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Parikh vectors
3ZUJ_1 4AIL_1 2XWH_1 Letter Amino acid
9 0 11 H Histidine
10 0 33 P Proline
14 0 25 Y Tyrosine
10 0 14 N Asparagine
16 0 25 D Aspartic acid
4 2 14 C Cysteine
20 4 33 G Glycine
19 0 29 K Lycine
10 0 15 F Phenylalanine
7 1 43 T Threonine
7 0 40 R Arginine
4 0 12 Q Glutamine
15 0 30 E Glutamic acid
12 0 25 I Isoleucine
13 0 55 L Leucine
6 0 15 M Methionine
12 0 47 S Serine
2 0 9 W Tryptophan
11 4 54 A Alanine
15 0 34 V Valine 

3ZUJ_1|Chains A, B, C, D, E, F|FLUORESCENT PROTEIN DRONPA|ECHINOPHYLLIA SP. SC22 (301887)
>4AIL_1|Chain A|5'-D(*AP*CP*GP*GP*GP*TP*AP*AP*GP*CP*AP)-3'|SYNTHETIC CONSTRUCT (32630)
>2XWH_1|Chain A|RNA DEPENDENT RNA POLYMERASE|HEPATITIS C VIRUS ISOLATE HC-J6 (11113)
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 \(w=f(c)\)
3ZUJ , Knot 102 216 0.84 40 159 209 
SVIKPDMKIKLRMEGAVNGHPFAIEGVGLGKPFEGKQSMDLKVKEGGPLPFAYDILTMAFCYGNRVFAKYPENIVDYFKQSFPEGYSWERSMIYEDGGICNATNDITLDGDCYIYEIRFDGVNFPANGPVMQKRTVKWELSTEKLYVRDGVLKSDGNYALSLEGGGHYRCDFKTTYKAKKVVQLPDYHSVDHHIEIKSHDKDYSNVNLHEHAEAHS
4AIL , Knot 7 11 0.50 8 9 9 
ACGGGTAAGCA
2XWH , Knot 223 563 0.83 40 253 529 
SMSYSWTGALITPCSPEEEKLPINPLSNSLLRYHNKVYCTTTKSASLRAKKVTFDRMQVLDSYYDSVLKDIKLAASKVTARLLTMEEACQLTPPHSARSKYGFGAKEVRSLSGRAVNHIKSVWKDLLEDSETPIPTTIMAKNEVFCVDPTKGGKKAARLIVYPDLGVRVCEKMALYDITQKLPQAVMGASYGFQYSPAQRVEFLLKAWAEKKDPMGFSYDTRCFDSTVTERDIRTEESIYRACSLPEEAHTAIHSLTERLYVGGPMFNSKGQTCGYRRCRASGVLTTSMGNTITCYVKALAACKAAGIIAPTMLVCGDDLVVISESQGTEEDERNLRAFTEAMTRYSAPPGDPPRPEYDLELITSCSSNVSVALGPQGRRRYYLTRDPTTPIARAAWETVRHSPVNSWLGNIIQYAPTIWARMVLMTHFFSILMAQDTLDQNLNFEMYGAVYSVSPLDLPAIIERLHGLDAFSLHTYTPHELTRVASALRKLGAPPLRAWKSRARAVRASLISRGGRAAVCGRYLFNWAVKTKLKLTPLPEARLLDLSSWFTVGAGGGDIYHS

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 \(c\).

\(|P_{f(3ZUJ_1)}(2) \setminus P_{f(4AIL_1)}(2)|=158\), \(|P_{f(4AIL_1)}(2) \setminus P_{f(3ZUJ_1)}(2)|=8\). 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 1:011010101010101110101111011111011010001010100111111100110111001001110010011001000110100100011000111001000101010001001010110111011110000101010000101001110001001101011100000100000100110110000100010100000000010100010100
Pair \(Z_2\) Length of longest common subsequence
3ZUJ_1,4AIL_1 166 3
3ZUJ_1,2XWH_1 196 4
4AIL_1,2XWH_1 248 3

Newick tree

 
[
	2XWH_1:11.82,
	[
		3ZUJ_1:83,4AIL_1:83
	]:36.82
]

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 \((0.85)(0.8)(\frac{227 }{\log_{20} 227}-\frac{11}{\log_{20}11})=75.8\)
Status Protein1 Protein2 d d1/2
Query variables 3ZUJ_1 4AIL_1 100 52.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]