# UNIVERSITA DEGLI STUDI DI NAPOLI FEDERICO II FACOLTA

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UNIVERSITA’ DEGLI STUDI DI NAPOLI FEDERICO II FACOLTA’ DI INGEGNERIA CORSO DI DOTTORATO IN INGEGNERIA DEI MATERIALI E DELLE STRUTTURE XXVIII-XXIX CICLO EVOLUTIONARY STABLE ANTI-CANCER THERAPIES BY AUTOLOGOUS CELL DEFECTION • CONCETTA DI NATALE, FLAVIA RUGGIERO, TERESA FOTTICCHIA, GIULIA IACCARINO, ANDREA PATRIZIA FALANGA ANNO ACCADEMICO 2013 -2014

Traditional anticancer therapy Identification and targeting of a molecule that is necessary for the development of the tumor. Molecule or pathway growth Tumor decrese Antiangiogenic therapy Angiogenesis is the physiological process through which new blood vessels form from pre-existing vessels. It is a fundamental step in the transition of tumors from a benign state to a malignant one. VEGF (Vascular Endothelial Growth Factor) is the primary responsible for angiogenesis. Bevacizumab is an anti-VEGF antibody Disadvantages The problem with most current anti-cancer treatments is that mutant cell lines that are resistant to treatments can spread and eventually confer resistance to the whole tumor. In other words the population reach a new equilibrium.

Aim: Identify a conditions that would make a treatment stable against mutant cell lines Starting Point: Evolutionary dynamics of cancer. In particular the attention was focused on the production of growth factors by cell cancer. Means: Evolutionary game theory of growth factor production. In the case of tumor progression, game theory is relevant to the study of growth factor production, because the growth factors produced by cancer cells could be used by neighboring modifided cells that don’t produce it leading to a tumor collapse.

The question What should a treatment do in order to impair the production of diffusible factors by tumor cells? The method There was described different methods that can lead to an evolutionarily stable therapy The method proposed here relies on autologous cell therapy: cells are harvested from the patient and genes coding for diffusible factors are knocked out; these modified cancer cells are then reintroduced in the tumor in order to modify the dynamics of the production of the factors coded by the knockout genes. The scope Show that, for certain parameters, such treatment is evolutionarily stable, that is, immune to the invasion of resistant cell lines.

Game theory is the breanch of mathematics that studies strategic interactions beetween players in which a player’s play-off depends not only on his own decisions, but also on the players’ decisions. Traditional game theory Rationality Evolutionary game theory Process of natural selection The individuals that are programmed to thake the best ‘decision’ leave more progeny and incrise in frequency within the population Evolutionary game theory can help us identify which state of population (of cancer cells ) are stable under which condition

Game theory definition Game theory is the study of interactions between players with same, different or conflictual aim. Game theory is widely applied in economics, engineering, political science and biology among the others Players or decision-makers the agents who compete in the game. Actions or strategies all the choices available to a decision-maker

Cooperative games: Cooperative game theory studies negotiations among rational agents who can make binding agreements about how to play the game. Now the emphasis is on the groups or coalitions of the players. The scope is to establish which coalition will form and how the agents will share the benefit that the coalition has, according to some ideas of fairness given by a set of desirable properties for the solution, called axioms. Non cooperative games In non-cooperative games the emphasis is mainly on the individual behavior: agents cannot commit themselves and perceive self-interest looking for their actions in order to achieve the most likely outcome of the game according to the rules of the game. Sometimes pre-play communication between agents is allowed, but in a non-cooperative game they are not able to make agreements except for those which are established by the rules of the game.

