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26R.Wallace/BioSystems103 (2011) 18–26

Two points are in the same orbit if they are similarly placed within their tiles or within the grout pattern.

(2)The isotropy group of x∈B consists of those g in G with

˛(g)=x=ˇ(g).It is trivial for every point except those in 1/2 ∩B,for which it is Z2×Z2,the direct product of integers modulo two with itself.

By contrast,embedding the tiled structure within a larger context permits definition of a much richer structure,i.e.,the iden-tification of local symmetries.

We construct a second groupoid as follows.Consider the plane R2as being decomposed as the disjoint union of P1=B∩X(the grout),P2=B\P1(the complement of P1in B,which is the tiles), and P3=R2/B(the exterior of the tiled room).Let E be the group of all euclidean motions of the plane,and define the local symmetry groupoid G loc as the set of triples(x, ,y)in B×E×B for which x= y, and for which y has a neighborhood u in R2such that (u∩P i)⊆P i for i=1–3.The composition is given by the same formula as for G( , R2).

For this groupoid-in-context there are only afinite number of orbits:

O1=interior points of the tiles.

O2=interior edges of the tiles.

O3=interior crossing points of the grout.

O4=exterior boundary edge points of the tile grout.

O5=boundary‘T’points.

O6=boundary corner points.

The isotropy group structure is,however,now very rich indeed: The isotropy group of a point in O1is now isomorphic to the entire rotation group O2.

It is Z2×Z2for O2.

For O3it is the eight-element dihedral group D4.

For O4,O5and O6it is simply Z2.

These are the‘local symmetries’of the tile-in-context. References

Anfinsen,C.,1973.Principles that govern the folding of protein chains.Science181, 223–230.

Astbury,W.,1935.The X-ray interpretation of the denaturation and the structure of the seed gobulins.Biochemistry29,2351–2360.

Atlan,H.,Cohen,I.,1998.Immune information,self-organization,and meaning.

International Immunology10,711–717.

Brown,R.,1987.From groups to groupoids:a brief survey.Bulletin of the London Mathematical Society19,113–134.

Bruce,M.,Dickinson,A.,1987.Biological evidence that scrapie agent has an inde-pendent genome.Journal of General Virology68,79–89.

Cannas Da Silva,A.,Weinstein,A.,1999.Geometric Models for Noncommutative Algebras.American Mathematical Society,Providence,RI.

Chiti,F.,Webster,P.,Taddei,N.,Clark,A.,Stefani,M.,Ramponi,G.,Dobson,C.,1999.

Designing conditions for in vitro formation of amyloid protofilaments andfibrils.

Proceedings of the National Academy of Sciences of America96,3590–3594. Chou,K.,Zhang,C.,1995.Prediction of protein structural classes.Critcal Reviews in Biochemistry and Molecular Biology30,275–349.

Chou,K.,Maggiora,G.,1998.Domain structural class prediction.Protein Engineering 11,523–538.

Collinge,J.,Clarke,A.,2007.A general model of prion strains and their pathogenicity.

Science318,930–936.

Dill,K.,Banu Ozkan,S.,Sweikl,T.,Chodera,J.,Voelz,V.,2007.The protein fold-ing problem:when will it be solved?Current Opinion in Structural Biology17, 342–346.

Eldredge,N.,Gould,S.,1972.Punctuated equilibrium:an alternative to phyletic gradualism.In:Schopf,T.(Ed.),Models in Paleobiology,pp.82–115.

Falsig,J.,Nilsson,K.,Knowles,T.,Aguzzi,A.,2008.Chemical and biophysical insights into the propagation of prion strains.HFSP Journal2,332–341.

Feynman,R.,2000.Lectures on Computation.Westview,New York. Goldschmidt,L.,Teng,P.,Riek,R.,Eisenberg,D.,2010.Identifying the amylome, proteins capable of forming amyloid-likefibrils.Proceedings of the National Academy of Sciences of America107,3487–3492.

Golubitsky,M.,Stewart,I.,2006.Nonlinear dynamics and networks:the groupoid formalism.Bulletin of the American Mathematical Society43,305–364.Gruebele,M.,2005.Downhill protein folding:evolution meets ptes Rendus Biologies328,701–712.

Hartl,F.,Hayer-Hartl,M.,2009.Converging concepts of protein folding in vitro and in vivo.Naturea Structural and Molecular Biology16,574–581.

Hecht,M.,Das,A.,Go,A.,Bradley,L.,Wei,Y.,2004.De novo proteins from designed combinatorial libraries.Protein Science13,1711–2173.

Huang,Y.,Liu,Z.,2009.Kinetic advantage of intrinsically disordered proteins in coupled folding-binding process:a critical assessment of the‘fly-casting’mech-anism.Journal of Molecular Biology393,1143–1159.

Ivankov,D.,Garbuzynskiy,S.,Alm,E.,Plaxco,K.,Baker,D.,Finkelstein,A.,2003.Con-tact order revisited:influence of protein size on the folding rate.Protein Science 12,2057–2062.

