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Parikh vectors
3KWQ_1 2PEE_1 7NGH_1 Letter Amino acid
5 21 54 V Valine
11 32 55 L Leucine
4 12 18 P Proline
4 27 18 S Serine
5 22 25 T Threonine
3 16 14 Y Tyrosine
14 4 15 R Arginine
4 29 12 D Aspartic acid
6 5 9 Q Glutamine
7 38 34 I Isoleucine
4 38 18 K Lycine
2 11 17 M Methionine
11 32 47 A Alanine
1 0 0 C Cysteine
2 3 8 H Histidine
4 22 20 F Phenylalanine
0 3 3 W Tryptophan
1 31 10 N Asparagine
8 28 13 E Glutamic acid
2 13 46 G Glycine 

3KWQ_1|Chains A, E|Histone H3.2|Xenopus laevis (8355)
>2PEE_1|Chains A, B|Serine protease inhibitor|Thermoanaerobacter tengcongensis (273068)
>7NGH_1|Chains A, B, C|Proton/glutamate symporter, SDF family|Thermococcus kodakarensis KOD1 (69014)
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 \(w=f(c)\)
3KWQ , Knot 52 98 0.81 38 83 94 
PHRYRPGTVALREIRRYQESTELLIRKLPFQRLVREIAQDFKTDLRFQSSAVMALQEASEAYLVALFEDTNLCAIHAKRVTIMPKDIQLARRIRGERA
2PEE , Knot 155 387 0.79 38 190 362 
VQAANLMDRIKANPVSEKNIDEKFIYNTADFSIELFKNSIDDKENSLISPLSAMLALAMTANGADNETLAQMEKALGKDISIEDLNKYLYTYMKKLPNEEKSKLTIANSIWFKENDFMPSKDFLQIIADYYKADIFKAAFDSSTVSDINNWVKSKTNGMIDKILNKIDPEDVMYLINAVAFDAEWETVYEKASVHEDIFTDVYGNRQKVEFMNSEENLYIEEENAIGFVKPYAKNHYSFVAILPDENISVNEYIKTLTGQKFIDLIKNAKITLVRASLPKFKYEYTIKMNETLESLGMTDAFLPDKADFSKLGKSDIGNLYISEVLHKTFISVDELGTKAGAVTSVDITAAGIPVNFKTVKLNRPFIFAIIDNSTNLPIFIGTVLSL
7NGH , Knot 166 436 0.77 38 197 394 
MGKSLLRRYLDYPVLWKILWGLVLGAVFGLIAGHFGYAGAVKTYIKPFGDLFVRLLKMLVMPIVLASLVVGAASISPARLGRVGVKIVVYYLATSAMAVFFGLIVGRLFNVGANVNLGSGTGKAIEAQPPSLVQTLLNIVPTNPFASLAKGEVLPVIFFAIILGIAITYLMNRNEERVRKSAETLLRVFDGLAEAMYLIVGGVMQYAPIGVFALIAYVMAEQGVRVVGPLAKVVGAVYTGLFLQIVITYFILLKVFGIDPIKFIRKAKDAMITAFVTRSSSGTLPVTMRVAEEEMGVDKGIFSFTLPLGATINMDGTALYQGVTVLFVANAIGHPLTLGQQLVVVLTAVLASIGTAGVPGAGAIMLAMVLQSVGLDLTPGSPVALAYAMILGIDAILDMGRTMVNVTGDLAGTVIVAKTEKELDESKWISHHHHHH

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 \(c\).

\(|P_{f(3KWQ_1)}(2) \setminus P_{f(2PEE_1)}(2)|=39\), \(|P_{f(2PEE_1)}(2) \setminus P_{f(3KWQ_1)}(2)|=146\). 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 1:10000110111001000000001110011100110011001000101000111110010010111110000101101001011100101100101001
Pair \(Z_2\) Length of longest common subsequence
3KWQ_1,2PEE_1 185 3
3KWQ_1,7NGH_1 176 3
2PEE_1,7NGH_1 153 4

Newick tree

 
[
	3KWQ_1:94.42,
	[
		7NGH_1:76.5,2PEE_1:76.5
	]:17.92
]

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 \((0.85)(0.8)(\frac{485 }{\log_{20} 485}-\frac{98}{\log_{20}98})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 3KWQ_1 2PEE_1 144 90
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]