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Parikh vectors
2ETN_1 4JYV_1 5AKQ_1 Letter Amino acid
14 13 29 A Alanine
2 9 18 H Histidine
7 11 35 I Isoleucine
18 28 34 L Leucine
11 8 31 K Lycine
5 16 21 T Threonine
0 9 6 C Cysteine
11 15 18 R Arginine
1 7 13 N Asparagine
10 12 35 D Aspartic acid
21 10 28 E Glutamic acid
12 25 32 G Glycine
12 16 28 S Serine
6 12 8 Q Glutamine
5 4 9 M Methionine
1 7 13 F Phenylalanine
5 14 12 P Proline
0 7 6 W Tryptophan
2 7 12 Y Tyrosine
14 24 35 V Valine 

2ETN_1|Chains A, B, C|anti-cleavage anti-GreA transcription factor Gfh1|Thermus aquaticus (271)
>4JYV_1|Chain A[auth H]|Factor VII light chain|Homo sapiens (9606)
>5AKQ_1|Chains A, B|N-ISOPROPYLAMMELIDE ISOPROPYL AMIDOHYDROLASE|PSEUDOMONAS SP. ADP (47660)
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 \(w=f(c)\)
2ETN , Knot 71 157 0.76 36 99 147 
MAREVKLTKAGYERLMKQLEQERERLQEATKILQELMESSDDYDDSGLEAAKQEKARIEARIDSLEDVLSRAVILEEGTGEVIGLGSVVELEDPATGERLSVQVVSPAEASVLENPMKISDASPMGKALLGHRVGDVLSLDTPKGKKEFRVVAIHGR
4JYV , Knot 114 254 0.82 40 184 245 
IVGGKVCPKGECPWQVLLLVNGAQLCGGTLINTIWVVSAAHCFDKIKNWRNLIAVLGEHDLSEHDGDEQSRRVAQVIIPSTYVPGTTNHDIALLRLHQPVVLTDHVVPLCLPERTFSERTLAFVRFSLVSGWGQLLDRGATALELMVLNVPRLMTQDCLQQSRKVGDSPNITEYMFCAGYSDGSKDSCKGDSGGPHATHYRGTWYLTGIVSWGQGCATVGHFGVYTRVSQYIEWLQKLMRSEPRPGVLLRAPFP
5AKQ , Knot 177 423 0.84 40 226 398 
MGSSHHHHHHSSGLVPRGSHMSKDFDLIIRNAYLSEKDSVYDIGIVGDRIIKIEAKIEGTVKDEIDAKGNLVSPGFVDAHTHMDKSFTSTGERLPKFWSRPYTRDAAIEDGLKYYKNATHEEIKRHVIEHAHMQVLHGTLYTRTHVDVDSVAKTKAVEAVLEAKEELKDLIDIQVVAFAQSGFFVDLESESLIRKSLDMGCDLVGGVDPATRENNVEGSLDLCFKLAKEYDVDIDYHIHDIGTVGVYSINRLAQKTIENGYKGRVTTSHAWCFADAPSEWLDEAIPLYKDSGMKFVTCFSSTPPTMPVIKLLEAGINLGCASDDIRDFWVPFGNGDMVQGALIETQRLELKTNRDLGLIWKMITSEGARVLGIEKNYGIEVGKKADLVVLNSLSPQWAIIDQAKRLCVIKNGRIIVKDEVIVA

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 \(c\).

\(|P_{f(2ETN_1)}(2) \setminus P_{f(4JYV_1)}(2)|=53\), \(|P_{f(4JYV_1)}(2) \setminus P_{f(2ETN_1)}(2)|=138\). 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 1:1100101001100011001000000100100110011000000000110110000101010100100110011110010101111101101001101001010110110101100110100101110111100110110100101000101111010
Pair \(Z_2\) Length of longest common subsequence
2ETN_1,4JYV_1 191 3
2ETN_1,5AKQ_1 169 5
4JYV_1,5AKQ_1 174 3

Newick tree

 
[
	4JYV_1:93.52,
	[
		2ETN_1:84.5,5AKQ_1:84.5
	]:9.02
]

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 \((0.85)(0.8)(\frac{411 }{\log_{20} 411}-\frac{157}{\log_{20}157})=75.8\)
Status Protein1 Protein2 d d1/2
Query variables 2ETN_1 4JYV_1 102 79.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]