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Parikh vectors
1CCZ_1 8CQX_1 5IZD_1 Letter Amino acid
11 14 22 T Threonine
9 1 22 Y Tyrosine
6 10 27 D Aspartic acid
2 3 11 H Histidine
16 9 35 S Serine
4 7 36 I Isoleucine
15 40 36 L Leucine
6 4 13 M Methionine
3 2 9 W Tryptophan
5 50 44 A Alanine
6 19 22 R Arginine
10 8 22 N Asparagine
18 20 37 E Glutamic acid
13 5 38 K Lycine
17 28 33 V Valine
8 9 19 F Phenylalanine
6 22 21 P Proline
4 0 1 C Cysteine
7 16 15 Q Glutamine
5 33 45 G Glycine 

1CCZ_1|Chain A|PROTEIN (CD58)|Homo sapiens (9606)
>8CQX_1|Chains A, B, C, D|Ribokinase|Thermus sp. (275)
>5IZD_1|Chains A, B, C, D, E, F, G, H|D-glyceraldehyde dehydrogenase (NADP(+))|Thermoplasma acidophilum (2303)
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 \(w=f(c)\)
1CCZ , Knot 81 171 0.81 40 125 165 
FSQQIYGVVYGNVTFHVPSNVPLKEVLWKKQKDKVAELENSEFRAFSSFKNRVYLDTVSGSLTIYNLTSSDEDEYEMESPNITDTMKFFLYVLEMVSKPMIYWECSNATLTCEVLEGTDVELKLYQGKEHLRSLRQKTMSYQWTNLRAPFKCKAVNRVSQESEMEVVNCPE
8CQX , Knot 123 300 0.78 38 141 264 
MILVVGSLNMDLVLRVKRLPRPGETVLGEDYQTHPGGKGANQAVAIARLGGKVRMLGRVGEDPFGQALKSGLAQEGVDVAWVLETPGPSGTGFILVDPEGQNQIAVAPGANARLVPEDLPATAFQGVGVVLLQLEIPLETVVRAAALGRKAGARILLNAAPAHALPSEILQSVDLLLVNEVEAAQLTEASPPRTPEEALALARQLRGRAPQAQVVLTLGAQGAVWSGTEESHFPAFPVRAVDTTAAGDAFAGALALGLAEGQNMRAALRFANAAGALATTRPGAQPSLPFRDEVEALLFG
5IZD , Knot 206 508 0.84 40 237 465 
MDTKLYIDGQWVNSSSGKTVDKYSPVTGQVIGRFEAATRDDVDRAIDAAEDAFWAWNDLGSVERSKIIYRAKELIEKNRAELENIIMEENGKPVKEAKEEVDGVIDQIQYYAEWARKLNGEVVEGTSSHRKIFQYKVPYGIVVALTPWNFPAGMVARKLAPALLTGNTVVLKPSSDTPGSAEWIVRKFVEAGVPKGVLNFITGRGSEIGDYIVEHKKVNLITMTGSTATGQRIMQKASANMAKLILELGGKAPFMVWKDADMDNALKTLLWAKYWNAGQSCIAAERLYVHEDIYDTFMSRFVELSRKLALGDPKNADMGPLINKGALQATSEIVEEAKESGAKILFGGSQPSLSGPYRNGYFFLPTIIGNADQKSKIFQEEIFAPVIGARKISSVEEMYDLANDSKYGLASYLFTKDPNIIFEASERIRFGELYVNMPGPEASQGYHTGFRMTGQAGEGSKYGISEYLKLKNIYVDYSGKPLHINTVRDDLFQSGRPVLGSSHHHHHH

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 \(c\).

\(|P_{f(1CCZ_1)}(2) \setminus P_{f(8CQX_1)}(2)|=77\), \(|P_{f(8CQX_1)}(2) \setminus P_{f(1CCZ_1)}(2)|=93\). 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 1:100010111010101011001110011100000011010000101100100010100101010100100000000010010100010111011011001110100001010001101001010100100010010000100010010111000110010000010110010
Pair \(Z_2\) Length of longest common subsequence
1CCZ_1,8CQX_1 170 3
1CCZ_1,5IZD_1 186 5
8CQX_1,5IZD_1 160 4

Newick tree

 
[
	1CCZ_1:91.92,
	[
		8CQX_1:80,5IZD_1:80
	]:11.92
]

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 \((0.85)(0.8)(\frac{471 }{\log_{20} 471}-\frac{171}{\log_{20}171})=88.1\)
Status Protein1 Protein2 d d1/2
Query variables 1CCZ_1 8CQX_1 110 87.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]