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Parikh vectors
2BUW_1 1GXN_1 3MTK_1 Letter Amino acid
22 34 8 G Glycine
6 24 9 S Serine
4 20 6 Y Tyrosine
7 16 12 K Lycine
10 11 3 P Proline
12 15 9 F Phenylalanine
13 25 4 T Threonine
5 8 1 W Tryptophan
14 19 7 V Valine
14 19 10 D Aspartic acid
0 1 3 C Cysteine
14 13 4 Q Glutamine
15 8 20 I Isoleucine
4 3 7 H Histidine
15 24 23 L Leucine
1 7 3 M Methionine
12 32 11 A Alanine
13 18 8 R Arginine
14 25 15 N Asparagine
14 10 15 E Glutamic acid 

2BUW_1|Chain A|PROTOCATECHUATE 3,4-DIOXYGENASE ALPHA CHAIN|ACINETOBACTER CALCOACETICUS (62977)
>1GXN_1|Chain A|PECTATE LYASE|CELLVIBRIO CELLULOSA (155077)
>3MTK_1|Chains A, B|Diguanylate cyclase/phosphodiesterase|Caldicellulosiruptor saccharolyticus (351627)
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 \(w=f(c)\)
2BUW , Knot 99 209 0.84 38 148 201 
MNGWNFQELKETPSQTGGPYVHIGLLPKQANIEVFEHNLDNNLVQDNTQGQRIRLEGQVFDGLGLPLRDVLIEIWQADTNGVYPSQADTQGKQVDPNFLGWGRTGADFGTGFWSFNTIKPGAVPGRKGSTQAPHISLIIFARGINIGLHTRVYFDDEAEANAKDPVLNSIEWATRRQTLVAKREERDGEVVYRFDIRIQGENETVFFDI
1GXN , Knot 142 332 0.82 40 197 311 
GLVPRGSHMTGRMLTLDGNPAANWLNNARTKWSASRADVVLSYQQNNGGWPKNLDYNSVGNGGGGNESGTIDNGATITEMVFLAEVYKSGGNTKYRDAVRKAANFLVNSQYSTGALPQFYPLKGGYSDHATFNDNGMAYALTVLDFAANKRAPFDTDVFSDNDRTRFKTAVTKGTDYILKAQWKQNGVLTVWCAQHGALDYQPKKARAYELESLSGSESVGVLAFLMTQPQTAEIEQAVRAGVAWFNSPRTYLEGYTYDSSLAATNPIVPRAGSKMWYRFYDLNTNRGFFSDRDGSKFYDITQMSLERRTGYSWGGNYGTSIINFAQKVGYL
3MTK , Knot 83 178 0.80 40 126 171 
SKLEFLAFYDELTGLPNKNSLIRWLNLKVSQMDCIDTYLIFLEVRDLEKLNVTYGYDLVDELIIHISKRIKDIAGEGNKAFKIGFDRFAIICKSENISDFIERMLSQLLLPYNVNGNLIRVNFNIGAAQIENSNEAAANLMRRCDLALIKAKEEGLNEYVIFKPSIEIQTLEHHHHHH

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 \(c\).

\(|P_{f(2BUW_1)}(2) \setminus P_{f(1GXN_1)}(2)|=65\), \(|P_{f(1GXN_1)}(2) \setminus P_{f(2BUW_1)}(2)|=114\). 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 1:10110100100010001110101111100101011000100011000001001010101101111110011101101000110100100010010101111100110110111010010111111001000110101111101101110001010001010100111001011000001110000001011001010101000011101
Pair \(Z_2\) Length of longest common subsequence
2BUW_1,1GXN_1 179 3
2BUW_1,3MTK_1 144 3
1GXN_1,3MTK_1 175 3

Newick tree

 
[
	1GXN_1:93.36,
	[
		2BUW_1:72,3MTK_1:72
	]:21.36
]

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 \((0.85)(0.8)(\frac{541 }{\log_{20} 541}-\frac{209}{\log_{20}209})=95.4\)
Status Protein1 Protein2 d d1/2
Query variables 2BUW_1 1GXN_1 122 99.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]