Quantum and Lattice Models of Biological Evolution

Noise in environments can cause mutation in genetic materials of biological systems. In this paper, I first introduce some molecular models of biological evolution, including Eigen model with connected mutation-selection scheme and Crow-Kimura (CK) model with parallel mutationselection scheme. Baake et al. mapped the CK model into a quantum spin model. Recently, Saakian and I did the similar mapping for the Eigen model. Using Suzuki-Trottere formalism, we studied statics and dynamics of the Eigen model and the CK model with the single-peak fitness function and found that the relaxation in the parallel model is faster than that in the connected model. We studied both models with rather general fitness functions and obtained error thresholds for various cases. We studied the Eigen model with multiple peaks which can represent virus or cancer cells attached by drug or the immune systems. Finally, we studied a lattice model for co-evolution of virus and immune system and found that the model shows self-organized behavior.