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Parikh vectors
8ACW_1 5YKD_1 6GCP_1 Letter Amino acid
24 25 12 P Proline
32 57 35 A Alanine
17 18 10 Q Glutamine
19 20 14 I Isoleucine
33 27 25 S Serine
3 9 1 W Tryptophan
20 31 13 R Arginine
13 12 17 N Asparagine
46 44 24 L Leucine
28 10 10 K Lycine
17 18 7 F Phenylalanine
28 27 13 T Threonine
11 12 6 Y Tyrosine
24 24 10 D Aspartic acid
30 31 15 E Glutamic acid
16 20 13 H Histidine
48 32 27 V Valine
5 6 7 C Cysteine
41 45 22 G Glycine
13 11 7 M Methionine 

8ACW_1|Chain A|Na(+)-translocating NADH-quinone reductase subunit A|Vibrio cholerae (666)
>5YKD_1|Chains A, B, C, D|D-hydantoinase/dihydropyrimidinase|Pseudomonas aeruginosa (strain ATCC 15692 / DSM 22644 / CIP 104116 / JCM 14847 / LMG 12228 / 1C / PRS 101 / PAO1) (208964)
>6GCP_1|Chains A, B, C, D|Pteridine reductase|Trypanosoma brucei brucei (5702)
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 \(w=f(c)\)
8ACW , Knot 188 468 0.82 40 228 436 
MGSSHHHHHHSSGLEVLFQGPHMITIKKGLDLPIAGTPSQVISDGKAIKKVALLGEEYVGMRPTMHVRVGDEVKKAQILFEDKKNPGVKFTSPVSGKVVEINRGAKRVLQSVVIEVAGDDQVTFDKFEANQLASLNRDAIKTQLVESGLWTAFRTRPFSKVPAIDSTSEAIFVTAMDTNPLAAEPTVVINEQSEAFVAGLDVLSALTTGKVYVCKKGTSLPRSQQPNVEEHVFDGPHPAGLAGTHMHFLYPVSADHVAWSINYQDVIAVGQLFLTGELYTQRVVSLAGPVVNKPRLVRTVMGASLEQLVDSEIMPGEVRIISGSVLSGTKATGPHAYLGRYHLQVSVLREGRDKELFGWAMPGKNKFSVTRSFLGHLFKGQVYNMTTTTNGSDRSMVPIGNYEKVMPLDMEPTLLLRDLCAGDSDSAVRLGALELDEEDLALCTFVCPGKYEYGQLLRECLDKIEKEG
5YKD , Knot 199 479 0.85 40 244 450 
MSLLIRGATVVTHEESYRADVLCANGLIQAIGENLETPSGCDVLDGGGQYLMPGGIDPHTHMQLPFMGTVASEDFFSGTAAGLAGGTTSIIDFVIPNPRQSLLEAFHTWRGWAQKSAADYGFHVAITWWSDEVAREMGELVAQHGVNSFKHFMAYKNAIMAADDTLVASFERCLELGAVPTVHAENGELVFHLQQKLLAQGLTGPEAHPLSRPPQVEGEAASRAIRIAETLGTPLYLVHISSREALDEIAYARAKGQPVYGEVLAGHLLLDDSVYRHPDWATAAGYVMSPPFRPVEHQEALWRGLQSGNLHTTATDHCCFCAEQKAMGRDDFSKIPNGTAGIEDRMALLWDAGVNSGRLSMHEFVALTSTNTAKIFNLFPRKGAIRVGADADLVLWDPQGSRTLSAATHHQRVDFNIFEGRTVRGIPSHTISQGKLLWAAGDLRAEPGAGRYVERPAYPSVYEVLGRRAERQRPVAVER
6GCP , Knot 127 288 0.83 40 179 275 
MGSSHHHHHHSSGLVPRGSHMEAPAAVVTGAAKRIGRAIAVKLHQTGYRVVIHYHNSAEAAVSLADELNKERSNTAVVCQADLTNSNVLPASCEEIINSCFRAFGRCDVLVNNASAFYPTPLVQGDHEDNSNGKTVETQVAELIGTNAIAPFLLTMSFAQRQKGTNPNCTSSNLSIVNLCDAMVDQPCMAFSLYNMGKHALVGLTQSAALELAPYGIRVNGVAPGVSLLPVAMGEEEKDKWRRKVPLGRREASAEQIADAVIFLVSGSAQYITGSIIKVDGGLSLVHA

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 \(c\).

\(|P_{f(8ACW_1)}(2) \setminus P_{f(5YKD_1)}(2)|=59\), \(|P_{f(5YKD_1)}(2) \setminus P_{f(8ACW_1)}(2)|=75\). 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 1:110000000000110111011011010011011111010011001011001111100011101010101100100101110000011101001101011010011001100111011100010100101001101000110001100111011000110011110000011110110001111010111000001111110110110010101000100110000101000110110111111001011011010011101000011111011101010000110111111001011001111010011000111101011010110100101101011000101011001000011111111000101000111011010100100000100001111100001111010101110010110000110111101000011100110110000101100010010001
Pair \(Z_2\) Length of longest common subsequence
8ACW_1,5YKD_1 134 5
8ACW_1,6GCP_1 155 14
5YKD_1,6GCP_1 165 4

Newick tree

 
[
	6GCP_1:83.93,
	[
		8ACW_1:67,5YKD_1:67
	]:16.93
]

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 \((0.85)(0.8)(\frac{947 }{\log_{20} 947}-\frac{468}{\log_{20}468})=126.\)
Status Protein1 Protein2 d d1/2
Query variables 8ACW_1 5YKD_1 161 156.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]