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Parikh vectors
4IFJ_1 6HNO_1 2QDC_1 Letter Amino acid
22 16 19 R Arginine
8 6 7 C Cysteine
9 2 16 H Histidine
11 13 19 I Isoleucine
2 8 12 K Lycine
21 15 15 S Serine
6 2 5 W Tryptophan
20 32 18 A Alanine
6 10 22 Q Glutamine
35 33 21 G Glycine
4 6 8 F Phenylalanine
16 10 11 D Aspartic acid
18 17 18 T Threonine
15 9 6 N Asparagine
13 16 28 E Glutamic acid
21 26 25 L Leucine
7 5 7 M Methionine
13 17 24 P Proline
17 6 11 Y Tyrosine
27 25 17 V Valine 

4IFJ_1|Chain A|Kelch-like ECH-associated protein 1|Homo sapiens (9606)
>6HNO_1|Chain A|17-beta-hydroxysteroid dehydrogenase 14|Homo sapiens (9606)
>2QDC_1|Chain A|Tyrosine-protein phosphatase non-receptor type 7|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 \(w=f(c)\)
4IFJ , Knot 116 291 0.75 40 158 246 
SSAPKVGRLIYTAGGYFRQSLSYLEAYNPSDGTWLRLADLQVPRSGLAGCVVGGLLYAVGGRNNSPDGNTDSSALDCYNPMTNQWSPCAPMSVPRNRIGVGVIDGHIYAVGGSHGCIHHNSVERYEPERDEWHLVAPMLTRRIGVGVAVLNRLLYAVGGFDGTNRLNSAECYYPERNEWRMITAMNTIRSGAGVCVLHNCIYAAGGYDGQDQLNSVERYDVETETWTFVAPMKHRRSALGITVHQGRIYVLGGYDGHTFLDSVECYDPDTDTWSEVTRMTSGRSGVGVAVT
6HNO , Knot 121 274 0.82 40 171 263 
GHMATGTRYAGKVVVVTGGGRGIGAGIVRAFVNSGARVVICDKDESGGRALEQELPGAVFILCDVTQEDDVKTLVSETIRRFGRLDCVVNNAGHAPPPQRPEETSAQGFRQLLELNLLGTYTLTKLALPYLRKSQGNVINISSLVGAIGQAQAVPYVATKGAVTAMTKALALDESPYGVRVNCISPGNIWTPLWEELAALMPDPRASIREGMLAQPLGRMGQPAEVGAAAVFLASEANFCTGIELLVTGGAELGYGCKASRSTPVDAPDIPSGS
2QDC , Knot 136 309 0.84 40 195 293 
MGSDKIHHHHHHMNTPREVTLHFLRTAGHPLTRWALQRQPPSPKQLEEEFLKIPSNFVSPEDLDIPGHASKDRYKTILPNPQSRVCLGRAQSQEDGDYINANYIRGYDGKEKVYIATQGPMPNTVSDFWEMVWQEEVSLIVMLTQLREGKEKCVHYWPTEEETYGPFQIRIQDMKECPEYTVRQLTIQYQEERRSVKHILFSAWPAHQTPESAGPLLRLVAEVEESPETAAHPGPIVVHCSAGIGRTGCFIATRIGCQQLKARGEVDILGIVCQLRLDRGGMIQTAEQYQFLHHTLALYAGQLPEEPSP

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 \(c\).

\(|P_{f(4IFJ_1)}(2) \setminus P_{f(6HNO_1)}(2)|=76\), \(|P_{f(6HNO_1)}(2) \setminus P_{f(4IFJ_1)}(2)|=89\). 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 1:001101101100111010001001010010010110110101100111101111110111100001010000011000011000101011101100011111101010111100101000010000100001011111100011111111001101111101000100100001000010110110010011110110001011110010001001000010000101111100000111101001010111100100110010000100001001001001001111110
Pair \(Z_2\) Length of longest common subsequence
4IFJ_1,6HNO_1 165 3
4IFJ_1,2QDC_1 187 4
6HNO_1,2QDC_1 158 3

Newick tree

 
[
	4IFJ_1:91.02,
	[
		6HNO_1:79,2QDC_1:79
	]:12.02
]

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 \((0.85)(0.8)(\frac{565 }{\log_{20} 565}-\frac{274}{\log_{20}274})=82.1\)
Status Protein1 Protein2 d d1/2
Query variables 4IFJ_1 6HNO_1 101 97.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]