The importance of how the information circulates in an interface has been first shown in [8]. The graphs *Saa i* show the presence of long range correlations that are not easily detected with approaches based on the contact surface (buried surface or Voronoi cells based interface). The motifs that allow this transfer are the zigzags, especially the long ones, and the vertices V, V3. On the contrary, a pure ladder topology, in BB or SC or both, does not have ways to correlate far apart atoms. The frequent presence of zigzag, V and V3 motifs implies the existence of constraints on the positions or the physico-chemical properties of non neighbouring atoms, and sometimes of very far apart atoms. Indeed, in Figure 27, the mere establishment of the path formed by the residues S2, E123, L5, L124, I7, V127 (where BB and SC are both present) requires to satisfy physical conditions due to the volume of the atoms and the distribution of electric charge. Each residue has its internal constraints, among which the fixed length of covalent bonds and the planarity of the surface *Cα*, *C*, 0, *N* that contains the peptide bond. The sequence of edges transfers constraints and establishes a correlation between the outer amino acids S2 and V127 even if they are not in physical contact. The exact microscopic description of the constrains is not easy to find, although. Notice that this example of information transfer along the interface has been detected thanks to a description guided by graph theory and based on edges.

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three rotations can be removed from the counting, that reduces the number of needed relative

As an example, consider the graph of 2ojw in Figure 19; there are *N* = 7 amino acids (even the amino acids that do not participate to the graph but are included within the considered regions must be counted) thus one needs 3*N* − 6 = 15 relative positions. The distances between two consecutive *Cα* is fixed in all polypeptides (as the distance TG in the graph). In the present case there are 5 of them. The distance between the first and the third of three consecutive *Cα* (distances TT and GI in the graph) is fixed by the general properties of polypeptides, that makes 3 distances. The graph has 3 edges, namely 3 other distances, thus one remains with 15 − 5 − 3 − 3 = 4 more distances to be fixed. This indicates that this graph is insufficient to reconstruct the shape. A more general counting is possible. In [8] the average number of amino acids (18) and of edges (12) in a *β* interface have been evaluated. Their ratio is very close to 3/2 = 1.5, thus it is reasonable to assume that if there are *N* residues in the interface, there will be nearly 2*N*/3 edges (actually, multiple edges should not be counted, here; this may further reduce the number of known edges). Also, we expects

*N* − 2 consecutive *Cα* and *N* − 4 groups of three consecutive *Cα* namely we have

known distances = (*N* − 2)+(*N* − 4) + 2

are left with the number of distances that are needed to fix the shape

information to fix the shape of the interface.

an incomplete set of distances is treated in [16].

V3, has been studied in detail in *Saa*

the atoms.

**4.2. Perspectives**

unknown distances = (3*N* − 6) −

We subtract the number of known distances to the number of degrees of freedhom and we

Thus a single level does not provide enough data but two levels provide sufficient

Of course, a full evaluation of the interface degrees of freedom needs a much more complex calculation with atoms but the present counting suggests that few lowest levels should be enough to provide an accurate description of the interface shape and the position of most of

A more complete account of the problem of reconstructing the shape of a set of points given

The result R5 is clearly indicative of the major complexity of the side chain by respect to the backbone. In the paper [8] the role of the residues with multiple interactions, namely V and

Following this observation, one could introduce a parametrization based on the length of the side chain. The discussion on the information flow in the interface points in the same

direction, of dedicating a new publication to the study of the SC graphs.

degrees of freedhom = 3*N* − 6 (4)

Geometry and Topology in Protein Interfaces -- Some Tools for Investigations

3

8 3 *N* − 6 = 1 3

*N* = 8 3

0 and has been correlated to the length of the side chains.

*N* − 6 (5)

*N* (6)

positions or distances to

In summary, the main difference that appears comparing buried surface and Voronoi cells based interface with SI is that the first two definitions focus on the spatial organization of the interface while the latter may suggest a temporal organization and allows to evaluate how the information circulate in the interface.

The result R1 clearly states that the level 1 graphs can have a BB graph of size comparable to that of level 0, thus still informative. At level 2 a smaller BB graph is observed. Preliminary results on levels higher than 2 indicate that the BB graphs are also present.

The result R2 indicates that the level 1 graphs have a structure similar to the one found at level 0, in other words that these two levels present several common elements. Instead, from R3 the level 2 graphs seem organized in a different way. The main structure of the level 2 graphs is the ladder one, in both the anti-parallel and parallel orientations, that indicates that these orientations are not distinguishable at this level, and possibly above. Preliminary results on levels higher than 2 indicate that the main distinctive motif of the parallel orientation, namely the zigzag5, is quite rare. Thus, it seems that level 0 and 1 are those that contain the most useful geometrical and topological information.

In [8] we have already used the properties in R2, R3, by implementing algorithms that, from the PDB structure, are able to characterize an interface and tell if it has a *β* structure, and which is its orientation. These algorithms are based on the level 0 only3. The analysis of level 1 graphs confirms and expands this possibility, because the information from both the levels can now be combined for a more accurate recognition.

The BB graphs at level 0 have been previously associated to structural hydrogen bonds that are present in the anti-parallel and in the parallel orientations of *β* strands.

It is possible that level 1 (for both the BB and SC graphs) doesn't describe proper chemical bonds but weaker dipole-dipole or Van der Waals interactions . This comparison for the level 1 has not yet been explored.

#### **4.1. Counting the degrees of freedhom**

The question that we address now is to evaluate if the description provided by the graphs is enough to reconstruct the shape of the interface or not.

To reconstruct a shape in three dimensions one needs 3 coordinates for each point: 3*N*, where *N* is the number of points. Actually, the absolute position of the centre of mass and the spatial orientation of the object in the space are totally irrelevant thus three overall translations and

3 Other interface arrangements are not yet recognized.

three rotations can be removed from the counting, that reduces the number of needed relative positions or distances to

$$\text{degrees of freedom} = 3N - 6 \tag{4}$$

As an example, consider the graph of 2ojw in Figure 19; there are *N* = 7 amino acids (even the amino acids that do not participate to the graph but are included within the considered regions must be counted) thus one needs 3*N* − 6 = 15 relative positions. The distances between two consecutive *Cα* is fixed in all polypeptides (as the distance TG in the graph). In the present case there are 5 of them. The distance between the first and the third of three consecutive *Cα* (distances TT and GI in the graph) is fixed by the general properties of polypeptides, that makes 3 distances. The graph has 3 edges, namely 3 other distances, thus one remains with 15 − 5 − 3 − 3 = 4 more distances to be fixed. This indicates that this graph is insufficient to reconstruct the shape. A more general counting is possible. In [8] the average number of amino acids (18) and of edges (12) in a *β* interface have been evaluated. Their ratio is very close to 3/2 = 1.5, thus it is reasonable to assume that if there are *N* residues in the interface, there will be nearly 2*N*/3 edges (actually, multiple edges should not be counted, here; this may further reduce the number of known edges). Also, we expects *N* − 2 consecutive *Cα* and *N* − 4 groups of three consecutive *Cα* namely we have

$$\text{known distances} = (N - 2) + (N - 4) + \frac{2}{3}N = \frac{8}{3}N - 6\tag{5}$$

We subtract the number of known distances to the number of degrees of freedhom and we are left with the number of distances that are needed to fix the shape

$$\text{unknown distance} = (3N - 6) - \left(\frac{8}{3}N - 6\right) = \frac{1}{3}N \tag{6}$$

Thus a single level does not provide enough data but two levels provide sufficient information to fix the shape of the interface.

Of course, a full evaluation of the interface degrees of freedom needs a much more complex calculation with atoms but the present counting suggests that few lowest levels should be enough to provide an accurate description of the interface shape and the position of most of the atoms.

A more complete account of the problem of reconstructing the shape of a set of points given an incomplete set of distances is treated in [16].

#### **4.2. Perspectives**

14 ime knjige

and based on edges.

the information circulate in the interface.

useful geometrical and topological information.

1 has not yet been explored.

**4.1. Counting the degrees of freedhom**

3 Other interface arrangements are not yet recognized.

can now be combined for a more accurate recognition.

enough to reconstruct the shape of the interface or not.

requires to satisfy physical conditions due to the volume of the atoms and the distribution of electric charge. Each residue has its internal constraints, among which the fixed length of covalent bonds and the planarity of the surface *Cα*, *C*, 0, *N* that contains the peptide bond. The sequence of edges transfers constraints and establishes a correlation between the outer amino acids S2 and V127 even if they are not in physical contact. The exact microscopic description of the constrains is not easy to find, although. Notice that this example of information transfer along the interface has been detected thanks to a description guided by graph theory

In summary, the main difference that appears comparing buried surface and Voronoi cells based interface with SI is that the first two definitions focus on the spatial organization of the interface while the latter may suggest a temporal organization and allows to evaluate how

The result R1 clearly states that the level 1 graphs can have a BB graph of size comparable to that of level 0, thus still informative. At level 2 a smaller BB graph is observed. Preliminary

The result R2 indicates that the level 1 graphs have a structure similar to the one found at level 0, in other words that these two levels present several common elements. Instead, from R3 the level 2 graphs seem organized in a different way. The main structure of the level 2 graphs is the ladder one, in both the anti-parallel and parallel orientations, that indicates that these orientations are not distinguishable at this level, and possibly above. Preliminary results on levels higher than 2 indicate that the main distinctive motif of the parallel orientation, namely the zigzag5, is quite rare. Thus, it seems that level 0 and 1 are those that contain the most

In [8] we have already used the properties in R2, R3, by implementing algorithms that, from the PDB structure, are able to characterize an interface and tell if it has a *β* structure, and which is its orientation. These algorithms are based on the level 0 only3. The analysis of level 1 graphs confirms and expands this possibility, because the information from both the levels

The BB graphs at level 0 have been previously associated to structural hydrogen bonds that

It is possible that level 1 (for both the BB and SC graphs) doesn't describe proper chemical bonds but weaker dipole-dipole or Van der Waals interactions . This comparison for the level

The question that we address now is to evaluate if the description provided by the graphs is

To reconstruct a shape in three dimensions one needs 3 coordinates for each point: 3*N*, where *N* is the number of points. Actually, the absolute position of the centre of mass and the spatial orientation of the object in the space are totally irrelevant thus three overall translations and

are present in the anti-parallel and in the parallel orientations of *β* strands.

results on levels higher than 2 indicate that the BB graphs are also present.

The result R5 is clearly indicative of the major complexity of the side chain by respect to the backbone. In the paper [8] the role of the residues with multiple interactions, namely V and V3, has been studied in detail in *Saa* 0 and has been correlated to the length of the side chains. Following this observation, one could introduce a parametrization based on the length of the side chain. The discussion on the information flow in the interface points in the same direction, of dedicating a new publication to the study of the SC graphs.

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A: K β 85 Y β E β

B: R β 77

A: L α 111

B: T - 340 G β S β P β L β V β T β M β L - 330

A: N - 11

B: K β 171 H β I

S - A β E β V β S β V β E β 19

**3\_1Y13\_AB\_9\_21\_161\_174\_level\_1**

β A β K β Q β T - 164

E -

**Figure 4.** Anti-parallel orientation of the *β*-strands.

**Figure 5.** Anti-parallel orientation of the *β*-strands.

**Figure 6.** Anti-parallel orientation of the *β*-strands

Y β Y β T - 91

**3\_1PM4\_AB\_82\_95\_64\_80\_level\_1**

E β G β W β G β N

> G - E β K β T β I β

**3\_1SNR\_AB\_109\_129\_326\_342\_level\_1**

Geometry and Topology in Protein Interfaces -- Some Tools for Investigations

R β F β K β 125

**Table 1.** Table of the proteins considered in this paper, from [6]. In summary, we have 24 antiparallel, 8 parallel and 7 oblique orientations.

380 Oligomerization of Chemical and Biological Compounds Geometry and Topology in Protein Interfaces – Some Tools for Investigations 17 10.5772/58420 Geometry and Topology in Protein Interfaces -- Some Tools for Investigations http://dx.doi.org/10.5772/58420 381

**Figure 4.**  Table of the proteins considered in this paper, from [6]. In summary, we have 24 antiparallel, 8 parallel and 7 oblique

*β* **interface**

**Figure 5.** Anti-parallel orientation of the *β*-strands.

**Figure 6.** Anti-parallel orientation of the *β*-strands

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A: P - 74

B: G - 177

A: D - 21

B: V α 38

> A: V - 491

> D: Y β 436

G -

H β F

Q β

**4\_1J8D\_AB\_19\_29\_30\_40\_level\_1**

L β

β S

**4\_1PVN\_AD\_489\_496\_432\_438\_level\_1**

I β

S β

D β 434

β K

β I

β A β 32

K - 494

H β Y β

D β 27

**Figure 10.** Anti-parallel orientation of the *β*-strands.

**Figure 11.** Anti-parallel orientation of the *β*-strands.

**Figure 12.** Anti-parallel orientation of the *β*-strands.

V β V β I β E -

**3\_2P90\_AB\_71\_81\_168\_180\_level\_1**

V β T β M β R

N - 81

Geometry and Topology in Protein Interfaces -- Some Tools for Investigations

**Figure 7.** Anti-parallel orientation of the *β*-strands.

**Figure 8.** Anti-parallel orientation of the *β*-strands.

**Figure 9.** Anti-parallel orientation of the *β*-strands.

382 Oligomerization of Chemical and Biological Compounds Geometry and Topology in Protein Interfaces – Some Tools for Investigations 19 10.5772/58420 Geometry and Topology in Protein Interfaces -- Some Tools for Investigations http://dx.doi.org/10.5772/58420 383

