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Parikh vectors
8HXU_1 5RKG_1 3GCS_1 Letter Amino acid
0 5 10 M Methionine
5 8 12 F Phenylalanine
7 5 14 N Asparagine
6 5 15 Q Glutamine
22 3 16 G Glycine
21 14 44 L Leucine
25 6 24 A Alanine
1 3 3 C Cysteine
12 8 22 I Isoleucine
23 5 22 V Valine
13 10 27 D Aspartic acid
4 8 19 K Lycine
3 1 5 W Tryptophan
9 8 15 Y Tyrosine
17 17 22 S Serine
15 7 19 T Threonine
11 10 19 R Arginine
15 12 23 E Glutamic acid
9 8 12 H Histidine
13 6 17 P Proline 

8HXU_1|Chains A, B|Beta-lactamase class B VIM-2|Pseudomonas aeruginosa (287)
>5RKG_1|Chain A|PH-interacting protein|Homo sapiens (9606)
>3GCS_1|Chain A|Mitogen-activated protein kinase 14|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 \(w=f(c)\)
8HXU , Knot 104 231 0.81 38 148 225 
EYPTVSEIPVGEVRLYQIADGVWSHIATQSFDGAVYPSNGLIVRDGDELLLIDTAWGAKNTAALLAEIEKQIGLPVTRAVSTHFHDDRVGGVDVLRAAGVATYASPSTRRLAEVEGNEIPTHSLEGLSSSGDAVRFGPVELFYPGAAHSTDNLVVYVPSASVLYGGCAIYELSRTSAGNVADADLAEWPTSIERIQQHYPEAQFVIPGHGLPGGLDLLKHTTNVVKAHTNR
5RKG , Knot 73 149 0.81 40 117 140 
MHHHHHHSSGVDLGTENLYFQSMSYDIQAWKKQCEELLNLIFQCEDSEPFRQPVDLLEYPDYRDIIDTPMDFATVRETLEAGNYESPMELCKDVRLIFSNSKAYTPSKRSRIYSMSLRLSAFFEEHISSVLSDYKSALRFHKRNTITKR
3GCS , Knot 155 360 0.84 40 220 344 
GSQERPTFYRQELNKTIWEVPERYQNLSPVGSGAYGSVCAAFDTKTGLRVAVKKLSRPFQSIIHAKRTYRELRLLKHMKHENVIGLLDVFTPARSLEEFNDVYLVTHLMGADLNNIVKSQKLTDDHVQFLIYQILRGLKYIHSADIIHRDLKPSNLAVNEDSELKILDFGLCRHTDDEMTGYVATRWYRAPEIMLNWMHYNQTVDIWSVGCIMAELLTGRTLFPGTDHIDQLKLILRLVGTPGAELLKKISSESARNYIQSLTQMPKMNFANVFIGANPLAVDLLEKMLVLDSDKRITAAQALAHAYFAQYHDPDDEPVADPYDQSLESRDLLIDEWKSLTYDEVISFVPPPLDQEEMES

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 \(c\).

\(|P_{f(8HXU_1)}(2) \setminus P_{f(5RKG_1)}(2)|=98\), \(|P_{f(5RKG_1)}(2) \setminus P_{f(8HXU_1)}(2)|=67\). 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 1:001010011110101001101110011000101110100111100100111100111100011111010001111100110001000011110110111110010100001101010011000101100010110111101101111000001110110101101101100100001101101011011001001000010101111101111110110000011010000
Pair \(Z_2\) Length of longest common subsequence
8HXU_1,5RKG_1 165 3
8HXU_1,3GCS_1 174 4
5RKG_1,3GCS_1 187 5

Newick tree

 
[
	3GCS_1:92.76,
	[
		8HXU_1:82.5,5RKG_1:82.5
	]:10.26
]

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 \((0.85)(0.8)(\frac{380 }{\log_{20} 380}-\frac{149}{\log_{20}149})=69.6\)
Status Protein1 Protein2 d d1/2
Query variables 8HXU_1 5RKG_1 88 73
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]