Kamtekar,S.,Schiffer,J.,Xiong,H.,Babik,J.,Hecht,M.,1993.Protein design by patterning of polar and nonpolar amino acids.Science262,1680–1685. Khinchin,A.,1957.Mathematical Foundations of Information Theory.Dover,New York.

Kim,W.,Hecht,M.,2006.Generic hydrophobic residues are sufficient to promote aggregation of the Alzheimer’s A␤42peptide.Proceedings of the Nattional Academy of Sciences of America103,15824–15829.

Krebs,M.,Domike,K.,Donald,A.,2009.Protein aggregation:more than justfibrils.

Biochemical Society Transactions37(part4),682–686.

Levitt,M.,Chothia,C.,1976.Structural patterns in goblular protiens.Nature261, 552–557.

Li,J.,Browning,S.,Mahal,S.,Oelschlegel,A.,Weissman,C.,2010.Darwinian evolution of prions in cell culture.Science327,869–872.

Matsumoto,Y.,2002.An introduction to Morse Theory.Translations of the American Mathematical Society208,Providence,RI.

Maury,C.,2009.Self-propagating␤-sheet polypeptide structures as prebiotic infor-mational molecular entities:the amylod world.Origins of Life and Evolution of Bioshperes39,141–150.

Mirny,L.,Shakhnovich,E.,2001.Protein folding theory:from lattice to all-atom models.Annual Reviews of Biophysics and Biomolecular Structure30,361–396.

Plaxco,K.,Simons,K.,Baker,D.,1998.Contact order,transition state placement and the refolding rates of single domain proteins.Journal of Molecular Biology277, 985–994.

Scheuner,D.,Kaufman,R.,2008.The unfolded protein response:a pathway that links insulin demand with␤-cell failure and diabetes.Endocrinology Reviews 29,317–333.

Sawaya,M.,Sambashivan,S.,Nelson,R.,Ivanova,M.,2007.Atomic structures of amyloid cross-␤spines reveal varied steric zippers.Nature447,453–457. Serdyuk,I.,2007.Structured proteins and proteins with intrinsic disorder.Molecular Biology41,297–313.

Thomas,D.,1992.Concepts in protein folding.Federation of European Biochemical Societies307,10–13.

Tlusty,T.,2007a.A model for the emergence of the genetic code as a transition in a noisy information channel.Journal of Theoretical Biology249,331–342. Tlusty,T.,2007b.A relation between the multiplicity of the second eigenvalue of a graph Laplacian.Courant’s nodal line theorem and the substantial dimension of tight polyhedral surfaces.Electrical Journal of Linear Algebra16,315–324. Tlusty,T.,2008a.Rate-distortion scenario for the emergence and evolution of noisy molecular codes.Physical Review Letters100,048101–048104.

Tlusty,T.,2008b.A simple model for the evolution of molecular codes driven by the interplay of accuracy,diversity and cost.Physical Biology5,016001.

Tlusty,T.,2008c.Casting polymer nets to optimize noisy molecular codes.Proceed-ings of the National Academy of Sciences of America105,8238–8243.

Tlusty,T.,2010a.A colorful origin for the genetic code:information theory,statis-tical mechanics and the emergence of molecular codes.Physics of Life Reviews, doi:10.1016/j.plrev.2010.06.002.

Tlusty,T.,2010b.Reply to comment.Physics of Life Reviews7,381–384.

Tycko,R.,2006.Molecular structure of amyloidfibrils:insights from solid-state NMR.

Quarterly Reviews of Biophysics39,1–55.

Uversky,V.,Oldfield,C.,Dunker,K.,2008.Intrinsically disordered proteins in human diseases:introducing the D2concept.Annual Reviews of Biophysics37,215–246.

Wallace,R.,Fullilove,M.,2008.Collective Consciousness and its Discontents:Institu-tional Distributed Cognition.Racial Policy and Public Health in the United States.

Springer,New York.

Wallace,R.,Wallace,D.,2008.Punctuated equilibrium in statistical models of gener-alized coevolutionary resilience:how sudden ecosystem transitions can entrain both phenotype expression and Darwinian selection.Transactions on Compu-tational Systems Biology IX,LNBI5121,23–85.

Wallace,R.,Wallace,D.,2009.Code,context,and epigenetic catalysis in gene expression.Transactions on Computational Systems Biology XI,LNBI5750,283–334.

Wallace,R.,Wallace,D.,in press.Cultural epigenetics:on the heritability of complex diseases.Transactions on Computational Systems Biology.

Wallace,R.,2010a.A rate distortion approach to protein symmetry.BioSystems101, 97–108.

Wallace,R.,2010b.A scientific open season.Physics of Life Reviews7,377–378. Wang,L.,Maji,S.,Sawaya,M.,Eisenberg,D.,Riek,R.,2008.Bacterial inclusion bodies contain amyloid-like structure.PLOS Biology6,e195.

Weinstein,A.,1996.Groupoids:unifying internal and external symmetry.Notices of the American Mathematical Society43,744–752.

Yang,W.,Gruebele,M.,2004.Folding␭-repressor at its speed limit.Biophysical Journal87,596–608.