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Parikh vectors
8GDW_1 1HVZ_1 4YAO_1 Letter Amino acid
7 1 24 I Isoleucine
3 0 20 H Histidine
6 0 16 M Methionine
13 0 48 A Alanine
3 5 35 R Arginine
8 0 38 D Aspartic acid
9 0 24 Q Glutamine
7 0 53 E Glutamic acid
5 2 42 G Glycine
0 0 8 W Tryptophan
2 0 31 Y Tyrosine
1 6 7 C Cysteine
3 1 25 F Phenylalanine
5 1 30 T Threonine
3 0 23 N Asparagine
6 1 55 L Leucine
5 0 34 K Lycine
4 0 28 P Proline
7 0 37 S Serine
6 1 43 V Valine 

8GDW_1|Chains A[auth AAA], B[auth BBB], C[auth CCC], D[auth DDD]|Ssr1698 protein|Synechocystis sp. PCC 6803 (1148)
>1HVZ_1|Chain A|THETA DEFENSIN 1|null
>4YAO_1|Chains A, B|NADPH--cytochrome P450 reductase|Rattus norvegicus (10116)
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 \(w=f(c)\)
8GDW , Knot 56 103 0.84 38 87 101 
MADPLTPAISDRICKHMNEDHASAIALYAQVFGQQTDVTMAQMQAIDPTGMDLVVESEGGSKTIRIEFEQPLKDSEDAHQVLIAMAKQARSVGKNSAENLYFQ
1HVZ , Knot 10 18 0.53 16 14 15 
GFCRCLCRRGVCRCICTR
4YAO , Knot 248 621 0.85 40 286 593 
IQTTAPPVKESSFVEKMKKTGRNIIVFYGSQTGTAEEFANRLSKDAHRYGMRGMSADPEEYDLADLSSLPEIDKSLVVFCMATYGEDPTDNAQDFYDWLQETDVDLTGVKFAVFGLGNKTYEHFNAMGKYVDQRLEQLGAQRIFELGLGDDDGNLEEDFITWREQFWPAVCEFFGVEATGEESSIRQYELVVHEDMDVAKVYTGEMGRLKSYENQKPPFDAKNPFLAAVTANRKLNQGTERHLMHLELDISDSKIRYESGDHVAVYPANDSALVNQIGEILGADLDVIMSLNNLDEESNKKHPFPCPTTYRTALTYYLDITNPPRTNVLYELAQYASEPSEQEHLHKMASSSGEGKELYLSWVVEARRHILAILQDYPSLRPPIDHLCELLPRLQARYYSIASSSKVHPNSVHICAVAVEYEAKSGRVNKGVATSWLRAKEPAGENGGRALVPMFVRKSQFRLPFKSTTPVIMVGPGTGIAPFMGFIQERAWLREQGKEVGETLLYYGCRRSDEDYLYREELARFHKDGALTQLNVAFSREQAHKVYVQHLLKRDREHLWKLIHEGGAHIYVCGDARNMAKDVQNTFYDIVAEFGPMEHTQAVDYVKKLMTKGRYSLDVWS

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 \(c\).

\(|P_{f(8GDW_1)}(2) \setminus P_{f(1HVZ_1)}(2)|=86\), \(|P_{f(1HVZ_1)}(2) \setminus P_{f(8GDW_1)}(2)|=13\). 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 1:1101101110001000100001011110101110000101101011010110111000110001010100110000010011111100100110001001010
Pair \(Z_2\) Length of longest common subsequence
8GDW_1,1HVZ_1 99 2
8GDW_1,4YAO_1 237 3
1HVZ_1,4YAO_1 282 3

Newick tree

 
[
	4YAO_1:14.64,
	[
		8GDW_1:49.5,1HVZ_1:49.5
	]:98.14
]

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 \((0.85)(0.8)(\frac{121 }{\log_{20} 121}-\frac{18}{\log_{20}18})=38.7\)
Status Protein1 Protein2 d d1/2
Query variables 8GDW_1 1HVZ_1 51 29.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]