Proteins are the workhorse molecules of the cell, which are obtained by folding long chains of amino acids. Since not all shapes are obtained as a folded chain of amino acids, there should be global geometrical constr...Proteins are the workhorse molecules of the cell, which are obtained by folding long chains of amino acids. Since not all shapes are obtained as a folded chain of amino acids, there should be global geometrical constraints on the shape. Moreover, since the function of a protein is largely determined by its shape, constraints on the shape should have some influence on its interaction with other proteins. In this paper, we consider global geometrical constraints on the shape of proteins. Using a mathematical toy model, in which proteins are represented as closed chains of tetrahedrons, we have identified not only global geometrical constraints on the shape of proteins, but also their influence on protein interactions. As an example, we show that a garlic-bulb like structure appears as a result of the constraints. Regarding the influence of global geometrical constraints on interactions, we consider their influence on the structural coupling of two distal sites in allosteric regulation. We then show the inseparable relationship between global geometrical constraints and protein interactions;i.e. they are different sides of the same coin. This finding could be important for the understanding of the basic mechanisms of allosteric regulation of protein functions.展开更多
This paper shows the usefulness of discrete differential geometry in global analysis. Using the discrete differential geometry of triangles, we could consider the global structure of closed trajectories (of triangles)...This paper shows the usefulness of discrete differential geometry in global analysis. Using the discrete differential geometry of triangles, we could consider the global structure of closed trajectories (of triangles) on a triangular mesh consisting of congruent isosceles triangles. As an example, we perform global analysis of an Escher-style trick art, i.e., a simpler version of “Ascending and Descending”. After defining the local structure on the trick art, we analyze its global structure and attribute its paradox to a singular point (i.e., a singular triangle) at the center. Then, the endless “Penrose stairs” is described as a closed trajectory around the isolated singular point. The approach fits well with graphical projection and gives a simple and intuitive example of the interaction between global and local structures. We could deal with higher dimensional objects as well by considering n-simplices (n > 2) instead of triangles.展开更多
This paper proposes a novel discrete differential geometry of n-simplices. It was originally developed for protein structure analysis. Unlike previous works, we consider connection between space-filling n-simplices. U...This paper proposes a novel discrete differential geometry of n-simplices. It was originally developed for protein structure analysis. Unlike previous works, we consider connection between space-filling n-simplices. Using cones of an integer lattice, we introduce tangent bundle-like structure on a collection of n-simplices naturally. We have applied the mathematical framework to analysis of protein structures. In this paper, we propose a simple encoding method which translates the conformation of a protein backbone into a 16-valued sequence.展开更多
This paper proposes a novel four-dimensional approach to the structural study of protein complexes. In the approach, the surface of a protein molecule is to be described using the intersection of a pair of four-dimens...This paper proposes a novel four-dimensional approach to the structural study of protein complexes. In the approach, the surface of a protein molecule is to be described using the intersection of a pair of four-dimensional triangular cones (with multiple top vertexes). As a mathematical toy model of protein complexes, we consider complexes of closed trajectories of n-simplices (n=2,3,4...), where the design problem of protein complexes corresponds to an extended version of the Hamiltonian cycle problem. The problem is to find “a set of” closed trajectories of n-simplices which fills the n-dimensional region defined by a given pair of n+1 -dimensional triangular cones. Here we give a solution to the extended Hamiltonian cycle problem in the case of n=2 using the discrete differential geometry of triangles (i.e., 2-simplices).展开更多
This paper proposes a novel category theoretic approach to describe protein’s shape, i.e., a description of their shape by a set of algebraic equations. The focus of the approach is on the relations between proteins,...This paper proposes a novel category theoretic approach to describe protein’s shape, i.e., a description of their shape by a set of algebraic equations. The focus of the approach is on the relations between proteins, rather than on the proteins themselves. Knowledge of category theory is not required as mathematical notions are defined concretely. In this paper, proteins are represented as closed trajectories (i.e., loops) of flows of triangles. The relations between proteins are defined using the fusion and fission of loops of triangles, where allostery occurs naturally. The shape of a protein is then described with quantities that are measurable with unity elements called “unit loops”. That is, protein’s shape is described with the loops that are obtained by the fusion of unit loops. Measurable loops are called “integral”. In the approach, the unit loops play a role similar to the role “1” plays in the set Z of integers. In particular, the author considers two categories of loops, the “integral” loops and the “rational” loops. Rational loops are then defined using algebraic equations with “integral loop” coefficients. Because of the approach, our theory has some similarities to quantum mechanics, where only observable quantities are admitted in physical theory. The author believes that this paper not only provides a new perspective on protein engineering, but also promotes further collaboration between biology and other disciplines.展开更多
