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.

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: precursor protein degradation, nonlinear accumulation rate, and a threshold size termed the commitment size. These models fit mean trends but predict different distributions given the birth size. To validate these models, we used the Akaike information criterion and compared them to open datasets of slow-growing E. coli cells in different media. the degradation model could explain the division strategy for media where cells are larger, while the commitment size model could account for smaller cells. The power-law model, finally, better fits the data at intermediate regimes.

Phosphate-Solubilizing Bacteria with Low-Solubility Fertilizer Improve Soil P Availability and Yield of Kikuyu Grass.

Inoculation with phosphate-solubilizing bacteria (PSB) and the application of phosphorus (P) sources can improve soil P availability, enhancing the sustainability and efficiency of agricultural systems. The implementation of this technology in perennial grasses, such as Kikuyu grass, for cattle feed in soils with high P retention, such as Andisols, has been little explored. The objective of this study was to evaluate the productive response of Kikuyu grass and soil P dynamics to BSF inoculation with different P sources. The experiment was conducted on a Kikuyu pasture, which was evaluated for 18 months (September 2020 to March 2022). Three P fertilizers with different solubility levels were applied: diammonium phosphate (DAP) (high-solubility), rock phosphate (RP), and compost (OM) (low-solubility). Moreover, the inoculation of a PSB consortium (Azospirillum brasilense D7, Rhizobium leguminosarum T88 and Herbaspirillum sp. AP21) was tested. Inoculation with PSB and fertilization with rock phosphate (RP) increased soil labile P and acid phosphomonoesterase activity. Increased grass yield and quality were related with higher soil inorganic P (Pi) availability. This study validated, under field conditions, the benefits of PSB inoculation for soil P availability and Kikuyu grass productivity.

The fission yeast cell size control system integrates pathways measuring cell surface area, volume, and time.

Eukaryotic cells tightly control their size, but the relevant aspect of size is unknown in most cases. Fission yeast divide at a threshold cell surface area (SA) due, in part, to the protein kinase Cdr2. We find that fission yeast cells only divide by SA under a size threshold. Mutants that divide at a larger size shift to volume-based divisions. Diploid cells divide at a larger size than haploid cells do, but they maintain SA-based divisions, and this indicates that the size threshold for changing from surface-area-based to volume-based control is set by ploidy. Within this size control system, we found that the mitotic activator Cdc25 accumulates like a volume-based sizer molecule, whereas the mitotic cyclin Cdc13 accumulates in the nucleus as a timer. We propose an integrated model for cell size control based on multiple signaling pathways that report on distinct aspects of cell size and growth, including cell SA (Cdr2), cell volume (Cdc25), and time (Cdc13). Combined modeling and experiments show how this system can generate both sizer- and adder-like properties.

PyEcoLib: a python library for simulating stochastic cell size dynamics.

Recently, there has been an increasing need for tools to simulate cell size regulation due to important applications in cell proliferation and gene expression. However, implementing the simulation usually presents some difficulties, as the division has a cycle-dependent occurrence rate. In this article, we gather a recent theoretical framework inPyEcoLib, a python-based library to simulate the stochastic dynamics of the size of bacterial cells. This library can simulate cell size trajectories with an arbitrarily small sampling period. In addition, this simulator can include stochastic variables, such as the cell size at the beginning of the experiment, the cycle duration timing, the growth rate, and the splitting position. Furthermore, from a population perspective, the user can choose between tracking a single lineage or all cells in a colony. They can also simulate the most common division strategies (adder, timer, and sizer) using the division rate formalism and numerical methods. As an example of PyecoLib applications, we explain how to couple size dynamics with gene expression predicting, from simulations, how the noise in protein levels increases by increasing the noise in division timing, the noise in growth rate and the noise in cell splitting position. The simplicity of this library and its transparency about the underlying theoretical framework yield the inclusion of cell size stochasticity in complex models of gene expression.

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.

Continuous rate modeling of bacterial stochastic size dynamics.

