Identifying stochastic oscillations in single-cell live imaging time series using Gaussian processes

GPMelt: A hierarchical Gaussian process framework to explore the dark meltome of thermal proteome profiling experiments

Thermal proteome profiling (TPP) is a proteome wide technology that enables unbiased detection of protein drug interactions as well as changes in post-translational state of proteins between different biological conditions. Statistical analysis of temperature range TPP (TPP-TR) datasets relies on comparing protein melting curves, describing the amount of non-denatured proteins as a function of temperature, between different conditions (e.g. presence or absence of a drug). However, state-of-the-art models are restricted to sigmoidal melting behaviours while unconventional melting curves, representing up to 50% of TPP-TR datasets, have recently been shown to carry important biological information. We present a novel statistical framework, based on hierarchical Gaussian process models and named GPMelt, to make TPP-TR datasets analysis unbiased with respect to the melting profiles of proteins. GPMelt scales to multiple conditions, and extension of the model to deeper hierarchies (i.e. with additional sub-levels) allows to deal with complex TPP-TR protocols. Collectively, our statistical framework extends the analysis of TPP-TR datasets for both protein and peptide level melting curves, offering access to thousands of previously excluded melting curves and thus substantially increasing the coverage and the ability of TPP to uncover new biology.

Cell coupling compensates for changes in single-cell Her6 dynamics and provides phenotypic robustness

This paper investigates the effect of altering the protein expression dynamics of the bHLH transcription factor Her6 at the single-cell level in the embryonic zebrafish telencephalon. Using a homozygote endogenous Her6:Venus reporter and 4D single-cell tracking, we show that Her6 oscillates in neural telencephalic progenitors and that the fusion of protein destabilisation (PEST) domain alters its expression dynamics, causing most cells to downregulate Her6 prematurely. However, counterintuitively, oscillatory cells increase, with some expressing Her6 at high levels, resulting in increased heterogeneity of Her6 expression in the population. These tissue-level changes appear to be an emergent property of coupling between single-cells, as revealed by experimentally disrupting Notch signalling and by computationally modelling alterations in Her6 protein stability. Despite the profound differences in the single-cell Her6 dynamics, the size of the telencephalon is only transiently altered and differentiation markers do not exhibit significant differences early on; however, a small increase is observed at later developmental stages. Our study suggests that cell coupling provides a compensation strategy, whereby an almost normal phenotype is maintained even though single-cell gene expression dynamics are abnormal, granting phenotypic robustness.

An improved rhythmicity analysis method using Gaussian Processes detects cell-density dependent circadian oscillations in stem cells

MOTIVATION

Detecting oscillations in time series remains a challenging problem even after decades of research. In chronobiology, rhythms (for instance in gene expression, eclosion, egg-laying, and feeding) tend to be low amplitude, display large variations amongst replicates, and often exhibit varying peak-to-peak distances (non-stationarity). Most currently available rhythm detection methods are not specifically designed to handle such datasets, and are also limited by their use of P-values in detecting oscillations.

RESULTS

We introduce a new method, ODeGP (Oscillation Detection using Gaussian Processes), which combines Gaussian Process regression and Bayesian inference to incorporate measurement errors, non-uniformly sampled data, and a recently developed non-stationary kernel to improve detection of oscillations. By using Bayes factors, ODeGP models both the null (non-rhythmic) and the alternative (rhythmic) hypotheses, thus providing an advantage over P-values. Using synthetic datasets, we first demonstrate that ODeGP almost always outperforms eight commonly used methods in detecting stationary as well as non-stationary symmetric oscillations. Next, by analyzing existing qPCR datasets, we demonstrate that our method is more sensitive compared to the existing methods at detecting weak and noisy oscillations. Finally, we generate new qPCR data on mouse embryonic stem cells. Surprisingly, we discover using ODeGP that increasing cell-density results in rapid generation of oscillations in the Bmal1 gene, thus highlighting our method's ability to discover unexpected and new patterns. In its current implementation, ODeGP is meant only for analyzing single or a few time-trajectories, not genome-wide datasets.

AVAILABILITY AND IMPLEMENTATION

ODeGP is available at https://github.com/Shaonlab/ODeGP.

