Nonequilibrium statistical mechanics of dividing cell populations

Equilibrium states corresponding to targeted hyperuniform nonequilibrium pair statistics

The Zhang-Torquato conjecture [G. Zhang and S. Torquato, Phys. Rev. E, 2020, 101, 032124.] states that any realizable pair correlation function g₂(r) or structure factor S(k) of a translationally invariant nonequilibrium system can be attained by an equilibrium ensemble involving only (up to) effective two-body interactions. To further test and study this conjecture, we consider two singular nonequilibrium models of recent interest that also have the exotic hyperuniformity property: a 2D "perfect glass" and a 3D critical absorbing-state model. We find that each nonequilibrium target can be achieved accurately by equilibrium states with effective one- and two-body potentials, lending further support to the conjecture. To characterize the structural degeneracy of such a nonequilibrium-equilibrium correspondence, we compute higher-order statistics for both models, as well as those for a hyperuniform 3D uniformly randomized lattice (URL), whose higher-order statistics can be very precisely ascertained. Interestingly, we find that the differences in the higher-order statistics between nonequilibrium and equilibrium systems with matching pair statistics, as measured by the "hole" probability distribution, provide measures of the degree to which a system is out of equilibrium. We show that all three systems studied possess the bounded-hole property and that holes near the maximum hole size in the URL are much rarer than those in the underlying simple cubic lattice. Remarkably, upon quenching, the effective potentials for all three systems possess local energy minima (i.e., inherent structures) with stronger forms of hyperuniformity compared to their target counterparts. Our methods are expected to facilitate the self-assembly of tunable hyperuniform soft-matter systems.

Contributions to the 'noise floor' in gene expression in a population of dividing cells

Experiments with cells reveal the existence of a lower bound for protein noise, the noise floor, in highly expressed genes. Its origins are still debated. We propose a minimal model of gene expression in a proliferating bacterial cell population. The model predicts the existence of a noise floor and it semi-quantitatively reproduces the curved shape of the experimental noise vs. mean protein concentration plots. When the cell volume increases in a different manner than does the mean protein copy number, the noise floor level is determined by the cell population’s age structure and by the dependence of the mean protein concentration on cell age. Additionally, the noise floor level may depend on a biological limit for the mean number of bursts in the cell cycle. In that case, the noise floor level depends on the burst size distribution width but it is insensitive to the mean burst size. Our model quantifies the contributions of each of these mechanisms to gene expression noise.

Exactly solvable model of gene expression in a proliferating bacterial cell population with stochastic protein bursts and protein partitioning

Many of the existing stochastic models of gene expression contain the first-order decay reaction term that may describe active protein degradation or dilution. If the model variable is interpreted as the molecule number, and not concentration, the decay term may also approximate the loss of protein molecules due to cell division as a continuous degradation process. The seminal model of that kind leads to gamma distributions of protein levels, whose parameters are defined by the mean frequency of protein bursts and mean burst size. However, such models (whether interpreted in terms of molecule numbers or concentrations) do not correctly account for the noise due to protein partitioning between daughter cells. We present an exactly solvable stochastic model of gene expression in cells dividing at random times, where we assume description in terms of molecule numbers with a constant mean protein burst size. The model is based on a population balance equation supplemented with protein production in random bursts. If protein molecules are partitioned equally between daughter cells, we obtain at steady state the analytical expressions for probability distributions similar in shape to gamma distributions, yet with quite different values of mean burst size and mean burst frequency than would result from fitting of the classical continuous-decay model to these distributions. For random partitioning of protein molecules between daughter cells, we obtain the moment equations for the protein number distribution and thus the analytical formulas for the squared coefficient of variation.

Universal protein distributions in a model of cell growth and division

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 calculated exactly, and a single parameter is found to determine the distribution shape and the variance-to-mean relation. Estimating this parameter from bacterial temporal data reproduces the measured distribution shape with high accuracy and leads to predictions for future experiments.

