Interplay between Markovianity and progressive quenching.

Progressive quenching (PQ) is a process in which we sequentially fix a system's degrees of freedom, which would otherwise evolve according to their stochastic dynamics. Previous studies have discovered what we refer to as the hidden martingale property in PQ. Here we first attribute this martingale property to the canonicity of the two-layer ensemble comprising quenched and thermal ensembles and demonstrate that the Markovian property, coupled with the detailed balance (DB) of the evolution dynamics, underpins this canonicity. We then expand the PQ to the Markovian dynamics on the transition network where the DB is not required. Additionally, we examine the PQ of the systems that evolve through non-Markovian dynamics between consecutive quenching. When non-Markovian dynamics ensure a trajectory-wise DB, such as in an equilibrium spin system with a hidden part, the PQ can occasionally maintain the canonical structure of the overall statistical ensemble but not always. Last, we analytically and numerically investigate the PQ of a non-Markovian spin system with delayed interaction and illustrate how the reduction of spin correlations due to the delay can be compensated by the PQ.

Martingale drift of Langevin dynamics and classical canonical spin statistics.

A martingale is a stochastic process that encodes a kind of fairness or unbiasedness, which is associated with a reference process. Here we show that, if the reference process x_{t} evolves according to the Langevin equation with drift a(x) and if a(x_{t}) is a martingale, then its amplitude is the Langevin function, which originally described the canonical response of a single classical Heisenberg spin under static field. Furthermore, the asymptotic limit of x_{t}/t obeys the ensemble statistics of such a Heisenberg spin.

Martingale-induced local invariance in progressive quenching.

Progressive quenching (PQ) is a stochastic process during which one fixes, one after another, the degrees of freedom of a globally coupled Ising spin system while letting it thermalize through a heat bath. It has previously been shown that during PQ, the mean equilibrium spin value follows a martingale process and this process can characterize the memory of the system. In the present study, we find that the aforementioned martingale implies a local invariance of the path weight for the total quenched magnetization, the Markovian process whose increment is the spin that is fixed last. Consequently, PQ lets the probability distribution for the total quenched magnetization evolve while keeping the Boltzmann-like factor, or a canonical structure, under constraint, which consists of a path-independent potential and a path-counting entropy. Moreover, when the PQ starts from full equilibrium, the probability distribution at each stage of PQ is found to be the limit distribution of what we call recycled quenching, the process in which a randomly chosen quenched spin is unquenched after a single step of PQ. The local invariance is directly derived from the martingale property, and not from other known theorems on martingale processes.

Heterogeneous nucleation of creases in swelling polymer gels.

Surface creasing is a common occurrence in gels under strong enough compression. The transition from smooth to creased surface has been well studied in equilibrium conditions and applied to achieve stimuli-responsive properties. Classical predictions of the creased state, assuming the gel is at equilibrium and homogeneous, are generally satisfactory, while the transient behavior in swelling gels is often far from equilibrium and is commonly heterogeneous. The short-time response is essential for materials in dynamic environments, but it remains unreported and largely unknown due to the limited temporal resolution of the techniques used so far. Here, we use spatially resolved multispeckle diffusing wave spectroscopy (MSDWS) with submicrosecond time resolution to measure the spatially dependent swelling and creasing of a constrained poly (vinyl alcohol) chemical gel in borax solutions of varying concentrations. Our high-speed imaging by MSDWS shows that the swelling behavior and mechanical response at the microscopic level can be highly heterogeneous in time and space, and is detectable hundreds of seconds before the corresponding macroscopic creasing transition. This unprecedented visualization of the heterogeneous and time-dependent behavior beyond equilibrium morphological changes unveils the full complexity of the transient material response after exposure to external stimuli and sheds light on the formation mechanism of metastable states in transient processes.

GTP-dependent formation of straight tubulin oligomers leads to microtubule nucleation.

