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Parikh vectors
1QOB_1 8TUW_1 3TVO_1 Letter Amino acid
1 10 9 N Asparagine
10 26 19 D Aspartic acid
5 15 9 I Isoleucine
0 5 1 M Methionine
8 29 12 T Threonine
5 15 7 Y Tyrosine
8 34 17 V Valine
9 25 13 A Alanine
10 26 13 E Glutamic acid
2 11 12 H Histidine
7 46 26 L Leucine
3 22 13 F Phenylalanine
6 35 17 S Serine
1 42 7 R Arginine
4 7 1 C Cysteine
5 20 24 K Lycine
0 9 7 W Tryptophan
5 8 12 Q Glutamine
6 34 22 G Glycine
3 29 17 P Proline 

1QOB_1|Chains A, B|FERREDOXIN|Nostoc sp. (103690)
>8TUW_1|Chain A[auth M1]|Maturation protein A|Pseudomonas phage PP7 (12023)
>3TVO_1|Chain A[auth X]|Carbonic anhydrase 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 \(w=f(c)\)
1QOB , Knot 55 98 0.85 36 82 96 
ATFKVTLINEAEGTKHEIEVPDDEYILDAAEEQGYDLPFSCRAGACSTCAGKLVSGTVDQSKQSFLDDDQIEAGYVLTCVAYPTSDVVIQTHKEEDLY
8TUW , Knot 187 448 0.85 40 229 424 
LEEHYYSRKTCGSGGGLSVQRTGYPQTDPPVFKAVNSFERAPLIVGDHVHSLPYWSRKITLECTPYPFTLRWEGGVHKDGIPDKVPHPSCDPGSELVSYTHSREVDCERFIGPVPSSFTDPKVTSAQWKFLEDMADMKALADLNKSLVNLPMLYKERRETLKMVGNRLGGLVRVAHAAQDRDLKRYFKARRKDRRKVAEEVANGHLELIFGWLPLIGELEGAIEYAELPDLDFIRCHGLHTLVLQSTPWDNSVDVRSYPNWERAAGTRITGSVRTRGVVESRASVRTALRFNLETSLAGDARRLGFEPISTTYDMIPLSFVVGWFSNFDKYVRTLAPLIGVTFETGSQNRRTTCELVGCTRFYPRTVSPPSGWFARWKDFPDGSLSEVSGLRRTDIRSVLSTLPDPDVRFHADVGLFEISAGISLLAQRYLKPLQRLLKRKSFFYGRT
3TVO , Knot 110 258 0.79 40 174 246 
HHWGFGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFQVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK

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 \(c\).

\(|P_{f(1QOB_1)}(2) \setminus P_{f(8TUW_1)}(2)|=30\), \(|P_{f(8TUW_1)}(2) \setminus P_{f(1QOB_1)}(2)|=177\). 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 1:10101011001010000101100001101100010011100011100001101101010000001100001011011001101000111000000010
Pair \(Z_2\) Length of longest common subsequence
1QOB_1,8TUW_1 207 3
1QOB_1,3TVO_1 172 3
8TUW_1,3TVO_1 167 4

Newick tree

 
[
	1QOB_1:98.73,
	[
		3TVO_1:83.5,8TUW_1:83.5
	]:15.23
]

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 \((0.85)(0.8)(\frac{546 }{\log_{20} 546}-\frac{98}{\log_{20}98})=132.\)
Status Protein1 Protein2 d d1/2
Query variables 1QOB_1 8TUW_1 168 101
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]