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Parikh vectors
1OXQ_1 3MMF_1 2PEJ_1 Letter Amino acid
0 10 6 N Asparagine
0 9 6 I Isoleucine
7 7 7 R Arginine
5 11 10 Q Glutamine
13 13 17 E Glutamic acid
15 22 5 G Glycine
9 12 3 F Phenylalanine
9 17 5 P Proline
3 1 0 C Cysteine
12 12 2 H Histidine
13 26 19 L Leucine
4 24 6 K Lycine
3 2 4 M Methionine
9 13 7 A Alanine
6 19 5 D Aspartic acid
14 18 14 S Serine
6 12 9 T Threonine
5 7 1 W Tryptophan
2 8 1 Y Tyrosine
5 17 7 V Valine 

1OXQ_1|Chains A, B, C, D, E|Baculoviral IAP repeat-containing protein 7|Homo sapiens (9606)
>3MMF_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>2PEJ_1|Chains A, B, C, D, E, F|ORF134|Synechococcus sp. (32049)
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 \(w=f(c)\)
1OXQ , Knot 67 140 0.78 36 104 131 
MGSSHHHHHHSSGLVPRGSHMLETEEEEEEGAGATLSRGPAFPGMGSEELRLASFYDWPLTAEVPPELLAAAGFFHTGHQDKVRCFFCYGGLQSWKRGDDPWTEHAKWFPSCQFLLRSKGRDFVHSVQETHSQLLGSWDP
3MMF , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
2PEJ , Knot 67 134 0.81 38 103 129 
MEFKKVAKETAITLQSALTLQAVRLISQQLSETNPGQAIWLGEFSKRHPIQESDLYLEAMMLENKELVLRILTVRENLAEGVLEFLPEMVLSQIKQSNGNHRRSLLERLTQVDSSSTDQTEPNPGESDTSEDSE

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 \(c\).

\(|P_{f(1OXQ_1)}(2) \setminus P_{f(3MMF_1)}(2)|=42\), \(|P_{f(3MMF_1)}(2) \setminus P_{f(1OXQ_1)}(2)|=114\). 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 1:11000000000011110100110000000011110100111111110001011010011101011101111111100100001001100111001001001100010111000111000100110010000001110101
Pair \(Z_2\) Length of longest common subsequence
1OXQ_1,3MMF_1 156 3
1OXQ_1,2PEJ_1 141 3
3MMF_1,2PEJ_1 165 4

Newick tree

 
[
	3MMF_1:83.28,
	[
		1OXQ_1:70.5,2PEJ_1:70.5
	]:12.78
]

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 \((0.85)(0.8)(\frac{400 }{\log_{20} 400}-\frac{140}{\log_{20}140})=78.2\)
Status Protein1 Protein2 d d1/2
Query variables 1OXQ_1 3MMF_1 96 73.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]