Global Geometrical Constraints on the Shape of Proteins and Their Influence on Allosteric Regulation

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.


Introduction
In this paper, we consider global geometrical constraints on the shape of proteins, using the mathematical toy model of proteins proposed in [1].Proteins are the workhorse molecules of the cell, which are obtained as a complex of folded chains of amino acids.Since the function of proteins depends primarily on their shape, structural studies are essential for understanding proteins.In our approach, protein molecules are represented as a complex of closed trajectories of tetrahedrons.Then, the surface of proteins is obtained as the intersection of a pair of four-dimensional cones [2].Interactions between proteins are defined (or mimicked) as "fusion and fission" of closed trajectories.
Previously, two types of geometrical constraints are known in the study of protein structures.One is a set of constraints on the backbone conformation due to collisions between atoms [3].The backbone conformation is determined by torsion angle pairs ( ) , φ ψ along the backbone, and their allowed values are shown in the Ramachandran map [4].The other is a set of constraints on relative distances between certain pairs of atoms, which are obtained from either physical experiments or theoretical estimates.The determination of protein structures which satisfy a set of constraints on inter-atomic distances, known as the distance geometry problem, is an important problem in structural biology [5].
In virology, another type of geometrical constraints, the symmetry of the virus structure, is also considered.Viruses are metastable macromolecular assemblies composed of the viral genome enclosed within the protein shells, called viral capsids [6].Virus capsids are highly specific assemblies that are formed from a large number of often identical subunits.Formulated in [7] is a set of structural constraints on the subunit arrangements, using an extension of the underlying symmetry group.On the other hand, [8] finds that some viruses allow their representation as two-dimensional monohedral tilings of a bound surface, where each tile represents a subunit.Note that viral molecules consist of separeated parts.Protein molecules are obtained by folding a chain of linked parts and it is impossible to describe the shape of proteins by symmetry alone nor to describe their surface by tiling of basic subunits.What we will consider below are global constraints on the shape of a complex of folded chains of basic blocks, such as triangles and tetrahedrons.One of the advantages of our model is the correspondence between "the shape of molecules" and "interaction between molecules".Since a protein's function is largely determined by its shape, constraints on the shape of a protein should have some influence on its interaction with other proteins.In our model, the geometrical constraints on the shape of a molecule correspond to the constraints on the interaction between three molecules, such as allosteric regulations.In the section before the conclusion, we will explain the correspondence between geometrical constraints on the shape and allosteric regulations using an example.An introduction to allosteric regulation is also given there.
Finally, Genocript (http://www.genocript.com) is the one-man bio-venture started by Naoto Morikawa in 2000 which is developing software tools for protein structure analysis.

Discrete Differential Geometry of Triangles
Now, let us consider the case of closed trajectories of triangles to explain the basic ideas behind our approach.For detailed description, see [1] and [2].
In the following, the coordinates of points in the N-dimensional Eucledean space N E ( 3 N = or 4) are represented by a monomial in N indeterminates 0 1 1 , , , N x x x −  for space saving purposes.For example, point ( ) 3 3   , , l m n E ∈ ⊂  is represented by 0 1 2 l m n x x x , where  denotes the set of all integers.( ) px , respectively.Note that i j j i x x x x = for all pairs of i and j.

Flows of Triangles
Flows of triangles are defined using unit cubes in E 3 .As shown in Figure 1(a), unit cubes are piled-up in the direction of ( ) 3 , where each of the three upper faces is divided into two triangles by the vertical diagonal (thick line).Then, a flow of triangles is obtained along the diagonals (Figure 1(b)).
That is, the piled-up cubes form a mountain range-like structure and the vertical diagonals on its surface determine a flow of "slant" triangles on the slope.
As an example, let us consider the unit cube with the eight corner points 0, x , 0 2 x x , 0 1 2 x x x , and 1 2 x x (Figure 1(c)).Let 0 1 P = , 1 0 P x x = , and 3 1 P x = .Then, the upper face 0 1 2 3 P P P P is divided into two "slant" triangles 0 1 2 P PP and 0 3 2 P P P .The triangle flow goes down (or up) along the edge 0 2 P P at 0 1 2 P PP and 0 3 2 P P P .
In the following, we give the mathematical definition of the mountain range-like structure and the associated flow of triangles.
Definition 2 (Slant Triangles) Let i.e., ( ) ( ) ( ) ( ) ( ) The line segment joining vertex a and vertex is called the diagonal edge of the slant triangle.The set 2 S of all slant triangles is defined by: ) 1 and 012 , 1 and 102 .

