Nunchaku: optimally partitioning data into piece-wise contiguous segments.

MOTIVATION: When analyzing 1D time series, scientists are often interested in identifying regions where one variable depends linearly on the other. Typically, they use an ad hoc and therefore often subjective method to do so. RESULTS: Here, we develop a statistically rigorous, Bayesian approach to infer the optimal partitioning of a dataset not only into contiguous piece-wise linear segments, but also into contiguous segments described by linear combinations of arbitrary basis functions. We therefore present a general solution to the problem of identifying discontinuous change points. Focusing on microbial growth, we use the algorithm to find the range of optical density where this density is linearly proportional to the number of cells and to automatically find the regions of exponential growth for both Escherichia coli and Saccharomyces cerevisiae. For budding yeast, we consequently are able to infer the Monod constant for growth on fructose. Our algorithm lends itself to automation and high throughput studies, increases reproducibility, and should facilitate data analyses for a broad range of scientists. AVAILABILITY AND IMPLEMENTATION: The corresponding Python package, entitled Nunchaku, is available at PyPI: https://pypi.org/project/nunchaku.

Determining growth rates from bright-field images of budding cells through identifying overlaps.

Much of biochemical regulation ultimately controls growth rate, particularly in microbes. Although time-lapse microscopy visualises cells, determining their growth rates is challenging, particularly for those that divide asymmetrically, like Saccharomyces cerevisiae, because cells often overlap in images. Here, we present the Birth Annotator for Budding Yeast (BABY), an algorithm to determine single-cell growth rates from label-free images. Using a convolutional neural network, BABY resolves overlaps through separating cells by size and assigns buds to mothers by identifying bud necks. BABY uses machine learning to track cells and determine lineages and estimates growth rates as the rates of change of volumes. Using BABY and a microfluidic device, we show that bud growth is likely first sizer- then timer-controlled, that the nuclear concentration of Sfp1, a regulator of ribosome biogenesis, varies before the growth rate does, and that growth rate can be used for real-time control. By estimating single-cell growth rates and so fitness, BABY should generate much biological insight.

Analysing and meta-analysing time-series data of microbial growth and gene expression from plate readers.

Responding to change is a fundamental property of life, making time-series data invaluable in biology. For microbes, plate readers are a popular, convenient means to measure growth and also gene expression using fluorescent reporters. Nevertheless, the difficulties of analysing the resulting data can be a bottleneck, particularly when combining measurements from different wells and plates. Here we present omniplate, a Python module that corrects and normalises plate-reader data, estimates growth rates and fluorescence per cell as functions of time, calculates errors, exports in different formats, and enables meta-analysis of multiple plates. The software corrects for autofluorescence, the optical density's non-linear dependence on the number of cells, and the effects of the media. We use omniplate to measure the Monod relationship for the growth of budding yeast in raffinose, showing that raffinose is a convenient carbon source for controlling growth rates. Using fluorescent tagging, we study yeast's glucose transport. Our results are consistent with the regulation of the hexose transporter (HXT) genes being approximately bipartite: the medium and high affinity transporters are predominately regulated by both the high affinity glucose sensor Snf3 and the kinase complex SNF1 via the repressors Mth1, Mig1, and Mig2; the low affinity transporters are predominately regulated by the low affinity sensor Rgt2 via the co-repressor Std1. We thus demonstrate that omniplate is a powerful tool for exploiting the advantages offered by time-series data in revealing biological regulation.

Multiple nutrient transporters enable cells to mitigate a rate-affinity tradeoff.

Eukaryotic genomes often encode multiple transporters for the same nutrient. For example, budding yeast has 17 hexose transporters (HXTs), all of which potentially transport glucose. Using mathematical modelling, we show that transporters that use either facilitated diffusion or symport can have a rate-affinity tradeoff, where an increase in the maximal rate of transport decreases the transporter's apparent affinity. These changes affect the import flux non-monotonically, and for a given concentration of extracellular nutrient there is one transporter, characterised by its affinity, that has a higher import flux than any other. Through encoding multiple transporters, cells can therefore mitigate the tradeoff by expressing those transporters with higher affinities in lower concentrations of nutrients. We verify our predictions using fluorescent tagging of seven HXT genes in budding yeast and follow their expression over time in batch culture. Using the known affinities of the corresponding transporters, we show that their regulation in glucose is broadly consistent with a rate-affinity tradeoff: as glucose falls, the levels of the different transporters peak in an order that mostly follows their affinity for glucose. More generally, evolution is constrained by tradeoffs. Our findings indicate that one such tradeoff often occurs in the cellular transport of nutrients.

