Section: New Results
Control and observation of continuous bioreactors
The team has worked these past years on several approaches for the control of perfectly stirred bioreactor. The stabilization is often achieved with the help of the dilution rate as a manipulated variable (which requires the use of an upstream tank).
Last year, the team has proposed to stabilize reactors with the help of by-pass and recirculation loops as control variables instead of the input dilution rate (see the main contribution  ). This year, the team has investigated the stabilization of an unstable process under constant dilution rate, considering the addition of a new species (”biological control”). A well chosen species can make the new system globally asymptotically stable on the positive orthant (i.e. the new species needs to be present at initial time, but is asymptotically washed-out). The team has obtained a new result under different removal rates, which typically occurs in reactors with membranes.
Most of the time, the concentration of input nutrient is supposed to be known (possibly time varying), which is not realistic from an applied point of view. This year, the team has proposed a new kind of unknown inputs observers for the reconstruction of the input concentration, when only the biomass or the substrate concentration inside the tank is measured. The unknown input is assumed to be piecewise constant or periodic with known period. The proposed observer possesses variable gains and its convergence is established with the help of Lyapunov transformations. This is a work pursued in cooperation with Professors G. Acuna (Universidad de Santiago, Chile) and D. Dochain (CESAME, Louvain-la-Neuve, Belgium).
The team has also worked on an important class of problems involving the tracking of prescribed trajectories in the chemostat model with one limiting substrate. In [Oops!] , a prescribed oscillatory behavior for a chemostat with one species is generated by an appropriate choice of a time-varying dilution rate, and the global uniform asymptotic stability of the behavior is proved by a Lyapunov approach. In [Oops!] , new tracking results for chemostats with two species , based on Lyapunov function methods, have been presented. In particular, in a first step, we used a linear feedback control of the dilution rate and an appropriate time-varying substrate input concentration to produce a locally exponentially stable oscillatory behavior. In a second step, we generated a globally stable oscillatory reference trajectory for the species concentrations, using a nonlinear feedback control depending on the dilution rate and the substrate input concentration. This guarantees that all trajectories for the closed loop chemostat dynamics are attracted to the reference trajectory. Finally, we constructed an explicit Lyapunov function for the corresponding global error dynamics.
Notice that our works differ from the earlier results because we use Lyapunov function methods to globally feedback stabilizing a predefined oscillating behavior. Notice also that our control laws possess the required properties of positiveness and boundedness.
Optimal control of fed-batch reactors
Participant : Alain Rapaport.
The time optimal control of fed-batch chemostats with one reaction involving one substrate and one biomass has been solved by J. Moreno in 1999 using a technique based on Green's theorem. The optimal trajectories correspond to “most rapid approach paths” toward
the target, when the growth function is a monotonic function,
a singular arc, when the growth presents an inhibition for large concentrations of nutrient.
The optimal controls are of two possible types: “bang-bang” (i.e. no feeding or feeding at the maximal rate) and singular ones.
When the growth function presents more than one inhibition, we have proposed a numerical technique for the approximation of the optimal trajectories. This technique is based on a regularization of the problem with an additional control such that the optimal solution admits a regular synthesis, that can be determined numerically by the Maximum Principle. The solution of the original problem may present several singular arcs [Oops!] .
We have also carried out a work initiated last year within the cooperation program with Chile (see Section 8.1.2 ). Two kinds of extensions have been considered in this work :
the consideration of unbounded or “impulsive”controls. A bounded measurable control can be assimilated to a device that tunes the speed of a pump over a certain range, while an unbounded control can be assimilated to an arbitrary fast dilution with respect to the biological time scale,
the consideration of several species of micro-organisms in competition for a single substrate.
Contrary to the one species case, we have shown that with more than one reaction, the optimal trajectories are not necessarily most rapid approach paths. This can be explained by the fact that the argumentation based of Green's theorem is valid only for planar systems. Instead, we have proposed a characterization of the optimal solution in terms of a set of two variational inequalities of Hamilton-Jacobi-Bellman type [Oops!] . This approach has been inspired by recent theoretical results on turnpike optimality in calculus of variations problems [Oops!] . For monotonic growth functions, the optimal solution consists of an “immediate one impulse” or a “delayed one impulse” strategy. As a particular case, we generalize the result of Moreno with one species to the impulsive framework, but the determination of an optimal synthesis for non-monotonic growth functions is still an open problem.
Optimal design of bio-processes
Optimal interconnections of bioreactors have been widely studied in the literature during these five last years, in terms of minimizing the total retention time at steady state. Most of the studies deals with single species.
The team investigates the link between optimal configurations and its ability to sustain a biodiversity, for instance in presence of an “invader”. More precisely, the coexistence of several species of microorganisms in an interconnected bioreactor composed of two tanks in series has been studied in [Oops!] . It has been shown that at most only one species could survive in an optimally designed bio-system. In the presence of an invader only two cases may arise: either the invader takes the place of the actual species or the invader simply cannot survive and is washed out. This result may have important consequences in medical or agro-food systems in which it is expected that the processes cannot be invaded by pathogen species.
Analysis of an SSCP profile
Fingerprinting profiles yield snapshots of the microbial community structure. Extracting quantitative information from them has turned out to be a non-trivial task. In particular, microbial communities with a large diversity, such as those used for wastewater processing, have complex profiles, consisting of a broad background with a number of sharp peaks on top of it. We established previously that the relative importance of the background signal contains crucial diversity information, although this background is often neglected in the current analysis of fingerprints. This result, obtained from a simulation study of fingerprinting profiles, was extended in different steps:
We found that different communities all with a given diversity index have a similar background signal. This index is the so-called Simpson diversity index, defined as the sum of the squares of the relative abundances, and which can be interpreted as the probability of finding individuals of the same species when randomly sampling two individuals from the community. This result suggests that the Simpson diversity index is clearly encoded in the profiles, and should be easy to extract.
