The 7 Bridges in K

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The 7 Bridges in Königsberg and Compositional
Representation of Protein Sequences

• Bailin Hao (???)
• (ITP BGI, CAS )
• Huimin Xie (???)
• (Math Dept. Suzhou U)
• Shuyu Zhang (???)
• (IP. Acad. Sinica)

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Compositional Approach inProkaryote Phylogeny

• Justification of using K-tuples instead of
  primary protein sequences.
• Problem of uniqueness of reconstruction of
  protein sequence from its constituent K-tuples.
• Picking up a special class of proteins without
  biological knowledge.

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7 bridges in Königsberg
Euler (1736) 4 odd
nodes No! Dénes König, Theory of
Finite and Infinite Graphs
1st ed.(1932). Birkhaüser (1990)
From Königsberg to Königs book, So runs the
graphic tale
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Basic Notions

• A Graph G(V, E), where V is a set of nodes
  (vertices), E is a set of edges (bonds)
• Edges undirected (u,v)(v,u), uand v adjacent
  directed (u,v) differs from (v,u), u incident to
  v.
• A weight may be associated with (u,v) cost,
  distance, transfer function, reaction rate, etc.
• Eulerian graph each edge appears once and only
  once in a path
• Hamiltonian graph each vertex appears once and
  only once in a path. Hamiltonian cycle of minimal
  weight --- Travel Salesman Problem (TSP)

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• An Euler path
• An Euler loop
• Euler grahp ?
  loop
• Semi-Euler praph ?
  path, no loop
• Problem of Eulerian loop simple, known solutions
• Problem of Hamiltonian paths much harder

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Hamiltonian Loops much harder

• 10 nodes
• 15 arcs
• di3 ? nodes
• Traveling Salesman Problem
• NP-hard problems

No!
Yes!
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• Graph nodes arcs
• Directed , Labeled arcs and nodes
• Simple graph
• No rings at nodes
• No repeated arcs

i
i
j
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• Indegree din(i)
• 
  Outdegree dout(i)
• Euler graph
• din(i) dout(i) ? di
  ?i

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Simple Euler GraphDiagonal matrix
Mdiag( d1, d2, dn )Adjacent
matrix Aaij
aij
aii
0 Kirchhoff matrix
CM-A ?Cij
? Cij0
det(C)0All minors of C are equal.
Denote this common minor by
1
n
i,j0
0
i
j
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• Number of Euler loops in simple Euler Graph
• N G de Bruijn
• T van Aardenrie Ehrenfest
• C A B Smith
• W T Tuite
• BEST Theorem
• e (G) ? ?(di-1)!

i
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• Number of Eulerian loops in general Eular G.
• some aii?0
  rings
• some aijgt1
  parallel arcs
• Putting auxiliary nodes on these rings and
  parallel arcs makes the graph simple.

i
13

• No need to work with bigger A matrix.
• Just let some aii?0, aijgt1 in original A.
• Eliminate redundancy caused by unlebeled arcs.
• Modified BEST Theorem
• e(G)
  

? ?(di-1)!
i
? aij!
ij
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MALS
K5
ALSL

• ANPA_PSEAM 82AA
• MALSLFTVGQLIFLFWTMRITEASPDPAAKAAPAAAAAPAAAAPDTASDA
  AAAAALTAANAKAAAELTAANAAAAAAATARG

LSLF
SLFT
LFTV
FTVG
TVGQ
VGQL
AKAA
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6 rings
auxiliary arc
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From pdb.seq-a special selection of
SWISSPROT2821-12820 proteins ( May 2000
)Rnumber of reconstructed AA
sequences from a given protein decomposition
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Compositional Representation of Proteins

• The collection W or W ,n j
  may be used as an equivalent representation of
  the original protein sequence.
• A seemingly trivial result upon further
  reflection random AA sequences have unique
  reconstruction as well.
• Compositional Representation works equally for
  random AA sequences and most of protein
  sequences.
• A given realization of a short random AA
  sequence is as specific as a real protein
  sequence.

M
K
L -k1
K
j
i
j1
i1
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• Nucleotide correlations in DNA/RNA
• Much studied
• K2 correlation functions 16
  9 6
• See Wentian Li, Computer Chem. 21(1997)
  257-271.
• Amino Acid correlations in Proteins
• Almost no study
• Hard to comprehend 400 correlation
  functions at K2
• Proteins too short to define correlation
  functions
• One should approach the problem from a more
  deterministic point of view
• Repeated AA segments in proteins are strong
  manifestation of correlations!

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• On-going study the other extreme
• Quit a few proteins have an enormous
• number of reconstructions.
• Transmembrane
• Antifreeze
• Fibrous collagens
• Coarse-graining closer to biology by reducing
  the number of AAs

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• Preprint
• NSF ITP 01 018
• LANL E-archive
  physics/0103028
• arxiv.org or
  cn.arxiv.org
• Cross-referenced in q-bio since 15 Sept 2003