Complexity Analysis And Mathematical Model Of DNA-Based Self-Assembly

Since Adleman solve the Hamilton path problem using DNA molecules in 1994, many mathematical problems were solved by self-assembly of DNA such as matrix multiplication, addition, symbolic determinants. What is morej in Nov. 2001 Y. Denenson simulated the transformation of states of finite autonomous. These facts demonstrate that DNA molecules computing by self-assembly according to Waston-Crick complement is powerful. Self-assembly of DNA is to use DNA strands to encode information and apply some operations to these strands, then the result strands will be generated automatically. Many researchers try to develop various self-assembly model based DNA molecules to solve the mathematical problem.In this paper, we developed a self-assembly model for DNA-based parallel addition. The central feature of this model is to apply the parallel logic. We make the complexity analysis of the algorithm used here. Compared it with the algorithm described by F. Guarnieri and Bancroft, this algorithm has some advantages.In addition, a mathematical model of self-assembly of DNA was developed. A state of self-assembly of DNA was specified by a vector (xi, X2...xn) . When a equilibrium is reached, there is a probability distribution of the states. Then a probability distribution of the states at the equilibrium corresponding a self-assembly model. All the possible can form a manifold called S, and the probability distribution of the states self-assembly system reached form a submanifold of 5", called A. So the difference of two self-assembly model is the division of two probability distribution at the manifold. Then the self-assembly system corresponding the probability distribution which is a given probability distribution was found. Futher, note that the natural DNA sequences are generated by self-assembly of A,C,G,T. The self-assembly model simulating the process generating the natural DNA sequences can be found.