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Parikh vectors
6ULV_1 6ARE_1 6NDD_1 Letter Amino acid
9 19 5 T Threonine
5 29 6 V Valine
5 56 9 A Alanine
12 16 7 N Asparagine
7 15 8 D Aspartic acid
2 6 8 C Cysteine
2 7 2 H Histidine
6 8 8 R Arginine
7 22 5 I Isoleucine
15 41 6 L Leucine
10 18 9 Q Glutamine
10 22 10 E Glutamic acid
4 6 2 M Methionine
16 22 5 P Proline
7 5 5 Y Tyrosine
5 45 8 G Glycine
12 27 7 K Lycine
6 10 7 F Phenylalanine
3 24 11 S Serine
3 1 7 W Tryptophan 

6ULV_1|Chains A, B, C, D|Bromodomain-containing protein 4|Homo sapiens (9606)
>6ARE_1|Chains A, B|Acetyl-CoA acetyltransferase, acetylated enzyme|Aspergillus fumigatus Af293 (330879)
>6NDD_1|Chains A, D, G, J, M, P|Snaclec rhodocetin subunit gamma|Calloselasma rhodostoma (8717)
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 \(w=f(c)\)
6ULV , Knot 74 146 0.84 40 124 141 
QGPLGSSTNPPPPETSNPNKPKRQTNQLQYLLRVVLKTLWKHQFAWPFQQPVDAVKLNLPDYYKIIKTPMDMGTIKKRLENNYYWNAQECIQDFNTMFTNCYIYNKPGDDIVLMAEALEKLFLQKINELPTEEPEFPGRLERPHRD
6ARE , Knot 167 399 0.83 40 195 371 
GMSSLPAVYIVSSARTPVGSFLGSLSSLTAPQLGAHAIKAALAKVDGLKPSDVQEVFFGNVISANVGQNPARQCALGAGLEESTICTTVNKVCASGLKAIILGAQTIMTGNADVVVAGGTESMSNAPHYLPNLRTGAKYGHQSLVDGIMKDGLTDAGKQELMGLQAEECAQDHGFSREQQDEYAIRTYEKAQAAQKAGLFDEEIAPIQLPGFRGKPDVTVTQDEEPKNLNPEKLRAIKPAFIPGSGTVTAPNSSPLNDGAAAVVLVSEAKLKELNLKPVAKILGWGDAAQQPSKFTTAPALAIPKALKHAGVGQDAIDAFEINEAFSVVALANMKLLGIPEEKVNLHGGAVAIGHPIGASGARILTTLLGVLKAKKGKLGCAGICNGGGGASALVVELL
6NDD , Knot 68 135 0.82 40 111 132 
DFNCLPGWSAYDQHCYQAFNEPKTWDEAERFCTEQAKRGHLVSIGSDGEADFVAQLVTNNIKRPELYVWIGLRDRRKEQQCSSEWSMSASIIYVNWNTGESQMCQGLARWTGFRKWDYSDCQAKNPFVCKFPSEC

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 \(c\).

\(|P_{f(6ULV_1)}(2) \setminus P_{f(6ARE_1)}(2)|=53\), \(|P_{f(6ARE_1)}(2) \setminus P_{f(6ULV_1)}(2)|=124\). 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 1:01111000011110000100100000010011011100110001111100110110101100001100110110100010000010100010010011000010001100111110110011100100110001011101001000
Pair \(Z_2\) Length of longest common subsequence
6ULV_1,6ARE_1 177 3
6ULV_1,6NDD_1 159 3
6ARE_1,6NDD_1 194 3

Newick tree

 
[
	6ARE_1:96.88,
	[
		6ULV_1:79.5,6NDD_1:79.5
	]:17.38
]

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 \((0.85)(0.8)(\frac{545 }{\log_{20} 545}-\frac{146}{\log_{20}146})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 6ULV_1 6ARE_1 143 98.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]