Understanding the discrete genetic toggle switch phenomena using a discrete 'nullcline' construct inspired by the Markov chain tree theorem

Nullclines provide a convenient way of characterising and understanding the behaviour of low dimensional nonlinear deterministic systems, but are, perhaps not unsurprisingly, a poor predictor of the behaviour of discrete state stochastic systems in the low numbers regime. Such models are appropriate in many biological systems. In this paper we propose a graphical discrete `nullcline-like' construction, inspired by the Markov chain tree theorem, and investigate its application to the original genetic toggle switch, which is a feedback interconnection of two mutually repressing genes. When the feedback gain (the `cooperativity') is sufficiently large, the deterministic system exhibits bistability, which shows itself as a bimodal stationary distribution in the discrete stochastic system for sufficiently large numbers. However, at small numbers a third mode appears corresponding to roughly equal numbers of each molecule. Without cooperativity, on the other hand (i.e. low feedback gain), the deterministic system has just one stable equilibrium. Nevertheless, the stochastic system can still exhibit bimodality. In this paper, we illustrate that the discrete `nullclines' proposed can, without the need to calculate the steady state distribution, provide an efficient graphical way of predicting the shape of the stationary probability distribution in different parameter regimes, thus allowing for greater insights in the observed behaviours

Measuring Distance between Systems under Bounded Power Excitation

This work suggests a way of measuring distance between two linear systems under a given bounded power excitation. The measure introduced can be used to bound from above and below the difference in closed-loop behavior of two plants with the same controller for a specified reference or disturbance spectrum. Given an unknown, single input real plant and its identified model, an upper bound on the distance between the plant and its model as expressed by this measure can be obtained from time domain data

Scalable Design of Heterogeneous Networks

A systematic approach to the analysis and design of a class of large dynamical systems is presented. The approach allows decentralised control laws to be designed independently using only local subsystem models. Design can be conducted using standard techniques, including loopshaping based on Nyquist and Popov plots, H$_\infty$ methods, and $\mu$-synthesis procedures. The approach is applied to a range of network models, including those for consensus, congestion control, electrical power systems, and distributed optimisation algorithms subject to delays.Engineering and Physical Sciences Research Council grant number EP/G066477/

Determining paediatric patient thickness from a single digital radiograph – a proof of principle

Objective: This work presents a proof of principle for a method of estimating the thickness of an attenuator from a single radiograph using the image, the exposure factors with which it was acquired and a priori knowledge of the characteristics of the X-ray unit and detector used for the exposure. It is intended this could be developed into a clinical tool to assist with paediatric patient dose audit, for which a measurement of patient size is required. Methods: The proof of principle used measured pixel value and effective linear attenuation coefficient to estimate the thickness of a Solid Water attenuator. The kerma at the detector was estimated using a measurement of pixel value on the image and measured detector calibrations. The initial kerma was estimated using a lookup table of measured output values. The effective linear attenuation coefficient was measured for Solid Water at varying kVp. 11 test images of known and varying thicknesses of Solid Water were acquired at 60, 70 and 81 kVp. Estimates of attenuator thickness were made using the model and the results compared to the known thickness. Results: Estimates of attenuator thickness made using the model differed from the known thickness by 3.8 mm (3.2%) on average, with a range of 0.5–10.8 mm (0.5–9%). Conclusion: A proof of principle is presented for a method of estimating the thickness of an attenuator using a single radiograph of the attenuator. The method has been shown to be accurate using a Solid Water attenuator, with a maximum difference between estimated and known attenuator thickness of 10.8 mm (9%). The method shows promise as a clinical tool for estimating abdominal paediatric patient thickness for paediatric patient dose audit, and is only contingent on the type of data routinely collected by Medical Physics departments. Advances in knowledge: A computational model has been created that is capable of accurately estimating the thickness of a uniform attenuator using only the radiographic image, the exposure factors with which it was acquired and a priori knowledge of the characteristics of the X-ray unit and detector used for the exposure. </jats:sec

Algorithms for worst case identification in H-infinity and the nu-gap metric

This paper considers two robustly convergent algorithms for the identification of a linear system from (possibly) noisy frequency response data. Both algorithms are based on the same principle; obtaining a good worst case fit to the data under a smoothness constraint on the obtained model. However they differ in their notions of distance and smoothness. The first algorithm yields an FIR model of a stable system and is optimal, in a certain sense for a finite model order. The second algorithm may be used for modelling unstable plants and yields a real rational approximation in the -gap. Given a model and a controller stabilising the true plant, a procedure for winding number correction is also suggested

Constraints on Fluctuations in Sparsely Characterized Biological Systems.

