Mathematical Modeling of HIV Dynamics After Antiretroviral Therapy Initiation: A Review.

This review shows the potential ground-breaking impact that mathematical tools may have in the analysis and the understanding of the HIV dynamics. In the first part, early diagnosis of immunological failure is inferred from the estimation of certain parameters of a mathematical model of the HIV infection dynamics. This method is supported by clinical research results from an original clinical trial: data just after 1 month following therapy initiation are used to carry out the model identification. The diagnosis is shown to be consistent with results from monitoring of the patients after 6 months. In the second part of this review, prospective research results are given for the design of individual anti-HIV treatments optimizing the recovery of the immune system and minimizing side effects. In this respect, two methods are discussed. The first one combines HIV population dynamics with pharmacokinetics and pharmacodynamics models to generate drug treatments using impulsive control systems. The second one is based on optimal control theory and uses a recently published differential equation to model the side effects produced by highly active antiretroviral therapy therapies. The main advantage of these revisited methods is that the drug treatment is computed directly in amounts of drugs, which is easier to interpret by physicians and patients.

Kinetic modelling of in vitro cell-based assays to characterize non-specific bindings and ADME processes in a static and a perfused fluidic system.

Recently, physiologically based perfusion in vitro systems have been developed to provide cell culture environment close to in vivo cell environment (e.g., fluidic conditions, organ interactions). In this work, we model and compare the fate of a chemical, benzo[a]pyrene (B[a]P), in a perfusion and a standard (static well-plate) system. These in vitro systems are composed of Caco-2 and HepG2 cells so as to mimic absorption across the small intestine and intestinal and hepatic metabolism. Compartmental models were developed and calibrated with B[a]P kinetics data in the culture medium to estimate the apparent permeability of Caco-2 cells, the in vitro biotransformation of B[a]P, as well as the different routes of loss by non-specific adsorption. Our results show that non-specific binding is the main process responsible for the depletion of B[a]P in the culture media: at steady state, only 40% and 24% of the total concentration of B[a]P are bioavailable in the static and perfused systems, respectively. We also showed that Caco-2 permeability in the perfused culture system is closer to in vivo conditions than the one obtained in the static system and that higher cellular metabolic activities are observed in static conditions. Perfused in vitro systems combined with kinetic modelling are promising tools for studying in vitro the different processes involved in the toxicokinetics of xenobiotics.

From structure to dynamics: frequency tuning in the p53-Mdm2 network. II Differential and stochastic approaches.

In Part I of this work, we carried out a logical analysis of a simple model describing the interplay between protein p53, its main negative regulator Mdm2 and DNA damage, and briefly discussed the corresponding differential model (Abou-Jaoudé et al., 2009). This analysis allowed us to reproduce several qualitative features of the kinetics of the p53 response to damage and provided an interpretation of the short and long characteristic periods of oscillation reported by Geva-Zatorsky et al. (2006) depending on the irradiation dose. Starting from this analysis, we focus here on more quantitative aspects of the dynamics of our network and combine the differential description of our system with stochastic simulations which take molecular fluctuations into account. We find that the amplitude of the p53 and Mdm2 oscillations is highly variable (to a degree that depends, however, on the bifurcation properties of the system). In contrast, peak width and timing remain more regular, consistent with the experimental data. Our simulations also show that noise can induce repeated pulses of p53 and Mdm2 that, at low damage, resemble the slow irregular fluctuations observed experimentally. Adding the stochastic dimension in our modeling further allowed us to account for an increase of the fraction of cells oscillating with a high frequency when the irradiation dose increases, as observed by Geva-Zatorsky et al. (2006).

From structure to dynamics: frequency tuning in the p53-Mdm2 network I. Logical approach.

We investigate the dynamical properties of a simple four-variable model describing the interactions between the tumour suppressor protein p53, its main negative regulator Mdm2 and DNA damage, a model inspired by the work of Ciliberto et al. [2005. Steady states and oscillations in the p53/Mdm2 network. Cell Cycle 4(3), 488-493]. Its core consists of an antagonist circuit between p53 and nuclear Mdm2 embedded in a three-element negative circuit involving p53, cytoplasmic and nuclear Mdm2. A major concern has been to develop an integrated approach in which various types of descriptions complement each other. Here we present the logical analysis of our network and briefly discuss the corresponding differential model. Introducing the new notion of "logical bifurcation diagrams", we show that the essential qualitative dynamical properties of our network can be summarized by a small number of bifurcation scenarios, which can be understood in terms of the balance between the positive and negative circuits of the core network. The model displays a wide variety of behaviours depending on the level of damage, the efficiency of damage repair and, importantly, the DNA-binding affinity and transcriptional activity of p53, which are both stress- and cell-type specific. Our results qualitatively account for several experimental observations such as p53 pulses after irradiation, failure to respond to irradiation, shifts in the frequency of the oscillations, or rapid dampening of the oscillations in a cell population. They also suggest a great variability of behaviour from cell to cell and between different cell-types on the basis of different post-translational modifications and transactivation properties of p53. Finally, our differential analysis provides an interpretation of the high and low frequency oscillations observed by Geva-Zatorsky et al. [2006. Oscillations and variability in the p53 system. Mol. Syst. Biol. 2, 2006.0033] depending on the irradiation dose. A more detailed analysis of our differential model as well as its stochastic analysis will be developed in a next paper.