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Parikh vectors
3BHF_1 7KMF_1 7EOV_1 Letter Amino acid
26 30 11 A Alanine
16 14 14 N Asparagine
22 16 20 F Phenylalanine
3 1 9 W Tryptophan
15 8 10 Y Tyrosine
26 23 15 V Valine
20 15 10 D Aspartic acid
8 10 12 Q Glutamine
33 37 27 E Glutamic acid
15 15 12 H Histidine
23 16 22 K Lycine
3 4 4 C Cysteine
37 23 16 G Glycine
23 25 25 I Isoleucine
32 34 25 L Leucine
18 15 12 P Proline
13 21 15 R Arginine
8 13 6 M Methionine
27 26 24 S Serine
22 21 13 T Threonine 

3BHF_1|Chains A, B|Monomeric sarcosine oxidase|Bacillus sp. (1409)
>7KMF_1|Chains A[auth C], B[auth D]|Translation initiation factor eIF-2B subunit beta|Homo sapiens (9606)
>7EOV_1|Chain A|cytosolic sulfotransferase SULT2A8|Mus musculus (10090)
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 \(w=f(c)\)
3BHF , Knot 166 390 0.84 40 218 371 
MSTHFDVIVVGAGSMGMAAGYQLAKQGVKTLLVDAFDPPHTNGSHHGDTKIIRHAYGEGREYVPLALRSQELWYELEKETHHKIFTKTGVLVFGPKGESAFVAETMEAAKEHSLTVDLLEGDEINKRWPGITVPENYNAIFEPNSGVLFSENCIRAYRELAEARGAKVLTHTRVEDFDISPDSVKIETANGSYTADKLIVSMGAWNSKLLSKLNLDIPLQPYRQVVGFFESDESKYSNDIDFPGFMVEVPNGIYYGFPSFGGCGLKLGYHTFGQKIDPDTINREFGVYPEDESNLRAFLEEYMPGANGELKRGAVCMYTKTLDEHFIIDLHPEHSNVVIAAGFSGHGFKFSSGVGEVLSQLALTGKTEHDISIFSINRPALKESLQKTTI
7KMF , Knot 155 367 0.83 40 219 353 
MDYKDDDDKENLYFQSMPGSAAKGSELSERIESFVETLKRGGGPRSSEEMARETLGLLRQIITDHRWSNAGELMELIRREGRRMTAAQPSETTVGNMVRRVLKIIREEYGRLHGRSDESDQQESLHKLLTSGGLNEDFSFHYAQLQSNIIEAINELLVELEGTMENIAAQALEHIHSNEVIMTIGFSRTVEAFLKEAARKRKFHVIVAECAPFCQGHEMAVNLSKAGIETTVMTDAAIFAVMSRVNKVIIGTKTILANGALRAVTGTHTLALAAKHHSTPLIVCAPMFKLSPQFPNEEDSFHKFVAPEEVLPFTEGDILEKVSVHCPVFDYVPPELITLFISNIGGNAPSYIYRLMSELYHPDDHVL
7EOV , Knot 135 302 0.85 40 193 290 
MGSSHHHHHHSSGLVPRGSHMTDEFLWIEGIPFPTVYYSQEIIREVRDRFVVRDEDTIIVTYPKSGTHWLNEIVCLILTKGDPTWVQSTIANERTPWIEFENNYRILNSKEGPRLMASLLPIQLFPKSFFSSKAKVIYLIRNPRDVLVSGYHYFNALKQGKEQVPWKIYFENFLQGKSYFGSWFEHACGWISLRKRENILVLSYEQLKKDTRNTIKKICEFLGENLESGELELVLKNISFQIMKERMISQSCLSNIEKHEFIMRKGITGDWKNHFTVAQAEAFDKAFQEKAADFPQELFSWE

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 \(c\).

\(|P_{f(3BHF_1)}(2) \setminus P_{f(7KMF_1)}(2)|=69\), \(|P_{f(7KMF_1)}(2) \setminus P_{f(3BHF_1)}(2)|=70\). 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 1:100010111111101111110011001100111011011000100010001100101010001111100001100100000001100011111110100111100101100001010110100100011110110000111010011110000101000110101101100001001010100101001010001001110111100011001010111010001111100000000001011111101101100111011101101100011001010010001110100000101110001111010100111010000100011101010000111111101011010011101100111010000010110100111000100001
Pair \(Z_2\) Length of longest common subsequence
3BHF_1,7KMF_1 139 4
3BHF_1,7EOV_1 175 3
7KMF_1,7EOV_1 174 4

Newick tree

 
[
	7EOV_1:92.41,
	[
		3BHF_1:69.5,7KMF_1:69.5
	]:22.91
]

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 \((0.85)(0.8)(\frac{757 }{\log_{20} 757}-\frac{367}{\log_{20}367})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 3BHF_1 7KMF_1 131 126.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]