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Parikh vectors
2PYE_1 6USN_1 4BIS_1 Letter Amino acid
23 16 19 R Arginine
17 11 11 Q Glutamine
22 13 31 E Glutamic acid
5 5 12 M Methionine
8 7 23 F Phenylalanine
12 12 21 P Proline
4 2 20 I Isoleucine
17 48 27 L Leucine
14 4 21 Y Tyrosine
16 19 17 V Valine
10 14 27 K Lycine
14 21 24 S Serine
23 8 23 T Threonine
10 2 8 W Tryptophan
24 26 23 A Alanine
2 10 15 N Asparagine
18 15 16 D Aspartic acid
4 5 5 C Cysteine
20 24 23 G Glycine
13 1 15 H Histidine 

2PYE_1|Chain A|HLA class I histocompatibility antigen, A-2 alpha chain|Homo sapiens (9606)
>6USN_1|Chains A, B, C, D|Sepiapterin reductase|Homo sapiens (9606)
>4BIS_1|Chains A, B|LYSINE-SPECIFIC DEMETHYLASE 4A|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 \(w=f(c)\)
2PYE , Knot 122 276 0.82 40 182 264 
GSHSMRYFFTSVSRPGRGEPRFIAVGYVDDTQFVRFDSDAASQRMEPRAPWIEQEGPEYWDGETRKVKAHSQTHRVDLGTLRGYYNQSEAGSHTVQRMYGCDVGSDWRFLRGYHQYAYDGKDYIALKEDLRSWTAADMAAQTTKHKWEAAHVAEQLRAYLEGTCVEWLRRYLENGKETLQRTDAPKTHMTHHAVSDHEATLRCWALSFYPAEITLTWQRDGEDQTQDTELVETRPAGDGTFQKWAAVVVPSGQEQRYTCHVQHEGLPKPLTLRWEP
6USN , Knot 113 263 0.79 40 146 245 
GSMEGGLGRAVCLLTGASRGFGRTLAPLLASLLSPGSVLVLSARNDEALRQLEAELGAERSGLRVVRVPADLGAEAGLQQLLGALRELPRPKGLQRLLLINNAGSLGDVSKGFVDLSDSTQVNNYWALNLTSMLCLTSSVLKAFPDSPGLNRTVVNISSLCALQPFKGWALYCAGKAARDMLFQVLALEEPNVRVLNYAPGPLDTDMQQLARETSVDPDMRKGLQELKAKGKLVDCKVSAQKLLSLLEKDEFKSGAHVDFYDK
4BIS , Knot 167 381 0.86 40 238 367 
MHHHHHHSSGVDLGTENLYFQSMASESETLNPSARIMTFYPTMEEFRNFSRYIAYIESQGAHRAGLAKVVPPKEWKPRASYDDIDDLVIPAPIQQLVTGQSGLFTQYNIQKKAMTVREFRKIANSDKYCTPRYSEFEELERKYWKNLTFNPPIYGADVNGTLYEKHVDEWNIGRLRTILDLVEKESGITIEGVNTPYLYFGMWKTSFAWHTEDMDLYSINYLHFGEPKSWYSVPPEHGKRLERLAKGFFPGSAQSCEAFLRHKMTLISPLMLKKYGIPFDKVTQEAGEFMITFPYGYHAGFNHGFNCAESTNFATRRWIEYGKQAVLCSCRKDMVKISMDVFVRKFQPERYKLWKAGKDNTVIDHTLPTPEAAEFLKESEL

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 \(c\).

\(|P_{f(2PYE_1)}(2) \setminus P_{f(6USN_1)}(2)|=102\), \(|P_{f(6USN_1)}(2) \setminus P_{f(2PYE_1)}(2)|=66\). 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 1:100010011001001101010111110100001101000110001010111100011001010000101000000101101010000001100010010100110010110100001001000111000100101101110000001011011001010101001011000100100010000110001000110000101001110101101010100010000000011000111010100111111101000000001000111011010101
Pair \(Z_2\) Length of longest common subsequence
2PYE_1,6USN_1 168 3
2PYE_1,4BIS_1 184 5
6USN_1,4BIS_1 172 3

Newick tree

 
[
	4BIS_1:90.67,
	[
		2PYE_1:84,6USN_1:84
	]:6.67
]

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 \((0.85)(0.8)(\frac{539 }{\log_{20} 539}-\frac{263}{\log_{20}263})=78.4\)
Status Protein1 Protein2 d d1/2
Query variables 2PYE_1 6USN_1 100 96
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]