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Parikh vectors
3WNX_1 4CQH_1 5SPT_1 Letter Amino acid
9 19 6 E Glutamic acid
5 4 0 W Tryptophan
17 20 22 V Valine
22 39 15 A Alanine
17 7 16 N Asparagine
25 23 13 G Glycine
9 17 6 H Histidine
12 46 19 L Leucine
15 9 5 F Phenylalanine
13 14 11 S Serine
10 21 2 R Arginine
8 8 7 Y Tyrosine
2 3 3 C Cysteine
11 18 3 Q Glutamine
14 8 6 I Isoleucine
12 2 13 K Lycine
4 7 2 M Methionine
12 29 5 P Proline
12 24 6 T Threonine
17 11 9 D Aspartic acid 

3WNX_1|Chain A|Protein ERGIC-53|Homo sapiens (9606)
>4CQH_1|Chain A|BACTERIOPHYTOCHROME|DEINOCOCCUS RADIODURANS (1299)
>5SPT_1|Chains A, B|Non-structural protein 3|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
3WNX , Knot 114 246 0.85 40 175 240 
MNHKVHMDGVGGDPAVALPHRRFEYKYSFKGPHLVQSDGTVPFWAHAGNAIPSSDQIRVAPSLKSQRGSVWTKTKAAFENWEVEVTFRVTGRGRIGADGLAIWYAENQGLEGPVFGSADLWNGVGIFFDSFDNDGKKNNPAIVIIGNNGQIHYDHQNDGASQALASCQRDFRNKPYPVRAKITYYQNTLTVMINNGFTPDKNDYEFCAKVENMIIPAQGHFGISAATGGLADDHDVLSFLTFQLTE
4CQH , Knot 136 329 0.79 40 178 306 
MARDPQPFFPPLYLGGPEITTENCEREPIHIPGSIQPHGALLTADGHSGEVLQVSLNAATFLGQEPTVLRGQTLAALLPDQWPALQTALPPGCQDALQYRATLDWPAAGHLSLTVHRVAELLILEFEPTEAWDSIGPHALRNAMFALESAPNLRALAEVATQTVRELSGFDRVMLYKFAPDATGEVIAEARREGMQAYLGHRFPASTTPAQARALYTRHLLRLTADTRAAAVPLDPVLNPQTNAPTPLGGAVLRATSPMHMQYLRNMGVGSSLSVSVVVGGQLWGLIVCHHQTPYVLPPDLRTTLEYLGRLLSLQVQRKEAEFHHHHHH
5SPT , Knot 80 169 0.81 38 119 162 
SMVNSFSGYLKLTDNVYIKNADIVEEAKKVKPTVVVNAANVYLKHGGGVAGALNKATNNAMQVESDDYIATNGPLKVGGSCVLSGHNLAKHCLHVVGPNVNKGEDIQLLKSAYENFNQHEVLLAPLLSAGIFGADPIHSLRVCVDTVRTNVYLAVFDKNLYDKLVSSFL

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 \(c\).

\(|P_{f(3WNX_1)}(2) \setminus P_{f(4CQH_1)}(2)|=85\), \(|P_{f(4CQH_1)}(2) \setminus P_{f(3WNX_1)}(2)|=88\). 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 1:100010101111011111100010000010110110001011111011011100001011101000010110000111001010101010101011101111101000110111110101101111110010001000011111110010100000001100111000001000101101010000001011100110100000010101001111101011101101111000011011010100
Pair \(Z_2\) Length of longest common subsequence
3WNX_1,4CQH_1 173 3
3WNX_1,5SPT_1 170 3
4CQH_1,5SPT_1 173 3

Newick tree

 
[
	4CQH_1:86.99,
	[
		3WNX_1:85,5SPT_1:85
	]:1.99
]

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 \((0.85)(0.8)(\frac{575 }{\log_{20} 575}-\frac{246}{\log_{20}246})=93.3\)
Status Protein1 Protein2 d d1/2
Query variables 3WNX_1 4CQH_1 115 102.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]