Are physiological oscillations physiological?

Despite widespread and striking examples of physiological oscillations, their functional role is often unclear. Even glycolysis, the paradigm example of oscillatory biochemistry, has seen questions about its oscillatory function. Here, we take a systems approach to argue that oscillations play critical physiological roles, such as enabling systems to avoid desensitization, to avoid chronically high and therefore toxic levels of chemicals, and to become more resistant to noise. Oscillation also enables complex physiological systems to reconcile incompatible conditions such as oxidation and reduction, by cycling between them, and to synchronize the oscillations of many small units into one large effect. In pancreatic β-cells, glycolytic oscillations synchronize with calcium and mitochondrial oscillations to drive pulsatile insulin release, critical for liver regulation of glucose. In addition, oscillation can keep biological time, essential for embryonic development in promoting cell diversity and pattern formation. The functional importance of oscillatory processes requires a re-thinking of the traditional doctrine of homeostasis, holding that physiological quantities are maintained at constant equilibrium values, a view that has largely failed in the clinic. A more dynamic approach will initiate a paradigm shift in our view of health and disease. A deeper look into the mechanisms that create, sustain and abolish oscillatory processes requires the language of nonlinear dynamics, well beyond the linearization techniques of equilibrium control theory. Nonlinear dynamics enables us to identify oscillatory ('pacemaking') mechanisms at the cellular, tissue and system levels.

An improved rhythmicity analysis method using Gaussian Processes detects cell-density dependent circadian oscillations in stem cells

Detecting oscillations in time series remains a challenging problem even after decades of research. In chronobiology, rhythms in time series (for instance gene expression, eclosion, egg-laying and feeding) datasets tend to be low amplitude, display large variations amongst replicates, and often exhibit varying peak-to-peak distances (non-stationarity). Most currently available rhythm detection methods are not specifically designed to handle such datasets. Here we introduce a new method, ODeGP ( O scillation De tection using G aussian P rocesses), which combines Gaussian Process (GP) regression with Bayesian inference to provide a flexible approach to the problem. Besides naturally incorporating measurement errors and non-uniformly sampled data, ODeGP uses a recently developed kernel to improve detection of non-stationary waveforms. An additional advantage is that by using Bayes factors instead of p-values, ODeGP models both the null (non-rhythmic) and the alternative (rhythmic) hypotheses. Using a variety of synthetic datasets we first demonstrate that ODeGP almost always outperforms eight commonly used methods in detecting stationary as well as non-stationary oscillations. Next, on analyzing existing qPCR datasets that exhibit low amplitude and noisy oscillations, we demonstrate that our method is more sensitive compared to the existing methods at detecting weak oscillations. Finally, we generate new qPCR time-series datasets on pluripotent mouse embryonic stem cells, which are expected to exhibit no oscillations of the core circadian clock genes. Surprisingly, we discover using ODeGP that increasing cell density can result in the rapid generation of oscillations in the Bmal1 gene, thus highlighting our method’s ability to discover unexpected patterns. In its current implementation, ODeGP (available as an R package) is meant only for analyzing single or a few time-trajectories, not genome-wide datasets.

Microtubules as a potential platform for energy transfer in biological systems: a target for implementing individualized, dynamic variability patterns to improve organ function

Variability characterizes the complexity of biological systems and is essential for their function. Microtubules (MTs) play a role in structural integrity, cell motility, material transport, and force generation during mitosis, and dynamic instability exemplifies the variability in the proper function of MTs. MTs are a platform for energy transfer in cells. The dynamic instability of MTs manifests itself by the coexistence of growth and shortening, or polymerization and depolymerization. It results from a balance between attractive and repulsive forces between tubulin dimers. The paper reviews the current data on MTs and their potential roles as energy-transfer cellular structures and presents how variability can improve the function of biological systems in an individualized manner. The paper presents the option for targeting MTs to trigger dynamic improvement in cell plasticity, regulate energy transfer, and possibly control quantum effects in biological systems. The described system quantifies MT-dependent variability patterns combined with additional personalized signatures to improve organ function in a subject-tailored manner. The platform can regulate the use of MT-targeting drugs to improve the response to chronic therapies. Ongoing trials test the effects of this platform on various disorders.

