Symmetry Breaking on a Chessboard: 50 Years After Eigen’s and Winkler’s “Laws of the Game”

This paper analyses several mathematical games developed 50 years ago by Manfred Eigen and Ruthild Winkler in their famous book “Laws of the Game: How the Principles of Nature Govern Chance,” published for the first time in German in 1975. These games are intended to represent the essential aspects of chemical selection processes via symmetry breaking in biological systems. Special attention is paid to games that model biochemical kinetics, in which a chessboard is used to represent different types of substrates. The time-dependent statistical outcomes of several games are studied by Monte Carlo techniques. Analytical stochastic models applied to these games relate game rules to partial differential equation problems with appropriate initial and boundary conditions: rationalizing their outcomes, they confirm the intuitions of the original authors and add new insights. The potential impact of game-based models on current research on biological homochirality is discussed.