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Parikh vectors
3MBW_1 6QFR_1 2RAG_1 Letter Amino acid
11 18 20 R Arginine
20 20 40 D Aspartic acid
22 15 21 P Proline
29 16 60 A Alanine
24 26 36 L Leucine
8 12 10 M Methionine
18 16 25 S Serine
11 18 20 I Isoleucine
28 19 18 E Glutamic acid
26 25 32 G Glycine
17 18 20 T Threonine
13 10 12 Y Tyrosine
20 20 23 V Valine
11 11 6 Q Glutamine
16 8 2 C Cysteine
12 13 14 H Histidine
12 30 29 K Lycine
17 12 13 F Phenylalanine
5 3 6 W Tryptophan
10 8 10 N Asparagine 

3MBW_1|Chain A|Ephrin type-A receptor 2|Homo sapiens (9606)
>6QFR_1|Chain A|Dual specificity mitogen-activated protein kinase kinase 7|Homo sapiens (9606)
>2RAG_1|Chains A, B, C, D|Dipeptidase|Caulobacter crescentus (190650)
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 \(w=f(c)\)
3MBW , Knot 143 330 0.83 40 214 316 
APEHHHHHHDYDIPTTENLYFQGAMDAAQGKEVVLLDFAAAGGELGWLTHPYGKGWDLMQNIMNDMPIYMYSVCNVMSGDQDNWLRTNWVYRGEAERIFIELKFTVRDCNSFPGGASSCKETFNLYYAESDLDYGTNFQKRLFTKIDTIAPDEITVSSDFEARHVKLNVEERSVGPLTRKGFYLAFQDIGACVALLSVRVYYKKCPELLQGLAHFPETIAGSDAPSLATVAGTCVDHAVVPPGGEEPRMHCAVDGEWLVPIGQCLCQAGYEKVEDACQACSPGFFKFEASESPCLECPEHTLPSPEGATSCECEEGFFRAPQDPASMPCT
6QFR , Knot 138 318 0.83 40 206 306 
MGHHHHHHSAKQTGYLTIGGQRYQAEINDLENLGEMGSGTCGQVWKMRFRKTGHVIAVKQMRRSGNKEENKRILMDLDVVLKSHDCPYIVQCFGTFITNTDVFIAMELMGTCAEKLKKRMQGPIPERILGKMTVAIVKALYYLKEKHGVIHRDVKPSNILLDERGQIKLCDFGISGRLVDSKAKTRSAGCAAYMAPERIDPPDPTKPDYDIRADVWSLGISLVELATGQFPYKNCKTDFEVLTKVLQEEPPLLPGHMGFSGDFQSFVKDCLTKDHRKRPKYNKLLEHSFIKRYETLEVDVASWFKDVMAKTESPRTSG
2RAG , Knot 168 417 0.81 40 216 387 
MSRPLLFALTALSLTAASLAHAADKKAADKPAPSAVSKADKALHDKFLTLDTHLDTPAHFGRPGWDIADHHEVEHDFSQVDLPRMNQGGLDGGFFVVYIGQGELTEKGYTYARDYALHRTIEIREMLAANPDTFEMALTSDDARRIAKAGKKFAFVSMENSWPVGEDLSLVETFYKEGLRMAGPVHFRNNQLADSSTDPKGKIWNGYSPLGLRWLAEANRLGIVIDVSHASDDVVDQSVALSKAPIIASHSGPKAVYDHPRNLDDARLKKIADAGGAICINSIYLTDTTPSPERKAALEALGRAPDMKTATPEAVKAYADKRAAIDKAHPAARGDFDLYMKSMLHVLKVAGPKGVCVGADWDGGGGMDGFEDITDLPKITARLKAEGYSDADIEAIWSGNVLRIVDAAQAYAKSVTK

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 \(c\).

\(|P_{f(3MBW_1)}(2) \setminus P_{f(6QFR_1)}(2)|=86\), \(|P_{f(6QFR_1)}(2) \setminus P_{f(3MBW_1)}(2)|=78\). 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 1:110000000000110000101011101101001111011111101111001010110110011001110100100110100001100011001010011101010100000111110000001010010001001001000110010011100101000101001010100001111000110111001110111101010000010110111011001110011011011100100111111100101001101011111100100110001001001001111010100010100100011010110000000111011001101100
Pair \(Z_2\) Length of longest common subsequence
3MBW_1,6QFR_1 164 6
3MBW_1,2RAG_1 150 4
6QFR_1,2RAG_1 164 4

Newick tree

 
[
	6QFR_1:84.20,
	[
		3MBW_1:75,2RAG_1:75
	]:9.20
]

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 \((0.85)(0.8)(\frac{648 }{\log_{20} 648}-\frac{318}{\log_{20}318})=91.4\)
Status Protein1 Protein2 d d1/2
Query variables 3MBW_1 6QFR_1 113 113
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]