Free Brownian Motion Research Articles

Escape Through a Small Opening: Receptor Trafficking in a Synaptic Membrane

We model the motion of a receptor on the membrane surface of a synapse as free Brownian motion in a planar domain with intermittent trappings in and escapes out of corrals with narrow openings. We compute the mean confinement time of the Brownian particle in the asymptotic limit of a narrow opening and calculate the probability to exit through a given small opening, when the boundary contains more than one. Using this approach, it is possible to describe the Brownian motion of a random particle in an environment containing domains with small openings by a coarse grained diffusion process. We use the results to estimate the confinement time as a function of the parameters and also the time it takes for a diffusing receptor to be anchored at its final destination on the postsynaptic membrane, after it is inserted in the membrane. This approach provides a framework for the theoretical study of receptor trafficking on membranes. This process underlies synaptic plasticity, which relates to learning and memory. In particular, it is believed that the memory state in the brain is stored primarily in the pattern of synaptic weight values, which are controlled by neuronal activity. At a molecular level, the synaptic weight is determined by the number and properties of protein channels (receptors) on the synapse. The synaptic receptors are trafficked in and out of synapses by a diffusion process. Following their synthesis in the endoplasmic reticulum, receptors are trafficked to their postsynaptic sites on dendrites and axons. In this model the receptors are first inserted into the extrasynaptic plasma membrane and then random walk in and out of corrals through narrow openings on their way to their final destination.

Large deviation bounds for matrix Brownian motion

We prove large deviation bounds for the convergence of Hermitian matrix valued Brownian motion towards free Brownian motion. As a consequence, we obtain upper and lower bounds on the microstates entropy introduced by Voiculescu [24].

Levy area for the free Brownian motion: existence and non-existence

This note extends the work of Capitaine (J. Funct. Anal. 179 (1) (2001) 153) on the Levy area process for the free Brownian motion in two directions. First, we reprove that a Levy area for the Free Brownian motion exists in the Von Neumann tensor product, by exhibiting a non-commutative Burkholder–Davis–Gundy type inequality. Then, we show that there does not exist a Levy area in the projective tensor product.

Generalised Brownian Motion and Second Quantisation

A new approach to the generalised Brownian motion introduced by M. Bożejko and R. Speicher is described, based on symmetry rather than deformation. The symmetrisation principle is provided by Joyal's notions of tensorial and combinatorial species. Any such species V gives rise to an endofunctor FV of the category of Hilbert spaces with contractions. A generalised Brownian motion is an algebra of creation and annihilation operators acting on FV(H) for arbitrary Hilbert spaces H and having a prescription for the calculation of vacuum expectations in terms of a function t on pair partitions. The positivity is encoded by a *-semigroup of broken pair partitions whose representation space with respect to t is V. The existence of the second quantisation as functor Γt from Hilbert spaces to noncommutative probability spaces is investigated for functions t with the multiplicative property. For a certain one parameter interpolation between the fermionic and the free Brownian motion it is shown that the field algebras Γ(K) are type II1 factors when K is infinite dimensional.

Free diffusions, free entropy and free Fisher information

Motivated by the stochastic quantization approach to large N matrix models, we study solutions to free stochastic differential equations dX t=dS t− 1 2 f(X t) dt where S t is a free brownian motion. We show existence, uniqueness and Markov property of solutions. We define a relative free entropy as well as a relative free Fisher information, and show that these quantities behave as in the classical case. Finally we show that, in contrast with classical diffusions, in general the asymptotic distribution of the free diffusion does not converge, as t→∞, towards the master field (i.e., the Gibbs state).

Restriction of secretory granule motion near the plasma membrane of chromaffin cells.