Nucleation of microtubules (MTs) is essential for cellular activities, but its mechanism is unknown because of the difficulty involved in capturing rare stochastic events in the early stage of polymerization. Here, combining rapid flush negative stain electron microscopy (EM) and kinetic analysis, we demonstrate that the formation of straight oligomers of critical size is essential for nucleation. Both GDP and GTP tubulin form single-stranded oligomers with a broad range of curvatures, but upon nucleation, the curvature distribution of GTP oligomers is shifted to produce a minor population of straight oligomers. With tubulin having the Y222F mutation in the ß subunit, the proportion of straight oligomers increases and nucleation accelerates. Our results support a model in which GTP binding generates a minor population of straight oligomers compatible with lateral association and further growth to MTs. This study suggests that cellular factors involved in nucleation promote it via stabilization of straight oligomers.

Memory through a hidden martingale process in progressive quenching.

Progressive quenching (PQ) is the stochastic process in which the system's degrees of freedom are sequentially fixed. While such a process does not satisfy the local detailed balance, it has been found that the some physical observable of a complete spin network exhibits the martingale property. We studied the system's response to the perturbation given at intermediate stages of the PQ. The response at the final stage reveals the persistent memory, and we show that this persistence is a direct consequence of the martingale process behind it. Not only the mean response, the shape of the probability distribution at the stage of perturbation is also memorized. Using the hidden martingale process we can predict the final bimodal distribution from the early-stage unimodal distribution in the regime where the unfrozen spins are paramagnetic. We propose a viewpoint that the martingale property is a stochastic conservation law which is supported by some stochastic invariance.

Progressive quenching: Globally coupled model.

We study the processes in which fluctuating elements of a system are progressively fixed (quenched) while keeping the interaction with the remaining unfixed elements. If the interaction is global among Ising spin elements and if the unfixed part is reequilibrated each time after fixing an element, the evolution of a large system is martingale about the equilibrium spin value of the unfixed spins. Due to this property the system starting from the critical point yields the final magnetization, whose distribution shows non-Gaussian and slow transient behavior with the system size.

Mesoscopic formulas of linear and angular momentum fluxes.

Many approaches of coarse graining have been developed under the names of Cosserat theory or polar-fluid theory for those materials in which some component elements undergo nonaffine deformations, such as elastic materials with inclusions or granular matters. For the complex elements such as living cells, however, the microscopic variables and their dynamics are often unknown, and there has been no systematic theory of coarse graining from the microscales nor the formulas like the Irving-Kirkwood formula that constitutes the macroscopic stress or couple stress in terms of some microscale quantities. We show that, for the quasi-steady states, the coarse-graining procedure must generally provide us with the Cosserat-type balance equations as long as the procedure keeps track of the conservation of linear and angular momenta, and that the fluxes of these conserved quantities should generally be expressed in the Irving-Kirkwood-type formulas, where the interparticle distance or forces and torques should be replaced by those associated to the pair of neighboring coarse-graining volumes. This framework, which refers to no particular microvariables or dynamics, is valid for active complex matters out of equilibrium and with any multibody interactions.

Elastic Anisotropy Scenario for Cooperative Binding of Kinesin-Coated Beads on Microtubules.

Muto et al. reported in 2005 an observation called cooperative binding, according to which the initial binding of a bead covered with active kinesins on a microtubule filament was capable of favoring the subsequent binding of similar beads on the same filament up to distances of the order of a few microns. This positive bias is stronger ahead of the initially bound bead than behind. We explain this effect by combining the recently proposed notion of shear screening length with the notion of localized tubulin conformational transition induced by motor binding. Elastic terms linked to the polarity of protofilaments, up to now ignored, provide adequate description to the long-range elastic shear generated by motor binding. The subsequent binding is favored when and where the shear displacement of protofilaments meets the requirement for specific strong binding. We propose experimental tests of our model, which open the way to a new type of spectroscopy for biomolecular processes.

Nonequilibrium statistical mechanics of the heat bath for two Brownian particles.

We propose a new look at the heat bath for two Brownian particles, in which the heat bath as a "system" is both perturbed and sensed by the Brownian particles. Nonlocal thermal fluctuations give rise to bath-mediated static forces between the particles. Based on the general sum rule of the linear response theory, we derive an explicit relation linking these forces to the friction kernel describing the particles' dynamics. The relation is analytically confirmed in the case of two solvable models and could be experimentally challenged. Our results point out that the inclusion of the environment as a part of the whole system is important for micron- or nanoscale physics.

