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Parikh vectors
3NST_1 6LBM_1 4ESM_1 Letter Amino acid
27 3 30 A Alanine
18 6 30 D Aspartic acid
3 1 3 C Cysteine
27 4 13 E Glutamic acid
10 1 7 H Histidine
26 3 14 T Threonine
24 9 16 R Arginine
11 8 11 N Asparagine
32 6 35 L Leucine
13 3 6 M Methionine
19 6 10 S Serine
11 2 7 W Tryptophan
25 2 18 V Valine
16 4 10 Q Glutamine
34 6 21 G Glycine
11 7 12 I Isoleucine
11 11 11 K Lycine
18 7 17 F Phenylalanine
26 8 20 P Proline
6 6 11 Y Tyrosine 

3NST_1|Chain A|Gentisate 1,2-dioxygenase|Pseudaminobacter salicylatoxidans (93369)
>6LBM_1|Chain A[auth C]|Forkhead box protein C2|Homo sapiens (9606)
>4ESM_1|Chain A|Phenylalanine-4-hydroxylase|Chromobacterium violaceum (243365)
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 \(w=f(c)\)
3NST , Knot 154 368 0.82 40 216 352 
MQNEKLDHESVTQAMQPKDTPELRALYKSFEEESIIPLWTQLGDLMPIHPKSKAVPHVWKWSTLLRLARKSGELVPVGRGGERRALGLANPGLGGNAYISPTMWAAIQYLGPRETAPEHRHSQNAFRFVVEGEGVWTVVNGDPVRMSRGDLLLTPGWCFHGHMNDTDQPMAWIDGLDIPFSQQMDVGFFEFGSDRVTDYATPNFSRGERLWCHPGLRPLSGLQNTVASPIGAYRWEFTDRALTEQLLLEDEGQPATVAPGHAAIRYVNPTTGGDVMPTLRCEFHRLRAGTETATRNEVGSTVFQVFEGAGAVVMNGETTKLEKGDMFVVPSWVPWSLQAETQFDLFRFSDAPIMEALSFMRTKIEGQK
6LBM , Knot 55 103 0.82 40 85 99 
GPKPPYSYIALITMAIQNAPEKKITLNGIYQFIMDRFPFYRENKQGWQNSIRHNLSLNECFVKVPRDDKKPGKGSYWTLDPDSYNMFENGSFLRRRRRFKKKD
4ESM , Knot 134 302 0.84 40 188 287 
GLPGSMNDRADFVVPDITTRKNVGLSHDANDFTLPQPLDRYSAEDHATWATLYQRQCKLLPGRACDEFLEGLERLEVDADRVPDFNKLNEKLMAATGWKIVAVPGLIPDDVFFEHLANRRFPVTWWLREPHQLDYLQEPDVFHDLFGHVPLLINPVFADALEAYGKGGVKAKALGALPMLARLYWYTVEFGLINTPAGMRIYGAGILSSKSESIYCLDSASPNRVGFDLMRIMNTRYRIDTFQKTYFVIDSFKQLFDATAPDFAPLYLQLADAQPWGAGDIAPDDLVLNAGDHQGWADTEDV

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 \(c\).

\(|P_{f(3NST_1)}(2) \setminus P_{f(6LBM_1)}(2)|=169\), \(|P_{f(6LBM_1)}(2) \setminus P_{f(3NST_1)}(2)|=38\). 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 1:10000100001001101000101011000100001111100110111101000111011010011011000101111101100011111011111010101011111001110001100000001101110101110110101101001011101110101010000011111011011100010111101100010001010100100110011101101100011011110010100011000111000101101111011100101001101110100010010110001000011001101101111111010000100101111101111010100010110100111101101100010100
Pair \(Z_2\) Length of longest common subsequence
3NST_1,6LBM_1 207 3
3NST_1,4ESM_1 166 4
6LBM_1,4ESM_1 181 3

Newick tree

 
[
	6LBM_1:10.51,
	[
		3NST_1:83,4ESM_1:83
	]:18.51
]

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 \((0.85)(0.8)(\frac{471 }{\log_{20} 471}-\frac{103}{\log_{20}103})=110.\)
Status Protein1 Protein2 d d1/2
Query variables 3NST_1 6LBM_1 140 90
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]