Roadmap on biology in time varying environments.

Biological organisms experience constantly changing environments, from sudden changes in physiology brought about by feeding, to the regular rising and setting of the Sun, to ecological changes over evolutionary timescales. Living organisms have evolved to thrive in this changing world but the general principles by which organisms shape and are shaped by time varying environments remain elusive. Our understanding is particularly poor in the intermediate regime with no separation of timescales, where the environment changes on the same timescale as the physiological or evolutionary response. Experiments to systematically characterize the response to dynamic environments are challenging since such environments are inherently high dimensional. This roadmap deals with the unique role played by time varying environments in biological phenomena across scales, from physiology to evolution, seeking to emphasize the commonalities and the challenges faced in this emerging area of research.

Resolving fluorescent species by their brightness and diffusion using correlated photon-counting histograms.

Fluorescence fluctuation spectroscopy (FFS) refers to techniques that analyze fluctuations in the fluorescence emitted by fluorophores diffusing in a small volume and can be used to distinguish between populations of molecules that exhibit differences in brightness or diffusion. For example, fluorescence correlation spectroscopy (FCS) resolves species through their diffusion by analyzing correlations in the fluorescence over time; photon counting histograms (PCH) and related methods based on moment analysis resolve species through their brightness by analyzing fluctuations in the photon counts. Here we introduce correlated photon counting histograms (cPCH), which uses both types of information to simultaneously resolve fluorescent species by their brightness and diffusion. We define the cPCH distribution by the probability to detect both a particular number of photons at the current time and another number at a later time. FCS and moment analysis are special cases of the moments of the cPCH distribution, and PCH is obtained by summing over the photon counts in either channel. cPCH is inherently a dual channel technique, and the expressions we develop apply to the dual colour case. Using simulations, we demonstrate that two species differing in both their diffusion and brightness can be better resolved with cPCH than with either FCS or PCH. Further, we show that cPCH can be extended both to longer dwell times to improve the signal-to-noise and to the analysis of images. By better exploiting the information available in fluorescence fluctuation spectroscopy, cPCH will be an enabling methodology for quantitative biology.

Estimating numbers of intracellular molecules through analysing fluctuations in photobleaching.

The impact of fluorescence microscopy has been limited by the difficulties of expressing measurements of fluorescent proteins in numbers of molecules. Absolute numbers enable the integration of results from different laboratories, empower mathematical modelling, and are the bedrock for a quantitative, predictive biology. Here we propose an estimator to infer numbers of molecules from fluctuations in the photobleaching of proteins tagged with Green Fluorescent Protein. Performing experiments in budding yeast, we show that our estimates of numbers agree, within an order of magnitude, with published biochemical measurements, for all six proteins tested. The experiments we require are straightforward and use only a wide-field fluorescence microscope. As such, our approach has the potential to become standard for those practising quantitative fluorescence microscopy.

Transition between fermentation and respiration determines history-dependent behavior in fluctuating carbon sources.

Cells constantly adapt to environmental fluctuations. These physiological changes require time and therefore cause a lag phase during which the cells do not function optimally. Interestingly, past exposure to an environmental condition can shorten the time needed to adapt when the condition re-occurs, even in daughter cells that never directly encountered the initial condition. Here, we use the molecular toolbox of Saccharomyces cerevisiae to systematically unravel the molecular mechanism underlying such history-dependent behavior in transitions between glucose and maltose. In contrast to previous hypotheses, the behavior does not depend on persistence of proteins involved in metabolism of a specific sugar. Instead, presence of glucose induces a gradual decline in the cells' ability to activate respiration, which is needed to metabolize alternative carbon sources. These results reveal how trans-generational transitions in central carbon metabolism generate history-dependent behavior in yeast, and provide a mechanistic framework for similar phenomena in other cell types.

Distributed and dynamic intracellular organization of extracellular information.

