Model Checking to Assess T-Helper Cell Plasticity

Computational modeling constitutes a crucial step toward the functional understanding of complex cellular networks. In particular, logical modeling has proven suitable for the dynamical analysis of large signaling and transcriptional regulatory networks. In this context, signaling input components are generally meant to convey external stimuli, or environmental cues. In response to such external signals, cells acquire specific gene expression patterns modeled in terms of attractors (e.g., stable states). The capacity for cells to alter or reprogram their differentiated states upon changes in environmental conditions is referred to as cell plasticity. In this article, we present a multivalued logical framework along with computational methods recently developed to efficiently analyze large models. We mainly focus on a symbolic model checking approach to investigate switches between attractors subsequent to changes of input conditions. As a case study, we consider the cellular network regulating the differentiation of T-helper (Th) cells, which orchestrate many physiological and pathological immune responses. To account for novel cellular subtypes, we present an extended version of a published model of Th cell differentiation. We then use symbolic model checking to analyze reachability properties between Th subtypes upon changes of environmental cues. This allows for the construction of a synthetic view of Th cell plasticity in terms of a graph connecting subtypes with arcs labeled by input conditions. Finally, we explore novel strategies enabling specific Th cell polarizing or reprograming events.LabEx MemoLife, Ecole Normale Supérieure, FCT grants: (PEst-OE/EEI/LA0021/2013, IF/01333/2013), Ph.D.program of the Agence National de Recherche sur Le Sida (ANRS), European Research Council consolidator grant

Logical Modeling and Analysis of Cellular Regulatory Networks With GINsim 3.0

The logical formalism is well adapted to model large cellular networks, in particular when detailed kinetic data are scarce. This tutorial focuses on this well-established qualitative framework. Relying on GINsim (release 3.0), a software implementing this formalism, we guide the reader step by step toward the definition, the analysis and the simulation of a four-node model of the mammalian p53-Mdm2 network

Logical Modeling and Dynamical Analysis of Cellular Networks

The logical (or logic) formalism is increasingly used to model regulatory and signaling networks. Complementing these applications, several groups contributed various methods and tools to support the definition and analysis of logical models. After an introduction to the logical modeling framework and to several of its variants, we review here a number of recent methodological advances to ease the analysis of large and intricate networks. In particular, we survey approaches to determine model attractors and their reachability properties, to assess the dynamical impact of variations of external signals, and to consistently reduce large models. To illustrate these developments, we further consider several published logical models for two important biological processes, namely the differentiation of T helper cells and the control of mammalian cell cycle.LabEx MemoLife; Ecole Normale Supérieure; French Plan Cancer (2014–2017) project: (Modeling cell communication networks in breast cancer - CoMET); FCT grants: (UID/CEC/50021/2013, IF/01333/2013); Fundação Calouste Gulbenkian

A Theoretical Exploration of Birhythmicity in the p53-Mdm2 Network

Experimental observations performed in the p53-Mdm2 network, one of the key protein modules involved in the control of proliferation of abnormal cells in mammals, revealed the existence of two frequencies of oscillations of p53 and Mdm2 in irradiated cells depending on the irradiation dose. These observations raised the question of the existence of birhythmicity, i.e. the coexistence of two oscillatory regimes for the same external conditions, in the p53-Mdm2 network which would be at the origin of these two distinct frequencies. A theoretical answer has been recently suggested by Ouattara, Abou-Jaoudé and Kaufman who proposed a 3-dimensional differential model showing birhythmicity to reproduce the two frequencies experimentally observed. The aim of this work is to analyze the mechanisms at the origin of the birhythmic behavior through a theoretical analysis of this differential model. To do so, we reduced this model, in a first step, into a 3-dimensional piecewise linear differential model where the Hill functions have been approximated by step functions, and, in a second step, into a 2-dimensional piecewise linear differential model by setting one autonomous variable as a constant in each domain of the phase space. We find that two features related to the phase space structure of the system are at the origin of the birhythmic behavior: the existence of two embedded cycles in the transition graph of the reduced models; the presence of a bypass in the orbit of the large amplitude oscillatory regime of low frequency. Based on this analysis, an experimental strategy is proposed to test the existence of birhythmicity in the p53-Mdm2 network. From a methodological point of view, this approach greatly facilitates the computational analysis of complex oscillatory behavior and could represent a valuable tool to explore mathematical models of biological rhythms showing sufficiently steep nonlinearities