**Figure 10.** Anti-parallel orientation of the *β*-strands.

18 ime knjige

B: I β 44

A: G - 24 V β Q β V β R β C - 19

A: L β 192 A β S β S β E - R - I β D β F - 200

B: F - 69

A: Y - 153

B: S β 31 T β I β S

T - S β I β T -

T - Q - E -

**3\_2I9D\_AB\_148\_166\_17\_33\_level\_1**

β N

R - 164

I β R - E - C β S β A β T β 61

**3\_2GVH\_AB\_189\_202\_59\_73\_level\_1**

**Figure 7.** Anti-parallel orientation of the *β*-strands.

**Figure 8.** Anti-parallel orientation of the *β*-strands.

**Figure 9.** Anti-parallel orientation of the *β*-strands.

G β V β T β A β L β

**3\_2BCM\_BA\_43\_53\_17\_26\_level\_1**

C - 51

**Figure 11.** Anti-parallel orientation of the *β*-strands.

**Figure 12.** Anti-parallel orientation of the *β*-strands.

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D: L - 25

H: N - 103

A:H - 126 D - E - M -

E: R α 235

> A: D β 76

B: C β 184 K β V

L β I β S β G - D - V - 82

**Figure 17.** Anti-parallel orientation of the *β*-strands.

**Figure 18.** Anti-parallel orientation of the *β*-strands.

**Figure 16.** Anti-parallel orientation of the *β*-strands.

S β Y β T β E β

> M β S β I β A β A β I β 96

I β V β K β D β I β 135

**5\_1FB1\_AE\_125\_138\_218\_237\_level\_1**

T β S β V β T β K β N - K - 220

**5\_1HI9\_AB\_75\_84\_175\_185\_level\_1**

β S

β I β 177

**5\_1EFI\_DH\_23\_33\_94\_103\_level\_1**

M - 31

Geometry and Topology in Protein Interfaces -- Some Tools for Investigations

**Figure 13.** Anti-parallel orientation of the *β*-strands.

**Figure 14.** Anti-parallel orientation of the *β*-strands.

**Figure 15.** Anti-parallel orientation of the *β*-strands.

384 Oligomerization of Chemical and Biological Compounds Geometry and Topology in Protein Interfaces – Some Tools for Investigations 21 10.5772/58420 Geometry and Topology in Protein Interfaces -- Some Tools for Investigations http://dx.doi.org/10.5772/58420 385

**Figure 16.** Anti-parallel orientation of the *β*-strands.

20 ime knjige

A: M - 1

D: R β 27 Y β G β V β G

A: H β 451 T β D β G - V -

B: Y β 188 L β G β N - T

D: I β 96 A β A β I β S β M β A β N - 103

E: L - 31

E β T β Y β S β F

**5\_1EEI\_DE\_94\_103\_21\_33\_level\_1**

**Figure 14.** Anti-parallel orientation of the *β*-strands.

**Figure 15.** Anti-parallel orientation of the *β*-strands.

**Figure 13.** Anti-parallel orientation of the *β*-strands.

A β S β V β R β 6

**4\_2H5X\_AD\_1\_8\_18\_29\_level\_1**

T -

**4\_3BF0\_AB\_451\_463\_178\_190\_level\_1**

> - K - 23

β 20

L α A α461

**Figure 17.** Anti-parallel orientation of the *β*-strands.

**Figure 18.** Anti-parallel orientation of the *β*-strands.

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A: S - 51

B: V β 63

A: H β 45 Y β C β

B: Y β 110

A: T β 7

B: S β 87 V β E

E β E

Q β I β V β C - T - V -

**6\_2Z9H\_AB\_5\_18\_77\_89\_level\_1**

β D

β G β I β 79

**Figure 22.** Anti-parallel orientation of the *β*-strands.

**Figure 23.** Anti-parallel orientation of the *β*-strands.

**Figure 24.** Anti-parallel orientation of the *β*-strands.

Q β

V β

**6\_2GJV\_AB\_43\_56\_102\_112\_level\_1**

M β

**6\_1U1S\_AB\_48\_55\_60\_66\_level\_1**

T β

S β

β L β W β 104

V β

S β 60

F β E - P - 54

> H α 16

Y β 55

Geometry and Topology in Protein Interfaces -- Some Tools for Investigations

**Figure 19.** Anti-parallel orientation of the *β*-strands.

**Figure 20.** Anti-parallel orientation of the *β*-strands.

**Figure 21.** Anti-parallel orientation of the *β*-strands.

386 Oligomerization of Chemical and Biological Compounds Geometry and Topology in Protein Interfaces – Some Tools for Investigations 23 10.5772/58420 Geometry and Topology in Protein Interfaces -- Some Tools for Investigations http://dx.doi.org/10.5772/58420 387

**Figure 22.** Anti-parallel orientation of the *β*-strands.

22 ime knjige

A: K - 187 Q -

B: E β 103

**Figure 19.** Anti-parallel orientation of the *β*-strands.

**Figure 20.** Anti-parallel orientation of the *β*-strands.

**Figure 21.** Anti-parallel orientation of the *β*-strands.

S - R β T β V β

**5\_1WUR\_AB\_186\_199\_93\_105\_level\_1**

G β K β V β V β V β E - 95

A: T β 44

B: T β 193

A: I β 74 G β I β I - D -

B: M α 19 G

I - R β N β T - S - V - L - 10

**5\_2RCF\_AB\_72\_83\_8\_21\_level\_1**

T β 46

**5\_2OJW\_AB\_42\_48\_188\_195\_level\_1**

β I

β 190

> N - G - 82

S β

M β 197

**Figure 23.** Anti-parallel orientation of the *β*-strands.

**Figure 24.** Anti-parallel orientation of the *β*-strands.

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R β 9

Geometry and Topology in Protein Interfaces -- Some Tools for Investigations

> E - 127

β Y

β 96

A: S - 2 T - T β L β A β I β

B: E - 120

A: M - 1 Q β I β K β I β K β 6

B: V - 120 D β L β I β E β V - D

A: G - 120 A β F β K β

D: P - 90 K β V β F

**Figure 28.** Parallel orientation of the *β*-strands.

**Figure 29.** Parallel orientation of the *β*-strands.

**Figure 30.** Parallel orientation of the *β*-strands.

V - E β L β V β E β V - 127

**3\_2BAZ\_AB\_1\_8\_116\_130\_level\_1**

**4\_1L3A\_AD\_118\_129\_88\_98\_level\_1**

**3\_1SJN\_AB\_1\_12\_118\_130\_level\_1**

**Figure 25.** Anti-parallel orientation of the *β*-strands.

**Figure 26.** Anti-parallel orientation of the *β*-strands.

**Figure 27.** Anti-parallel orientation of the *β*-strands.

388 Oligomerization of Chemical and Biological Compounds Geometry and Topology in Protein Interfaces – Some Tools for Investigations 25 10.5772/58420 Geometry and Topology in Protein Interfaces -- Some Tools for Investigations http://dx.doi.org/10.5772/58420 389

**Figure 28.** Parallel orientation of the *β*-strands.

24 ime knjige

A: I - 5

G: V β 96

> A: M - 1 I β K β

G: Q - 94 L β V β A β L β L

V β N β I β T β L β 41

V - T

**7\_2RAQ\_AB\_33\_46\_76\_91\_level\_1**

β V β E β D β V β 79

A: E - 35

B: E - 91

**Figure 25.** Anti-parallel orientation of the *β*-strands.

**Figure 26.** Anti-parallel orientation of the *β*-strands.

**Figure 27.** Anti-parallel orientation of the *β*-strands.

K β

A β

**7\_1WNR\_AG\_1\_10\_87\_96\_level\_1**

P β

**7\_1HX5\_AG\_3\_13\_92\_99\_level\_1**

L β

L -

L - 8

V β 93

> R β 8

β R - 87

**Figure 29.** Parallel orientation of the *β*-strands.

**Figure 30.** Parallel orientation of the *β*-strands.

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A: M - 48

B: R α 517 I β D

A: V β 142

B: G - 171 A β

D - S β

between or inside proteins, protein-DNA or protein-RNA complexes and so on.

aspects that have been systematically negletted in all the previous publications.

**Figure 34.** Parallel orientation of the *β*-strands.

**Figure 35.** Parallel orientation of the *β*-strands.

orientations are now more precise.

models for prediction with some input from level 1.

**5. Conclusion**

D β K β M β

**8\_1Q3S\_AB\_46\_57\_515\_527\_level\_1**

V β

β V β I β A β A - 524

> V β

The whole analysis presented so far has been triggered by the problem of investigating biological interfaces, namely interfaces that form during the biochemical activity in a cell,

In the introduction, motivations are given for the possibility of using geometry to understand the interactions. In the Results and Discussion sections, the topological, but intrinsically geometrical, properties of the interaction graphs have been presented with the first goal of learning how to distinguish the two main orientations and the second goal of estimating

Both these aspects have been clearly addressed in this paper. The tools to distinguish the two

The results are the consistency of the information from level 1 and level 0 and the limited amount of information from level 2. Both the results suggest to enrich previous statistics and

**8\_2V9U\_AB\_140\_148\_170\_177\_level\_1**

S β

V β

N β 146

T β 175

S - 55

Geometry and Topology in Protein Interfaces -- Some Tools for Investigations

**Figure 31.** Parallel orientation of the *β*-strands.

**Figure 32.** Parallel orientation of the *β*-strands.

**Figure 33.** Parallel orientation of the *β*-strands.

390 Oligomerization of Chemical and Biological Compounds Geometry and Topology in Protein Interfaces – Some Tools for Investigations 27 10.5772/58420 Geometry and Topology in Protein Interfaces -- Some Tools for Investigations http://dx.doi.org/10.5772/58420 391

**Figure 34.** Parallel orientation of the *β*-strands.

26 ime knjige

A: H β 3

D: T - 331

> A: M - 1

E: I β 48

A: N β 37

G: T α 516

**Figure 31.** Parallel orientation of the *β*-strands.

**Figure 32.** Parallel orientation of the *β*-strands.

**Figure 33.** Parallel orientation of the *β*-strands.

T β F

> Q β

T β

V β

T

V β

L β

D - 41

I β

**5\_1NQU\_AE\_1\_6\_46\_54\_level\_1**

L β

**7\_1OEL\_AG\_34\_43\_511\_524\_level\_1**

Y β

V β

E - 5

R β 52

D β

**4\_2A7R\_AD\_1\_12\_330\_339\_level\_1**

β I

β R β 335

N β

V β K β 9

**Figure 35.** Parallel orientation of the *β*-strands.

#### **5. Conclusion**

The whole analysis presented so far has been triggered by the problem of investigating biological interfaces, namely interfaces that form during the biochemical activity in a cell, between or inside proteins, protein-DNA or protein-RNA complexes and so on.

In the introduction, motivations are given for the possibility of using geometry to understand the interactions. In the Results and Discussion sections, the topological, but intrinsically geometrical, properties of the interaction graphs have been presented with the first goal of learning how to distinguish the two main orientations and the second goal of estimating aspects that have been systematically negletted in all the previous publications.

Both these aspects have been clearly addressed in this paper. The tools to distinguish the two orientations are now more precise.

The results are the consistency of the information from level 1 and level 0 and the limited amount of information from level 2. Both the results suggest to enrich previous statistics and models for prediction with some input from level 1.

These results are confirmed by the naive counting of the degrees of freedhom: at least in a coarse-grained view, the two levels *Saa* 0 and *Saa* 1 provide enough information to reconstruct the shape of the interface (completeness).

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[6] Feverati G, Achoch M, Zrimi J, Vuillon L, Lesieur C. *β*-strand interfaces of non-dimeric protein oligomers are characterized by scattered charge residues pattern. PLoS ONE 7(4): e32558 (2012). The article and the Supporting Information are freely downloadable from Plos One: http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0032558#s5.

Geometry and Topology in Protein Interfaces -- Some Tools for Investigations

[7] Zrimi J, Ng Ling A, Giri-Rachman Arifint E, Feverati G, Lesieur C. Cholera toxin B subunits assemble into pentamers: proposition of a fly-casting mechanism. Plos One

[8] Feverati G, Achoch M, Vuillon L, Lesieur C. Residue interaction networks of *β*-strand interfaces from healthy protein oligomers have few HUBs (multiple contact residues) and low interconnectedness: a potential protection mechanism against network

[10] Janin J, Bahadur R P, Chakrabarti P. Protein-protein interaction and quaternary

[11] Feverati G, Lesieur C. Oligomeric interfaces under the lens: Gemini. PLoS ONE 5(3):

[12] Feverati G, Lesieur C, Vuillon L. Symmetrization: ranking and clustering in protein interfaces. Referred proceeding of the conference "Mathematics of distances and applications", MDA 2012, Varna. Publisher: ITHEA, Sofia. Editors: M. Deza, M.

[13] Lance G N, Williams W T. A General Theory of Classificatory Sorting Strategies. 1.

[14] Murtagh F and Contreras P. Methods of Hierarchical Clustering. Data Mining and Knowledge Discovery. Wiley-Interscience, Vol. 2, No. 1, pp. 86âA ¸S97 (2012). ˘

[15] Martin A J M, Vidotto M, Boscariol F, Di Domenico T, Walsh I, Tosatto S C E. Residue Interaction Network Generator (RING). http://protein.bio.unipd.it/ring/

[16] Liberti L, Lavor C, Maculan N, Mucherino A. Euclidean Distance Geometry and

Hierarchical Systems The Computer Journal (1967) 9 (4): 373-380.

Applications SIAM Review 56[1]:3-69. doi 10.1137/120875909

rewiring and chain dissociation. Under revision by Plos One.

http://www.expasy.org/proteomics/protein\_structure

5(12) e15347 (2010).

e9897 (2010).

Petitjean, K. Markov.

[9] ExPASy: Bioinformatics resource portal,

structure. Q Rev Biophys 41: 133-180 (2008).

In Methods, the notion of *Saa* 0 as the minimal description of the interface, or framework, already used in all our previous publications, has been presented here with solid mathematical arguments: as all buds are present in *Saa* 0 and no bud can appear later, it is legitimate to call *Saa* 0 a framework because a smallest set would miss some buds, namely some parts of the interface. We find that this and the completeness of the interface add important values to the validity of the methods.

