Symetries and Asymetries of the Immune System Response: a approach

Jean-François Mascari Damien Giacchero Nikolaos Sfakianakis IAC - CNR Centre Lacassagne Institute of Applied Mathematics Rome, Italy Nice, France University of Heidelberg [email protected] [email protected] [email protected]

in collaboration with Ludovica Mascari

Abstract—A new modeling approach and conceptual framework With respect to , Petri Nets have been also to the immune system response and its dual role with respect to studied via Theory, so there exists the possibility to cancer is proposed based on Applied . States of further develop our approach as a foundation for a new cells and pathogenes are structured as mathematical structures computational model of the Immune System dealing explicitly (categories), the interactions, at a given phase, between cells of with the modularity of its interactions. the immune system and pathogenes, correspond to a pair of adjunctions (adjoint ), the interaction process consisting of the sequential composition of an identification phase, a II. BEHAVIORS AS CATEGORIES AND BEHAVIORS preparation phase and an activation phase is modeled by the TRANSFORMATIONS AS FUNCTORS composition of maps of adjunctions: the approach is illustrated by considering the Cancer-Immunity Cycle. A third dimension is needed to model Cancer Immunoediting. The categorical A. Cells Behaviors as Categories foundations of our approach is based on Marco Grandis and The first step of our modeling consists in representing the Rober Paré theory of Intercategories. dynamics of cells and pathogenes by categories such that the states of cells/pathogenes are the objects of the corresponding Keywords—immune system; cancer; category theory; cancer- category and the state transitions are composable immunity cycle; cancer immunoediting of such categories. I. INTRODUCTION The behavior of the Immune cell during the phase k of an Modeling approaches to the Immune System range between a interaction is represented by the category of states of the top-down (differential equations) and a bottom-up (agent- Immune cell during the phase k of the interaction such that based) models. Recently an intermediate level approach based - objects: states of during the phase k of the interaction on Colored Petri Nets[1] has been proposed. This promising - morphisms: state transitions of during the phase k. approach needs further improvements in order to deal in particular with the modularity of the representation of The behavior of the pathogene during the phase k of an complex interactions. In this perspective we propose a new interaction is represented by the category of states of the framework based on Category Theory allowing a structured way of thinking and modeling. The idea is to map the pathogene during the phase k of the interaction such that structure, function and evolution of the Immune System with - objects: states of during the phase k algebraic (categorical) constructions. The advantage our - morphisms: state transitions of during the phase k. proposal consists in offering a flexible interface between the three worlds: Biology, Mathematics and Computer Science. With respect to Biology we represent the Immune System by considering the roles of its components in order to found the symmetries and asymmetries with respect to its interactions with pathogenes and tumors. Fig. 1. Categories of Behaviors With respect to Mathematics, Category Theory is usually considered as Pure Mathematics but its applications to Biology is not new: its application to the modeling of the Immune System, as presented in the paper is new. B. Transformations of Cells Behaviors as Functors between - for every -object and -object ’, a bijection Categories between The Behavior Transformation the Immune cell from the natural in and . phase h of the interaction to the phase k is represented by a from the category to the category .

The Interaction ( on ) of the Behavior of the Immune cell Fig. 2. Behaviors transformations of immune cells during the phase k with the Behavior of the pathogene during the phase k is represented as follows: The Behavior Transformation the pathogene from the  the Immune cell , in a state ), acts on the state phase h of the interaction to the phase k of the interaction is )) of the pathogene represented by a functor from the category to the category  the pathogene then makes a transition from the . state )) to a state ( )  state ( ) acts on the Immune cell proompted to ( ( )  the Immune cell has then made a transition from a state ) to the state ( ( ).

Fig. 3. Behaviors transformations of patogenes

Behaviors transformations can be composed.

Fig. 4. Composition of Behaviors transformations of the Immune System cells

Fig. 6. Adjunction Interaction between Immunne System cells and pathogenes

IV. TRANSFORMATIONS OF INTERACTIONS AS MAPS OF ADJUNCTIONS

The third step consists in modeling the phase transition of the Fig. 5. Composition of Behaviors transformations of pathogenes interactions of the Immune System with the pathogenes as maps of the adjunctions associated to each phase during the transition. III. INTERACTIONS AS An adjunctions map (transformation of adjunctions) ( , ‘) The second step consists in modeling, for each phase of the from a -adjunction to a -adjunction interactions of the Immune System with the pathogenes, the consists of:a functor from to , a interactions as a pair of functors between a category of cells of Immune System and a category of pathogenes satisfying a functor from to duality condition expressed by an adjunction (a generalization of adjoint operarators). Given two categories and a adjunction consists of: - a functor from to , a functor from to and such that i) the following diagrams commute

ii) for every -object and -object ’ the following diagram commute (“the transformation of the adjunction between two categories corresponds to the adjunction between the transformed categories”

Fig. 8. Composition of Transformation of Adjunction Interaction between Immunne System cells and pathogenes

VI. THE CANCER-IMMUNITY CYCLE REVISITED Our model can be applied to the innate and the adaptive Immune System as well as their interactions. We illustrate our approach by considering the well known Cancer-Immunity Cycle.

Fig. 8. Categorification of the Cancer-Immunity Cycle

Fig. 8. Transformation of Adjunction Interaction between Immunne System cells and pathogenes

V. INTERACTIONS PROCESSES AS COMPOSITION OF MAPS OF ADJUNCTION The fourth step consists in modeling the overall process of interactions, through the three phases of identification, preparation and activation, as composition of maps of adjunctions.

Fig. 8. Categorification of the Cancer-Immunity Cycle

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