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Parikh vectors
4FFD_1 1TUH_1 4NRM_1 Letter Amino acid
19 19 15 G Glycine
22 7 12 K Lycine
9 2 15 P Proline
8 8 5 T Threonine
2 3 1 W Tryptophan
15 9 21 V Valine
13 11 12 A Alanine
11 13 21 R Arginine
24 12 21 E Glutamic acid
3 12 7 H Histidine
10 4 8 Y Tyrosine
2 2 5 C Cysteine
3 2 13 Q Glutamine
25 7 15 L Leucine
14 5 14 S Serine
7 7 4 N Asparagine
16 11 12 D Aspartic acid
12 9 10 F Phenylalanine
12 9 17 I Isoleucine
7 4 2 M Methionine 

4FFD_1|Chain A|PF00702 DOMAIN PROTEIN|Pyrococcus horikoshii (53953)
>1TUH_1|Chain A|hypothetical protein EGC068|uncultured bacterium (77133)
>4NRM_1|Chain A|RNA demethylase ALKBH5|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 \(w=f(c)\)
4FFD , Knot 107 234 0.83 40 153 225 
MIRAVFFDSLGTLNSVEGAAKSHLKIMEEVLGDYPLNPKTLLDEYEKLTREAFSNYAGKPYRPLRDILEEVMRKLAEKYGFKYPENFWEISLRMSQRYGELYPEVVEVLKSLKGKYHVGMITDSDTEQAMAFLDALGIKDLFDSITTSEEAGFFKPHPRIFELALKKAGVKGEEAVYVGDNPVKDCGGSKNLGMTSILLDRKGEKREFWDKCDFIVSDLREVIKIVDELNGQGS
1TUH , Knot 76 156 0.82 40 121 147 
MHHHHHHMGIVTGPIIDHSKENDVMNEAEQNAETVRRGYAAFNSGDMKTLTELFDENASWHTPGRSRIAGDHKGREAIFAQFGRYGGETGGTFKAVLLHVLKSDDGRVIGIHRNTAERGGKRLDVGCCIVFEFKNGRVIDGREHFYDLYAWDEFWR
4NRM , Knot 105 230 0.82 40 160 220 
GHMPERSDYEEQQLQKEEEARKVKSGIRQMRLFSQDECAKIEARIDEVVSRAEKGLYNEHTVDRAPLRNKYFFGEGYTYGAQLQKRGPGQERLYPPGDVDEIPEWVHQLVIQKLVEHRVIPEGFVNSAVINDYQPGGCIVSHVDPIHIFERPIVSVSFFSDSALCFGCKFQFKPIRVSEPVLSLPVRRGSVTVLSGYAADEITHCIRPQDIKERRAVIILRKTRLDAPRL

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 \(c\).

\(|P_{f(4FFD_1)}(2) \setminus P_{f(1TUH_1)}(2)|=99\), \(|P_{f(1TUH_1)}(2) \setminus P_{f(4FFD_1)}(2)|=67\). 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 1:110111100110100101110001011001110011010011000001000110001101001100110011001100011001001101010100001010101101100101000111100000001111101111001100100000111101010110111001110100110110011000110001110011100010000110000111001001101100101010
Pair \(Z_2\) Length of longest common subsequence
4FFD_1,1TUH_1 166 4
4FFD_1,4NRM_1 159 3
1TUH_1,4NRM_1 159 4

Newick tree

 
[
	1TUH_1:81.84,
	[
		4FFD_1:79.5,4NRM_1:79.5
	]:2.34
]

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 \((0.85)(0.8)(\frac{390 }{\log_{20} 390}-\frac{156}{\log_{20}156})=70.2\)
Status Protein1 Protein2 d d1/2
Query variables 4FFD_1 1TUH_1 88 74.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]