Mechanisms of cell size regulation in slow-growing Escherichia coli cells: discriminating models beyond the adder.

Under ideal conditions, Escherichia coli cells divide after adding a fixed cell size, a strategy known as the adder. This concept applies to various microbes and is often explained as the division that occurs after a certain number of stages, associated with the accumulation of precursor proteins at a rate proportional to cell size. However, under poor media conditions, E. coli cells exhibit a different size regulation. They are smaller and follow a sizer-like division strategy where the added size is inversely proportional to the size at birth. We explore three potential causes for this deviation: degradation of the precursor protein and two models where the propensity for accumulation depends on the cell size: a nonlinear accumulation rate, and accumulation starting at a threshold size termed the commitment size. These models fit the mean trends but predict different distributions given the birth size. To quantify the precision of the models to explain the data, we used the Akaike information criterion and compared them to open datasets of slow-growing E. coli cells in different media. We found that none of the models alone can consistently explain the data. However, the degradation model better explains the division strategy when cells are larger, whereas size-related models (power-law and commitment size) account for smaller cells. Our methodology proposes a data-based method in which different mechanisms can be tested systematically.

A cell-based model for size control in the multiple fission alga Chlamydomonas reinhardtii.

Understanding how population-size homeostasis emerges from stochastic individual cell behaviors remains a challenge in biology.1,2,3,4,5,6,7 The unicellular green alga Chlamydomonas reinhardtii (Chlamydomonas) proliferates using a multiple fission cell cycle, where a prolonged G1 phase is followed by n rounds of alternating division cycles (S/M) to produce 2n daughters. A "Commitment" sizer in mid-G1 phase ensures sufficient cell growth before completing the cell cycle. A mitotic sizer couples mother-cell size to division number (n) such that daughter size distributions are uniform regardless of mother size distributions. Although daughter size distributions were highly robust to altered growth conditions, ∼40% of daughter cells fell outside of the 2-fold range expected from a "perfect" multiple fission sizer.7,8 A simple intuitive power law model with stochastic noise failed to reproduce individual division behaviors of tracked single cells. Through additional iterative modeling, we identified an alternative modified threshold (MT) model, where cells need to cross a threshold greater than 2-fold their median starting size to become division-competent (i.e., Committed), after which their behaviors followed a power law model. The Commitment versus mitotic size threshold uncoupling in the MT model was likely a key pre-adaptation in the evolution of volvocine algal multicellularity. A similar experimental approach was used in size mutants mat3/rbr and dp1 that are, respectively, missing repressor or activator subunits of the retinoblastoma tumor suppressor complex (RBC). Both mutants showed altered relationships between Commitment and mitotic sizer, suggesting that RBC functions to decouple the two sizers.

Arf6 anchors Cdr2 nodes at the cell cortex to control cell size at division.

Fission yeast cells prevent mitotic entry until a threshold cell surface area is reached. The protein kinase Cdr2 contributes to this size control system by forming multiprotein nodes that inhibit Wee1 at the medial cell cortex. Cdr2 node anchoring at the cell cortex is not fully understood. Through a genomic screen, we identified the conserved GTPase Arf6 as a component of Cdr2 signaling. Cells lacking Arf6 failed to divide at a threshold surface area and instead shifted to volume-based divisions at increased overall size. Arf6 stably localized to Cdr2 nodes in its GTP-bound but not GDP-bound state, and its guanine nucleotide exchange factor (GEF), Syt22, was required for both Arf6 node localization and proper size at division. In arf6Δ mutants, Cdr2 nodes detached from the membrane and exhibited increased dynamics. These defects were enhanced when arf6Δ was combined with other node mutants. Our work identifies a regulated anchor for Cdr2 nodes that is required for cells to sense surface area.

Efficient computation of stochastic cell-size transient dynamics.

BACKGROUND: How small, fast-growing bacteria ensure tight cell-size distributions remains elusive. High-throughput measurement techniques have propelled efforts to build modeling tools that help to shed light on the relationships between cell size, growth and cycle progression. Most proposed models describe cell division as a discrete map between size at birth and size at division with stochastic fluctuations assumed. However, such models underestimate the role of cell size transient dynamics by excluding them. RESULTS: We propose an efficient approach for estimation of cell size transient dynamics. Our technique approximates the transient size distribution and statistical moment dynamics of exponential growing cells following an adder strategy with arbitrary precision. CONCLUSIONS: We approximate, up to arbitrary precision, the distribution of division times and size across time for the adder strategy in rod-shaped bacteria cells. Our approach is able to compute statistical moments like mean size and its variance from such distributions efficiently, showing close match with numerical simulations. Additionally, we observed that these distributions have periodic properties. Our approach further might shed light on the mechanisms behind gene product homeostasis.

