Institute of Fundamental Technological Research

Cells respond dynamically to pulsatile cytokine stimulation. Here we report that single, or well-spaced pulses of TNFα (>100 min apart) give a high probability of NF-κB activation. However, fewer cells respond to shorter pulse intervals (<100 min) suggesting a heterogeneous refractory state. This refractory state is established in the signal transduction network downstream of TNFR and upstream of IKK, and depends on the level of the NF-κB system negative feedback protein A20. If a second pulse within the refractory phase is IL-1β instead of TNFα, all of the cells respond. This suggests a mechanism by which two cytokines can synergistically activate an inflammatory response. Gene expression analyses show strong correlation between the cellular dynamic response and NF-κB-dependent target gene activation. These data suggest that refractory states in the NF-κB system constitute an inherent design motif of the inflammatory response and we suggest that this may avoid harmful homogenous cellular activation.

Motivation: cis-regulatory DNA sequence elements, such as enhancers and silencers, function to control the spatial and temporal expression of their target genes. Although the overall levels of gene expression in large cell populations seem to be precisely controlled, transcription of individual genes in single cells is extremely variable in real time. It is, therefore, important to understand how these cis-regulatory elements function to dynamically control transcription at single-cell resolution. Recently, statistical methods have been proposed to back calculate the rates involved in mRNA transcription using parameter estimation of a mathematical model of transcription and translation. However, a major complication in these approaches is that some of the parameters, particularly those corresponding to the gene copy number and transcription rate, cannot be distinguished; therefore, these methods cannot be used when the copy number is unknown.

Results: Here, we develop a hierarchical Bayesian model to estimate biokinetic parameters from live cell enhancer–promoter reporter measurements performed on a population of single cells. This allows us to investigate transcriptional dynamics when the copy number is variable across the population. We validate our method using synthetic data and then apply it to quantify the function of two known developmental enhancers in real time and in single cells.

A central challenge in computational modeling of dynamic biological systems is parameter inference from experimental time course measurements. However, one would not only like to infer kinetic parameters but also study their variability from cell to cell. Here we focus on the case where single-cell fluorescent protein imaging time series data are available for a population of cells. Based on van Kampen’s linear noise approximation, we derive a dynamic state space model for molecular populations which is then extended to a hierarchical model. This model has potential to address the sources of variability relevant to single-cell data, namely, intrinsic noise due to the stochastic nature of the birth and death processes involved in reactions and extrinsic noise arising from the cell-to-cell variation of kinetic parameters. In order to infer such a model from experimental data, one must also quantify the measurement process where one has to allow for nonmeasurable molecular species as well as measurement noise of unknown level and variance. The availability of multiple single-cell time series data here provides a unique testbed to fit such a model and quantify these different sources of variation from experimental data.

Linear noise approximation, kinetic parameter estimation, intrinsic and extrinsic noise, state space model and Kalman filter, Bayesian hierarchical modeling

We present a novel and simple method to numerically calculate Fisher information matrices for stochastic chemical kinetics models. The linear noise approximation is used to derive model equations and a likelihood function that leads to an efficient computational algorithm. Our approach reduces the problem of calculating the Fisher information matrix to solving a set of ordinary differential equations. This is the first method to compute Fisher information for stochastic chemical kinetics models without the need for Monte Carlo simulations. This methodology is then used to study sensitivity, robustness, and parameter identifiability in stochastic chemical kinetics models. We show that significant differences exist between stochastic and deterministic models as well as between stochastic models with time-series and time-point measurements. We demonstrate that these discrepancies arise from the variability in molecule numbers, correlations between species, and temporal correlations and show how this approach can be used in the analysis and design of experiments probing stochastic processes at the cellular level. The algorithm has been implemented as a Matlab package and is available from the authors upon request.

