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Parikh vectors
7FBM_1 1IHP_1 5IXY_1 Letter Amino acid
10 30 26 G Glycine
3 9 7 H Histidine
2 18 8 Y Tyrosine
6 28 18 A Alanine
3 18 15 N Asparagine
14 36 38 L Leucine
7 18 23 I Isoleucine
14 15 28 K Lycine
3 4 8 M Methionine
6 25 7 F Phenylalanine
6 47 24 S Serine
11 29 18 D Aspartic acid
3 18 12 Q Glutamine
8 22 18 E Glutamic acid
2 4 6 W Tryptophan
19 39 14 T Threonine
10 28 34 V Valine
4 18 11 R Arginine
1 10 5 C Cysteine
1 22 11 P Proline 

7FBM_1|Chain A|Fatty acid-binding protein, heart|Homo sapiens (9606)
>1IHP_1|Chain A|PHYTASE|Aspergillus ficuum (5058)
>5IXY_1|Chains A, B, C, D|L-lactate dehydrogenase A chain|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 \(w=f(c)\)
7FBM , Knot 66 133 0.81 40 99 125 
MVDAFLGTWKLVDSKNFDDYMKSLGVGFATRQVASMTKPTTIIEKNGDILTLKTHSTFKNTEISFKLGVEFDETTADDRKVKSIVTLDGGKLVHLQKWDGQETTLVRELIDGKLILTLTHGTAVCTRTYEKEA
1IHP , Knot 187 438 0.86 40 231 419 
SCDTVDQGYQCFSETSHLWGQYAPFFSLANESVISPEVPAGCRVTFAQVLSRHGARYPTDSKGKKYSALIEEIQQNATTFDGKYAFLKTYNYSLGADDLTPFGEQELVNSGIKFYQRYESLTRNIVPFIRSSGSSRVIASGKKFIEGFQSTKLKDPRAQPGQSSPKIDVVISEASSSNNTLDPGTCTVFEDSELADTVEANFTATFVPSIRQRLENDLSGVTLTDTEVTYLMDMCSFDTISTSTVDTKLSPFCDLFTHDEWINYDYLQSLKKYYGHGAGNPLGPTQGVGYANELIARLTHSPVHDDTSSNHTLDSSPATFPLNSTLYADFSHDNGIISILFALGLYNGTKPLSTTTVENITQTDGFSSAWTVPFASRLYVEMMQCQAEQEPLVRVLVNDRVVPLHGCPVDALGRCTRDSFVRGLSFARSGGDWAECFA
5IXY , Knot 142 331 0.83 40 197 322 
ATLKDQLIYNLLKEEQTPQNKITVVGVGAVGMACAISILMKDLADELALVDVIEDKLKGEMMDLQHGSLFLRTPKIVSGKDYNVTANSKLVIITAGARQQEGESRLNLVQRNVNIFKFIIPNVVKYSPNCKLLIVSNPVDILTYVAWKISGFPKNRVIGSGCNLDSARFRYLMGERLGVHPLSCHGWVLGEHGDSSVPVWSGMNVAGVSLKTLHPDLGTDKDKEQWKEVHKQVVESAYEVIKLKGYTSWAIGLSVADLAESIMKNLRRVHPVSTMIKGLYGIKDDVFLSVPCILGQNGISDLVKVTLTSEEEARLKKSADTLWGIQKELQF

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 \(c\).

\(|P_{f(7FBM_1)}(2) \setminus P_{f(1IHP_1)}(2)|=25\), \(|P_{f(1IHP_1)}(2) \setminus P_{f(7FBM_1)}(2)|=157\). 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 1:1101111010110000100010011111100011010010011000101101000001000010101110100001000010011010110110100101000011001101011101001011000000001
Pair \(Z_2\) Length of longest common subsequence
7FBM_1,1IHP_1 182 3
7FBM_1,5IXY_1 176 4
1IHP_1,5IXY_1 164 4

Newick tree

 
[
	7FBM_1:91.88,
	[
		5IXY_1:82,1IHP_1:82
	]:9.88
]

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 \((0.85)(0.8)(\frac{571 }{\log_{20} 571}-\frac{133}{\log_{20}133})=127.\)
Status Protein1 Protein2 d d1/2
Query variables 7FBM_1 1IHP_1 163 104
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]