# Reaction kinetics in the few-encounter limit

## TL;DR

This paper develops a spectral approach to analyze first passage time distributions in reaction kinetics, especially in the few-encounter limit where traditional mean-based models are insufficient.

## Contribution

It introduces a spectral representation of FPT statistics that captures the full distribution, crucial for understanding reactions triggered by few encounters.

## Key findings

- Spectral method accurately determines FPT distributions over wide time scales.
- Narrowing of FPT density occurs in the few-encounter limit.
- Applicable to biological processes like protein aggregation and gene regulation.

## Abstract

The classical theory of chemical reactions can be understood in terms of diffusive barrier crossing, where the rate of a reaction is determined by the inverse of the mean first passage time (FPT) to cross a free energy barrier. Whenever a few reaction events suffice to trigger a response or the energy barriers are not high, the mean first passage time alone does not suffice to characterize the kinetics, i.e., the kinetics do not occur on a single time-scale. Instead, the full statistics of the FPT are required. We present a spectral representation of the FPT statistics that allows us to understand and accurately determine FPT distributions over several orders of magnitudes in time. A canonical narrowing of the first passage density is shown to emerge whenever several molecules are searching for the same target, which was termed the 'few-encounter limit'. The few-encounter limit is essential in all situations, in which already the first encounter triggers a response, such as misfolding-triggered aggregation of proteins or protein transcription regulation.