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Parikh vectors
1DYK_1 7RXD_1 7OYP_1 Letter Amino acid
32 28 17 V Valine
17 7 7 R Arginine
23 33 19 D Aspartic acid
6 0 1 C Cysteine
32 42 26 L Leucine
22 21 12 F Phenylalanine
9 19 8 Y Tyrosine
13 20 11 Q Glutamine
39 52 22 G Glycine
9 9 2 M Methionine
23 26 17 P Proline
2 9 7 W Tryptophan
27 60 13 A Alanine
17 33 10 N Asparagine
9 10 12 H Histidine
21 29 9 I Isoleucine
31 32 12 T Threonine
17 39 13 E Glutamic acid
21 50 24 K Lycine
24 26 18 S Serine 

1DYK_1|Chain A|LAMININ ALPHA 2 CHAIN|MUS MUSCULUS (10090)
>7RXD_1|Chain A[auth B]|Maltodextrin-binding protein,Immunoglobulin G-binding protein A,Immunoglobulin G-binding protein G|Escherichia coli (562)
>7OYP_1|Chain A|Carbonic anhydrase 2|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 \(w=f(c)\)
1DYK , Knot 162 394 0.82 40 216 364 
APLASVPTPAFPFPVPTMVHGPCVAESEPALLTGSKQFGLSRNSHIAIAFDDTKVKNRLTIELEVRTEAESGLLFYMARINHADFATVQLRNGFPYFSYDLGSGDTSTMIPTKINDGQWHKIKIVRVKQEGILYVDDASSQTISPKKADILDVVGILYVGGLPINYTTRRIGPVTYSLDGCVRNLHMEQAPVDLDQPTSSFHVGTCFANAESGTYFDGTGFAKAVGGFKVGLDLLVEFEFRTTRPTGVLLGVSSQKMDGMGIEMIDEKLMFHVDNGAGRFTAIYDAEIPGHMCNGQWHKVTAKKIKNRLELVVDGNQVDAQSPNSASTSADTNDPVFVGGFPGGLNQFGLTTNIRFRGCIRSLKLTKGTGKPLEVNFAKALELRGVQPVSCPTT
7RXD , Knot 205 545 0.79 38 230 469 
MKIEEGKLVIWINGDKGYNGLAEVGKKFEKDTGIKVTVEHPDKLEEKFPQVAATGDGPDIIFWAHDRFGGYAQSGLLAEITPDKAFQDKLYPFTWDAVRYNGKLIAYPIAVEALSLIYNKDLLPNPPKTWEEIPALDKELKAKGKSALMFNLQEPYFTWPLIAADGGYAFKYENGKYDIKDVGVDNAGAKAGLTFLVDLIKNKHMNADTDYSIAEAAFNKGETAMTINGPWAWSNIDTSKVNYGVTVLPTFKGQPSKPFVGVLSAGINAASPNKELAKEFLENYLLTDEGLEAVNKDKPLGAVALKSYEEELAKDPRIAATMENAQKGEIMPNIPQMSAFWYAVRTAVINAASGRQTVDQALAFAQILIMPNLTEEQRNGFIQSLKDDPSVSKEILAEAKKLNEHQAPKGGSGGAGSGDQQSAFYEILNMPNLNEAQRNGFIQSLKDDPSQSTNVLGEAKKLNESQAGGGSGGGSGGSAVTTYKLVINGKTLKGETTTKAVDAETAEKAFKQYANDNGVDGVWTYDDATKTFTVTEGSGHHHHHH
7OYP , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK

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 \(c\).

\(|P_{f(1DYK_1)}(2) \setminus P_{f(7RXD_1)}(2)|=71\), \(|P_{f(7RXD_1)}(2) \setminus P_{f(1DYK_1)}(2)|=85\). 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 1:1111011011111111011011011000111101000111000001111100001000101010100010011110110100101101010011101000110100001110010010100101101000111010010000101001011011111011111100000011110001010100101001110100100010110011010010010101110111110111011101010000101111110000101111011000111010011101011001011101001010010100100010111010010100100100010000111111111110011100010101010010100101011010110110101101100100
Pair \(Z_2\) Length of longest common subsequence
1DYK_1,7RXD_1 156 4
1DYK_1,7OYP_1 156 4
7RXD_1,7OYP_1 152 5

Newick tree

 
[
	1DYK_1:78.65,
	[
		7RXD_1:76,7OYP_1:76
	]:2.65
]

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 \((0.85)(0.8)(\frac{939 }{\log_{20} 939}-\frac{394}{\log_{20}394})=145.\)
Status Protein1 Protein2 d d1/2
Query variables 1DYK_1 7RXD_1 176 153.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]