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

Periodic solutions of piecewise affine gene network models: the case of a negative feedback loop

In this paper the existence and unicity of a stable periodic orbit is proven, for a class of piecewise affine differential equations in dimension 3 or more, provided their interaction structure is a negative feedback loop. It is also shown that the same systems converge toward a unique stable equilibrium point in dimension 2. This extends a theorem of Snoussi, which showed the existence of these orbits only. The considered class of equations is usually studied as a model of gene regulatory networks. It is not assumed that all decay rates are identical, which is biologically irrelevant, but has been done in the vast majority of previous studies. Our work relies on classical results about fixed points of monotone, concave operators acting on positive variables. Moreover, the used techniques are very likely to apply in more general contexts, opening directions for future work

Symbolic numeric analysis of attractors in randomly generated piecewise affine models of gene networks

International audienceThe goal of this paper is to present and experiment the computer aided analysis of phase portraits of some ordinary differential equations. The latter are piecewise affine, and have been primitively introduced as coarse-grained models of gene regulatory networks. Their simple formulation allows for numerical investigation, but their typical phase portrait is still largely unknown. They have been shown to present all the main aspects of nonlinear dynamics, including chaos. But it is still of interest to simulate random versions of these models, and to count and classify their attractors. This paper presents algorithms that allow for an automatic treatment of this kind, and apply it to four-dimensional sample systems. Contrary to previous studies, the latter have several thresholds in each direction, a fact whose consequences on the number and nature of attractors is discussed

How to control a biological switch: a mathematical framework for the control of piecewise affine models of gene networks

This article introduces preliminary results on the control of gene networks, in the context of piecewise-affine models. We propose an extension of this well-documented class of models, where some input variables can affect the main terms of the equations, with a special focus on the case of affine dependence on inputs. This class is illustrated with the example of two genes inhibiting each other. This example has been observed on real biological systems, and is known to present a bistable switch for some parameter values. Here, the parameters can be controlled. Some generic control problems are proposed, which are qualitative, respecting the coarse-grained nature of piecewise-affine models. Piecewise constant feedback laws that solve these control problems are characterized in terms of affine inequalities, and can even be computed explicitly for a subclass of inputs. The latter is characterized by the condition that each state variable of the system is affected by at most one input variable. These general feedback laws are then applied to the two dimensional example, showing how to control this system toward various behaviours, including the usual bi-stability, as well as situations involving a unique global equilibrium

Qualitative control of periodic solutions in piecewise affine systems; application to genetic networks

Hybrid systems, and especially piecewise affine (PWA) systems, are often used to model gene regulatory networks. In this paper we elaborate on previous work about control problems for this class of models, using also some recent results guaranteeing the existence and uniqueness of limit cycles, based solely on a discrete abstraction of the system and its interaction structure. Our aim is to control the transition graph of the PWA system to obtain an oscillatory behaviour, which is indeed of primary functional importance in numerous biological networks; we show how it is possible to control the appearance or disappearance of a unique stable limit cycle by hybrid qualitative action on the degradation rates of the PWA system, both by static and dynamic feedback, i.e. the adequate coupling of a controlling subnetwork. This is illustrated on two classical gene network modules, having the structure of mixed feedback loops

An introduction to modelling flower primordium initiation

International audienceIn this chapter we present models of processes involved in the initiation and development of a flower. In the first section, we briefly present models of hormonal transport. We focus on two key aspects of flower development, namely the initiation, due to the periodic local accumulation of auxin (a plant hormone) near at the plant apex, and the genetic regulation of its development. In the first section, we described the main assumptions about auxin transport that have been proposed and tested in the literature. We show how the use of models make it possible to test assumptions expressed in terms of local cell-to-cell interaction rules and to check if they lead to patterning in the growing tissue consistent with observation.Then, we investigated gene regulatory networks that controls the initial steps of flower development and differentiation. In a simplified form, this network contains a dozen of actors interacting with each other in space and time. The understanding of such a complex system here also requires a modeling approach in order to quantify these interactions and analyze their properties. We briefly present the two main formalisms that are used to model GRN: the Boolean and the ODE formalisms. We illustrate on a sub-module of the flower GRN both types of models and discuss their main advantages and drawbacks. We show how manipulations of the network models can be used to make predictions corresponding to possible biological manipulations of the GRN (e.g. loss-of-function mutants).Throughout the chapter, we highlight specific mathematical topics of particular interest to the development of the ideas developed in the different sections in separated boxes (called Math-boxes). The reading of these boxes is relatively independent of the main text

Plant hormone signaling during development: insights from computational models

International audienceRecent years have seen an impressive increase in our knowledge of the topology of plant hormone signaling networks. The complexity of these topologies has motivated the development of models for several hormones to aid understanding of how signaling networks process hormonal inputs. Such work has generated essential insights into the mechanisms of hormone perception and of regulation of cellular responses such as transcription in response to hormones. In addition, modeling approaches have contributed significantly to exploring how spatio-temporal regulation of hormone signaling contributes to plant growth and patterning. New tools have also been developed to obtain quantitative information on hormone distribution during development and to test model predictions, opening the way for quantitative understanding of the developmental roles of hormones. ⺠Plant hormone signaling pathways exhibit complex topologies. ⺠Computational models predict the dynamics of hormone signaling. ⺠Modeling provides key insights on the role of hormones during growth and development. ⺠New tools allow for a quantitative understanding of hormone signaling

Transition probabilities for piecewise affine models of genetic networks

International audienceIn the piecewise affine framework, trajectories evolve among hyperrectangles in the state space. A qualitative description of the dynamics - useful for models of genetic networks - can be obtained by viewing each hyperrectangle as a node in a discrete system, so that trajectories follow a path in a transition graph. In this paper, a probabilistic interpretation is given for the transition between two nodes A and B, based on the volume of the initial conditions on hyperrectangle A whose trajectories cross to B. In an example consisting of two intertwinned negative loops, this probabilistic interpretation is used to predict the most likely periodic orbit given a set of parameters, or to find parameters such that the system yields a desired periodic orbit with a high probability

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

Understanding Microwave Heating in Biomass-Solvent Systems

A new mechanism is proposed to provide a viable physical explanation for the action of microwaves in solvent extraction processes. The key innovation is Temperature-Induced Diffusion, a recently-demonstrated phenomenon that results from selective heating using microwaves. A mechanism is presented which incorporates microwave heating, cellular expansion, heat transfer and mass transfer, all of which affect the pressure of cell structures within biomass. The cell-pressure is modelled with time across a range of physical and process variables, and compared with the expected outputs from the existing steam-rupture theory. It is shown that steam-rupture is only possible at the extreme fringes of realistic physical parameters, but Temperature-Induced Diffusion is able to explain cell-rupture across a broad and realistic range of physical parameters and heating conditions. Temperature-Induced Diffusion is the main principle that governs microwave-assisted extraction, and this paves the way to being able to select processing conditions and feedstocks based solely on their physical properties. Graphical abstract Keywords Microwave processing, heat transfer, mass transfer, plant cell rupture, cellular expansion mechanics, solvent extractio