Quantitative Performance Bounds in Biomolecular Circuits due to Temperature Uncertainty

Performance of biomolecular circuits is affected by changes in temperature, due to its influence on underlying reaction rate parameters. While these performance variations have been estimated using Monte Carlo simulations, how to analytically bound them is generally unclear. To address this, we apply control-theoretic representations of uncertainty to examples of different biomolecular circuits, developing a framework to represent uncertainty due to temperature. We estimate bounds on the steady-state performance of these circuits due to temperature uncertainty. Through an analysis of the linearised dynamics, we represent this uncertainty as a feedback uncertainty and bound the variation in the magnitude of the input-output transfer function, providing a estimate of the variation in frequency-domain properties. Finally, we bound the variation in the time trajectories, providing an estimate of variation in time-domain properties. These results should enable a framework for analytical characterisation of uncertainty in biomolecular circuit performance due to temperature variation and may help in estimating relative performance of different controllers

Dynamical Consequences of Bandpass Feedback Loops in a Bacterial Phosphorelay

Under conditions of nutrient limitation, Bacillus subtilis cells terminally differentiate into a dormant spore state. Progression to sporulation is controlled by a genetic circuit consisting of a phosphorelay embedded in multiple transcriptional feedback loops, which is used to activate the master regulator Spo0A by phosphorylation. These transcriptional regulatory interactions are “bandpass”-like, in the sense that activation occurs within a limited band of Spo0A~P concentrations. Additionally, recent results show that the phosphorelay activation occurs in pulses, in a cell-cycle dependent fashion. However, the impact of these pulsed bandpass interactions on the circuit dynamics preceding sporulation remains unclear. In order to address this question, we measured key features of the bandpass interactions at the single-cell level and analyzed them in the context of a simple mathematical model. The model predicted the emergence of a delayed phase shift between the pulsing activity of the different sporulation genes, as well as the existence of a stable state, with elevated Spo0A activity but no sporulation, embedded within the dynamical structure of the system. To test the model, we used time-lapse fluorescence microscopy to measure dynamics of single cells initiating sporulation. We observed the delayed phase shift emerging during the progression to sporulation, while a re-engineering of the sporulation circuit revealed behavior resembling the predicted additional state. These results show that periodically-driven bandpass feedback loops can give rise to complex dynamics in the progression towards sporulation

A Kalman Filter Approach for Biomolecular Systems with Noise Covariance Updating

An important part of system modeling is determining parameter values, particularly for biomolecular systems, where direct measurements of individual parameters are typically hard. While Extended Kalman Filters have been used for this purpose, the choice of the process noise covariance is generally unclear. In this chapter, we address this issue for biomolecular systems using a combination of Monte Carlo simulations and experimental data, exploiting the dependence of the process noise covariance on the states and parameters, as given in the Langevin framework. We adapt a Hybrid Extended Kalman Filtering technique by updating the process noise covariance at each time step based on estimates. We compare the performance of this framework with different fixed values of process noise covariance in biomolecular system models, including an oscillator model, as well as in experimentally measured data for a negative transcriptional feedback circuit. We find that the Extended Kalman Filter with such process noise covariance update is closer to the optimality condition in the sense that the innovation sequence becomes white and in achieving a balance between the mean square estimation error and parameter convergence time. The results of this chapter may help in the use of Extended Kalman Filters for systems where process noise covariance depends on states and/or parameters.Comment: 23 pages, 9 figure

Negative Feedback Facilitates Temperature Robustness in Biomolecular Circuit Dynamics

Temporal dynamics in many biomolecular circuits can change with temperature because of the temperature dependence of underlying reaction rate parameters. It is generally unclear what circuit mechanisms can inherently facilitate robustness in the dynamics to variations in temperature. Here, we address this issue using a combination of mathematical models and experimental measurements in a cell-free transcription-translation system. We find that negative transcriptional feedback can reduce the effect of temperature variation on circuit dynamics. Further, we find that effective negative feedback due to first-order degradation mechanisms can also enable such a temperature robustness effect. Finally, we estimate temperature dependence of key parameters mediating such negative feedback mechanisms. These results should be useful in the design of temperature robust circuit dynamics

Quantitative Performance Bounds in Biomolecular Circuits due to Temperature Uncertainty

Performance of biomolecular circuits is affected by changes in temperature, due to its influence on underlying reaction rate parameters. While these performance variations have been estimated using Monte Carlo simulations, how to analytically bound them is generally unclear. To address this, we apply control-theoretic representations of uncertainty to examples of different biomolecular circuits, developing a framework to represent uncertainty due to temperature. We estimate bounds on the steady-state performance of these circuits due to temperature uncertainty. Through an analysis of the linearised dynamics, we represent this uncertainty as a feedback uncertainty and bound the variation in the magnitude of the input-output transfer function, providing a estimate of the variation in frequency-domain properties. Finally, we bound the variation in the time trajectories, providing an estimate of variation in time-domain properties. These results should enable a framework for analytical characterisation of uncertainty in biomolecular circuit performance due to temperature variation and may help in estimating relative performance of different controllers

Designing Robustness to Temperature in a Feedforward Loop Circuit

'Incoherent feedforward loops' represent important biomolecular circuit elements capable of a rich set of dynamic behavior including adaptation and pulsed responses. Temperature can modulate some of these properties through its effect on the underlying reaction rate parameters. It is generally unclear how to design a circuit where these properties are robust to variations in temperature. Here, we address this issue using a combination of tools from control and dynamical systems theory as well as preliminary experimental measurements towards such a design. Using a structured uncertainty representation, we analyze a standard incoherent feedforward loop circuit, noting mechanisms that intrinsically confer temperature robustness to some of its properties. Further, we study design variants that can enhance this robustness to temperature, including different negative feedback configurations as well as conditions for perfect temperature compensation. Finally, we find that the response of an incoherent feedforward loop circuit in cells can change with temperature. These results present groundwork for the design of a temperature-robust incoherent feedforward loop circuit