Rewriting game theory as a foundation for state-based models of gene regulation

TLDR: A game-theoretic foundation for gene regulatory analysis based on the recent formalism of rewriting game theory is presented and it is shown that their models are specific instances of a C/P game deduced from the K parameter.

Abstract: We present a game-theoretic foundation for gene regulatory analysis based on the recent formalism of rewriting game theory. Rewriting game theory is discrete and comes with a graph-based framework for understanding compromises and interactions between players and for computing Nash equilibria. The formalism explicitly represents the dynamics of its Nash equilibria and, therefore, is a suitable foundation for the study of steady states in discrete modelling. We apply the formalism to the discrete analysis of gene regulatory networks introduced by R. Thomas and S. Kauffman. Specifically, we show that their models are specific instances of a C/P game deduced from the K parameter.

Figures

Fig. 4. Change-of-mind equilibrium for our running strategic-form example

Fig. 1. Example of sequential(extensive form) and strategic game(normal form)

Fig. 5. Two-variable regulatory network for phage λ ( is wildcard)

Fig. 7. λ-phage C/P game.

Fig. 6. Kauffman (left) and Thomas (right) analysis of two-variable phage λ

Fig. 3. The (free) change-of-mind relation for agent a in Gcp

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Rewriting game theory as a foundation for state-based models of gene regulation" ?

The authors present a game-theoretic foundation for gene regulatory analysis based on the recent formalism of rewriting game theory. The formalism explicitly represents the dynamics of its Nash equilibria and, therefore, is a suitable foundation for the study of steady states in discrete modelling. The authors apply the formalism to the discrete analysis of gene regulatory networks introduced by R. Thomas and S. Kauffman. Specifically, the authors show that their models are specific instances of a C/P game deduced from the K parameter. 

Q2. What have the authors stated for future works in "Rewriting game theory as a foundation for state-based models of gene regulation" ?

Their future work concerns similar treatments of the various abstraction levels in between, namely chemical, biochemical, cellular, multi-cellular and environmental level. 

Q3. What is the purpose of C/P games?

Conversion/preference (C/P) games have been designed as an abstraction over strategic-form games and as a game formalism that introduces as few concepts as possible. 

Q4. what is the level of protein binding and catalysis captured in expression games?

In other words, the authors have moved from the chemical abstraction level of protein binding and catalysis captured in expression games, up to the biochemical abstraction level of, e.g., phage λ’s lysogenic and lytic states. 

Q5. What is the definition of a strategic-form game?

Clearly strategic-form games are instances of C/P games, conversions are one dimension move (for instance along a line or a column), while preferences are given by comparisons over payoffs: a synopsis is preferred by an agent over another if his payoff is larger in the former. 

Q6. What is the relevance of state-based analysis?

The relevance of state-based analysis comes from the fact that the state in question has been observed to be (self-)sustainable: it is phage λ’s lysogenic state that “involves integration of the phage DNA into the bacterial chromosome [of its host] where it is passively replicated at each cell division — just as though it were a legitimate part of the bacterial genome” [21] 

Q7. What is the biological justification for why the states are asynchronous?

s ′) – s ⊳g s ′ , K-Approxg(s, s ′) ∧ (∀g′ . g′ 6= g ⇒ πg′(s) = πg′(s′))be the synchronous respectively g-asynchronous preference relations. 

Q8. What is the definition of abstract Nash equilibrium?

The notion of abstract Nash equilibrium specialises to Nash’s concrete form in the presence of the discussed structural constraints on strategic-form games. 

Q9. Why is the phase transition not considered atomically?

The rationale for the latter is that moving from a state to another involves a phase transition, which is costly in terms of energy, and two phase transitions should therefore not be considered atomically. 

Q10. What is the meaning of a play of the game on the left?

A play of the game on the left is a path from the root to a leaf, where the first (second) number indicates the payoff to agent a1 (a2). 

Q11. What is the simplest way to describe a change of mind equilibri?

In this paper, for example, the authors have shown that changeof-mind equilibria can be used to predict what gene expression will take place. 

Q12. What is the definition of a C/P game?

For linear order g0 < . . . < gn, let gi ⊖ gj = i − j, and let s // ±1 s′ , ∀g ∈ G . | πg(s) ⊖ πg(s′) | ≤ 1A C/P game whose conversion fulfills the previous definition is called 1-restrained. 

Q13. What is the only probabilistic Nash equilibrium?

In the example, the only probabilistic Nash equilibrium arises if both agents choose between their two options with equal probability for expected payoffs of a half to each. 

Q14. What is the main difference between the two perspectives?

The game-theoretic perspective the authors provide is technically and conceptually beneficial because non-cooperative game theory is the embodiment of the compete-andcoexist reality of genes and because it allows us to leverage the independently developed theory of dynamic equilibria in rewriting game theory. 

Q15. What is the definition of a Nash equilibrium?

The following definition says that a synopsis s, i.e., their abstraction over (combined) strategies, is an abstract Nash equilibrium if and only if all agents are happy, meaning that whenever an agent can convert s to s′ then it is not the case that he prefers s′ to s. 

Q16. What is the biological justification for why the states are inescapable?

The cycle between 〈cI 0, cro0〉 and 〈cI 1, cro1〉 is a known false positive of Kauffman’s model.cial, i.e., that they are inescapable, is the exact the justification for why they are both change-of-mind equilibria.