Modeling the bioconversion of polysaccharides in a continuous reactor: A case study of the production of oligogalacturonates by Dickeya dadantii.

In the quest for a sustainable economy of the Earth's resources and for renewable sources of energy, a promising avenue is to exploit the vast quantity of polysaccharide molecules contained in green wastes. To that end, the decomposition of pectin appears to be an interesting target because this polymeric carbohydrate is abundant in many fruit pulps and soft vegetables. To quantitatively study this degradation process, here we designed a bioreactor that is continuously fed with de-esterified pectin (PGA). Thanks to the pectate lyases produced by bacteria cultivated in the vessel, the PGA is depolymerized into oligogalacturonates (UGA), which are continuously extracted from the tank. A mathematical model of our system predicted that the conversion efficiency of PGA into UGA increases in a range of coefficients of dilution until reaching an upper limit where the fraction of UGA that is extracted from the bioreactor is maximized. Results from experiments with a continuous reactor hosting a strain of the plant pathogenic bacterium Dickeya dadantii and in which the dilution coefficients were varied quantitatively validated the predictions of our model. A further theoretical analysis of the system enabled an a priori comparison of the efficiency of eight other pectate lyase-producing microorganisms with that of D. dadantii Our findings suggest that D. dadantii is the most efficient microorganism and therefore the best candidate for a practical implementation of our scheme for the bioproduction of UGA from PGA.

Intrinsic feedbacks in MAPK signaling cascades lead to bistability and oscillations.

Previous studies have demonstrated that double phosphorylation of a protein can lead to bistability if some conditions are fulfilled. It was also shown that the signaling behavior of a covalent modification cycle can be quantitatively and, more importantly, qualitatively modified when this cycle is coupled to a signaling pathway as opposed to being isolated. This property was named retroactivity. These two results are studied together in this paper showing the existence of interesting phenomena--oscillations and bistability--in signaling cascades possessing at least one stage with a double-phosphorylation cycle as in MAPK cascades.

A nested bistable module within a negative feedback loop ensures different types of oscillations in signaling systems.

In this article, we consider a double phosphorylation cycle, a ubiquitous signaling component, having the ability to display bistability, a behavior strongly related to the existence of positive feedback loops. If this component is connected to other signaling elements, it very likely undergoes some sort of protein-protein interaction. In several cases, these interactions result in a non-explicit negative feedback effect, leading to interlinked positive and negative feedbacks. This combination was studied in the literature as a way to generate relaxation-type oscillations. Here, we show that the two feedbacks together ensure two types of oscillations, the relaxation-type ones and a smoother type of oscillations functioning in a very narrow range of frequencies, in such a way that outside that range, the amplitude of the oscillations is severely compromised. Even more, we show that the two feedbacks are essential for both oscillatory types to emerge, and it is their hierarchy what determines the type of oscillation at work. We used bifurcation analyses and amplitude vs. frequency curves to characterize and classify the oscillations. We also applied the same ideas to another simple model, with the goal of generalizing what we learned from signaling models. The results obtained display the wealth of oscillatory dynamics that exists in a system with a bistable module nested within a negative feedback loop, showing how to transition between different types of oscillations and other dynamical behaviors such as excitability. Our work provides a framework for the study of other oscillatory systems based on bistable modules, from simple two-component models to more complex examples like the MAPK cascade and experimental cases like cell cycle oscillators.

Pseudo-nullclines enable the analysis and prediction of signaling model dynamics.

A powerful method to qualitatively analyze a 2D system is the use of nullclines, curves which separate regions of the plane where the sign of the time derivatives is constant, with their intersections corresponding to steady states. As a quick way to sketch the phase portrait of the system, they can be sufficient to understand the qualitative dynamics at play without integrating the differential equations. While it cannot be extended straightforwardly for dimensions higher than 2, sometimes the phase portrait can still be projected onto a 2-dimensional subspace, with some curves becoming pseudo-nullclines. In this work, we study cell signaling models of dimension higher than 2 with behaviors such as oscillations and bistability. Pseudo-nullclines are defined and used to qualitatively analyze the dynamics involved. Our method applies when a system can be decomposed into 2 modules, mutually coupled through 2 scalar variables. At the same time, it helps track bifurcations in a quick and efficient manner, key for understanding the different behaviors. Our results are both consistent with the expected dynamics, and also lead to new responses like excitability. Further work could test the method for other regions of parameter space and determine how to extend it to three-module systems.

Toward a quantitative modeling of the synthesis of the pectate lyases, essential virulence factors in Dickeya dadantii.

A dynamic mathematical model has been developed and validated to describe the synthesis of pectate lyases (Pels), the major virulence factors in Dickeya dadantii. This work focuses on the simultaneous modeling of the metabolic degradation of pectin by Pel enzymes and the genetic regulation of pel genes by 2-keto-3-deoxygluconate (KDG), a catabolite product of pectin that inactivates KdgR, one of the main repressors of pel genes. This modeling scheme takes into account the fact that the system is composed of two time-varying compartments: the extracellular medium, where Pel enzymes cleave pectin into oligomers, and the bacterial cytoplasm where, after internalization, oligomers are converted to KDG. Using the quasi-stationary state approximations, the model consists of some nonlinear differential equations for which most of the parameters could be estimated from the literature or from independent experiments. The few remaining unknown parameters were obtained by fitting the model equations against a set of Pel activity data. Model predictions were verified by measuring the time courses of bacterial growth, Pel production, pel mRNA accumulation, and pectin consumption under various growth conditions. This work reveals that pectin is almost totally consumed before the burst of Pel production. This paradoxical behavior can be interpreted as an evolutionary strategy to control the diffusion process so that as soon as a small amount of pectin is detected by the bacteria in its surroundings, it anticipates more pectin to come. The model also predicts the possibility of bistable steady states in the presence of constant pectin compounds.