Although cells respond specifically to environments, how environmental identity is encoded intracellularly is not understood. Here, we study this organization of information in budding yeast by estimating the mutual information between environmental transitions and the dynamics of nuclear translocation for 10 transcription factors. Our method of estimation is general, scalable, and based on decoding from single cells. The dynamics of the transcription factors are necessary to encode the highest amounts of extracellular information, and we show that information is transduced through two channels: Generalists (Msn2/4, Tod6 and Dot6, Maf1, and Sfp1) can encode the nature of multiple stresses, but only if stress is high; specialists (Hog1, Yap1, and Mig1/2) encode one particular stress, but do so more quickly and for a wider range of magnitudes. In particular, Dot6 encodes almost as much information as Msn2, the master regulator of the environmental stress response. Each transcription factor reports differently, and it is only their collective behavior that distinguishes between multiple environmental states. Changes in the dynamics of the localization of transcription factors thus constitute a precise, distributed internal representation of extracellular change. We predict that such multidimensional representations are common in cellular decision-making.

Morphologically constrained and data informed cell segmentation of budding yeast.

Motivation: Although high-content image cytometry is becoming increasingly routine, processing the large amount of data acquired during time-lapse experiments remains a challenge. The majority of approaches for automated single-cell segmentation focus on flat, uniform fields of view covered with a single layer of cells. In the increasingly popular microfluidic devices that trap individual cells for long term imaging, these conditions are not met. Consequently, most techniques for segmentation perform poorly. Although potentially constraining the generalizability of software, incorporating information about the microfluidic features, flow of media and the morphology of the cells can substantially improve performance. Results: Here we present DISCO (Data Informed Segmentation of Cell Objects), a framework for using the physical constraints imposed by microfluidic traps, the shape based morphological constraints of budding yeast and temporal information about cell growth and motion to allow tracking and segmentation of cells in microfluidic devices. Using manually curated datasets, we demonstrate substantial improvements in both tracking and segmentation when compared with existing software. Availability and implementation: The MATLAB code for the algorithm and for measuring performance is available at https://github.com/pswain/segmentation-software and the test images and the curated ground-truth results used for comparing the algorithms are available at http://datashare.is.ed.ac.uk/handle/10283/2002. Contact: mcrane2@uw.edu. Supplementary information: Supplementary data are available at Bioinformatics online.

Predicting metabolic adaptation from networks of mutational paths.

Competition for substrates is a ubiquitous selection pressure faced by microbes, yet intracellular trade-offs can prevent cells from metabolizing every type of available substrate. Adaptive evolution is constrained by these trade-offs, but their consequences for the repeatability and predictability of evolution are unclear. Here we develop an eco-evolutionary model with a metabolic trade-off to generate networks of mutational paths in microbial communities and show that these networks have descriptive and predictive information about the evolution of microbial communities. We find that long-term outcomes, including community collapse, diversity, and cycling, have characteristic evolutionary dynamics that determine the entropy, or repeatability, of mutational paths. Although reliable prediction of evolutionary outcomes from environmental conditions is difficult, graph-theoretic properties of the mutational networks enable accurate prediction even from incomplete observations. In conclusion, we present a novel methodology for analyzing adaptive evolution and report that the dynamics of adaptation are a key variable for predictive success.The structure and dynamics of microbial communities reflect trade-offs in the ability to use different resources. Here, Josephides and Swain incorporate metabolic trade-offs into an eco-evolutionary model to predict networks of mutational paths and the evolutionary outcomes for microbial communities.

Distributing tasks via multiple input pathways increases cellular survival in stress.

Improving in one aspect of a task can undermine performance in another, but how such opposing demands play out in single cells and impact on fitness is mostly unknown. Here we study budding yeast in dynamic environments of hyperosmotic stress and show how the corresponding signalling network increases cellular survival both by assigning the requirements of high response speed and high response accuracy to two separate input pathways and by having these pathways interact to converge on Hog1, a p38 MAP kinase. Cells with only the less accurate, reflex-like pathway are fitter in sudden stress, whereas cells with only the slow, more accurate pathway are fitter in increasing but fluctuating stress. Our results demonstrate that cellular signalling is vulnerable to trade-offs in performance, but that these trade-offs can be mitigated by assigning the opposing tasks to different signalling subnetworks. Such division of labour could function broadly within cellular signal transduction.

