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Parikh vectors
2FOA_1 1JXI_1 4JJM_1 Letter Amino acid
17 37 11 A Alanine
8 4 4 C Cysteine
4 16 8 E Glutamic acid
10 12 7 I Isoleucine
18 30 6 L Leucine
3 6 12 F Phenylalanine
7 12 9 P Proline
22 17 12 S Serine
19 19 11 T Threonine
7 5 1 W Tryptophan
7 13 7 D Aspartic acid
15 12 5 Q Glutamine
25 23 28 G Glycine
2 7 6 M Methionine
27 20 16 V Valine
12 17 5 R Arginine
17 7 7 N Asparagine
6 21 5 H Histidine
3 7 13 K Lycine
11 3 2 Y Tyrosine 

2FOA_1|Chain A|elastase-1|Sus scrofa (9823)
>1JXI_1|Chains A, B|PHOSPHOMETHYLPYRIMIDINE KINASE|Salmonella typhimurium (90371)
>4JJM_1|Chains A, B|Peptidyl-prolyl cis-trans isomerase|Citrus sinensis (2711)
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 \(w=f(c)\)
2FOA , Knot 108 240 0.82 40 162 230 
VVGGTEAQRNSWPSQISLQYRSGSSWAHTCGGTLIRQNWVMTAAHCVDRELTFRVVVGEHNLNQNNGTEQYVGVQKIVVHPYWNTDDVAAGYDIALLRLAQSVTLNSYVQLGVLPRAGTILANNSPCYITGWGLTRTNGQLAQTLQQAYLPTVDYAICSSSSYWGSTVKNSMVCAGGDGVRSGCQGDSGGPLHCLVNGQYAVHGVTSFVSRLGCNVTRKPTVFTRVSAYISWINNVIASN
1JXI , Knot 123 288 0.80 40 172 264 
MGHHHHHHHHHHSSGYHIQGRHMQRINALTIAGTDPSGGAGIQADLKTFSALGAYGCSVITALVAENTCGVQSVYRIEPDFVAAQLDSVFSDVRIDTTKIGMLAETDIVEAVAERLQRHHVRNVVLDTVMLAKSGDPLLSPSAIETLRVRLLPQVSLITPNLPEAAALLDAPHARTEQEMLAQGRALLAMGCEAVLMKGGHLEDAQSPDWLFTREGEQRFSAPRVNTKNTHGTGCTLSAALAALRPRHRSWGETVNEAKAWLSAALAQADTLEVGKGIGPVHHFHAWW
4JJM , Knot 84 175 0.82 40 123 169 
GSHMPNPKVFFDMTVGGQPAGRIVMELFADVTPRTAENFRALCTGEKGIGKSGKPLHYKGSSFHRVIPGFMCQGGDFTAGNGTGGESIYGSKFADENFVKKHTGPGILSMANAGPGTNGSQFFVCTAKTEWLDGKHVVFGQVVEGMDVVKAIEKVGSSSGRTNKPVVIADCGQLS

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 \(c\).

\(|P_{f(2FOA_1)}(2) \setminus P_{f(1JXI_1)}(2)|=73\), \(|P_{f(1JXI_1)}(2) \setminus P_{f(2FOA_1)}(2)|=83\). 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 1:111100100001100101000010011000110110001110110010001010111100010000100001110011101010000111100111101100101000101111101101110001001011110000101100100101101001100000011001000110111011001001001111001101001101100110011001000101100101010110011100
Pair \(Z_2\) Length of longest common subsequence
2FOA_1,1JXI_1 156 3
2FOA_1,4JJM_1 181 3
1JXI_1,4JJM_1 165 4

Newick tree

 
[
	4JJM_1:89.27,
	[
		2FOA_1:78,1JXI_1:78
	]:11.27
]

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 \((0.85)(0.8)(\frac{528 }{\log_{20} 528}-\frac{240}{\log_{20}240})=82.3\)
Status Protein1 Protein2 d d1/2
Query variables 2FOA_1 1JXI_1 99 92
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]