We used total internal reflection fluorescence microscopy to study quantitatively the motion and distribution of secretory granules near the plasma membrane (PM) of living bovine chromaffin cells. Within the approximately 300-nm region measurably illuminated by the evanescent field resulting from total internal reflection, granules are preferentially concentrated close to the PM. Granule motion normal to the substrate (the z direction) is much slower than would be expected from free Brownian motion, is strongly restricted over tens of nanometer distances, and tends to reverse directions within 0.5 s. The z-direction diffusion coefficients of granules decrease continuously by two orders of magnitude within less than a granule diameter of the PM as granules approach the PM. These analyses suggest that a system of tethers or a heterogeneous matrix severely limits granule motion in the immediate vicinity of the PM. Transient expression of the light chains of tetanus toxin and botulinum toxin A did not disrupt the restricted motion of granules near the PM, indicating that SNARE proteins SNAP-25 and VAMP are not necessary for the decreased mobility. However, the lack of functional SNAREs on the plasma or granule membranes in such cells reduces the time that some granules spend immediately adjacent to the PM.

Quantum fluctuation-dissipation theorem: a time-domain formulation

A time-domain formulation of the equilibrium quantum fluctuation–dissipation theorem (FDT) in the whole range of temperatures is presented. In the classical limit, the FDT establishes a proportionality relation between the dissipative part of the linear response function and the derivative of the corresponding equilibrium correlation function. At zero temperature, the FDT takes the form of Hilbert transform relations between the dissipative part of the response function and the corresponding symmetrized equilibrium correlation function, and thus allows to establish a connection with the analytic signal theory. The harmonic oscillator is treated as a basic example. The time-domain formulation of the relation between dissipation and fluctuations is especially valuable in out-of-equilibrium dynamics. We show on the specific example of the free particle quantum Brownian motion that it is the case in the discussion of the aging phenomena.

The Lévy Area Process for the Free Brownian Motion

In this paper, we construct a Lévy area process for the free Brownian motion and in this way, a typical geometric rough path (in the sense of T. Lyons (1998, Rev. Mat. Iberoamer.14, 215–310)), lying above the free Brownian path. Thus, the general results of Lyons on differential equations driven by rough paths may be applied to the free Brownian motion case.

Aging effects in free quantum Brownian motion

The two-time correlation function C xx ( t, t′) of the displacement x( t)− x( t 0) of a free quantum Brownian particle with respect to its position at a given time t 0 is calculated analytically in the framework of the Caldeira and Leggett ohmic dissipation model. As a result, at any temperature T, C xx(t,t′) exhibits aging, i.e. it depends explicitly on both times t and t′ and not only on the time difference τ= t− t′, even in the limit of large age t′ (t 0⩽t′⩽t) , in contrast with a dynamic variable in equilibrium such as the particle velocity. The equilibrium quantum fluctuation-dissipation theorem (QFDT) has to be modified in order to relate the response function χ xx ( t, t′) to C xx ( t, t′), since this latter quantity takes into account even those fluctuations of the displacement which take place during the waiting time t w = t′− t 0. We describe the deviation from QFDT in terms of an effective inverse temperature β eff ( τ, t w ). The behaviour of this quantity as a function of τ for given values of T and t w is analysed. In the classical limit it is shown that β eff ( τ, t w )= βD( τ)/[ D( τ)+ D( t w )], where D( t) denotes the time-dependent diffusion coefficient.

Binary collision model for quantum brownian motion

A binary collision model for phenomena of quantum dissipation is developed. Unlike the harmonic oscillator model, widely used for over thirty years, this model assumes nonlinear coupling between system and environment, and is applicable to both bosonic and fermionic baths. The system interacts with an ideal bath through binary collisions only. Solutions for the classical and quantum-mechanical problems in the case of free Brownian motion are presented, and the quantum-classical correspondence for nonequilibrium processes is established. It is shown that in the Brownian motion limit the two models lead to identical dynamical behavior, provided the coupling coefficients in the harmonic oscillator Hamiltonian are temperature dependent. For cases of bath particles of finite mass and number the two models lead to different results. Linear response theory for the model is developed, and the results are compared with those for the harmonic oscillator model. At the end, possible applications of the model are suggested.

