compared to traditional methods focusing on the mean-squared displacement we show that the MME analysis performs better in the determination of the anomalous diffusion exponent. We also demonstrate that combination of regular moments with moments of the MME method provides additional criteria to determine the exact physical nature of the underlying stochastic subdiffusion processes. We put the methods to test using experimental data as well as simulated time series from different models for normal and anomalous dynamics such as diffusion on fractals, continuous time random walks, and fractional Brownian motion.     

Deviation from the Einstein relation of the single-file diffusion coefficient

The deviation of the diffusion coefficient D of a one-dimensional fluid from its tracer diffusion coefficient D^★ is calculated from Gürsey's equation of State, using a square-well interaction potential. Enormous departures from the Einstein relation D = D^* are noted.     

Hydrophobic and electrostatic interactions modulate protein escape at the ribosomal exit tunnel

After translation, nascent proteins must escape the ribosomal exit tunnel to attain complete folding to their native states. This escape process also frees up the ribosome tunnel for a new translation job. In this study, we investigate the impacts of energetic interactions between the ribosomal exit tunnel and nascent proteins on the protein escape process by molecular dynamics simulations using partially coarse-grained models that incorporate hydrophobic and electrostatic interactions of the ribosome tunnel of Haloarcula marismortui with nascent proteins. We find that, in general, attractive interactions slow down the protein escape process, whereas repulsive interactions speed it up. For the small globular proteins considered, the median escape time correlates with both the number of hydrophobic residues, N_h, and the net charge, Q, of a nascent protein. A correlation coefficient exceeding 0.96 is found for the relation between the median escape time and a combined quantity of N_h + 5.9Q, suggesting that it is ∼6 times more efficient to modulate the escape time by changing the total charge than the number of hydrophobic residues. The estimated median escape times are found in the submillisecond-to-millisecond range, indicating that the escape does not delay the ribosome recycling. For various types of the tunnel model, with and without hydrophobic and electrostatic interactions, the escape time distribution always follows a simple diffusion model that describes the escape process as a downhill drift of a Brownian particle, suggesting that nascent proteins escape along barrier-less pathways at the ribosome tunnel.     

Genome-wide survey of DNA-binding proteins in Arabidopsis thaliana: analysis of distribution and functions

The interaction of proteins with their respective DNA targets is known to control many high-fidelity cellular processes. Performing a comprehensive survey of the sequenced genomes for DNA-binding proteins (DBPs) will help in understanding their distribution and the associated functions in a particular genome. Availability of fully sequenced genome of Arabidopsis thaliana enables the review of distribution of DBPs in this model plant genome. We used profiles of both structure and sequence-based DNA-binding families, derived from PDB and PFam databases, to perform the survey. This resulted in 4471 proteins, identified as DNA-binding in Arabidopsis genome, which are distributed across 300 different PFam families. Apart from several plant-specific DNA-binding families, certain RING fingers and leucine zippers also had high representation. Our search protocol helped to assign DNA-binding property to several proteins that were previously marked as unknown, putative or hypothetical in function. The distribution of Arabidopsis genes having a role in plant DNA repair were particularly studied and noted for their functional mapping. The functions observed to be overrepresented in the plant genome harbour DNA-3-methyladenine glycosylase activity, alkylbase DNA N-glycosylase activity and DNA-(apurinic or apyrimidinic site) lyase activity, suggesting their role in specialized functions such as gene regulation and DNA repair.     

Analysis of fluorophore diffusion by continuous distributions of diffusion coefficients: application to photobleaching measurements of multicomponent and anomalous diffusion.

Fluorescence recovery after photobleaching (FRAP) is widely used to measure fluorophore diffusion in artificial solutions and cellular compartments. Two new strategies to analyze FRAP data were investigated theoretically and applied to complex systems with anomalous diffusion or multiple diffusing species: 1) continuous distributions of diffusion coefficients, alpha(D), and 2) time-dependent diffusion coefficients, D(t). A regression procedure utilizing the maximum entropy method was developed to resolve alpha(D) from fluorescence recovery curves, F(t). The recovery of multi-component alpha(D) from simulated F(t) with random noise was demonstrated and limitations of the method were defined. Single narrow Gaussian alpha(D) were recovered for FRAP measurements of thin films of fluorescein and size-fractionated FITC-dextrans and Ficolls, and multi-component alpha(D) were recovered for defined fluorophore mixtures. Single Gaussian alpha(D) were also recovered for solute diffusion in viscous media containing high dextran concentrations. To identify anomalous diffusion from FRAP data, a theory was developed to compute F(t) and alpha(D) for anomalous diffusion models defined by arbitrary nonlinear mean-squared displacement <x2> versus time relations. Several characteristic alpha(D) profiles for anomalous diffusion were found, including broad alpha(D) for subdiffusion, and alpha(D) with negative amplitudes for superdiffusion. A method to deduce apparent D(t) from F(t) was also developed and shown to provide useful complementary information to alpha(D). alpha(D) and D(t) were determined from photobleaching measurements of systems with apparent anomalous subdiffusion (nonuniform solution layer) and superdiffusion (moving fluid layer). The results establish a practical strategy to characterize complex diffusive phenomena from photobleaching recovery measurements.     

Protein Sliding along DNA: Dynamics and Structural Characterization

Efficient search of DNA by proteins is fundamental to the control of cellular regulatory processes. It is currently believed that protein sliding, hopping, and transfer between adjacent DNA segments, during which the protein nonspecifically interacts with DNA, are central to the speed of their specific recognition. In this study, we focused on the structural and dynamic features of proteins when they scan the DNA. Using a simple computational model that represents protein-DNA interactions by electrostatic forces, we identified that the protein makes use of identical binding interfaces for both nonspecific and specific DNA interactions. Accordingly, in its one-dimensional diffusion along the DNA, the protein is bound at the major groove and performs a helical motion, which is stochastic and driven by thermal diffusion. A microscopic structural insight into sliding from our model, which is governed by electrostatic forces, corroborates previous experimental studies suggesting that the active site of some regulatory proteins continually faces the interior of the DNA groove while sliding along sugar-phosphate rails. The diffusion coefficient of spiral motion along the major groove of the DNA is not affected by salt concentration, but the efficiency of the search can be significantly enhanced by increasing salt concentration due to a larger number of hopping events. We found that the most efficient search comprises ∼ 20% sliding along the DNA and ∼ 80% hopping and three-dimensional diffusion. The presented model that captures various experimental features of facilitated diffusion has the potency to address other questions regarding the nature of DNA search, such as the sliding characteristics of oligomeric and multidomain DNA-binding proteins that are ubiquitous in the cell.