Tradeoffs in the design of RNA thermometers.

The synthesis of RNA thermometers is aimed at achieving temperature responses with desired thresholds and sensitivities. Although previous works have generated thermometers with a variety of thresholds and sensitivities as well as guidelines for design, possible constraints in the achievable thresholds and sensitivities remain unclear. We addressed this issue using a two-state model and its variants, as well as melt profiles generated from thermodynamic computations. In the two-state model, we found that the threshold was inversely proportional to the sensitivity, in the case of a fixed energy difference between the two states. Notably, this constraint could persist in variations of the two-state model with sequentially unfolding states and branched parallel pathways. Furthermore, the melt profiles generated from a library of thermometers exhibited a similar constraint. These results should inform the design of RNA thermometers as well as other responses that are mediated in a similar fashion.

Design of a Toolbox of RNA Thermometers.

RNA thermometers are RNA regulatory elements that convert temperature into a functional biological response through a temperature-induced conformational change. These regulatory elements have been investigated in numerous natural contexts and have been designed for synthetic biology as well. A basic challenge has been the design of an RNA thermometer whose final activity in response to temperature matches a prespecified response, in terms of its sensitivity, threshold, and leakiness. This chapter provides a methodology for the design of a toolbox of RNA thermometers. We describe considerations for the conceptual design, a computational assessment, and strategies for experimental synthesis and measurement.

Experimental evidence for constraints in amplitude-timescale co-variation of a biomolecular pulse generating circuit design.

Understanding constraints on the functional properties of biomolecular circuit dynamics, such as the possible variations of amplitude and timescale of a pulse, is an important part of biomolecular circuit design. While the amplitude-timescale co-variations of the pulse in an incoherent feedforward loop have been investigated computationally using mathematical models, experimental support for any such constraints is relatively unclear. Here, the authors address this using experimental measurement of an existing pulse generating incoherent feedforward loop circuit realisation in the context of a standard mathematical model. They characterise the trends of co-variation in the pulse amplitude and rise time computationally by randomly exploring the parameter space. They experimentally measured the co-variation by varying inducers and found that larger amplitude pulses have a slower rise time. They discuss the gap between the experimental measurements and predictions of the standard model, highlighting model additions and other biological factors that might bridge the gap.

On the length scale dependence of DNA conformational change under local perturbation.

Conformational change of a DNA molecule is frequently observed in multiple biological processes and has been modeled using a chain of strongly coupled oscillators with a nonlinear bistable potential. While the mechanism and properties of conformational change in the model have been investigated and several reduced order models developed, the conformational dynamics as a function of the length of the oscillator chain is relatively less clear. To address this, we use a modified Lindstedt-Poincare method and numerical computations. We calculate a perturbation expansion of the frequency of the model's nonzero modes, finding that approximating these modes with their unperturbed dynamics, as in a previous reduced order model, may not hold when the length of the DNA model increases. We investigate the conformational change to the local perturbation in models of varying lengths, finding that for the chosen input and parameters, there are two regions of DNA length in the model - first, where the minimum energy required to undergo the conformational change increases with the DNA length; and second, where it is almost independent of the length of the DNA model. We analyze the conformational change in these models by adding randomness to the local perturbation, finding that the tendency of the system to remain in a stable conformation against random perturbation decreases with increase in DNA length. These results should help to understand the role of the length of a DNA molecule in influencing its conformational dynamics.

Assessment of Robustness to Temperature in a Negative Feedback Loop and a Feedforward Loop.

Robustness to temperature variation is an important specification in biomolecular circuit design. While the cancellation of parametric temperature dependencies has been shown to improve the temperature robustness of the period in a synthetic oscillator design, the performance of other biomolecular circuit designs in different temperature conditions is relatively unclear. Using a combination of experimental measurements and mathematical models, we assessed the temperature robustness of two biomolecular circuit motifs-a negative feedback loop and a feedforward loop. We found that the measured responses of both the circuits changed with temperature, both in the amplitude and in the transient response. We also found that, in addition to the cancellation of parametric temperature dependencies, certain parameter regimes could facilitate the temperature robustness of the negative feedback loop, although at a performance cost. We discuss these parameter regimes in the context of the measured data for the negative feedback loop. These results should help develop a framework for assessing and designing temperature robustness in biomolecular circuits.

