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Parikh vectors
6QFQ_1 6GKR_1 4HCU_1 Letter Amino acid
6 8 9 Q Glutamine
13 28 26 E Glutamic acid
9 21 12 I Isoleucine
15 23 26 L Leucine
7 11 15 F Phenylalanine
3 8 6 W Tryptophan
2 15 9 Y Tyrosine
8 26 15 A Alanine
15 28 15 R Arginine
4 12 6 N Asparagine
10 8 6 H Histidine
4 9 9 M Methionine
14 23 22 V Valine
14 19 19 G Glycine
10 8 12 K Lycine
10 20 10 T Threonine
11 16 12 D Aspartic acid
1 4 8 C Cysteine
3 13 9 P Proline
11 16 23 S Serine 

6QFQ_1|Chain A|Induced myeloid leukemia cell differentiation protein Mcl-1|Homo sapiens (9606)
>6GKR_1|Chains A, B, C|Branched-chain-amino-acid aminotransferase|Thermobaculum terrenum ATCC BAA-798 (525904)
>4HCU_1|Chain A|Tyrosine-protein kinase ITK/TSK|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 \(w=f(c)\)
6QFQ , Knot 83 170 0.83 40 129 164 
MHHHHHHLVPRGSEDELYRQSLEIISRYLREQATGAKDTKPMGRSGATSRKALETLRRVGDGVQRNHETAFQGMLRKLDIKNEDDVKSLSRVMIHVFSDGVTNWGRIVTLISFGAFVAKHLKTINQESCIEPLAESITDVLVRTKRDWLVKQRGWDGFVEFFHVEDLEGG
6GKR , Knot 140 316 0.85 40 202 304 
MSQQENPKYLWWNHRIIPWEEATVHLTDYWWASVTAVFEGIRGYWNNAEGEMYIFRLEDHARRLEQSMQLIRMPKEFTVDEICQATIDLVRANDYRGDVYIMPLAYAVGNKAFSVVGDRTTEMFIYSRPAVSRLEEDFSLHACYSSWTRINERVLPPRIKALANYRNSQLASSEAAMNGYDTALFLNPEGKVAEGTGSCVFFVRKGKLITPDITSGILESITRDTVIHLAREVLGLEVEERVVDRTETYLADEAFLCGTHAEITPIASIDRHEMKHGAPGPITRQLRDIYREVVYGRDFRYRNWLTPVGMGVRAEQ
4HCU , Knot 125 269 0.86 40 189 260 
GSGKWVIDPSELTFVQEIGSGQFGLVHLGYWLNKDKVAIKTIREGAMSEEDFIEEAEVMMKLSHPKLVQLYGVCLEQAPICLVFEFMEHGCLSDYLRTQRGLFAAETLLGMCLDVCEGMAYLEEACVIHRDLAARNCLVGENQVIKVSDFGMTRFVLDDQYTSSTGTKFPVKWASPEVFSFSRYSSKSDVWSFGVLMWEVFSEGKIPYENRSNSEVVEDISTGFRLYKPRLASTHVYQIMNHCWRERPEDRPAFSRLLRQLAEIAESGL

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 \(c\).

\(|P_{f(6QFQ_1)}(2) \setminus P_{f(6GKR_1)}(2)|=48\), \(|P_{f(6GKR_1)}(2) \setminus P_{f(6QFQ_1)}(2)|=121\). 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 1:10000001110100001000010110001000101100001110011000011001001101100000011011100101000001001001110110011001101101101111110010010000010111001001110000011100011011101101001011
Pair \(Z_2\) Length of longest common subsequence
6QFQ_1,6GKR_1 169 4
6QFQ_1,4HCU_1 170 3
6GKR_1,4HCU_1 165 4

Newick tree

 
[
	6QFQ_1:85.48,
	[
		6GKR_1:82.5,4HCU_1:82.5
	]:2.98
]

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 \((0.85)(0.8)(\frac{486 }{\log_{20} 486}-\frac{170}{\log_{20}170})=92.6\)
Status Protein1 Protein2 d d1/2
Query variables 6QFQ_1 6GKR_1 118 89.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]