Stategic games A stratecic game is a game in which each player takes his decision only once at the same time The ingredients: • N = {1, 2, …, n} is a set of n players • Xi is the set of possible actions or strategies for player • x = [x 1, …, xn]T is the action profile • fi : X R it can represent a cost (to minimize) or a profit (to maximize). Finite game If each player has a finite number of available strategies, we say that the game is a finite game

Basics in ESS Theory Ø Evolutionary Games (non Cooperative) Ø Def. ESS Ø Large Population Ø Interaction between 2 individuals

Basics in ESS Theory Ø Symmetric Game Ø Hp. 1 Ø Hp. 2 Ø u(x, x’) Payoff Fitness

Basics in ESS Theory Ø Mixed Strategies Ø Evolutionary Stable Strategy x = ESS for invasion barrier if

Basics in ESS Theory Ø Restatement of ESS strategy 1) an ESS is a NE 2) and x’ in some neighbourhood of x

Methodology Producer of diffusible factors (+/+) Ø 2 possible strategies non Producer of diffusible factors (-/-) Ø (+/+) cells pay a cost 0 < c < 1 Ø each cell benefits from the diffusible factors produced by all the cells in its group (of size n). Ø cell benefit b(j) is given by the logistic function j<k j>k j = n° of cellule in a group of size n k = inflection point of the benefit curve s = parameter controlling the steepness of the curve

Methodology Ø Payoff functions of (+/+) cells and (-/-) cells where 0 < x < is the fraction of (+/+) cells within the population

EVOLUTIONARY DYNAMICS Motivations Ø The ESS may not exist; anyway, its analysis tells us nothing about the evolution of the system Ø The ESS definition deals only with monomorphic populations: all the population individuals use the same strategy Goal Ø We want to study whether a polymorphic population with the same profile as that generated by the ESS can also be stable A specific type of evolutionary dynamics: replicator dynamics

EVOLUTIONARY DYNAMICS The idea: to study the evolution of groups of individuals, depending on the strategy adopted by each of them Ø We are going to consider a population in which individuals, called replicators, exist in several different ways - the producer cells are replicators with strategy (+/+) and nonproducer cells are replicators with strategy (-/-) Ø Each type of individual passes its behavior to its descendants without any modifications Replicator Dynamics: if a strategy is performing better than the average in terms of fitness, the agents who play it will see their fraction increase in the total population

EVOLUTIONARY DYNAMICS Fundamentals of a dynamic system Ø Dynamic process in continuous time: system of first order ordinary differential equations state vector (1) state space Ø Stationary state (or equilibrium or fixed point): Ø A convergent trajectory is a trajectory, described by the ODE (1), starting from the initial state vector and converging to a stationary state (2)

EVOLUTIONARY DYNAMICS Fundamentals of a dynamic system

EVOLUTIONARY DYNAMICS Replicator dynamics: the equation (3) (4)

EVOLUTIONARY DYNAMICS Replicator dynamics Ø A fixed point of the replicator dynamics is a population that satisfies the conditions Fixed points describe populations that are no longer evolving Theorem: A strategy is an ESS if and only if the corresponding fixed point in the replicator dynamic is asymptotically stable Evolutionary Equilibrium (EE): any asymptotically stable fixed point of the dynamical process of evolution of the population of replicators

EVOLUTIONARY DYNAMICS Summary of equilibria concepts Remarks: Theorem:

EVOLUTIONARY DYNAMICS Methodology Ø A public good game is used in the framework of evolutionary game theory Ø For the case studied, the replicator dynamics is given by: (5)

Tumor: Public good 23

RESULTS 2 equilibria 1°) +/+ and - / - cells coexist 2°) - / - replace +/+ cells Figure 1. Difference between therapies that target growth factors and autologous cell therapy. (A) Targeting diffusible factors directly increases the threshold (h) of the PGG; as a consequence the system has new internal equilibria (empty circle: unstable; filled circle: stable). The therapy is successful (dotted line: the+/+cells go extinct) only if the new unstable equilibrium (dark blue; h = 0. 8) is above the previous stable equilibrium (gray; h = 0. 4); if this is not the case the system will move to the new internal equilibrium (light blue; h = 0. 6). (B) Autologous cell therapy does not rely on changing the benefit function of the diffusible factor, but must introduce a critical amount of -/ -cells in order to destabilize the equilibrium and move the fraction of+/+below the unstable equilibrium, after which the+/+cells will go extinct. 24