Also, it is important to stress the ability of the symmetric minimization to detect the BB hydrogen bonds from the knowledge of the positions of non-hydrogen atoms only: in this case, geometry intrinsically reveals the chemical interactions, without making use of a cut off or other external scales.

#### **Acknowledgements**

This work is founded by the region Rhone-Alpes.

It's a pleasure to thanks Claire Lesieur for most valuable comments and suggestions, and Laurent Vuillon, for his help.

With sadness, the author remembers the friend and colleague Laurent Fournier, recently deceased, for his useful suggestions to improve the C++ codes used in the analyses.

#### **Author details**

Giovanni Feverati

Fédération de recherche MSIF, University of Savoie and CNRS, Annecy-le-Vieux, France

#### **References**

28 ime knjige

coarse-grained view, the two levels *Saa*

In Methods, the notion of *Saa*

legitimate to call *Saa*

or other external scales.

**Acknowledgements**

Laurent Vuillon, for his help.

Res 69: 567-577 (2002).

689-697 (1953).

**Author details** Giovanni Feverati

**References**

the shape of the interface (completeness).

mathematical arguments: as all buds are present in *Saa*

important values to the validity of the methods.

This work is founded by the region Rhone-Alpes.

These results are confirmed by the naive counting of the degrees of freedhom: at least in a

already used in all our previous publications, has been presented here with solid

some parts of the interface. We find that this and the completeness of the interface add

Also, it is important to stress the ability of the symmetric minimization to detect the BB hydrogen bonds from the knowledge of the positions of non-hydrogen atoms only: in this case, geometry intrinsically reveals the chemical interactions, without making use of a cut off

It's a pleasure to thanks Claire Lesieur for most valuable comments and suggestions, and

With sadness, the author remembers the friend and colleague Laurent Fournier, recently

deceased, for his useful suggestions to improve the C++ codes used in the analyses.

Fédération de recherche MSIF, University of Savoie and CNRS, Annecy-le-Vieux, France

of attack. Biochim Biophys Acta, vol. 1778, num. 7-8, p. 1611-23 (2008).

The strategies of toxins (review). Mol Membr Biol 14: 45-64 (1997).

review. Rev Physiol Biochem Pharmacol 159: 1-77 (2007).

[1] Iacovache I, van der Goot G F, Pernot L. Pore formation: An ancient yet complex form

[2] Lesieur C, Vecsey-Semjen B, Abrami L, Fivaz M, van der Goot G F. Membrane insertion:

[3] Kirkitadze M D, Bitan G, Teplow D B. Paradigm shifts in Alzheimer's disease and other neurodegenerative disorders: the emerging role of oligomeric assemblies. J Neurosci

[4] Harrison R S, Sharpe P C, Singh Y, Fairlie D P. Amyloid peptides and proteins in

[5] Crick F H C. The packing of alpha-helices: simple coiled-coils. Acta Crystallogr 6:

1 provide enough information to reconstruct

0 and no bud can appear later, it is

0 as the minimal description of the interface, or framework,

0 a framework because a smallest set would miss some buds, namely

0 and *Saa*

**Chapter 13**

**Provisional chapter**

**From Tilings to Fibers – Bio-mathematical Aspects of**

Protein oligomers are made by the association of protein chains via intermolecular amino acid interactions (interaction between subunits) forming so called protein interfaces. This chapter proposes mathematical concepts to investigate the shape constraints on the protein interfaces in order to promote oligomerization. First, we focus on tiling the plane (2 dimensions) by translation with abstract shapes. Using the fundamental Theorem of Beauquier-Nivat, we show that the shapes of the tiles must be either like a square or like a hexagon to tile the whole plane. Second, we look in more details at the tiling of a cylinder and discuss its relevancy in constructing protein fibers. The universality of such "building" properties are investigated through biological examples. This chapter is written four-hand by a mathematician and a biologist in order to present bio-mathematical aspects of fiber

Proteins are made by polymerization of 20 different amino acids via a covalent bond called the peptide bond. The ordered amino acids read along the peptide bonds constitute the protein backbone and are referred to as the primary structure or the primary sequence. The secondary structure involves hydrogen bonding to form *α*-helix or pleated-sheet structures. The intricate folding of the polypeptide chain in a globular protein via long range atomic interactions is referred to as the tertiary structure. Some proteins (hemoglobin, for example) have quaternary structure— several polypeptide chains are nested together. These proteins are called protein oligomers. The different chains associate by the formation of contact zones called protein interfaces, made of contacts between atoms of the amino acids (Fig. 1A). The majority of proteins are organized as oligomers and not as single monomers (for a review see chapter in this book by Gotte). Protein oligomers adopt different symmetries and different stoichiometries (number of repeated monomers). According to the PDB (Protein Data Bank [30] ) where all available atomic structures of proteins are stored, protein oligomers exist in cyclic, dihedral and cubic point group symmetries. Most proteins have *Cn* or *Dn* symmetry (see [17, 22, 24]) and almost only viral proteins adopt cubic symmetry [35]. High stoichiometry complexes in eukaryotes have most often a *C*1 symmetry (identity). Some

> ©2012 Lesieur and Vuillon et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2014 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

proteins adopt helical symmetry and construct fibers (see [24]).

**From Tilings to Fibers – Bio-mathematical Aspects**

**Fold Plasticity**

C. Lesieur and L. Vuillon

C. Lesieur1 and L. Vuillon2∗

**of Fold Plasticity**

http://dx.doi.org/10.5772/58577

10.5772/58577

**1. Introduction**

constructions.

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

**Provisional chapter**

#### **From Tilings to Fibers – Bio-mathematical Aspects of Fold Plasticity From Tilings to Fibers – Bio-mathematical Aspects of Fold Plasticity**

C. Lesieur and L. Vuillon C. Lesieur1 and L. Vuillon2∗

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/58577 10.5772/58577

#### **1. Introduction**

Protein oligomers are made by the association of protein chains via intermolecular amino acid interactions (interaction between subunits) forming so called protein interfaces. This chapter proposes mathematical concepts to investigate the shape constraints on the protein interfaces in order to promote oligomerization. First, we focus on tiling the plane (2 dimensions) by translation with abstract shapes. Using the fundamental Theorem of Beauquier-Nivat, we show that the shapes of the tiles must be either like a square or like a hexagon to tile the whole plane. Second, we look in more details at the tiling of a cylinder and discuss its relevancy in constructing protein fibers. The universality of such "building" properties are investigated through biological examples. This chapter is written four-hand by a mathematician and a biologist in order to present bio-mathematical aspects of fiber constructions.

Proteins are made by polymerization of 20 different amino acids via a covalent bond called the peptide bond. The ordered amino acids read along the peptide bonds constitute the protein backbone and are referred to as the primary structure or the primary sequence. The secondary structure involves hydrogen bonding to form *α*-helix or pleated-sheet structures. The intricate folding of the polypeptide chain in a globular protein via long range atomic interactions is referred to as the tertiary structure. Some proteins (hemoglobin, for example) have quaternary structure— several polypeptide chains are nested together. These proteins are called protein oligomers. The different chains associate by the formation of contact zones called protein interfaces, made of contacts between atoms of the amino acids (Fig. 1A). The majority of proteins are organized as oligomers and not as single monomers (for a review see chapter in this book by Gotte). Protein oligomers adopt different symmetries and different stoichiometries (number of repeated monomers). According to the PDB (Protein Data Bank [30] ) where all available atomic structures of proteins are stored, protein oligomers exist in cyclic, dihedral and cubic point group symmetries. Most proteins have *Cn* or *Dn* symmetry (see [17, 22, 24]) and almost only viral proteins adopt cubic symmetry [35]. High stoichiometry complexes in eukaryotes have most often a *C*1 symmetry (identity). Some proteins adopt helical symmetry and construct fibers (see [24]).

the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2014 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Lesieur and Vuillon et al., licensee InTech. This is an open access chapter distributed under the terms of

In this chapter, we investigate the construction of biological fibers by using a mathematical model for fibers. Essentially the formation of the fiber is considered as the tiling of an infinite height cylinder derived from an initial two dimensional tiling of the plane followed by the translation of a single tile. This single tile could be replaced by a *n*-mer in order to construct more complex fibers. Of course proteins have tridimensional shapes (3D-structure) and the biological fiber could have cross section and more complex internal organization not considered here because the idea is to construct a complete mathematical model for fibers applicable for any 3D-structure of the protein. This "simplification" is supported biologically because there exists many severe human diseases associated with the formation of fibers by proteins structurally and functionally unrelated. Notorious examples, are Alzheimer (*Aβ*-amyloid), Parkinson (synuclein), cerebral amyloid angiopathy (cystatin C-amyloidosis) and type II diabetes (IAPP, amylin). It is important to realize that fiber formation is also observed in cancer (p53), cardiovascular (transthyretin, serpin) and inflammatory diseases (serpin) (reviewed in [2, 6, 9, 26, 27]). These proteins (indicated in bracket next to the disease) have the fold plasticity to undergo a transition from an oligomeric state to a fiber state, change that leads to the loss of the protein function and the pathologies, called conformational diseases (Fig. 1B). Because this transition is shared by unrelated proteins, it is reasonable to assume that the change is based on a generic binding properties of the interface. The idea of the work is to define the shape constraints of the interfaces of a fiber (infinite height cylinder) compared to the interfaces of an oligomer (finite height cylinder) to ultimately address the problem of conformational diseases. The 2D tiling is relevant as the answer lies in the properties of surfaces. As mentioned, we are particularly interested in determining invariant properties, i.e. property of the surface that are true for any 3D-structure of the protein. In other words, the goal is to trace the local properties (the interface) necessary to carry out a global change. The relation between local properties and quaternary structures is discussed in details in the work by Claverie, Hofunung and Monod and the Monod, Changeux, Wyman (MCW) model ([8, 28]).

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**Figure 1. Protein oligomers and fibers. A. Protein oligomer.** As an example, the x-ray structure of the cholera toxin B pentamer (*CtxB*5) is shown (PDB code 3CHB). Each chain is indicated in a different ribbon color. The top and bottom images represent a top view and a side view of the pentamer, respectively.The atomic interactions involved in a protein interface are highlighted in space fill representation showing all atoms. Each chain participates for one domain of the interface, also called segment or side. (*CtxB*5) has a *C*5 symmetry that is well described by a cylinder (bottom image). **B. Oligomer to fiber**

impose the use of either 2 interfaces or 3 interfaces. (5.2) We extend our construction to tile the fiber by *n*-mers instead of a single chain. (5.3) We explain how the fold plasticity is able to make a transition from finite cylinder to fiber in the p53 case. (6.1) We investigate the case of non regular tilings in order to design fibers with only 1 direction of periodicity and this implies strong constraints on the shape of the tile. (6.2) We also investigate transition from fiber to tiling of the whole three dimensional space. (7) We give a conclusion by comparing the fiber case to the whole tiling of the 3D space. All along this chapter biological illustrations

Oligomers with *Cn* symmetry can be constructed using a single chain replicated with a rotational axe of order *n* that is by a rotation of angle 360/*n* degrees (in other terms *n* equals to the number of monomers in the oligomers). Such oligomer adopts a cylindrical surface with each chain associating with 2 adjacent chains via a single interface (Fig. 1A). The interface is made of 2 interacting domains also called segments or sides, each one provided by one chain (Fig. 1A). It is important to clearly distinguish the notion of having one interface from the notion of having multiple regions or multiple patches of interfaces. Interfaces can be made of amino acids that are not contiguous along the backbone making up several "regions" (Fig. 2A). Yet the different regions still constitute a single interface because they are only able to bind to a single side. As an analogy, let's consider a poster to be stuck on a wall. You may choose to put several patches of glue on the reverse side of the poster or only one : it will

**transition.** Each monomer is indicated by a different color.

are presented in parallel to the mathematical descriptions.

**2.** *Cn* **and** *Dn* **symmetries on oligomerization**

Clearly the transition to a fiber involves other determinants besides those of the protein interface, as for example the spatial positioning of the interface relative to the main body of the protein, a typical folding problem for which 3D-tiling would be more appropriate. Nevertheless this problem is not addressed in the present chapter which focus essentially on the transition from an oligomer to a fiber seen from the protein interface view point. For more information on the mechanisms of assembly and 3D determinant, one can read the chapter by Claire Lesieur titled "The Assembly of Protein Oligomers: Old Stories and New Perspectives with Graph Theory". For fiber formation, readers can referred to the work by Gebauer [11, 12].

This chapter proposes a mathematical approach of the problem using the principles of tiling and considering the protein interfaces as boundaries of a tile. The chapter is divided into 9 parts: (1) After this introduction. (2) We focus on usual *Cn* and *Dn* symmetries that build finite height cylinders and we explain the construction of interfaces between chains. In paragraph (3) we give a formal description of the boundary of a tile by using abstract geometrical objects called polyominoes, we state the fundamental Theorem of Beauquier-Nivat which explains the properties of the boundary of a tile associated with its capacity to tile a plan. (4) We present the regular tilings of the plane which are tilings constructed by translation of all integral combinations of 2 vectors. (5.1) These two directions of periodicity on the tilings allow us to construct fibers (that are infinite cylinders) and to

**Figure 1. Protein oligomers and fibers. A. Protein oligomer.** As an example, the x-ray structure of the cholera toxin B pentamer (*CtxB*5) is shown (PDB code 3CHB). Each chain is indicated in a different ribbon color. The top and bottom images represent a top view and a side view of the pentamer, respectively.The atomic interactions involved in a protein interface are highlighted in space fill representation showing all atoms. Each chain participates for one domain of the interface, also called segment or side. (*CtxB*5) has a *C*5 symmetry that is well described by a cylinder (bottom image). **B. Oligomer to fiber transition.** Each monomer is indicated by a different color.

impose the use of either 2 interfaces or 3 interfaces. (5.2) We extend our construction to tile the fiber by *n*-mers instead of a single chain. (5.3) We explain how the fold plasticity is able to make a transition from finite cylinder to fiber in the p53 case. (6.1) We investigate the case of non regular tilings in order to design fibers with only 1 direction of periodicity and this implies strong constraints on the shape of the tile. (6.2) We also investigate transition from fiber to tiling of the whole three dimensional space. (7) We give a conclusion by comparing the fiber case to the whole tiling of the 3D space. All along this chapter biological illustrations are presented in parallel to the mathematical descriptions.

