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Parikh vectors
5PQN_1 6ZJH_1 7TQN_1 Letter Amino acid
14 27 14 R Arginine
13 27 8 E Glutamic acid
7 29 16 G Glycine
6 15 8 F Phenylalanine
11 27 27 A Alanine
4 12 3 N Asparagine
5 32 6 I Isoleucine
6 26 6 K Lycine
7 16 21 S Serine
5 15 15 T Threonine
0 6 3 W Tryptophan
5 16 5 Y Tyrosine
13 25 11 D Aspartic acid
1 1 8 C Cysteine
8 13 11 Q Glutamine
10 12 8 H Histidine
18 33 32 L Leucine
7 8 5 M Methionine
6 24 24 P Proline
10 36 12 V Valine 

5PQN_1|Chains A, B|Bromodomain-containing protein 1|Homo sapiens (9606)
>6ZJH_1|Chain A[auth AAA]|Trehalose phosphorylase/synthase|Thermoproteus uzoniensis (strain 768-20) (999630)
>7TQN_1|Chain A|Three-prime repair exonuclease 1|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 \(w=f(c)\)
5PQN , Knot 74 156 0.79 38 120 151 
MHHHHHHSSGVDLGTENLYFQSMEQVAMELRLTELTRLLRSVLDQLQDKDPARIFAQPVSLKEVPDYLDHIKHPMDFATMRKRLEAQGYKNLHEFEEDFDLIIDNCMKYNARDTVFYRAAVRLRDQGGVVLRQARREVDSIGLEEASGMHLPERPA
6ZJH , Knot 171 400 0.85 40 221 389 
MIERYVEFVGEHEIDAIFKYAEKLKDLSILHVNSTAAGGGVAEILHRLVPLMRELGLNAEWKVIRGSQDFFTVTKSFHNALQTGKGEIPDEYFKIYDEWQEINAGEIPLDYDVVFIHDPQPAGLVKYRKKGTWIWRCHIDISNPHPKVWGFLRGYISKYDGMIVSIPEFARDDLDIPQIAIPPSIDPLSPKNMPLPQTTVDRIVDKFGVDRERPIILQVSRYDRAKDPVGVIESFRLAKRHVPDAQLVYLGSPATDDPEGEVVYRETVEAARGEKDVHLLMLPPNSHVEVNAFQRAATVVMQKSIKEGFGLTVSEALWKRKPVIGGNTGGIRIQVINGVTGFLVDSPKAAAYYLVYLLKNKKVREEMGEAGRDHVRRNFLITQQLRRYLMAILYLTKRHA
7TQN , Knot 109 243 0.82 40 157 231 
SMGSQTLPHGHMQTLIFFDMEATGLPFSQPKVTELCLLAVHRCALESPPTSQGPPPTVPPPPRVVDKLSLCVAPGKACSPAASEITGLSTAVLAAHGRQCFDDNLANLLLAFLRRQPQPWCLVAHNGDRYDFPLLQAELAMLGLTSALDGAFCVDSITALKALERASSPSEHGPRKSYSLGSIYTRLYGQSPPDSHTAEGDVLALLSICQWRPQALLRWVDAHARPFGTIRPMYGVTASARTK

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 \(c\).

\(|P_{f(5PQN_1)}(2) \setminus P_{f(6ZJH_1)}(2)|=40\), \(|P_{f(6ZJH_1)}(2) \setminus P_{f(5PQN_1)}(2)|=141\). 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 1:100000000110110001010010011101010010011001100100001101110110100110010010011011010001010100010010001011100010001000110011101000111110010001001110010110110011
Pair \(Z_2\) Length of longest common subsequence
5PQN_1,6ZJH_1 181 3
5PQN_1,7TQN_1 165 4
6ZJH_1,7TQN_1 192 3

Newick tree

 
[
	6ZJH_1:96.61,
	[
		5PQN_1:82.5,7TQN_1:82.5
	]:14.11
]

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 \((0.85)(0.8)(\frac{556 }{\log_{20} 556}-\frac{156}{\log_{20}156})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 5PQN_1 6ZJH_1 150 102.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]