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Field theory of molecular cooperators

It has been suggested that major transitions in evolution require the emergence of novelties, often associated to the cooperative behaviour of previously existing objects or agents. A key innovation involves the first cooperative interactions among molecules in a prebiotic biosphere. One of the simplest scenarios includes two molecular species capable of helping each other forming a catalytic loop or hypercycle. The second order kinetics of the hypercycle implies a hyperbolic growth dynamics, capable of overcoming some selection barriers associated to non-cooperative molecular systems. Moreover, it has been suggested that molecular replicators might have benefited from a limited diffusion associated to their attachment to surfaces: evolution and escape from extinction might have been tied to living on a surface. In this paper we propose a field theoretical model of the hypercycle involving reaction and diffusion through the use of a many-body Hamiltonian. This treatment allows a characterisation of the spatially correlated dynamics of the system, where the critical dimension is found to be $d_c=2$. We discuss the role of surface dynamics as a selective advantage for the system's survival.