Links between topology of the transition graph and limit cycles in a two-dimensional piecewise affine biological model

International audienceA class of piecewise affine differential (PWA) models, initially proposed by Glass and Kauffman (in J Theor Biol 39:103-129, 1973), has been widely used for the modelling and the analysis of biological switch-like systems, such as genetic or neural networks. Its mathematical tractability facilitates the qualitative analysis of dynamical behaviors, in particular periodic phenomena which are of prime importance in biology. Notably, a discrete qualitative description of the dynamics, called the transition graph, can be directly associated to this class of PWA systems. Here we present a study of periodic behaviours (i.e. limit cycles) in a class of two-dimensional piecewise affine biological models. Using concavity and continuity properties of Poincaré maps, we derive structural principles linking the topology of the transition graph to the existence, number and stability of limit cycles. These results notably extend previous works on the investigation of structural principles to the case of unequal and regulated decay rates for the 2-dimensional case. Some numerical examples corresponding to minimal models of biological oscillators are treated to illustrate the use of these structural principles

Dynamics of the Goodwin model combined with a “zero-order ultrasensitivity” phosphorylation module.

<p>The time series have been obtained by numerical integration of Eqs. (13), with the following parameter values: , , , , , . The kinetics rates of the phosphorylation/dephosphorylation reactions are as in Fig. 8 but multiplied by a factor 100. For clarity and to facilitate the comparison with the 3-variable model, variable has been multiplied by a factor 100. In panel B, a comparison of the limit cycle obtained for the present model (blue curve) and for the original 3-variable Goodwin model (black curve) shows a reasonable agreement between the two models. The period, about 40.7 a.u., is very close the period of the original Goodwin model. In panel D, the thin curve corresponds to the steady state of the phosphorylation module (cf. Fig. 8) and the thick curve is the trajectory of the present system.</p

Scheme of the Goodwin model combined with a single phosphorylation/dephosphorylation module.

<p>The phosphorylation/dephosphorylation module can generate sharp thresholds through “zero-order ultrasensitivity”.</p

Dynamics of the Goodwin model combined with the multisite phosphorylation module.

<p>These results have been obtained by numerical integration of Eqs. (11). Parameter values for the Goodwin model are as in Fig. 2. Kinetic parameter values for the phosphorylation module are as in Fig. 3 for but multiplied by a factor 100. Conservation parameter values, and , are the same as in Fig. 3. In panel A, the thin blue curve corresponds to at steady state, as obtained in Fig. 3, while the thick line denotes the trajectory of the present system. The inset is a zoom on the lower part of that curve. The oscillations are in very good agreement with the oscillations generated by the 3-variable model (Fig. 2, see also Fig. 7 for a comparison of the limit cycles). The period of the oscillations, about 40 a.u., is also consistent with the 3-variable model.</p

Derivation of Qualitative Dynamical Models from Biochemical Networks

International audienceAs technological advances allow a better identification of cellular networks, more and more molecular data are produced allowing the construction of detailed molecular interaction maps. One strategy to get insights into the dynamical properties of such systems is to derive compact dynamical models from these maps, in order to ease the analysis of their dynamics. Starting from a case study, we present a methodology for the derivation of qualitative dynamical models from biochemical networks. Properties are formalised using abstract interpretation. We first abstract states and traces by quotienting the number of instances of chemical species by intervals. Since this abstraction is too coarse to reproduce the properties of interest, we refine it by introducing additional constraints. The resulting abstraction is able to identify the dynamical properties of interest in our case study

Scheme of the Goodwin model combined with the multisite phosphorylation module.

<p>In this model, variable is the kinase E which can be found in its free form or in a complex with any phosphoform of S. We also assume that all forms of free S, except , can induce the synthesis of X.</p