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Parikh vectors
5CLR_1 3KBF_1 8WDC_1 Letter Amino acid
28 16 25 A Alanine
22 25 22 G Glycine
37 8 58 I Isoleucine
23 3 37 F Phenylalanine
11 2 36 Y Tyrosine
24 15 37 V Valine
14 5 31 R Arginine
25 8 42 D Aspartic acid
33 9 22 E Glutamic acid
11 1 20 M Methionine
21 12 42 T Threonine
3 1 1 W Tryptophan
19 5 19 Q Glutamine
13 7 10 H Histidine
57 9 56 L Leucine
16 6 26 P Proline
16 7 46 N Asparagine
6 2 7 C Cysteine
39 9 36 K Lycine
33 7 41 S Serine 

5CLR_1|Chains A, B|LegK4|Legionella pneumophila (strain Lens) (297245)
>3KBF_1|Chain A|Superoxide dismutase [Cu-Zn]|Caenorhabditis elegans (6239)
>8WDC_1|Chains A, B, C|Core protein OPG136|Vaccinia virus (strain Western Reserve) (10254)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
5CLR , Knot 180 451 0.81 40 225 418 
GIDPFTMKLLRFHELKSLPGMDEKALELLIKVLGNKGIRKLIKSADGKPISREIMIHEFGIDCQILFITTEASLKPIIVPTENKISGGGKSYCEQFKVYALDDGKTYFLKSVKIDAESLTEFTNEKDTLSKLGRLVGTFFNEQTQVHYILTTFIKGIDLSRYKNALPLNVNLKHFWEVLGIMISVCHQVKQFHELGLIHRDLKPGNIMLDADMQCHLVDFGSSSSDKEPKPASWGTASYLAPELNAQEDFIAFSQVSDLFALAYSLDELFNPFRQVKFAKVDIGIKNKHLVLLHAEIEACITGLMSNETSVRTLYFSRILQLQRVPESFKSRPEAFTYLIMLLTQWKSCYEAPEMNKELDEIIAEIKVAYENHEQDAVKIITLLEQLSKADGLLNSHKALLSVLIKSLANVQQQELGHDDILPRRFESDVVSRLIKTPTAKMMAAIKQVSD
3KBF , Knot 78 157 0.83 40 116 151 
SNRAVAVLRGETVTGTIWITQKSENDQAVIEGEIKGLTPGLHGFHVHQYGDSTNGCISAGPHFNPFGKTHGGPKSEIRHVGDLGNVEAGADGVAKIKLTDTLVTLYGPNTVVGRSMVVHAGQDDLGEGVGDKAEESKKTGNAGARAACGVIALAAPQ
8WDC , Knot 241 614 0.84 40 264 562 
MMPIKSIVTLDQLEDSEYLFRIVSTVLPHLCLDYKVCDQLKTTFVHPFDILLNNSLGSVTKQDELQAAISKLGINYLIDTTSRELKLFNVTLNAGNIDIINTPINISSETNPIINTHSFYDLPPFTQHLLNIRLTDTEYRARFIGGYIKPDGSDSMDVLAEKKYPDLNFDNTYLFNILYKDVINAPIKEFKAKIVNGVLSRQDFDNLIGVRQYITIQDRPRFDDAYNIADAARHYGVNLNTLPLPNVDLTTMPTYKHLIMFEQYFIYTYDRVDIYYNGNKMLFDDEIINFTISMRYQSLIPRLVDFFPDIPVNNNIVLHTRDPQNAAVNVTVALPNVQFVDINRNNKFFINFFNLLAKEQRSTAIKVTKSMFWDGMDYEEYKSKNLQDMMFINSTCYVFGLYNHNNTTYCSILSDIISAEKTPIRVCLLPRVVGGKTVTNLISETLKSISSMTIREFPRKDKSIMHIGLSETGFMRFFQLLRLMADKPHETAIKEVVMAYVGIKLGDKGSPYYIRKESYQDFIYLLFASMGFKVTTRRSIMGSNNISIISIRPRVTKQYIVATLMKTSCSKNEAEKLITSAFDLLNFMVSVSDFRDYQSYRQYRNYCPRYFYAG

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(5CLR_1)}(2) \setminus P_{f(3KBF_1)}(2)|=155\), \(|P_{f(3KBF_1)}(2) \setminus P_{f(5CLR_1)}(2)|=46\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:1101101011010010011110001101110111001100110010101100011100111000111100010101111100001011100000010101100100011001010100100100000010011011101100000100110011011010000011110101001101111110100010010011110001011011101010001101100000001011011010011101010001111001001111100100110110010110101110000111101010101011100000100101001101001100100010110011111001000001101000100111010110000000110110110010010111000011101110011010000110001110010001100110010101111100100
Pair \(Z_2\) Length of longest common subsequence
5CLR_1,3KBF_1 201 4
5CLR_1,8WDC_1 143 4
3KBF_1,8WDC_1 202 4

Newick tree

 
[
	3KBF_1:10.76,
	[
		5CLR_1:71.5,8WDC_1:71.5
	]:37.26
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{608 }{\log_{20} 608}-\frac{157}{\log_{20}157})=129.\)
Status Protein1 Protein2 d d1/2
Query variables 5CLR_1 3KBF_1 161 108
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]