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Parikh vectors
5KLB_1 1GXT_1 2EUI_1 Letter Amino acid
9 6 4 P Proline
18 3 4 T Threonine
4 2 1 W Tryptophan
12 5 8 D Aspartic acid
0 3 1 C Cysteine
8 8 7 Q Glutamine
17 6 15 E Glutamic acid
4 2 4 H Histidine
11 2 8 K Lycine
28 10 11 V Valine
12 8 14 R Arginine
10 3 3 N Asparagine
29 9 21 L Leucine
18 3 13 S Serine
27 5 7 F Phenylalanine
9 1 8 Y Tyrosine
10 3 11 A Alanine
11 7 3 G Glycine
36 4 6 I Isoleucine
12 1 4 M Methionine 

5KLB_1|Chains A, B, C, D|Ion transport protein|Arcobacter butzleri (strain RM4018) (367737)
>1GXT_1|Chain A|HYDROGENASE MATURATION PROTEIN HYPF|ESCHERICHIA COLI (562)
>2EUI_1|Chains A, B, C[auth D], D[auth E]|Probable acetyltransferase|Pseudomonas aeruginosa PAO1 (208964)
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 \(w=f(c)\)
5KLB , Knot 125 285 0.82 38 169 264 
MDYKDDDDKGSLVPRGSHMYLRITNIVESSFFTKFIIYLIVLNGITMGLETSKTFMQSFGVYTTLFNQIVITIFTIEIILRIYVHRISFFKDPWSLFDFFVVAISLVPTSSGFEILRVLRVLRLFRLVTAVPQMRKIVSALISVIPGMLSVIALMTLFFYIFAIMATQLFGERFPEWFGTLGESFYTLFQVMTLDDWSNGIVRPLMEVYPYAWVFFIPFIFVVTFVMINLVVAIIVDAMAILNQKEEQHIIDEVQSHEDNINNEIIKLREEIVELKELIKTSLKN
1GXT , Knot 50 91 0.82 40 72 88 
MAKNTSCGVQLRIRGKVQGVGFRPFVWQLAQQLNLHGDVCNDGDGVEVRLREDPEVFLVQLYQHCPPLARIDSVEREPFIWSQLPTEFTIR
2EUI , Knot 76 153 0.83 40 117 151 
MRIVQATLEHLDLLAPLFVKYREFYGMLSYPESSRKFLEKRLRRKESVIYLALADEEDRLLGFCQLYPSFSSLSLKRVWILNDIYVAEEARRQLVADHLLQHAKQMARETHAVRMRVSTSVDNEVAQKVYESIGFREDQEFKNYTLPISDELS

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 \(c\).

\(|P_{f(5KLB_1)}(2) \setminus P_{f(1GXT_1)}(2)|=133\), \(|P_{f(1GXT_1)}(2) \setminus P_{f(5KLB_1)}(2)|=36\). 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 1:100000000101110100101010011000110011101111011011100000110011100011001110110101110101001011001101101111110111000110110110110110110111010011011101111110111110111011111100111001101110110010011011010010011101110101011111111111101111011111110111110000000110010000001000110100011010011000100
Pair \(Z_2\) Length of longest common subsequence
5KLB_1,1GXT_1 169 3
5KLB_1,2EUI_1 156 4
1GXT_1,2EUI_1 125 3

Newick tree

 
[
	5KLB_1:86.68,
	[
		2EUI_1:62.5,1GXT_1:62.5
	]:24.18
]

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 \((0.85)(0.8)(\frac{376 }{\log_{20} 376}-\frac{91}{\log_{20}91})=88.0\)
Status Protein1 Protein2 d d1/2
Query variables 5KLB_1 1GXT_1 109 71.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]