Robustness of a biomolecular oscillator to pulse perturbations.

Biomolecular oscillators can function robustly in the presence of environmental perturbations, which can either be static or dynamic. While the effect of different circuit parameters and mechanisms on the robustness to steady perturbations has been investigated, the scenario for dynamic perturbations is relatively unclear. To address this, the authors use a benchmark three protein oscillator design - the repressilator - and investigate its robustness to pulse perturbations, computationally as well as use analytical tools of Floquet theory. They found that the metric provided by direct computations of the time it takes for the oscillator to settle after pulse perturbation is applied, correlates well with the metric provided by Floquet theory. They investigated the parametric dependence of the Floquet metric, finding that the parameters that increase the effective delay enhance robustness to pulse perturbation. They found that the structural changes such as increasing the number of proteins in a ring oscillator as well as adding positive feedback, both of which increase effective delay, facilitates such robustness. These results highlight such design principles, especially the role of delay, for designing an oscillator that is robust to pulse perturbation.

Dependence of bacterial growth rate on dynamic temperature changes.

Temperature is an important determinant of bacterial growth. While the dependence of bacterial growth on different temperatures has been well studied for many bacterial species, prediction of bacterial growth rate for dynamic temperature changes is relatively unclear. Here, the authors address this issue using a combination of experimental measurements of the growth, at the resolution of 5 min, of Escherichia coli and mathematical models. They measure growth curves at different temperatures and estimate model parameters to predict bacterial growth profiles subject to dynamic temperature changes. They compared these predicted growth profiles for various step-like temperature changes with experimental measurements using the coefficient of determination and mean square error and based on this comparison, ranked the different growth models, finding that the generalised logistic growth model gave the smallest error. They note that as the maximum specific growth increases the duration of this growth predominantly decreases. These results provide a basis to compute the dependence of the growth rate parameter in biomolecular circuits on dynamic temperatures and may be useful for designing biomolecular circuits that are robust to temperature.

Non-normality can facilitate pulsing in biomolecular circuits.

Non-normality can underlie pulse dynamics in many engineering contexts. However, its role in pulses generated in biomolecular contexts is generally unclear. Here, the authors address this issue using the mathematical tools of linear algebra and systems theory on simple computational models of biomolecular circuits. They find that non-normality is present in standard models of feedforward loops. They used a generalised framework and pseudospectrum analysis to identify non-normality in larger biomolecular circuit models, finding that it correlates well with pulsing dynamics. Finally, they illustrate how these methods can be used to provide analytical support to numerical screens for pulsing dynamics as well as provide guidelines for design.

Describing function-based approximations of biomolecular systems.

Mathematical methods provide useful framework for the analysis and design of complex systems. In newer contexts such as biology, however, there is a need to both adapt existing methods as well as to develop new ones. Using a combination of analytical and computational approaches, the authors adapt and develop the method of describing functions to represent the input-output responses of biomolecular signalling systems. They approximate representative systems exhibiting various saturating and hysteretic dynamics in a way that is better than the standard linearisation. Furthermore, they develop analytical upper bounds for the computational error estimates. Finally, they use these error estimates to augment the limit cycle analysis with a simple and quick way to bound the predicted oscillation amplitude. These results provide system approximations that can add more insight into the local behaviour of these systems than standard linearisation, compute responses to other periodic inputs and to analyse limit cycles.

Period-amplitude co-variation in biomolecular oscillators.

The period and amplitude of biomolecular oscillators are functionally important properties in multiple contexts. For a biomolecular oscillator, the overall constraints in how tuning of amplitude affects period, and vice versa, are generally unclear. Here, the authors investigate this co-variation of the period and amplitude in mathematical models of biomolecular oscillators using both simulations and analytical approximations. The authors computed the amplitude-period co-variation of 11 benchmark biomolecular oscillators as their parameters were individually varied around a nominal value, classifying the various co-variation patterns such as a simultaneous increase/decrease in period and amplitude. Next, the authors repeated the classification using a power norm-based amplitude metric, to account for the amplitudes of the many biomolecular species that may be part of the oscillations, finding largely similar trends. Finally, the authors calculate 'scaling laws' of period-amplitude co-variation for a subset of these benchmark oscillators finding that as the approximated period increases, the upper bound of the amplitude increases, or reaches a constant value. Based on these results, the authors discuss the effect of different parameters on the type of period-amplitude co-variation as well as the difficulty in achieving an oscillation with large amplitude and small period.