Self-assembling molecules are ubiquitous in nature, among which are proteins, nucleic acids (DNA and RNA), peptides and lipids. Recognizing the ability of biomolecules to self-assemble into various 3D shapes at the na...Self-assembling molecules are ubiquitous in nature, among which are proteins, nucleic acids (DNA and RNA), peptides and lipids. Recognizing the ability of biomolecules to self-assemble into various 3D shapes at the nanoscale, researchers are mimicking the self-assembly strategy for engineering of complex nanostructures. However, the general principles underlying the design of self-assembled molecules have not yet been identified. The question is “How to obtain a well-defined shape with desired properties by folding a chain of subunits (such as amino acids and nucleic acids)”, where properties are determined by the precise spatial arrangement of the subunits on the surface. In this paper, we consider the question from the viewpoint of the discrete differential geometry of n-simplices. Self-assembling molecules are then represented as a union of trajectories of 3-simplices (i.e., tetrahedrons), and the question is rephrased as a “boundary value problem” for flows on a space of tetrahedrons. Also considered is a characterization of two types of surface flows of n-simplices. It is a rough classification of surface flows, but may be essential in characterizing important properties of biomolecules such as allosteric regulation. The author believes this paper not only provides a new perspective for the engineering of self-assembling molecules, but also promotes further collaboration between mathematics and other disciplines in life science.展开更多
Proteins perform a variety of functions in living organisms and their functions are largely determined by their shape. In this paper, we propose a novel mathematical method for designing protein-like molecules of a gi...Proteins perform a variety of functions in living organisms and their functions are largely determined by their shape. In this paper, we propose a novel mathematical method for designing protein-like molecules of a given shape. In the mathematical model, molecules are represented as loops of n-simplices (2-simplices are triangles and 3-simplices are tetrahedra). We design a new molecule of a given shape by patching together a set of smaller molecules that cover the shape. The covering set of small molecules is defined using a binary relation between sets of molecules. A new molecule is then obtained as a sum of the smaller molecules, where addition of molecules is defined using transformations acting on a set of (n + 1)-dimensional cones. Due to page limitations, only the two-dimensional case (i.e., loops of triangles) is considered. No prior knowledge of Sheaf Theory, Category Theory, or Protein Science is required. The author hopes that this paper will encourage further collaboration between Mathematics and Protein Science.展开更多
This paper proposes a novel application of cohomology to protein structure analysis. Since proteins interact each other by forming transient protein complexes, their shape (e.g., shape complementarity) plays an import...This paper proposes a novel application of cohomology to protein structure analysis. Since proteins interact each other by forming transient protein complexes, their shape (e.g., shape complementarity) plays an important role in their functions. In our mathematical toy models, proteins are represented as a loop of triangles (2D model) or tetrahedra (3D model), where their interactions are defined as fusion of loops. The purpose of this paper is to describe the conditions for loop fusion using the language of cohomology. In particular, this paper uses cohomology to describe the conditions for “allosteric regulation”, which has been attracted attention in safer drug discovery. I hope that this paper will provide a new perspective on the mechanism of allosteric regulation. Advantages of the model include its topological nature. That is, we can deform the shape of loops by deforming the shape of triangles (or tetrahedra) as long as their folded structures are preserved. Another advantage is the simplicity of the “allosteric regulation” mechanism of the model. Furthermore, the effect of the “post-translational modification” can be understood as a resolution of singularities of a flow of triangles (or tetrahedra). No prior knowledge of either protein science, exterior calculus, or cohomology theory is required. The author hopes that this paper will facilitate the interaction between mathematics and protein science.展开更多
文摘Proteins are the workhorse molecules of the cell, which are obtained by folding long chains of amino acids. Since not all shapes are obtained as a folded chain of amino acids, there should be global geometrical constraints on the shape. Moreover, since the function of a protein is largely determined by its shape, constraints on the shape should have some influence on its interaction with other proteins. In this paper, we consider global geometrical constraints on the shape of proteins. Using a mathematical toy model, in which proteins are represented as closed chains of tetrahedrons, we have identified not only global geometrical constraints on the shape of proteins, but also their influence on protein interactions. As an example, we show that a garlic-bulb like structure appears as a result of the constraints. Regarding the influence of global geometrical constraints on interactions, we consider their influence on the structural coupling of two distal sites in allosteric regulation. We then show the inseparable relationship between global geometrical constraints and protein interactions;i.e. they are different sides of the same coin. This finding could be important for the understanding of the basic mechanisms of allosteric regulation of protein functions.
文摘This paper shows the usefulness of discrete differential geometry in global analysis. Using the discrete differential geometry of triangles, we could consider the global structure of closed trajectories (of triangles) on a triangular mesh consisting of congruent isosceles triangles. As an example, we perform global analysis of an Escher-style trick art, i.e., a simpler version of “Ascending and Descending”. After defining the local structure on the trick art, we analyze its global structure and attribute its paradox to a singular point (i.e., a singular triangle) at the center. Then, the endless “Penrose stairs” is described as a closed trajectory around the isolated singular point. The approach fits well with graphical projection and gives a simple and intuitive example of the interaction between global and local structures. We could deal with higher dimensional objects as well by considering n-simplices (n > 2) instead of triangles.