P P P x x a
Their diagonal edges are the line segment 0 2 P P .
Definition 3 (Gradient of Slant Triangles) Let ( ) ( ) gradient Ds of s is defined by A flow of "flat" triangles is defined on the hyperplane ) ( ) .    2(a)).By patching "consistent" local trajectories together, we will obtain a flow of flat triangles as shown in Figure 1(a).

Vector Fields of Triangles
As shown in Figure 1(a), a mountain range-like structure induces a flow of triangles on 2 B .We can define a "tangent space" structure on the space 2 B of flat triangles, where each flat triangle assume one of the three gradient vectors x x , 0 2 x x , and 0 1 x x .[ ] .  ( )

P P P x x =
).

Contour of Closed Trajectories of Triangles
The ridge lines of tangent cones are given by three vectors ( ) 1, 0, 0 , ( ) 0,1, 0 , and ( ) 0, 0,1 .To compute the contour of the region of 2 D H swept by a set of closed trajectories of flat triangles, we will consider another type of triangular cones whose ridge lines are given by the slopes of slant triangles, i.e., ( )

( ) (
) ( ) , and Let c be a tangent cone and w be a cotangent cone.Then, we can divide all the slant triangles of the flow induced by c into three groups: 1) inside w, 2) outside w, and 3) on the surface of w (Figure 2 Suppose that φ is the set of 3 L * lattice points on the intersection of the surface of 0 c and the surface of ( ) with respect to 0 c is defined by We also call ( ) the (one-dimensional) contour with respect to 0 c .Definition 21 Let 0 c be a three-dimensional tangent cone.Set | intersects with the surface of .
Let c be a three-dimensional tangent cone.Let c V be the vector field induced by c.We define the region .   , , , ,   , ,  ,   , , , , , , , , , , , , , , , , c Cone P P P P P w c Cone K K K c P P P P P P P P P P P P P P P P

2.4..Constraints on the Contour of Closed Trajectories
Let c be a three-dimensional tangent cone.We have computed the contour of ( ) t R c using c and the associated cotangent roof We denote the top vertices of an inverted cotangent cone iv by ( ) , ., : .( ) Example 10 In the case of Figure 3, IRoof P P P P P P P P P P P P IRoof P P P P In Theorem 2, we have computed the "contour" of ( ) t R c for a give tangent cone c (Figure 5(a)).Now, we will compute the "contour" of ( ) ( ) , , Starting with some definitions, we will consider the correspondence between the two types of regions of .
It is not difficult to show that the maps are well-defined.By Theorem 2, we have ( ) 5 for the correlation between the relevant maps.
Theorem 4 2 ι is not surjective.Proof.Let us consider the case of Figure 6(a).We have w Cone K iv ICone K w iv P P P P P P P P P P R c w iv gets dented on the bottom, where

P x
= , and ) Corollary 2 (Self-eclipse of Rc(w, iv)) There exists a contour pair ( ) That is, not all closed polygonal lines defined by contour pairs correspond to a closed trajectory of triangles induced by a tangent cone.In other words, there exist global geometrical constraints on the contour of closed trajectories of triangles.
Definition 36 (Self-eclipsed closed trajectory complexes) A contour pair ( )  In the next section, we will consider geometrical constraints on the shape of closed trajectories of tetrahedrons.As an example, it will be shown that a garlic bulb-like structure appears as a result of the constraints, where a flattened dodecahedron gets dented on the bottom and has vertical linear grooves on the side (Figure 11).

