The biochemical materials are described in terms of the opportune Hierarchical Markov-State Model and of the originating chain(s). The time evolution of the equations of motion of the Markov chain is controlled; to this aim, the transitions form the unrestrained simulations and those between the local Markov-State Models are compared. The formalisms of quantum-mechanical systems are applied in the opportune measure spaces. The ergodicity of the Markov chains is controlled. The numerical simulations, the properties to be requested on numerical approximations are studied. As a result, the ergodicity of the Mar