Signaling cascades transmit information downstream and upstream but unlikely simultaneously.

BACKGROUND: Signal transduction is the process through which cells communicate with the external environment, interpret stimuli and respond to them. This mechanism is controlled by signaling cascades, which play the role of intracellular transmitter, being able to transmit biochemical information between cell membrane and nucleus. In theory as well as in practice, it has been shown that a perturbation can propagate upstream (and not only downstream) a cascade, by a mechanism known as retroactivity. This study aims to compare the conditions on biochemical parameters which favor one or the other direction of signaling in such a cascade. RESULTS: From a mathematical point of view, we show that the steady states of a cascade of arbitrary length n are described by an iterative map of second order, meaning that the cascade tiers are actually coupled three-by-three. We study the influence of the biochemical parameters in the control of the direction of transmission - upstream and/or downstream - along a signaling cascade. A numerical and statistical approach, based on the random scan of parameters describing a 3-tier signaling cascade, provides complementary findings to the analytical study. In particular, computing the likelihood of parameters with respect to various signaling regimes, we identify conditions on biochemical parameters which enhance a specific direction of propagation corresponding to forward or retro-signaling regimes. A compact graphical representation is designed to relay the gist of these conditions. CONCLUSIONS: The values of biochemical parameters such as kinetic rates, Michaelis-Menten constants, total concentrations of kinases and of phosphatases, determine the propensity of a cascade to favor or impede downstream or upstream signal transmission. We found that generally there is an opposition between parameter sets favoring forward and retro-signaling regimes. Therefore, on one hand our study supports the idea that in most cases, retroactive effects can be neglected when a cascade which is efficient in forward signaling, is perturbed by an external ligand inhibiting the activation at some tier of the cascade. This result is relevant for therapeutic methodologies based on kinase inhibition. On the other hand, our study highlights a less-known part of the parameter space where, although the forward signaling is inefficient, the cascade can interestingly act as a retro-signaling device.

Bistable reaction-diffusion systems can have robust zero-velocity fronts.

We show that for a class of bistable reaction-diffusion systems, zero-velocity fronts can be robust in the singular limit where one of the diffusion coefficients vanishes. In this case, stationary fronts can persist along variations of the system parameters. This property contrasts with the standard result that the front velocity v(&mgr;), expressed as a function of a control parameter &mgr;, is zero only at some isolated values &mgr;(0), and thus not giving robustness to zero-velocity fronts when &mgr; is varied. (c) 2000 American Institute of Physics.

Modelling the onset of virulence in pathogenic bacteria.

Bacterial virulence is a multifactorial process. In this chapter, we review some known mechanisms used by bacteria to trigger their production of virulence factors. We develop the idea that although the onset of virulence shows up an abrupt transition, the modelling of this dynamics can be classified in two qualitatively distinct infectious transitions which are respectively called "shift" or "switch." We review methods enabling one to determine the types of behaviour that can be exhibited by a given model and we consider applications in three cases of virulence factor regulation. We conclude that in most cases a "successful" infection would require that the onset of virulence follows an irreversible switch behaviour.

Retroactive signaling in short signaling pathways.

In biochemical signaling pathways without explicit feedback connections, the core signal transduction is usually described as a one-way communication, going from upstream to downstream in a feedforward chain or network of covalent modification cycles. In this paper we explore the possibility of a new type of signaling called retroactive signaling, offered by the recently demonstrated property of retroactivity in signaling cascades. The possibility of retroactive signaling is analysed in the simplest case of the stationary states of a bicyclic cascade of signaling cycles. In this case, we work out the conditions for which variables of the upstream cycle are affected by a change of the total amount of protein in the downstream cycle, or by a variation of the phosphatase deactivating the same protein. Particularly, we predict the characteristic ranges of the downstream protein, or of the downstream phosphatase, for which a retroactive effect can be observed on the upstream cycle variables. Next, we extend the possibility of retroactive signaling in short but nonlinear signaling pathways involving a few covalent modification cycles.

Simple molecular networks that respond optimally to time-periodic stimulation.

BACKGROUND: Bacteria or cells receive many signals from their environment and from other organisms. In order to process this large amount of information, Systems Biology shows that a central role is played by regulatory networks composed of genes and proteins. The objective of this paper is to present and to discuss simple regulatory network motifs having the property to maximize their responses under time-periodic stimulations. In elucidating the mechanisms underlying these responses through simple networks the goal is to pinpoint general principles which optimize the oscillatory responses of molecular networks. RESULTS: We took a look at basic network motifs studied in the literature such as the Incoherent Feedforward Loop (IFFL) or the interlerlocked negative feedback loop. The former is also generalized to a diamond pattern, with network components being either purely genetic or combining genetic and signaling pathways. Using standard mathematics and numerical simulations, we explain the types of responses exhibited by the IFFL with respect to a train of periodic pulses. We show that this system has a non-vanishing response only if the inter-pulse interval is above a threshold. A slight generalisation of the IFFL (the diamond) is shown to work as an ideal pass-band filter. We next show a mechanism by which average of oscillatory response can be maximized by bursting temporal patterns. Finally we study the interlerlocked negative feedback loop, i.e. a 2-gene motif forming a loop where the nodes respectively activate and repress each other, and show situations where this system possesses a resonance under periodic stimulation. CONCLUSION: We present several simple motif designs of molecular networks producing optimal output in response to periodic stimulations of the system. The identified mechanisms are simple and based on known network motifs in the literature, so that that they could be embodied in existing organisms, or easily implementable by means of synthetic biology. Moreover we show that these designs can be studied in different contexts of molecular biology, as for example in genetic networks or in signaling pathways.