Flows of Tetrahedrons
Now let us consider the case of tetrahedrons.To define a flow of tetrahedrons, we use unit cubes in the four-dimensional Euclidean space E 4 .By piling up unit cubes in the direction of ( ) | , , , .
, , ,  The line segment joining vertex a and vertex , i.e., the cube-diagonal, is called the diagonal edge of the slant tetrahedron.Then, the four upper faces of each unit cube are divided into six tetrahedrons along the diagonal edge as shown in Example 12.The set 3 S of all slant tetrahedrons is defined by: Example 12 Shown in Figure 7(a) is a four-dimensional unit cube at the origin 1 P .The upper face 1 y yz z x xy xyz xx PP P P P P P P of the cube is divided into six tetrahedrons along the cube-diagonal  Definition 39 (Gradient of Slant Tetrahedrons) Let

PP P P x x x x x x x x x PP P P x x x x x x x x x PP P P x x x x x x x x x PP P P x x x x x x x x x PP P P x x x x x x x x x PP P P x x x
The gradient Ds of s is defined by .

Ds x x x
.

xy xyz H y xy xyz H y yz xyz H z yz xyz H z xz xyz H x xz xyz H
Remark.Note that each tetrahedron has two long edges and four short edges, where the diagonal edge correspond to a short edge.Flows of tetrahedrons go along the diagonal edge at each tetrahedron.( )

By projecting slant tetrahedrons onto
The local trajectory at s is given by either , and Thick

Vector Fields of Tetrahedrons
The tangent space on the space 3 B of flat tetrahedrons is defined in the same way as the tangent space

TB t x x x x x x x x x x x x
Tangent cones are also defined similarly for The set of all the top vertices of a cone c is denoted by   ( ) The flow of tetrahedrons determined by c V is called the flow of tetrahedrons induced by c.

Shape of Closed Trajectories of Tetrahedrons
To compute the surface (2-faces) of the region of 3D H swept by a set of closed trajectories of flat tetrahedrons, we will consider another type of cones whose ridge lines are given by the "slopes" of slant tetrahedrons, i.e., ( ) and ( ) We denote the top vertices of a cotangent cone c by ( ) In the case of Figure 8, , .
As in the case of slant triangles, all the slant tetrahedrons of the flow induced by a tangent cone are divided into three groups by a cotangent cone: 1) inside the cotangent cone, 2) outside the cotangent cone, and3) on the surface (3-faces) of the cotangent cone (Figure 9).Unlike the case of triangles, multiple types of slant tetrahedrons are on the surface (3-faces) of the cotangent cone (tetrahedrons B, C, D, E, F in Figure 9).Among them, only two types of tetrahedrons (B and F) flow through the surface (2-faces). .
with respect to 0 c is defined by We also call ( ) (two-dimensional) surface vein with respect to 0 c .
Definition 57 Let 0 c be a four-dimensional tangent cone.Set | intersects with the 3-faces of .
c be a four-dimensional tangent cone.We define the region ( ) (  ( ) R c is the rhombic dodecahedron surrounded by the black thick lines, which consists of four closed trajectories of length six, i.e., consists of 24 tetrahedrons.
All the 24 tetrahedrons of ( ) , , t top c L * ∂ ⊂ / implies not only the existence of loopholes but also dents and bulges on the surface.That is, the tetrahedrons of type B and F correspond to loopholes, the tetrahedrons of type C to dents, and the tetrahedrons of type E to bulges (Figure 9).
To define a four-dimensional cotangent roof * Roof A for any R c is the rhombic dodecahedron surrounded by the black and grey thick lines, which consists of a closed trajectories of length 24.In the figure, the tetrahedrons with the grey diagonal edge are type E. The tetrahedrons with the black diagonal edge are type D.

