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Parikh vectors
1PPZ_1 9DFN_1 4AHY_1 Letter Amino acid
3 6 4 W Tryptophan
7 8 20 Y Tyrosine
5 10 9 Q Glutamine
1 29 7 E Glutamic acid
13 18 14 I Isoleucine
3 26 7 K Lycine
15 14 23 T Threonine
24 14 15 A Alanine
34 19 28 G Glycine
13 24 13 L Leucine
38 21 36 S Serine
2 14 8 H Histidine
1 10 7 M Methionine
5 19 11 F Phenylalanine
9 18 2 R Arginine
12 17 13 N Asparagine
7 18 15 D Aspartic acid
6 9 8 C Cysteine
9 5 17 P Proline
17 22 13 V Valine 

1PPZ_1|Chain A|Trypsin|Fusarium oxysporum (5507)
>9DFN_1|Chain A|Radical SAM core domain-containing protein|Trichoderma virens (413071)
>4AHY_1|Chain A|FLOCCULATION PROTEIN FLO5|SACCHAROMYCES CEREVISIAE S288C (559292)
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 \(w=f(c)\)
1PPZ , Knot 99 224 0.79 40 128 206 
IVGGTSASAGDFPFIVSISRNGGPWCGGSLLNANTVLTAAHCVSGYAQSGFQIRAGSLSRTSGGITSSLSSVRVHPSYSGNNNDLAILKLSTSIPSGGNIGYARLAASGSDPVAGSSATVAGWGATSEGGSSTPVNLLKVTVPIVSRATCRAQYGTSAITNQMFCAGVSSGGKDSCQGDSGGPIVDSSNTLIGAVSWGNGCARPNYSGVYASVGALRSFIDTYA
9DFN , Knot 145 321 0.87 40 209 310 
MHHHHHHSSGVDLGTENLYFQSMGQIPVSVNYFFTRKCNKSCGFCFHTAKTSHMEDISRAKRGLQLLQRAGMKKINFAGGEPFLYPKFLGELVDFCKEDLHLESVSIITNGSLVREEWVRKHAKNIDILACSCDSFDENMNIEIGRGTGNQVEILYRIAKWCRKNEIKFKLNTVVTRLNYEEDMNEHIDTLQPFRWKVFQVLIVEGENDSEKTLRDARRFTISDKQFEVFCSKHRHHKSFVAEPNRLMASSYLLVDEYMRFIDKDGNKLTKSILDVGVEAAMKEIKWDVDAFQERGGVYEWTKENEQVEACSTGMSEKLVW
4AHY , Knot 119 270 0.82 40 165 258 
MGSSHHHHHHSSGLVPRGSHMSGATEACLPAGQRKSGMNINFYQYSLKDSSTYSNAAYMAYGYASKTKLGSVGGQTDISIDYNIPCVSSSGTFPCPQEDSYGNWGCKGMGACSNSQGIAYWSTDLFGFYTTPTNVTLEMTGYFLPPQTGSYTFSFATVDDSAILSVGGSIAFECCAQEQPPITSTNFTINGIKPWDGSLPDNITGTVYMYAGYYYPLKVVYSNAVSWGTLPISVELPDGTTVSDNFEGYVYSFDDDLSQSNCTIPDPSIH

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 \(c\).

\(|P_{f(1PPZ_1)}(2) \setminus P_{f(9DFN_1)}(2)|=51\), \(|P_{f(9DFN_1)}(2) \setminus P_{f(1PPZ_1)}(2)|=132\). 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 1:11110010110111110100011110110110100110110010101001101011010000111000100101010001000011110100011011011010111010011110010111111000110001101101011110010001001001100011011100110000010011111000001111101101010100011010111100110001
Pair \(Z_2\) Length of longest common subsequence
1PPZ_1,9DFN_1 183 3
1PPZ_1,4AHY_1 153 5
9DFN_1,4AHY_1 192 9

Newick tree

 
[
	9DFN_1:98.86,
	[
		1PPZ_1:76.5,4AHY_1:76.5
	]:22.36
]

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 \((0.85)(0.8)(\frac{545 }{\log_{20} 545}-\frac{224}{\log_{20}224})=91.8\)
Status Protein1 Protein2 d d1/2
Query variables 1PPZ_1 9DFN_1 121 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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]