Temporal control of the integrated stress response by a stochastic molecular switch

Stress granules (SGs) are formed in the cytosol as an acute response to environmental cues and activation of the integrated stress response (ISR), a central signaling pathway controlling protein synthesis. Using chronic virus infection as stress model, we previously uncovered a unique temporal control of the ISR resulting in recurrent phases of SG assembly and disassembly. Here, we elucidate the molecular network generating this fluctuating stress response by integrating quantitative experiments with mathematical modeling and find that the ISR operates as a stochastic switch. Key elements controlling this switch are the cooperative activation of the stress-sensing kinase PKR, the ultrasensitive response of SG formation to the phosphorylation of the translation initiation factor eIF2α, and negative feedback via GADD34, a stress-induced subunit of protein phosphatase 1. We identify GADD34 messenger RNA levels as the molecular memory of the ISR that plays a central role in cell adaptation to acute and chronic stress.

Differential phase register of Hes1 oscillations with mitoses underlies cell-cycle heterogeneity in ER+ breast cancer cells

Tumors exhibit heterogeneities that are not due to mutations, including cancer stem cells with different potencies. We show that the cancer stem-cell state predisposed to dormancy in vivo has a highly variable and long cell cycle. Using single-cell live imaging for the transcriptional repressor Hes1 (a key molecule in cancer), we show a type of circadian-like oscillatory expression of Hes1 in all cells in the population. The most potent cancer stem cells tend to divide around the trough of the Hes1 oscillatory wave, a feature predictive of a long cell cycle. A concept proposed here is that the position of cell division with respect to the Hes1 wave is predictive of its prospective cell-cycle length and cancer cellular substate.

Here, we study the dynamical expression of endogenously labeled Hes1, a transcriptional repressor implicated in controlling cell proliferation, to understand how cell-cycle length heterogeneity is generated in estrogen receptor (ER)⁺ breast cancer cells. We find that Hes1 shows oscillatory expression with ∼25 h periodicity and during each cell cycle has a variable peak in G1, a trough around G1–S transition, and a less variable second peak in G2/M. Compared to other subpopulations, the cell cycle in CD44^HighCD24^Low cancer stem cells is longest and most variable. Most cells divide around the peak of the Hes1 expression wave, but preceding mitoses in slow dividing CD44^HighCD24^Low cells appear phase-shifted, resulting in a late-onset Hes1 peak in G1. The position, duration, and shape of this peak, rather than the Hes1 expression levels, are good predictors of cell-cycle length. Diminishing Hes1 oscillations by enforcing sustained expression slows down the cell cycle, impairs proliferation, abolishes the dynamic expression of p21, and increases the percentage of CD44^HighCD24^Low cells. Reciprocally, blocking the cell cycle causes an elongation of Hes1 periodicity, suggesting a bidirectional interaction of the Hes1 oscillator and the cell cycle. We propose that Hes1 oscillations are functionally important for the efficient progression of the cell cycle and that the position of mitosis in relation to the Hes1 wave underlies cell-cycle length heterogeneity in cancer cell subpopulations.

A digital health platform for assisting the diagnosis and monitoring of COVID-19 progression: An adjuvant approach for augmenting the antiviral response and mitigating the immune-mediated target organ damage

Coronavirus disease 2019 (COVID-19), which is a respiratory illness associated with high mortality, has been classified as a pandemic. The major obstacles for the clinicians to contain the disease are limited information availability, difficulty in disease diagnosis, predicting disease prognosis, and lack of disease monitoring tools. Additionally, the lack of valid therapies has further contributed to the difficulties in containing the pandemic. Recent studies have reported that the dysregulation of the immune system leads to an ineffective antiviral response and promotes pathological immune response, which manifests as ARDS, myocarditis, and hepatitis. In this study, a novel platform has been described for disseminating information to physicians for the diagnosis and monitoring of patients with COVID-19. An adjuvant approach using compounds that can potentiate antiviral immune response and mitigate COVID-19-induced immune-mediated target organ damage has been presented. A prolonged beneficial effect is achieved by implementing algorithm-based individualized variability measures in the treatment regimen.

