Corrigendum: Rare cell variability and drug-induced reprogramming as a mode of cancer drug resistance.

This corrects the article DOI: 10.1038/nature22794.

Rare cell variability and drug-induced reprogramming as a mode of cancer drug resistance.

Therapies that target signalling molecules that are mutated in cancers can often have substantial short-term effects, but the emergence of resistant cancer cells is a major barrier to full cures. Resistance can result from secondary mutations, but in other cases there is no clear genetic cause, raising the possibility of non-genetic rare cell variability. Here we show that human melanoma cells can display profound transcriptional variability at the single-cell level that predicts which cells will ultimately resist drug treatment. This variability involves infrequent, semi-coordinated transcription of a number of resistance markers at high levels in a very small percentage of cells. The addition of drug then induces epigenetic reprogramming in these cells, converting the transient transcriptional state to a stably resistant state. This reprogramming begins with a loss of SOX10-mediated differentiation followed by activation of new signalling pathways, partially mediated by the activity of the transcription factors JUN and/or AP-1 and TEAD. Our work reveals the multistage nature of the acquisition of drug resistance and provides a framework for understanding resistance dynamics in single cells. We find that other cell types also exhibit sporadic expression of many of these same marker genes, suggesting the existence of a general program in which expression is displayed in rare subpopulations of cells.

Exact lower and upper bounds on stationary moments in stochastic biochemical systems.

In the stochastic description of biochemical reaction systems, the time evolution of statistical moments for species population counts is described by a linear dynamical system. However, except for some ideal cases (such as zero- and first-order reaction kinetics), the moment dynamics is underdetermined as lower-order moments depend upon higher-order moments. Here, we propose a novel method to find exact lower and upper bounds on stationary moments for a given arbitrary system of biochemical reactions. The method exploits the fact that statistical moments of any positive-valued random variable must satisfy some constraints that are compactly represented through the positive semidefiniteness of moment matrices. Our analysis shows that solving moment equations at steady state in conjunction with constraints on moment matrices provides exact lower and upper bounds on the moments. These results are illustrated by three different examples-the commonly used logistic growth model, stochastic gene expression with auto-regulation and an activator-repressor gene network motif. Interestingly, in all cases the accuracy of the bounds is shown to improve as moment equations are expanded to include higher-order moments. Our results provide avenues for development of approximation methods that provide explicit bounds on moments for nonlinear stochastic systems that are otherwise analytically intractable.

Intercellular Variability in Protein Levels from Stochastic Expression and Noisy Cell Cycle Processes.

Inside individual cells, expression of genes is inherently stochastic and manifests as cell-to-cell variability or noise in protein copy numbers. Since proteins half-lives can be comparable to the cell-cycle length, randomness in cell-division times generates additional intercellular variability in protein levels. Moreover, as many mRNA/protein species are expressed at low-copy numbers, errors incurred in partitioning of molecules between two daughter cells are significant. We derive analytical formulas for the total noise in protein levels when the cell-cycle duration follows a general class of probability distributions. Using a novel hybrid approach the total noise is decomposed into components arising from i) stochastic expression; ii) partitioning errors at the time of cell division and iii) random cell-division events. These formulas reveal that random cell-division times not only generate additional extrinsic noise, but also critically affect the mean protein copy numbers and intrinsic noise components. Counter intuitively, in some parameter regimes, noise in protein levels can decrease as cell-division times become more stochastic. Computations are extended to consider genome duplication, where transcription rate is increased at a random point in the cell cycle. We systematically investigate how the timing of genome duplication influences different protein noise components. Intriguingly, results show that noise contribution from stochastic expression is minimized at an optimal genome-duplication time. Our theoretical results motivate new experimental methods for decomposing protein noise levels from synchronized and asynchronized single-cell expression data. Characterizing the contributions of individual noise mechanisms will lead to precise estimates of gene expression parameters and techniques for altering stochasticity to change phenotype of individual cells.

Modeling homeostasis mechanisms that set the target cell size.

How organisms maintain cell size homeostasis is a long-standing problem that remains unresolved, especially in multicellular organisms. Recent experiments in diverse animal cell types demonstrate that within a cell population, cellular proliferation is low for small and large cells, but high at intermediate sizes. Here we use mathematical models to explore size-control strategies that drive such a non-monotonic profile resulting in the proliferation capacity being maximized at a target cell size. Our analysis reveals that most models of size control yield proliferation capacities that vary monotonically with cell size, and non-monotonicity requires two key mechanisms: (1) the growth rate decreases with increasing size for excessively large cells; and (2) cell division occurs as per the Adder model (division is triggered upon adding a fixed size from birth), or a Sizer-Adder combination. Consistent with theory, Jurkat T cell growth rates increase with size for small cells, but decrease with size for large cells. In summary, our models show that regulation of both growth and cell-division timing is necessary for size control in animal cells, and this joint mechanism leads to a target cell size where cellular proliferation capacity is maximized.

