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Parikh vectors
4ZOS_1 7WDB_1 7BTC_1 Letter Amino acid
5 40 13 K Lycine
3 40 7 F Phenylalanine
5 33 9 T Threonine
5 49 9 A Alanine
6 42 5 R Arginine
10 102 14 L Leucine
13 54 14 E Glutamic acid
3 19 9 H Histidine
4 34 7 Y Tyrosine
8 40 15 V Valine
4 44 7 N Asparagine
6 29 8 D Aspartic acid
4 27 9 Q Glutamine
2 21 4 M Methionine
6 32 4 P Proline
7 49 15 S Serine
1 16 1 W Tryptophan
3 12 0 C Cysteine
3 29 10 G Glycine
8 52 17 I Isoleucine 

4ZOS_1|Chains A, B, C, D|protein YE0340 from Yersinia enterocolitica subsp. enterocolitica 8081|Yersinia enterocolitica (393305)
>7WDB_1|Chains A, B, C, D|Short transient receptor potential channel 5|Homo sapiens (9606)
>7BTC_1|Chains A, B, C, D|GTP-binding protein Rheb|Homo sapiens (9606)
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 \(w=f(c)\)
4ZOS , Knot 57 106 0.83 40 89 103 
SNAMQKIKSEERHIICELRCEPENRERVKELVLKFVEPARLETGCLYYDLYQKIDEPDTFYIIDGWVNQEAVTSHAENPHVAEVMSDLQPLLTFGPSISLITRVSD
7WDB , Knot 292 764 0.84 40 305 695 
MAQLYYKKVNYSPYRDRIPLQIVRAETELSAEEKAFLNAVEKGDYATVKQALQEAEIYYNVNINCMDPLGRSALLIAIENENLEIMELLLNHSVYVGDALLYAIRKEVVGAVELLLSYRRPSGEKQVPTLMMDTQFSEFTPDITPIMLAAHTNNYEIIKLLVQKRVTIPRPHQIRCNCVECVSSSEVDSLRHSRSRLNIYKALASPSLIALSSEDPILTAFRLGWELKELSKVENEFKAEYEELSQQCKLFAKDLLDQARSSRELEIILNHRDDHSEELDPQKYHDLAKLKVAIKYHQKEFVAQPNCQQLLATLWYDGFPGWRRKHWVVKLLTCMTIGFLFPMLSIAYLISPRSNLGLFIKKPFIKFICHTASYLTFLFMLLLASQHIVRTDLHVQGPPPTVVEWMILPWVLGFIWGEIKEMWDGGFTEYIHDWWNLMDFAMNSLYLATISLKIVAYVKYNGSRPREEWEMWHPTLIAEALFAISNILSSLRLISLFTANSHLGPLQISLGRMLLDILKFLFIYCLVLLAFANGLNQLYFYYETRAIDEPNNCKGIRCEKQNNAFSTLFETLQSLFWSVFGLLNLYVTNVKARHEFTEFVGATMFGTYNVISLVVLLNMLIAMMNNSYQLIADHADIEWKFARTKLWMSYFDEGGTLPPPFNIIPSPKSFLYLGNWFNNTFCPKRDPDGRRRRRNLRSFTERNADSLIQNQHYQEVIRNLVKRYVAAMIRNSKTHEGLTEENFKELKQDISSFRYEVLDLLGNR
7BTC , Knot 81 177 0.79 38 130 169 
MPQSKSRKIAILGYRSVGKSSLTIQFVEGQFVDSNDPTIENTFTKLITVNGQEYHLQLVDTAGQDEYSIFPQTYSIDINGYILVYSVTSIKSFEVIKVIHGKLLDMVGKVQIPIMLVGNKKDLHMERVISYEEGKALAESWNAAFLESSAKENQTAVDVFRRIILEAEKLEHHHHHH

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 \(c\).

\(|P_{f(4ZOS_1)}(2) \setminus P_{f(7WDB_1)}(2)|=9\), \(|P_{f(7WDB_1)}(2) \setminus P_{f(4ZOS_1)}(2)|=225\). 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 1:0011001000000110010001000001001110110110100101000100010010010110111000110001001011011001011101110101100100
Pair \(Z_2\) Length of longest common subsequence
4ZOS_1,7WDB_1 234 4
4ZOS_1,7BTC_1 143 3
7WDB_1,7BTC_1 213 4

Newick tree

 
[
	7WDB_1:12.40,
	[
		4ZOS_1:71.5,7BTC_1:71.5
	]:50.90
]

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 \((0.85)(0.8)(\frac{870 }{\log_{20} 870}-\frac{106}{\log_{20}106})=215.\)
Status Protein1 Protein2 d d1/2
Query variables 4ZOS_1 7WDB_1 270 152
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]