HES1 protein oscillations are necessary for neural stem cells to exit from quiescence

Quiescence is a dynamic process of reversible cell cycle arrest. High-level persistent expression of the HES1 transcriptional repressor, which oscillates with an ultradian periodicity in proliferative neural stem cells (NSCs), is thought to mediate quiescence. However, it is not known whether this is due to a change in levels or dynamics. Here, we induce quiescence in embryonic NSCs with BMP4, which does not increase HES1 level, and we find that HES1 continues to oscillate. To assess the role of HES1 dynamics, we express persistent HES1 under a moderate strength promoter, which overrides the endogenous oscillations while maintaining the total HES1 level within physiological range. We find that persistent HES1 does not affect proliferation or entry into quiescence; however, exit from quiescence is impeded. Thus, oscillatory expression of HES1 is specifically required for NSCs to exit quiescence, a finding of potential importance for controlling reactivation of stem cells in tissue regeneration and cancer.

A functionally divergent intrinsically disordered region underlying the conservation of stochastic signaling

Stochastic signaling dynamics expand living cells' information processing capabilities. An increasing number of studies report that regulators encode information in their pulsatile dynamics. The evolutionary mechanisms that lead to complex signaling dynamics remain uncharacterized, perhaps because key interactions of signaling proteins are encoded in intrinsically disordered regions (IDRs), whose evolution is difficult to analyze. Here we focused on the IDR that controls the stochastic pulsing dynamics of Crz1, a transcription factor in fungi downstream of the widely conserved calcium signaling pathway. We find that Crz1 IDRs from anciently diverged fungi can all respond transiently to calcium stress; however, only Crz1 IDRs from the Saccharomyces clade support pulsatility, encode extra information, and rescue fitness in competition assays, while the Crz1 IDRs from distantly related fungi do none of the three. On the other hand, we find that Crz1 pulsing is conserved in the distantly related fungi, consistent with the evolutionary model of stabilizing selection on the signaling phenotype. Further, we show that a calcineurin docking site in a specific part of the IDRs appears to be sufficient for pulsing and show evidence for a beneficial increase in the relative calcineurin affinity of this docking site. We propose that evolutionary flexibility of functionally divergent IDRs underlies the conservation of stochastic signaling by stabilizing selection.

Inferring kinetic parameters of oscillatory gene regulation from single cell time-series data

Gene expression dynamics, such as stochastic oscillations and aperiodic fluctuations, have been associated with cell fate changes in multiple contexts, including development and cancer. Single cell live imaging of protein expression with endogenous reporters is widely used to observe such gene expression dynamics. However, the experimental investigation of regulatory mechanisms underlying the observed dynamics is challenging, since these mechanisms include complex interactions of multiple processes, including transcription, translation and protein degradation. Here, we present a Bayesian method to infer kinetic parameters of oscillatory gene expression regulation using an auto-negative feedback motif with delay. Specifically, we use a delay-adapted nonlinear Kalman filter within a Metropolis-adjusted Langevin algorithm to identify posterior probability distributions. Our method can be applied to time-series data on gene expression from single cells and is able to infer multiple parameters simultaneously. We apply it to published data on murine neural progenitor cells and show that it outperforms alternative methods. We further analyse how parameter uncertainty depends on the duration and time resolution of an imaging experiment, to make experimental design recommendations. This work demonstrates the utility of parameter inference on time course data from single cells and enables new studies on cell fate changes and population heterogeneity.

Stochastic models for single-cell data: Current challenges and the way forward

Although the quantity and quality of single-cell data have progressed rapidly, making quantitative predictions with single-cell stochastic models remains challenging. The stochastic nature of cellular processes leads to at least three challenges in building models with single-cell data: (a) because variability in single-cell data can be attributed to multiple different sources, it is difficult to rule out conflicting mechanistic models that explain the same data equally well; (b) the distinction between interesting biological variability and experimental variability is sometimes ambiguous; (c) the nonstandard distributions of single-cell data can lead to violations of the assumption of symmetric errors in least-squares fitting. In this review, we first discuss recent studies that overcome some of the challenges or set up a promising direction and then introduce some powerful statistical approaches utilized in these studies. We conclude that applying and developing statistical approaches could lead to further progress in building stochastic models for single-cell data.

