top of page
THEORY FOR SINGLE-X SCIENCE 

Single-molecule theory of enzymatic inhibition

​

The classical theory of enzymatic inhibition takes a deterministic, bulk-based, approach to quantitatively describe how inhibitors affect the progression of enzymatic reactions. Catalysis at the single-enzyme level is, however, inherently stochastic which could lead to strong deviations from classical predictions. To study this, we collaborated with Tal Robin and Michael Urbakh. Taking a single-enzyme perspective, we rebuilt the theory of enzymatic inhibition from the bottom up [Nat. Commun. 9(1), 779, 2018]. We discovered that stochastic fluctuations at the single-enzyme level could unexpectedly make inhibitors act as activators, which should fundamentally change the way we think about enzymatic inhibition.

​

High-order Michaelis-Menten equations 

​

Single-molecule measurements provide a platform for investigating the dynamical properties of enzymatic reactions. To this end, the single-molecule Michaelis-Menten equation was instrumental as it asserts that the first moment of the enzymatic turnover time depends linearly on the reciprocal of the substrate concentration. This, in turn, provided robust and convenient means to determine the maximal turnover rate and the Michaelis-Menten constant. Yet, the information provided by these parameters is incomplete and does not allow access to key

observables such as the lifetime of the enzyme-substrate complex, the rate of substrate-enzyme binding, and the probability of successful product formation. We collaborated with Michael Urbakh and Tal Robin to derive a set of high-order Michaelis-Menten equations [Nat. Commun., 15, 240, 2025]. These equations capture universal linear relations between the reciprocal of the substrate concentration and distinguished combinations of turnover time moments, essentially generalizing the Michaelis-Menten equation to moments of any order. We demonstrated how key observables such as the lifetime of the enzyme-substrate complex, the rate of substrate-enzyme binding, and the probability of successful product formation, can all be inferred using these high-order Michaelis-Menten equations. We tested our inference procedure to show that it is robust, producing accurate results with only several thousand turnover events per substrate concentration.

​

MM.png

Microscopic theory of adsorption kinetics

​

Adsorption is the accumulation of a solute at an interface that is formed between a solution and an additional gas, liquid, or solid phase. The macroscopic theory of adsorption dates back more than a century and is now well-established. Yet, despite recent advancements, a detailed and self-contained theory of single-particle adsorption was lacking. We bridged this gap by developing a microscopic theory of adsorption kinetics, from which

macroscopic properties follow directly [J. Chem. Phys. 158, 094107, 2023]. One of our central achievements is the derivation of the microscopic version of the seminal Ward–Tordai relation, which connects the surface and subsurface adsorbate concentrations via a universal equation that holds for arbitrary adsorption dynamics. We furthermore presented a microscopic interpretation of the Ward–Tordai relation that, in turn, allowed us to generalize it to arbitrary dimension, geometry, and initial conditions. The power of our approach was showcased on a set of unsolved adsorption problems to which we presented the first exact analytical solutions. The developed framework sheds fresh light on the fundamentals of adsorption kinetics, opening new research avenues in surface science with possible applications to artificial and biological sensing and to the design of nano-scale devices.

​

Escape of a sticky particle

​

Adsorption to a surface, reversible binding, and trapping are all prevalent scenarios where particles exhibit “stickiness.” Escape and first-passage times are known to be drastically affected, but a detailed understanding of this phenomenon remains elusive. To tackle this problem, we collaborated with Denis Grebenkov to develop an  

analytical approach to the escape of a diffusing particle from a domain of arbitrary shape, size, and surface reactivity [Phys. Rev. Research 5, 043196, 2023]. We used this approach to elucidate the effect of stickiness on the escape time from a slab domain, revealing how adsorption and desorption rates affect the mean and variance and providing a novel way to infer these rates from measurements. Moreover, as any smooth boundary is locally flat, the results we have obtained for slab domains were leveraged to devise a numerically efficient scheme for simulating sticky boundaries in arbitrary domains. Our work thus offers a starting point for analytical and numerical studies of stickiness and its role in escape, first-passage, and diffusion-controlled reactions.

