Clive G. Bowsher

Identifying sources of variation and the flow of information in biochemical networks

To understand how cells control and exploit biochemical fluctuations, we must identify the sources of stochasticity, quantify their effects, and distinguish informative variation from confounding “noise.” We present an analysis that allows fluctuations of biochemical networks to be decomposed into multiple components, gives conditions for the design of experimental reporters to measure all components, and provides a technique to predict the magnitude of these components from models. Further, we identify a particular component of variation that can be used to quantify the efficacy of information flow through a biochemical network. By applying our approach to osmosensing in yeast, we can predict the probability of the different osmotic conditions experienced by wild-type yeast and show that the majority of variation can be informational if we include variation generated in response to the cellular environment. Our results are fundamental to quantifying sources of variation and thus are a means to understand biological “design.”

The Dynamics of Economic Functions: Modeling and Forecasting the Yield Curve

The class of functional signal plus noise (FSN) models is introduced that provides a new, general method for modeling and forecasting time series of economic functions. The underlying, continuous economic function (or “signal”) is a natural cubic spline whose dynamic evolution is driven by a cointegrated vector autoregression for the ordinates (or “y-values”) at the knots of the spline. The natural cubic spline provides flexible cross-sectional fit and results in a linear state-space model. This FSN model achieves dimension reduction, provides a coherent description of the observed yield curve and its dynamics as the cross-sectional dimension N becomes large, and can be feasibly estimated and used for forecasting when N is large. The integration and cointegration properties of the model are derived. The FSN models are then applied to forecasting 36-dimensional yield curves for U.S. Treasury bonds at the 1-month-ahead horizon. The method consistently outperforms the dynamic Nelson–Siegel and random walk forecasts on the basis of both mean squared forecast error criteria and economically relevant loss functions derived from the realized profits of pairs trading algorithms. The analysis also highlights in a concrete setting the dangers of attempting to infer the relative economic value of model forecasts on the basis of their associated mean squared forecast errors.

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.

Information transfer by leaky, heterogeneous, protein kinase signaling systems.

Significance Extracellular concentrations convey information to cells about their environment. To sense these signals, cells use biomolecular networks that exhibit inevitable cell-to-cell variability and basal activity. Basal activity is widespread under physiological conditions (with phenotypic consequences), is often raised in disease, and can eradicate the transfer of information. In an experimental study of ERK signaling by single cells exhibiting heterogeneous ERK expression and basal activity, we verify our central theoretical prediction: Negative feedback substantially increases information transfer to the nucleus by preventing a near-flat average response curve and reducing sensitivity to variation in the ERK expression level. Our results reveal an important role for negative feedback mechanisms in protecting information transfer by saturable cell signaling systems from basal activity. Cells must sense extracellular signals and transfer the information contained about their environment reliably to make appropriate decisions. To perform these tasks, cells use signal transduction networks that are subject to various sources of noise. Here, we study the effects on information transfer of two particular types of noise: basal (leaky) network activity and cell-to-cell variability in the componentry of the network. Basal activity is the propensity for activation of the network output in the absence of the signal of interest. We show, using theoretical models of protein kinase signaling, that the combined effect of the two types of noise makes information transfer by such networks highly vulnerable to the loss of negative feedback. In an experimental study of ERK signaling by single cells with heterogeneous ERK expression levels, we verify our theoretical prediction: In the presence of basal network activity, negative feedback substantially increases information transfer to the nucleus by both preventing a near-flat average response curve and reducing sensitivity to variation in substrate expression levels. The interplay between basal network activity, heterogeneity in network componentry, and feedback is thus critical for the effectiveness of protein kinase signaling. Basal activity is widespread in signaling systems under physiological conditions, has phenotypic consequences, and is often raised in disease. Our results reveal an important role for negative feedback mechanisms in protecting the information transfer function of saturable, heterogeneous cell signaling systems from basal activity.

The fidelity of dynamic signaling by noisy biomolecular networks.