Computationally enhanced quantitative phase microscopy reveals autonomous oscillations in mammalian cell growth

The fine balance of growth and division is a fundamental property of the physiology of cells, and one of the least understood. Its study has been thwarted by difficulties in the accurate measurement of cell size and the even greater challenges of measuring growth of a single cell over time. We address these limitations by demonstrating a computationally enhanced methodology for quantitative phase microscopy for adherent cells, using improved image processing algorithms and automated cell-tracking software. Accuracy has been improved more than twofold and this improvement is sufficient to establish the dynamics of cell growth and adherence to simple growth laws. It is also sufficient to reveal unknown features of cell growth, previously unmeasurable. With these methodological and analytical improvements, in several cell lines we document a remarkable oscillation in growth rate, occurring throughout the cell cycle, coupled to cell division or birth yet independent of cell cycle progression. We expect that further exploration with this advanced tool will provide a better understanding of growth rate regulation in mammalian cells.

Dynamic properties of noise and Her6 levels are optimized by miR-9, allowing the decoding of the Her6 oscillator

Noise is prevalent in biology and has been widely quantified using snapshot measurements. This static view obscures our understanding of dynamic noise properties and how these affect gene expression and cell state transitions. Using a CRISPR/Cas9 Zebrafish her6::Venus reporter combined with mathematical and in vivo experimentation, we explore how noise affects the protein dynamics of Her6, a basic helix-loop-helix transcriptional repressor. During neurogenesis, Her6 expression transitions from fluctuating to oscillatory at single-cell level. We identify that absence of miR-9 input generates high-frequency noise in Her6 traces, inhibits the transition to oscillatory protein expression and prevents the downregulation of Her6. Together, these impair the upregulation of downstream targets and cells accumulate in a normally transitory state where progenitor and early differentiation markers are co-expressed. Computational modelling and double smFISH of her6 and the early neurogenesis marker, elavl3, suggest that the change in Her6 dynamics precedes the downregulation in Her6 levels. This sheds light onto the order of events at the moment of cell state transition and how this is influenced by the dynamic properties of noise. Our results suggest that Her/Hes oscillations, facilitated by dynamic noise optimization by miR-9, endow progenitor cells with the ability to make a cell state transition.

Order Through Disorder: The Characteristic Variability of Systems

Randomness characterizes many processes in nature, and therefore its importance cannot be overstated. In the present study, we investigate examples of randomness found in various fields, to underlie its fundamental processes. The fields we address include physics, chemistry, biology (biological systems from genes to whole organs), medicine, and environmental science. Through the chosen examples, we explore the seemingly paradoxical nature of life and demonstrate that randomness is preferred under specific conditions. Furthermore, under certain conditions, promoting or making use of variability-associated parameters may be necessary for improving the function of processes and systems.

Intrinsic noise, Delta-Notch signalling and delayed reactions promote sustained, coherent, synchronized oscillations in the presomitic mesoderm

Using a stochastic individual-based modelling approach, we examine the role that Delta-Notch signalling plays in the regulation of a robust and reliable somite segmentation clock. We find that not only can Delta-Notch signalling synchronize noisy cycles of gene expression in adjacent cells in the presomitic mesoderm (as is known), but it can also amplify and increase the coherence of these cycles. We examine some of the shortcomings of deterministic approaches to modelling these cycles and demonstrate how intrinsic noise can play an active role in promoting sustained oscillations, giving rise to noise-induced quasi-cycles. Finally, we explore how translational/transcriptional delays can result in the cycles in neighbouring cells oscillating in anti-phase and we study how this effect relates to the propagation of noise-induced stochastic waves.

