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Parikh vectors
6ZUV_1 4ZIO_1 7JYN_1 Letter Amino acid
11 10 1 F Phenylalanine
21 12 3 T Threonine
13 2 0 W Tryptophan
29 8 7 R Arginine
16 6 2 N Asparagine
17 6 8 H Histidine
13 11 3 Y Tyrosine
18 12 6 D Aspartic acid
12 10 5 I Isoleucine
40 12 10 L Leucine
30 23 4 G Glycine
20 12 5 P Proline
30 11 10 S Serine
20 11 2 A Alanine
14 0 1 C Cysteine
15 8 4 Q Glutamine
25 14 3 V Valine
22 21 13 E Glutamic acid
13 22 7 K Lycine
4 7 2 M Methionine 

6ZUV_1|Chain A|Palmitoleoyl-protein carboxylesterase NOTUM|Homo sapiens (9606)
>4ZIO_1|Chains A, B, C|mCherry|Discosoma (86599)
>7JYN_1|Chain A|Bromodomain-containing protein 3|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 \(w=f(c)\)
6ZUV , Knot 162 383 0.83 40 219 364 
ETGSAQQLNEDLRLHLLLNTSVTCNDGSPAGYYLKESRGSRRWLLFLEGGWYCFNRENCDSRYDTMRRLMSSRDWPRTRTGTGILSSQPEENPYWWNANMVFIPYCSSDVWSGASSKSEKNEYAFMGALIIQEVVRELLGRGLSGAKVLLLAGSSAGGTGVLLNVDRVAEQLEKLGYPAIQVRGLADSGWFLDNKQYRHTDCVDTITCAPTEAIRRGIRYWNGVVPERCRRQFQEGEEWNCFFGYKVYPTLRSPVFVVQWLFDEAQLTVDNVHLTGQPVQEGLRLYIQNLGRELRHTLKDVPASFAPACLSHEIIIRSHWTDVQVKGTSLPRALHCWDRSLHDSHKASKTPLKGCPVHLVDSCPWPHCNPSCPTGTKHHHHHH
4ZIO , Knot 101 219 0.82 40 147 210 
AIIKEFMRFKVHMEGSVNGHEFEIEGEGEGRPYEGTQTAKLKVTKGGPLPFAWDILSPQFMYGSKAYVKHPADIPDYLKLSFPEGFKWERVMNFEDGGVVTVTQDSSLQDGEFIYKVKLRGTNFPSDGPVMQKKTMGXEASSERMYPEDGALKGEIKQRLKLKDGGHYDAEVKTTYKAKKPVQLPGAYNVNIKLDITSHNEDYTIVEQYERAEGRHSTG
7JYN , Knot 48 96 0.76 38 76 88 
HHHHHHSHMGKQASASYDSEEEEEGLPMSYDEKRQLSLDINRLPGEKLGRVVHIIQSREPSLRDSNPDEIEIDFETLKPTTLRELERYVKSCLQKK

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 \(c\).

\(|P_{f(6ZUV_1)}(2) \setminus P_{f(4ZIO_1)}(2)|=131\), \(|P_{f(4ZIO_1)}(2) \setminus P_{f(6ZUV_1)}(2)|=59\). 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 1:00101001000101011100010000101110010000100011111011100100000000000100110000110000101110001000101101011111000001101100000000011111111001100111011011011111100111011110100110010011011101011100111100000000001001001100110011001011110000001001001001110010101001111101110010101001010101100110101001100100010011101111010001110001001010100110110010001000001000110101101100011100010010100000000
Pair \(Z_2\) Length of longest common subsequence
6ZUV_1,4ZIO_1 190 3
6ZUV_1,7JYN_1 193 6
4ZIO_1,7JYN_1 143 3

Newick tree

 
[
	6ZUV_1:10.57,
	[
		4ZIO_1:71.5,7JYN_1:71.5
	]:31.07
]

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 \((0.85)(0.8)(\frac{602 }{\log_{20} 602}-\frac{219}{\log_{20}219})=108.\)
Status Protein1 Protein2 d d1/2
Query variables 6ZUV_1 4ZIO_1 142 110
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]