Robustness analysis of a nucleic acid controller for a dynamic biomolecular process using the structured singular value

In the field of synthetic biology, theoretical frameworks and software tools are now available that allow control systems represented as chemical reaction networks to be translated directly into nucleic acid-based chemistry, and hence implement embedded control circuitry for biomolecular processes. However, the development of tools for analysing the robustness of such controllers is still in its infancy. An interesting feature of such control circuits is that, although the transfer function of a linear system can be easily implemented via a chemical network of catalysis, degradation and annihilation reactions, this introduces additional nonlinear dynamics, due to the annihilation kinetics. We exemplify this problem for a dynamical biomolecular feedback system, and demonstrate how the structured singular value (μ) analysis framework can be extended to rigorously analyse the robustness of this class of system. We show that parametric uncertainty in the system affects the location of its equilibrium, and that this must be taken into account in the analysis. We also show that the parameterisation of the system can be scaled for experimental feasibility without affecting its robustness properties, and that a statistical analysis via Monte Carlo simulation fails to uncover the worst-case uncertainty combination found by μ-analysis.</p

Modelling of signal uncertainty and control objectives in robust controller design

This work develops a new paradigm for optimal robust controller synthesis in the frequency domain. A detailed examination is made of the engineering motivation and engineering efficacy underlying the various strands of robust control theory. The modelling of (a) signal uncertainty and (b) control system objectives in both Tioo and C\ control theories is considered in particular detail. Based on this examination, a theory which can fa irly be described as ‘a m odified 7ioo control theory’ or ‘a frequency domain C\ control theory’ is proposed. New signal sets for the modelling of uncertain signals are introduced. It is argued that these models more faithfully capture the way in which uncertain signals act on real physical systems. It is shown that by adopting these new models for uncertain signals, control theory can be used to non-conservatively minimise maximum tracking errors in the time domain, in the SISO case. In the MIMO case, the problem of optimally synthesising a controller to non-conservatively minimise tracking errors in the time domain leads to a modest variation on existing control theory, requiring the usual norm to be modified slightly. It is argued th a t the proposed paradigm in general achieves a better quality of control and more fa ith fu lly expresses the true objectives of feedback control systems. The proposed development is seen to also extend naturally to Ti.2 control theory, and indeed provides a new deterministic justification for the 7^2 control problem in the MIMO case. The question of design transparency in the synthesis of optimal robust controllers for multivariable systems is considered in detail. The implications of the proposed paradigm for transparency of design and weighting function selection are detailed. A decoupling design procedure for robust controller synthesis is proposed which, under certain restrictive conditions, allows the calculation of super-optimal robust controllers on a loop by loop basis. The usefulness of a classical decoupling approach to MIMO control system design in the context of multivariable robust control theory is demonstrated. A number of design examples are presented which show how the ideas and methods developed in this work can be applied to realistic control problems

Stochastic noise and synchronisation during Dictyostelium aggregation make cAMP oscillations robust

The molecular network, which underlies the oscillations in the concentration of adenosine 3′, 5′-cyclic monophosphate (cAMP) during the aggregation phase of starvation-induced development in Dictyostelium discoideum, achieves remarkable levels of robust performance in the face of environmental variations and cellular heterogeneity. However, the reasons for this robustness remain poorly understood. Tools and concepts from the field of control engineering provide powerful methods for uncovering the mechanisms underlying the robustness of these types of biological systems. Using such methods, two important factors contributing to the robustness of cAMP oscillations in Dictyostelium are revealed. First, stochastic fluctuations in the molecular interactions of the intracellular network, arising from random or directional noise and biological sources, play an important role in preserving stable oscillations in the face of variations in the kinetics of the network. Second, synchronisation of the aggregating cells through the diffusion of extracellular cAMP appears to be a key factor in ensuring robustness to cell-to-cell variations of the oscillatory waves of cAMP observed in Dictyostelium cell cultures. The conclusions have important general implications for the robustness of oscillating biomolecular networks (whether seen at organism, cell, or intracellular levels and including circadian clocks or Ca2+ oscillations, etc.), and suggest that such analysis can be conducted more reliably by using models including stochastic simulations, even in the case where molecular concentrations are very high

Elucidating the mechanisms of cooperative calcium-calmodulin interactions: a structural systems biology approach

BACKGROUND: Calmodulin is an important multifunctional molecule that regulates the activities of a large number of proteins in the cell. Calcium binding induces conformational transitions in calmodulin that make it specifically active to particular target proteins. The precise mechanisms underlying calcium binding to calmodulin are still, however, quite poorly understood. RESULTS: In this study, we adopt a structural systems biology approach and develop a mathematical model to investigate various types of cooperative calcium-calmodulin interactions. We compare the predictions of our analysis with physiological dose-response curves taken from the literature, in order to provide a quantitative comparison of the effects of different mechanisms of cooperativity on calcium-calmodulin interactions. The results of our analysis reduce the gap between current understanding of intracellular calmodulin function at the structural level and physiological calcium-dependent calmodulin target activation experiments. CONCLUSION: Our model predicts that the specificity and selectivity of CaM target regulation is likely to be due to the following factors: variations in the target-specific Ca2+ dissociation and cooperatively effected dissociation constants, and variations in the number of Ca2+ ions required to bind CaM for target activation

A simplified modelling framework facilitates more complex representations of plant circadian clocks

