Evaluation of stochastic effects on biomolecular networks using the generalised Nyquist stability criterion

Abstract—Stochastic differential equations are now commonly used to model biomolecular networks in systems biology, and much recent research has been devoted to the development of methods to analyse their stability properties. Stability analysis of such systems may be performed using the Laplace transform, which requires the calculation of the exponential matrix involving time symbolically. However, the calculation of the symbolic exponential matrix is not feasible for problems of even moderate size, as the required computation time increases exponentially with the matrix order. To address this issue, we present a novel method for approximating the Laplace transform which does not require the exponential matrix to be calculated explicitly. The calculation time associated with the proposed method does not increase exponentially with the size of the system, and the approximation error is shown to be of the same order as existing methods. Using this approximation method, we show how a straightforward application of the generalized Nyquist stability criterion provides necessary and sufficient conditions for the stability of stochastic biomolecular networks. The usefulness and computational efficiency of the proposed method is illustrated through its application to the problem of analysing a model for limit-cycle oscillations in cAMP during aggregation of Dictyostelium cells

Robustness analysis of magnetic torquer controlled spacecraft attitude dynamics

This paper describes a systematic approach to the robustness analysis of linear periodically time-varying (LPTV) systems. The method uses the technique known as Lifting to transform the original time-varying uncertain system into linear fractional transformation (LFT) form. The stability and performance robustness of the system to structured parametric uncertainty can then be analysed non-conservatively using the structured singular value μ. The method is applied to analyse the stability robustness of an attitude control law for a spacecraft controlled by magnetic torquer bars, whose linearised dynamics can naturally be written in linear periodically time-varying form. The proposed method allows maximum allowable levels of uncertainty, as well as worst-case uncertainty combinations to be computed. The destabilising effect of these uncertain parameter combinations is verified in time-domain simulations

In vitro Growth of the Landschütz Ascites Tumour with Retention of High Mouse Virulence

ALTHOTJGH ascites tumours have been propagated for many years by regular passage in their natural host, only recently has it been possible to maintain some of these cells in continuous passage in tissue culture. The ease of adaptation to in vitro growth, the stability of cells adapted in this way with respect to morphology and chromosome constitution, and their virulence for mice have been variousl

Least-squares methods for identifying biochemical regulatory networks from noisy measurements

<b>Background</b>: We consider the problem of identifying the dynamic interactions in biochemical networks from noisy experimental data. Typically, approaches for solving this problem make use of an estimation algorithm such as the well-known linear Least-Squares (LS) estimation technique. We demonstrate that when time-series measurements are corrupted by white noise and/or drift noise, more accurate and reliable identification of network interactions can be achieved by employing an estimation algorithm known as Constrained Total Least Squares (CTLS). The Total Least Squares (TLS) technique is a generalised least squares method to solve an overdetermined set of equations whose coefficients are noisy. The CTLS is a natural extension of TLS to the case where the noise components of the coefficients are correlated, as is usually the case with time-series measurements of concentrations and expression profiles in gene networks. <b>Results</b>: The superior performance of the CTLS method in identifying network interactions is demonstrated on three examples: a genetic network containing four genes, a network describing p53 activity and <i>mdm2</i> messenger RNA interactions, and a recently proposed kinetic model for interleukin (IL)-6 and (IL)-12b messenger RNA expression as a function of ATF3 and NF-κB promoter binding. For the first example, the CTLS significantly reduces the errors in the estimation of the Jacobian for the gene network. For the second, the CTLS reduces the errors from the measurements that are corrupted by white noise and the effect of neglected kinetics. For the third, it allows the correct identification, from noisy data, of the negative regulation of (IL)-6 and (IL)-12b by ATF3. <b>Conclusion</b>: The significant improvements in performance demonstrated by the CTLS method under the wide range of conditions tested here, including different levels and types of measurement noise and different numbers of data points, suggests that its application will enable more accurate and reliable identification and modelling of biochemical networks

Validation of a model of regulation in the tryptophan operon against multiple experiment data using global optimisation

This paper is concerned with validating a mathematical model of regulation in the tryptophan operon using global optimization. Although a number of models for this biochemical network are proposed, in many cases only qualitative agreement between the model output and experimental data was demonstrated, since very little information is currently available to guide the selection of parameter values for the models. This paper presents a model validating method using both multiple experimental data and global optimization

A geometrical formulation of the μ-lower bound problem

A new problem formulation for the structured singular value μ in the case of purely real (possibly repeated) uncertainties is presented. The approach is based on a geometrical interpretation of the singularity constraint arising in the μ lower bound problem. An interesting feature of this problem formulation is that the resulting parametric search space is independent of the number of times any parameter is repeated in the structured uncertainty matrix. A corresponding lower bound algorithm combining randomisation and optimisation methods is developed, and some probabilistic performance guarantees are derived. The potential usefulness of the proposed approach is demonstrated on two high-order real μ analysis problems from the aerospace and systems biology literature

Identifcation of the hydraulic model from operational measurements for supervisory pressure control

The operational pressure control is a cost-e®ective way to leakage reduction and many pressure control methods and algorithms have been developed. Whilst the pres- sure control algorithm is model-based, the hydraulic model of the considered distribu- tion network is not always available. Therefore, this paper will focus on the development of an aggregated hydraulic model of the network considered, in particular, identi¯ca- tion of a leakage enhanced model using the operational measurements or the available historical data. This will enable a pressure optimisation algorithm to calculate the optimal pressure schedules for the implementation of a pressure control scheme. The identi¯cation problem is formulated as a parameter estimation problem in this paper and a least-square based method is derived for estimating the parameters in the model. A case study provided by a UK water company is performed to illustrate the use of the method and the identi¯cation results from real operational data are presented

Resonance bifurcations from robust homoclinic cycles

We present two calculations for a class of robust homoclinic cycles with symmetry Z_n x Z_2^n, for which the sufficient conditions for asymptotic stability given by Krupa and Melbourne are not optimal. Firstly, we compute optimal conditions for asymptotic stability using transition matrix techniques which make explicit use of the geometry of the group action. Secondly, through an explicit computation of the global parts of the Poincare map near the cycle we show that, generically, the resonance bifurcations from the cycles are supercritical: a unique branch of asymptotically stable period orbits emerges from the resonance bifurcation and exists for coefficient values where the cycle has lost stability. This calculation is the first to explicitly compute the criticality of a resonance bifurcation, and answers a conjecture of Field and Swift in a particular limiting case. Moreover, we are able to obtain an asymptotically-correct analytic expression for the period of the bifurcating orbit, with no adjustable parameters, which has not proved possible previously. We show that the asymptotic analysis compares very favourably with numerical results.Comment: 24 pages, 3 figures, submitted to Nonlinearit