Next, we constructed an estimator for the Simpson diversity index from fingerprinting profiles. Its performance was tested on simulated fingerprints, and compared to existing diversity estimators (for the number of species, for the Shannon diversity index and for the Simpson diversity index). Whereas the latter estimators all exhibit saturation characteristics, the Simpson diversity index estimator has linear characteristics up to the highest diversity levels observed experimentally (see next item).
We applied the Simpson diversity index estimator to several hundreds of experimental fingerprinting profiles. The estimated diversity values allow to distinctly classify microbial communities based on their origin: (from low to high diversity:) aquatic systems, digestive systems (human, animal), bioreactors, microbial systems in soil. Moreover, the logarithmic scale of the Simpson diversity index spreads evenly the classes of communities, indicating that this scale is particularly practical.
We developed an application of the Simpson diversity index for metagenomic studies. Metagenomics looks at a microbial community as a huge pool of genes with possibly interesting functionality. By repeatedly sampling genes and checking their properties, one tries to encounter these interesting genes. We showed that the Simpson diversity index allows us to estimate the number of samples where the probability of finding previously checked genes becomes appreciable. The Simpson diversity index estimator could therefore be used for the judicious selection of the most interesting communities for metagenomic studies.
Neutral community model for microbial ecology
Participant : Bart Haegeman.
Hubbell's neutral model describes the dynamics of an ecological community in terms of random birth, death and immigration events, attributing equivalent characteristics to all species. Although the absurd simplicity of these assumptions, the model predictions of species-abundance and species-area relationships are remarkably accurate for some ecosystems. The neutral model seems to capture some essential features of community dynamics, without requiring quantitative information about species behavior, unlike other ecological models (Lotka-Volterra equations, chemostat model, ...). This is particularly attractive for microbial ecology, as quantitative data (interaction strengths, growth functions, ...) about the thousands of species making up the community is completely missing.
However, Hubbell's model as such cannot be applied in microbial ecology. First, the model assumes that the number of individuals does not vary as a function of time. This would exclude, e.g., the description of a microbial community recovering after the addition of a toxic substance. We have proposed a model extension that deals with this shortcoming. Next, as species abundances cannot be estimated for microbial communities, models should be expressed in terms of global static or dynamical properties, like diversity or divergence. We have computed the dynamics of the Simpson diversity index under neutral dynamics. In this way, we have obtained a theoretical framework based on Hubbell's model, but appropriately modified to be testable on microbial communities.
Models of competition for one resource
We are interested by the system :
which represents the competition of N consumers on one resource. The variables xi represent the concentration of the consumers and s the concentration of the resource. The growth rate i( s, x) is “density dependent" which means that it depends not only of the resource s but also of the concentration of the various consumers ; the function si( s, x) is increasing and the functions xji( s, x1, x2, ..., xj, ... xN) are decreasing ; this last assumption expresses some kind of competition exerted by the species j on the species i . The function f represents the dynamics of the resource alone and the di 's are “disparition" terms caused either by mortality and/or migration out of the system under consideration. Under these general assumptions not much is known on system ( 1 ).
In the note  we proved that, in the particular case (corresponding to the chemostat model) :
there exists a unique Globally Asymptotically Stable (G.A.S.) equilibrium [ se, xe] and :
where the “growth threshold" si* is defined as the first value of s such that the supremum of the function xii( s, xi) is greater than d .
We conjecture that this result is still true under the weakest assumption that :
the function f(·) is strictly decreasing,
the “growth threshold" si* is the first value of s such that the supremum of the function xii( s, xi) is greater than di
and this conjecture is strongly supported by simulations. In the paper [Oops!] we give sufficient conditions for a unique G.A.S. equilibrium which relax some assumptions (the function f is just decreasing, but assumptions on the i 's are not very natural) and show that f decreasing is essential. The paper [Oops!] gives a new set of sufficient conditions which confirms the conjecture. Paper [Oops!] gives an alternative proof for G.A.S. under the hypothesis of the note  .
We are currently studying for the model :
where the vi 's represent a species of virus specific of each species of bacteria. Some results are already present in the literature but we are mainly interested in the case where the number N of species is very large, which seems to be new.
Markov chain Monte Carlo
Markov chain Monte Carlo (MCMC) algorithms allow us to draw samples from a probability distribution known up to a multiplicative constant. They consist in sequentially simulating a single Markov chain whose limit distribution is . Many techniques exist to speed up the convergence towards the target distribution by improving the mixing properties of the chain. An alternative is to run many Markov chains in parallel. The simplest multiple chain algorithm would be to make use of parallel independent chains. The recommendations concerning this idea seem contradictory in the literature, as shown by the many short runs vs. one long run debate. It can be noted that independent parallel chains may be a poor idea: among these chains some may not converge. Therefore one long chain could be preferable to many short ones. Moreover, many parallel independent chains can artificially exhibit a robuster behavior, which does not correspond to a real convergence of the algorithm.
In practice, one however makes use of several chains in parallel. It is then tempting to exchange information between these chains to improve mixing properties of the MCMC samplers. A general framework of population Monte Carlo (PMC) has been proposed in this respect. In the present work [Oops!] we propose an interacting method between parallel chains, which provides an independent sample from the target distribution. Contrary to PMC, the proposal law in our work is given and does not adapt itself to the previous simulations. Hence, the problem of the choice of this law still remains.