Momentum transfer in nonequilibrium steady states.

When a Brownian object interacts with noninteracting gas particles under nonequilibrium conditions, energy dissipation associated with Brownian motion causes an additional force on the object as a "momentum transfer deficit." This principle is demonstrated first by a new nonequilibrium steady state model and then applied to several known models such as an adiabatic piston for which a simple explanation has been lacking.

From the stochasticity of molecular processes to the variability of synaptic transmission.

The variability of the postsynaptic response following a single action potential arises from two sources: the neurotransmitter release is probabilistic, and the postsynaptic response to neurotransmitter release has variable timing and amplitude. At individual synapses, the number of molecules of a given type that are involved in these processes is small enough that the stochastic (random) properties of molecular events cannot be neglected. How the stochasticity of molecular processes contributes to the variability of synaptic transmission, its sensitivity and its robustness to molecular fluctuations has important implications for our understanding of the mechanistic basis of synaptic transmission and of synaptic plasticity.

Fibroblast growth factor and insulin-like growth factor rescue growth cones of sensory neurites from collapse after tetracaine-induced injury.

BACKGROUND: Basic fibroblast growth factor (bFGF) and insulin-like growth factor (IGF)-1 have multiple effects on cells, including proliferation, differentiation, and survival. In this study, we investigated the effects of different concentrations of IGF and bFGF on the morphology of growth cones of the developing sensory neurons after tetracaine-induced injury in vitro. METHODS: Dorsal root ganglia were isolated from chick embryos on embryonic day 7 or 8 and cultured for 24 hours. Tissues were then exposed to 100 mumol/L tetracaine for 60 minutes. The media were replaced by tetracaine-free media containing different concentrations of IGF, bFGF, or combination of IGF 50 ng/mL and bFGF 5 ng/mL and incubated for a further 24 hours. Growth cone collapse assays were then performed to assess regeneration of neurons. RESULTS: Exposure of dorsal root ganglia explants to tetracaine 100 mumol/L for 1 hour caused significant growth cone collapse 24 hours after washing out tetracaine (P < 0.01). It was found that adding bFGF (5, 10, 20, and 50 ng/mL) or IGF (50 and 100 ng/mL) to the replacement media significantly decreased growth cone collapse percentage at 24 hours after washout (P < 0.01); however, the low concentrations of bFGF (2 ng/mL) or IGF (25 ng/mL) did not cause significant change. Growth cone collapse after simultaneous addition of 5 ng/mL bFGF and 50 ng/mL IGF was statistically lower than the values after adding 5 ng/mL bFGF (P < 0.01), and it was marginally lower than 50 ng/mL IGF. CONCLUSION: bFGF and bIGF decreased growth cone collapse after tetracaine-induced injury in vitro.

One-dimensional Brownian motion of charged nanoparticles along microtubules: a model system for weak binding interactions.

Various proteins are known to exhibit one-dimensional Brownian motion along charged rodlike polymers, such as microtubules (MTs), actin, and DNA. The electrostatic interaction between the proteins and the rodlike polymers appears to be crucial for one-dimensional Brownian motion, although the underlying mechanism has not been fully clarified. We examined the interactions of positively-charged nanoparticles composed of polyacrylamide gels with MTs. These hydrophilic nanoparticles bound to MTs and displayed one-dimensional Brownian motion in a charge-dependent manner, which indicates that nonspecific electrostatic interaction is sufficient for one-dimensional Brownian motion. The diffusion coefficient decreased exponentially with an increasing particle charge (with the exponent being 0.10 kBT per charge), whereas the duration of the interaction increased exponentially (exponent of 0.22 kBT per charge). These results can be explained semiquantitatively if one assumes that a particle repeats a cycle of binding to and movement along an MT until it finally dissociates from the MT. During the movement, a particle is still electrostatically constrained in the potential valley surrounding the MT. This entire process can be described by a three-state model analogous to the Michaelis-Menten scheme, in which the two parameters of the equilibrium constant between binding and movement, and the rate of dissociation from the MT, are derived as a function of the particle charge density. This study highlights the possibility that the weak binding interactions between proteins and rodlike polymers, e.g., MTs, are mediated by a similar, nonspecific charge-dependent mechanism.