General calibration of microbial growth in microplate readers.

Optical density (OD) measurements of microbial growth are one of the most common techniques used in microbiology, with applications ranging from studies of antibiotic efficacy to investigations of growth under different nutritional or stress environments, to characterization of different mutant strains, including those harbouring synthetic circuits. OD measurements are performed under the assumption that the OD value obtained is proportional to the cell number, i.e. the concentration of the sample. However, the assumption holds true in a limited range of conditions, and calibration techniques that determine that range are currently missing. Here we present a set of calibration procedures and considerations that are necessary to successfully estimate the cell concentration from OD measurements.

Inferring time derivatives including cell growth rates using Gaussian processes.

Often the time derivative of a measured variable is of as much interest as the variable itself. For a growing population of biological cells, for example, the population's growth rate is typically more important than its size. Here we introduce a non-parametric method to infer first and second time derivatives as a function of time from time-series data. Our approach is based on Gaussian processes and applies to a wide range of data. In tests, the method is at least as accurate as others, but has several advantages: it estimates errors both in the inference and in any summary statistics, such as lag times, and allows interpolation with the corresponding error estimation. As illustrations, we infer growth rates of microbial cells, the rate of assembly of an amyloid fibril and both the speed and acceleration of two separating spindle pole bodies. Our algorithm should thus be broadly applicable.

A geometric analysis of fast-slow models for stochastic gene expression.

Stochastic models for gene expression frequently exhibit dynamics on several different scales. One potential time-scale separation is caused by significant differences in the lifetimes of mRNA and protein; the ratio of the two degradation rates gives a natural small parameter in the resulting chemical master equation, allowing for the application of perturbation techniques. Here, we develop a framework for the analysis of a family of 'fast-slow' models for gene expression that is based on geometric singular perturbation theory. We illustrate our approach by giving a complete characterisation of a standard two-stage model which assumes transcription, translation, and degradation to be first-order reactions. In particular, we present a systematic expansion procedure for the probability-generating function that can in principle be taken to any order in the perturbation parameter, allowing for an approximation of the corresponding propagator probabilities to that same order. For illustrative purposes, we perform this expansion explicitly to first order, both on the fast and the slow time-scales; then, we combine the resulting asymptotics into a composite fast-slow expansion that is uniformly valid in time. In the process, we extend, and prove rigorously, results previously obtained by Shahrezaei and Swain (Proc Natl Acad Sci USA 105(45):17256-17261, 2008) and Bokes et al. (J Math Biol 64(5):829-854, 2012; J Math Biol 65(3):493-520, 2012). We verify our asymptotics by numerical simulation, and we explore its practical applicability and the effects of a variation in the system parameters and the time-scale separation. Focussing on biologically relevant parameter regimes that induce translational bursting, as well as those in which mRNA is frequently transcribed, we find that the first-order correction can significantly improve the steady-state probability distribution. Similarly, in the time-dependent scenario, inclusion of the first-order fast asymptotics results in a uniform approximation for the propagator probabilities that is superior to the slow dynamics alone. Finally, we discuss the generalisation of our geometric framework to models for regulated gene expression that involve additional stages.

BioJazz: in silico evolution of cellular networks with unbounded complexity using rule-based modeling.

Systems biologists aim to decipher the structure and dynamics of signaling and regulatory networks underpinning cellular responses; synthetic biologists can use this insight to alter existing networks or engineer de novo ones. Both tasks will benefit from an understanding of which structural and dynamic features of networks can emerge from evolutionary processes, through which intermediary steps these arise, and whether they embody general design principles. As natural evolution at the level of network dynamics is difficult to study, in silico evolution of network models can provide important insights. However, current tools used for in silico evolution of network dynamics are limited to ad hoc computer simulations and models. Here we introduce BioJazz, an extendable, user-friendly tool for simulating the evolution of dynamic biochemical networks. Unlike previous tools for in silico evolution, BioJazz allows for the evolution of cellular networks with unbounded complexity by combining rule-based modeling with an encoding of networks that is akin to a genome. We show that BioJazz can be used to implement biologically realistic selective pressures and allows exploration of the space of network architectures and dynamics that implement prescribed physiological functions. BioJazz is provided as an open-source tool to facilitate its further development and use. Source code and user manuals are available at: http://oss-lab.github.io/biojazz and http://osslab.lifesci.warwick.ac.uk/BioJazz.aspx.