Bacterial division is an inherently stochastic process with effects on fluctuations of protein concentration and phenotype variability. Current modeling tools for the stochastic short-term cell-size dynamics are scarce and mainly phenomenological. Here we present a general theoretical approach based on the Chapman-Kolmogorov equation incorporating continuous growth and division events as jump processes. This approach allows us to include different division strategies, noisy growth, and noisy cell splitting. Considering bacteria synchronized from their last division, we predict oscillations in both the central moments of the size distribution and its autocorrelation function. These oscillations, barely discussed in past studies, can arise as a consequence of the discrete time displacement invariance of the system with a period of one doubling time, and they do not disappear when including stochasticity on either division times or size heterogeneity on the starting population but only after inclusion of noise in either growth rate or septum position. This result illustrates the usefulness of having a solid mathematical description that explicitly incorporates the inherent stochasticity in various biological processes, both to understand the process in detail and to evaluate the effect of various sources of variability when creating simplified descriptions.

Disentangling the Complexity of the Rumen Microbial Diversity Through Fractionation Using a Sucrose Density Gradient.

The ruminal microbial community is an important element in health, nutrition, livestock productivity, and climate impact. Despite the historic and current efforts to characterize this microbial diversity, many of its members remain unidentified, making it challenging to associate microbial groups with functions. Here we present a low-cost methodology for rumen sample treatment that separates the microbial community based on cell size, allowing for the identification of subtle compositional changes. In brief, the sample is centrifuged through a series of sucrose density gradients, and cells migrate to their corresponding density fraction. From each fraction, DNA is extracted and 16S rRNA gene amplicons are sequenced. We tested our methodology on four animals under two different conditions, fasting, and post-feeding. Each fraction was examined by confocal microscopy showing that the same sucrose fraction consistently separated similar cell-sized microorganisms independent of the animal or treatment. Microbial composition analysis using metabarcoding showed that our methodology detected low abundance bacterial families and population changes between fasting and post-feeding treatments that could not be observed by bulk DNA analysis. In conclusion, the sucrose-based method is a powerful low-cost approximation to untwine, enrich, and potentially isolate uncharacterized members of the ruminal microbiome.

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.

Correlation between protein concentration and bacterial cell size can reveal mechanisms of gene expression.

Classically, gene expression is modeled as a chemical process with reaction rates dependent on the concentration of the reactants (typically, DNA loci, plasmids, RNA, enzymes, etc). Other variables like cell size are in general ignored. Size dynamics can become an important variable due to the low number of many of these reactants, imperfectly symmetric cell partitioning and molecule segregation. In this work we measure the correlation between size and protein concentration by observing the gene expression of the RpOD gene from a low-copy plasmid in Escherichia coli during balanced growth in different media. A positive correlation was found, and we used it to examine possible models of cell size dynamics and plasmid replication. We implemented a previously developed model describing the full gene expression process including transcription, translation, loci replication, cell division and molecule segregation. By comparing with the observed correlation, we determine that the transcription rate must be proportional to the size times the number of plasmids. We discuss how fluctuations in plasmid segregation, due to the low copy number, can impose limits in this correlation.

Unification of cell division control strategies through continuous rate models.

Recent experiments support the adder model for E. coli division control. This model posits that bacteria grow, on average, a fixed size before division. It also predicts decorrelation between the noise in the added size and the size at birth. Here we develop a theory based on stochastic hybrid systems which could explain the main division strategies, including not only the adder strategy but the whole range from sizer to timer. We use experiments to explore the division control of E. coli growing with glycerol as carbon source. In this medium, the division strategy is sizerlike, which means that the added size decreases with the size at birth. We found, as our theory predicts, that in a sizerlike strategy the mean added size decreases with the size at birth while the noise in added size increases. We discuss possible molecular mechanisms underlying this strategy and propose a general model that encompasses the different division strategies.

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.

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.

Synaptic transmission may provide an evolutionary benefit to HIV through modulation of latency.

Transmission of HIV is known to occur by two mechanisms in vivo: the free virus pathway, where viral particles bud off an infected cell before attaching to an uninfected cell, and the cell-cell pathway, where infected cells form virological synapses through close contact with an uninfected cell. It has also been shown that HIV replication includes a positive feedback loop controlled by the viral protein Tat, which may act as a stochastic switch in determining whether an infected cell enters latency. In this paper, we introduce a simple mathematical model of HIV replication containing both the free virus and cell-cell pathways. Using this model, we demonstrate that the high multiplicity of infection in cell-cell transmission results in a suppression of latent infection, and that this modulation of latency through balancing the two transmission mechanisms can provide an evolutionary benefit to the virus. This benefit increases with decreasing overall viral fitness, which may provide a within-host evolutionary pressure toward more cell-cell transmission in late-stage HIV infection.

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.

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.

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.

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.