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Parikh vectors
4KQK_1 5XDM_1 2KYA_1 Letter Amino acid
22 5 2 P Proline
38 9 2 G Glycine
13 4 2 N Asparagine
15 5 0 D Aspartic acid
7 2 0 C Cysteine
10 6 5 Q Glutamine
8 2 2 K Lycine
1 1 0 W Tryptophan
17 7 1 R Arginine
8 8 0 H Histidine
35 12 6 L Leucine
19 3 1 M Methionine
6 2 0 F Phenylalanine
14 6 3 S Serine
5 4 0 Y Tyrosine
15 7 3 E Glutamic acid
18 8 2 I Isoleucine
14 7 1 T Threonine
35 8 1 V Valine
56 13 3 A Alanine 

4KQK_1|Chains A, B|Nicotinate-nucleotide--dimethylbenzimidazole phosphoribosyltransferase|Salmonella enterica subsp. enterica serovar Typhimurium (99287)
>5XDM_1|Chains A, B|Septum site-determining protein MinC|Escherichia coli K-12 (83333)
>2KYA_1|Chain A|Patellamide protein|Prochloron didemni (1216)
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 \(w=f(c)\)
4KQK , Knot 148 356 0.81 40 188 338 
MQTLHALLRDIPAPDAEAMARAQQHIDGLLKPPGSLGRLETLAVQLAGMPGLNGTPQVGEKAVLVMCADHGVWDEGVAVYPKIVTAIMAANMTRGTTGVCVLAAQAGAKVHVIDVGIDAEPIPGVVNMRVARGCGNIAVGPAMSRLQAEALLLEVSRYTCDLAQRGVTLFGVGEMGMANTTPAAAMVSVFTGSDAKEVVGIGANLPPSRIDNKVDVVRRAIAINQPNPRDGIDVLSKVGGFDLVGMTGVMLGAARCGLPVLLDGFLSYSAALAACQIAPAVRPYLIPSHFSAEKGARIALAHLSMEPYLHMAMRLGEGSGAALAMPIVEAACAMFHNMGELAASNIVLPEGNANAT
5XDM , Knot 61 119 0.81 40 94 112 
PVTKTRLIDTPVRSGQRIYAPQCDLIVTSHVSAGAELIADGNIHVYGMMRGRALAGASGDRETQIFCTNLMAELVSIAGEYWLSDQIPAEFYGKAARLQLVENALTVQPLNLEHHHHHH
2KYA , Knot 22 34 0.76 28 30 32 
MNKKNILPQQGQPVIRLTAGQLSSQLAELSEEAL

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 \(c\).

\(|P_{f(4KQK_1)}(2) \setminus P_{f(5XDM_1)}(2)|=129\), \(|P_{f(5XDM_1)}(2) \setminus P_{f(4KQK_1)}(2)|=35\). 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 1:10010111001111010111010001011101110110100111011111110101011001111101001110011110101101111101001001101111011101011011101011111101011010101111111001010111101000000110011011111011110001111110110100100111111011100100010110011110010100110110011110111101111111001111110111000111110011111010111001010011011110101010101110110101111111110110111001101110011110101010
Pair \(Z_2\) Length of longest common subsequence
4KQK_1,5XDM_1 164 3
4KQK_1,2KYA_1 182 3
5XDM_1,2KYA_1 96 3

Newick tree

 
[
	4KQK_1:96.10,
	[
		5XDM_1:48,2KYA_1:48
	]:48.10
]

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 \((0.85)(0.8)(\frac{475 }{\log_{20} 475}-\frac{119}{\log_{20}119})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 4KQK_1 5XDM_1 131 86
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]