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Parikh vectors
6MCJ_1 3LHX_1 9FMD_1 Letter Amino acid
7 22 7 V Valine
8 9 3 N Asparagine
5 16 3 I Isoleucine
8 22 13 S Serine
2 5 1 W Tryptophan
5 12 3 Y Tyrosine
0 5 0 C Cysteine
10 23 9 G Glycine
5 15 26 K Lycine
5 6 8 M Methionine
10 18 5 D Aspartic acid
19 30 11 L Leucine
10 9 16 P Proline
9 20 4 T Threonine
11 10 2 H Histidine
6 8 4 F Phenylalanine
13 36 18 A Alanine
7 17 15 R Arginine
14 12 12 Q Glutamine
9 24 24 E Glutamic acid 

6MCJ_1|Chains A, B|Orange carotenoid-binding protein|Tolypothrix sp. PCC 7601 (1188)
>3LHX_1|Chains A, B|Ketodeoxygluconokinase|Shigella flexneri (623)
>9FMD_1|Chain A[auth 1]|PRKR-interacting protein 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 \(w=f(c)\)
6MCJ , Knot 77 163 0.80 38 115 150 
MTYTVDESTRPALERFQRFDVDTQLALLWYGYLDLKPQLNPAPPNSVDTPARAVFDHIQDLSQQEQLQAQRDLIKGGSGEINRGYNALSPNAKLEVWLLLAQGMENGTIIPMPSDYQLPNGTEEFTAQVKKLEFDQRLNFMLTAVQAMGGSSAHHHHHHHHHH
3LHX , Knot 135 319 0.81 40 179 301 
MSLSKKIAVIGECMIELSEKGADVKRGFGGDTLNTSVYIARQVDPAALTVHYVTALGTDSFSQQMLDAWHGENVDTSLTQRMENRLPGLYYIETDSTGERTFYYWRNEAAAKFWLASEQSAAICEELANFDYLYLSGISLAILSPTSREKLLSLLRECRAKGGKVIFDNNYRPRLWASKEETQQVYQQMLECTDIAFLTLDDEDALWGQQPVEDVIARTHNAGVKEVVVKRGADSCLVSIAGEALVDVPAVKLPKEKVIDTTAAGDSFSAGYLAVRLTGGSAENAAKRGHLTASTVIQYRGAIIPREAMPAEGHHHHHH
9FMD , Knot 78 184 0.73 38 108 167 
MASPAASSVRPPRPKKEPQTLVIPKNAAEEQKLKLERLMKNPDKAVPIPEKMSEWAPRPPPEFVRDVMGSSAGAGSGEFHVYRHLRRREYQRQDYMDAMAEKQKLDAEFQKRLEKNKIAAEEQTAKRRKKRQKLKEKKLLAKKMKLEQKKQEGPGQPKEQGSSSSAEASGTEEEEEVPSFTMGR

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 \(c\).

\(|P_{f(6MCJ_1)}(2) \setminus P_{f(3LHX_1)}(2)|=51\), \(|P_{f(3LHX_1)}(2) \setminus P_{f(6MCJ_1)}(2)|=115\). 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 1:1000100000111001001010001111101010101010111100100110111001001000001010001101101010010011010101011111101100101111100001101000101010010100010111011011110010000000000
Pair \(Z_2\) Length of longest common subsequence
6MCJ_1,3LHX_1 166 6
6MCJ_1,9FMD_1 131 4
3LHX_1,9FMD_1 161 3

Newick tree

 
[
	3LHX_1:86.50,
	[
		6MCJ_1:65.5,9FMD_1:65.5
	]:21.00
]

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 \((0.85)(0.8)(\frac{482 }{\log_{20} 482}-\frac{163}{\log_{20}163})=93.7\)
Status Protein1 Protein2 d d1/2
Query variables 6MCJ_1 3LHX_1 116 87
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]