Constraints on the Shape of Protein Molecules
In this paper, we consider the shape of complexes of closed trajectories of  Definition 66 (Surface Pairs) Let w be a three-dimensional cotangent cone.
Let iv be a three-dimensional inverted cotangent cone.A pair ( ) , w iv of w and iv is called a four-dimensional surface pair.
perform their function.In high-throughput proteomics, proteins are characterized using a interaction network between proteins and intermediate protein complexes.Since the function of a protein is primarily determined by the three-dimensional shape, it is the shape of proteins that is characterized by the interaction network.
In this section, we will consider "interaction" between closed trajectories of triangles as a simplified geometrical description of protein interactions.Despite its simplicity, the closed trajectory model of protein interaction gives a novel geometrical interpretation of the difference between direct interactions of two proteins and cooperative interactions of three proteins (such as allosteric regulation).

Fusion and Fission of Closed Trajectories of Triangles
We have seen in the previous sections that vector fields of triangles are associated with three-dimensional tangent cones.Here we will define "fusion and fission" of closed trajectories of triangles using the tangent cone structure.For the sake of simplicity, we only consider the case of flows of triangles.Let V respectively, where a N and b N is the numbers of the closed trajectories.Then, addition of closed trajectories of triangles is defined by Example 34 Shown in Figure 12(a) is a flow of triangles consisting of 36 closed trajectories of length six and infinitely many open trajectories of various lengths, where each closed trajectory sweeps a hexagonal region.By putting unit cubes on the associated tangent cone, we obtain another decomposition of the same region into a set of closed trajectories as shown in Figure 12(b).Then, we have  Note that m 0 is the closed trajectory given in Figure 1.That is, we obtained m 0 as a result of "fusion and fission" of 16 hexagons.Then, m 0 appears as a "factor" of a longer closed trajectory m 6 .
In the above examples, closed trajectories x i s (of length six) are given first.Then, m 0 is obtained as a result of interactions of the x i s.The challenge we propose is to give a set of equations of m 0 on variables x i s first, and solve the system of simultaneous equations.
Open Problem 1 (Simultaneous equations for shape) Let { } , , , M m m m  be a finite set of closed trajectories of length longer than six.Suppose that we are given a finite set of addition equations with respect to x i s and m j s: ) ( ) Find three-dimensional tangent cones for the variables x i s that make the addition equations true, where x i s are assigned the closed trajectory induced by the corresponding tangent cone.Then, m i s are obtained as intermediate products of the interactions between x i s.
Since the interaction of closed trajectories is primarily determined by their contours, it is their contours which are characterized by a set of simultaneous equations.In the case of Open Problem 1, a closed trajectory m 0 is characterized N. Morikawa DOI: 10.4236/am.2018.9100761148 Applied Mathematics using interactions between closed trajectories of length six and other closed trajectories.Therefore, the set of equations is nothing but a specification of the shape of m 0 if m 0 is uniquely determined.

Allosteric Regulation of Interactions
Now let us consider the difference between direct interactions of two proteins and cooperative interactions of three proteins (such as allosteric regulation).In our closed trajectory model, allosteric regulation corresponds to the complex of self-eclipsed closed trajectories of triangles (Definition 36).We will start with a brief introduction to allosteric regulation.