Analysis of Noise Mechanisms in Cell-Size Control.

At the single-cell level, noise arises from multiple sources, such as inherent stochasticity of biomolecular processes, random partitioning of resources at division, and fluctuations in cellular growth rates. How these diverse noise mechanisms combine to drive variations in cell size within an isoclonal population is not well understood. Here, we investigate the contributions of different noise sources in well-known paradigms of cell-size control, such as adder (division occurs after adding a fixed size from birth), sizer (division occurs after reaching a size threshold), and timer (division occurs after a fixed time from birth). Analysis reveals that variation in cell size is most sensitive to errors in partitioning of volume among daughter cells, and not surprisingly, this process is well regulated among microbes. Moreover, depending on the dominant noise mechanism, different size-control strategies (or a combination of them) provide efficient buffering of size variations. We further explore mixer models of size control, where a timer phase precedes/follows an adder, as has been proposed in Caulobacter crescentus. Although mixing a timer and an adder can sometimes attenuate size variations, it invariably leads to higher-order moments growing unboundedly over time. This results in a power-law distribution for the cell size, with an exponent that depends inversely on the noise in the timer phase. Consistent with theory, we find evidence of power-law statistics in the tail of C. crescentus cell-size distribution, although there is a discrepancy between the observed power-law exponent and that predicted from the noise parameters. The discrepancy, however, is removed after data reveal that the size added by individual newborns in the adder phase itself exhibits power-law statistics. Taken together, this study provides key insights into the role of noise mechanisms in size homeostasis, and suggests an inextricable link between timer-based models of size control and heavy-tailed cell-size distributions.

Conditional Moment Closure Schemes for Studying Stochastic Dynamics of Genetic Circuits.

Inside individual cells, stochastic expression drives random fluctuations in gene product copy numbers, which corrupts functioning of both natural and synthetic genetic circuits. Dynamic models of genetic circuits are formulated stochastically using the chemical master equation framework. Since obtaining probability distributions can be computationally expensive in these models, noise is typically investigated through lower-order statistical moments (mean, variance, correlation, skewness, etc.) of mRNA/proteins levels. However, due to the nonlinearities in genetic circuits, this moment dynamics is typically not closed, in the sense that the time derivative of the lower-order statistical moments depends on high-order moments. Moment equations are closed by expressing higher-order moments as nonlinear functions of lower-order moments, a technique commonly referred to as moment closure. We provide a new moment closure scheme for studying stochastic dynamics of genetic circuits, where genes randomly toggle between transcriptionally active and inactive states. The method is based on conditioning protein levels on active states of genes and then expressing higher-order moments as functions of lower-order conditional moments. The conditional closure scheme is illustrated on different circuit motifs and found to outperform existing closure techniques. Rapid computation of stochasticity through closure methods will enable improved characterization and design of synthetic circuits that exhibit robust performance in spite of noisy expression of underlying genes.

Balance de Flujos Metabólicos en Saccharomyces cerevisiae basado en Compartimentalización Intracelular / Metabolic Flows Balance in Saccharomyces cerevisiae based on Intracellular Compartmentalization

Una de las técnicas más utilizadas para la predicción de producción de bioproductos y distribución intracelular de flujos de microorganismos es el Análisis de Balance de Flujos - FBA por sus siglas en inglés. El FBA requiere de una función objetivo que represente el objetivo biológico del microorganismo estudiado. En este trabajo se propone un nuevo tipo de funciones objetivo basada en la combinación de objetivos de compartimentos físicos presentes en el microorganismo estudiado. Este tipo de funciones objetivo son examinadas junto con un modelo estequiométrico extraído de la reconstrucción iMM904 del microorganismo S. cerevisiae. Su desempeño se compara con la función objetivo más usada en la literatura, la maximización de biomasa, en condiciones experimentales anaeróbicas en cultivos continuos y aeróbicas en cultivos tipo lote. La función objetivo propuesta en este trabajo mejora las predicciones de crecimiento en un 10% y las predicciones de producción de etanol en un 75% respecto a las obtenidas por la función objetivo de maximización de biomasa, en condiciones anaeróbicas. En condiciones aeróbicas tipo lote la función objetivo propuesta mejora en un 98% las predicciones de crecimiento y en un 70% las predicciones de etanol con respecto a la función objetivo de biomasa.

