Role of substrate unbinding in Michaelis-Menten enzymatic reactions.

The Michaelis-Menten equation provides a hundred-year-old prediction by which any increase in the rate of substrate unbinding will decrease the rate of enzymatic turnover. Surprisingly, this prediction was never tested experimentally nor was it scrutinized using modern theoretical tools. Here we show that unbinding may also speed up enzymatic turnover--turning a spotlight to the fact that its actual role in enzymatic catalysis remains to be determined experimentally. Analytically constructing the unbinding phase space, we identify four distinct categories of unbinding: inhibitory, excitatory, superexcitatory, and restorative. A transition in which the effect of unbinding changes from inhibitory to excitatory as substrate concentrations increase, and an overlooked tradeoff between the speed and efficiency of enzymatic reactions, are naturally unveiled as a result. The theory presented herein motivates, and allows the interpretation of, groundbreaking experiments in which existing single-molecule manipulation techniques will be adapted for the purpose of measuring enzymatic turnover under a controlled variation of unbinding rates. As we hereby show, these experiments will not only shed first light on the role of unbinding but will also allow one to determine the time distribution required for the completion of the catalytic step in isolation from the rest of the enzymatic turnover cycle.

Test for determining a subdiffusive model in ergodic systems from single trajectories.

Experiments on particle motion show that it is often subdiffusive. This subdiffusion may be due to trapping, percolationlike structures, or viscoelastic behavior of the medium. While the models based on trapping (leading to continuous-time random walks) can easily be distinguished from the rest by testing their nonergodicity, the latter two cases are harder to distinguish. We propose a statistical test for distinguishing between these two based on the space-filling properties of trajectories, and prove its feasibility and specificity using synthetic data. We moreover present a flow chart for making a decision on a type of subdiffusion for a broader class of models.

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))(2)>≳1, S(k,t) effectively decays as a stretched exponential S(k,t)≃S(k)e(-(Γ(k)t)(ß)) with ß≃ν, where the relaxation rate is Γ(k)~(k(B)T/mω(o)(2))(1/ß)k(2/ß), 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 ß 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.

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.

Unusual response to a localized perturbation in a generalized elastic model.

The generalized elastic model encompasses several physical systems such as polymers, membranes, single-file systems, fluctuating surfaces, and rough interfaces. We consider the case of an applied localized potential, namely, an external force acting only on a single (tagged) probe, leaving the rest of the system unaffected. We derive the fractional Langevin equation for the tagged probe, as well as for a generic (untagged) probe, where the force is not directly applied. Within the framework of the fluctuation-dissipation relations, we discuss the unexpected physical scenarios arising when the force is constant and time periodic, whether or not the hydrodynamic interactions are included in the model. For short times, in the case of the constant force, we show that the average drift is linear in time for long-range hydrodynamic interactions and behaves ballistically or exponentially for local hydrodynamic interactions. Moreover, it can be opposite to the direction of the external disturbance for some values of the model's parameters. When the force is time periodic, the effects are macroscopic: the system splits into two distinct spatial regions whose size is proportional to the value of the applied frequency. These two regions are characterized by different amplitudes and phase shifts in the response dynamics.

Accurate quantification of diffusion and binding kinetics of non-integral membrane proteins by FRAP.

Non-integral membrane proteins frequently act as transduction hubs in vital signaling pathways initiated at the plasma membrane (PM). Their biological activity depends on dynamic interactions with the PM, which are governed by their lateral and cytoplasmic diffusion and membrane binding/unbinding kinetics. Accurate quantification of the multiple kinetic parameters characterizing their membrane interaction dynamics has been challenging. Despite a fair number of approximate fitting functions for analyzing fluorescence recovery after photobleaching (FRAP) data, no approach was able to cope with the full diffusion-exchange problem. Here, we present an exact solution and matlab fitting programs for FRAP with a stationary Gaussian laser beam, allowing simultaneous determination of the membrane (un)binding rates and the diffusion coefficients. To reduce the number of fitting parameters, the cytoplasmic diffusion coefficient is determined separately. Notably, our equations include the dependence of the exchange kinetics on the distribution of the measured protein between the PM and the cytoplasm, enabling the derivation of both k(on) and k(off) without prior assumptions. After validating the fitting function by computer simulations, we confirm the applicability of our approach to live-cell data by monitoring the dynamics of GFP-N-Ras mutants under conditions with different contributions of lateral diffusion and exchange to the FRAP kinetics.