#### **2.** *Cn* **and** *Dn* **symmetries on oligomerization**

2

(MCW) model ([8, 28]).

Gebauer [11, 12].

In this chapter, we investigate the construction of biological fibers by using a mathematical model for fibers. Essentially the formation of the fiber is considered as the tiling of an infinite height cylinder derived from an initial two dimensional tiling of the plane followed by the translation of a single tile. This single tile could be replaced by a *n*-mer in order to construct more complex fibers. Of course proteins have tridimensional shapes (3D-structure) and the biological fiber could have cross section and more complex internal organization not considered here because the idea is to construct a complete mathematical model for fibers applicable for any 3D-structure of the protein. This "simplification" is supported biologically because there exists many severe human diseases associated with the formation of fibers by proteins structurally and functionally unrelated. Notorious examples, are Alzheimer (*Aβ*-amyloid), Parkinson (synuclein), cerebral amyloid angiopathy (cystatin C-amyloidosis) and type II diabetes (IAPP, amylin). It is important to realize that fiber formation is also observed in cancer (p53), cardiovascular (transthyretin, serpin) and inflammatory diseases (serpin) (reviewed in [2, 6, 9, 26, 27]). These proteins (indicated in bracket next to the disease) have the fold plasticity to undergo a transition from an oligomeric state to a fiber state, change that leads to the loss of the protein function and the pathologies, called conformational diseases (Fig. 1B). Because this transition is shared by unrelated proteins, it is reasonable to assume that the change is based on a generic binding properties of the interface. The idea of the work is to define the shape constraints of the interfaces of a fiber (infinite height cylinder) compared to the interfaces of an oligomer (finite height cylinder) to ultimately address the problem of conformational diseases. The 2D tiling is relevant as the answer lies in the properties of surfaces. As mentioned, we are particularly interested in determining invariant properties, i.e. property of the surface that are true for any 3D-structure of the protein. In other words, the goal is to trace the local properties (the interface) necessary to carry out a global change. The relation between local properties and quaternary structures is discussed in details in the work by Claverie, Hofunung and Monod and the Monod, Changeux, Wyman

Clearly the transition to a fiber involves other determinants besides those of the protein interface, as for example the spatial positioning of the interface relative to the main body of the protein, a typical folding problem for which 3D-tiling would be more appropriate. Nevertheless this problem is not addressed in the present chapter which focus essentially on the transition from an oligomer to a fiber seen from the protein interface view point. For more information on the mechanisms of assembly and 3D determinant, one can read the chapter by Claire Lesieur titled "The Assembly of Protein Oligomers: Old Stories and New Perspectives with Graph Theory". For fiber formation, readers can referred to the work by

This chapter proposes a mathematical approach of the problem using the principles of tiling and considering the protein interfaces as boundaries of a tile. The chapter is divided into 9 parts: (1) After this introduction. (2) We focus on usual *Cn* and *Dn* symmetries that build finite height cylinders and we explain the construction of interfaces between chains. In paragraph (3) we give a formal description of the boundary of a tile by using abstract geometrical objects called polyominoes, we state the fundamental Theorem of Beauquier-Nivat which explains the properties of the boundary of a tile associated with its capacity to tile a plan. (4) We present the regular tilings of the plane which are tilings constructed by translation of all integral combinations of 2 vectors. (5.1) These two directions of periodicity on the tilings allow us to construct fibers (that are infinite cylinders) and to Oligomers with *Cn* symmetry can be constructed using a single chain replicated with a rotational axe of order *n* that is by a rotation of angle 360/*n* degrees (in other terms *n* equals to the number of monomers in the oligomers). Such oligomer adopts a cylindrical surface with each chain associating with 2 adjacent chains via a single interface (Fig. 1A). The interface is made of 2 interacting domains also called segments or sides, each one provided by one chain (Fig. 1A). It is important to clearly distinguish the notion of having one interface from the notion of having multiple regions or multiple patches of interfaces. Interfaces can be made of amino acids that are not contiguous along the backbone making up several "regions" (Fig. 2A). Yet the different regions still constitute a single interface because they are only able to bind to a single side. As an analogy, let's consider a poster to be stuck on a wall. You may choose to put several patches of glue on the reverse side of the poster or only one : it will still bind to only one surface. The *Cn* oligomers have only 1 interface and they polymerize following a single direction that is why they can describe a cylindrical surface (fig. 1A). An important notion is the number of adjacent chains in the oligomerization. In the *Cn* symmetry, each chain has exactly 2 adjacent chains (namely the *M*th chain has for adjacent chains the (*M* − 1)th chain and the (*M* + 1)th chain (this calculation is done modulo *n*)) (see fig. 3). To summarize *Cn* is an oligomerization with 1 interface (possibly with distinct regions) and with 2 adjacent chains at each chain (see Fig. 3 and Fig. 4).

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I M

From Tilings to Fibers – Bio-mathematical Aspects of Fold Plasticity

I I I I

I

I

**Figure 3. Abstract view of** *C*4 **symmetry.** A tetramer formed by a single interface between the part *I* and ¯*I*. Each chain is

**Figure 4. Heptamer with 4H56 PDB code with each chain adjacent to 2 other chains.** Remark that in the beta barrel the adjacency between pairs of chains is the same as in the other part of the heptamer. This implies an oligomerization with exactly

A *polyomino* is a fundamental object in theory of tilings introduced by Golomb [16]. Polyominoes are building blocks for tiling and will be for us an abstract version of a single

A polyomino is composed of juxtaposition of unit squares with corners on the **Z**2 grid (the corners of each unit square are integer points) such that every 2 adjacent squares in the polyomino share a unit segment; and also such that we can reach every pair of squares of the polyomino by a path inside the polyomino moving by unitary steps on adjacent unit squares

There are 2 distinct polyominoes constructed by 2 adjacent unit squares and are called "domino". Remark that a domino can be horizontal or vertical this is why we find 2 dominoes

The most simple polyomino has only one unit square and is called "mino"(Fig. 6).

3

I

adjacent via the interface to exactly 2 other chains

**3. Definitions and notation**

chain in an oligomerization process.

1 interface.

(see Fig. 5).

(Fig. 7).

I

1

I

2

4

**Figure 2. Protein interfaces. A. Left, rectangular schematic of the backbone of a protein oligomer.** Two interacting chains are indicated, chain *M* and *M* + 1. Chain *M* participates in one side of the interface indicated by the red boxes and chain *M* + 1 participates in the complementary sides indicated in green. In the example the protein has 3 distinct regions of interface. Right, Illustration of the 3 regions of interface on the x-ray structure of CtxB5 (see fig. 1A). **B.** *Dn* **symmetry oligomer.** Example with the protein 3GVF (PDB code), a *D*3 symmetry oligomer made of 6 chains organized as 3 dimers.

A slightly more complicated symmetry is the *Dn* symmetry oligomer which is reconstructed by a rotational axe plus 2 axes of symmetry perpendicular to the rotational axe. Such symmetry might look like to polymerize in 2 directions because it has one region of interface orthogonal to another one so the former can bind in a perpendicular direction. However, the "orthogonal" interface cannot grow besides forming a dimer so a *Dn* symmetry is in fact similar to a *Cn* symmetry considering the polymerization of a dimer instead of a monomer (Fig. 2B).

**Figure 3. Abstract view of** *C*4 **symmetry.** A tetramer formed by a single interface between the part *I* and ¯*I*. Each chain is adjacent via the interface to exactly 2 other chains

**Figure 4. Heptamer with 4H56 PDB code with each chain adjacent to 2 other chains.** Remark that in the beta barrel the adjacency between pairs of chains is the same as in the other part of the heptamer. This implies an oligomerization with exactly 1 interface.

### **3. Definitions and notation**

4

(Fig. 2B).

still bind to only one surface. The *Cn* oligomers have only 1 interface and they polymerize following a single direction that is why they can describe a cylindrical surface (fig. 1A). An important notion is the number of adjacent chains in the oligomerization. In the *Cn* symmetry, each chain has exactly 2 adjacent chains (namely the *M*th chain has for adjacent chains the (*M* − 1)th chain and the (*M* + 1)th chain (this calculation is done modulo *n*)) (see fig. 3). To summarize *Cn* is an oligomerization with 1 interface (possibly with distinct

**Figure 2. Protein interfaces. A. Left, rectangular schematic of the backbone of a protein oligomer.** Two interacting chains are indicated, chain *M* and *M* + 1. Chain *M* participates in one side of the interface indicated by the red boxes and chain *M* + 1 participates in the complementary sides indicated in green. In the example the protein has 3 distinct regions of interface. Right, Illustration of the 3 regions of interface on the x-ray structure of CtxB5 (see fig. 1A). **B.** *Dn* **symmetry oligomer.** Example with the protein 3GVF (PDB code), a *D*3 symmetry oligomer made of 6 chains organized as 3 dimers.

A slightly more complicated symmetry is the *Dn* symmetry oligomer which is reconstructed by a rotational axe plus 2 axes of symmetry perpendicular to the rotational axe. Such symmetry might look like to polymerize in 2 directions because it has one region of interface orthogonal to another one so the former can bind in a perpendicular direction. However, the "orthogonal" interface cannot grow besides forming a dimer so a *Dn* symmetry is in fact similar to a *Cn* symmetry considering the polymerization of a dimer instead of a monomer

regions) and with 2 adjacent chains at each chain (see Fig. 3 and Fig. 4).

A *polyomino* is a fundamental object in theory of tilings introduced by Golomb [16]. Polyominoes are building blocks for tiling and will be for us an abstract version of a single chain in an oligomerization process.

A polyomino is composed of juxtaposition of unit squares with corners on the **Z**2 grid (the corners of each unit square are integer points) such that every 2 adjacent squares in the polyomino share a unit segment; and also such that we can reach every pair of squares of the polyomino by a path inside the polyomino moving by unitary steps on adjacent unit squares (see Fig. 5).

The most simple polyomino has only one unit square and is called "mino"(Fig. 6).

There are 2 distinct polyominoes constructed by 2 adjacent unit squares and are called "domino". Remark that a domino can be horizontal or vertical this is why we find 2 dominoes (Fig. 7).

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104592937, 400795844,<PHONE_NUMBER>,<PHONE_NUMBER>,<PHONE_NUMBER>0,<PHONE_NUMBER>3,<PHONE_NUMBER>78,<PHONE_NUMBER>524,<PHONE_NUMBER>268,<PHONE_NUMBER>6011,<PHONE_NUMBER>7108,<PHONE_NUMBER>98244,<PHONE_NUMBER>826973, ... We recognize the beginning of the sequence namely 1 mino, 2

We don't know if it possible to find a closed formula (i.e. a mathematical formula depending on *n* that counts the number of distinct polyominoes with *n* unit squares) and this is a

We have defined polyominoes and now we would like to tile the whole plane by translation

Let *P* be a polyomino. A *tiling by translation* of *P* is a covering of the whole plane **R**2 by translated images of *P* such that there is no hole in the tiling and no overlapping. A

Remark, that we use only translated images of *P* and neither rotation nor reflection nor glide

In order to find a characterization of polyominoes that tile the plane by translation we focus on the boundary of a polyomino *P*. We code the paths of the boundary by 4 letters: *a*

Now, we take a starting point *o* on the boundary and we turn clockwise considering the sequence of steps in order to make a path constituted by unit segments from *o* to the first return on *o*. Let **b**(*P*) be a *boundary word of P* that is the path from *o* to *o* in clockwise that codes the boundary of the polyomino *P* in the following way : starting from an origin on the boundary *o*, the boundary word **b**(*P*) is the concatenation of labels of boundary unit segments read clockwise. Remark that as we turn clockwise we can't have in the step *aa*¯, *aa*¯ ,

*bb* otherwise the considered geometric figure is not a union of unit squares and thus

*b* a down step.

polyomino that tiles the plane by translation is called a *tile* (Fig. 10).

dominoes, 6 triominoes, 19 tetraminoes and so on.

research problem for combinatorists.

of a single polyomino.

**Figure 10. Tilings of the plane**

represents a left step, *b* an up step, *a*¯ a right step and ¯

the coded object could not be a polyomino.

reflection.

*b*¯ *b* or ¯

**Figure 5. Path inside a polyomino**

**Figure 6. The most simple polyomino: the mino**

#### **Figure 7. The 2 dominoes**

There are 6 polyominoes constructed by 3 2-by-2 adjacent unit squares: an horizontal bar of 3 unit squares, a vertical bar of 3 unit squares and four "triominoes *L*" with 3 adjacent unit squares that form an "*L*" shape and the 3 rotations by 90, 180 and 270 degrees (Fig. 8).

#### **Figure 8. The 6 triominoes**

In fact, the number of polyominoes with four unit squares (namely the tetraminoes) is 19.

In addition, the usual definition adds that there is no hole in the interior of a polyomino (Fig. 9).

**Figure 9. Union of unit squares with a hole in the interior**

The number of polyominoes with *n* unit squares is given by the sequence number A001168 according to the Sloane Integer Sequences Encyclopedia (http://oeis.org/A001168): 1, 2, 6, 19, 63, 216, 760, 2725, 9910, 36446, 135268, 505861, 1903890, 7204874, 27394666, 104592937, 400795844,<PHONE_NUMBER>,<PHONE_NUMBER>,<PHONE_NUMBER>0,<PHONE_NUMBER>3,<PHONE_NUMBER>78,<PHONE_NUMBER>524,<PHONE_NUMBER>268,<PHONE_NUMBER>6011,<PHONE_NUMBER>7108,<PHONE_NUMBER>98244,<PHONE_NUMBER>826973, ... We recognize the beginning of the sequence namely 1 mino, 2 dominoes, 6 triominoes, 19 tetraminoes and so on.