Fluorescent and luminescent proteins are often used as reporters of transcriptional activity. Given the prevalence of noise in biochemical systems, the time-series data arising from these is of significant interest in efforts to calibrate stochastic models of gene expression and obtain information about sources of nongenetic variability. We present a statistical inference framework that can be used to estimate kinetic parameters of gene expression, as well as the strength and half-life of extrinsic noise from single fluorescent-reporter-gene time-series data. The method takes into account stochastic variability in a fluorescent signal resulting from intrinsic noise of gene expression, kinetics of fluorescent protein maturation, and extrinsic noise, which is assumed to arise at transcriptional level. We use the linear noise approximation and derive an explicit formula for the likelihood of observed fluorescent data. The method is embedded in a Bayesian paradigm, so that certain parameters can be informed from other experiments allowing portability of results across different studies. Inference is performed using Markov chain Monte Carlo. Fluorescent reporters are primary tools to observe dynamics of gene expression and the correct interpretation of fluorescent data is crucial to investigating these fundamental processes of cellular life. As both magnitude and frequency of the noise may have a dramatic effect on the cell fitness, the quantification of stochastic fluctuation is essential to the understanding of how genes are regulated. Our method provides a framework that addresses this important question.

Heterogeneity between individual cells is a common feature of dynamic cellular processes, including signaling, transcription, and cell fate; yet the overall tissue level physiological phenotype needs to be carefully controlled to avoid fluctuations. Here we show that in the NF-κB signaling system, the precise timing of a dual-delayed negative feedback motif [involving stochastic transcription of inhibitor κB (IκB)-α and -ε] is optimized to induce heterogeneous timing of NF-κB oscillations between individual cells. We suggest that this dual-delayed negative feedback motif enables NF-κB signaling to generate robust single cell oscillations by reducing sensitivity to key parameter perturbations. Simultaneously, enhanced cell heterogeneity may represent a mechanism that controls the overall coordination and stability of cell population responses by decreasing temporal fluctuations of paracrine signaling. It has often been thought that dynamic biological systems may have evolved to maximize robustness through cell-to-cell coordination and homogeneity. Our analyses suggest in contrast, that this cellular variation might be advantageous and subject to evolutionary selection. Alternative types of therapy could perhaps be designed to modulate this cellular heterogeneity.

Nuclear factor kappa B (NF-kappaB) signalling is activated by cellular stress and inflammation and regulates cytokine expression. We applied single-cell imaging to investigate dynamic responses to different doses of tumour necrosis factor alpha (TNFalpha). Lower doses activated fewer cells and those responding showed an increasingly variable delay in the initial NF-kappaB nuclear translocation and associated IkappaBalpha degradation. Robust 100 minute nuclear:cytoplasmic NF-kappaB oscillations were observed over a wide range of TNFalpha concentrations. The result is supported by computational analyses, which identified a limit cycle in the system with a stable 100 minute period over a range of stimuli, and indicated no co-operativity in the pathway activation. These results suggest that a stochastic threshold controls functional all-or-nothing responses in individual cells. Deterministic and stochastic models simulated the experimentally observed activation threshold and gave rise to new predictions about the structure of the system and open the way for better mechanistic understanding of physiological TNFalpha activation of inflammatory responses in cells and tissues.

NF-

The nuclear factor κB (NF-κB) transcription factor regulates cellular stress responses and the immune response to infection. NF-κB activation results in oscillations in nuclear NF-κB abundance. To define the function of these oscillations, we treated cells with repeated short pulses of tumor necrosis factor–α at various intervals to mimic pulsatile inflammatory signals. At all pulse intervals that were analyzed, we observed synchronous cycles of NF-κB nuclear translocation. Lower frequency stimulations gave repeated full-amplitude translocations, whereas higher frequency pulses gave reduced translocation, indicating a failure to reset. Deterministic and stochastic mathematical models predicted how negative feedback loops regulate both the resetting of the system and cellular heterogeneity. Altering the stimulation intervals gave different patterns of NF-κB–dependent gene expression, which supports the idea that oscillation frequency has a functional role.