Challenges in determining anomalous diffusion in crowded fluids.

Anomalous diffusion in crowded fluids, e.g. in the cytoplasm of living cells, is a frequent phenomenon. Despite manifold observations of anomalous diffusion with several experimental techniques, a thorough understanding of the underlying microscopic causes is still lacking. Here, we have quantitatively compared two popular techniques with which anomalous diffusion is typically assessed. Using extensive computer simulations of two prototypical random walks with stationary increments, i.e. fractional brownian motion and obstructed diffusion, we find that single particle tracking (SPT) yields results for the diffusion anomaly that are equivalent to those obtained by fluorescence correlation spectroscopy (FCS). We also show that positional uncertainties, inherent to SPT experiments, lead to a systematic underestimation of the diffusion anomaly, regardless of the underlying random walk and measurement technique. This effect becomes particularly relevant when the position uncertainty is larger than the average positional displacement between two successive frames.

Distribution of first-passage times to specific targets on compactly explored fractal structures.

The distribution of the first passage times (FPT) of a one-dimensional random walker to a target site follows a power law F(t)~t(-3/2). We generalize this result to another situation pertinent to compact exploration and consider the FPT of a random walker with specific source and target points on an infinite fractal structure with spectral dimension d(s)<2. We show that the probability density of the first return to the origin has the form F(t)~t(d(s)/2-2), and the FPT to a specific target at distance r follows the law F(r,t)~r(d(w)-d(f)) t(d(s)/2-2), where d(w) and d(f) are the walk dimension and the fractal dimension of the structure, respectively. The distance dependence of F(r,t) reproduces the one of the mean FPT of a random walk in a confined domain.

Unequal twins: probability distributions do not determine everything.

It is the common lore to assume that knowing the equation for the probability distribution function (PDF) of a stochastic model as a function of time tells the whole picture defining all other characteristics of the model. We show that this is not the case by comparing two exactly solvable models of anomalous diffusion due to geometric constraints: the comb model and the random walk on a random walk. We show that though the two models have exactly the same PDFs, they differ in other respects, like their first passage time distributions, their autocorrelation functions, and their aging properties.

Hopping around an entropic barrier created by force.

We use Langevin dynamics to investigate the role played by the recently discovered force-induced entropic energy barrier on the two-state hopping phenomena that has been observed in single RNA, DNA and protein molecules placed under a stretching force. Simple considerations about the free energy of a molecule readily show that the application of force introduces an entropic barrier separating the collapsed state of the molecule, from a force-driven extended conformation. A notable characteristic of the force induced barrier is its long distances to transition state, up to tens of nanometers, which renders the kinetics of crossing this barrier highly sensitive to an applied force. Langevin dynamics across such force induced barriers readily demonstrates the hopping behavior observed for a variety of single molecules placed under force. Such hopping is frequently interpreted as a manifestation of two-state folding/unfolding reactions observed in bulk experiments. However, given that such barriers do not exist at zero force these reactions do not take place at all in bulk.

Universal self-similarity of propagating populations.