Closed-form stochastic solutions for non-equilibrium dynamics and inheritance of cellular components over many cell divisions

Stochastic dynamics govern many important processes in cellular biology, and an underlying theoretical approach describing these dynamics is desirable to address a wealth of questions in biology and medicine. Mathematical tools exist for treating several important examples of these stochastic processes, most notably gene expression and random partitioning at single-cell divisions or after a steady state has been reached. Comparatively little work exists exploring different and specific ways that repeated cell divisions can lead to stochastic inheritance of unequilibrated cellular populations. Here we introduce a mathematical formalism to describe cellular agents that are subject to random creation, replication and/or degradation, and are inherited according to a range of random dynamics at cell divisions. We obtain closed-form generating functions describing systems at any time after any number of cell divisions for binomial partitioning and divisions provoking a deterministic or random, subtractive or additive change in copy number, and show that these solutions agree exactly with stochastic simulation. We apply this general formalism to several example problems involving the dynamics of mitochondrial DNA during development and organismal lifetimes.

Weak correlation of starch and volume in synchronized photosynthetic cells

In cultures of unicellular algae, features of single cells, such as cellular volume and starch content, are thought to be the result of carefully balanced growth and division processes. Single-cell analyses of synchronized photoautotrophic cultures of the unicellular alga Chlamydomonas reinhardtii reveal, however, that the cellular volume and starch content are only weakly correlated. Likewise, other cell parameters, e.g., the chlorophyll content per cell, are only weakly correlated with cell size. We derive the cell size distributions at the beginning of each synchronization cycle considering growth, timing of cell division and daughter cell release, and the uneven division of cell volume. Furthermore, we investigate the link between cell volume growth and starch accumulation. This work presents evidence that, under the experimental conditions of light-dark synchronized cultures, the weak correlation between both cell features is a result of a cumulative process rather than due to asymmetric partition of biomolecules during cell division. This cumulative process necessarily limits cellular similarities within a synchronized cell population.

Gene expression noise facilitates adaptation and drug resistance independently of mutation

We show that the effect of stress on the reproductive fitness of noisy cell populations can be modeled as a first-passage time problem, and demonstrate that even relatively short-lived fluctuations in gene expression can ensure the long-term survival of a drug-resistant population. We examine how this effect contributes to the development of drug-resistant cancer cells, and demonstrate that permanent immunity can arise independently of mutations.

Stochastic model for tumor growth with immunization

We analyze a stochastic model for tumor cell growth with both multiplicative and additive colored noises as well as nonzero cross correlations in between. Whereas the death rate within the logistic model is altered by a deterministic term characterizing immunization, the birth rate is assumed to be stochastically changed due to biological motivated growth processes leading to a multiplicative internal noise. Moreover, the system is subjected to an external additive noise which mimics the influence of the environment of the tumor. The stationary probability distribution P_{s} is derived depending on the finite correlation time, the immunization rate, and the strength of the cross correlation. P_{s} offers a maximum which becomes more pronounced for increasing immunization rate. The mean-first-passage time is also calculated in order to find out under which conditions the tumor can suffer extinction. Its characteristics are again controlled by the degree of immunization and the strength of the cross correlation. The behavior observed can be interpreted in terms of a biological model of tumor evolution.

Cellular properties and population asymptotics in the population balance equation

Proliferating cell populations at steady-state growth often exhibit broad protein distributions with exponential tails. The sources of this variation and its universality are of much theoretical interest. Here we address the problem by asymptotic analysis of the population balance equation. We show that the steady-state distribution tail is determined by a combination of protein production and cell division and is insensitive to other model details. Under general conditions this tail is exponential with a dependence on parameters consistent with experiment. We discuss the conditions for this effect to be dominant over other sources of variation and the relation to experiments.

Multiple mutant clones in blood rarely coexist

Leukemias arise due to mutations in the genome of hematopoietic (blood) cells. Hematopoiesis has a multicompartment architecture, with cells exhibiting different rates of replication and differentiation. At the root of this process, one finds a small number of stem cells, and hence the description of the mutation-selection dynamics of blood cells calls for a stochastic approach. We use stochastic dynamics to investigate to which extent acquired hematopoietic disorders are associated with mutations of single or multiple genes within developing blood cells. Our analysis considers the appearance of mutations both in the stem cell compartment as well as in more committed compartments. We conclude that in the absence of genomic instability, acquired hematopoietic disorders due to mutations in multiple genes are most likely very rare events, as multiple mutations typically require much longer development times compared to those associated with a single mutation.