We don't know if it possible to find a closed formula (i.e. a mathematical formula depending on *n* that counts the number of distinct polyominoes with *n* unit squares) and this is a research problem for combinatorists.

We have defined polyominoes and now we would like to tile the whole plane by translation of a single polyomino.

Let *P* be a polyomino. A *tiling by translation* of *P* is a covering of the whole plane **R**2 by translated images of *P* such that there is no hole in the tiling and no overlapping. A polyomino that tiles the plane by translation is called a *tile* (Fig. 10).

**Figure 10. Tilings of the plane**

6

**Figure 5. Path inside a polyomino**

**Figure 7. The 2 dominoes**

**Figure 8. The 6 triominoes**

**Figure 9. Union of unit squares with a hole in the interior**

(Fig. 9).

**Figure 6. The most simple polyomino: the mino**

There are 6 polyominoes constructed by 3 2-by-2 adjacent unit squares: an horizontal bar of 3 unit squares, a vertical bar of 3 unit squares and four "triominoes *L*" with 3 adjacent unit squares that form an "*L*" shape and the 3 rotations by 90, 180 and 270 degrees (Fig. 8).

In fact, the number of polyominoes with four unit squares (namely the tetraminoes) is 19. In addition, the usual definition adds that there is no hole in the interior of a polyomino

The number of polyominoes with *n* unit squares is given by the sequence number A001168 according to the Sloane Integer Sequences Encyclopedia (http://oeis.org/A001168): 1, 2, 6, 19, 63, 216, 760, 2725, 9910, 36446, 135268, 505861, 1903890, 7204874, 27394666, Remark, that we use only translated images of *P* and neither rotation nor reflection nor glide reflection.

In order to find a characterization of polyominoes that tile the plane by translation we focus on the boundary of a polyomino *P*. We code the paths of the boundary by 4 letters: *a* represents a left step, *b* an up step, *a*¯ a right step and ¯ *b* a down step.

Now, we take a starting point *o* on the boundary and we turn clockwise considering the sequence of steps in order to make a path constituted by unit segments from *o* to the first return on *o*. Let **b**(*P*) be a *boundary word of P* that is the path from *o* to *o* in clockwise that codes the boundary of the polyomino *P* in the following way : starting from an origin on the boundary *o*, the boundary word **b**(*P*) is the concatenation of labels of boundary unit segments read clockwise. Remark that as we turn clockwise we can't have in the step *aa*¯, *aa*¯ , *b*¯ *b* or ¯ *bb* otherwise the considered geometric figure is not a union of unit squares and thus the coded object could not be a polyomino.

For example, the boundary word of the mino is *aba*¯¯ *b* and all cyclic permutations of letters *ba*¯¯ *ba*, *a*¯¯ *bab* and ¯ *baba*¯ leads to different boundary words depending where the starting point is taken. Nevertheless each cyclic permutation of letters define the same polyomino (Fig. 11).

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*b* by using the point (i) we

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*b* and by simplification

*ba*¯¯ *b* 403

In fact, if *w* is a part of a boundary word read clockwise then *w* is the same path read anti clockwise ! In Fig. 13 we see an example of a path clockwise associated with *w* = *ba*¯¯

*b*.

w

a

*ba*¯¯

a

*ba*¯¯¯¯ *ba*¯¯¯

From Tilings to Fibers – Bio-mathematical Aspects of Fold Plasticity

b

*b*. In fact a mino tiles in a unique way and

*b* one in pseudo square by

*b* and given a tiling like a "non robust

and the same path anti clockwise that is associated with *w* = *ba*¯¯

according to the point (ii) we finally have *w* = *baba*¯

a

w

word of *P* we could recover the possible surrounding of *P*.

*X* · *Y* · *Z where one of the variables in the factorization may be empty.*

**Examples of factorizations and associated tilings of the plane.**

*b* thus *X* = *a* and *Y* = *b* and *X* = *a*¯ and *Y* = ¯

X=a

**Figure 14. Tiling by a mino and the 4 adjacent tiles of the grey mino**

taking *X* = *aa* and *Y* = *b* and *X* = *a*¯*a*¯ and *Y* = ¯

X=a

X=a

Y=b

X=a

*aba*¯¯

if it is equal to *X* · *Y* · *X* · *Y* such a polyomino is called *pseudo square*.

has the property that each tile is surrounded by 4 adjacent tiles (Fig. 14).

Y=b

Y=b Y=b

A horizontal domino has exactly 2 factorizations **b**(*P*) = *aaba*¯*a*¯¯

brickwall". In this case each tile is surrounded by 4 adjacent tiles (Fig. 15).

**Figure 13. Coding of a finite path clockwise and anti clockwise**

reverse the word and take the bar of each letter and thus find *w* = ¯¯

a

b

b

In a tiling of the plane each tile are surrounded by a certain number of adjacent copies of *P*. The spirit of the Beauquier-Nivat's Theorem [1] is that by investigating carefully a boundary

**Theorem 3.1** (Beauquier, Nivat)**.** *A polyomino P tiles the plane by translation if and only if the boundary word* **b**(*P*) *is equal up to a cyclic permutation of the symbols to the factorization X* · *Y* · *Z* ·

If the boundary word is equal to *X* · *Y* · *Z* · *X* · *Y* · *Z* such a polyomino is called *pseudo hexagon*,

A mino has a unique factorization on pseudo square, indeed the boundary word is **b**(*P*) =

b b b

**Figure 11. Coding of the boundary of the mino**

Thus for a boundary word *w* = *w*1*w*2 ··· *wn* where all the *wi* are letters that is *wi* ∈ Σ = {*a*, *b*, *a*¯, ¯ *b*} we define the conjugate of a boundary word *w* if *w* = *uv* then a conjugate is defined by *vu*. We read on Fig. 11 the following boundary word of a mino *a*¯¯ *bab* and we construct the 3 other possible boundary words (by changing the origin of the reading) by the notion of conjugate : we construct *ba*¯¯ *ba* by taking *u* = *a*¯¯ *ba* and *v* = *b*, we construct *aba*¯¯ *b* by taking *u* = *a*¯¯ *b* and *v* = *ab* and we construct ¯ *baba*¯ by taking *u* = *a*¯ and *v* = ¯ *bab*.

This notion of boundary word is efficient because each polyomino can be characterized by it's boundary word.

For example the *L*-triomino could be defined directly by it's boundary word *ba*¯¯ *ba*¯¯ *baab* (Fig. 12). Remark that each conjugate of the word *ba*¯¯ *ba*¯¯ *baab* leads to the same *L*-triomino shape.

**Figure 12. Coding of the boundary of the** *L***-triomino**

We know define a formal operation on words over the alphabet Σ = {*a*, *b*, *a*¯, ¯ *b*}. This operation will be crucial to define interfaces between tiles and for us abstract interfaces between chains.

We define the *u* operator on the word over the alphabet Σ by

if a word *w* of length *n* is written *w* = *w*1*w*2 ··· *wn* where all the *wi* are letters that is *wi* ∈ Σ = {*a*, *b*, *a*¯, ¯ *b*}

$$(i)\overline{w} = \overline{w\_n}\,\overline{w\_{n-1}}\,\cdots\,\overline{w\_1}$$

$$(ii)\overline{\overline{a}} = a \text{ and } \overline{\overline{b}} = b.$$

The point (i) means that to use the bar operator on a word *w*, we bar each letter of the word *w* read in reverse order. The point (ii) means that 2 times the operator bar on a letter is equal to this letter.

In fact, if *w* is a part of a boundary word read clockwise then *w* is the same path read anti clockwise ! In Fig. 13 we see an example of a path clockwise associated with *w* = *ba*¯¯ *ba*¯¯ *b* and the same path anti clockwise that is associated with *w* = *ba*¯¯ *ba*¯¯ *b* by using the point (i) we reverse the word and take the bar of each letter and thus find *w* = ¯¯ *ba*¯¯¯¯ *ba*¯¯¯ *b* and by simplification according to the point (ii) we finally have *w* = *baba*¯ *b*.

**Figure 13. Coding of a finite path clockwise and anti clockwise**

8

*ba*¯¯ *ba*, *a*¯¯

{*a*, *b*, *a*¯, ¯

shape.

Σ = {*a*, *b*, *a*¯, ¯

to this letter.

*b*}

(*i*)*w* = *wn wn*−1 ··· *w*1

(*ii*)*a* = *a* and *b* = *b*.

taking *u* = *a*¯¯

it's boundary word.

*bab* and ¯

**Figure 11. Coding of the boundary of the mino**

notion of conjugate : we construct *ba*¯¯

**Figure 12. Coding of the boundary of the** *L***-triomino**

*b* and *v* = *ab* and we construct ¯

(Fig. 12). Remark that each conjugate of the word *ba*¯¯

For example, the boundary word of the mino is *aba*¯¯

*b* and all cyclic permutations of letters

*ba* and *v* = *b*, we construct *aba*¯¯

*bab*.

*baab* leads to the same *L*-triomino

*bab* and we

*ba*¯¯ *baab*

*b*}. This operation

*b* by

*baba*¯ leads to different boundary words depending where the starting point is

taken. Nevertheless each cyclic permutation of letters define the same polyomino (Fig. 11).

a

b b

a

Thus for a boundary word *w* = *w*1*w*2 ··· *wn* where all the *wi* are letters that is *wi* ∈ Σ =

construct the 3 other possible boundary words (by changing the origin of the reading) by the

This notion of boundary word is efficient because each polyomino can be characterized by

a a

will be crucial to define interfaces between tiles and for us abstract interfaces between chains.

if a word *w* of length *n* is written *w* = *w*1*w*2 ··· *wn* where all the *wi* are letters that is *wi* ∈

The point (i) means that to use the bar operator on a word *w*, we bar each letter of the word *w* read in reverse order. The point (ii) means that 2 times the operator bar on a letter is equal

For example the *L*-triomino could be defined directly by it's boundary word *ba*¯¯

a

a b b

b

We know define a formal operation on words over the alphabet Σ = {*a*, *b*, *a*¯, ¯

We define the *u* operator on the word over the alphabet Σ by

*ba* by taking *u* = *a*¯¯

defined by *vu*. We read on Fig. 11 the following boundary word of a mino *a*¯¯

*b*} we define the conjugate of a boundary word *w* if *w* = *uv* then a conjugate is

*baba*¯ by taking *u* = *a*¯ and *v* = ¯

*ba*¯¯

b

In a tiling of the plane each tile are surrounded by a certain number of adjacent copies of *P*. The spirit of the Beauquier-Nivat's Theorem [1] is that by investigating carefully a boundary word of *P* we could recover the possible surrounding of *P*.

**Theorem 3.1** (Beauquier, Nivat)**.** *A polyomino P tiles the plane by translation if and only if the boundary word* **b**(*P*) *is equal up to a cyclic permutation of the symbols to the factorization X* · *Y* · *Z* · *X* · *Y* · *Z where one of the variables in the factorization may be empty.*

If the boundary word is equal to *X* · *Y* · *Z* · *X* · *Y* · *Z* such a polyomino is called *pseudo hexagon*, if it is equal to *X* · *Y* · *X* · *Y* such a polyomino is called *pseudo square*.

#### **Examples of factorizations and associated tilings of the plane.**

A mino has a unique factorization on pseudo square, indeed the boundary word is **b**(*P*) = *aba*¯¯ *b* thus *X* = *a* and *Y* = *b* and *X* = *a*¯ and *Y* = ¯ *b*. In fact a mino tiles in a unique way and has the property that each tile is surrounded by 4 adjacent tiles (Fig. 14).

**Figure 14. Tiling by a mino and the 4 adjacent tiles of the grey mino**

A horizontal domino has exactly 2 factorizations **b**(*P*) = *aaba*¯*a*¯¯ *b* one in pseudo square by taking *X* = *aa* and *Y* = *b* and *X* = *a*¯*a*¯ and *Y* = ¯ *b* and given a tiling like a "non robust brickwall". In this case each tile is surrounded by 4 adjacent tiles (Fig. 15).

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*b*¯ *ba*¯*a*¯¯ *baa*¯ *b*¯ *babb* it

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From Tilings to Fibers – Bio-mathematical Aspects of Fold Plasticity

*baab* has only one factorization in pseudo

*ba*¯¯ *ba*¯ *bab* has

For the thin cross with 9 unit squares with boundary word equals to *aaba*¯*abb* ¯ *a*¯¯

anymore (the square with the symbol "?" cannot be covered) (Fig. 17).

**Figure 17. A thin cross that doesn't tile the plane**

The triomino L with boundary word equals to *ba*¯¯

**Figure 18. The** *L***-triomino tiles the plane like a pseudo hexagon**

the plane.

While the little thin cross with 5 unit squares with boundary words *abab*¯ *a*¯¯

factorizations with *X* = *aba*,*Y* = *bab*¯ and with *X* = *bab*,*Y* = *ab*¯ *a*¯ (Fig. 19).

2 factorizations in pseudo squares and thus 2 distinct tiling ways according to the 2

To summarize, for each polymino we are able to decide if it tiles the plane by translation by considering the factorization of it's boundary word. Now, we investigate regular tilings of

hexagon, thus it tiles the plane in a unique way (Fig. 18).

is impossible to find a factorization and this proves that the thin cross doesn't tile the plane

?

*ba*¯¯

**Figure 15.  In this case each tile is surrounded by 6 adjacent tiles (Fig. 16).

**Figure 16. Tiling of the plane by a domino like a pseudo hexagon and the 6 adjacent tiles of the grey domino**

Remark that the difference between the 2 tilings by a horizontal domino is the number of adjacent copies of polyominoes that surround the domino at the origin. Thus a domino tiles either like a pseudo square if the number of adjacent tiles of each tile is 4 or like a pseudo hexagon if the number of adjacent tiles of each tile is 6.

