Two dimensionless parameters and a mechanical analogue for the HKB model of motor coordination.

The widely cited Haken-Kelso-Bunz (HKB) model of motor coordination is used in an enormous range of applications. In this paper, we show analytically that the weakly damped, weakly coupled HKB model of two oscillators depends on only two dimensionless parameters; the ratio of the linear damping coefficient and the linear coupling coefficient and the ratio of the combined nonlinear damping coefficients and the combined nonlinear coupling coefficients. We illustrate our results with a mechanical analogue. We use our analytic results to predict behaviours in arbitrary parameter regimes and show how this led us to explain and extend recent numerical continuation results of the full HKB model. The key finding is that the HKB model contains a significant amount of behaviour in biologically relevant parameter regimes not yet observed in experiments or numerical simulations. This observation has implications for the development of virtual partner interaction and the human dynamic clamp, and potentially for the HKB model itself.

On the regularization of impact without collision: the Painlevé paradox and compliance.

We consider the problem of a rigid body, subject to a unilateral constraint, in the presence of Coulomb friction. We regularize the problem by assuming compliance (with both stiffness and damping) at the point of contact, for a general class of normal reaction forces. Using a rigorous mathematical approach, we recover impact without collision (IWC) in both the inconsistent and the indeterminate Painlevé paradoxes, in the latter case giving an exact formula for conditions that separate IWC and lift-off. We solve the problem for arbitrary values of the compliance damping and give explicit asymptotic expressions in the limiting cases of small and large damping, all for a large class of rigid bodies.

Time-Delayed Models of Gene Regulatory Networks.

We discuss different mathematical models of gene regulatory networks as relevant to the onset and development of cancer. After discussion of alternative modelling approaches, we use a paradigmatic two-gene network to focus on the role played by time delays in the dynamics of gene regulatory networks. We contrast the dynamics of the reduced model arising in the limit of fast mRNA dynamics with that of the full model. The review concludes with the discussion of some open problems.

Discontinuity-induced bifurcations of piecewise smooth dynamical systems.

This paper presents an overview of the current state of the art in the analysis of discontinuity-induced bifurcations (DIBs) of piecewise smooth dynamical systems, a particularly relevant class of hybrid dynamical systems. Firstly, we present a classification of the most common types of DIBs involving non-trivial interactions of fixed points and equilibria of maps and flows with the manifolds in phase space where the system is non-smooth. We then analyse the case of limit cycles interacting with such manifolds, presenting grazing and sliding bifurcations. A description of possible classification strategies to predict and analyse the scenarios following such bifurcations is also discussed, with particular attention to those methodologies that can be applied to generic n-dimensional systems.

Comparing different ODE modelling approaches for gene regulatory networks.

A fundamental step in synthetic biology and systems biology is to derive appropriate mathematical models for the purposes of analysis and design. For example, to synthesize a gene regulatory network, the derivation of a mathematical model is important in order to carry out in silico investigations of the network dynamics and to investigate parameter variations and robustness issues. Different mathematical frameworks have been proposed to derive such models. In particular, the use of sets of nonlinear ordinary differential equations (ODEs) has been proposed to model the dynamics of the concentrations of mRNAs and proteins. These models are usually characterized by the presence of highly nonlinear Hill function terms. A typical simplification is to reduce the number of equations by means of a quasi-steady-state assumption on the mRNA concentrations. This yields a class of simplified ODE models. A radically different approach is to replace the Hill functions by piecewise-linear approximations [Casey, R., de Jong, H., Gouze , J.-L., 2006. Piecewise-linear models of genetic regulatory networks: equilibria and their stability. J. Math. Biol. 52 (1), 27-56]. A further modelling approach is the use of discrete-time maps [Coutinho, R., Fernandez, B., Lima, R., Meyroneinc, A., 2006. Discrete time piecewise affine models of genetic regulatory networks. J. Math. Biol. 52, 524-570] where the evolution of the system is modelled in discrete, rather than continuous, time. The aim of this paper is to discuss and compare these different modelling approaches, using a representative gene regulatory network. We will show that different models often lead to conflicting conclusions concerning the existence and stability of equilibria and stable oscillatory behaviours. Moreover, we shall discuss, where possible, the viability of making certain modelling approximations (e.g. quasi-steady-state mRNA dynamics or piecewise-linear approximations of Hill functions) and their effects on the overall system dynamics.

