Protein is a
sequence of amino acids
of length typically less than
1,000, where there are 20 kinds of amino acids. In nature, each protein
is folded into a well-defined three dimensional structure, the
and its functional properties are largely determined by the structure.
(Click the right image to view larger image.)
We approximate native protein structures by a folded sequence of tetrahedrons, where the shape of the tetrahedron sequence is given as a trajectory of some discrete differential vector field of tetrahedron tiles. To absorb irregularity of actual protein structures, we allow rotation and translation of tetrahedron tiles.
Let's consider unit cubes and pile them up in the direction from (1,1,1) to (-1,-1,-1). And view the resulting surface from above. Then, if one prints diagonal lines on the upper faces of each cube, he/she obtains a drawing made up of the diagonal lines, which defines a flow of triangle tiles (2-simplices). If we consider unit cubes in the four-dimensional space, we would obtain a flow of tetrahedron tiles (3-simplices). Moreover, any trajectory of tetrahedrons could be implemented by "origami" folding. (Click the right image to view larger image.)
See the PROGRAMS section for HeteroNumberViewer to view closed tetrahedron trajectories of affine vector fields.
flows on a cube, using a collection of overlapping local
"flat" charts which cover the cube as shown on the right. Note that
the bottom flow of triangles (example 3) in the figure could
be extended to
a flow of
which is obtained by partitioning a
rhombic dodecahedron into four identical parts as shown in the
figure of section (4) below. (Click the right image to view larger
[NOTE] A trajectory of d-simplices induces a flow of (d-1)-simplices on its surfaces.
(2-2-1) "Triangle mesh" representation
Fig. A (c) shows a division of the object into 72 triangle pieces, where we should solve the following two problems: (i) could we connect all the triangles contained in the object to form a chain which visits each triangle exactly once and also returns to the starting triangle (Hamiltonian path)?, and (ii) how should we decompose the object to obtain a set of Hamiltonian paths of triangles which cover the object.
Because it is extremely time-consuming to search a Hamiltonian path, we often end up with a set of short chains as shown in Fig. A (d), where chains are terminated when they come to the boundary edges.
(2-2-2) "Flow of triangles" representationAccording to the discrete differential geometry of triangles, once the boundary is given (Fig. B (a)), connections over all the triangles contained in the object are uniquely determined. In the case of Fig. A (a), the object is represented by a flow of triangles with one hole which is consisted of 10 triangles as shown in Fig. B (b).
that we use weights in scales for weighing objects and a ruler
to measure their length. Biological processes, such as signal
regulation, transcription, and so on, are not performed by freely
diffusing and occasionally colliding proteins. Instead, proteins
usually do their jobs by forming structured ensemble of proteins, i.e. protein
complexes. The schematic figures on the
right show two notable features of the
formation of protein
complexes. (Click the right image to view larger image.)
Assigning hetero numbers to proteins and their complexes, we could describe the formation of protein complexes as additon of the corresponding hetero numbers.
Recall that we use weights in scales for weighing objects and a ruler to measure their length. Hetero numbers are a system of units for measuring shape of objects such as proteins. See the PROGRAMS section for HeteroNumberViewer to view the 3-dim'l hetero numbers.
Roughly speaking, d-dim'l hetero numbers are a subset of closed affine trajectories of d-simplex tiles, where "fusion and fission" of closed trajectories is described algebraically. (Click the right image to view larger image.)