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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
native structure,
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.

- DDG
of tetrahedron tiles and local protein structure

(The slides and manuscript of the lecture given at Plovdiv2007)

- Algorithm
of the 5-tile encoding (or "D
^{2}encoding")

[NOTE] "D

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.

One
could describe
triangle
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
tetrahedrons,
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
image.)

[NOTE] A trajectory of d-simplices
induces a flow of (d-1)-simplices
on its surfaces.

Let’s consider the two-dimensional object of Fig.A (a), which is approximated by a polygon of 16 vertices as shown in Fig. A (b). (Click the right image to view larger image.)

(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" representation

According 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).Putting one more unit cube (colored red), we obtain another representation as shown in Fig. B (c). Now, the object is decomposed into two parts, each of which is represented by a flow of triangles without any hole.

Recall
that we use weights in scales for weighing objects and a ruler
to measure their length. Biological processes, such as signal
transmission, cell-fate
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.)

- Plan A: Define simultaneous equations, where a given hetro number is specified as a root of equations.
- Plan B: Design a parallel programmming language, where a given hetero number is specified as the semactics of a program.

- DARPA's Programmable Matter program [article1, article2]
- DARPA News Release (April 17, 2009)
- Article: "Universal ‘Rubik’s Cube’ Could Become Pentagon Shapeshifter," June 5, 2009)
- Right-Angle Tetrahedra [Milli-biology
project]

(Slide from E.Demaine, "The Theory and Practice of Origami," FAB5: The Fifth International Fab Lab Forum and Symposium on Digital Fabrication, Aug. 20, 2009)