 

Escape from textured adsorbing surfaces

​

The escape dynamics of sticky particles from textured surfaces is poorly understood despite its importance to various scientific and technological domains. In collaboration with Denis Grebenkov, we investigated the escape

time of adsorbates from prevalent surface topographies, including holes/pits, pillars, and grooves [J. Chem. Phys. 160, 184105, 2024]. For these cases, we derived analytical expressions for the probability density functions and the means of the escape times. A particularly interesting scenario is that of very deep and narrow confining spaces within the surface. In this limit, the joint effect of the entrapment and stickiness prolongs the escape time, resulting in an effective desorption rate that is dramatically lower than that of the untextured surface. This rate is shown to abide a universal scaling law, which couples the equilibrium constants of adsorption with the relevant confining length scales. While our results are analytical and exact, we also presented an approximation for deep and narrow cavities based on an effective description of one-dimensional diffusion that is punctuated by motionless adsorption events. This simple and physically motivated approximation provides high-accuracy predictions within its range of validity and works relatively well even for cavities of intermediate depth. 

 

Pillars.png

A unified approach to gated reactions on networks

​

For two molecules to react they must first meet. This condition is, however, insufficient since stochastic transitions between reactive and non-reactive states result in an effective fluctuating molecular “gate” that can prevent reactions from occurring despite the spatial proximity of the reactants involved. To better understand this phenomenon, we developed a general approach to gated reactions on networks [Phys. Rev. Lett. 127, 018301, 2021; J. Chem. Phys. 155, 234112, 2021]. The analysis gives a practical way of calculating the mean and distribution of gated reaction times from the corresponding, and often known, ungated reaction times. The theory also sheds light on universal features of gated reactions, as well as on novel and exotic kinetics that arise due to molecular gating. 

Gated first-passage processes in continuous space

​

Gated first-passage processes, where completion depends on both hitting a target and satisfying additional constraints, are prevalent across various fields. Despite their significance, analytical solutions to basic problems remain unknown, e.g. the detection time of a diffusing particle by a gated interval, disk, or sphere. We collaborated with Aanjaneya Kumar and M. S. Santhanam to elucidate the challenges posed by gated 

first-passage processes which take place in continuous space, and to present a renewal framework to overcome them [Rep. Prog. Phys. 87, 108101, 2024]. The presented framework offers a unified approach for a wide range of problems, including those with single-point, half-line, and interval targets. The latter have so far evaded exact solutions. Our analysis revealed that solutions to gated problems can be obtained directly from the ungated dynamics. This, in turn, reveals universal properties and asymptotic behaviors, shedding light on cryptic intermediate-time regimes and refining the notion of high-crypticity for continuous-space gated processes. We also extended our formalism to higher dimensions, showcasing its versatility and applicability. Our work provides valuable insights into the dynamics of continuous gated first-passage processes and offers analytical tools for studying them across diverse domains.

 

Torri-CGPT.png

Inference from gated first-passage times

​

When does a stochastic time-series cross a predefined threshold for the first time? The answer to this question relies on the first-passage time — a concept that has attracted interdisciplinary research interest for several decades. However, in applications ranging from time-series analysis to chemical reactions, various forms of “gating” prevent first-passage times from being observed directly. The challenge is then to infer the statistics of

first-passage times from the gated observations, i.e., from actual detection times, which can be strikingly different from the true first-passage times. Motivated by this inference challenge, and the complete lack of results on this front, we collaborated with Aanjaneya Kumar and M. S. Santhanam to develop a model-free framework that allows for the reconstruction of first-passage time statistics from observed detection times [Phys. Rev. Research 5, L032043, 2023]. The universality of the proposed framework remarkably allows for the inference of unknown first-passage statistics without needing to know or assume the underlying laws of stochastic motion. In addition, when the underlying laws of motions are known, our framework provides a way to infer physically meaningful parameters, e.g., diffusion coefficients and gating rates. Being the first of its kind, our approach opens new avenues connecting experimental and theoretical research by allowing practitioners to infer keystone quantities, which are often concealed from direct observation.

​

Reuveni Group | Tel Aviv University, Tel Aviv 6997801, Israel | Phone: +972-3-640-8694 | Email: shlomire@tauex.tau.ac.il 

bottom of page