Structure and dynamics of the ‘protein folding code’ infe

Tlusty’s topological rate distortion analysis of the genetic code is applied to protein symmetries and protein folding rates.Unlike the genetic case,numerous thermodynamically accessible‘protein folding codes’can be identiﬁed from empirical classiﬁcations.Folding rates follow from a topologically driven rate distortion argument,a model that can,in principle,be extended to intrinsically disordered proteins.The elaborate cellular regulatory machinery of the endoplasmic reticulum and heat shock proteins is needed to prevent transition between the various thermodynamically‘natural’sets of hydrophobic-core protein conformations,and its corrosion by aging would account for the subsequent onset of many protein folding disorders.These results imply markedly different evolutionary trajectories for the genetic and protein folding codes,and suggest that the‘protein folding code’is really a complicated composite,distributed across protein production and a cellular,or higher,regulatory apparatus acting as a canalizing catalyst that drives the system to converge on particular transitive components within a signiﬁcantly larger‘protein folding groupoid’.

1.Introduction

It is obvious,from the great spectrum of protein folding dis-orders,that the amyloidﬁbril and other,less well-characterized geometric forms,must compete thermodynamically and kinetically with three-dimensional globular,and unfolded monomeric states. This suggests,we will show,the existence of numerous underly-ing‘protein folding codes’whose ultimate structures are,perhaps, a somewhat debatable matter of formal taxonomy.Fig.1,from Hartl and Hayer-Hartl(2009),schematically expands the spectrum of foldedﬁnal conformations according to an in vivo‘folding fun-nel’model dispersed across a measure of intra-vs.inter-molecular contact for hydrophobic-core proteins forming tertiary structure. Intra-molecular conformations involve three-dimensional assem-blages of␣-helices and␤-sheets,while the most densely packed inter-molecular form is,perhaps,the semicrystalline amyloidﬁbril.

At a later stage we will examine intrinsically disordered pro-teins(IDP)that lack the hydrophobic-core,do not have inherent tertiary structures,and reach a conformation only in partnership with another chemical species(e.g.,Uversky et al.,2008;Serdyuk, 2007).

The basic spectrum of Fig.1for proteins having a hydrophobic core,in general,explains the necessity of the elaborate regulatory structures associated with the endoplasmic reticulum and its atten-E-mail address:wallace@pi.cpmc.columbia.edu.dant spectrum of chaperone proteins(e.g.,Scheuner and Kaufman, 2008),and the evolutionary pattern of protein sequences inferred by Goldschmidt et al.(2010).The inevitable corrosion of the cellular regulatory apparatus with age would then explain the subsequent onset of amyloidﬁbril and other aggregation disorders.

Most particularly,the spectrum of valleys in Fig.1characterizes a set of equivalence classes that deﬁnes a‘protein folding groupoid’, in the sense of Weinstein(1996).As we will argue below,both the native state and amyloidﬁbril have structured subdivisions, internal equivalence classes,that deﬁne a nested set of groupoids. See Mathematical Appendix for a summary of standard material on groupoids.

With regard to the disjunction between‘native’and‘amyloid’protein forms,very early on,Astbury(1935)conjectured that glob-ular proteins could also have a linear state,based on pioneering X-ray studies.Chiti et al.(1999)argue that

...[P]rovided appropriate conditions are maintained over pro-longed periods of time,the formation of ordered amyloid protoﬁlaments andﬁbrils could be an intrinsic property of many polypeptide chains,rather than being a phenomenon limited to

a very few aberrant sequences.

Wang et al.(2008),in a an elegant series of experiments on bacterial inclusion bodies,conclude that

...[A]myloid aggregation appears to be a common property of protein segments and consequently is observed in both eukaryotes and prokaryotes...[Thus]there must be evolved

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