Control of spatiotemporal patterns in the Gray-Scott model.

This paper studies the effects of a time-delayed feedback control on the appearance and development of spatiotemporal patterns in a reaction-diffusion system. Different types of control schemes are investigated, including single-species, diagonal, and mixed control. This approach helps to unveil different dynamical regimes, which arise from chaotic state or from traveling waves. In the case of spatiotemporal chaos, the control can either stabilize uniform steady states or lead to bistability between a trivial steady state and a propagating traveling wave. Furthermore, when the basic state is a stable traveling pulse, the control is able to advance stationary Turing patterns or yield the above-mentioned bistability regime. In each case, the stability boundary is found in the parameter space of the control strength and the time delay, and numerical simulations suggest that diagonal control fails to control the spatiotemporal chaos.

The growth rate of metastatic non-seminomatous germ cell testicular tumours measured by marker production doubling time--I. Theoretical basis and practical application.

Changes in serum tumour marker levels in non-seminomatous germ cell testicular tumours (NSGCTT) are used to monitor tumour growth and response to treatment. A novel method of calculating the actual tumour marker production (TMP) per day is reported; estimation of the rate of change of TMP is a measure of the tumour growth or regression rate. TMP is calculated mathematically from the rate of increase in serum marker level and its natural half life. TMP is assumed to be proportional to the number of marker producing cells in the tumour. TMP was calculated over the time between orchidectomy and the start of chemotherapy. The rate of increase in TMP with time is expressed as the marker production doubling time (MPDT) and is a measure of the growth rate. In a group of 51 patients with metastatic NSGCTT, TMP varied from 0.012 to 5985 iu/l/day (AFP) and 0.08-5404 iu/l/day (HCG). MPDT varied from 0.5 to greater than 80 days (45 cases) for AFP + ve patients and from 1.8 to greater than 80 days (34 cases) for HCG + ve patients; greater than 80% of cases had a MPDT less than or equal to 32 days. In 45/51 (88%) patients, there was no discrepancy in MPDT between markers. The use of changes in serum marker level to follow tumour progression and regression is simple, but the calculation of actual TMP provides clearer information about the change in number of marker producing cells and can be used as non-invasive method for measuring the tumour growth rate of metastatic disease and response to treatment.

The growth rate of metastatic nonseminomatous germ cell testicular tumours measured by marker production doubling time--II. Prognostic significance in patients treated by chemotherapy.

Tumour growth rates have been measured in metastatic non-seminomatous germ cell testicular tumours (NSGCTT) by estimating the rate of rise of tumour marker production (TMP). TMP was calculated for the time between orchidectomy and the start of chemotherapy in a group of 58 patients with metastatic NSGCTT treated with BEP combination chemotherapy (bleomycin, etoposide and cisplatin). Calculation of TMP (iu/l/day) took account of the continuing clearance of marker from the serum. TMP increased with time in 51 patients and this rise generally appeared to be exponential. The rate of this increase was expressed as the marker production doubling time (MPDT) and is a measure of the tumour growth rate. MPDT varied from 0.5 to greater than 80 days (45 cases) for AFP + ve patients and from 1.8 to greater than 80 days (34 cases) for HCG + ve patients. Patients who failed BEP first line therapy had shorter MPDTs than those who responded (AFP P = 0.08, HCG P = 0.003). It was found that patients with a MPDT less than or equal to 4 days were more likely to fail treatment than those who had a MPDT greater than 4 days (AFP P = 0.009, HCG P = 0.005). MPDTs were independent of initial serum marker concentration. Patients with small volume disease had longer MPDTs than patients with large volume disease (AFP P = 0.02, HCG P = 0.04). Rapid tumour growth rate reflected by short MPDT carries a poor prognosis in patients with NSGCTT treated by BEP chemotherapy.