DNA computing is a method of applying interactions among biological macromolecules to realize parallel computation, which has the extensive foreground of application. However, it is usually time consuming and expensive to validate the accuracy of DNA computing algorithms experimentally. Therefore, to reduce the cost of experiments, it is necessary to develop mathematical models for computer simulations before carrying out the corresponding DNA computing experimental work. With the mathematical theory of Yuanjian, the paper firstly models the experimental processes of DNA computing, namely the Yuanjian model, to solve the Hamiltonian Path Problem (HPP). Then, an encoding rule is given from the view of mathematics. With the improvement of the Yuanjian model presented above, a generalized Yuanjian model for solving the multi-node HPP is derived. The simulations verify the effectiveness of the proposed model which can be applied as a new bionic algorithm to solve the HPP for any size digraph. Furthermore, the presented Yuanjian model provides a model reference for DNA computing technology, which highlights the prospect of mathematical models in reducing the experiment cost for DNA computing algorithms.