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Parikh vectors
1SIS_1 4XTO_1 9QCU_1 Letter Amino acid
0 8 12 H Histidine
0 22 15 I Isoleucine
3 11 12 K Lycine
3 10 13 M Methionine
2 23 10 F Phenylalanine
3 17 8 N Asparagine
3 5 16 P Proline
0 24 23 S Serine
0 15 7 Y Tyrosine
0 21 25 V Valine
1 19 51 A Alanine
8 0 8 C Cysteine
1 10 7 Q Glutamine
4 21 40 G Glycine
1 36 32 L Leucine
2 15 20 T Threonine
0 7 2 W Tryptophan
2 4 28 R Arginine
2 7 27 D Aspartic acid
0 13 17 E Glutamic acid 

1SIS_1|Chain A|SCORPION INSECTOTOXIN I5A|Mesobuthus eupeus (34648)
>4XTO_1|Chains A, B, C, D, E|Sodium pumping rhodopsin|Dokdonia eikasta (308116)
>9QCU_1|Chains A, B|L-asparaginase II|Rhizobium etli (29449)
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 \(w=f(c)\)
1SIS , Knot 20 36 0.66 28 33 34 
MCMPCFTTDPNMAKKCRDCCGGNGKCFGPQCLCNRX
4XTO , Knot 126 288 0.82 38 168 273 
MTQELGNANFENFIGATEGFSEIAYQFTSHILTLGYAVMLAGLLYFILTIKNVDKKFQMSNILSAVVMVSAFLLLYAQAQNWTSSFTFNEEVGRYFLDPSGDLFNNGYRYLNWLIDVPMLLFQILFVVSLTTSKFSSVRNQFWFSGAMMIITGYIGQFYEVSNLTAFLVWGAISSAFFFHILWVMKKVINEGKEGISPAGQKILSNIWILFLISWTLYPGAYLMPYLTGVDGFLYSEDGVMARQLVYTIADVSSKVIYGVLLGNLAITLSKNKELVEANSLEHHHHHH
9QCU , Knot 154 373 0.81 40 193 346 
GIDPFTMTPSEDFVVTDRGGIVENSHRVHAAVVDAKGRLLYALGNPTRMTLARAAAKPAQALAILETEGVAGYGFDDADIALMCASHSSEDRHIARTRAMLSKIKAEEADLRCGGHPSLSEMVNRSWIKQDFIPTAVCSNCSGKHVGMLAGARAIGAGTDGYHLPDHPMQGRVKRTVAELCDLDAGDVEWGTDGCNLPTPAFPLDRLGRIYAKLASAADGSDAGEGQSTRCAALAHIFRAMARHPEMVAGEGRYCTMLMRAFDGALVGKLGADASYAIGVRASDATRQLGTDGALGISVKIEDGNLEMLYAVVTELLERLGIGSPDVRSQLASFHHPQRVNTMGVTTGGVSFPFKLRGSKSNVDDPRLAAVAR

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 \(c\).

\(|P_{f(1SIS_1)}(2) \setminus P_{f(4XTO_1)}(2)|=24\), \(|P_{f(4XTO_1)}(2) \setminus P_{f(1SIS_1)}(2)|=159\). 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 1:101101000101100000001101001110010000
Pair \(Z_2\) Length of longest common subsequence
1SIS_1,4XTO_1 183 2
1SIS_1,9QCU_1 196 3
4XTO_1,9QCU_1 175 4

Newick tree

 
[
	1SIS_1:97.11,
	[
		4XTO_1:87.5,9QCU_1:87.5
	]:9.61
]

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 \((0.85)(0.8)(\frac{324 }{\log_{20} 324}-\frac{36}{\log_{20}36})=93.7\)
Status Protein1 Protein2 d d1/2
Query variables 1SIS_1 4XTO_1 118 66.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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