Introduction to Allosteric Regulation
In biological systems, all proteins bind to other molecules to carry out their functions.For example, enzymes bind to one or more reactant molecules to catalyze chemical reactions in our body.The region on the surface to which other molecules bind is called the active site.
The binding of a molecule at an active site is often controlled by the binding of another molecule at a distant site other than the active site.This type of regulation of protein function is called allosteric regulation.The distant site is called an allosteric site.
Allosteric regulation, which is known as "the second secret of life"', second only to the genetic code [9] [10], is ubiquitous in biological processes.But we still lack general understanding of the mechanisms underlying the coupling between allosteric and active sites [11]  Currently almost all the drugs modify the actions of proteins by directly binding to their active sites.On the other hand, gaining increasing attention recently in drug discovery is another type of drugs, called allosteric drugs, which bind to the allosteric sites on their target proteins [16] [17] [18].This is because allosteric drugs have several advantages over traditional drugs, such as higher specificity, fewer side effects, and lower toxicity.
However, allosteric drug discovery is more challenging than traditional drug discovery due to difficulties in identification of allosteric sites, prediction of drug modulatory effects, and others.For example, allosteric sites may have features we are not yet aware of because of our insufficient understanding of how coupling between the active site and the allosteric site occurs.
In the past ten years, various computational approaches have been developed is obtained by putting two unit cubes on the tangent cone of (a) (Figure 13(b)).
We also obtain the interaction of the three closed trajectories m 0 , m 2 , and x 22 by putting one more unit cube on the tangent cone of (b) (Figure 13(c)).However, m 0 and m 2 do not interact without the binding of x 22 because of the overlap of the slopes of the tangent cone, i.e., self-eclipse.That is, ( ) In this case, x 22 activates the interaction between m 0 and m 2 .

Conclusions
We have proposed a novel simplified geometrical description of the shape of protein molecules and their interactions.Using the model, we have identified not only global geometrical constraints on the shape of proteins, but also their influence on protein interactions.As an example of the global constraints, a "garlic-bulb like structure" was shown.As an example of their influence on interactions, the structural coupling between active and allosteric sites was considered.In particular, our model gives a novel geometrical interpretation of the long-distance regulation of protein interactions, which could be important for the understanding of the basic mechanisms of allosteric regulation of protein functions.
As for future research questions, we have already proposed an open problem in the text, i.e., the problem of simultaneous equations for shape.Just as the function (i.e., shape) of a protein is determined by its interaction with other molecules, the shape of a complex of closed trajectories may be determined uniquely by its interaction (i.e., fusion and fission) with other closed trajectories.
triangle defined by three points a,

Figure 1 .
Figure 1.Flow of triangles: (a) A mountain range-like structure obtained by piling up unit cubes in the direction of ( ) 1, 1, 1 − − − , whose peaks are

Example 2
In the case of Example 1, the slope of 0 Flows of slant triangles along the diagonal edges are defined as follows.Definition 4 (Local Trajectories of Slant Triangles) Let 2 s S ∈ .The local trajectory of slant triangles at s is a set of three consecutive slant triangles, consisting of s and two adjacent slant triangles which do not include the diagonal edge of s.By patching "consistent" local trajectories together, we will obtain a flow of slant triangles as shown in Figure 1(a).Let( ) ( )

2 B of all flat triangles on 2
a flat triangle.The line segment joining diagonal edge of the flat triangle (Figure 1(c)).The set

Figure 2 .
Figure 2. Slant triangles: (a) The four local trajectories of slant triangles at

−.−. 2 TB on 2 B
Shown above is a schematic diagram of the relationship between the three vertices of a slant triangle and a 2-face of In the diagram, the diagonal edges of slant triangles are drawn with thick line, where the diagonal edges on the 2-face are colored black and the others are colored grey.Note that all slant triangles are projected onto the same flat triangle Triangle D intersects the 2-face of the cotangent cone.Triangles E, F,and G are located outside the cotangent cone.Definition 7 (Tangent Space) The tangent space is defined by

)Example 4
Let c be a three-dimensional tangent cone.The surface lattice points ( ) pt c ∂ of c is the set of all the L 3 lattice points on the surface of c, i.e., The surface lattice points of the three-dimensional tangent cone

Definition 12 (Example 5
Let c be a three-dimensional tangent cone.Then, S d c is the set of all the slant triangles on the surface of c, i.e.Vector Fields) Let c be a three-dimensional tangent cone.The vector field c V induced by c on 2 B is defined by The flow of triangles determined by c V is called the flow of triangles induced by c.In the case of Figure1(c),

.
Let w be a three-dimensional cotangent cone.The surface lattice points ( ) pt w ∂ of w is the set of all the 3 L * lattice points on the surface of w, i.e.,

