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Parikh vectors
1BKJ_1 1EJZ_1 1SZA_1 Letter Amino acid
11 0 7 D Aspartic acid
3 4 3 C Cysteine
6 0 1 H Histidine
6 0 9 F Phenylalanine
9 0 5 P Proline
1 0 1 W Tryptophan
17 0 5 V Valine
18 0 3 Q Glutamine
15 0 11 E Glutamic acid
24 1 10 A Alanine
11 0 5 R Arginine
13 0 10 N Asparagine
14 0 12 I Isoleucine
19 0 9 S Serine
13 0 11 T Threonine
8 0 7 Y Tyrosine
15 2 3 G Glycine
24 0 18 L Leucine
7 0 11 K Lycine
6 0 3 M Methionine 

1BKJ_1|Chains A, B|NADPH-FLAVIN OXIDOREDUCTASE|Vibrio harveyi (669)
>1EJZ_1|Chain A|RNA (5'-R(*CP*GP*CP*UP*AP*CP*GP*C)-3')|
>1SZA_1|Chains A, B, C|PCF11 protein|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
1BKJ , Knot 110 240 0.83 40 161 231 
MNNTIETILAHRSIRKFTAVPITDEQRQTIIQAGLAASSSSMLQVVSIVRVTDSEKRNELAQFAGNQAYVESAAEFLVFCIDYQRHATINPDVQADFTELTLIGAVDSGIMAQNCLLAAESMGLGGVYIGGLRNSAAQVDELLGLPENSAVLFGMCLGHPDQNPEVKPRLPAHVVVHENQYQELNLDDIQSYDQTMQAYYASRTSNQKLSTWSQEVTGKLAGESRPHILPYLNSKGLAKR
1EJZ , Knot 5 8 0.43 8 5 5 
CGCUACGC
1SZA , Knot 69 144 0.79 40 111 139 
MDHDTEVIVKDFNSILEELTFNSRPIITTLTKLAEENISCAQYFVDAIESRIEKCMPKQKLYAFYALDSICKNVGSPYTIYFSRNLFNLYKRTYLLVDNTTRTKLINMFKLWLNPNDTGLPLFEGSALEKIEQFLIKASAAALE

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 \(c\).

\(|P_{f(1BKJ_1)}(2) \setminus P_{f(1EJZ_1)}(2)|=161\), \(|P_{f(1EJZ_1)}(2) \setminus P_{f(1BKJ_1)}(2)|=5\). 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 1:100010011100010010111100000001101111100001101101101000000001101110010100110111101000001010101010100101111100111100011110011111101111000110100111110001111110110100010101011101110000000101001000000101001000000010010001010111000101110100011100
Pair \(Z_2\) Length of longest common subsequence
1BKJ_1,1EJZ_1 166 1
1BKJ_1,1SZA_1 164 3
1EJZ_1,1SZA_1 116 1

Newick tree

 
[
	1BKJ_1:89.18,
	[
		1SZA_1:58,1EJZ_1:58
	]:31.18
]

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 \((0.85)(0.8)(\frac{248 }{\log_{20} 248}-\frac{8}{\log_{20}8})=83.7\)
Status Protein1 Protein2 d d1/2
Query variables 1BKJ_1 1EJZ_1 107 55.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]