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Parikh vectors
7VAP_1 5ILF_1 7DVE_1 Letter Amino acid
10 14 8 N Asparagine
26 7 53 D Aspartic acid
3 8 4 C Cysteine
6 1 17 H Histidine
19 2 11 M Methionine
8 6 5 W Tryptophan
38 11 42 R Arginine
55 12 50 G Glycine
47 8 41 L Leucine
24 6 7 K Lycine
19 3 15 F Phenylalanine
27 7 28 T Threonine
21 3 10 Y Tyrosine
56 12 69 A Alanine
53 2 24 E Glutamic acid
23 10 30 S Serine
17 3 14 Q Glutamine
35 6 17 I Isoleucine
35 2 31 P Proline
56 6 52 V Valine 

7VAP_1|Chains A, B, C|V-type ATP synthase alpha chain|Thermus thermophilus HB8 (300852)
>5ILF_1|Chain A|Lysozyme C|Gallus gallus (9031)
>7DVE_1|Chains A, B|6'''-hydroxyparomomycin C oxidase|Microbacterium trichothecenolyticum (69370)
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 \(w=f(c)\)
7VAP , Knot 224 578 0.82 40 243 526 
MIQGVIQKIAGPAVIAKGMLGARMYDICKVGEEGLVGEIIRLDGDTAFVQVYEDTSGLKVGEPVVSTGLPLAVELGPGMLNGIYDGIQRPLERIREKTGIYITRGVVVHALDREKKWAWTPMVKPGDEVRGGMVLGTVPEFGFTHKILVPPDVRGRVKEVKPAGEYTVEEPVVVLEDGTELKMYHTWPVRRARPVQRKLDPNTPFLTGMRILDVLFPVAMGGTAAIPGPFGAGKSVTQQSLAKWSNADVVVYVGCGERGNEMTDVLVEFPELTDPKTGGPLMHRTVLIANTSNMPVAAREASIYVGVTIAEYFRDQGFSVALMADSTSRWAEALREISSRLEEMPAEEGYPPYLAARLAAFYERAGKVITLGGEEGAVTIVGAVSPPGGDMSEPVTQSTLRIVGAFWRLDASLAFRRHFPAINWNGSYSLFTSALDPWYRENVAEDYPELRDAISELLQREAGLQEIVQLVGPDALQDAERLVIEVGRIIREDFLQQNAYHEVDAYCSMKKAYGIMKMILAFYKEAEAAIKRGVSIDEILQLPVLERIGRARYVSEEEFPAYFEEAMKEIQGAFKALA
5ILF , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
7DVE , Knot 204 528 0.80 40 207 463 
MSTRVYPAQVDVAIVGSGPAGATYARILSERASSATIAMFEVGPTVSDPPGAHVKNIADADERAHAQRRSEGPHAREDDDRVGGIVKSAQRRARPGTYLLESGYQADGEDGLPVAAFSSNVGGMAAHWTGACPRPNDSERIGFLDETGELDELLSEGERLLGVTTDAFDASPYAGIVRERLAAVEDAHRDADERVQRMPLAVHRRDDGPLVWSGSDVVLGDITRGNPNFTLFDESLVTRVLVEDGRAAGVVVTDVRTGERRDVRARFVVVAADALRTPQLLWASGIRPDALGRYLNDQAQIVFAVRMRDFTPVVDADGVPQTGLSEYTGVTWVPFTDDMPFHGQVMQLDASPVKLADDDPAAPGSIVGLGLFCAKDLQASDRVAFSDSDVDGYGMPAMQLHYTLSDRDHATIDRAKAEIVRLGKAIGDPLDDRPFVMPLGASLHYQGTVRMGLADDGASVCSPDSEVWGAPGLFVAGNGVIPTATACNPTLTGVALAVRGARHIADEITADLASVKLAAALEHHHHHH

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 \(c\).

\(|P_{f(7VAP_1)}(2) \setminus P_{f(5ILF_1)}(2)|=174\), \(|P_{f(5ILF_1)}(2) \setminus P_{f(7VAP_1)}(2)|=35\). 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 1:11011100111111110111110100100110011110110101001110100000110110111001111110111111011001100110010000110100111101100000111011101100101111110110111000111110101010010111000100111110010010100011100101100010100111011011011111111101111111111001000011010010111011010010010011101101001001111100011110000111110010101110110010001101111100000110110010001001110010110111011110001101101110011101111101111010011000010111111010101110001111010100011001101100001100010100110011000111001101111011001001110110110001100010001010001001011101111100010111001101001101111001101001000011101001100101110111
Pair \(Z_2\) Length of longest common subsequence
7VAP_1,5ILF_1 209 3
7VAP_1,7DVE_1 148 5
5ILF_1,7DVE_1 185 4

Newick tree

 
[
	5ILF_1:10.63,
	[
		7VAP_1:74,7DVE_1:74
	]:31.63
]

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 \((0.85)(0.8)(\frac{707 }{\log_{20} 707}-\frac{129}{\log_{20}129})=165.\)
Status Protein1 Protein2 d d1/2
Query variables 7VAP_1 5ILF_1 204 125
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]