This is the final version. Available on open access from Public Library of Science via the DOI in this recordData Availability: All MATLAB files used to generate the results presented in the study are available from https://github.com/mathiasfoo/essystemplantcircadian.The circadian clock orchestrates biological processes so that they occur at specific times of the day, thereby facilitating adaptation to diurnal and seasonal environmental changes. In plants, mathematical modelling has been comprehensively integrated with experimental studies to gain a better mechanistic understanding of the complex genetic regulatory network comprising the clock. However, with an increasing number of circadian genes being discovered, there is a pressing need for methods facilitating the expansion of computational models to incorporate these newly-discovered components. Conventionally, plant clock models have comprised differential equation systems based on Michaelis-Menten kinetics. However, the difficulties associated with modifying interactions using this approach—and the concomitant problem of robustly identifying regulation types—has contributed to a complexity bottleneck, with quantitative fits to experimental data rapidly becoming computationally intractable for models possessing more than ≈50 parameters. Here, we address these issues by constructing the first plant clock models based on the S-System formalism originally developed by Savageau for analysing biochemical networks. We show that despite its relative simplicity, this approach yields clock models with comparable accuracy to the conventional Michaelis-Menten formalism. The S-System formulation also confers several key advantages in terms of model construction and expansion. In particular, it simplifies the inclusion of new interactions, whilst also facilitating the modification of regulation types, thereby making it well-suited to network inference. Furthermore, S-System models mitigate the issue of parameter identifiability. Finally, by applying linear systems theory to the models considered, we provide some justification for the increased use of aggregated protein equations in recent plant clock modelling, replacing the separate cytoplasmic/nuclear protein compartments that were characteristic of the earlier models. We conclude that as well as providing a simplified framework for model development, the S-System formalism also possesses significant potential as a robust modelling method for designing synthetic gene circuits.Royal SocietyEngineering and Physical Sciences Research Council (EPSRC)Biotechnology and Biological Sciences Research Council (BBSRC

Crosstalk between G-protein and Ca2+ pathways switches intracellular cAMP levels

Cyclic adenosine monophosphate and cyclic guanosine monophosphate are universal intracellular messengers whose concentrations are regulated by molecular networks comprised of different isoforms of the synthases adenylate cyclase or guanylate cyclase and the phosphodiesterases which degrade these compounds. In this paper, we employ a systems biology approach to develop mathematical models of these networks that, for the first time, take into account the different biochemical properties of the isoforms involved. To investigate the mechanisms underlying the joint regulation of cAMP and cGMP, we apply our models to analyse the regulation of cilia beat frequency in Paramecium by Ca(2+). Based on our analysis of these models, we propose that the diversity of isoform combinations that occurs in living cells provides an explanation for the huge variety of intracellular processes that are dependent on these networks. The inclusion of both G-protein receptor and Ca(2+)-dependent regulation of AC in our models allows us to propose a new explanation for the switching properties of G-protein subunits involved in nucleotide regulation. Analysis of the models suggests that, depending on whether the G-protein subunit is bound to AC, Ca(2+) can either activate or inhibit AC in a concentration-dependent manner. The resulting analysis provides an explanation for previous experimental results that showed that alterations in Ca(2+) concentrations can either increase or decrease cilia beat frequency over particular Ca(2+) concentration ranges

Exploiting the dynamic properties of covalent modification cycle for the design of synthetic analog biomolecular circuitry

Background: Cycles of covalent modification are ubiquitous motifs in cellular signalling. Although such signalling cycles are implemented via a highly concise set of chemical reactions, they have been shown to be capable of producing multiple distinct input-output mapping behaviours – ultrasensitive, hyperbolic, signal-transducing and threshold-hyperbolic. Results: In this paper, we show how the set of chemical reactions underlying covalent modification cycles can be exploited for the design of synthetic analog biomolecular circuitry. We show that biomolecular circuits based on the dynamics of covalent modification cycles allow (a) the computation of nonlinear operators using far fewer chemical reactions than purely abstract designs based on chemical reaction network theory, and (b) the design of nonlinear feedback controllers with strong performance and robustness properties. Conclusions: Our designs provide a more efficient route for translation of complex circuits and systems from chemical reactions to DNA strand displacement-based chemistry, thus facilitating their experimental implementation in future Synthetic Biology applications

Computational modelling suggests dynamic interactions between Ca2+, IP3 and G protein-coupled modules are key to robust Dictyostelium aggregation

Under conditions of starvation, Dictyostelium cells begin a programme of development during which they aggregate to form a multicellular structure by chemotaxis, guided by propagating waves of cyclic AMP that are relayed robustly from cell to cell. In this paper, we develop and analyse a new model for the intracellular and extracellular cAMP dependent processes that regulate Dictyostelium migration. The model allows, for the first time, a quantitative analysis of the dynamic interactions between calcium, IP(3) and G protein-dependent modules that are shown to be key to the generation of robust cAMP oscillations in Dictyostelium cells. The model provides a mechanistic explanation for the transient increase in cytosolic free Ca(2+) concentration seen in recent experiments with the application of the calmodulin inhibitor calmidazolium (R24571) to Dictyostelium cells, and also allows elucidation of the effects of varying both the conductivity of stretch-activated channels and the concentration of external phosphodiesterase on the oscillatory regime of an individual cell. A rigorous analysis of the robustness of the new model shows that interactions between the different modules significantly reduce the sensitivity of the resulting cAMP oscillations to variations in the kinetics of different Dictyostelium cells, an essential requirement for the generation of the spatially and temporally synchronised chemoattractant cAMP waves that guide Dictyostelium aggregation