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Parikh vectors
3MYQ_1 5WNA_1 2NTU_1 Letter Amino acid
12 20 18 T Threonine
7 5 8 W Tryptophan
17 23 21 V Valine
19 9 10 D Aspartic acid
17 11 11 P Proline
11 5 4 Q Glutamine
26 18 36 L Leucine
8 7 11 Y Tyrosine
13 13 29 A Alanine
1 5 0 C Cysteine
13 10 9 E Glutamic acid
12 6 0 H Histidine
9 4 15 I Isoleucine
24 14 7 K Lycine
2 3 9 M Methionine
7 8 7 R Arginine
10 8 3 N Asparagine
18 32 13 S Serine
22 22 25 G Glycine
12 8 13 F Phenylalanine 

3MYQ_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>5WNA_1|Chains A[auth C], D[auth H]|mAb 3D3 Fab heavy chain|Homo sapiens (9606)
>2NTU_1|Chain A|Bacteriorhodopsin|Halobacterium salinarum (2242)
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 \(w=f(c)\)
3MYQ , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
5WNA , Knot 101 231 0.79 40 150 218 
EEQLVESGGGLVQPGRSLRLSCVGSGLRFEEHAMHWVRQAPGRGLEWVSGISWNSGSVGYADSVKGRFTTSRDNAKDILFLEMNTLRSEDTALYFCAIMVATTKNDFHYYKDVWGKGTTVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPKSCDKTH
2NTU , Knot 113 249 0.83 36 153 240 
QAQITGRPEWIWLALGTALMGLGTLYFLVKGMGVSDPDAKKFYAITTLVPAIAFTMYLSMLLGYGLTMVPFGGEQNPIYWARYADWLFTTPLLLLDLALLVDADQGTILALVGADGIMIGTGLVGALTKVYSYRFVWWAISTAAMLYILYVLFFGFTSKAESMRPEVASTFKVLRNVTVVLWSAYPVVWLIGSEGAGIVPLNIETLLFMVLDVSAKVGFGLILLRSRAIFGEAEAPEPSAGDGAAATSD

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 \(c\).

\(|P_{f(3MYQ_1)}(2) \setminus P_{f(5WNA_1)}(2)|=83\), \(|P_{f(5WNA_1)}(2) \setminus P_{f(3MYQ_1)}(2)|=57\). 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 1:10001101000110010001111010000110100001000101011010000100101100101101010000001110111101000110101011010101000010000011010110100001011011001011111111101101011100110110010001001010010101111001001001101001111001011110011010000110100101010101001110010110110000101010
Pair \(Z_2\) Length of longest common subsequence
3MYQ_1,5WNA_1 140 4
3MYQ_1,2NTU_1 163 3
5WNA_1,2NTU_1 153 4

Newick tree

 
[
	2NTU_1:81.83,
	[
		3MYQ_1:70,5WNA_1:70
	]:11.83
]

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 \((0.85)(0.8)(\frac{491 }{\log_{20} 491}-\frac{231}{\log_{20}231})=74.9\)
Status Protein1 Protein2 d d1/2
Query variables 3MYQ_1 5WNA_1 95 87.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]