Flux Balance Analysis - FBA - is one of the most used techniques in prediction of microorganism bioproducts. It requires an objective function that represents biological objective of the studied microorganism. This paper presents a new kind of objective functions based on individual physical compartment objetives in the studied microorganism. These kind of functions was tested with a stoichiometric model extracted from iMM904 reconstruction of S. cerevisiae and its performance is compared with the most used objective function in literature, growth maximization, in anaerobic and aerobic batch conditions. The presented objective function outperform growth predictions in 10% and ethanol predictions in 75% compared with obtained by maximization of growth objective function, in anaerobic conditions. In aerobic batch conditions the presented objective function outperforms in 98% growth preditions and 70% ethanol predictions compared with growth maximization.

Predictive potential of flux balance analysis of Saccharomyces cerevisiae using as optimization function combinations of cell compartmental objectives.

BACKGROUND: The main objective of flux balance analysis (FBA) is to obtain quantitative predictions of metabolic fluxes of an organism, and it is necessary to use an appropriate objective function to guarantee a good estimation of those fluxes. METHODOLOGY: In this study, the predictive performance of FBA was evaluated, using objective functions arising from the linear combination of different cellular objectives. This approach is most suitable for eukaryotic cells, owing to their multiplicity of cellular compartments. For this reason, Saccharomyces cerevisiae was used as model organism, and its metabolic network was represented using the genome-scale metabolic model iMM904. As the objective was to evaluate the predictive performance from the FBA using the kind of objective function previously described, substrate uptake and oxygen consumption were the only input data used for the FBA. Experimental information about microbial growth and exchange of metabolites with the environment was used to assess the quality of the predictions. CONCLUSIONS: The quality of the predictions obtained with the FBA depends greatly on the knowledge of the oxygen uptake rate. For the most of studied classifications, the best predictions were obtained with "maximization of growth", and with some combinations that include this objective. However, in the case of exponential growth with unknown oxygen exchange flux, the objective function "maximization of growth, plus minimization of NADH production in cytosol, plus minimization of NAD(P)H consumption in mitochondrion" gave much more accurate estimations of fluxes than the obtained with any other objective function explored in this study.

Predicción a escala genómica de Componentes de Saccharomyces cerevisiae mediante Análisis de Balance de Flujos / Prediction of genome scale of Saccharomyces cerevisiae by flux balance analysis

El microorganismo Saccharomyces cerevisiae cuenta con gran número de modelos biológicos conocidos como reconstrucciones, las cuales pueden ser a escala genómica. De estas reconstrucciones a escala genómica provienen los modelos matemáticos, también llamados modelos estequiométricos. Una de las técnicas más usadas para estudiar estos modelos es el Análisis de Balance de Flujos (FBA). El proposito del FBA es predecir el crecimiento del microorganismo bajo estudio, y la producción y consumo de componentes como el etanol, CO2 glicerol, sucinato, acetato y piruvato. Para determinar si las predicciones obtenidas mediante FBA son únicas se utiliza la técnica de Análisis de Variabilidad Flujos (FVA). El presente trabajo muestra los resultados de aplicar el FBA a la reconstrucción reciente del microorganismo S. cerevisiae, la denominada iMM904 y los compara con un conjunto de datos experimentales presente en la literatura. Este trabajo también estudia la existencia de múltiples predicciones FBA utilizando la técnica FVA. Los resultados ilustran que es posible predecir el crecimiento del microorganimo S. cerevisiae, con errores entre el 11% y 28%; la producción de CO2, con errores entre el 0.3% y 4.5% y la producción de etanol, con errores entre el 11% y 13%.

Several biological models, named reconstructions, are used for the study of the S. cerevisiae microorganism. The reconstructions can be genomic scaled. Mathematical models are generated from the reconstructions and they are called stoichiometric models. The flux balance analysis (FBA) is one of the tools used for the analysis of these models. The FBA attempts to predict the evolution of the microorganism and the consumption and production of components like glucose, ethanol, glycerol, succinate, acetate and pyruvate. A Flux variability analysis (FVA) is used to determine the uniqueness of the FBA predictions. This paper shows the results of applying FBA to the iMM904 reconstruction of S. cerevisiae and compares them with experimental data from literature. The results in this paper show that it is possible to predict the evolution with errors between 11% and 28% ; the production of CO2 with errors between 0.3% and 4.5%; and the production of ethanol with errors between 11% and 13%, using FBA for the iMM904 model.