RESULTS Figure 2. Introducing a critical amount of-/cells can lead the population to collapse. The plots show frequencies and fitness over time (the bold line is the average of 10 simulations). If the initial fraction of+/+cells is locally below the unstable internal equilibrium (case b), clonal selection will spontaneously lead to the increase in frequency of -/- cells and to the consequent collapse of the tumor for lack of essential diffusible factors; if not (case a), the original equilibrium frequencies will persist. 25

Under certain conditions the system can evolve to a stable equilibrium in which +/+ cells persist Additional strategies: • Extend diffusion range of the diffusible factor Ø Disrupting the binding molecules on the ECM or the binding domains on the diffusible factors Ø Increasing long range isoforms of the factors • Temporary provison of exogenous diffusible factors

• • • CONCLUSIONS Gene therapy is the application of genetic principles in the treatment of human disease. It can be applied as Therapy for Cancers, inherited disorders, infectious diseases, immune system disorders Gene therapy = Introduction of genetic material into normal cells in order to: – counteract the effect of a disease gene – introduce a new function Altered cells not produce protein

Different Delivery Systems are Available • In vivo versus ex vivo – In vivo = delivery of genes takes place in the body – Ex vivo = delivery takes place out of the body, and then cells are placed back into the body

BASIC IDEA • . . IS TO FIGHT CANCER USING MODIFIED CANCER CELLS THAT ARE DEFECTIVE FOR THE PRODUCTION OF ESSENTIAL DIFFUSIBLE FACTOR MODIFIED CELLS (-/-) NO COST (c) FREE RIDER CANCER CELLS (+/+) COST(c) TUMOR COLLAPSE

THE GOAL • . . IS TO USE MODIFIED CANCER CELLS AS FREE RIDERS TO INDUCE A. . . . “TRAGEDY OF COMMONS” In cancer population

Theoric concepts TRAGEDY OF COMMONS The tragedy of the commons is an economics theory by Garrett Hardin, according to which individuals, acting independently and rationally according to each one's self-interest, behave contrary to the whole group's long-term best interests by depleting some common resource. The concept is often cited in connection with sustainable development, meshing economic growth and environmental protection, as well as in the debate over global warming. Application to evolutionary biology The tragedy of the commons is referred to in studies of evolutionary biology, social evolution, sociobiology and behavioral ecology. A tragedy of the commons is brought about by selfish individuals whose genes for selfish behaviour would therefore come to predominate. A parallel was drawn recently between the tragedy of the commons and the competing behaviour of parasites that through acting selfishly eventually diminish or destroy their common host.

Application to GAME THEORY • NASH EQUILIBRIUM • PARETO OPTIMUM

Collapsing and Component tragedies We deﬁne a ‘collapsing’ tragedy as a situation in which selﬁsh individual behaviour results in the entire resource vanishing For example, if the resource is a social good formed by cooperation, collapse would mean that the group loses the cooperative behaviour in question, and the social good ceases to exist. Unlike a component tragedy, is a tragedy in which selfishness reduces the resource but not to the extent that it is lost completely.

• • • Advantages No drug resistance ( evolutionary is desidered) Cancer cells (+/+) have a lower fitness in a population of modified cells (-/-) No toxicity Disadvantages Proliferation stroma cells(diffusible factor provided by the stroma may still enable the tumor to grow) Receptor/Its downstream expression constitutively active ( Herceptin). Only metastatic cancer. FUTURE PERSPECTIVES EARLY CANCER STAGE and PRE- METASTATIC STAGE

THANKS FOR YOUR ATTENTION Il miglior risultato si ottiene quando ogni componente del gruppo fa ciò che è meglio per sé e per il gruppo… Dinamiche dominanti, signori, dinamiche dominanti… (J. NASH)