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Parikh vectors
4WUX_1 5TWQ_1 8OOI_1 Letter Amino acid
8 30 26 Q Glutamine
11 44 69 L Leucine
10 19 32 T Threonine
8 28 65 A Alanine
4 7 12 C Cysteine
7 6 59 I Isoleucine
4 18 31 F Phenylalanine
9 22 53 V Valine
5 27 58 R Arginine
13 27 68 E Glutamic acid
6 12 47 K Lycine
5 4 20 M Methionine
4 20 43 P Proline
8 3 32 N Asparagine
10 13 62 D Aspartic acid
14 27 63 G Glycine
2 15 10 H Histidine
8 22 40 S Serine
2 5 3 W Tryptophan
10 5 13 Y Tyrosine 

4WUX_1|Chain A|Pulmonary surfactant-associated protein A|Rattus norvegicus (10116)
>5TWQ_1|Chain A|human DNA Polymerase Mu|Homo sapiens (9606)
>8OOI_1|Chains A, B, C, D, E, F|Transitional endoplasmic reticulum ATPase|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 \(w=f(c)\)
4WUX , Knot 71 148 0.80 40 118 143 
AYLDEELQTELYEIKHQILQTMGVLSLQGSMLSVGDKVFSTNGQSVNFDTIKEMCTRAGGNIAVPRTPEENEAIASIAKKYNNYVYLGMIDDQTEGDFHYLDGASVSYTNWYPGEPRGQGKEDCVEMYTDGTWNDRGCLQYRLAVCEF
5TWQ , Knot 146 354 0.80 40 198 341 
GSAAASPAWMPAYACQRPTPLTHHNTGLSEALEILAEAAGFEGSEGRLLTFCRAASVLKALPSPVTTLSQLQGLPHFGEHSSRVVQELLEHGVCEEVERVRRSERYQTMKLFTQIFGVGVKTADRWYREGLRTLDDLREQPQKLTQQQKAGLQHHQDLSTPVLRSDVDALQQVVEEAVGQALPGATVTLTGGFRRGKLQGHDVDFLITHPKEGQEAGLLPRVMCRLQDQGLILYHQHQHSCCESPTRLAQQSHMDAFERSFCIFRLPQPGSWKAVRVDLVVAPVSQFPFALLGWTGSKLFQRELRRFSRKEKGLWLNSHGLFDPEQKTFFQAASEEDIFRHLGLEYLPPEQRNA
8OOI , Knot 295 806 0.81 40 279 705 
MASGADSKGDDLSTAILKQKNRPNRLIVDEAINEDNSVVSLSQPKMDELQLFRGDTVLLKGKKRREAVCIVLSDDTCSDEKIRMNRVVRNNLRVRLGDVISIQPCPDVKYGKRIHVLPIDDTVEGITGNLFEVYLKPYFLEAYRPIRKGDIFLVRGGMRAVEFKVVETDPSPYCIVAPDTVIHCEGEPIKREDEEESLNEVGYDDIGGCRKQLAQIKEMVELPLRHPALFKAIGVKPPRGILLYGPPGTGKTLIARAVANETGAFFFLINGPEIMSKLAGESESNLRKAFEEAEKNAPAIIFIDELDAIAPKREKTHGEVERRIVSQLLTLMDGLKQRAHVIVMAATNRPNSIDPALRRFGRFDREVDIGIPDATGRLEILQIHTKNMKLADDVDLEQVANETHGHVGADLAALCSEAALQAIRKKMDLIDLEDETIDAEVMNSLAVTMDDFRWALSQSNPSALRETVVEVPQVTWEDIGGLEDVKRELQELVQYPVEHPDKFLKFGMTPSKGVLFYGPPGCGKTLLAKAIANECQANFISIKGPELLTMWFGESEANVREIFDKARQAAPCVLFFDELDSIAKARGGNIGDGGGAADRVINQILTEMDGMSTKKNVFIIGATNRPDIIDPAILRPGRLDQLIYIPLPDEKSRVAILKANLRKSPVAKDVDLEFLAKMTNGFSGADLTEICQRACKLAIRESIESEIRRERERQTNPSAMEVEEDDPVPEIRRDHFEEAMRFARRSVSDNDIRKYEMFAQTLQQSRGFGSFRFPSGNQGGAGPSQGSGGGTGGSVYTEDNDDDLYG

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 \(c\).

\(|P_{f(4WUX_1)}(2) \setminus P_{f(5TWQ_1)}(2)|=60\), \(|P_{f(5TWQ_1)}(2) \setminus P_{f(4WUX_1)}(2)|=140\). 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 1:1010001000100100011001111010101101100110001001010010010001110111100100001110110000001011110000010100101101000010110101010000101000101000101000111001
Pair \(Z_2\) Length of longest common subsequence
4WUX_1,5TWQ_1 200 3
4WUX_1,8OOI_1 211 4
5TWQ_1,8OOI_1 167 4

Newick tree

 
[
	4WUX_1:10.45,
	[
		5TWQ_1:83.5,8OOI_1:83.5
	]:24.95
]

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 \((0.85)(0.8)(\frac{502 }{\log_{20} 502}-\frac{148}{\log_{20}148})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 4WUX_1 5TWQ_1 132 94
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]