Limit cycles in piecewise-affine gene network models with multiple interaction loops

In this paper we consider piecewise affine differential equations modeling gene networks. We work with arbitrary decay rates, and under a local hypothesis expressed as an alignment condition of successive focal points. The interaction graph of the system may be rather complex (multiple intricate loops of any sign, multiple thresholds...). Our main result is an alternative theorem showing that, if a sequence of region is periodically visited by trajectories, then under our hypotheses, there exists either a unique stable periodic solution, or the origin attracts all trajectories in this sequence of regions. This result extends greatly our previous work on a single negative feedback loop. We give several examples and simulations illustrating different cases

Cyclic negative feedback systems: what is the chance of oscillation?

International audienceMany biological oscillators have a cyclic structure consisting of negative feedback loops. In this paper, we analyze the impact that the addition of a positive or a negative self-feedback loop has on the oscillatory behaviour of the three negative feedback oscillators proposed by Tsai et al (Science 231:126-129, 2008) where, in contast with numerous oscillator models, the interactions between elements occur via the modulation of the degradation rates. Through analytical and computational studies we show that an additional self-feedback affects the dynamical behaviour. In the high cooperativity limit, i.e. for large Hill coefficients, we derive exact analytical conditions for oscillations and show that the relative location between the dissociation constants of the Hill functions and the ratio of kinetic parameters determines the possibility of oscillatory activities. We compute analytically the probability of oscillations for the three models and show that the smallest domain of periodic behaviour is obtained for the negative-plus-negative feedback system whereas the additional positive self-feedback loop does not modify significantly the chance to oscillate. We numerically investigate to what extent the properties obtained in the sharp situation applied in the smooth case. Results suggest that a switch-like coupling behaviour, a time-scale separation and a repressilator-type architecture with an even number of elements facilitate the emergence of sustained oscillations in biological systems. An additional positive self-feedback loop produces robustness and adaptability whereas an additional negative self-feedback loop reduces the chance to oscillate

Qualitative modeling in computational systems biology

The human body is composed of a large collection of cells,\the building blocks of life". In each cell, complex networks of biochemical processes contribute in maintaining a healthy organism. Alterations in these biochemical processes can result in diseases. It is therefore of vital importance to know how these biochemical networks function. Simple reasoning is not su±cient to comprehend life's complexity. Mathematical models have to be used to integrate information from various sources for solving numerous biomedical research questions, the so-called systems biology approach, in which quantitative data are scarce and qualitative information is abundant. Traditional mathematical models require quantitative information. The lack in ac- curate and su±cient quantitative data has driven systems biologists towards alternative ways to describe and analyze biochemical networks. Their focus is primarily on the anal- ysis of a few very speci¯c biochemical networks for which accurate experimental data are available. However, quantitative information is not a strict requirement. The mutual interaction and relative contribution of the components determine the global system dy- namics; qualitative information is su±cient to analyze and predict the potential system behavior. In addition, mathematical models of biochemical networks contain nonlinear functions that describe the various physiological processes. System analysis and parame- ter estimation of nonlinear models is di±cult in practice, especially if little quantitative information is available. The main contribution of this thesis is to apply qualitative information to model and analyze nonlinear biochemical networks. Nonlinear functions are approximated with two or three linear functions, i.e., piecewise-a±ne (PWA) functions, which enables qualitative analysis of the system. This work shows that qualitative information is su±cient for the analysis of complex nonlinear biochemical networks. Moreover, this extra information can be used to put relative bounds on the parameter values which signi¯cantly improves the parameter estimation compared to standard nonlinear estimation algorithms. Also a PWA parameter estimation procedure is presented, which results in more accurate parameter estimates than conventional parameter estimation procedures. Besides qualitative analysis with PWA functions, graphical analysis of a speci¯c class of systems is improved for a certain less general class of systems to yield constraints on the parameters. As the applicability of graphical analysis is limited to a small class of systems, graphical analysis is less suitable for general use, as opposed to the qualitative analysis of PWA systems. The technological contribution of this thesis is tested on several biochemical networks that are involved in vascular aging. Vascular aging is the accumulation of changes respon- sible for the sequential alterations that accompany advancing age of the vascular system and the associated increase in the chance of vascular diseases. Three biochemical networks are selected from experimental data, i.e., remodeling of the extracellular matrix (ECM), the signal transduction pathway of Transforming Growth Factor-¯1 (TGF-¯1) and the unfolded protein response (UPR). The TGF-¯1 model is constructed by means of an extensive literature search and con- sists of many state equations. Model reduction (the quasi-steady-state approximation) reduces the model to a version with only two states, such that the procedure can be visual- ized. The nonlinearities in this reduced model are approximated with PWA functions and subsequently analyzed. Typical results show that oscillatory behavior can occur in the TGF-¯1 model for speci¯c sets of parameter values. These results meet the expectations of preliminary experimental results. Finally, a model of the UPR has been formulated and analyzed similarly. The qualitative analysis yields constraints on the parameter values. Model simulations with these parameter constraints agree with experimental results