For the thin cross with 9 unit squares with boundary word equals to *aaba*¯*abb* ¯ *a*¯¯ *b*¯ *ba*¯*a*¯¯ *baa*¯ *b*¯ *babb* it is impossible to find a factorization and this proves that the thin cross doesn't tile the plane anymore (the square with the symbol "?" cannot be covered) (Fig. 17).

**Figure 17. A thin cross that doesn't tile the plane**

10

Y=b

tiles (Fig. 16).

X=a a

X=aa

X=aa

Y=a

a a

Z=b Z=b

Y=a X=a

a a

X=a

Y=a

hexagon if the number of adjacent tiles of each tile is 6.

X=a

X=a

Y=a

Y=b

**Figure 15. Tiling of the plane by a domino like a pseudo square and the 4 adjacent tiles of the grey domino**

One factorization in pseudo hexagon by taking *X* = *a*, *Y* = *a*, *Z* = *b* and *X* = *a*¯, *Y* = *a*¯, *Z* = ¯

which give a tiling like a "robust brickwall". In this case each tile is surrounded by 6 adjacent

Z=b Z=b

**Figure 16. Tiling of the plane by a domino like a pseudo hexagon and the 6 adjacent tiles of the grey domino**

Remark that the difference between the 2 tilings by a horizontal domino is the number of adjacent copies of polyominoes that surround the domino at the origin. Thus a domino tiles either like a pseudo square if the number of adjacent tiles of each tile is 4 or like a pseudo

*b*

X=a a

a a

Y=b Y=b a a

> The triomino L with boundary word equals to *ba*¯¯ *ba*¯¯ *baab* has only one factorization in pseudo hexagon, thus it tiles the plane in a unique way (Fig. 18).

**Figure 18. The** *L***-triomino tiles the plane like a pseudo hexagon**

While the little thin cross with 5 unit squares with boundary words *abab*¯ *a*¯¯ *ba*¯¯ *ba*¯ *bab* has 2 factorizations in pseudo squares and thus 2 distinct tiling ways according to the 2 factorizations with *X* = *aba*,*Y* = *bab*¯ and with *X* = *bab*,*Y* = *ab*¯ *a*¯ (Fig. 19).

To summarize, for each polymino we are able to decide if it tiles the plane by translation by considering the factorization of it's boundary word. Now, we investigate regular tilings of the plane.

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adjacent tiles for each tile; be careful that while a regular tiling is always given by integral combinations of 2 vectors *v*1 and *v*2, the boundary of each tile in the pseudo hexagon case use 6 sides in correspondence 2-by-2 with the translations *v*1, *v*2 and *v*3). Are you able to recover the 6 adjacent polyominoes of the grey polyomino of Fig. 20 ? Hint: in Fig. 20 one adjacency is only by a single vertical step (for the left regular tiling) or horizontal (for the

From now on, we are dealing with the form of the oligomerization of proteins either like a cylinder or like a fiber. For us a cylinder has finite height (Fig. 1A) and a fiber is a possibly

In our model we would like to make the construction of a fiber (that is a non finite height cylinder) by transformation on regular tilings. As the tiling is regular, the whole tiling is invariant by translation of the 2 non null and non collinear vectors *v*1 and *v*2 defined at the previous paragraph. *v*1 and *v*2 are the vectors of translations that respectively send *X* to *X* and send *Y* to *Y*. In mathematics the parallelogram constructed by *v*1 and *v*2 at a given integer point is called the fundamental domain that is the minimal part of the tiling used to

Let *e*1 be a horizontal unit vector. We choose a direction parallel to *e*1 namely *m*.*v*1 + *nv*2 = *k*.*e*1 with *m* and *n* given integers. Then we construct the little circle (that is by usual definition the horizontal circle of the cylinder) whose perimeter is equal to *k* by superposing the tiles of the original tiling and the tiles translated by *k*.*e*1. Indeed as the tiling is invariant by translation of integral combination of *v*1 and *v*2 we are able recover the same shape by using the translation *m*.*v*1 + *nv*2 (which is by construction equal to *k*.*e*1). Fig. 21 shows an example of a cylinder constructed with a regular tiling by a mino using the identification of the two bold vertical borders for the translation 8 *v*1 (here for a mino *v*1=*e*1). In the associated cylinder

right regular tiling) ...

**5.1. Regular tiling case**

non finite height object (Fig. 25).

**Figure 20. Two regular tilings of the plane by the same polyomino**

**5. From tilings of the plane to tilings of a fiber**

construct the whole tiling by integral combinations of *v*1 and *v*2.

**Figure 19. The 2 tilings like a pseudo square by a thin cross with 5 unit squares**

#### **4. Regular tilings by translation**

The theorem of Beauquier-Nivat gives a correspondence between the way of tiling and the factorization that is the number of copies of the polyomino that surrounds a given polyomino. For the mino, we have 4 copies to cover the whole boundary (because a unit square is a pseudo square that tiles like a square). For the triomino L we have 6 copies to cover the whole boundary (because a triomino L is a pseudo hexagon that tiles like a hexagon). In fact we extend the local correspondences between the coding of part of the boundary word to a geometrical translations namely *v*1 translates *X* to *X* and *v*2 translates *Y* to *Y* (and possibly *v*3 translates *Z* to *Z*). Remark that we have 2 or 3 translations according to the pseudo square or pseudo hexagon, respectively, and by translation to the whole tiling each polyomino has the same local surrounding.

It is interesting to introduce the notion of regular tiling. In a regular tiling the surrounding by adjacent tiles is the same for each polyomino. The mino and the triomino *L* tile only in a regular way by extending the local vectors to the whole tiling.

A *regular tiling* is a tiling by translation of a polyomino *P* such that each tile in the tiling has the same surrounding by translated copies of the tile *P* according to a given factorization of its boundary word. Each factorization leads to a regular tiling of the plane by translation by extending the local translation to the whole plane in the following way.

If *P* is a pseudo square, the factorization **b**(*P*) = *X* · *Y* · *X* · *Y* defines 4 sides of the tile where the sides in correspondence are identified by the pairings (*X*, *X*) and (*Y*,*Y*). The translations *v*1, *v*2 corresponding to these pairings allow us to tile the whole plane in a regular way by using integral combinations of the translations *v*1 and *v*2 to *P* in order to generate the whole tiling (Fig. 14 or Fig. 15).

In the case of a pseudo hexagon the construction with 6 sides is similar. If *P* is a pseudo hexagon, the factorization **b**(*P*) = *X* · *Y* · *Z* · *X* · *Y* · *Z* defines 6 sides of the tile where the sides in correspondence are identified by the pairings (*X*, *X*), (*Y*,*Y*) and (*Z*, *Z*). The translations *v*1, *v*2 and *v*3 corresponding to these pairings allow us to tile the whole plane in a regular way by using integral combination of the translations *v*1 and *v*2 to *P* in order to generate the whole tiling (Fig. 16 or Fig. 18) . Remark that the third vector of translation *v*3 is an integral combination of *v*1 and *v*2 (*v*3 = *v*1 −*v*2 in Fig. 16).

Observe that 2 distinct factorizations of the boundary word of *P* give 2 distinct regular tilings of the plane (Fig. 19 in the case of 4 adjacent tiles for each tile or Fig. 20 in the case of 6 adjacent tiles for each tile; be careful that while a regular tiling is always given by integral combinations of 2 vectors *v*1 and *v*2, the boundary of each tile in the pseudo hexagon case use 6 sides in correspondence 2-by-2 with the translations *v*1, *v*2 and *v*3). Are you able to recover the 6 adjacent polyominoes of the grey polyomino of Fig. 20 ? Hint: in Fig. 20 one adjacency is only by a single vertical step (for the left regular tiling) or horizontal (for the right regular tiling) ...

**Figure 20. Two regular tilings of the plane by the same polyomino**

#### **5. From tilings of the plane to tilings of a fiber**

#### **5.1. Regular tiling case**

12

**Figure 19. The 2 tilings like a pseudo square by a thin cross with 5 unit squares**

regular way by extending the local vectors to the whole tiling.

combination of *v*1 and *v*2 (*v*3 = *v*1 −*v*2 in Fig. 16).

extending the local translation to the whole plane in the following way.

The theorem of Beauquier-Nivat gives a correspondence between the way of tiling and the factorization that is the number of copies of the polyomino that surrounds a given polyomino. For the mino, we have 4 copies to cover the whole boundary (because a unit square is a pseudo square that tiles like a square). For the triomino L we have 6 copies to cover the whole boundary (because a triomino L is a pseudo hexagon that tiles like a hexagon). In fact we extend the local correspondences between the coding of part of the boundary word to a geometrical translations namely *v*1 translates *X* to *X* and *v*2 translates *Y* to *Y* (and possibly *v*3 translates *Z* to *Z*). Remark that we have 2 or 3 translations according to the pseudo square or pseudo hexagon, respectively, and by translation to the whole tiling each polyomino has

It is interesting to introduce the notion of regular tiling. In a regular tiling the surrounding by adjacent tiles is the same for each polyomino. The mino and the triomino *L* tile only in a

A *regular tiling* is a tiling by translation of a polyomino *P* such that each tile in the tiling has the same surrounding by translated copies of the tile *P* according to a given factorization of its boundary word. Each factorization leads to a regular tiling of the plane by translation by

If *P* is a pseudo square, the factorization **b**(*P*) = *X* · *Y* · *X* · *Y* defines 4 sides of the tile where the sides in correspondence are identified by the pairings (*X*, *X*) and (*Y*,*Y*). The translations *v*1, *v*2 corresponding to these pairings allow us to tile the whole plane in a regular way by using integral combinations of the translations *v*1 and *v*2 to *P* in order to generate the whole

In the case of a pseudo hexagon the construction with 6 sides is similar. If *P* is a pseudo hexagon, the factorization **b**(*P*) = *X* · *Y* · *Z* · *X* · *Y* · *Z* defines 6 sides of the tile where the sides in correspondence are identified by the pairings (*X*, *X*), (*Y*,*Y*) and (*Z*, *Z*). The translations *v*1, *v*2 and *v*3 corresponding to these pairings allow us to tile the whole plane in a regular way by using integral combination of the translations *v*1 and *v*2 to *P* in order to generate the whole tiling (Fig. 16 or Fig. 18) . Remark that the third vector of translation *v*3 is an integral

Observe that 2 distinct factorizations of the boundary word of *P* give 2 distinct regular tilings of the plane (Fig. 19 in the case of 4 adjacent tiles for each tile or Fig. 20 in the case of 6

**4. Regular tilings by translation**

the same local surrounding.

tiling (Fig. 14 or Fig. 15).

From now on, we are dealing with the form of the oligomerization of proteins either like a cylinder or like a fiber. For us a cylinder has finite height (Fig. 1A) and a fiber is a possibly non finite height object (Fig. 25).

In our model we would like to make the construction of a fiber (that is a non finite height cylinder) by transformation on regular tilings. As the tiling is regular, the whole tiling is invariant by translation of the 2 non null and non collinear vectors *v*1 and *v*2 defined at the previous paragraph. *v*1 and *v*2 are the vectors of translations that respectively send *X* to *X* and send *Y* to *Y*. In mathematics the parallelogram constructed by *v*1 and *v*2 at a given integer point is called the fundamental domain that is the minimal part of the tiling used to construct the whole tiling by integral combinations of *v*1 and *v*2.

Let *e*1 be a horizontal unit vector. We choose a direction parallel to *e*1 namely *m*.*v*1 + *nv*2 = *k*.*e*1 with *m* and *n* given integers. Then we construct the little circle (that is by usual definition the horizontal circle of the cylinder) whose perimeter is equal to *k* by superposing the tiles of the original tiling and the tiles translated by *k*.*e*1. Indeed as the tiling is invariant by translation of integral combination of *v*1 and *v*2 we are able recover the same shape by using the translation *m*.*v*1 + *nv*2 (which is by construction equal to *k*.*e*1). Fig. 21 shows an example of a cylinder constructed with a regular tiling by a mino using the identification of the two bold vertical borders for the translation 8 *v*1 (here for a mino *v*1=*e*1). In the associated cylinder the two vertical borders are collapsed in a single border and this transformation is general as soon as we identify in the tiling two infinite borders in correspondence by a translation of vector *m*.�*v*1 + *n*�*v*2.

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v2 v1

v1

v1

the abstract view and Fig. 24 for a part of a real fiber) and the polymerization in fiber with name 3J1R has 6 adjacent chains to each chain (see Fig. 25 for the abstract view and Fig.26

This chapter is an investigation of the combinatorial possibilities on the construction of the fiber. Either a fiber is composed by a regular tiling by a single chain (Fig. 23 for the abstract view and Fig. 24 for a part of a real fiber with pseudo square tiling and Fig. 25 for the abstract view and Fig.26 for a part of real fiber with pseudo hexagon tiling) or the regular tiling of the fiber is composed by a *n*-mer (in mathematics we called this object a meta tile). Thus globally the fiber is a tiling by a meta-tile and this meta-tile could be a single chain or a *n*-mer.

In real life we find fibers tile either by a single chain or by a *n*-mer. In Fig. 27 we see a real fiber with name 3J2U constructed with a tetramer of three chains namely A, B and 2 times the chain C. The tetramer is used to tile the fiber of pseudo hexagon type like in Fig. 25 by

Another example is the construction using oligonucleotides of DNA inspired by the seminal work of Winfree [37]. These technics lead to nano tubes composed by dimers surrounded by

Conway has a criterium for tiling the plane in the case of a tile and a rotation of the tile by 180 degrees. In his article, he forms a meta-tile using the tile *T* and it's rotation Rot180(*T*) and this meta-tile is able to tile the plane by translation. Guy Cousineau wrote an article to

about p53, the final construction of the fiber uses tiling of dimers (see Fig. 33D).

*C C* . You will notice in the next section

v2

v1

v2

**Figure 22. Two boundaries in correspondence by the translation** 4*v*1 + 2*v*2)

substituting each pseudo hexagon by the tetramer *A B*

either 4 adjacent dimers [23] or 6 adjacent dimers [39].

for a part of real fiber).