Design of a Toolbox of RNA Thermometers.

Biomolecular temperature sensors can be used for efficient control of large-volume bioreactors, for spatiotemporal imaging and control of gene expression, and to engineer robustness to temperature in biomolecular circuit design. Although RNA-based sensors, called "thermometers", have been investigated in both natural and synthetic contexts, an important challenge is to design diverse responses to temperature differing in sensitivity and threshold. We address this issue by constructing a library of RNA thermometers based on thermodynamic computations and experimentally measuring their activities in cell-free biomolecular "breadboards". Using free energies of the minimum free energy structures as well as melt profile computations, we estimated that a diverse set of temperature responses were possible. We experimentally found a wide range of responses to temperature in the range 29-37 °C with fold-changes varying over 3-fold around the starting thermometer. The sensitivities of these responses ranged over 10-fold around the starting thermometer. We correlated these measurements with computational expectations, finding that although there was no strong correlation for the individual thermometers, overall trends of diversity, fold-changes, and sensitivities were similar. These results present a toolbox of RNA-based circuit elements with diverse temperature responses.

Transient response characteristics in a biomolecular integral controller.

The cellular behaviour of perfect adaptation is achieved through the use of an integral control element in the underlying biomolecular circuit. It is generally unclear how integral action affects the important aspect of transient response in these biomolecular systems, especially in light of the fact that it typically deteriorates the transient response in engineering contexts. To address this issue, the authors investigated the transient response in a computational model of a simple biomolecular integral control system involved in bacterial signalling. They find that the transient response can actually speed up as the integral gain parameter increases. On further analysis, they find that the underlying dynamics are composed of slow and fast modes and the speed-up of the transient response is because of the speed-up of the slow-mode dynamics. Finally, they note how an increase in the integral gain parameter also leads to a decrease in the amplitude of the transient response, consistent with the overall improvement in the transient response. These results should be useful in understanding the overall effect of integral action on system dynamics, particularly for biomolecular systems.

Synthetic circuit for exact adaptation and fold-change detection.

Biological organisms use their sensory systems to detect changes in their environment. The ability of sensory systems to adapt to static inputs allows wide dynamic range as well as sensitivity to input changes including fold-change detection, a response that depends only on fold changes in input, and not on absolute changes. This input scale invariance underlies an important strategy for search that depends solely on the spatial profile of the input. Synthetic efforts to reproduce the architecture and response of cellular circuits provide an important step to foster understanding at the molecular level. We report the bottom-up assembly of biochemical systems that show exact adaptation and fold-change detection. Using a malachite green aptamer as the output, a synthetic transcriptional circuit with the connectivity of an incoherent feed-forward loop motif exhibits pulse generation and exact adaptation. A simple mathematical model was used to assess the amplitude and duration of pulse response as well as the parameter regimes required for fold-change detection. Upon parameter tuning, this synthetic circuit exhibits fold-change detection for four successive rounds of two-fold input changes. The experimental realization of fold-change detection circuit highlights the programmability of transcriptional switches and the ability to obtain predictive dynamical systems in a cell-free environment for technological applications.

Characterization of tradeoffs in biomolecular signaling.

Systems-level tradeoffs are fundamental in engineering, and recent work has highlighted an analogous role for them in biology. However, the extent of validity of these tradeoffs, especially for biomolecular systems, is generally unclear. Here, we address this issue for signaling tradeoffs that can constrain, for a fixed concentration of the signaling protein, a simultaneous enhancement of the gain and range of an amplifier or of the gain and threshold of a switch. We find that these gain-related tradeoffs persist in mathematical models of biomolecular reaction mechanisms that are at the core of large classes of signaling systems. Further, we find that these tradeoffs are also prevalent in the parametric functional forms commonly used to describe input-output curves in experimental analyses. Finally, we find that these tradeoffs can persist even in the presence of transcriptional feedback mechanisms that can change the concentration of the signaling protein. These results present a systematic characterization of these tradeoffs in biomolecular signaling systems.

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.