Mechanistic links between cellular trade-offs, gene expression, and growth.

Intracellular processes rarely work in isolation but continually interact with the rest of the cell. In microbes, for example, we now know that gene expression across the whole genome typically changes with growth rate. The mechanisms driving such global regulation, however, are not well understood. Here we consider three trade-offs that, because of limitations in levels of cellular energy, free ribosomes, and proteins, are faced by all living cells and we construct a mechanistic model that comprises these trade-offs. Our model couples gene expression with growth rate and growth rate with a growing population of cells. We show that the model recovers Monod's law for the growth of microbes and two other empirical relationships connecting growth rate to the mass fraction of ribosomes. Further, we can explain growth-related effects in dosage compensation by paralogs and predict host-circuit interactions in synthetic biology. Simulating competitions between strains, we find that the regulation of metabolic pathways may have evolved not to match expression of enzymes to levels of extracellular substrates in changing environments but rather to balance a trade-off between exploiting one type of nutrient over another. Although coarse-grained, the trade-offs that the model embodies are fundamental, and, as such, our modeling framework has potentially wide application, including in both biotechnology and medicine.

A microfluidic system for studying ageing and dynamic single-cell responses in budding yeast.

Recognition of the importance of cell-to-cell variability in cellular decision-making and a growing interest in stochastic modeling of cellular processes has led to an increased demand for high density, reproducible, single-cell measurements in time-varying surroundings. We present ALCATRAS (A Long-term Culturing And TRApping System), a microfluidic device that can quantitatively monitor up to 1000 cells of budding yeast in a well-defined and controlled environment. Daughter cells are removed by fluid flow to avoid crowding allowing experiments to run for over 60 hours, and the extracellular media may be changed repeatedly and in seconds. We illustrate use of the device by measuring ageing through replicative life span curves, following the dynamics of the cell cycle, and examining history-dependent behaviour in the general stress response.

Environmental sensing, information transfer, and cellular decision-making.

The recognition that gene expression can be substantially stochastic poses the question of how cells respond to dynamic environments using biochemistry that itself fluctuates. The study of cellular decision-making aims to solve this puzzle by focusing on quantitative understanding of the variation seen across isogenic populations in response to extracellular change. This behaviour is complex, and a theoretical framework within which to embed experimental results is needed. Here we review current approaches, with an emphasis on information theory, sequential data processing, and optimality arguments. We conclude by highlighting some limitations of these techniques and the importance of connecting both theory and experiment to measures of fitness.

Unmixing of fluorescence spectra to resolve quantitative time-series measurements of gene expression in plate readers.

BACKGROUND: To connect gene expression with cellular physiology, we need to follow levels of proteins over time. Experiments typically use variants of Green Fluorescent Protein (GFP), and time-series measurements require specialist expertise if single cells are to be followed. Fluorescence plate readers, however, a standard in many laboratories, can in principle provide similar data, albeit at a mean, population level. Nevertheless, extracting the average fluorescence per cell is challenging because autofluorescence can be substantial. RESULTS: Here we propose a general method for correcting plate reader measurements of fluorescent proteins that uses spectral unmixing and determines both the fluorescence per cell and the errors on that fluorescence. Combined with strain collections, such as the GFP fusion collection for budding yeast, our methodology allows quantitative measurements of protein levels of up to hundreds of genes and therefore provides complementary data to high throughput studies of transcription. We illustrate the method by following the induction of the GAL genes in Saccharomyces cerevisiae for over 20 hours in different sugars and argue that the order of appearance of the Leloir enzymes may be to reduce build-up of the toxic intermediate galactose-1-phosphate. Further, we quantify protein levels of over 40 genes, again over 20 hours, after cells experience a change in carbon source (from glycerol to glucose). CONCLUSIONS: Our methodology is sensitive, scalable, and should be applicable to other organisms. By allowing quantitative measurements on a per cell basis over tens of hours and over hundreds of genes, it should increase our understanding of the dynamic changes that drive cellular behaviour.