In this article, we introduce three 3D graphical representations of DNA primary sequences, which we call RY-curve, MK-curve and SW-curve, based on three classifications of the DNA bases. The advantages of our representations are that (i) these 3D curves are strictly non-degenerate and there is no loss of information when transferring a DNA sequence to its mathematical representation and (ii) the coordinates of every node on these 3D curves have clear biological implication. Two applications of these 3D curves are presented: (a) a simple formula is derived to calculate the content of the four bases (A, G, C and T) from the coordinates of nodes on the curves; and (b) a 12-component characteristic vector is constructed to compare similarity among DNA sequences from different species based on the geometrical centers of the 3D curves. As examples, we examine similarity among the coding sequences of the first exon of beta-globin gene from eleven species and validate similarity of cDNA sequences of beta-globin gene from eight species.

In this paper, we propose two four-base related 2D curves of DNA primary sequences (termed as F-B curves) and their corresponding single-base related 2D curves (termed as A-related, G-related, T-related and C-related curves). The constructions of these graphical curves are based on the assignments of individual base to four different sinusoidal (or tangent) functions; then by connecting all these points on these four sinusoidal (tangent) functions, we can get the F-B curves; similarly, by connecting the points on each of the four sinusoidal (tangent) functions, we get the single-base related 2D curves. The proposed 2D curves are all strictly non degenerate. Then, a 8-component characteristic vector is constructed to compare similarity among DNA sequences from different species based on a normalized geometrical centers of the proposed curves. As examples, we examine similarity among the coding sequences of the first exon of beta-globin gene from eleven species, similarity of cDNA sequences of beta-globin gene from eight species, and similarity of the whole mitochondrial genomes of 18 eutherian mammals. The experimental results well demonstrate the effectiveness of the proposed method.

The replication of the foot-and-mouth disease virus (FMDV) genome is critically dependent upon the activity of a virally encoded RNA-dependent RNA polymerase (RdRp). In this study, four mutant RdRps of FMDV were isolated from viral quasi-species treated with ribavirin, of which two were single mutants (L123F and T381A) and two were double mutants (T291I/T381I and L123F/F244L). The mutant proteins were expressed in Escherichia coli and purified by His-bind resin chromatography. In combination with real-time RT-PCR, an in vitro RNA replication system that uses genome RNA/VPg as template-primers was used to determine polymerase activity. Mutant L123F exhibited a 0.6-fold decrease (p < 0.001) in polymerase activity relative to wild-type RdRp, whereas the activity of L123F/F244L and T381A was undetectable. Surprisingly, the activity of T291I/T381I yielded a 0.7-fold increase (p < 0.001) as compared to wild-type. In order to study the structure-function relationship of RdRp, all structures of the RdRp-RNA template-primer complex were obtained through homology modeling and molecular docking. The VPg1 orientation in the RdRp-VPg1 complexes was determined and analyzed with mathematical methods. Our results reveal that the orientation of VPg after binding to the polymerase determines the FMDV RdRp catalytic activity, which provides a basis for the rational design of novel antiviral agents.

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In this work we report a simple way to measure the similarity between two nucleotide sequences by using graph theory and information theory. This method reported allows for theoretical comparisons of naturally occurring nucleotide sequences.