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Parikh vectors
4QUL_1 3WTQ_1 6FHM_1 Letter Amino acid
14 12 7 R Arginine
15 7 11 N Asparagine
20 21 17 D Aspartic acid
10 10 5 M Methionine
14 8 10 F Phenylalanine
26 28 21 S Serine
16 11 14 T Threonine
13 15 13 V Valine
20 14 5 E Glutamic acid
8 11 10 H Histidine
19 17 5 I Isoleucine
7 17 8 P Proline
8 3 0 C Cysteine
4 14 18 Q Glutamine
22 16 12 K Lycine
10 8 3 Y Tyrosine
12 12 15 A Alanine
16 12 15 G Glycine
21 34 12 L Leucine
3 1 5 W Tryptophan 

4QUL_1|Chains A, B[auth C]|Caspase-3|Homo sapiens (9606)
>3WTQ_1|Chain A|Vitamin D3 receptor|Rattus norvegicus (10116)
>6FHM_1|Chains A, B|Outer-membrane lipoprotein carrier protein|Escherichia coli (strain K12) (83333)
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 \(w=f(c)\)
4QUL , Knot 124 278 0.83 40 184 273 
MENTENSVDSKSIKNLEPKIIHGSESMDSGISLDNSYKMDYPEMGLCIIINNKNWHKSTGMTSRSGTDVDAANLRETFRNLKYEVRNKNDLTREEIVELMRDVSKEDHSKRSSFVCVLLSHGEEGIIFGTNGPVDLKKITNFFRGDRCRSLTGKPKLFIIQACRGTELDCGIETDSGVDDDMACHKIPVEADFLYAYSTAPGYYSWRNSKDGSWFIQSLCAMLKQYADKLEFMHILTRVNRKVATEFESFSFDATFHAKKQIPCIVSMLTKELYFYHL
3WTQ , Knot 119 271 0.82 40 172 256 
GSHMGSPNSPLKDSLRPKLSEEQQHIIAILLDAHHKTYDPTYADFRDFRPPVRMDGSTGSVTLDLSPLSMLPHLADLVSYSIQKVIGFAKMIPGFRDLTSDDQIVLLKSSAIEVIMLRSNQSFTMDDMSWDCGSQDYKYDVTDVSKAGHTLELIEPLIKFQVGLKKLNLHEEEHVLLMAICIVSPDRPGVQDAKLVEAIQDRLSNTLQTYIRCRHPPPGSHQLYAKMIQKLADLRSLNEEHSKQYRSLSFQPENSMKLTPLVLEVFGNEIS
6FHM , Knot 91 206 0.78 38 138 195 
MGSSHHHHHHSSGLVPRGSHMASMDAASDLKSRLDKVSSFHASFTQKVTDGSGAAVQEGQGDLWVKRPNLENWHMTQPDESILVSDGKTLWFYNPFVEQATATWLKDATGNTPFMLIARNQSSDWQQYNIKQNGDDFVLTPKASNGNLKQFTINVGRDGTIHQFSAVEQDDQRSSYQLKSQQNGAVDAAKFTFTPPQGVTVDDQRK

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 \(c\).

\(|P_{f(4QUL_1)}(2) \setminus P_{f(3WTQ_1)}(2)|=88\), \(|P_{f(3WTQ_1)}(2) \setminus P_{f(4QUL_1)}(2)|=76\). 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 1:10000001000010010101101000100110100000100101110111000010000110000100101101000100100010000010000110110010000000000110111001001111100111010010011010000010101011110100100100110000110001100011101011010001110001000001011100101110001001011011001000110010010101010100011011011000101001
Pair \(Z_2\) Length of longest common subsequence
4QUL_1,3WTQ_1 164 4
4QUL_1,6FHM_1 174 4
3WTQ_1,6FHM_1 158 4

Newick tree

 
[
	4QUL_1:86.30,
	[
		3WTQ_1:79,6FHM_1:79
	]:7.30
]

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 \((0.85)(0.8)(\frac{549 }{\log_{20} 549}-\frac{271}{\log_{20}271})=78.7\)
Status Protein1 Protein2 d d1/2
Query variables 4QUL_1 3WTQ_1 96 94
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]