Example 7
The surface lattice points of the three-dimensional cotangent cone

Figure 3 .
Figure 3.The closed trajectory of Figure 1 (a) and the associated contour pair: (a) Tangent cone (b)).In particular, we can compute the contour of closed trajectories induced by a tangent cone using a cotangent cone as shown below.c be a three-dimensional tangent cone.

the polygonal line obtained by joining the adjacent 3 L
the surface of the associated cotangent roof.Note that ( )

Theorem 1
Let c be a three-dimensional tangent cone.Suppose that all the slant triangles of S d c (i.e., slant triangles on the surface of c) into two groups: inside the roof and outside the roof.That is, the region swept by all the closed trajectories of c V .Proof.consists of the diagonal edges of slant triangles on the surface of c.Since flows of slant triangles go along the diagonal edge at each slant triangle, there is no slant triangle crossing of the slant triangles on the surface of c into two parts:

Example 8
. Let c a three-dimensional tangent cone.follows immediately from the theorem.□ In the case of Figure3(a), the closed polygonal line vertices and 12 line segments, where Figure4).Now, we will compute regions of

.
Let iv be a three-dimensional inverted cotangent cone.The surface lattice points ( ) pt iv ∂ of iv is the set of all the 3 L * lattice points on the surface of iv, i.e.,

Figure 4 .
Figure 4. Schematic diagram showing the procedure for computing the contour of a region of 2 D H .The upper row shows the procedure for a region

φ is the set of 3 L
* lattice points on the intersection of the surface of w and the surface of iv.be a three-dimensional contour pair.Let ( ) 3 , c w iv E Φ ⊂ be the polygonal line obtained by joining the adjacent 3 L * lattice points of ( ) , c w iv φ (Figure 3(c)).Since all the points of ( ) , c w iv φ are on the surface of w (or iv), the points of ( ) , c w iv φ are connected along the surface of w (or iv).Note that ( ) , c w iv φ forms a closed polygonal line if ( ) top w or ( ) itop iv is finite (Figure 3(c)).Definition 29 .Let ( ) , w iv be a three-dimensional contour pair.The one-dimensional surface mesh ( ) , c m w iv with respect to ( )

Theorem 2
the (one-dimensional) contour with respect to ( ) , w iv .Remark Note that we have two types of one-dimensional surface meshes, i.e., the contour ( ) t m c with respect to a tangent cone c and the contour ( ) .Let ( ) , w iv be a three-dimensional contour pair.Let c be a three-dimensional tangent cone.Suppose that ( ) top c is finite and

.
slope inclination of cotangent cones is steeper than that of tangent cones.On the other hand, That is, p resides on the part of the surface of ( ) c iv c which is expanded by the "roof" operation.But the expanded part of ( ) c iv c is strictly contained in ( ) c w c , i.e.,( ) , c R w iv for a given contour pair ( ) , w iv (Figure 5(b)).Definition 32 Let ( ) , w iv be a three-dimensional contour pair.The tangent cone ( ) , w iv be a three-dimensional contour pair.Then, s. Definition 33 Sets of there-dimensional cones are defined by N. Morikawa DOI: 10.4236/am.2018.9100761129 Applied Mathematics

Figure 5
Figure 5. Maps between , t n R and , c n R and the relevant maps between sets of 1 n + forms a flattened hexagon as shown in the figure.

