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 alternativemodelling 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

THE SOLOW¡¯S MODEL WITH ENDOGENOUS POPULATION: A NEOCLASSICAL GROWTH CYCLE MODEL

It is shown here that the Solow (1956) neo-classical growth paradigm not only explains the ¡°first¡± stylised fact of economic growth, namely the existence of a globally stable state of balanced growth, but, once endowed with a demographically founded formulation of the labour supply, is also capable to endogenously explain a second main stylised fact of growth, i.e., the generation of globally stable oscillations around the path of balanced growth.Solow¡¯s Balanced Growth Model, Endogenous Population, Neoclassical Growth-Cycle Model

Oscillations and temporal signalling in cells

The development of new techniques to quantitatively measure gene expression in cells has shed light on a number of systems that display oscillations in protein concentration. Here we review the different mechanisms which can produce oscillations in gene expression or protein concentration, using a framework of simple mathematical models. We focus on three eukaryotic genetic regulatory networks which show "ultradian" oscillations, with time period of the order of hours, and involve, respectively, proteins important for development (Hes1), apoptosis (p53) and immune response (NFkB). We argue that underlying all three is a common design consisting of a negative feedback loop with time delay which is responsible for the oscillatory behaviour

Adaptive Detection of Instabilities: An Experimental Feasibility Study

We present an example of the practical implementation of a protocol for experimental bifurcation detection based on on-line identification and feedback control ideas. The idea is to couple the experiment with an on-line computer-assisted identification/feedback protocol so that the closed-loop system will converge to the open-loop bifurcation points. We demonstrate the applicability of this instability detection method by real-time, computer-assisted detection of period doubling bifurcations of an electronic circuit; the circuit implements an analog realization of the Roessler system. The method succeeds in locating the bifurcation points even in the presence of modest experimental uncertainties, noise and limited resolution. The results presented here include bifurcation detection experiments that rely on measurements of a single state variable and delay-based phase space reconstruction, as well as an example of tracing entire segments of a codimension-1 bifurcation boundary in two parameter space.Comment: 29 pages, Latex 2.09, 10 figures in encapsulated postscript format (eps), need psfig macro to include them. Submitted to Physica

Design and implementation of a mammalian synthetic gene oscillator

The core goal of synthetic biology as a discipline is to design, develop and characterize biological parts in order to precisely control cellular behaviour. Much of the research in this field has been focused on the development of gene regulatory networks, namely switches and oscillators. The study of synthetic gene oscillators has attracted significant attention in the past decade due to their intriguing dynamics and relevance in controlling inflammatory, metabolic and circadian signalling pathways. Additionally, the precise expression dynamics and molecular mechanisms that underlie the mammalian circadian clock structure are not fully understood. The work presented herein regards the design and implementation of a tuneable mammalian synthetic gene oscillator with a novel biological structure. To this end, an approach based on a combination of in silico design and in vivo part validation, in conjunction with a comparative analysis of previously implemented synthetic gene oscillators, was taken when assembling the proposed system. The topology of the system relies on a delayed negative feedback loop, consisting of the coupled regulatory activities of the transcription regulators LacI, tTA, and Gal4. The numerical solution and stability analysis of an ODE-based model describing the dynamics of the system are indicative that the proposed system is capable of generating sustained oscillations across a wide range of parameter values. The biological parts that comprise the system have been monitored and validated in HEK293T cells through time-lapse fluorescence microscopy and image analysis. The in vivo performance of the proposed mammalian synthetic gene oscillator was also assessed in the HEK293T cell line, and monitored using time-lapse fluorescence microscopy. Damped fluorescence oscillations were observed: these could be tuned by a differential IPTG concentration gradient and abolished by doxycycline. The proposed mammalian synthetic gene oscillator provides valuable insight into the gene expression regulatory processes leading to oscillatory behaviour, and has the potential to foster progress in future synthetic biology-based therapies.Open Acces

Mapping dynamical systems with distributed time delays to sets of ordinary differential equations

Real-world dynamical systems with retardation effects are described in general not by a single, precisely defined time delay, but by a range of delay times. It is shown that an exact mapping onto a set of $N+1$ ordinary differential equations exists when the respective delay distribution is given in terms of a gamma distribution with discrete exponents. The number of auxiliary variables one needs to introduce, $N$, is inversely proportional to the variance of the delay distribution. The case of a single delay is therefore recovered when $N\to\infty$. Using this approach, denoted the kernel series framework, we examine systematically how the bifurcation phase diagram of the Mackey-Glass system changes under the influence of distributed delays. We find that local properties, f.i. the locus of a Hopf bifurcation, are robust against the introduction of broadened memory kernels. Period-doubling transitions and the onset of chaos, which involve non-local properties of the flow, are found in contrast to be more sensible to distributed delays. Our results indicate that modeling approaches of real-world processes should take the effects of distributed delay times into account.Comment: 16 pages, 6 figure

A mathematical model of the sterol regulatory element binding protein 2 cholesterol biosynthesis pathway

Cholesterol is one of the key constituents for maintaining the cellular membrane and thus the integrity of the cell itself. In contrast high levels of cholesterol in the blood are known to be a major risk factor in the development of cardiovascular disease. We formulate a deterministic nonlinear ordinary differential equation model of the sterol regulatory element binding protein 2 (SREBP-2) cholesterol genetic regulatory pathway in an hepatocyte. The mathematical model includes a description of genetic transcription by SREBP-2 which is subsequently translated to mRNA leading to the formation of 3-hydroxy-3-methylglutaryl coenzyme A reductase (HMGCR), a main precursor of cholesterol synthesis. Cholesterol synthesis subsequently leads to the regulation of SREBP-2 via a negative feedback formulation. Parameterised with data from the literature, the model is used to understand how SREBP-2 transcription and regulation affects cellular cholesterol concentration. Model stability analysis shows that the only positive steady-state of the system exhibits purely oscillatory, damped oscillatory or monotic behaviour under certain parameter conditions. In light of our findings we postulate how cholesterol homestasis is maintained within the cell and the advantages of our model formulation are discussed with respect to other models of genetic regulation within the literature

Dynamics of moving average rules in a continuous-time financial market model

Within a continuous-time framework, this paper proposes a stochastic heterogeneous agent model (HAM) of financial markets with time delays to unify various moving average rules used in discrete-time HAMs. The time delay represents a memory length of a moving average rule in discrete-time HAMs. Intuitive conditions for the stability of the fundamental price of the deterministic model in terms of agents' behavior parameters and memory length are obtained. It is found that an increase in memory length not only can destabilize the market price, resulting in oscillatory market price characterized by a Hopf bifurcation, but also can stabilize an otherwise unstable market price, leading to stability switching as the memory length increases. Numerical simulations show that the stochastic model is able to characterize long deviations of the market price from its fundamental price and excess volatility and generate most of the stylized facts observed in financial markets. © 2010