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Parikh vectors
3GBF_1 3UAM_1 6HBW_1 Letter Amino acid
59 21 21 A Alanine
34 15 3 R Arginine
39 16 7 G Glycine
47 14 18 L Leucine
5 3 2 M Methionine
30 15 9 T Threonine
13 9 3 Y Tyrosine
2 0 1 C Cysteine
17 7 10 H Histidine
27 17 11 S Serine
15 7 4 N Asparagine
32 16 8 D Aspartic acid
7 10 1 Q Glutamine
7 7 11 K Lycine
10 7 7 F Phenylalanine
26 10 4 E Glutamic acid
12 8 0 I Isoleucine
27 16 7 P Proline
11 7 1 W Tryptophan
23 11 13 V Valine 

3GBF_1|Chain A|PhpD|Streptomyces viridochromogenes (1938)
>3UAM_1|Chains A, B, C, D, E, F|Chitin binding domain|Burkholderia pseudomallei (320372)
>6HBW_1|Chains A, C|PROTEIN (HEMOGLOBIN ALPHA 1)|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 \(w=f(c)\)
3GBF , Knot 177 443 0.81 40 214 405 
MRIDPFKLAHWMNARKYTAAQTADLAGLPLDDLRRLLGDEANEPDPAAATALAEALSVEPSQLAADAHRNLTVVHKSAEEMHASRRPIQRDGIHFYNYYTLAAPEGRVAPVVLDILCPSDRLPALNNGHLEPAITVNLGPGDINGRWGEEITPQTWRVLHANHGGDRWITGDSYVEPSYCPHSYSLAGDAPARIVSYTAQSNISPLMTEANNWSTGAFEEALKALSGKVSAGSVLDLFLARRAHTRTSAAEAAGVPPADLEAALRSPASETGLTVLRTLGRALGFDYRVLLPADDQHDGVGKTWTTIEDSRRSRRTFGTYEAASMASAAHLPDLVGSFLRVDADGRGADLIDHAENHYVVTEGRLTLEWDGPDGPASVELEPDGSAWTGPFVRHRWHGTGTVLKFGSGAHLGYQDWLELTNTFEPAATLRRGRRDLAGWGYDN
3UAM , Knot 100 216 0.83 38 155 209 
GSMSTTDSRSATKSSSISPRHGRVITPESRAVYLYEAGRLDFGQVNELEGGKFFPATQSGLRDPDAPDDVANGMPPRDGEIASGGRTADARAQLNEPDSVAHWQKHAVRSGQSLQISWSYSMPHKTRRWTYWITKPGWDTQARLARAHFEPDPLKVYLNTYQPYWGPDADKELIPQGETIHEFNLPTRTGYHVLLAVWDVADTANAFYQVIDLNFA
6HBW , Knot 69 141 0.80 38 99 137 
VLSPADKTNVKAAWGKVGAHAGEYGAEALERMFLSFPTTKTYFPHFDLSHGSAQVKGHGKKVADALTNAVAHVDDMPNALSALSDLHAHKLRVDPVNFKLLSHCLLVTLAAHLPAEFTPAVHASLDKFLASVSTVLTSKYR

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 \(c\).

\(|P_{f(3GBF_1)}(2) \setminus P_{f(3UAM_1)}(2)|=106\), \(|P_{f(3UAM_1)}(2) \setminus P_{f(3GBF_1)}(2)|=47\). 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 1:10101101101101000011001011111100100111001001011110111011010100111010001011000100101000110001101000001111010111111011010001111001010111010111101010110010100101101001100110100010100010000111011101100010001011100100100111001101101010110110111100100000110111111101011100110001101100110111100011111000001110010010000000001100011011011011011101101010101101100100001100101010101101110101010101101111000101010110110110110001101000101110100100011111000
Pair \(Z_2\) Length of longest common subsequence
3GBF_1,3UAM_1 153 4
3GBF_1,6HBW_1 169 5
3UAM_1,6HBW_1 150 3

Newick tree

 
[
	3GBF_1:82.38,
	[
		3UAM_1:75,6HBW_1:75
	]:7.38
]

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 \((0.85)(0.8)(\frac{659 }{\log_{20} 659}-\frac{216}{\log_{20}216})=124.\)
Status Protein1 Protein2 d d1/2
Query variables 3GBF_1 3UAM_1 153 113.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]