**5.2. Tiling fibers by** *n***-mers**

e1

**Figure 21. From tiling to cylinder by using the translation of** 8 **times the vector** �*e*1**.**

In the next example (Fig. 22), the vectors �*v*1 and �*v*2 send respectively *X* = *bab*¯ to *X* = ¯ *ba*¯ *b* and *Y* = *aba* to *Y* = *a*¯¯ *ba*¯. In order to find a horizontal direction we take 2 times the vector �*v*1 and 1 time the vector �*v*2. Remark that the distance between the point *O* and the point *O* translated by 2�*v*1 +�*v*2 is exactly 5. If we want a little circle with perimeter 10 we consider the translation 2 ∗ (2�*v*1 +�*v*2) that is 4 times �*v*1 and 2 times �*v*1. Now we make a fiber by rolling up the plane by identification the 2 sides in bold (this is exactly the border of the tiling given by a staircase in correspondence by the translation with vector 4�*v*1 + 2�*v*2) (Fig. 22). In order to have a more general construction and in particular to construct a twist in the fiber, we could use a rational coordinate vector �*r*1 instead of �*e*1 and find a direction parallel to �*r*1 given by the equation *m*�*v*1 + *n*�*v*2 = *k*�*r*1.

Conversely, if a polyomino tiles a fiber by translation we recover the whole tiling by finding the little circle and the invariance of the fiber by translation by *k*.�*e*1 and considering in **R**2 the strip at the origin and adding all the strips on **R**2 by translation ℓ.*k*.�*e*1 with ℓ ∈ **Z**.

To summarize, in order to construct a model of fiber using a regular tiling by a single polyomino, the polyomino must tile the plane in a regular way and thus must have at least a factorization as a pseudo square or as a pseudo hexagon. In the fiber, if the tiling is regular then the number of adjacent copies that surround the border is either 4 or 6. This implies that the number of distinct interfaces is 2 or 3.

Of course the model is very flat and in real fibers the molecules are in a 3 dimensional space nevertheless this model allows us to make hypothesis about the number of interface sites and we try to find either 2 or 3 interface sites in the fiber.

We have explained that to build a fiber with a non finite height cylinder shape, it is necessary to have 2 vectors, namely 2 directions of polymerization growth. We have also seen that each tile of fiber will have 4 or 6 adjacent chains formed via 2 or 3 interfaces, respectively. There are examples of such fibers tiled with a single protein: the famous tobacco mosaic virus which was the first virus to be discovered has 4 adjacent chains to each chain (see Fig. 23 for

**Figure 22. Two boundaries in correspondence by the translation** 4*v*1 + 2*v*2)

the abstract view and Fig. 24 for a part of a real fiber) and the polymerization in fiber with name 3J1R has 6 adjacent chains to each chain (see Fig. 25 for the abstract view and Fig.26 for a part of real fiber).

#### **5.2. Tiling fibers by** *n***-mers**

14

vector *m*.�*v*1 + *n*�*v*2.

and *Y* = *aba* to *Y* = *a*¯¯

the equation *m*�*v*1 + *n*�*v*2 = *k*�*r*1.

that the number of distinct interfaces is 2 or 3.

and we try to find either 2 or 3 interface sites in the fiber.

8 e1

**Figure 21. From tiling to cylinder by using the translation of** 8 **times the vector** �*e*1**.**

the two vertical borders are collapsed in a single border and this transformation is general as soon as we identify in the tiling two infinite borders in correspondence by a translation of

In the next example (Fig. 22), the vectors �*v*1 and �*v*2 send respectively *X* = *bab*¯ to *X* = ¯

�*v*1 and 1 time the vector �*v*2. Remark that the distance between the point *O* and the point *O* translated by 2�*v*1 +�*v*2 is exactly 5. If we want a little circle with perimeter 10 we consider the translation 2 ∗ (2�*v*1 +�*v*2) that is 4 times �*v*1 and 2 times �*v*1. Now we make a fiber by rolling up the plane by identification the 2 sides in bold (this is exactly the border of the tiling given by a staircase in correspondence by the translation with vector 4�*v*1 + 2�*v*2) (Fig. 22). In order to have a more general construction and in particular to construct a twist in the fiber, we could use a rational coordinate vector �*r*1 instead of �*e*1 and find a direction parallel to �*r*1 given by

Conversely, if a polyomino tiles a fiber by translation we recover the whole tiling by finding the little circle and the invariance of the fiber by translation by *k*.�*e*1 and considering in **R**2 the

To summarize, in order to construct a model of fiber using a regular tiling by a single polyomino, the polyomino must tile the plane in a regular way and thus must have at least a factorization as a pseudo square or as a pseudo hexagon. In the fiber, if the tiling is regular then the number of adjacent copies that surround the border is either 4 or 6. This implies

Of course the model is very flat and in real fibers the molecules are in a 3 dimensional space nevertheless this model allows us to make hypothesis about the number of interface sites

We have explained that to build a fiber with a non finite height cylinder shape, it is necessary to have 2 vectors, namely 2 directions of polymerization growth. We have also seen that each tile of fiber will have 4 or 6 adjacent chains formed via 2 or 3 interfaces, respectively. There are examples of such fibers tiled with a single protein: the famous tobacco mosaic virus which was the first virus to be discovered has 4 adjacent chains to each chain (see Fig. 23 for

strip at the origin and adding all the strips on **R**2 by translation ℓ.*k*.�*e*1 with ℓ ∈ **Z**.

*ba*¯. In order to find a horizontal direction we take 2 times the vector

*ba*¯ *b*

> This chapter is an investigation of the combinatorial possibilities on the construction of the fiber. Either a fiber is composed by a regular tiling by a single chain (Fig. 23 for the abstract view and Fig. 24 for a part of a real fiber with pseudo square tiling and Fig. 25 for the abstract view and Fig.26 for a part of real fiber with pseudo hexagon tiling) or the regular tiling of the fiber is composed by a *n*-mer (in mathematics we called this object a meta tile). Thus globally the fiber is a tiling by a meta-tile and this meta-tile could be a single chain or a *n*-mer.

> In real life we find fibers tile either by a single chain or by a *n*-mer. In Fig. 27 we see a real fiber with name 3J2U constructed with a tetramer of three chains namely A, B and 2 times the chain C. The tetramer is used to tile the fiber of pseudo hexagon type like in Fig. 25 by substituting each pseudo hexagon by the tetramer *A B C C* . You will notice in the next section about p53, the final construction of the fiber uses tiling of dimers (see Fig. 33D).

> Another example is the construction using oligonucleotides of DNA inspired by the seminal work of Winfree [37]. These technics lead to nano tubes composed by dimers surrounded by either 4 adjacent dimers [23] or 6 adjacent dimers [39].

> Conway has a criterium for tiling the plane in the case of a tile and a rotation of the tile by 180 degrees. In his article, he forms a meta-tile using the tile *T* and it's rotation Rot180(*T*) and this meta-tile is able to tile the plane by translation. Guy Cousineau wrote an article to

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**Figure 25. Fiber with a pseudo hexagon shape.** Each tile is surrounded by 6 tiles. Remark that pseudo hexagon shapes

**Figure 26. Fiber with 3J1R PDB code: an example of tiling of a fiber with 6 adjacent chains.**

For higher stoichiometry, we construct a *n*-mer invariant by rotation of 360/*n* degrees and we copy this meta-tile in order to tile the whole fiber (see Fig. 29 for an abstract view of the

According to the previous tiling constraints, as the number of sites is 2 or 3 it is more easy to make a fiber with a monomer, dimer, trimer, tetramer and hexamer because we combine the shape of the *n*-mers with the form of the interfaces in order to build the whole fiber. We think that the better values are 1, 2, 3, 4, 6 because these values are also in accordance with the crystallographic constraints (the laws of crystallography [31] allow only periodical tilings with rotations of order 1,2,3,4 and 6, that is by rotations of 360, 180, 120, 90 or 60

appear in particular when there is a tilt on the fiber.

replacement by a tetramer).

**Figure 23. Fiber with a pseudo square shape.** Each tile is surrounded by 4 tiles.

**Figure 24. Tobacco Mosaic Virus with 2TMV PDB code: an example of tiling of a fiber with 4 adjacent chains.**

explain the combinatorics of tilings with axial symmetry and translation (see the chapter of Cousineau in this book). In real life, fibers could be constructed for example with dimers. In general the dimer in the fiber is either invariant by a rotation of 180 degrees or invariant by an axial symmetry and then the dimer is used to tile by translation a fiber (see Fig. 28). Thus our model allows us to first make an interface for forming the dimer. And eventually to make a fiber by translation of the dimer according to the tiling constraints that is by having exactly 2 or 3 interfaces for the tiling of the fiber by the dimer. To recap we have constructed either 3 (one for the dimer and 2 for a tiling like a pseudo square) or 4 interfaces (one for the dimer and 3 for a tiling like a pseudo hexagon).

410 Oligomerization of Chemical and Biological Compounds From Tilings to Fibers – Bio-mathematical Aspects of Fold Plasticity 17 10.5772/58577 From Tilings to Fibers – Bio-mathematical Aspects of Fold Plasticity http://dx.doi.org/10.5772/58577 411

16

**Figure 23. Fiber with a pseudo square shape.** Each tile is surrounded by 4 tiles.

dimer and 3 for a tiling like a pseudo hexagon).

**Figure 24. Tobacco Mosaic Virus with 2TMV PDB code: an example of tiling of a fiber with 4 adjacent chains.**

explain the combinatorics of tilings with axial symmetry and translation (see the chapter of Cousineau in this book). In real life, fibers could be constructed for example with dimers. In general the dimer in the fiber is either invariant by a rotation of 180 degrees or invariant by an axial symmetry and then the dimer is used to tile by translation a fiber (see Fig. 28). Thus our model allows us to first make an interface for forming the dimer. And eventually to make a fiber by translation of the dimer according to the tiling constraints that is by having exactly 2 or 3 interfaces for the tiling of the fiber by the dimer. To recap we have constructed either 3 (one for the dimer and 2 for a tiling like a pseudo square) or 4 interfaces (one for the

**Figure 25. Fiber with a pseudo hexagon shape.** Each tile is surrounded by 6 tiles. Remark that pseudo hexagon shapes appear in particular when there is a tilt on the fiber.

**Figure 26. Fiber with 3J1R PDB code: an example of tiling of a fiber with 6 adjacent chains.**

For higher stoichiometry, we construct a *n*-mer invariant by rotation of 360/*n* degrees and we copy this meta-tile in order to tile the whole fiber (see Fig. 29 for an abstract view of the replacement by a tetramer).

According to the previous tiling constraints, as the number of sites is 2 or 3 it is more easy to make a fiber with a monomer, dimer, trimer, tetramer and hexamer because we combine the shape of the *n*-mers with the form of the interfaces in order to build the whole fiber. We think that the better values are 1, 2, 3, 4, 6 because these values are also in accordance with the crystallographic constraints (the laws of crystallography [31] allow only periodical tilings with rotations of order 1,2,3,4 and 6, that is by rotations of 360, 180, 120, 90 or 60

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**Figure 29. Replacement of each pseudo square by a tetramer.**

**5.3. From fold plasticity to fibers: the P53 case**

(PDB 3KIF, Fig. 32) (see [38]).

constructed in synthetic biology.

fiber is given by symmetrical *n*-mers. On the other hand, dimers of pentamer or heptamers are able to fit with some deformations a hexagonal tile ie not all monomers are identical (Fig. 31). Such a dimer of pentamer has been observed in nature e.g. synthetic tachylectin

Likewise the pore forming toxin aerolysin can form a dimer of heptamer after a single amino acid mutation (see [33]). It remains to be established whether the dimeric versions of the pentamer and heptamer are more prompt to form fibers than their single oligomeric counterpart. Of course the combinatoric is more and more complicated for *n* > 7 and to construct the fiber the position of the interfaces on the pseudo square or on the pseudo hexagon is split in different parts for *n* > 7 and this construction is very constrained and maybe this phenomenon is difficult to see in real life. Nevertheless such fibers could be

We give some examples of replacements of a pseudo square by a dimer, by a tetramer and by an octamer and of a pseudo hexagon by a dimer, by a trimer and by a hexamer (see Fig. 30).

Notice that the main difference between tiling a finite height cylinder and tiling a fiber (an infinite height cylinder) is the number of directions of the polymerization growth, one in the former case and two in the latter case. Thus a transition from a cylinder to a fiber implies a transition from a single interface to 2 (4 adjacent chains) or 3 interfaces (6 adjacent chains). For this to occur one simple possibility is to start from an oligomer which has 2 regions of interfaces and loosen the constraints on one so it opens to bind in another direction. Common

**Figure 27. Fiber with 3J2U PBD code containing a tetramer inside each pseudo hexagon**

**Figure 28. Fiber with a dimer inside each pseudo hexagon**

degrees). Of course accordingly a transition from pentamer or heptamer (*n* = 5 or 7) to fiber would be expected to be more difficult at least with the mechanism described in this chapter. In this direction quasicrystals come from aperiodical tilings of the whole plane and non periodical structures could appear in viruses [35], nevertheless non periodical ways to form a fiber by translation of a single tile is described in the next section. Indeed, to form Penrose like tilings we must have two tiles with aperiodic arrangements [5, 31] and it is still an open problem to find an aperiodic tiling of the whole plane with a single tile [15] (we expect by the theorem of Beauquier-Nivat that the boundary of such tile (if it exists) must be fractal...). In our fiber model the rules seem to be restrictive because such symmetry fits directly without deformation into a square or hexagonal tile. Of course in real life, the *n*-mers could be symmetrical ([17, 22, 24]) but also non-symmetrical ([4, 32]). We think that for tiling constraints on the border of each *n*-mers the less expansive solution in order to construct a

**Figure 29. Replacement of each pseudo square by a tetramer.**

18

**Figure 27. Fiber with 3J2U PBD code containing a tetramer inside each pseudo hexagon**

degrees). Of course accordingly a transition from pentamer or heptamer (*n* = 5 or 7) to fiber would be expected to be more difficult at least with the mechanism described in this chapter. In this direction quasicrystals come from aperiodical tilings of the whole plane and non periodical structures could appear in viruses [35], nevertheless non periodical ways to form a fiber by translation of a single tile is described in the next section. Indeed, to form Penrose like tilings we must have two tiles with aperiodic arrangements [5, 31] and it is still an open problem to find an aperiodic tiling of the whole plane with a single tile [15] (we expect by the theorem of Beauquier-Nivat that the boundary of such tile (if it exists) must be fractal...). In our fiber model the rules seem to be restrictive because such symmetry fits directly without deformation into a square or hexagonal tile. Of course in real life, the *n*-mers could be symmetrical ([17, 22, 24]) but also non-symmetrical ([4, 32]). We think that for tiling constraints on the border of each *n*-mers the less expansive solution in order to construct a

**Figure 28. Fiber with a dimer inside each pseudo hexagon**

fiber is given by symmetrical *n*-mers. On the other hand, dimers of pentamer or heptamers are able to fit with some deformations a hexagonal tile ie not all monomers are identical (Fig. 31). Such a dimer of pentamer has been observed in nature e.g. synthetic tachylectin (PDB 3KIF, Fig. 32) (see [38]).