Allosteric model of an ion pump.

We present a simple model of a free-energy transducer made of allosterically coupled two ratchet subsystems. Each of the subsystems transports particles from one particle reservoir to another. The coupling of the subsystems imposes correlated transitions of the potential profiles of the two subsystems. As a result, a downhill flux in one subsystem with higher chemical-potential difference drives an uphill flux in the other subsystem with lower chemical-potential difference. The direction of the driven flux inverts depending on the direction of the driving flux. The ratio between the fluxes conveyed by the two subsystems is variable and nonstoichiometric. By selecting appropriate parameters, the maximum ratio of the driven flux to driving flux and maximum free-energy transducing efficiency reaches some 90 and 40%, respectively. At a stalled state, the driven flux vanishes while the driving flux remains finite. The allosteric model enables explicit analysis of the timing between binding-unbinding of particles and transitions of potential profile. The behavior of the model is similar to but different from that of the alternate access model, which is a biochemical model for active transport proteins. Our model works also as a regulatory system. We suggest that the correlated transitions of the subsystems (subunits or domains) through allosteric interaction are the origin of the diverse functions of the protein machineries.

Newton's cradle versus nonbinary collisions.

Newton's cradle is a classical example of a one-dimensional impact problem. In the early 1980s the naive perception of its behavior was corrected: For example, the impact of a particle does not exactly cause the release of the farthest particle of the target particle train, if the target particles have been just in contact with their own neighbors. It is also known that the naive picture would be correct if the whole process consisted of purely binary collisions. Our systematic study of particle systems with truncated power-law repulsive force shows that the quasibinary collision is recovered in the limit of hard core repulsion, or a very large exponent. In contrast, a discontinuous steplike repulsive force mimicking a hard contact, or a very small exponent, leads to a completely different process: the impacting cluster and the targeted cluster act, respectively, as if they were nondeformable blocks.

Influence of mechanical properties of nanoparticles on macrocrack formation.

We present an experimental investigation of drying suspensions of both hard and soft nanolatex spheres. The crack formation is examined as a function of the proportion of hard and soft deformable particles, leading to tunable elastic properties of the drying film. In our experimental systems, no crack formation could be observed below an onset value of the proportion in hard spheres phi approximately 0.45 . During the drying process, the mass of films with various compositions in hard and soft spheres is measured as a function of time. The results suggest that the soft particles undergo deformation that releases the internal stresses.

Compatibility between itinerant synaptic receptors and stable postsynaptic structure.

The density of synaptic receptors in front of presynaptic release sites is stabilized in the presence of scaffold proteins, but the receptors and scaffold molecules have local exchanges with characteristic times shorter than that of the receptor-scaffold assembly. We propose a mesoscopic model to account for the regulation of the local density of receptors as quasiequilibrium. It is based on two zones (synaptic and extrasynaptic) and multilayer (membrane, submembrane, and cytoplasmic) topological organization. The model includes the balance of chemical potentials associated with the receptor and scaffold protein concentrations in the various compartments. The model shows highly cooperative behavior including a "phase change" resulting in the formation of well-defined postsynaptic domains. This study provides theoretical tools to approach the complex issue of synaptic stability at the synapse, where receptors are transiently trapped yet rapidly diffuse laterally on the plasma membrane.

Diffusion trajectory of an asymmetric object: information overlooked by the mean square displacement.

Diffusion of an asymmetric object is characterized by its translational and rotational diffusion coefficients. Until now, anisotropic diffusion studies have been based on ensemble averages. Here we present a theoretical basis for the analysis of the trajectories of a single particle with anisotropic diffusion coefficients. We discuss the relevance of this method for motion of biomolecules in the membrane of living cells.

Microscopic heat from the energetics of stochastic phenomena.

The energetics of the stochastic process has shown the balance of energy on the mesoscopic level. The heat and the energy defined there are, however, generally different from their macroscopic counterpart. We show that this discrepancy can be removed by adding to these quantities the reversible heat associated with the mesoscopic free energy.