Piecewise smooth systems near a co-dimension 2 discontinuity manifold: can one say what should happen?

We consider a piecewise smooth system in the neighborhood of a co-dimension 2 discontinuity manifold $\Sigma$. Within the class of Filippov solutions, if $\Sigma$ is attractive, one should expect solution trajectories to slide on $\Sigma$. It is well known, however, that the classical Filippov convexification methodology is ambiguous on $\Sigma$. The situation is further complicated by the possibility that, regardless of how sliding on $\Sigma$ is taking place, during sliding motion a trajectory encounters so-called generic first order exit points, where $\Sigma$ ceases to be attractive. In this work, we attempt to understand what behavior one should expect of a solution trajectory near $\Sigma$ when $\Sigma$ is attractive, what to expect when $\Sigma$ ceases to be attractive (at least, at generic exit points), and finally we also contrast and compare the behavior of some regularizations proposed in the literature. Through analysis and experiments we will confirm some known facts, and provide some important insight: (i) when $\Sigma$ is attractive, a solution trajectory indeed does remain near $\Sigma$, viz. sliding on $\Sigma$ is an appropriate idealization (of course, in general, one cannot predict which sliding vector field should be selected); (ii) when $\Sigma$ loses attractivity (at first order exit conditions), a typical solution trajectory leaves a neighborhood of $\Sigma$; (iii) there is no obvious way to regularize the system so that the regularized trajectory will remain near $\Sigma$ as long as $\Sigma$ is attractive, and so that it will be leaving (a neighborhood of) $\Sigma$ when $\Sigma$ looses attractivity. We reach the above conclusions by considering exclusively the given piecewise smooth system, without superimposing any assumption on what kind of dynamics near $\Sigma$ (or sliding motion on $\Sigma$) should have been taking place.Comment: 19 figure

The effect of scale-free topology on the robustness and evolvability of genetic regulatory networks

We investigate how scale-free (SF) and Erdos-Renyi (ER) topologies affect the interplay between evolvability and robustness of model gene regulatory networks with Boolean threshold dynamics. In agreement with Oikonomou and Cluzel (2006) we find that networks with SFin topologies, that is SF topology for incoming nodes and ER topology for outgoing nodes, are significantly more evolvable towards specific oscillatory targets than networks with ER topology for both incoming and outgoing nodes. Similar results are found for networks with SFboth and SFout topologies. The functionality of the SFout topology, which most closely resembles the structure of biological gene networks (Babu et al., 2004), is compared to the ER topology in further detail through an extension to multiple target outputs, with either an oscillatory or a non-oscillatory nature. For multiple oscillatory targets of the same length, the differences between SFout and ER networks are enhanced, but for non-oscillatory targets both types of networks show fairly similar evolvability. We find that SF networks generate oscillations much more easily than ER networks do, and this may explain why SF networks are more evolvable than ER networks are for oscillatory phenotypes. In spite of their greater evolvability, we find that networks with SFout topologies are also more robust to mutations than ER networks. Furthermore, the SFout topologies are more robust to changes in initial conditions (environmental robustness). For both topologies, we find that once a population of networks has reached the target state, further neutral evolution can lead to an increase in both the mutational robustness and the environmental robustness to changes in initial conditions.Comment: 16 pages, 15 figure

In vivo single cell analysis reveals Gata2 dynamics in cells transitioning to hematopoietic fate

Cell fate is established through coordinated gene expression programs in individual cells. Regulatory networks that include the Gata2 transcription factor play central roles in hematopoietic fate establishment. Although Gata2 is essential to the embryonic development and function of hematopoietic stem cells that form the adult hierarchy, little is known about the in vivo expression dynamics of Gata2 in single cells. Here, we examine Gata2 expression in single aortic cells as they establish hematopoietic fate in Gata2Venus mouse embryos. Time-lapse imaging reveals rapid pulsatile level changes in Gata2 reporter expression in cells undergoing endothelial-to-hematopoietic transition. Moreover, Gata2 reporter pulsatile expression is dramatically altered in Gata2+/- aortic cells, which undergo fewer transitions and are reduced in hematopoietic potential. Our novel finding of dynamic pulsatile expression of Gata2 suggests a highly unstable genetic state in single cells concomitant with their transition to hematopoietic fate. This reinforces the notion that threshold levels of Gata2 influence fate establishment and has implications for transcription factor-related hematologic dysfunctions

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

Synchronizability of piecewise-linear maps

In this paper we discuss the phenomenon of synchronization of chaotic systems in the case of coupled piecewise linear (PWL) continuous and discontinuous one-dimensional maps. We present numerical results for two examples of coupled systems consisting of two PWL maps. We illustrate how the coupled system can achieve synchronization and discuss the nature of the bifurcation that occurs at a critical value of the coupling strength. We then determine this critical coupling using linear stability analysis. We discuss the effects of variation of the parameters of the PWL maps on the critical coupling and present different bifurcation scenarios obtained for different sets of values of these parameters. Finally, we discuss an extension of our work to the synchronizability of networks consisting of two or more PWL maps. We show how the synchronizability of a network of PWL maps can be improved by tuning the map parameters

Design and construction of a versatile synthetic network for bistable gene expression in mammalian systems

We constructed and modeled a novel synthetic network which may be able to exhibit bistable expression of a reporter gene in mammalian cells. This network is based on an aptamer-fused short-hairpin RNA (shRNA) directed against a single mRNA encoding both a EGFP reporter gene and the repressor tTR-KRAB, which, in turn, represses transcription of the shRNA. The activity of the shRNA can be controlled by an inducer molecule (theophylline) which prevents the aptamer-fused shRNA to be properly processed. Repression of the tTR-KRAB can be relieved by treatment with doxycyline. This reciprocal negative feed-back loop can exhibit a bistable response, as shown through the mathematical analysis performed here. Specifically, the network can be controlled to induce sustained expression of a shRNA, or the reporter gene, with a transient input of two different inducer molecules

Modeling RNA interference in mammalian cells

Our model has a simple mathematical form, amenable to analytical investigations and a small set of parameters with an intuitive physical meaning, that makes it a unique and reliable mathematical tool. The findings here presented will be a useful instrument for better understanding RNAi biology and as modelling tool in Systems and Synthetic Biology