Stochastic synthesis and degradation of arbitrary diffusive processes
Abstract
Target search processes are often carried out by diffusive elements that are introduced and removed at random times, causing fluctuations in the number of searchers in the system over time. For instance, proteins are synthesized and degraded stochastically in the cell, and susceptible to react with specific binding sites during their lifetime. Here we derive the density profile generated by non-interacting particles subject to stochastic synthesis and degradation (SSD) at constant rates, given a source location and an arbitrary propagator for the single, non-degrading particle. We also study the first-passage properties of these many-searcher processes, which can be deduced from the knowledge of the first-passage time distribution of a single particle. We analyse the behaviours of the characteristic reaction time and of the mean first-passage time in the limits of small and large rates. Taking into account the costs associated with synthesis leads to the existence of an optimal synthesis rate. We further generalise these results to bursty synthesis, when production events involve more than one particle at a time, as often observed in cell biology. The mean first-passage time in these situations can be significantly reduced at low synthesis or large degradation rates, as it is divided by the mean burst size in these regimes. The results presented here are applied to one-dimensional Brownian searches. Some general connections between SSD and resetting processes are pointed out as well as important differences, since SSD is associated with a distinct queueing protocol.
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BibTeX
@misc{boyer2026stochastic,
title = {{Stochastic synthesis and degradation of arbitrary diffusive processes}},
author = {Denis Boyer and Gabriel Mercado-Vásquez},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04843},
eprint = {2611.04843},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04843},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stochastic synthesis and degradation of arbitrary diffusive processes
AU - Denis Boyer
AU - Gabriel Mercado-Vásquez
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04843
UR - https://arxiv.org/abs/2611.04843
AB - Target search processes are often carried out by diffusive elements that are introduced and removed at random times, causing fluctuations in the number of searchers in the system over time. For instance, proteins are synthesized and degraded stochastically in the cell, and susceptible to react with specific binding sites during their lifetime. Here we derive the density profile generated by non-interacting particles subject to stochastic synthesis and degradation (SSD) at constant rates, given a source location and an arbitrary propagator for the single, non-degrading particle. We also study the first-passage properties of these many-searcher processes, which can be deduced from the knowledge of the first-passage time distribution of a single particle. We analyse the behaviours of the characteristic reaction time and of the mean first-passage time in the limits of small and large rates. Taking into account the costs associated with synthesis leads to the existence of an optimal synthesis rate. We further generalise these results to bursty synthesis, when production events involve more than one particle at a time, as often observed in cell biology. The mean first-passage time in these situations can be significantly reduced at low synthesis or large degradation rates, as it is divided by the mean burst size in these regimes. The results presented here are applied to one-dimensional Brownian searches. Some general connections between SSD and resetting processes are pointed out as well as important differences, since SSD is associated with a distinct queueing protocol.
ER -