arXiv subjects

Sarah Kostinski

Publications and source records attributed to Sarah Kostinski.

7 recordsLinked to original sources

arXiv2026

Transcription rate dynamics and RNA copy number noise: General relations and data-driven predictions

Stochastic models of gene expression typically begin with a microscopic model of transcription and propagate its statistics to the RNA distribution. Here we develop a doubly stochastic framework in which RNA production is Poisson conditional on a time-varying transcription rate $\lambda(t)$. For RNA lifetimes drawn independently from an arbitrary finite-mean distribution, we derive the RNA Fano factor in terms of the lifetime survival function and the autocovariance of $\lambda(t)$. For a Poisson degradation process with rate $\mu$, the survival kernel provides an exponential temporal filter, and the result depends only on the mean, variance, and normalized autocorrelation of $\lambda(t)$. These quantities may be calculated from an explicit rate model or, under ergodicity and adequate sampling, estimated from a sufficiently long rate trajectory. We validate the data-driven estimator using simulated transcription-rate trajectories, without supplying the known autocorrelation to the estimator. We also obtain exact analytical results for transcription-rate dynamics modeled by a discrete M/M/1 process and by drift-diffusion with reflecting, periodic, or first-passage-reset boundaries, and verify each result by direct simulation of the coupled transcription-rate and RNA copy-number processes.

physics.bio-ph↗

arXiv2025

Closed-form survival probabilities for biased random walks at arbitrary step number

We present a closed-form expression for the survival probability of a biased random walker to first reach a target site on a 1D lattice. The expression holds for any step number $N$ and is computationally faster than non-closed-form results in the literature. Because our result is exact even in the intermediate step number range, it serves as a tool to study convergence to the large $N$ limit. We also obtain a closed-form expression for the probability of last passage. In contrast to predictions of the large $N$ approximation, the new expression reveals a critical value of the bias beyond which the tail of the last-passage probability decays monotonically.

cond-mat.stat-mech↗

arXiv2025

Least Squares as Random Walks

Linear least squares (LLS) is perhaps the most common method of data analysis, dating back to Legendre, Gauss and Laplace. Framed as linear regression, LLS is also a backbone of mathematical statistics. Here we report on an unexpected new connection between LLS and random walks. To that end, we introduce the notion of a random walk based on a discrete sequence of data samples (data walk). We show that the slope of a straight line which annuls the net area under a residual data walk equals the one found by LLS. For equidistant data samples this result is exact and holds for an arbitrary distribution of steps.

stat.ME↗

arXiv2024

Exact closed-form recurrence probabilities for biased random walks at any step number

We report on a closed-form expression for the survival probability of a discrete 1D biased random walk to not return to its origin after N steps. Our expression is exact for any N, including the elusive intermediate range, thereby allowing one to study its convergence to the large N limit. In that limit we recover Polya's recurrence probability, i.e. the survival probability equals the magnitude of the bias. We then obtain a closed-form expression for the probability of last return. In contrast to the bimodal behavior for the unbiased case, we show that the probability of last return decays monotonically throughout the walk beyond a critical bias. We obtain a simple expression for the critical bias as a function of the walk length, and show that it saturates at $1/\sqrt{3}$ for infinitely long walks. This property is missed when using expressions developed for the large N limit. Finally, we discuss application to molecular motors' biased random walks along microtubules, which are of intermediate step number.

cond-mat.stat-mech↗

arXiv2021

The inspection paradox in stochastic resetting

The remaining travel time of a plane shortens with every minute that passes from its departure, and a flame diminishes a candle with every second it burns. Such everyday occurrences bias us to think that processes which have already begun will end before those which have just started. Yet, the inspection paradox teaches us that the converse can also happen when randomness is at play. The paradox comes from probability theory, where it is often illustrated by measuring how long passengers wait upon arriving at a bus stop at a random time. Interestingly, such passengers may on average wait longer than the mean time between bus arrivals -- a counter-intuitive result, since one expects to wait less when coming some time after the previous bus departed. In this \textit{viewpoint}, we review the inspection paradox and its origins. The insight gained is then used to explain why, in some situations, stochastic resetting expedites the completion of random processes. Importantly, this is done with elementary mathematical tools which help develop a probabilistic intuition for stochastic resetting and how it works. This viewpoint can thus be used as an accessible introduction to the subject.

cond-mat.stat-mech↗

arXiv2020

Growth-laws and invariants from ribosome biogenesis in lower Eukarya

Eukarya and Bacteria are the most evolutionarily distant domains of life, which is reflected by differences in their cellular structure and physiology. For example, Eukarya feature membrane-bound organelles such as nuclei and mitochondria, whereas Bacteria have none. The greater complexity of Eukarya renders them difficult to study from both an experimental and theoretical perspective. However, encouraged by a recent experimental result showing that budding yeast (a unicellular eukaryote) obeys the same proportionality between ribosomal proteome fractions and cellular growth rates as Bacteria, we derive a set of relations describing eukaryotic growth from first principles of ribosome biogenesis. We recover the observed ribosomal protein proportionality, and then continue to obtain two growth-laws for the number of RNA polymerases synthesizing ribosomal RNA per ribosome in the cell. These growth-laws, in turn, reveal two invariants of eukaryotic growth, i.e. quantities predicted to be conserved by Eukarya regardless of growth conditions. The invariants, which are the first of their kind for Eukarya, clarify the coordination of transcription and translation kinetics as required by ribosome biogenesis, and link these kinetic parameters to cellular physiology. We demonstrate application of the relations to the yeast S. cerevisiae and find the predictions to be in good agreement with currently available data. We then outline methods to quantitatively deduce several unknown kinetic and physiological parameters. The analysis is not specific to S. cerevisiae and can be extended to other lower (unicellular) Eukarya when data become available. The relations may also have relevance to certain cancer cells which, like bacteria and yeast, exhibit rapid cell proliferation and ribosome biogenesis.

physics.bio-ph↗

arXiv2015

An elementary derivation of first and last return times of 1D random walks

Random walks, and in particular, their first passage times, are ubiquitous in nature. Using direct enumeration of paths, we find the first return time distribution of a 1D random walker, which is a heavy-tailed distribution with infinite mean. Using the same method we find the last return time distribution, which follows the arcsine law. Both results have a broad range of applications in physics and other disciplines. The derivation presented here is readily accessible to physics undergraduates, and provides an elementary introduction into random walks and their intriguing properties.

cond-mat.stat-mech↗