Cells live in changing, dynamic environments. To understand cellular decision-making, we must therefore understand how fluctuating inputs are processed by noisy biomolecular networks. Here we present a general methodology for analyzing the fidelity with which different statistics of a fluctuating input are represented, or encoded, in the output of a signaling system over time. We identify two orthogonal sources of error that corrupt perfect representation of the signal: dynamical error, which occurs when the network responds on average to other features of the input trajectory as well as to the signal of interest, and mechanistic error, which occurs because biochemical reactions comprising the signaling mechanism are stochastic. Trade-offs between these two errors can determine the systems fidelity. By developing mathematical approaches to derive dynamics conditional on input trajectories we can show, for example, that increased biochemical noise (mechanistic error) can improve fidelity and that both negative and positive feedback degrade fidelity, for standard models of genetic autoregulation. For a group of cells, the fidelity of the collective output exceeds that of an individual cell and negative feedback then typically becomes beneficial. We can also predict the dynamic signal for which a given system has highest fidelity and, conversely, how to modify the network design to maximize fidelity for a given dynamic signal. Our approach is general, has applications to both systems and synthetic biology, and will help underpin studies of cellular behavior in natural, dynamic environments.

The magnitude and colour of noise in genetic negative feedback systems

The comparative ability of transcriptional and small RNA-mediated negative feedback to control fluctuations or ‘noise’ in gene expression remains unexplored. Both autoregulatory mechanisms usually suppress the average (mean) of the protein level and its variability across cells. The variance of the number of proteins per molecule of mean expression is also typically reduced compared with the unregulated system, but is almost never below the value of one. This relative variance often substantially exceeds a recently obtained, theoretical lower limit for biochemical feedback systems. Adding the transcriptional or small RNA-mediated control has different effects. Transcriptional autorepression robustly reduces both the relative variance and persistence (lifetime) of fluctuations. Both benefits combine to reduce noise in downstream gene expression. Autorepression via small RNA can achieve more extreme noise reduction and typically has less effect on the mean expression level. However, it is often more costly to implement and is more sensitive to rate parameters. Theoretical lower limits on the relative variance are known to decrease slowly as a measure of the cost per molecule of mean expression increases. However, the proportional increase in cost to achieve substantial noise suppression can be different away from the optimal frontier—for transcriptional autorepression, it is frequently negligible.

Stochastic Simulation of Biomolecular Networks in Dynamic Environments

Simulation of biomolecular networks is now indispensable for studying biological systems, from small reaction networks to large ensembles of cells. Here we present a novel approach for stochastic simulation of networks embedded in the dynamic environment of the cell and its surroundings. We thus sample trajectories of the stochastic process described by the chemical master equation with time-varying propensities. A comparative analysis shows that existing approaches can either fail dramatically, or else can impose impractical computational burdens due to numerical integration of reaction propensities, especially when cell ensembles are studied. Here we introduce the Extrande method which, given a simulated time course of dynamic network inputs, provides a conditionally exact and several orders-of-magnitude faster simulation solution. The new approach makes it feasible to demonstrate—using decision-making by a large population of quorum sensing bacteria—that robustness to fluctuations from upstream signaling places strong constraints on the design of networks determining cell fate. Our approach has the potential to significantly advance both understanding of molecular systems biology and design of synthetic circuits.

Information Transfer in Gonadotropin-releasing Hormone (GnRH) Signaling: EXTRACELLULAR SIGNAL-REGULATED KINASE (ERK)-MEDIATED FEEDBACK LOOPS CONTROL HORMONE SENSING.

Cell signaling pathways are noisy communication channels, and statistical measures derived from information theory can be used to quantify the information they transfer. Here we use single cell signaling measures to calculate mutual information as a measure of information transfer via gonadotropin-releasing hormone (GnRH) receptors (GnRHR) to extracellular signal-regulated kinase (ERK) or nuclear factor of activated T-cells (NFAT). This revealed mutual information values <1 bit, implying that individual GnRH-responsive cells cannot unambiguously differentiate even two equally probable input concentrations. Addressing possible mechanisms for mitigation of information loss, we focused on the ERK pathway and developed a stochastic activation model incorporating negative feedback and constitutive activity. Model simulations revealed interplay between fast (min) and slow (min-h) negative feedback loops with maximal information transfer at intermediate feedback levels. Consistent with this, experiments revealed that reducing negative feedback (by expressing catalytically inactive ERK2) and increasing negative feedback (by Egr1-driven expression of dual-specificity phosphatase 5 (DUSP5)) both reduced information transfer from GnRHR to ERK. It was also reduced by blocking protein synthesis (to prevent GnRH from increasing DUSP expression) but did not differ for different GnRHRs that do or do not undergo rapid homologous desensitization. Thus, the first statistical measures of information transfer via these receptors reveals that individual cells are unreliable sensors of GnRH concentration and that this reliability is maximal at intermediate levels of ERK-mediated negative feedback but is not influenced by receptor desensitization.