Anomalous diffusion of fluorescent probes inside living cell nuclei investigated by spatially-resolved fluorescence correlation spectroscopy

We have investigated spatial variations of the diffusion behavior of the green fluorescent protein mutant EGFP (F64L/S65T) and of the EGFP-β-galactosidase fusion protein in living cells with fluorescence correlation spectroscopy. Our fluorescence correlation spectroscopy device, in connection with a precision x-y translation stage, provides submicron spatial resolution and a detection volume smaller than a femtoliter. The fluorescence fluctuations in cell lines expressing EGFP are caused by molecular diffusion as well as a possible internal and a pH-dependent external protonation process of the EGFP chromophore. The latter processes result in two apparent nonfluorescent states that have to be taken into account when evaluating the fluorescence correlation spectroscopy data. The diffusional contribution deviates from ideal behavior and depends on the position in the cell. The fluorescence correlation spectroscopy data can either be evaluated as a two component model with one fraction of the molecules undergoing free Brownian motion with a diffusion coefficient approximately five times smaller than in aqueous solution, and another fraction diffusing one or two orders of magnitude slower. This latter component is especially noticeable in the nuclei. Alternatively, we can fit the data to an anomalous diffusion model where the time dependence of the diffusion serves as a measure for the degree of obstruction, which is large especially in nuclei. Possible mechanisms for this long tail behavior include corralling, immobile obstacles, and binding with a broad distribution of binding affinities. The results are consistent with recent numerical models of the chromosome territory structure in the cell nucleus.

Free Brownian Motion of Individual Lipid Molecules in Biomembranes

The mobility of phospolipids in free-standing and supported membranes was investigated on the level of individual molecules. For the analysis of trajectories a new statistical treatment was developed that permitted us to clearly distinguish different types of diffusional motion. A freely diffusing subfraction of lipids within supported membranes was identified. Its mobility was characterized by a mean lateral diffusion constant of D supp = 4.6 μm 2/s. In comparison, the mobility of lipids embedded in “free-standing” planar membranes yielded an increase in the mean diffusion constant by a factor of 4.5, D free = 20.6 μm 2/s. This increase is attributed to the ultrathin (≤1 nm) lubricating water layer between membranes and glass support.

Influence of inertial and memory friction effects on the rotational Brownian motion of noninteracting particles : application to the dielectric relaxation response and comparison with previous theories

Abstract A theoretical approach of free rotational Brownian motion of noninteracting particles compelled to rotate in two dimensions (disk model) and experiencing exponentially varying memory friction is presented both in time and frequency domains. This leads to obtain exact analytic expressions for the orientational correlation functions appropriate to dielectric relaxation. It is found that three regimes are possible for the angular velocity depending on the ratio β / γ, where β is the reciprocal of the frictional time and γ is the reciprocal of the charateristic time of the heat bath. The results so obtained are discussed with the help of plots giving the time evolution of the correlation function and allowing one to show only small differences with the small collision model of Sack [Proc.Phys Soc.B70, 402 (1957)]. These differences appear to be much more pronounced when the problem is analyzed in frequency as illustrated in Cole-Cole plots for the dynamic susceptibility. A formal connection with Madden and Kivelson's model is also proposed.

Stochastic calculus with respect to free Brownian motion and analysis on Wigner space

We define stochastic integrals with respect to free Brownian motion, and show that they satisfy Burkholder-Gundy type inequalities in operator norm. We prove also a version of Ito's predictable representation theorem, as well as product form and functional form of Ito's formula. Finally we develop stochastic analysis on the free Fock space, in analogy with stochastic analysis on the Wiener space.

Central limit theorem for the Edwards model

The Edwards model in one dimension is a transformed path measure for standard Brownian motion discouraging self-intersections. We prove a central limit theorem for the endpoint of the path, extending a law of large numbers proved by Westwater. The scaled variance is characterized in terms of the largest eigenvalue of a one-parameter family of differential operators, introduced and analyzed by van der Hofstad and den Hollander. Interestingly, the scaled variance turns out to be independent of the strength of self-repellence and to be strictly smaller than one (the value for free Brownian motion).

