Marcelle Kaufman

Towards a logical analysis of the immune response.

We present a new way to conceive, formalize and analyse models of the immune network. The models proposed are minimal ones, based essentially on the well-established negative feedback loop between helper and suppressor T cells. The occurrence of T-T interactions in both helper and suppressor circuits. These T-T interactions are represented here by autocatalytic feedback loops on TH and TS. The fact that immature B cells are sensitive to negative signaling, as was originally suggested by Lederberg (1959). There is a functional inactivation of immature B cells encountering antigen or anti-idiotypic antibody. This prevents further differentiation to a stage where the B cells become fully responsive. We describe the role of a logical method in the generation and analysis of the models, and the complementarity between this logical method and the more classical description by continuous differential equations. Logical analysis and numerical simulations of the differential equations show that the emerging model accounts for, the occurrence of multiple steady states (a virgin state, a memory state and a non-responsive state) in the absence of antigen, the kinetics of primary and secondary responses, high dose paralysis, low dose of paralysis. Its fit with real situations is surprisingly good for a model of this simplicity. Nevertheless, we give it as an example of what can now be done in the field rather than as a stable model.

Bifurcation analysis of nonlinear reaction-diffusion equations-II. Steady state solutions and comparison with numerical simulations

The steady state spatial patterns arising in nonlinear reaction-diffusion systems beyond an instability point of the thermodynamic branch are studied on a simple model network. A detailed comparison between the analytical solutions of the kinetic equations, obtained by bifurcation theory, and the results of computer simulations is presented for different boundary conditions. The characteristics of the dissipative structures are discussed and it is shown that the observed behavior depends strongly on both the boundary and initial conditions. The theoretical expressions are limited to the neighborhood of the marginal stability point. Computer simulations allow not only the verification of their predictions but also the investigation of the behavior of the system for larger deviations from the instability point. It is shown that new features such as multiplicity of solutions and secondary bifurcations can appear in this region.

Model analysis of the bases of multistationarity in the humoral immune response.

A formal analysis of the regulation of antibody production has been developed. It comprises two complementary approaches: a logical analysis in terms of discrete (boolean) variables and functions and a more classical analysis in terms of differential equations. A first paper dealt mostly with the logical description which provided global information on how complex the network needs to be in order to account for some main aspects of the immune response, without having to specify the details of the cellular interactions or to introduce a great number of parameters. Here we present the continuous approach and, in particular, a detailed study of the steady states and a discussion of their role in the dynamics of the immune response. The model subject to this analysis is a minimal one, which takes into account a small number of well-established facts concerning lymphocyte interactions and some reasonable assumptions. The core of the model is a negative feedback loop between the helper (TH) and suppressor (TS) T lymphocytes on which autocatalytic loops of the TH and TS populations on themselves are grafted. The salient feature of this minimal scheme is the prediction, for given environmental and parametrical conditions, of a multiplicity of steady states. This multistationarity occurs both in the absence of antigen or for constant antigen levels. Variations in the external constraints provoke switches among the steady states which might be related to the various modes of the humoral immune response, and depend on the doses of antigen injected and on the previous antigenic history of the system. In particular, high and low dose paralysis appear to be associated with two distinct steady state branches.

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).

On circuit functionality in boolean networks.

It has been proved, for several classes of continuous and discrete dynamical systems, that the presence of a positive (resp. negative) circuit in the interaction graph of a system is a necessary condition for the presence of multiple stable states (resp. a cyclic attractor). A positive (resp. negative) circuit is said to be functional when it “generates” several stable states (resp. a cyclic attractor). However, there are no definite mathematical frameworks translating the underlying meaning of “generates.” Focusing on Boolean networks, we recall and propose some definitions concerning the notion of functionality along with associated mathematical results.

The role of antigen presentation in the regulation of class-specific (Th1/Th2) immune responses

To prevent progressive and fatal expansion of an infectious agent, it is critical for the host to develop the appropriate immune response. The induction of a given set of effector cells is determined by the activation of T helper lymphocytes. These CD4 T cells can be classified into TH1 and TH2 lines according to their cytokine release pattern and their regulatory functions. Each subclass regulates a different set of immune effectors and they are, most of the time, mutually exclusive. Since not all immune effectors are effective against a given pathogen, the choice of a class response is thus a crucial event. We propose a qualitative model for understanding the dynamics of the TH1-TH2 network. In our model, the induction of a TH1 or a TH2 response is a function of the physical features of the pathogen, of the site of infection and of the efficiency of the ongoing response. One of our predictions is that anergy, in addition to its role in the mechanisms of peripheral tolerance, could have an important influence on the regulation of the TH1-TH2 balance. In summary, our model accounts for several important experimental observations in terms of transitions between multiple steady states. It emphazises the role of antigen presentation in directing the choice of a class-specific response.

Conceptual tools for the integration of data.

We propose to start the analysis of complex systems by systematically identifying the feedback circuits that govern their dynamics. These circuits can be identified without any ambiguity by examining the Jacobian matrix of the system. They provide precious information regarding the number and nature of steady states. Logical descriptions use variables and functions that can take only a limited number of discrete values (in simple cases, only two, 0 or 1). We developed an asynchronous method with continuous time, generalized by using variables with more than two levels and logical parameters. Reverse logics is a synthetic, inductive method. It aims at proceeding rationally from the experimental facts towards models rather than from models to predictions.

Theoretical insight into antigen-induced T-cell unresponsiveness

It is now well established that immune responses involve intricate, highly regulated interactions between functionally different sets and subsets of lymphocytes. The available evidences indicate that helper T lymphocytes, in particular, play an essential role in the initiation of most antibody or cell mediated responses. Activation of mature T helper cells leads to the production of regulatory factors that have an effect on the activation and function of other cells of the immune system. Numerous studies therefore are devoted to the conditions of activation and inactivation of these lymphocytes. In the present work we focus more specifically on the induction of long-lasting unresponsiveness after interaction of Interleukin-2 (I1–2) producing T helper cells with antigen. This phenomenon is important for the downregulation of a normal immune response, and might also participate in the mechanisms of self-tolerance.

NUMERICAL STUDY OF TRAVELLING WAVES IN A REACTION-DIFFUSION SYSTEM: RESPONSE TO A SPATIOTEMPORAL FORCING

We investigate numerically the effect of a spatiotemporal forcing on travelling waves produced in a model reaction-diffusion system operating under periodic boundary conditions. It is shown that, for suitable parameter values, this external forcing can stabilize the progressive wave regime and impose its directionality through wave inversion. When the field fails to reverse waves travelling in an opposite direction, chaotic-like spatiotemporal patterns may be observed.

A Dilute Bootstrap Percolation: Lattice Model for Unresponsiveness in T-Cell Immunology

The biased majority rule of cellular automata takes a spin up if and only if at least two of its four nearest neighbors on the square lattice are up. We generalize this type of bootstrap percolation by introducing quenched site dilution as well as a random birth and decay process. Our Monte Carlo simulations then give first-order transitions qualitatively similar to our results from meanfield reaction equations describing the induction of T-cell unresponsiveness in the immune system.