The field usually referred to as mathematical biology is a highly interdisciplinary area that lies at the intersection of mathematics and biology. Classical illustrations include the development of stochastic processes and statistical methods to solve problems in genetics and epidemiology. As the name used to describe work in this field indicates, with “biology” the noun, and “mathematical” the modifying adjective, the relationship between mathematics and biology has so far been one–way. Typically, mathematical results have emerged from or have been used to solve biological problems (see [34] for a comprehensive survey). In contrast, Leonard Adleman, [1], succeeded in solving an instance of the directed Hamiltonian path problem solely by manipulating DNA strings. This marks the first instance of the connection being reversed: a mathematical problem is the end toward which the tools of biology are used. To be semantically correct, instead of categorizing the research in DNA computing as belonging to mathematical biology, we should be employing the mirror–image term biological mathematics for the field born in November 1994. Despite the complexity of the technology involved, the idea behind biological mathematics is the simple observation that the following two processes, one biological and one mathematical, are analogous: (a) the very complex structure of a living being is the result of applying simple operations (copying, splicing, etc.) to initial information encoded in a DNA sequence, (b) the result f(w) of applying a computable function to an argument w can be obtained by applying a combination of basic simple functions to w (see Section 4 or [65] for details). If noticing this analogy were the only ingredient necessary to cook a computing DNA soup, we would have been playing computer games on our DNA

DNA computing: Arrival of biological mathematics

The stimulus for the development of the theory presented in this chapter is the string behaviors exhibited by the group of molecules often referred to collectively as the informational macromolecules. These include the molecules that play central roles in molecular biology and genetics: DNA, RNA, and the polypeptides. The discussion of the motivation for the generative systems is focused here on the recombinant behaviors of double stranded DNA molecules made possible by the presence of specific sets of enzymes. The function of this introduction is to provide richness to the reading of this chapter. It indicates the potential for productive interaction between the systems discussed and molecular biology, biotechnology, and DNA computing. However, the theory developed in this chapter can stand alone. It does not require a concern for its origins in molecular phenomena. Accordingly, only the most central points concerning the molecular connection are given here. An appendix to this chapter is included for those who wish to consider the molecular connection and possible applications in the biosciences. Here we present only enough details to motivate each term in the definition of the concept of a splicing rule that is given in the next section. The splicing rule concept is the foundation for the present chapter.

Language theory and molecular genetics: generative mechanisms suggested by DNA recombination

From a biological motivation of the interactions between linear and circular DNA sequences, we propose a new type of splicing model called "circular H systems" and show that they have the same computational power as Turing machines. It is also shown that there effectively exists a universal circular H system which can simulate any circular H system with the same terminal alphabet, which strongly suggests a feasible design for a DNA computer based on circular splicing.

On the power of circular splicing systems and DNA computability

Splicing representations of strictly locally testable languages

The field usually referred to as mathematical biology is a highly interdisciplinary area that lies at the intersection of mathematics and biology Classical illustrations include the development of stochastic processes and statistical methods to solve problems in genetics and epidemiology As the name used to describe work in this field indicates, with “biology” the noun, and “mathematical” the modifying adjective, the relationship between mathematics and biology has so far been one–way Typically, mathematical results have emerged from or have been used to solve biological problems (see [34] for a comprehensive survey) In contrast, Leonard Adleman, [1], succeeded in solving an instance of the directed Hamiltonian path problem solely by manipulating DNA strings This marks the first instance of the connection being reversed: a mathematical problem is the end toward which the tools of biology are used To be semantically correct, instead of categorizing the research in DNA computing as belonging to mathematical biology, we should be employing the mirror–image term biological mathematics for the field born in November 1994 Despite the complexity of the technology involved, the idea behind biological mathematics is the simple observation that the following two processes, one biological and one mathematical, are analogous: (a) the very complex structure of a living being is the result of applying simple operations (copying, splicing, etc) to initial information encoded in a DNA sequence, (b) the result f(w) of applying a computable function to an argument w can be obtained by applying a combination of basic simple functions to w (see Section 4 or [65] for details) If noticing this analogy were the only ingredient necessary to cook a computing DNA soup, we would have been playing computer games on our DNA

DNA computing: the arrival of biological mathematics

This paper concerns the formal study on the generative powers of extended splicing (H) systems. First, using a classical result by Post which characterizes the recursively enumerable languages in terms of his Post Normal systems, we establish several new characterizations of extended H systems which not only allow us to have very simple alternative proof methods for the previous results mentioned above, but also give a new insight into the relationships between families of extended H systems. We show a kind of normal form for extended H systems exactly characterizing the class of regular languages. We also show a new representation result for the family of context-free languages in terms of extended H systems.

Kobayashi S

Papers

DNA splicing systems and post systems.