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The two-dimensional cell-sorting problem is found to be mathematicallyequivalent to the one-dimensional random walk problem with pair creations andannihilations, i.e. the adhesion probabilities in the cell-sorting model relateanalytically to the expectation values in the random walk problem. This is anexample demonstrating that two completely different biological systems aregoverned by a common mathematical structure. This result is obtained throughthe equivalences of these systems with lattice spin models. It is also shownthat arbitrary generation operators can be written by the spin operators, andhence all biological stochastic problems can in principle be analyzed utilizingthe techniques and knowledge previously obtained in the study of lattice spinsystems.

Equivalence between two-dimensional cell-sorting and one-dimensional generalized random walk -- spin representations of generating operators