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Parikh vectors
8HEG_1 1VDN_1 1PKD_1 Letter Amino acid
26 10 19 A Alanine
10 7 17 E Glutamic acid
14 23 16 G Glycine
7 12 21 K Lycine
3 3 5 M Methionine
8 9 15 S Serine
1 1 4 W Tryptophan
8 5 10 Y Tyrosine
16 15 23 V Valine
8 7 16 I Isoleucine
6 12 16 F Phenylalanine
23 8 17 T Threonine
15 5 15 R Arginine
11 7 8 N Asparagine
8 10 16 D Aspartic acid
1 2 3 C Cysteine
8 4 9 Q Glutamine
18 8 38 L Leucine
12 10 19 P Proline
8 4 10 H Histidine 

8HEG_1|Chain A|VP1 of capsid protein|Foot-and-mouth disease virus (12110)
>1VDN_1|Chain A|Cyclophilin A|Saccharomyces cerevisiae (4932)
>1PKD_1|Chains A, C|Cell division protein kinase 2|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 \(w=f(c)\)
8HEG , Knot 102 211 0.86 40 153 207 
TTSAGESADPVTTTVENYGGETQVQRRHHTDVGFIMDRFVKINNTNPTHVIDLMQTHQHGLVGALLRAATYYFSDLEIVVRHEGNLTWVPNGAPEAALSNAGNPTAYNKAPFTRLALPYTAPHRVLATVYNGTSKYSTTGERTRGDLGALAARVATQLPASFNFGAIRATDISELLVRMKRAELYCPRPLLAVEVTAQDRHKQKIIAPAKQ
1VDN , Knot 78 162 0.81 40 123 155 
MSQVYFDVEADGQPIGRVVFKLYNDIVPKTAENFRALCTGEKGFGYAGSPFHRVIPDFMLQGGDFTAGNGTGGKSIYGGKFPDENFKKHHDRPGLLSMANAGPNTNGSQFFITTVPCPWLDGKHVVFGEVVDGYDIVKKVESLGSPSGATKARIVVAKSGEL
1PKD , Knot 134 297 0.85 40 190 283 
SMENFQKVEKIGEGTYGVVYKARNKLTGEVVALKKIRLDTETEGVPSTAIREISLLKELNHPNIVKLLDVIHTENKLYLVFEFLHQDLKKFMDASALTGIPLPLIKSYLFQLLQGLAFCHSHRVLHRDLKPQNLLINTEGAIKLADFGLARAFGVPVRTYTHEVVTLWYRAPEILLGCKYYSTAVDIWSLGCIFAEMVTRRALFPGDSEIDQLFRIFRTLGTPDEVVWPGVTSMPDYKPSFPKWARQDFSKVVPPLDEDGRSLLSQMLHYDPNKRISAKAALAHPFFQDVTKPVPHL

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 \(c\).

\(|P_{f(8HEG_1)}(2) \setminus P_{f(1VDN_1)}(2)|=92\), \(|P_{f(1VDN_1)}(2) \setminus P_{f(8HEG_1)}(2)|=62\). 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 1:0001100101100010001100010000000111110011010000100110110000011111110110001001011100010101110111011100110101000111001111001100111010010000000100001011111101100111010111101001001110100101001011111010100000001111100
Pair \(Z_2\) Length of longest common subsequence
8HEG_1,1VDN_1 154 3
8HEG_1,1PKD_1 159 4
1VDN_1,1PKD_1 171 4

Newick tree

 
[
	1PKD_1:84.32,
	[
		8HEG_1:77,1VDN_1:77
	]:7.32
]

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 \((0.85)(0.8)(\frac{373 }{\log_{20} 373}-\frac{162}{\log_{20}162})=63.4\)
Status Protein1 Protein2 d d1/2
Query variables 8HEG_1 1VDN_1 80 68.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]