Figure 6 .
Figure 6.Self-eclipsed close trajectories: (a) Contour pair ( ) 0 0 , w iv and trajectories of triangles induced by a three-dimensional tangent cone c is called a self-eclipsed closed trajectory complex (abbreviated as SECT) if there exists a self-eclipsed contour pair ( ) the case of Figure6(b), = (Figure 6(c)).That is, the closed trajectory of Figure 6(b) is a self-eclipsed closed trajectory complex.
we will obtain a flow of "slant" tetrahedrons as in the case of flows of triangles.Definition 37 (Standard Lattice) The four-dimensional standard lattice L We denote the group of all permutations of the four-element set { }

Figure 7 .
Figure 7. Slant tetrahedrons: (a) A four-dimensional unit cube shown in the Schlegel diagram (below) and its projection image on hyperplane 3D H (above), where Flows of slant tetrahedrons along the diagonal edges are defined as follows.Definition 40 (Local Trajectories of Slant Tetrahedrons) Let 3 s S ∈ .A local trajectory of slant tetrahedrons at s is a set of three consecutive slant tetrahedrons, consisting of s and two adjacent slant tetrahedrons which do not include the diagonal edge of s.By patching "consistent" local trajectories together, we obtain a flow of slant tetrahedrons.

Example 15
In the case of Example 12 (Figure 7(a)), the projection image 1 x xy y z xz xyz yz "consistent" local trajectories together, we obtain a flow of flat tetrahedrons as shown in Figure8.

Figure 8 .
Figure 8. Flows of tetrahedrons: (a) Closed trajectories of flat tetrahedrons induced by { } polygonal lines (black and grey) indicate the diagonal edges of flat tetrahedrons, where the black line indicate the polygonal line passing through xyz Q , xzw Q , and xyw Q .Note that there exist infinitely many closed trajectories of length six and length twelve; (b) Closed trajectories of flat tetrahedrons induced by = .Thick polygonal lines (black and grey) indicate the diagonal edges of flat tetrahedrons.Black lines indicate the polygonal lines passing through xyz Q , xzw Q , xyw Q , or yzw Q .Grey polygonal lines correspond to closed trajectories of length six and length twelve.
tangent space at t is denoted by [ ] 3 TB t .Note that there exists a one-to-one correspondence be a four-dimensional tangent cone.The peaks on the boundary of c is defined by

.
Let c be a four-dimensional tangent cone.The surface lattice points ( ) pt c ∂ of c is the set of all the L 4 lattice points included in the 3-faces of c, i.e., Example 17 The surface lattice points of the four-dimensional tangent cone d S c) Let c be a four-dimensional tangent cone.Then, S d c is the set of all the slant tetrahedrons included in the 3-faces of c, i.e.Definition 48 (Vector Fields) Let c be a four-dimensional tangent cone.The vector field c V induced by c on 3 B is defined by of flows of triangles, infinitely many closed trajectories are induced by a tangent cone.Example 19 Shown in Figure 8(a) is the closed trajectories of the flow induced by x x x x x x x .Two types of closed trajectories, one is length 6 and the other is length 12, are alternately stacked infinitely.Example 20 By putting another top vertex 1 2 3 x x x on the tangent cone of Figure 8(a), we obtain a decomposition of a rhombic dodecahedron into four closed trajectories of tetrahedrons (Figure 8(b)).Then, each triplet of the four top vertices of Cone x x x x x x x x x x x x induces infinitely many closed N. Morikawa DOI: 10.4236/am.2018.9100761136 Applied Mathematics trajectories outside the rhombic dodecahedron (grey polygonal lines).
be a four-dimensional cotangent cone.The surface lattice points ( ) pt w ∂ of w is the set of all the 4 L * lattice points included in the 3-faces of w, i.e.,

is the set of 4 L
* lattice points on the intersection of the 3-faces of 0 c and the 3-faces of of the polygonal lines obtained by joining the adjacent 4 L * lattice points of surface vertices Let 0 c be a four-dimensional tangent cone.The two-dimensional surface mesh ( )

Figure 9 .
Figure 9. Positional relationship of slant triangles of

V
of tetrahedrons, we have the following result.Theorem 5 There exist a four-dimensional tangent cone c such that induces infinitely many closed trajectories of tetrahedrons.However, we can not construct a cotangent cone which covers all the closed trajectories because more than three vertices are required to construct a "roof" on a tangent cone.)In particular, As in the case of flows of triangles, we can compute the case of Figure8(b) (or Figure10(a)),

