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Parikh vectors
2ZAF_1 9KUF_1 2KNE_1 Letter Amino acid
17 14 6 Q Glutamine
43 18 9 L Leucine
14 2 8 F Phenylalanine
20 9 17 D Aspartic acid
10 5 1 H Histidine
21 19 8 I Isoleucine
23 16 2 P Proline
32 15 21 E Glutamic acid
19 10 6 R Arginine
7 3 0 C Cysteine
14 10 9 M Methionine
29 15 4 S Serine
29 9 12 T Threonine
9 8 2 Y Tyrosine
31 15 7 V Valine
46 21 11 A Alanine
29 15 11 G Glycine
28 9 8 K Lycine
7 1 0 W Tryptophan
11 5 6 N Asparagine 

2ZAF_1|Chains A, B, C, D|Nitroalkane oxidase|Fusarium oxysporum (5507)
>9KUF_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N|ATP-dependent Clp protease proteolytic subunit, mitochondrial|Homo sapiens (9606)
>2KNE_1|Chain A|Calmodulin|Homo sapiens (9606)
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 \(w=f(c)\)
2ZAF , Knot 185 439 0.85 40 242 417 
MVDFKLSPSQLEARRHAQAFANTVLTKASAEYSTQKDQLSRFQATRPFYREAVRHGLIKAQVPIPLGGTMESLVHESIILEELFAVEPATSITIVATALGLMPVILCDSPSLQEKFLKPFISGEGEPLASLMHSEPNGTANWLQKGGPGLQTTARKVGNEWVISGEKLWPSNSGGWDYKGADLACVVCRVSDDPSKPQDPNVDPATQIAVLLVTRETIANNKKDAYQILGEPELAGHITTSGPHTRFTEFHVPHENLLCTPGLKAQGLVETAFAMSAALVGAMAIGTARAAFEEALVFAKSDTRGGSKHIIEHQSVADKLIDCKIRLETSRLLVWKAVTTLEDEALEWKVKLEMAMQTKIYTTDVAVECVIDAMKAVGMKSYAKDMSFPRLLNEVMCYPLFDGGNIGLKRRQMQRVMALEDYEPWAATYGSSKVDKSRL
9KUF , Knot 104 219 0.85 40 157 215 
IPIVVEQTGRGERAYDIYSRLLRERIVCVMGPIDDSVASLVIAQLLFLQSESNKKPIHMYINSPGGVVTAGLAIYDTMQYILNPICTWCVGQAASMGSLLLAAGTPGMRHSLPNSRIMIHQPSGGARGQATDIAIQAEEIMKLKKQLYNIYAKHTKQSLQVIESAMERDRYMSPMEAQEFGILDKVLVHPPQDGEDEPTLVQKEPVEAAPAAEPVPAST
2KNE , Knot 68 148 0.76 36 97 133 
ADQLTEEQIAEFKEAFSLFDKDGDGTITTKELGTVMRSLGQNPTEAELQDMINEVDADGNGTIDFPEFLTMMARKMKDTDSEEEIREAFRVFDKDGNGYISAAELRHVMTNLGEKLTDEEVDEMIREADIDGDGQVNYEEFVQMMTAK

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 \(c\).

\(|P_{f(2ZAF_1)}(2) \setminus P_{f(9KUF_1)}(2)|=131\), \(|P_{f(9KUF_1)}(2) \setminus P_{f(2ZAF_1)}(2)|=46\). 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 1:1101010100101000101110011001010000000010010100110001100111010111111101001100011100111101100101110111111111000101000110111010101110110001010101100111110001001100111010011100011100011011011001000100100101011001111110000110000010011101011101000110001001011000110011101011100111101111111111010111001111100000110001100001100110001010000111101100100011010101011100010000111001101101111000100101101100110011101101110000100111100001111001000100001
Pair \(Z_2\) Length of longest common subsequence
2ZAF_1,9KUF_1 177 4
2ZAF_1,2KNE_1 197 4
9KUF_1,2KNE_1 160 3

Newick tree

 
[
	2ZAF_1:97.75,
	[
		9KUF_1:80,2KNE_1:80
	]:17.75
]

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 \((0.85)(0.8)(\frac{658 }{\log_{20} 658}-\frac{219}{\log_{20}219})=123.\)
Status Protein1 Protein2 d d1/2
Query variables 2ZAF_1 9KUF_1 159 118
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]