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Parikh vectors
2BZM_1 8GGK_1 5SHQ_1 Letter Amino acid
6 32 21 A Alanine
10 18 14 R Arginine
1 9 12 M Methionine
3 8 7 W Tryptophan
3 21 15 D Aspartic acid
8 18 18 K Lycine
7 17 12 Y Tyrosine
9 44 36 L Leucine
10 14 14 P Proline
6 22 14 N Asparagine
10 22 24 E Glutamic acid
8 29 16 G Glycine
6 34 21 I Isoleucine
11 31 24 S Serine
8 27 22 T Threonine
5 39 16 V Valine
8 12 12 C Cysteine
5 23 17 Q Glutamine
2 11 13 H Histidine
3 29 15 F Phenylalanine 

2BZM_1|Chain A|COMPLEMENT FACTOR H|HOMO SAPIENS (9606)
>8GGK_1|Chain A[auth R]|Beta-2 adrenergic receptor|Homo sapiens (9606)
>5SHQ_1|Chains A, B, C, D|cAMP and cAMP-inhibited cGMP 3',5'-cyclic phosphodiesterase 10A|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 \(w=f(c)\)
2BZM , Knot 64 129 0.80 40 101 125 
EAEFGKCGPPPPIDNGDITSFPLSVYAPASSVEYQCQNLYQLEGNKRITCRNGQWSEPPKCLHPCVISREIMENYNIALRWTAKQKLYSRTGESVEFVCKRGYRLSSRSHTLRTTCWDGKLEYPTCAKR
8GGK , Knot 192 460 0.85 40 252 435 
MKTIIALSYIFCLVFADYKDDDDAMGQPGNGSAFLLAPNRSHAPDHDVENLYFQGTQQRDEVWVVGMGIVMSLIVLAIVFGNVLVITAIAKFERLQTVTNYFITSLACADLVMGLAVVPFGAAHILTKTWTFGNFWCEFWTSIDVLCVTASIETLCVIAVDRYFAITSPFKYQSLLTKNKARVIILMVWIVSGLTSFLPIQMHWYRATHQEAINCYAEETCCDFFTNQAYAIASSIVSFYVPLVIMVFVYSRVFQEAKRQLQKIDKSEGRFHVQNLSQVEQDGRTGHGLRRSSKFCLKEHKALKTLGIIMGTFTLCWLPFFIVNIVHVIQDNLIRKEVYILLNWIGYVNSGFNPLIYCRSPDFRIAFQELLCLRRSSLKAYGNGYSSNGNTGEQSGLEVLFQGPYHVEQEKENKLLAEDLPGTEDFVGHQGTVPSDNIDSQGRNASTNDSLLETSQVAPA
5SHQ , Knot 152 343 0.86 40 221 334 
GSSICTSEEWQGLMQFTLPVRLCKEIELFHFDIGPFENMWPGIFVYMVHRSCGTSCFELEKLCRFIMSVKKNYRRVPYHNWKHAVTVAHCMYAILQNNHTLFTDLERKGLLIACLCHDLDHRGFSNSYLQKFDHPLAALYSTSTMEQHHFSQTVSILQLEGHNIFSTLSSSEYEQVLEIIRKAIIATDLALYFGNRKQLEEMYQTGSLNLNNQSHRDRVIGLMMTACDLCSVTKLWPVTKLTANDIYAEFWAEGDEMKKLGIQPIPMMDRDKKDEVPQGQLGFYNAVAIPCYTTLTQILPPTEPLLKACRDNLSQWEKVIRGEETATWISSPSVAQKAAASED

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 \(c\).

\(|P_{f(2BZM_1)}(2) \setminus P_{f(8GGK_1)}(2)|=34\), \(|P_{f(8GGK_1)}(2) \setminus P_{f(2BZM_1)}(2)|=185\). 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 1:010110011111100101001110101110010000001001010001000010100110010101100011000011101010001000010010110001001000000100001010100100100
Pair \(Z_2\) Length of longest common subsequence
2BZM_1,8GGK_1 219 3
2BZM_1,5SHQ_1 194 4
8GGK_1,5SHQ_1 177 4

Newick tree

 
[
	2BZM_1:10.96,
	[
		5SHQ_1:88.5,8GGK_1:88.5
	]:19.46
]

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 \((0.85)(0.8)(\frac{589 }{\log_{20} 589}-\frac{129}{\log_{20}129})=134.\)
Status Protein1 Protein2 d d1/2
Query variables 2BZM_1 8GGK_1 174 110
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]