Patrick De Leenheer

VIRUS DYNAMICS: A GLOBAL ANALYSIS*

Exploiting the fact that standard models of within-host viral infections of target cell populations by HIV, developed by Perelson and Nelson [SIAM Rev., 41 (1999), pp. 3--44] and Nowak and May [Vir...

Stabilization of positive linear systems

We consider stabilization of equilibrium points of positive linear systems which are in the interior of the first orthant. The existence of an interior equilibrium point implies that the system matrix does not possess eigenvalues in the open right half plane. This allows to transform the problem to the stabilization problem of compartmental systems, which is known and for which a solution has been proposed already. We provide necessary and sufficient conditions to solve the stabilization problem by means of affine state feedback. A class of stabilizing feedbacks is given explicitly.

Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates

This paper derives new results for certain classes of chemical reaction networks, linking structural to dynamical properties. In particular, it investigates their monotonicity and convergence under the assumption that the rates of the reactions are monotone functions of the concentrations of their reactants. This is satisfied for, yet not restricted to, the most common choices of the reaction kinetics such as mass action, Michaelis-Menten and Hill kinetics. The key idea is to find an alternative representation under which the resulting system is monotone. As a simple example, the paper shows that a phosphorylation/dephosphorylation process, which is involved in many signaling cascades, has a global stability property. We also provide a global stability result for a more complicated example that describes a regulatory pathway of a prevalent signal transduction module, the MAPK cascade.

Quorum activation at a distance: spatiotemporal patterns of gene regulation from diffusion of an autoinducer signal.

Quorum sensing (QS) bacteria regulate gene expression collectively by exchanging diffusible signal molecules known as autoinducers. Although QS is often studied in well-stirred laboratory cultures, QS bacteria colonize many physically and chemically heterogeneous environments where signal molecules are transported primarily by diffusion. This raises questions of the effective distance range of QS and the degree to which colony behavior can be synchronized over such distances. We have combined experiments and modeling to investigate the spatiotemporal patterns of gene expression that develop in response to a diffusing autoinducer signal. We embedded a QS strain in a narrow agar lane and introduced exogenous autoinducer at one terminus of the lane. We then measured the expression of a QS reporter as a function of space and time as the autoinducer diffused along the lane. The diffusing signal readily activates the reporter over distances of ~1 cm on time scales of ~10 h. However, the patterns of activation are qualitatively unlike the familiar spreading patterns of simple diffusion, as the kinetics of response are surprisingly insensitive to the distance the signal has traveled. We were able to reproduce these patterns with a mathematical model that combines simple diffusion of the signal with logistic growth of the bacteria and cooperative activation of the reporter. In a wild-type QS strain, we also observed the propagation of a unique spatiotemporal excitation. Our results show that a chemical signal transported only by diffusion can be remarkably effective in synchronizing gene expression over macroscopic distances.

Extension of the Perron--Frobenius Theorem to Homogeneous Systems

This paper deals with a particular class of positive systems. The state components of a positive system are positive or zero for all positive times. These systems are often encountered in applied areas such as chemical engineering or biology. It is shown that for this particular class the first orthant contains an invariant ray in its interior. An invariant ray generalizes the concept of an eigenvector of linear systems to nonlinear homogeneous systems. Then sufficient conditions for uniqueness of this ray are given. The main result states that the vector field on an invariant ray determines the stability properties of the zero solution with respect to initial conditions in the first orthant. The asymptotic behavior of the solutions is examined. Finally, we compare our results to the Perron--Frobenius theorem, which gives a detailed picture of the dynamical behavior of positive linear systems.

Multistrain virus dynamics with mutations: a global analysis

We consider within-host virus models with n >or= 2 strains and allow mutation between the strains. If there is no mutation, a Lyapunov function establishes global stability of the steady state corresponding to the fittest strain. For small perturbations, this steady state persists, perhaps with small concentrations of some or all other strains, depending on the connectivity of the graph describing all possible mutations. Moreover, using a perturbation result due to Smith & Waltman (1999), we show that this steady state also preserves global stability.

Brief Stabilization of positive systems with first integrals

Positive systems possessing first integrals are considered. These systems frequently occur in applications. This paper is devoted to two stabilization problems. The first is concerned with the design of feedbacks to stabilize a given level set. Secondly, it is shown that the same feedback allows to globally stabilize an equilibrium point if it is asymptotically stable with respect to initial conditions in its level set. Two examples are provided and the results are compared with those in the literature.

Dynamical Models Explaining Social Balance and Evolution of Cooperation

Social networks with positive and negative links often split into two antagonistic factions. Examples of such a split abound: revolutionaries versus an old regime, Republicans versus Democrats, Axis versus Allies during the second world war, or the Western versus the Eastern bloc during the Cold War. Although this structure, known as social balance, is well understood, it is not clear how such factions emerge. An earlier model could explain the formation of such factions if reputations were assumed to be symmetric. We show this is not the case for non-symmetric reputations, and propose an alternative model which (almost) always leads to social balance, thereby explaining the tendency of social networks to split into two factions. In addition, the alternative model may lead to cooperation when faced with defectors, contrary to the earlier model. The difference between the two models may be understood in terms of the underlying gossiping mechanism: whereas the earlier model assumed that an individual adjusts his opinion about somebody by gossiping about that person with everybody in the network, we assume instead that the individual gossips with that person about everybody. It turns out that the alternative model is able to lead to cooperative behaviour, unlike the previous model.

Failure of antibiotic treatment in microbial populations

The tolerance of bacterial populations to biocidal or antibiotic treatment has been well documented in both biofilm and planktonic settings. However, there is still very little known about the mechanisms that produce this tolerance. Evidence that small, non-mutant subpopulations of bacteria are not affected by an antibiotic challenge has been accumulating and provides an attractive explanation for the failure of typical dosing protocols. Although a dosing challenge can kill the susceptible bacteria, the remaining persister cells can serve as a source of population regrowth. We give a condition for the failure of a periodic dosing protocol for a general chemostat model, which supports the simulations of an earlier, more specialized batch model. Our condition implies that the treatment protocol fails globally, in the sense that a mixed bacterial population will ultimately persist above a level that is independent of the initial composition of the population. We also give a sufficient condition for treatment success, at least for initial population compositions near the steady state of interest, corresponding to bacterial washout. Finally, we investigate how the speed at which the bacteria are wiped out depends on the duration of administration of the antibiotic. We find that this dependence is not necessarily monotone, implying that optimal dosing does not necessarily correspond to continuous administration of the antibiotic. Thus, genuine periodic protocols can be more advantageous in treating a wide variety of bacterial infections.

A Petri Net Approach to Persistence Analysis in Chemical Reaction Networks

A positive dynamical system is said to be persistent if every solution that starts in the interior of the positive orthant does not approach the boundary of this orthant. For chemical reaction networks and other models in biology, persistence represents a non-extinction property: if every species is present at the start of the reaction, then no species will tend to be eliminated in the course of the reaction. This paper provides checkable necessary as well as sufficient conditions for persistence of chemical species in reaction networks, and the applicability of these conditions is illustrated on some examples of relatively high dimension which arise in molecular biology. More specific results are also provided for reactions endowed with mass-action kinetics. Overall, the results exploit concepts and tools from Petri net theory as well as ergodic and recurrence theory.