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Parikh vectors
9MLK_1 4RYQ_1 6YRR_1 Letter Amino acid
9 10 2 H Histidine
11 14 12 I Isoleucine
39 10 6 T Threonine
10 7 4 W Tryptophan
16 8 4 D Aspartic acid
11 1 0 C Cysteine
30 9 10 K Lycine
55 16 14 S Serine
19 10 9 A Alanine
27 5 13 G Glycine
35 23 11 L Leucine
36 6 7 P Proline
11 1 4 R Arginine
18 4 6 Q Glutamine
22 2 6 E Glutamic acid
4 6 6 M Methionine
14 19 8 F Phenylalanine
20 13 5 Y Tyrosine
42 9 9 V Valine
20 8 12 N Asparagine 

9MLK_1|Chains A[auth H], C[auth I], E[auth J]|Kenv-4 Fab Heavy Chain|Mus musculus (10090)
>4RYQ_1|Chain A|Integral membrane protein|Bacillus cereus (226900)
>6YRR_1|Chain A|Profilin|Candidatus Heimdallarchaeota archaeon LC_3 (1841598)
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 \(w=f(c)\)
9MLK , Knot 185 449 0.83 40 223 415 
DVQLQESGPGLVKPSQSLSLTCTVTGYSITSDYAWNWIRQFPGNELEWMGYISYSGSTSYNPSLKSRISITRDTSKNQFFLQLNPVTTEDTATYYCARSVILGAWFAYWGQGTLVTVSAASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKRVEPKSCDKTHTCPPCPAPELLGGPSVFLFPPKPKDTLMISRTPEVTCVVVDVSHEDPEVKFNWYVDGVEVHNAKTKPREEQYNSTYRVVSVLTVLHQDWLNGKEYKCKVSNKALPAPIEKTISKAKGQPREPQVYTLPPSREEMTKNQVSLTCLVKGFYPSDIAVEWESNGQPENNYKTTPPVLDSDGSFFLYSKLTVDKSRWQQGNVFSCSVMHEALHNHYTQKSLSLSPGK
4RYQ , Knot 79 181 0.75 40 121 165 
MDYKDDDDKHHHHHHHHHHENLYFQSYVMFMKKSSIIVFFLTYGLFYVSSVLFPIDRTWYDALEKPSWTPPGMTIGMIWAVLFGLIALSVAIIYNNYGFKPKTFWFLFLLNYIFNQAFSYFQFSQKNLFLATVDCLLVAITTLLLIMFSSNLSKVSAWLLIPYFLWSAFATYLSWTIYSIN
6YRR , Knot 76 148 0.85 38 119 144 
MKDGFILKKKKIPKNIFRIGNKEMSYSDDLNSLIQSQYVSYVVILDPNGAIYWTNNENWQVNGSEVLRQWMGSAPSITVAGTKFSSFRNEPGVSFVGRNMAGGGLIILQKAPNGYVFLSWTSHEFLTTSGLPPLNIHAEIAMMAAKFQ

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 \(c\).

\(|P_{f(9MLK_1)}(2) \setminus P_{f(4RYQ_1)}(2)|=143\), \(|P_{f(4RYQ_1)}(2) \setminus P_{f(9MLK_1)}(2)|=41\). 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 1:01010001111101000101000101001000011011001110010111010001000001010001010000000011101011000001000010011111111011010110101100011011111100000011011110110001101101010011100110011111000110010011011000110000100100010000100010100000000011011101111101111110100011100010100111010000101010101011010010001000000000011011011000110100000010001111110001001010100101001110000100001010011011010011101000101000000011110001011100010100001001011000110011000000001010110
Pair \(Z_2\) Length of longest common subsequence
9MLK_1,4RYQ_1 184 3
9MLK_1,6YRR_1 182 3
4RYQ_1,6YRR_1 148 4

Newick tree

 
[
	9MLK_1:96.63,
	[
		6YRR_1:74,4RYQ_1:74
	]:22.63
]

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 \((0.85)(0.8)(\frac{630 }{\log_{20} 630}-\frac{181}{\log_{20}181})=128.\)
Status Protein1 Protein2 d d1/2
Query variables 9MLK_1 4RYQ_1 163 111.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]