Figure 10 .
Figure 10.Rhombic dodecahedrons: (a) Four closed trajectories of the flow induced by { } 1 0 1 2 0 2 3 0 1 3 1 2 3 , , , c Cone x x x x x x x x x x x x = (See also Figure 8 (b)); (b) The closed trajectory of length 24 induced by In the case of flows of tetrahedrons, we should also consider the case of more unit cubes on the tangent cone 1 c of Figure10(a), we obtain rhombic dodecaherons consisting of a closed trajectory of length 24 (Figure10(b) and Figure10(c)).However, we can not compute the shape of the rhombic dodecaherons using cotangent roofs.Cotangent roofs are not defined because the contour, i.e., the existence of the triangles of type D (Figure2(b)).On the other hand, in the four-dimensional case, ( )4

4 L
* lattice points to a" for each a A ∈ .Definition 59 (STAND) Let the case of Figure 10(b), the rhombic dodecahedron surrounded by the black and grey thick lines, which consists of a closed trajectories of length 24.Note that 0 2 x x , 1 2 x x , and 1 3 x x are outside { } * 1 Cone .The rhombic dodecahedron consists of not only type D but also type E tetrahedrons of Figure 9.In the figure, the tetrahedron with the grey diagonal edge are type E. The tetrahedrons with the black diagonal edge are type D. Example 26 In the case of Figure 10(c),

.
simplified geometrical model of protein molecules.As in the case of flows of triangles, we will specify the shape of regions of 3D HWe denote the top vertices of an inverted cotangent cone iv by Let iv be a four-dimensional inverted cotangent cone.The surface lattice points ( ) pt iv ∂ of iv is the set of all the 4 L * lattice points included in the 3-faces of iv, i.e., φis the set of 4 L * lattice points on the intersection of the 3-faces of w and the 3-faces of iv.Φ .Let ( ) , w iv be a four-dimensional surface pair.

VV
be two vector fields of triangles induced by three-dimensional tangent cones a c and b c respectively.Then, the vector field a c V can be obtained from the other b cV by "putting unit cubes on" and/or "taking unit cubes from" the tangent cone b c .Suppose that the closed trajectories of give two different decompositions of the same region into a set of closed trajectories of triangles.Definition 77 (Addition of closed trajectories) Addition is defined between sets of all the closed trajectories of vector fields.Given two three-dimensional tangent cones a c and b c

Figure 12 .
Figure 12.Addition of closed trajectories: (a) A flow of triangles consisting of 36 closed trajectories of length six.Shown below is the corresponding tangent cone (top view); (b) Closed trajectories of triangles obtained by putting unit cubes on the tangent cone of (a); (c) Closed trajectories of triangles obtained by putting unit cubes on the tangent cone of (b).

Example 35
By putting more unit cubes on the tangent cone of Figure12( closed trajectories of length six, i.e., hexagons.Let { } 0 1 s and g k s are finite sets of terms separated by addition sign (i.e., addition expressions with coefficients one).
[12].Allosteric regulation is typically triggered by the binding of a small molecule, but also triggered by the binding of another protein.When proteins bind to other molecules or proteins, changes in conformation and/or dynamics occur within the protein.Classically, allosteric regulation was considered to be induced through a change in conformation of the protein.Today, it is believed that allostery can take place through a change in the dynamic fluctuations (i.e., internal motions and vibrations) of the protein even without obvious conformational changes[13] [14][15].

Figure 13 .
Figure 13.Allosteric regulation and complexes of self-eclipsed closed trajectories: (a) An active site and an allosteric site of a self-eclipsed closed trajectory m 0 .Shown below is the corresponding three-dimensional tangent cone (top view); (b) Interaction of m 0 and x 22 , i.e., 0 22 6 7 m x m m + = + ; (c) Interaction of m 0 , m 2 , and x 22 , i.e., 0 22 ,

tangent space at t is denoted by
there exists an3 , w iv s.t. , .