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20R.Wallace /BioSystems 103 (2011) 18–26

rge scale structure

Broadly,Fig.1embraces a fourfold classification:

1.The ‘native state’determined,at low concentrations,entirely by the amino acid sequence in the classic sense of Anfinsen (1973).

2.Amorphous aggregates.

3.Semi-structured oligomers,as explored by Krebs et al.(2009).

4.Amyloid/amyloid-like one-dimensional fibrils.

Generalizing Table 1of Tlusty (2007a,b,2010a)according to the genus of the underlying graph,that is,the number of holes in the error network associated with the proposed code,we can apply Heawood’s graph genus formula for the coloring number that iden-tifies the maximal number of first excited modes of the coding graph Laplacian,chr ( )=Int

1

2

(7+

1+48 )

.(1)

where Int is the integer value of the enclosed expression and itself is defined from Euler’s formula (Tlusty,2007a,b,2010a )as =1−

1

2

(V −E +F ),(2)

where V is the number of code network vertices,E the number of network edges,and F the number of enclosed faces.Eq.(1)produces the table

(#network holes)chr( )(#prot.syms.)041728394105116,7128,9

13

In Tlusty’s scheme,the second column represents the maximal possible number of product classes that can be reliably produced by error-prone codes having holes in the underlying coding error network.As stated,in this context we have coarse-grained the rela-tion between DNA sequence space and tertiary protein structures.

From Tlusty’s perspective,then,our fourfold classification for Fig.1produces a the simplest possible large-scale ‘protein folding code’,a sphere limited by the four-color problem,and the simplest cognitive cellular regulatory system would thus be constrained to pass/fail on four basic flavors,as it were,of folded proteins.

Within the funnel leading to the native state,however,chaper-one processes would face far more difficult choices.

This suggests a possible twofold cellular regulatory structure,and next we consider the two most fully characterized geometric structures in more detail,the normal and amyloid forms.3.Normal globular proteins

Normal irregular protein symmetries were first classified by Levitt and Chothia (1976),following a visual study of polypep-tide chain topologies in a limited dataset of globular proteins.Four major classes emerged;all ␣-helices;all ␤-sheets;␣/␤;and ␣+␤,as illustrated in Fig.2.

While this scheme strongly dominates observed irregular pro-tein forms,Chou and Maggiora (1998),using a much larger data set,recognize three more ‘minor’symmetry equivalence classes;␮(multi-domain);␴(small protein);and ␳(peptide),and a possible three more ‘subminor’groupings.

We infer that the normal globular ‘protein folding code error network’is,essentially,a large connected ‘sphere’–producing the four dominant structural modes of Fig.2–having one minor,

and

Fig.2.From Chou and Zhang (1995).Standard equivalence classes for inexact pro-tein symmetries according to Levitt and Chothia (1976):(a)all-␣helices;(b)all-␤sheets;(c)␣+␤;(d)␣/␤.More recent work identifies a minimum of seven,and possibly as many as ten,such classes (Chou and Maggiora,1998).

possibly as many as three more ‘subminor’attachment handles,in the Morse Theory sense (Matsumoto,2002),a matter opening up other analytic approaches.4.Amyloid fibrils

As described above,Kim and Hecht (2006)suggest that over-all amyloid fibril geometry is very much driven by the underlying ␤-sheet coding 1010101,although the rate of fibril formation may be determined by exact chemical constitution.Work by Sawaya et al.(2007)parses some of those subtleties:they identify an eight-fold ‘steric zipper’symmetry necessarily associated with the linear amyloid fibrils that characterize a vast spectrum of protein folding disorders.Fig.3,adapted from their work,shows those symmetries.In essence,two identical sheets can be classified by the orientation of their faces (face-to-face/face-to-back),the orientation of their strands (with both sheets having the same edge of the strand up or one up and the other down),and whether the strands within the sheets are parallel or antiparallel.Five of the eight symmetry possibilities have been observed.This suggests,from the text table above,that the ‘amyloid folding code error network’is a double donut,that is,has two,different sized,interior holes,resembling,perhaps,a toroid with a smaller attachment handle.5.Amyloid self-replication

Maury (2009)has recently proposed an ‘amyloid world’model for the emergence of prebiotic informational entities,based on the extraordinary stability of amyloid structures in the face of the harsh conditions of the prebiotic world.From this perspec-tive,the synthesis of RNA,and the evolution of the RNA-protein world,were later,but necessary events for further biomolucu-lar evolution.Maury further argues that,in the contemporary DNA ⇔RNA ⇒protein world,the primordial ␤-conformation-based information system is preserved in the form of a cytoplasmic epigenetic memory.