Phenomenology of stochastic exponential growth

We discuss several models in order to shed light on the origin of power-law distributions and power-law correlations in financial time series. From an empirical point of view, the exponents describing the tails of the price increments distribution and the decay of the volatility correlations are rather robust and suggest universality. However, many of the models that appear naturally (for example, to account for the distribution of wealth) contain some multiplicative noise, w...

Protein distributions measured under a broad set of conditions in bacteria and yeast were shown to exhibit a common skewed shape, with variances depending quadratically on means. For bacteria these properties were reproduced by temporal measurements of protein content, showing accumulation and division across generations. Here we present a stochastic growth-and-division model with feedback which captures these observed properties. The limiting copy number distribution is calc...

This paper theoretically analyzes a phenomenological stochastic model for bacterial growth. This model comprises cell division and the linear growth of cells, where growth rates and cell cycles are drawn from lognormal distributions. We find that the cell size is expressed as a sum of independent lognormal variables. We show numerically that the quality of the lognormal approximation greatly depends on the distributions of the growth rate and cell cycle. Furthermore, we show ...

The transition density of a stochastic, logistic population growth model with multiplicative intrinsic noise is analytically intractable. Inferring model parameter values by fitting such stochastic differential equation (SDE) models to data therefore requires relatively slow numerical simulation. Where such simulation is prohibitively slow, an alternative is to use model approximations which do have an analytically tractable transition density, enabling fast inference. We int...

Recent methods have been developed to map single-cell lineage statistics to population growth. Because population growth selects for exponentially rare phenotypes, these methods inherently depend on sampling large deviations from finite data, which introduces systematic errors. A comprehensive understanding of these errors in the context of finite data remains elusive. To address this gap, we study the error in growth rate estimates across different models. We show that under...

What are the signatures of the onset of catastrophe? Here we present the rich system physics characterizing the stochasticity driven transition from homeostasis to breakdown in an experimentally motivated and analytically tractable minimal model. Recent high-precision experiments on individual bacterial cells, growing and dividing repeatedly in a variety of environments, have revealed a previously unknown intergenerational scaling law which not only uniquely determines the st...

Several simple growth and decay models use concepts and measures from fractal geometry. Kinetic models relevant to condensed matter observations arerecalled. They should be specifically adapted with appropriate degrees of freedom in order to reproduce power law behaviors, e.g. of species extinctions, cloud fracture, financial crashes. This allows one to illustrate scientific problems relevant to the growth and control of epidemias, and other event or entity propagation in bio...

In a growth-fragmentation system, cells grow in size slowly and split apart at random. Typically, the number of cells in the system grows exponentially and the distribution of the sizes of cells settles into an equilibrium 'asymptotic profile'. In this work we introduce a new method to prove this asymptotic behaviour for the growth-fragmentation equation, and show that the convergence to the asymptotic profile occurs at exponential rate. We do this by identifying an associate...

Originally motivated by the morphogenesis of bacterial microcolonies, the aim of this article is to explore models through different scales for a spatial population of interacting, growing and dividing particles. We start from a microscopic stochastic model, write the corresponding stochastic differential equation satisfied by the empirical measure, and rigorously derive its mesoscopic (mean-field) limit. Under smoothness and symmetry assumptions for the interaction kernel, w...

Single-cell experiments revealed substantial variability in generation times, growth rates but also in birth and division sizes between genetically identical cells. Understanding how these fluctuations determine the fitness of the population, i.e. its growth rate, is necessary in any quantitative theory of evolution. Here, we develop a biologically-relevant model which accounts for the stochasticity in single-cell growth rates, birth sizes and division sizes. We derive expres...