Dynamic structure factor of vibrating fractals

Salt induced slowdown of kinetics and dynamics during thermal gelation of egg-yolk

We investigated the effect of the NaCl concentration (0.3-2M) on the structure and dynamics of hen egg yolk at room temperature and during thermal gelation at temperatures in the range of 66-90 °C utilizing low-dose x-ray photon correlation spectroscopy in ultra-small angle x-ray scattering geometry. With an increase in the salt concentration, we observe progressive structural and dynamic changes at room temperature, indicating the disruption of yolk components such as yolk-granules and yolk-plasma proteins. Temperature- and salt-dependent structural and dynamic investigations suggest a delay in the gel formation and aggregation of yolk low-density lipoproteins with increasing ionic strength. However, the time-temperature superposition relationship observed in all samples suggests an identical mechanism underlying protein aggregation-gelation with a temperature-dependent reaction rate. The sol-gel transition time extracted from kinetic and dynamic information follows Arrhenius's behavior, and the activation energy (460 kJ/mol) is found to be independent of the salt concentration.

Emergence of Multiscale Dynamics in Colloidal Gels

To gain insight into the kinetics of colloidal gel evolution at low particle volume fractions ϕ, we utilize differential dynamic microscopy to investigate particle aggregation, geometric percolation, and the subsequent transition to nonergodic dynamics. We report the emergence of unexpectedly rich multiscale dynamics upon the onset of nonergodicity, which separates the wave vectors q into three different regimes. In the high-q domain, the gel exhibits ϕ-independent internal vibrations of fractal clusters. The intermediate-q domain is dominated by density fluctuations at the length scale of the clusters, as evidenced by the q independence of the relaxation time τ. In the low-q domain, the scaling of τ as q^{-3} suggests that the network appears homogeneous. The transitions between these three regimes introduce two characteristic length scales, distinct from the cluster size.

First Passage under Restart with Branching

First passage under restart with branching is proposed as a generalization of first passage under restart. Strong motivation to study this generalization comes from the observation that restart with branching can expedite the completion of processes that cannot be expedited with simple restart; yet a sharp and quantitative formulation of this statement is still lacking. We develop a comprehensive theory of first passage under restart with branching. This reveals that two widely applied measures of statistical dispersion-the coefficient of variation and the Gini index-come together to determine how restart with branching affects the mean completion time of an arbitrary stochastic process. The universality of this result is demonstrated and its connection to extreme value theory is also pointed out and explored.

Structural arrest and dynamic localization in biocolloidal gels

Casein micelles interacting via an entropic intermediate-ranged depletion attraction exhibit a fluid-to-gel transition due to arrested spinodal decomposition. The bicontinuous networked structure of the gel freezes shortly after formation. We determine the timescales of structural arrest from the build-up of network rigidity after pre-shear rejuvenation, and find that the arrest time as well as the plateau elastic modulus of the gel diverge as a function of the volume fraction and interaction potential. Moreover, we show using scaling from naïve mode coupling theory that their mechanical properties are dictated by their microscopic dynamics rather than their heterogeneous large scale structure.

Local orientational mobility in regular hyperbranched polymers

We study the dynamics of local bond orientation in regular hyperbranched polymers modeled by Vicsek fractals. The local dynamics is investigated through the temporal autocorrelation functions of single bonds and the corresponding relaxation forms of the complex dielectric susceptibility. We show that the dynamic behavior of single segments depends on their remoteness from the periphery rather than on the size of the whole macromolecule. Remarkably, the dynamics of the core segments (which are most remote from the periphery) shows a scaling behavior that differs from the dynamics obtained after structural average. We analyze the most relevant processes of single segment motion and provide an analytic approximation for the corresponding relaxation times. Furthermore, we describe an iterative method to calculate the orientational dynamics in the case of very large macromolecular sizes.

Optimal Stochastic Restart Renders Fluctuations in First Passage Times Universal

Stochastic restart may drastically reduce the expected run time of a computer algorithm, expedite the completion of a complex search process, or increase the turnover rate of an enzymatic reaction. These diverse first-passage-time (FPT) processes seem to have very little in common but it is actually quite the other way around. Here we show that the relative standard deviation associated with the FPT of an optimally restarted process, i.e., one that is restarted at a constant (nonzero) rate which brings the mean FPT to a minimum, is always unity. We interpret, further generalize, and discuss this finding and the implications arising from it.

Dynamics of comb-of-comb networks

The dynamics of complex networks, a current hot topic in many scientific fields, is often coded through the corresponding Laplacian matrix. The spectrum of this matrix carries the main features of the networks' dynamics. Here we consider the deterministic networks which can be viewed as "comb-of-comb" iterative structures. For their Laplacian spectra we find analytical equations involving Chebyshev polynomials whose properties allow one to analyze the spectra in deep. Here, in particular, we find that in the infinite size limit the corresponding spectral dimension goes as d(s) → 2. The d(s) leaves its fingerprint on many dynamical processes, as we exemplarily show by considering the dynamical properties of polymer networks, including single monomer displacement under a constant force, mechanical relaxation, and fluorescence depolarization.

Michaelis-Menten reaction scheme as a unified approach towards the optimal restart problem

We study the effect of restart, and retry, on the mean completion time of a generic process. The need to do so arises in various branches of the sciences and we show that it can naturally be addressed by taking advantage of the classical reaction scheme of Michaelis and Menten. Stopping a process in its midst-only to start it all over again-may prolong, leave unchanged, or even shorten the time taken for its completion. Here we are interested in the optimal restart problem, i.e., in finding a restart rate which brings the mean completion time of a process to a minimum. We derive the governing equation for this problem and show that it is exactly solvable in cases of particular interest. We then continue to discover regimes at which solutions to the problem take on universal, details independent forms which further give rise to optimal scaling laws. The formalism we develop, and the results obtained, can be utilized when optimizing stochastic search processes and randomized computer algorithms. An immediate connection with kinetic proofreading is also noted and discussed.

Effective degrees of freedom of a random walk on a fractal

We argue that a non-Markovian random walk on a fractal can be treated as a Markovian process in a fractional dimensional space with a suitable metric. This allows us to define the fractional dimensional space allied to the fractal as the ν-dimensional space F(ν) equipped with the metric induced by the fractal topology. The relation between the number of effective spatial degrees of freedom of walkers on the fractal (ν) and fractal dimensionalities is deduced. The intrinsic time of random walk in F(ν) is inferred. The Laplacian operator in F(ν) is constructed. This allows us to map physical problems on fractals into the corresponding problems in F(ν). In this way, essential features of physics on fractals are revealed. Particularly, subdiffusion on path-connected fractals is elucidated. The Coulomb potential of a point charge on a fractal embedded in the Euclidean space is derived. Intriguing attributes of some types of fractals are highlighted.

Contact Kinetics in Fractal Macromolecules

We consider the kinetics of first contact between two monomers of the same macromolecule. Relying on a fractal description of the macromolecule, we develop an analytical method to compute the mean first contact time for various molecular sizes. In our theoretical description, the non-Markovian feature of monomer motion, arising from the interactions with the other monomers, is captured by accounting for the nonequilibrium conformations of the macromolecule at the very instant of first contact. This analysis reveals a simple scaling relation for the mean first contact time between two monomers, which involves only their equilibrium distance and the spectral dimension of the macromolecule, independently of its microscopic details. Our theoretical predictions are in excellent agreement with numerical stochastic simulations.

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.