Signaling to Extracellular Signal-regulated Kinase from ErbB1 Kinase and Protein Kinase C: FEEDBACK, HETEROGENEITY, AND GATING

Background: The mechanisms underlying acute ERK signaling are poorly understood. Results: Feedback influences basal and acutely stimulated ERK responses but does not render signaling kinetics robust to ERK concentration. Conclusion: Acute ERK response kinetics depend on ERK concentration and activation mechanism as well as feedback. Significance: ERK responses to transient stimulation can be gated by ERK concentration, and short-term activation appears distributive rather than processive. Many extracellular signals act via the Raf/MEK/ERK cascade in which kinetics, cell-cell variability, and sensitivity of the ERK response can all influence cell fate. Here we used automated microscopy to explore the effects of ERK-mediated negative feedback on these attributes in cells expressing endogenous ERK or ERK2-GFP reporters. We studied acute rather than chronic stimulation with either epidermal growth factor (ErbB1 activation) or phorbol 12,13-dibutyrate (PKC activation). In unstimulated cells, ERK-mediated negative feedback reduced the population-average and cell-cell variability of the level of activated ppERK and increased its robustness to changes in ERK expression. In stimulated cells, negative feedback (evident between 5 min and 4 h) also reduced average levels and variability of phosphorylated ERK (ppERK) without altering the “gradedness” or sensitivity of the response. Binning cells according to total ERK expression revealed, strikingly, that maximal ppERK responses initially occur at submaximal ERK levels and that this non-monotonic relationship changes to an increasing, monotonic one within 15 min. These phenomena occur in HeLa cells and MCF7 breast cancer cells and in the presence and absence of ERK-mediated negative feedback. They were best modeled assuming distributive (rather than processive) activation. Thus, we have uncovered a novel, time-dependent change in the relationship between total ERK and ppERK levels that persists without negative feedback. This change makes acute response kinetics dependent on ERK level and provides a “gating” or control mechanism in which the interplay between stimulus duration and the distribution of ERK expression across cells could modulate the proportion of cells that respond to stimulation.

Signaling to ERK from ErbB1 and PKC: Feedback, Heterogeneity and Gating.

Background: The mechanisms underlying acute ERK signaling are poorly understood. Results: Feedback influences basal and acutely stimulated ERK responses but does not render signaling kinetics robust to ERK concentration. Conclusion: Acute ERK response kinetics depend on ERK concentration and activation mechanism as well as feedback. Significance: ERK responses to transient stimulation can be gated by ERK concentration, and short-term activation appears distributive rather than processive. Many extracellular signals act via the Raf/MEK/ERK cascade in which kinetics, cell-cell variability, and sensitivity of the ERK response can all influence cell fate. Here we used automated microscopy to explore the effects of ERK-mediated negative feedback on these attributes in cells expressing endogenous ERK or ERK2-GFP reporters. We studied acute rather than chronic stimulation with either epidermal growth factor (ErbB1 activation) or phorbol 12,13-dibutyrate (PKC activation). In unstimulated cells, ERK-mediated negative feedback reduced the population-average and cell-cell variability of the level of activated ppERK and increased its robustness to changes in ERK expression. In stimulated cells, negative feedback (evident between 5 min and 4 h) also reduced average levels and variability of phosphorylated ERK (ppERK) without altering the “gradedness” or sensitivity of the response. Binning cells according to total ERK expression revealed, strikingly, that maximal ppERK responses initially occur at submaximal ERK levels and that this non-monotonic relationship changes to an increasing, monotonic one within 15 min. These phenomena occur in HeLa cells and MCF7 breast cancer cells and in the presence and absence of ERK-mediated negative feedback. They were best modeled assuming distributive (rather than processive) activation. Thus, we have uncovered a novel, time-dependent change in the relationship between total ERK and ppERK levels that persists without negative feedback. This change makes acute response kinetics dependent on ERK level and provides a “gating” or control mechanism in which the interplay between stimulus duration and the distribution of ERK expression across cells could modulate the proportion of cells that respond to stimulation.