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Parikh vectors
4OST_1 3TXN_1 4BUF_1 Letter Amino acid
13 5 5 C Cysteine
30 8 8 P Proline
75 52 15 L Leucine
15 36 11 K Lycine
2 1 1 W Tryptophan
53 25 12 V Valine
17 17 14 R Arginine
56 24 9 Q Glutamine
24 5 19 H Histidine
50 19 25 G Glycine
14 21 14 I Isoleucine
0 12 11 F Phenylalanine
0 14 13 Y Tyrosine
66 39 11 A Alanine
10 6 12 N Asparagine
9 26 7 D Aspartic acid
28 18 9 T Threonine
21 32 17 E Glutamic acid
1 9 8 M Methionine
15 25 19 S Serine 

4OST_1|Chains A, B|Hax3|Xanthomonas campestris pv. armoraciae (329463)
>3TXN_1|Chain A|26S proteasome regulatory complex subunit p42B|Drosophila melanogaster (7227)
>4BUF_1|Chains A, B|TANKYRASE-2|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 \(w=f(c)\)
4OST , Knot 79 499 0.32 36 106 143 
MQWSGARALEALLTVAGELRGPPLQLDTGQLLKIAKRGGVTAVEAVHAWRNALTGAPLNLTPEQVVAIASHDGGKQALETVQRLLPVLCQAHGLTPQQVVAIASHDGGKQALETVQRLLPVLCQAHGLTPEQVVAIASHDGGKQALETVQALLPVLCQAHGLTPEQVVAIASNGGGKQALETVQRLLPVLCQAHGLTPQQVVAIASNGGGKQALETVQRLLPVLCQAHGLTPQQVVAIASNGGGKQALETVQRLLPVLCQAHGLTPQQVVAIASNCGGKQALETVQRLLPVLCQAHGLTPQQVVAIASNGGGKQALETVQRLLPVLCQAHGLTPQQVVAIASHDGGKQALETVQRLLPVLCQAHGLTPEQVVAIASNGGGKQALETVQRLLPVLCQAHGLTPEQVVAIASHDGGKQALETVQRLLPVLCQAHGLTPQQVVAIASNGGGRPALESIVAQLSRPDPALAALTNDHLVALACLGGRPALDAVKKLEHHHHHH
3TXN , Knot 159 394 0.80 40 197 364 
GVRDQEGAENDEERIRIKEQGILQQGELYKQEGKAKELADLIKVTRPFLSSISKAKAAKLVRSLVDMFLDMDAGTGIEVQLCKDCIEWAKQEKRTFLRQSLEARLIALYFDTALYTEALALGAQLLRELKKLDDKNLLVEVQLLESKTYHALSNLPKARAALTSARTTANAIYCPPKVQGALDLQSGILHAADERDFKTAFSYFYEAFEGFDSVDSVKALTSLKYMLLCKIMLGQSDDVNQLVSGKLAITYSGRDIDAMKSVAEASHKRSLADFQAALKEYKKELAEDVIVQAHLGTLYDTMLEQNLCRIIEPYSRVQVAHVAESIQLPMPQVEKKLSQMILDKKFSGILDQGEGVLIVFEETPVDKTYERVLETIQSMGKVVDTLYQKAKKLS
4BUF , Knot 112 240 0.85 40 176 232 
MHHHHHHSSGVDLGTENLYFQSMLNTSGSGTILIDLSPDDKEFQSVEEEMQSTVREHRDGGHAGGIFNRYNILKIQKVCNKKLWERYTHRRKEVSEENHNHANERMLFHGSPFVNAIIHKGFDERHAYIGGMFGAGIYFAENSSKSNQYVYGIGGGTGCPVHKDRSCYICHRQLLFCRVTLGKSFLQFSAMKMAHSPPGHHSVTGRPSVNGLALAEYVIYRGEQAYPEYLITYQIMRPEG

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 \(c\).

\(|P_{f(4OST_1)}(2) \setminus P_{f(3TXN_1)}(2)|=34\), \(|P_{f(3TXN_1)}(2) \setminus P_{f(4OST_1)}(2)|=125\). 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 1:1010110110111011101011110100101101100111011011011001101111010100111110001100110010011111001011010011111000110011001001111100101101001111100011001100101111110010110100111110011100110010011111001011010011111001110011001001111100101101001111100111001100100111110010110100111110001100110010011111001011010011111001110011001001111100101101001111100011001100100111110010110100111110011100110010011111001011010011111000110011001001111100101101001111100111011100111010010111111000011111011101110110010000000
Pair \(Z_2\) Length of longest common subsequence
4OST_1,3TXN_1 159 4
4OST_1,4BUF_1 190 6
3TXN_1,4BUF_1 177 3

Newick tree

 
[
	4BUF_1:95.55,
	[
		4OST_1:79.5,3TXN_1:79.5
	]:16.05
]

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 \((0.85)(0.8)(\frac{893 }{\log_{20} 893}-\frac{394}{\log_{20}394})=133.\)
Status Protein1 Protein2 d d1/2
Query variables 4OST_1 3TXN_1 140 100
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]