This paper explores the universal self-similarity of propagating populations. The following general propagation model is considered: particles are randomly emitted from the origin of a d-dimensional Euclidean space and propagate randomly and independently of each other in space; all particles share a statistically common--yet arbitrary--motion pattern; each particle has its own random propagation parameters--emission epoch, motion frequency, and motion amplitude. The universally self-similar statistics of the particles' displacements and first passage times (FPTs) are analyzed: statistics which are invariant with respect to the details of the displacement and FPT measurements and with respect to the particles' underlying motion pattern. Analysis concludes that the universally self-similar statistics are governed by Poisson processes with power-law intensities and by the Fréchet and Weibull extreme-value laws.

Detecting origins of subdiffusion: P-variation test for confined systems.

In this paper, we propose a method to distinguish between mechanisms leading to single molecule subdiffusion in confinement. We show that the method of p-variation, introduced in the recent paper [M. Magdziarz, Phys. Rev. Lett. 103, 180602 (2009)], can be successfully applied also for confined systems. We propose a test which allows distinguishing between heavy-tailed continuous-time random walk and fractional Brownian motion in the presence of binding potentials and reflecting boundaries. We apply our test to the experimental data describing motion of mRNA molecules inside E. coli cells. The results of the test show that it is more likely that fractional Brownian motion is the underlying process.

Universal generation of 1/f noises.

This paper establishes a universal mechanism for the generation of 1/f noises. The mechanism is based on a signal-superposition model, which superimposes signals transmitted from independent sources. All sources transmit a statistically common-yet arbitrary-stochastic signal pattern; each source has its own random transmission parameters-amplitude, frequency, and initiation epoch. We explore randomizations of the transmission parameters which render the power spectrum of the super-imposed signal invariant with respect of the stochastic signal pattern transmitted. Analysis shows that the aforementioned randomizations yield superimposed signals which are 1/f noises. The result obtained is, in effect, a randomized central limit theorem for 1/f noises.

Randomized central limit theorems: A unified theory.

The central limit theorems (CLTs) characterize the macroscopic statistical behavior of large ensembles of independent and identically distributed random variables. The CLTs assert that the universal probability laws governing ensembles' aggregate statistics are either Gaussian or Lévy, and that the universal probability laws governing ensembles' extreme statistics are Fréchet, Weibull, or Gumbel. The scaling schemes underlying the CLTs are deterministic-scaling all ensemble components by a common deterministic scale. However, there are "random environment" settings in which the underlying scaling schemes are stochastic-scaling the ensemble components by different random scales. Examples of such settings include Holtsmark's law for gravitational fields and the Stretched Exponential law for relaxation times. In this paper we establish a unified theory of randomized central limit theorems (RCLTs)-in which the deterministic CLT scaling schemes are replaced with stochastic scaling schemes-and present "randomized counterparts" to the classic CLTs. The RCLT scaling schemes are shown to be governed by Poisson processes with power-law statistics, and the RCLTs are shown to universally yield the Lévy, Fréchet, and Weibull probability laws.

Anomalies in the vibrational dynamics of proteins are a consequence of fractal-like structure.

Proteins have been shown to exhibit strange/anomalous dynamics displaying non-Debye density of vibrational states, anomalous spread of vibrational energy, large conformational changes, nonexponential decay of correlations, and nonexponential unfolding times. The anomalous behavior may, in principle, stem from various factors affecting the energy landscape under which a protein vibrates. Investigating the origins of such unconventional dynamics, we focus on the structure-dynamics interplay and introduce a stochastic approach to the vibrational dynamics of proteins. We use diffusion, a method sensitive to the structural features of the protein fold and them alone, in order to probe protein structure. Conducting a large-scale study of diffusion on over 500 Protein Data Bank structures we find it to be anomalous, an indication of a fractal-like structure. Taking advantage of known and newly derived relations between vibrational dynamics and diffusion, we demonstrate the equivalence of our findings to the existence of structurally originated anomalies in the vibrational dynamics of proteins. We conclude that these anomalies are a direct result of the fractal-like structure of proteins. The duality between diffusion and vibrational dynamics allows us to make, on a single-molecule level, experimentally testable predictions. The time dependent vibrational mean square displacement of an amino acid is predicted to be subdiffusive. The thermal variance in the instantaneous distance between amino acids is shown to grow as a power law of the equilibrium distance. Mean first passage time analysis is offered as a practical tool that may aid in the identification of amino acid pairs involved in large conformational changes.