Sticky dumbbells: from Hookean dumbbells to transient network

The Green-Tobolsky theory of transient networks is merged to the Hookean dumbbell model by considering Hookean sticky dumbbells, whose beads can randomly be stuck to a network submitted to affine deformation, or be set free from the network and undergo a free diffusive Brownian motion in the solvent. Sticking to and releasing from the network is treated as an instantaneous chemical reaction. This model has a closed-form solution, in which the stress is the sum of two (resp. three) Maxwellian codeformational relaxations for dumbbells with one (resp. two) sticking beads. When Brownian diffusion is faster than the chemical kinetics, one of the modes of two-sticking beads dumbbells is the Green-Tobolsky network relaxation, whereas the other modes correspond to fast configurational relaxations. In the opposite limit of fast chemical kinetics compared to Brownian relaxation, the effect of the network is to slow down the configurational response of Hookean dumbbells. Sticky dumbbells thus realise a continuous transition from Hookean dumbbells to transient networks.

Constrained diffusion or immobile fraction on cell surfaces: a new interpretation

Protein lateral mobility in cell membranes is generally measured using fluorescence photobleaching recovery (FPR). Since the development of this technique, the data have been interpreted by assuming free Brownian diffusion of cell surface receptors in two dimensions, an interpretation that requires that a subset of the diffusing species remains immobile. The origin of this so-called immobile fraction remains a mystery. In FPR, the motions of thousands of particles are inherently averaged, inevitably masking the details of individual motions. Recently, tracking of individual cell surface receptors has identified several distinct types of motion (Gross and Webb, 1988; Ghosh and Webb, 1988, 1990, 1994; Kusumi et al. 1993; Qian et al. 1991; Slattery, 1995), thereby calling into question the classical interpretation of FPR data as free Brownian motion of a limited mobile fraction. We have measured the motion of fluorescently labeled immunoglobulin E complexed to high affinity receptors (Fc epsilon RI) on rat basophilic leukemia cells using both single particle tracking and FPR. As in previous studies, our tracking results show that individual receptors may diffuse freely, or may exhibit restricted, time-dependent (anomalous) diffusion. Accordingly, we have analyzed FPR data by a new model to take this varied motion into account, and we show that the immobile fraction may be due to particles moving with the anomalous subdiffusion associated with restricted lateral mobility. Anomalous subdiffusion denotes random molecular motion in which the mean square displacements grow as a power law in time with a fractional positive exponent less than one. These findings call for a new model of cell membrane structure.

A Lévy theorem for free noises

We prove a Levy type characterization theorem for the free Brownian motion and the free Poisson process using martingale and convariance conditions and some assumption on fourth order conditional moments.

Free Brownian motion of a particle driven by a dichotomous random force.

Dynamic behavior of free Brownian motion of a particle driven by dichotomous (random telegraphic) colored noise has been treated based on the ordinary Langevin equation. The treatment makes direct use of the characteristic function without using the Fokker-Planck equation. The conditional probability density in the case where inertial effects are neglected has been obtained exactly as a function of the position of the particle, and it is shown that starting from the Dirac \ensuremath{\delta} function at t=0, the probability density exhibits \ensuremath{\delta} functions at both extremes at early times, and the amplitudes of the \ensuremath{\delta} functions gradually vanish as time goes on. It is shown that this presence of \ensuremath{\delta} functions at the early stage is associated with dichotomous noise. The characteristic function for the Langevin equation with complete consideration of inertial effects has also been obtained exactly.

On the solution of Kramers’ equation by Trotter’s formula

By considering the cases of free Brownian motion, and Brownian motion in a harmonic potential, we show that Trotter’s formula could be applied to solve Fokker–Planck equations with singular diffusion matrices.