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Parikh vectors
6UBI_1 4NCQ_1 8DPO_1 Letter Amino acid
15 8 19 L Leucine
21 7 20 V Valine
7 3 10 R Arginine
6 8 14 E Glutamic acid
1 1 5 M Methionine
4 2 4 W Tryptophan
15 16 11 A Alanine
4 2 18 N Asparagine
5 2 2 C Cysteine
7 5 17 H Histidine
33 5 16 S Serine
20 5 19 T Threonine
9 4 8 Y Tyrosine
11 9 7 D Aspartic acid
9 7 9 Q Glutamine
22 4 19 G Glycine
6 5 7 I Isoleucine
15 13 14 K Lycine
7 4 8 F Phenylalanine
12 7 16 P Proline 

6UBI_1|Chains A, D|VRC34.05 heavy chain|Homo sapiens (9606)
>4NCQ_1|Chains A, B, C|Superoxide dismutase [Ni]|Streptomyces coelicolor (1902)
>8DPO_1|Chains A, B[auth C], C[auth E], D[auth G]|Carbonic anhydrase|Neisseria gonorrhoeae (485)
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 \(w=f(c)\)
6UBI , Knot 102 229 0.80 40 144 221 
SQYLAQSGSEVKKPGASVKVSCRLHGDKFGAHYIHWVRQAPGQGLEWLGFFHPGTGDAVSAQKFQGRISLTRDTSISTGYLEVRRLSSDDTAVYYCARDKTYSDEATGMDIWGQGTTVIVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPKSCDKTH
4NCQ , Knot 60 117 0.81 40 97 112 
ACDLPCGVYDPAQARIEAESVKAVQEKMAGNDDPHFQTRATVIKEQRAELAKHHVSVLWSDYFKPPHFEKYPELHQLVNDTLKALSAAKGSKDPATGQKALDYIAQIDKIFWETKKA
8DPO , Knot 106 243 0.79 40 164 229 
HHHHHHDSGLVPRGSHMHGNHTHWGYTGHDSPESWGNLSEEFRLCSTGKNQSPVNITETVSGKLPAIKVNYKPSMVDVENNGHTIQVNYPEGGNTLTVNGRTYTLKQFHFHVPSENQIKGRTFPMEAHFVHLDENKQPLVLAVLYEAGKTNGRLSSIWNVMPMTAGKVKLNQPFDASTLLPKRLKYYRFAGSLTTPPCTEGVSWLVLKTYDHIDQAQAEKFTRAVGSENNRPVQPLNARVVIE

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 \(c\).

\(|P_{f(6UBI_1)}(2) \setminus P_{f(4NCQ_1)}(2)|=98\), \(|P_{f(4NCQ_1)}(2) \setminus P_{f(6UBI_1)}(2)|=51\). 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 1:0001100100100111010100010100111001011001110110111110110101101001010101000001001010100100000110001000000001011011101001110010001101111110000001101111011000110110101001110011001111100011001001101100011000010010001000010001010000000
Pair \(Z_2\) Length of longest common subsequence
6UBI_1,4NCQ_1 149 3
6UBI_1,8DPO_1 166 4
4NCQ_1,8DPO_1 169 3

Newick tree

 
[
	8DPO_1:86.61,
	[
		6UBI_1:74.5,4NCQ_1:74.5
	]:12.11
]

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 \((0.85)(0.8)(\frac{346 }{\log_{20} 346}-\frac{117}{\log_{20}117})=70.5\)
Status Protein1 Protein2 d d1/2
Query variables 6UBI_1 4NCQ_1 88 66.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]