Probing static disorder in Arrhenius kinetics by single-molecule force spectroscopy.

The widely used Arrhenius equation describes the kinetics of simple two-state reactions, with the implicit assumption of a single transition state with a well-defined activation energy barrier DeltaE, as the rate-limiting step. However, it has become increasingly clear that the saddle point of the free-energy surface in most reactions is populated by ensembles of conformations, leading to nonexponential kinetics. Here we present a theory that generalizes the Arrhenius equation to include static disorder of conformational degrees of freedom as a function of an external perturbation to fully account for a diverse set of transition states. The effect of a perturbation on static disorder is best examined at the single-molecule level. Here we use force-clamp spectroscopy to study the nonexponential kinetics of single ubiquitin proteins unfolding under force. We find that the measured variance in DeltaE shows both force-dependent and independent components, where the force-dependent component scales with F(2), in excellent agreement with our theory. Our study illustrates a novel adaptation of the classical Arrhenius equation that accounts for the microscopic origins of nonexponential kinetics, which are essential in understanding the rapidly growing body of single-molecule data.

Collapse dynamics of single proteins extended by force.

Single-molecule force spectroscopy has opened up new approaches to the study of protein dynamics. For example, an extended protein folding after an abrupt quench in the pulling force was shown to follow variable collapse trajectories marked by well-defined stages that departed from the expected two-state folding behavior that is commonly observed in bulk. Here, we explain these observations by developing a simple approach that models the free energy of a mechanically extended protein as a combination of an entropic elasticity term and a short-range potential representing enthalpic hydrophobic interactions. The resulting free energy of the molecule shows a force-dependent energy barrier of magnitude, DeltaE =epsilon(F - F(c))(3/2), separating the enthalpic and entropic minima that vanishes at a critical force F(c). By solving the Langevin equation under conditions of a force quench, we generate folding trajectories corresponding to the diffusional collapse of an extended polypeptide. The predicted trajectories reproduce the different stages of collapse, as well as the magnitude and time course of the collapse trajectories observed experimentally in ubiquitin and I27 protein monomers. Our observations validate the force-clamp technique as a powerful approach to determining the free-energy landscape of proteins collapsing and folding from extended states.

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.

Generalized elastic model yields a fractional Langevin equation description.

Starting from a generalized elastic model which accounts for the stochastic motion of several physical systems such as membranes, (semi)flexible polymers, and fluctuating interfaces among others, we derive the fractional Langevin equation (FLE) for a probe particle in such systems, in the case of thermal initial conditions. We show that this FLE is the only one fulfilling the fluctuation-dissipation relation within a new family of fractional Brownian motion equations. The FLE for the time-dependent fluctuations of the donor-acceptor distance in a protein is shown to be recovered. When the system starts from nonthermal conditions, the corresponding FLE, which does not fulfill the fluctuation-dissipation relation, is derived.

Subdiffusion of mixed origins: when ergodicity and nonergodicity coexist.

Single particle trajectories are investigated assuming the coexistence of two subdiffusive processes: diffusion on a fractal structure modeling spatial constraints on motion and heavy-tailed continuous time random walks representing energetic or chemical traps. The particles' mean squared displacement is found to depend on the way the mean is taken: temporal averaging over single-particle trajectories differs from averaging over an ensemble of particles. This is shown to stem from subordinating an ergodic anomalous process to a nonergodic one. The result is easily generalized to the subordination of any other ergodic process (i.e., fractional Brownian motion) to a nonergodic one. For certain parameters the ergodic diffusion on the underlying fractal structure dominates the transport yet displaying ergodicity breaking and aging.