文摘This paper proposes a novel discrete differential geometry of n-simplices. It was originally developed for protein structure analysis. Unlike previous works, we consider connection between space-filling n-simplices. Using cones of an integer lattice, we introduce tangent bundle-like structure on a collection of n-simplices naturally. We have applied the mathematical framework to analysis of protein structures. In this paper, we propose a simple encoding method which translates the conformation of a protein backbone into a 16-valued sequence.
文摘This paper proposes a novel four-dimensional approach to the structural study of protein complexes. In the approach, the surface of a protein molecule is to be described using the intersection of a pair of four-dimensional triangular cones (with multiple top vertexes). As a mathematical toy model of protein complexes, we consider complexes of closed trajectories of n-simplices (n=2,3,4...), where the design problem of protein complexes corresponds to an extended version of the Hamiltonian cycle problem. The problem is to find “a set of” closed trajectories of n-simplices which fills the n-dimensional region defined by a given pair of n+1 -dimensional triangular cones. Here we give a solution to the extended Hamiltonian cycle problem in the case of n=2 using the discrete differential geometry of triangles (i.e., 2-simplices).
文摘This paper proposes a novel category theoretic approach to describe protein’s shape, i.e., a description of their shape by a set of algebraic equations. The focus of the approach is on the relations between proteins, rather than on the proteins themselves. Knowledge of category theory is not required as mathematical notions are defined concretely. In this paper, proteins are represented as closed trajectories (i.e., loops) of flows of triangles. The relations between proteins are defined using the fusion and fission of loops of triangles, where allostery occurs naturally. The shape of a protein is then described with quantities that are measurable with unity elements called “unit loops”. That is, protein’s shape is described with the loops that are obtained by the fusion of unit loops. Measurable loops are called “integral”. In the approach, the unit loops play a role similar to the role “1” plays in the set Z of integers. In particular, the author considers two categories of loops, the “integral” loops and the “rational” loops. Rational loops are then defined using algebraic equations with “integral loop” coefficients. Because of the approach, our theory has some similarities to quantum mechanics, where only observable quantities are admitted in physical theory. The author believes that this paper not only provides a new perspective on protein engineering, but also promotes further collaboration between biology and other disciplines.
文摘Self-assembling molecules are ubiquitous in nature, among which are proteins, nucleic acids (DNA and RNA), peptides and lipids. Recognizing the ability of biomolecules to self-assemble into various 3D shapes at the nanoscale, researchers are mimicking the self-assembly strategy for engineering of complex nanostructures. However, the general principles underlying the design of self-assembled molecules have not yet been identified. The question is “How to obtain a well-defined shape with desired properties by folding a chain of subunits (such as amino acids and nucleic acids)”, where properties are determined by the precise spatial arrangement of the subunits on the surface. In this paper, we consider the question from the viewpoint of the discrete differential geometry of n-simplices. Self-assembling molecules are then represented as a union of trajectories of 3-simplices (i.e., tetrahedrons), and the question is rephrased as a “boundary value problem” for flows on a space of tetrahedrons. Also considered is a characterization of two types of surface flows of n-simplices. It is a rough classification of surface flows, but may be essential in characterizing important properties of biomolecules such as allosteric regulation. The author believes this paper not only provides a new perspective for the engineering of self-assembling molecules, but also promotes further collaboration between mathematics and other disciplines in life science.
文摘Proteins perform a variety of functions in living organisms and their functions are largely determined by their shape. In this paper, we propose a novel mathematical method for designing protein-like molecules of a given shape. In the mathematical model, molecules are represented as loops of n-simplices (2-simplices are triangles and 3-simplices are tetrahedra). We design a new molecule of a given shape by patching together a set of smaller molecules that cover the shape. The covering set of small molecules is defined using a binary relation between sets of molecules. A new molecule is then obtained as a sum of the smaller molecules, where addition of molecules is defined using transformations acting on a set of (n + 1)-dimensional cones. Due to page limitations, only the two-dimensional case (i.e., loops of triangles) is considered. No prior knowledge of Sheaf Theory, Category Theory, or Protein Science is required. The author hopes that this paper will encourage further collaboration between Mathematics and Protein Science.
文摘This paper proposes a novel application of cohomology to protein structure analysis. Since proteins interact each other by forming transient protein complexes, their shape (e.g., shape complementarity) plays an important role in their functions. In our mathematical toy models, proteins are represented as a loop of triangles (2D model) or tetrahedra (3D model), where their interactions are defined as fusion of loops. The purpose of this paper is to describe the conditions for loop fusion using the language of cohomology. In particular, this paper uses cohomology to describe the conditions for “allosteric regulation”, which has been attracted attention in safer drug discovery. I hope that this paper will provide a new perspective on the mechanism of allosteric regulation. Advantages of the model include its topological nature. That is, we can deform the shape of loops by deforming the shape of triangles (or tetrahedra) as long as their folded structures are preserved. Another advantage is the simplicity of the “allosteric regulation” mechanism of the model. Furthermore, the effect of the “post-translational modification” can be understood as a resolution of singularities of a flow of triangles (or tetrahedra). No prior knowledge of either protein science, exterior calculus, or cohomology theory is required. The author hopes that this paper will facilitate the interaction between mathematics and protein science.