Likewise the pore forming toxin aerolysin can form a dimer of heptamer after a single amino acid mutation (see [33]). It remains to be established whether the dimeric versions of the pentamer and heptamer are more prompt to form fibers than their single oligomeric counterpart. Of course the combinatoric is more and more complicated for *n* > 7 and to construct the fiber the position of the interfaces on the pseudo square or on the pseudo hexagon is split in different parts for *n* > 7 and this construction is very constrained and maybe this phenomenon is difficult to see in real life. Nevertheless such fibers could be constructed in synthetic biology.

We give some examples of replacements of a pseudo square by a dimer, by a tetramer and by an octamer and of a pseudo hexagon by a dimer, by a trimer and by a hexamer (see Fig. 30).

#### **5.3. From fold plasticity to fibers: the P53 case**

Notice that the main difference between tiling a finite height cylinder and tiling a fiber (an infinite height cylinder) is the number of directions of the polymerization growth, one in the former case and two in the latter case. Thus a transition from a cylinder to a fiber implies a transition from a single interface to 2 (4 adjacent chains) or 3 interfaces (6 adjacent chains). For this to occur one simple possibility is to start from an oligomer which has 2 regions of interfaces and loosen the constraints on one so it opens to bind in another direction. Common

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biological examples of such transition are domain swapping ([9, 25]) where the presence of

Another possibility is to start from a dimer with 2 regions of interfaces and again loosen one of the region but this time via a mutation. If the modified residue controlled the interactions between the 2 regions, its modification may loosen one region opening it to form a second interface. Since the initial molecule is a dimer, the mutation will free 2 new interfaces enabling the growth in 2 directions. The formation of the fiber will depend on the symmetry of the dimer. To illustrate this possibility, let's consider the p53 tetramer (see Fig. 33). There are several familial point mutations such as G334V or R337 that are responsible for the transition of the p53 into fibers or into a non-native oligomeric state, leading to impairment of the protein function and cancer development [27]. p53 tetramer has a *D*2 symmetry and is a dimer of a dimer. Each monomer is made of a *β*-strand, a small helix and a long *α*-helix (Fig. 33B). Contacts exist between the residue R337 of the small helix and the residues E349 and D352 of the long helix of the adjacent chain within a dimer of p53 (Fig. 33A). There are contacts between the residues L350 and K351 of one monomer of a dimer and the same residues on another monomer of another dimer (Fig. 33A). The mutation of R337 might inhibit the formation of the small helix and replace it by a more flexible linker which would be too mobile to allow the contacts with the residues E349 and D352 and might also subsequently loosen the contacts between L350 and K351 (Fig. 33B and C). As a consequence the relative position of the *β*-strand and the long *α*-helix would be

a flexible linker frees one interface for fiber formation.

altered allowing the formation of 2 new interfaces (Fig. 33D).

**6.1. From 1-periodical tiling to fibers**

a polyomino *P* and to map it on a fiber.

*k* ∈ **N**.

*V* = *Ur*

**6. Non periodical fiber case and tiling of the whole space case**

In fact, all the preceding constructions are based on regular tilings of the plane. We are able to relax this constraint and to investigate non regular tilings (Fig. 34). Non regular tilings of the plane are constructed by perturbation of regular tilings. We push some strips in the direction *v*1 in order to break the periodicity in direction *v*2. In the seminal article of Beauquier-Nivat [1] an important proposition is proved that a tiling (regular or not) by translation of a polyomino is always 1-periodic that is invariant by a translation by a non null vector. This proposition is crucial for us because it allows to take a non regular tiling by

We take a given tiling *T* by translation of a given polyomino *P* and as the tiling *T* is 1-periodic it is invariant by vector *v*1. Thus we have the property that the translation by *v*1 of the tiling *T* noted Trans*v*1 (*T*) is equal to the tiling *T* (here to be more formal and to define the equality between tilings we must introduce equivalent classes of tilings up to translation operator but it is too formal for this chapter written for biologists). By composition of the translations, we have that the translation of *k* times the vector *v*1 of the tiling *T* is also equal to the tiling *T*. That is Trans*k*.*v*1 (*T*) = *T*. Thus globally the tiling *T* is invariant by the translations *k*.*v*1 for

This global property of invariance of the tiling *T* implies a strong property on the boundary word of *P*. In fact in the original article of Beauquier-Nivat the following proposition: if the tiling *T* is 1-periodic then the boundary word is of the form *XWXV* with *W* = *Ur* and

. Thus locally in the boundary word we are able to read the periodicity property.

**Figure 30. Replacement of a pseudo square and of a pseudo hexagon.**

**Figure 31. Replacement of pseudo square by 2 non regular pentamers.**

**Figure 32. Double asymmetrical pentamer.**

biological examples of such transition are domain swapping ([9, 25]) where the presence of a flexible linker frees one interface for fiber formation.

Another possibility is to start from a dimer with 2 regions of interfaces and again loosen one of the region but this time via a mutation. If the modified residue controlled the interactions between the 2 regions, its modification may loosen one region opening it to form a second interface. Since the initial molecule is a dimer, the mutation will free 2 new interfaces enabling the growth in 2 directions. The formation of the fiber will depend on the symmetry of the dimer. To illustrate this possibility, let's consider the p53 tetramer (see Fig. 33). There are several familial point mutations such as G334V or R337 that are responsible for the transition of the p53 into fibers or into a non-native oligomeric state, leading to impairment of the protein function and cancer development [27]. p53 tetramer has a *D*2 symmetry and is a dimer of a dimer. Each monomer is made of a *β*-strand, a small helix and a long *α*-helix (Fig. 33B). Contacts exist between the residue R337 of the small helix and the residues E349 and D352 of the long helix of the adjacent chain within a dimer of p53 (Fig. 33A). There are contacts between the residues L350 and K351 of one monomer of a dimer and the same residues on another monomer of another dimer (Fig. 33A). The mutation of R337 might inhibit the formation of the small helix and replace it by a more flexible linker which would be too mobile to allow the contacts with the residues E349 and D352 and might also subsequently loosen the contacts between L350 and K351 (Fig. 33B and C). As a consequence the relative position of the *β*-strand and the long *α*-helix would be altered allowing the formation of 2 new interfaces (Fig. 33D).

#### **6. Non periodical fiber case and tiling of the whole space case**

#### **6.1. From 1-periodical tiling to fibers**

20

**Figure 30. Replacement of a pseudo square and of a pseudo hexagon.**

**Figure 31. Replacement of pseudo square by 2 non regular pentamers.**

**Figure 32. Double asymmetrical pentamer.**

In fact, all the preceding constructions are based on regular tilings of the plane. We are able to relax this constraint and to investigate non regular tilings (Fig. 34). Non regular tilings of the plane are constructed by perturbation of regular tilings. We push some strips in the direction *v*1 in order to break the periodicity in direction *v*2. In the seminal article of Beauquier-Nivat [1] an important proposition is proved that a tiling (regular or not) by translation of a polyomino is always 1-periodic that is invariant by a translation by a non null vector. This proposition is crucial for us because it allows to take a non regular tiling by a polyomino *P* and to map it on a fiber.

We take a given tiling *T* by translation of a given polyomino *P* and as the tiling *T* is 1-periodic it is invariant by vector *v*1. Thus we have the property that the translation by *v*1 of the tiling *T* noted Trans*v*1 (*T*) is equal to the tiling *T* (here to be more formal and to define the equality between tilings we must introduce equivalent classes of tilings up to translation operator but it is too formal for this chapter written for biologists). By composition of the translations, we have that the translation of *k* times the vector *v*1 of the tiling *T* is also equal to the tiling *T*. That is Trans*k*.*v*1 (*T*) = *T*. Thus globally the tiling *T* is invariant by the translations *k*.*v*1 for *k* ∈ **N**.

This global property of invariance of the tiling *T* implies a strong property on the boundary word of *P*. In fact in the original article of Beauquier-Nivat the following proposition: if the tiling *T* is 1-periodic then the boundary word is of the form *XWXV* with *W* = *Ur* and *V* = *Ur* . Thus locally in the boundary word we are able to read the periodicity property.

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**Figure 34. Non regular tiling with dominoes.** The grey domino is surrounded by 5 dominoes thus this tiling is certainly not regular. Remark that the horizontal strips are invariant by the horizontal vector *v*1 and also by the vector 3*v*1. Thus we are able

the second is *Z* because *W* = *U*3 = *YZ*. Remark in the picture the periodic part in a staircase

This condition on the border is a necessary condition to find a possible periodicity. In particular, if we want to make the tiling 1-periodic we must push the strip along this direction

In order to construct a fiber we take the tiling and map it on a fiber by superposing tiles according to the direction of periodicity given by *v*1. Now we choose a non null value for *k*, we are able to construct a fiber because the whole tiling is invariant by the translation *kv*1 and we superpose the tiles of the original tiling and the tiles translated by *kv*1. Thus we make a fiber that contains exactly *k* polyominoes along the direction *v*1. An important implication

to map the left border to the right border in order to form a non finite height cylinder.

shape (Fig. 35).

of periodicity.

**Figure 35. Non regular tiling and stair case shape.**

**Figure 33. Transition from oligomer to fiber, the example of the p53 case. A. The p53 tetrameric domain is made of 2 dimers.** Each monomer is made of a *β*-strand followed by a small helix ended by a long *α*-helix parallel to the *β*-strand (1SAK, PDB code). The backbone of each monomer is indicated by a different color. The spacefill amino acids are the residues in contact with R337, which is a residue sensitive to mutation related to some cancer development. **B. Backbone representation of the p53 dimer.** The model proposes that upon mutation of R337, connections between the 2 monomers within a dimer are loosen, enabling a free movement of the long *α*-helix relative to the *β*-strand. C. Schematic of the open-p53 dimer. D. As the structural change takes place in the four monomers, it offers enough new interfaces for growing in 2 directions *V*1 and *V*2.

For example for a bar of 5 unit squares with boundary word *abbbbba*¯¯ *b*¯ *b*¯ *b*¯ *b* ¯ *b*, we have many factorizations one in pseudo square by taking *X* = *a* and *Y* = *bbbbb* and *X* = *a*¯ and *Y* = ¯ *b*¯ *b*¯ *b*¯ *b*¯ *b* and 4 four factorisations in pseudo hexagon by taking *X* = *a* and *Y* = *b* and *Z* = *bbbb* and *X* = *a*¯ and *Y* = ¯ *b* and *Z* = ¯ *b*¯ *b*¯ *b*¯ *b* or *X* = *a* and *Y* = *bb* and *Z* = *bbb* and *X* = *a*¯ and *Y* = ¯ *b*¯ *b* and *Z* = ¯ *b*¯ *b*¯ *b* or ... or *X* = *a* and *Y* = *bbbb* and *Z* = *b* and *X* = *a*¯ and *Y* = ¯ *b*¯ *b*¯ *b*¯ *b* and *Z* = ¯ *b*. In this example the periodic part on the boundary word is *U*5 = *bbbbb* with *U* = *b* and *U*5 = ¯ *b*¯ *b*¯ *b*¯ *b*¯ *b*. Remark that in this case *U*5 = *bbbbb* appears in the factorization in pseudo square.

Another example without factorization in pseudo square is a polyomino constructed by union of 3 triomino L and forming a polyomino with 9 unit squares with boundary word ¯ *baabababab*¯ *a*¯¯ *ba*¯¯ *ba*¯¯ *b* here the factorizations are

*X* = ¯ *ba Y* = *a Z* = *babab X* = *ab*¯ *Y* = *a*¯ *Z* = ¯ *ba*¯¯ *ba*¯¯ *b* or *X* = ¯ *ba Y* = *aba Z* = *bab X* = *ab*¯ *Y* = *a*¯¯ *ba*¯ *Z* = ¯ *ba*¯¯ *b* or *X* = ¯ *ba Y* = *ababa Z* = *b X* = *ab*¯ *Y* = *a*¯¯ *ba*¯¯ *ba*¯ *Z* = ¯ *b*. In this case *U*3 = *ababab* with *r* = 3 and *U* = *ab* and *U*3 = ¯ *ba*¯¯ *ba*¯¯ *ba*¯ with *r* = 3 and *U* = ¯ *ba*¯. Remark that in this case *U*3 does not appear in the boundary word factorization and is splited in 2 parts, one is *Y* and 416 Oligomerization of Chemical and Biological Compounds From Tilings to Fibers – Bio-mathematical Aspects of Fold Plasticity 23 10.5772/58577 From Tilings to Fibers – Bio-mathematical Aspects of Fold Plasticity http://dx.doi.org/10.5772/58577 417