arXiv subjects

Naoto Morikawa

Publications and source records attributed to Naoto Morikawa.

7 recordsLinked to original sources

arXiv2026

A novel mathematical model of protein interactions (2): The mathematics behind the model

This article is a sequel to ``A novel mathematical model of protein interactions from the perspective of electron delocalization (2025)'', where protein molecules are modelded as loops of triangles with no singular holes inside. Here we consider two questions: (1) How can we define the shape of a molecule? (i.e., how can we define the shape of a loop with no singular holes inside?), and (2) Is the given loop a molecule? (i.e., are the holes within a loop are singular?). These questions are answered by (1) giving a defining system of equations for the shape of a molecule using concepts from category theory, and (2) giving a equation that determines whether holes within a loop are singular using concepts from cohomology theory, respectively. No prior knowledge of the previous paper, category theory, or cohomology theory is required.

math.DG↗

arXiv2025

A Novel Mathematical Model of Protein Interactions from the Perspective of Electron Delocalization

Proteins are the workhorse molecules of the cell and perform their biological functions by binding to other molecules through physical contact. Protein function is then regulated through coupling of bindings on the protein (allosteric regulation). Just as the genetic code provides the blueprint for protein synthesis, the coupling is thought to provide the basis for protein communication and interaction. However, it is not yet fully understood how binding of a molecule at one site affects binding of another molecule at another distal site on a protein, even more than $60$ years after its discovery in 1961. In this paper, I propose a simple mathematical model of protein interactions, using a ``quantized'' version of differential geometry, i.e., the discrete differential geometry of $n$-simplices. The model is based on the concept of electron delocalization, one of the main features of quantum chemistry, Allosteric regulation then follows tautologically from the definition of interactions. No prior knowledge of conventional discrete differential geometry, protein science, or quantum chemistry is required. I hope this paper will provide a starting point for many mathematicians to study chemistry and molecular biology.

q-bio.BM↗

arXiv2011

A novel method for identification of local conformational changes in proteins

Motivation: Proteins are known to undergo conformational changes in the course of their functions. The changes in conformation are often attributable to a small fraction of residues within the protein. Therefore identification of these variable regions is important for an understanding of protein function. Results: We propose a novel method for identification of local conformational changes in proteins. In our method, backbone conformations are encoded into a sequence of letters from a 16-letter alphabet (called D2 codes) to perform structural comparison. Since we do not use clustering analysis to encode local structures, the D2 codes not only provides a intuitively understandable description of protein structures, but also covers wide varieties of distortions. This paper shows that the D2 codes are better correlated with changes in the dihedral angles than a structural alphabet and a secondary structure description. In the case of the N37S mutant of HIV-1 protease, local conformational changes were captured by the D2 coding method more accurately than other methods. The D2 coding also provided a reliable representation of the difference between NMR models of an HIV-1 protease mutant.

q-bio.BM↗

arXiv2007

Discrete differential geometry of tetrahedrons and encoding of local protein structure

Local protein structure analysis is informative to protein structure analysis and has been used successfully in protein structure prediction and others. Proteins have recurring structural features, such as helix caps and beta turns, which often have strong amino acid sequence preferences. And the challenges for local structure analysis have been identification and assignment of such common short structural motifs. This paper proposes a new mathematical framework that can be applied to analysis of the local structure of proteins, where local conformations of protein backbones are described using differential geometry of folded tetrahedron sequences. Using the framework, we could capture the recurring structural features without any structural templates, which makes local structure analysis not only simpler, but also more objective. Programs and examples are available from http://www.genocript.com .

math.CO↗

arXiv2006

Number sequence representation of protein structures based on the second derivative of a folded tetrahedron sequence

This paper proposes a new mathematical approach to characterize native protein structures based on the discrete differential geometry of tetrahedron tiles. In the approach, local structure of proteins is classified into finite types according to shape. And one would obtain a number sequence representation of protein structures automatically. As a result, it would become possible to quantify structural preference of amino-acids objectively. And one could use the wide variety of sequence alignment programs to study protein structures since the number sequence has no internal structure. The programs and this paper with clear figures are available from http://www.genocript.com.

q-bio.BM↗

arXiv2006

Discrete differential geometry of triangle tiles and algebra of closed trajectories

This paper proposes a new mathematical framework that can be applied to biological problems such as analysis of the structures of proteins and protein complexes. In particular, it gives a new method for encoding the three-dimensional structure of a protein into a binary sequence, where proteins are approximated by a folded tetrahedron sequence. It also gives a new algebraic framework for describing molecular complexes and their interactions. For simplicity, we shall explain the framework in the case of two-dimensional objects. Then, the binary code of a plane curve is obtained as the ``second derivative'' of the curve and ``fusion and fission'' of closed trajectories is described algebraically.

math.CO↗

arXiv2005

Discrete differential geometry of proteins: a new method for encoding three-dimensional structures of proteins

In nature the three-dimensional structure of a protein is encoded in the corresponding gene. In this paper we describe a new method for encoding the three-dimensional structure of a protein into a binary sequence. The feature of the method is the correspondence between protein-folding and ``integration''. A protein is approximated by a folded tetrahedron sequence. And the binary code of a protein is obtained as the ``second derivative'' of the shape of the folded tetrahedron sequence. With this method at hand, we can extract static structural information of a protein from its gene. And we can describe the distribution of three-dimensional structures of proteins without any subjective hierarchical classification.

math.CO↗