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Parikh vectors
1KWZ_1 6FWA_1 8GZG_1 Letter Amino acid
11 10 14 T Threonine
10 6 15 N Asparagine
4 7 2 C Cysteine
14 21 32 E Glutamic acid
10 38 18 G Glycine
4 16 14 P Proline
11 16 23 V Valine
6 27 18 D Aspartic acid
3 18 9 H Histidine
14 21 15 K Lycine
7 17 9 F Phenylalanine
2 18 8 Y Tyrosine
13 39 26 A Alanine
7 23 24 I Isoleucine
10 44 34 L Leucine
2 8 1 W Tryptophan
4 25 17 R Arginine
3 19 19 Q Glutamine
4 10 7 M Methionine
10 34 23 S Serine 

1KWZ_1|Chains A, B, C|MANNOSE-BINDING PROTEIN A|Rattus norvegicus (10116)
>6FWA_1|Chains A, B|Flavin-dependent L-tryptophan oxidase VioA|Chromobacterium violaceum (strain ATCC 12472 / DSM 30191 / JCM 1249 / NBRC 12614 / NCIMB 9131 / NCTC 9757) (243365)
>8GZG_1|Chains A, B|DNA-directed RNA polymerase subunit alpha|Synechocystis sp. PCC 6803 (1148)
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 \(w=f(c)\)
1KWZ , Knot 72 149 0.80 40 121 146 
AIEVKLANMEAEINTLKSKLELTNKLHAFSMGKKSGKKFFVTNHERMPFSKVKALCSELRGTVAIPRNAEENKAIQEVAKTSAFLGITDEVTEGQFMYVTGGRLTYSNWKKDEPNDVGSGEDCVTIVDNGLWNDISCQASHTAVCEFPA
6FWA , Knot 173 417 0.83 40 230 398 
KHSSDICIVGAGISGLTCASHLLDSPACRGLSLRIFDMQQEAGGRIRSKMLDGKASIELGAGRYSPQLHPHFQSAMQHYSQKSEVYPFTQLKFKSHVQQKLKRAMNELSPRLKEHGKESFLQFVSRYQGHDSAVGMIRSMGYDALFLPDISAEMAYDIVGKHPEIQSVTDNDANQWFAAETGFAGLIQGIKAKVKAAGARFSLGYRLLSVRTDGDGYLLQLAGDDGWKLEHRTRHLILAIPPSAMAGLNVDFPEAWSGARYGSLPLFKGFLTYGEPWWLDYKLDDQVLIVDNPLRKIYFKGDKYLFFYTDSEMANYWRGCVAEGEDGYLEQIRTHLASALGIVRERIPQPLAHVHKYWAHGVEFCRDSDIDHPSALSHRDSGIIACSDAYTEHCGWMEGGLLSAREASRLLLQRIAA
8GZG , Knot 140 328 0.82 40 195 312 
MGSSHHHHHHSQDPMAQFQIECVESSTRKNQQQYSKFSLEPLDRGQGTTVGNALRRVLLSNLPGAAVTAIRIAGVNHEFATILGVREDVLEIMLNMKELVLKSYTDQPQIGRLTAIGPGTVTAAQFEVPSEVEVIDPNQYIATLAEGAKLEMEFRVERGVGYRVIERGKDENSSLDFLQIDSVFMPVTKVNYTVEDIRADGMSPKDRLILDIWTNGSIQPREALSEASDIIANLFIPLKDLNELEAAHSDYQDEVNPESQIPIEELQLSVRAYNCLKRAQINSVADLLEYSQEDLLEIKNFGLKSAEEVIEALQKRLGITLPHEKAKA

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 \(c\).

\(|P_{f(1KWZ_1)}(2) \setminus P_{f(6FWA_1)}(2)|=45\), \(|P_{f(6FWA_1)}(2) \setminus P_{f(1KWZ_1)}(2)|=154\). 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 1:11010110101010010001010001011011000100111000001110010110001010111100100001100110001111100010010110101101000010000100110100010110011100100010001100111
Pair \(Z_2\) Length of longest common subsequence
1KWZ_1,6FWA_1 199 4
1KWZ_1,8GZG_1 160 4
6FWA_1,8GZG_1 177 4

Newick tree

 
[
	6FWA_1:98.42,
	[
		1KWZ_1:80,8GZG_1:80
	]:18.42
]

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 \((0.85)(0.8)(\frac{566 }{\log_{20} 566}-\frac{149}{\log_{20}149})=121.\)
Status Protein1 Protein2 d d1/2
Query variables 1KWZ_1 6FWA_1 153 103.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]