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cTDY uITDNsuKU uKLONQRSB · 2001. 4. 27. · t 0/ *- *) & { + { / " / {/ * * loj * * [+ { * + qtdr : ;0< =?>a@ g ! #" &_ %$& ' ( g ) +* `_-, ' .&/, ' -_

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Page 1: cTDY uITDNsuKU uKLONQRSB · 2001. 4. 27. · t 0/ *- *) & { + { / " / {/ * * loj * * [+ { * + qtdr : ;0< =?>a@ g ! #" &_ %$& ' ( g ) +* `_-, ' .&/, ' -_ <; =">/'"@? ~

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CH COOHisoleucine (ileu)

ALIPHATIC AMINO ACIDS(major amino acids contributed to a hydrophobic region) 2 dim. blocks

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dimension

2 dim.

r 1 r 2 r 3a r 3b

3 dim.

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1

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r = 13a

r = 13b

one block of: term:

3x3 square

with blocks:

Two dimensional case: square

block of1 cell:cell center

r = 11

r = 12

r = 13a

r = 13b

one block of: term:

r = 1

Three-dimensional case: cube

with blocks:

block of1 cell:cell center

3x3x3 cubeThe planes of the schematic cube are laid through the cell centers.

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block 1

1

block 2

3

4

5

now block 4 and 5 fit in the cube

2

block 3 block 4 block 5

3:

2:

block 1, 2, 3 fit in the cube

4:

5:

steps: 1:

there is no way to place a block of the pool at the lowest cell position

block 3 gets removed from the cube

next higher block orientation than that of block 3 is tried

:�;0<�=?>A@ RD � � � I�glkWOuSUYlVaG� �HYhMXn9I�ghY �NKNj1m`q I m9giO&e s7VaSrKNK

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r

y = zr

x = yr

z = x

x

rotation with rotation number 3:

x

z

y

rz

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b

e

(2)

(3)

(1)

One can see that there exist sequences like (1) that cannot fold as well as (in this special case)

sequences like (2) that fold to every possible microconfiguration!

a: b: e:

With C={set of all possible chains through a, b, and e}

Every foldable sequence is represented by a chain of the set C.

There is no one to one correlation between sequences and microconfigurations.:

By rotation and reflexion of these microconfigurations one gets three not further reduceable microconfigurations.

Such a set of microconfigurations is:

a: e: i:

a

b:

With p={1, 1, 0, 1} the possible sequences s are: s:

f: j:

k:

l:h:

g:c:

d:

there are 12 microconfigurations in a 2x3 rectangle:

(The numbers are the indices i of p and givethe block types along the chain.)(2): 1, 4, 2

(3): 2, 1, 4

(4): 2, 4, 1 = (2)

(5): 4, 1, 2 = (3)

(6): 4, 2, 1 = (1)

(1): 1, 2, 4

i: 1 2 3 4

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V�b�O�e]KNghghM �

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e

a

b

c

d

fG (X) with X as block set of all blocks in MC x :

a

d

c

d

a

a

e

b

bb

b

e

a

G (X)

a

a

d

c

d

x

y

X={a, b, c, d, e} G(X)

G (S) with S as subset of X :f

c

e

b

S ={b, c, e}:2

G (S ):f 2G (S ):f 1

ce

. . . is connected

d

1S ={c, d, e}:

. . . is not connected

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ba c1

1

1

(i)

a b c

1

1

(ii)1

structure of move type:

a b c

1

2

1 (iii)

a b c

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min. block size:

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c

smin

a b c

1i 1 1

2 10o

-1 01∆

smin 32 2

3 2 2

smin

smin

a

a

b c

i

o

/

1 2 1

0 2 2

1 0 -1

1

2

3

3

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with

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i

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0

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2

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3

1

1

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ba1

1

1 c

(i)

cwith

(ii) as inverse of (i) !

(same move type as (i))

with

min. block size:

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Page 68: cTDY uITDNsuKU uKLONQRSB · 2001. 4. 27. · t 0/ *- *) & { + { / " / {/ * * loj * * [+ { * + qtdr : ;0< =?>a@ g ! #" &_ %$& ' ( g ) +* `_-, ' .&/, ' -_ <; =">/'"@? ~

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��N 4 YU6cLONQR 4 Y�YZYr6 � � � � � � ��[tTDJZJ�UkX$r2qsTIu\r� {��uk]Qr�JZRQU5uK]QTtEKTD� rOE\T�uK]QT E\TDU�u\EKY�[tuKY�r�N � � � � � � � ) � � � �)]QT<EKTDU\CQJ uKYZNQPSTDJZT�qsTDN JZY�NQT LWEC4 NQrOu`Tuv�CQY�qWLOJZTtN}u`6 X$r2qsT<u *3��TDUvLWEKT<P�Y�qsTDN�YZN +yP�CFEKT9� {

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Page 72: cTDY uITDNsuKU uKLONQRSB · 2001. 4. 27. · t 0/ *- *) & { + { / " / {/ * * loj * * [+ { * + qtdr : ;0< =?>a@ g ! #" &_ %$& ' ( g ) +* `_-, ' .&/, ' -_ <; =">/'"@? ~

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1

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� ��� ��� ' � � � ' ����� { � { 3 � 3 � { � 3 � 3 { � ������� ��� ������� { � �����i�� � 3(' � { 3 Q� � { 3 3 Q� '-'-'-'� ��� ��� 3 � 3 � 3 � ��1� ' ���2� { ���}� 3 { � ����� �� ��� ��� { � � � { ���W�& � �2� 3 �� �}� 3 Q� '-'-'-'� ��� ���6Q� ' � 3 � s���� ��� � � � � 3 { � ��� ' � � � ��� ' � { � � � � 3(' �� { { � { � 3 Q� '-'-'-'� � � ������� 3 � ���> { ' �< 3 �}� ���> '-' 3 { � �������� � � ��� 3 � � � �< � �}� ' { � 3 F�2� ' {R{ 3 Q� '-'-'-'� � � ��� { � ' � � ' ��� � �< ' �}�1� 3 3 { � ���

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lokal maximum at MC 560:

a b c d e f g h i j k

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