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Parikh vectors
9CST_1 8PJI_1 9CYD_1 Letter Amino acid
0 5 10 C Cysteine
8 5 18 Q Glutamine
4 14 16 E Glutamic acid
4 3 11 I Isoleucine
6 2 2 W Tryptophan
24 4 22 A Alanine
10 7 16 N Asparagine
7 5 13 D Aspartic acid
3 5 8 M Methionine
2 11 12 F Phenylalanine
3 12 12 P Proline
6 4 23 Y Tyrosine
5 10 7 R Arginine
2 12 9 H Histidine
8 11 26 L Leucine
9 14 20 V Valine
20 8 33 T Threonine
21 9 19 G Glycine
5 11 24 K Lycine
12 2 17 S Serine 

9CST_1|Chain A|Streptavidin|Streptomyces avidinii (1895)
>8PJI_1|Chain A|Protein ENL|Homo sapiens (9606)
>9CYD_1|Chain A|Papain-like protease|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
9CST , Knot 74 159 0.78 38 111 154 
MASMTGGQQMGRDEAGITGTWYNQLGSTFIVTAGADGALTGTYESAVGNAESRYVLTGRYDSAPATDGSGTALGWTVAWKNNYRNAHSATTWSGQYVGGAQARINTQWLLTSGTTEANAWASTLVGHDTFTKVKPSAASIDAAKKAGVNNGNPLDAVQQ
8PJI , Knot 73 154 0.79 40 112 147 
MDNQCTVQVRLELGHRAQLRKKPTTEGFTHDWMVFVRGPEQCDIQHFVEKVVFWLHDSFPKPRRVCKEPPYKVEESGYAGFIMPIEVHFKNKEEPRKVCFTYDLFLNLEGNPPVNHLRCEKLTFNNPTTEFRYKLLRAGGVMVMPEGAHHHHHH
9CYD , Knot 141 318 0.85 40 201 302 
SNAEVRTIKVFTTVDNINLHTQVVDMSMTYGQQFGPTYLDGADVTKIKPHNSHEGKTFYVLPNDDTLRVEAFEYYHTTDPSFLGRYMSALNHTKKWKYPQVNGLTSIKWADNNSYLATALLTLQQIELKFNPPALQDAYYRARAGEAANFCALILAYCNKTVGELGDVRETMSYLFQHANLDSCKRVLNVVCKTCGQQQTTLKGVEAVMYMGTLSYEQFKKGVQIPCTCGKQATKYLVQQESPFVMMSAPPAQYELKHGTFTCASEYTGNYQCGHYKHITSKETLYCIDGALLTKSSEYKGPITDVFYKENSYTTTIK

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 \(c\).

\(|P_{f(9CST_1)}(2) \setminus P_{f(8PJI_1)}(2)|=76\), \(|P_{f(8PJI_1)}(2) \setminus P_{f(9CST_1)}(2)|=77\). 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 1:110101100110001110101000110011101110111010000111010000110100001110010101111011100000010010010100111101010001110010001011100111000100101011010110011100101101100
Pair \(Z_2\) Length of longest common subsequence
9CST_1,8PJI_1 153 3
9CST_1,9CYD_1 164 4
8PJI_1,9CYD_1 191 3

Newick tree

 
[
	9CYD_1:92.80,
	[
		9CST_1:76.5,8PJI_1:76.5
	]:16.30
]

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 \((0.85)(0.8)(\frac{313 }{\log_{20} 313}-\frac{154}{\log_{20}154})=48.6\)
Status Protein1 Protein2 d d1/2
Query variables 9CST_1 8PJI_1 62 61
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]