Quantitative single-cell live imaging links HES5 dynamics with cell-state and fate in murine neurogenesis

During embryogenesis cells make fate decisions within complex tissue environments. The levels and dynamics of transcription factor expression regulate these decisions. Here, we use single cell live imaging of an endogenous HES5 reporter and absolute protein quantification to gain a dynamic view of neurogenesis in the embryonic mammalian spinal cord. We report that dividing neural progenitors show both aperiodic and periodic HES5 protein fluctuations. Mathematical modelling suggests that in progenitor cells the HES5 oscillator operates close to its bifurcation boundary where stochastic conversions between dynamics are possible. HES5 expression becomes more frequently periodic as cells transition to differentiation which, coupled with an overall decline in HES5 expression, creates a transient period of oscillations with higher fold expression change. This increases the decoding capacity of HES5 oscillations and correlates with interneuron versus motor neuron cell fate. Thus, HES5 undergoes complex changes in gene expression dynamics as cells differentiate.

Topological time-series analysis with delay-variant embedding

Identifying the qualitative changes in time-series data provides insights into the dynamics associated with such data. Such qualitative changes can be detected through topological approaches, which first embed the data into a high-dimensional space using a time-delay parameter and subsequently extract topological features describing the shape of the data from the embedded points. However, the essential topological features that are extracted using a single time delay are considered to be insufficient for evaluating the aforementioned qualitative changes, even when a well-selected time delay is used. We therefore propose a delay-variant embedding method that constructs the extended topological features by considering the time delay as a variable parameter instead of considering it as a single fixed value. This delay-variant embedding method reveals multiple-timescale patterns in a time series by allowing the observation of the variations in topological features, with the time delay serving as an additional dimension in the topological feature space. We theoretically prove that the constructed topological features are robust when the time series is perturbed by noise. Furthermore, we combine these features with the kernel technique in machine learning algorithms to classify the general time-series data. We demonstrate the effectiveness of our method for classifying the synthetic noisy biological and real time-series data. Our method outperforms a method that is based on a single time delay and, surprisingly, achieves the highest classification accuracy on an average among the standard time-series analysis techniques.

A Noisy Analog-to-Digital Converter Connects Cytosolic Calcium Bursts to Transcription Factor Nuclear Localization Pulses in Yeast

Several examples of transcription factors that show stochastic, unsynchronized pulses of nuclear localization have been described. Here we show that under constant calcium stress, nuclear localization pulses of the transcription factor Crz1 follow stochastic variations in cytosolic calcium concentration. We find that the size of the stochastic calcium bursts is positively correlated with the number of subsequent Crz1 pulses. Based on our observations, we propose a simple stochastic model of how the signaling pathway converts a constant external calcium concentration into a digital number of Crz1 pulses in the nucleus, due to the time delay from nuclear transport and the stochastic decoherence of individual Crz1 molecule dynamics. We find support for several additional predictions of the model and suggest that stochastic input to nuclear transport may produce noisy digital responses to analog signals in other signaling systems.

Pitx2 and nodal as conserved early markers of the anterior-posterior axis in the rabbit embryo

Attaining molecular and morphological axial polarity during gastrulation is a fundamental early requirement for normal development of the embryo. In mammals, the first morphological sign of the anterior-posterior axis appears anteriorly in the form of the anterior marginal crescent (or anterior visceral endoderm) while in the avian the first such sign is the Koller's sickle at the posterior pole of the embryonic disc. Despite this inverse mode of axis formation many genes and molecular pathways involved in various steps of this process seem to be evolutionarily conserved amongst amniotes, the nodal gene being a well-known example with its functional involvement prior and during gastrulation. The pitx2 gene, however, is a new candidate described in the chick as an early marker for anterior-posterior polarity and as a regulator of axis formation including twinning. To find out whether pitx2 has retained its inductive and early marker function during the evolution of mammals this study analyses pitx2 and nodal expression at parallel stages during formation of the anterior-posterior polarity in the early rabbit embryo using whole-mount in situ hybridization and serial light-microscopical sections. At a late pre-gastrulation stage a localized reduction of nodal expression presages the position of the anterior pole of the embryonic disc and thus serves as the earliest molecular marker of anterior-posterior polarity known so far. Pitx2 is expressed in a polarized manner in the anterior marginal crescent and in the posterior half of the embryonic disc during further development. In the anterior segment of the posterior pitx2 expression domain, the anterior streak domain (ASD) is defined by nodal expression as a hypothetical progenitor region of the anterior half of the primitive streak. The expression patterns of both genes thus serve as signs of a conserved involvement in early axis formation in amniotes and, possibly, in twinning in mammals as well.