Cell size control and gene expression homeostasis in single-cells.

Growth of a cell and its subsequent division into daughters is a fundamental aspect of all cellular living systems. During these processes, how do individual cells correct size aberrations so that they do not grow abnormally large or small? How do cells ensure that the concentration of essential gene products are maintained at desired levels, in spite of dynamic/stochastic changes in cell size during growth and division? Both these questions have fascinated researchers for over a century. We review how advances in singe-cell technologies and measurements are providing unique insights into these questions across organisms from prokaryotes to human cells. More specifically, diverse strategies based on timing of cell-cycle events, regulating growth, and number of daughters are employed to maintain cell size homeostasis. Interestingly, size homeostasis often results in size optimality - proliferation of individual cells in a population is maximized at an optimal cell size. We further discuss how size-dependent expression or gene-replication timing can buffer concentration of a gene product from cell-to-cell size variations within a population. Finally, we speculate on an intriguing hypothesis that specific size control strategies may have evolved as a consequence of gene-product concentration homeostasis.

HIV 2-LTR experiment design optimization.

Clinical trials are necessary in order to develop treatments for diseases; however, they can often be costly, time consuming, and demanding to the patients. This paper summarizes several common methods used for optimal design that can be used to address these issues. In addition, we introduce a novel method for optimizing experiment designs applied to HIV 2-LTR clinical trials. Our method employs Bayesian techniques to optimize the experiment outcome by maximizing the Expected Kullback-Leibler Divergence (EKLD) between the a priori knowledge of system parameters before the experiment and the a posteriori knowledge of the system parameters after the experiment. We show that our method is robust and performs equally well if not better than traditional optimal experiment design techniques.

Implications of Measurement Assay Type in Design of HIV Experiments.

Time series measurements of circular viral episome (2-LTR) concentrations enable indirect quantification of persistent low-level Human Immunodeficiency Virus (HIV) replication in patients on Integrase-Inhibitor intensified Combined Antiretroviral Therapy (cART). In order to determine the magnitude of these low level infection events, blood has to be drawn from a patients at a frequency and volume that is strictly regulated by the Institutional Review Board (IRB). Once the blood is drawn, the 2-LTR concentration is determined by quantifying the amount of HIV DNA present in the sample via a PCR (Polymerase Chain Reaction) assay. Real time quantitative Polymerase Chain Reaction (qPCR) is a widely used method of performing PCR; however, a newer droplet digital Polymerase Chain Reaction (ddPCR) method has been shown to provide more accurate quantification of DNA. Using a validated model of HIV viral replication, this paper demonstrates the importance of considering DNA quantification assay type when optimizing experiment design conditions. Experiments are optimized using a Genetic Algorithm (GA) to locate a family of suboptimal sample schedules which yield the highest fitness. Fitness is defined as the expected information gained in the experiment, measured by the Kullback-Leibler Divergence (KLD) between the prior and posterior distributions of the model parameters. We compare the information content of the optimized schedules to uniform schedules as well as two clinical schedules implemented by researchers at UCSF and the University of Melbourne. This work shows that there is a significantly greater gain information in experiments using a ddPCR assay vs. a qPCR assay and that certain experiment design considerations should be taken when using either assay.

A mechanistic stochastic framework for regulating bacterial cell division.

How exponentially growing cells maintain size homeostasis is an important fundamental problem. Recent single-cell studies in prokaryotes have uncovered the adder principle, where cells add a fixed size (volume) from birth to division, irrespective of their size at birth. To mechanistically explain the adder principle, we consider a timekeeper protein that begins to get stochastically expressed after cell birth at a rate proportional to the volume. Cell-division time is formulated as the first-passage time for protein copy numbers to hit a fixed threshold. Consistent with data, the model predicts that the noise in division timing increases with size at birth. Intriguingly, our results show that the distribution of the volume added between successive cell-division events is independent of the newborn cell size. This was dramatically seen in experimental studies, where histograms of the added volume corresponding to different newborn sizes collapsed on top of each other. The model provides further insights consistent with experimental observations: the distribution of the added volume when scaled by its mean becomes invariant of the growth rate. In summary, our simple yet elegant model explains key experimental findings and suggests a mechanism for regulating both the mean and fluctuations in cell-division timing for controlling size.