Biochemical processes are inherently stochastic, creating molecular fluctuations in otherwise identical cells. Such "noise" is widespread but has proven difficult to analyze because most systems are sparsely characterized at the single cell level and because nonlinear stochastic models are analytically intractable. Here, we exactly relate average abundances, lifetimes, step sizes, and covariances for any pair of components in complex stochastic reaction systems even when the dynamics of other components are left unspecified. Using basic mathematical inequalities, we then establish bounds for whole classes of systems. These bounds highlight fundamental trade-offs that show how efficient assembly processes must invariably exhibit large fluctuations in subunit levels and how eliminating fluctuations in one cellular component requires creating heterogeneity in another.The work was supported by grant 1137676 from the Division of Mathematical Sciences at the National Science Foundation, and grant GM081563 from the National Institutes of Health.This is the final version of the article. It first appeared from the American Physical Society via http://dx.doi.org/10.1103/PhysRevLett.116.05810

Kinetic Uncertainty Relations for the Control of Stochastic Reaction Networks.

Nonequilibrium stochastic reaction networks are commonly found in both biological and nonbiological systems, but have remained hard to analyze because small differences in rate functions or topology can change the dynamics drastically. Here, we conjecture exact quantitative inequalities that relate the extent of fluctuations in connected components, for various network topologies. Specifically, we find that regardless of how two components affect each other's production rates, it is impossible to suppress fluctuations below the uncontrolled equivalents for both components: one must increase its fluctuations for the other to be suppressed. For systems in which components control each other in ringlike structures, it appears that fluctuations can only be suppressed in one component if all other components instead increase fluctuations, compared to the case without control. Even the general N-component system-with arbitrary connections and parameters-must have at least one component with increased fluctuations to reduce fluctuations in others. In connected reaction networks it thus appears impossible to reduce the statistical uncertainty in all components, regardless of the control mechanisms or energy dissipation

Point Process Analysis of Noise in Early Invertebrate Vision

Noise is a prevalent and sometimes even dominant aspect of many biological processes. While many natural systems have adapted to attenuate or even usefully integrate noise, the variability it introduces often still delimits the achievable precision across biological functions. This is particularly so for visual phototransduction, the process responsible for converting photons of light into usable electrical signals (quantum bumps). Here, randomness of both the photon inputs (regarded as extrinsic noise) and the conversion process (intrinsic noise) are seen as two distinct, independent and significant limitations on visual reliability. Past research has attempted to quantify the relative effects of these noise sources by using approximate methods that do not fully account for the discrete, point process and time ordered nature of the problem. As a result the conclusions drawn from these different approaches have led to inconsistent expositions of phototransduction noise performance. This paper provides a fresh and complete analysis of the relative impact of intrinsic and extrinsic noise in invertebrate phototransduction using minimum mean squared error reconstruction techniques based on Bayesian point process (Snyder) filters. An integrate-fire based algorithm is developed to reliably estimate photon times from quantum bumps and Snyder filters are then used to causally estimate random light intensities both at the front and back end of the phototransduction cascade. Comparison of these estimates reveals that the dominant noise source transitions from extrinsic to intrinsic as light intensity increases. By extending the filtering techniques to account for delays, it is further found that among the intrinsic noise components, which include bump latency (mean delay and jitter) and shape (amplitude and width) variance, it is the mean delay that is critical to noise performance. Consequently, if one wants to increase visual fidelity, reducing the photoconversion lag is much more important than improving the regularity of the electrical signal.This work was supported by the Gates Cambridge Trust (PhD studentship for research) https://www.gatescambridge.org/. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript

Fundamental performance similarities between individual pitch control strategies for wind turbines.

The use of blade individual pitch control (IPC) offers a means of reducing the harmful turbine structural loads that arise from the uneven and unsteady forcing from the oncoming wind. In recent years two different and competing IPC techniques have emerged that are characterised by the specific loads that they are primarily designed to attenuate. In the first instance, methodologies such as single-blade control and Clarke Transform-based control have been developed to reduce the unsteady loads on the rotating blades, whilst tilt-yaw control and its many variants instead target load reductions in the non rotating turbine structures, such as the tower and main bearing. Given the seeming disparities between these controllers, the aim of this paper is to show the fundamental performance similarities that exist between them and hence unify research in this area. Specifically, we show that single-blade controllers are equivalent to a particular class of tilt-yaw controller, which itself is equivalent to Clarke~Transform-based control. This means that three architecturally dissimilar IPC controllers exist that yield exactly the same performance in terms of load reductions on fixed and rotating turbine structures. We further demonstrate this outcome by presenting results obtained from high-fidelity closed-loop turbine simulations