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Parikh vectors
8GYM_1 4DNA_1 9EOS_1 Letter Amino acid
1 16 24 P Proline
9 19 28 S Serine
6 28 29 T Threonine
3 30 27 R Arginine
4 10 20 Q Glutamine
0 12 16 H Histidine
4 17 35 F Phenylalanine
0 2 2 W Tryptophan
3 12 19 Y Tyrosine
8 23 36 D Aspartic acid
8 37 62 E Glutamic acid
5 31 9 I Isoleucine
2 12 7 M Methionine
5 11 17 N Asparagine
4 6 35 C Cysteine
10 37 65 L Leucine
5 57 63 A Alanine
5 48 13 G Glycine
8 21 60 K Lycine
3 34 43 V Valine 

8GYM_14|Chains N[auth 3T], RF[auth 3t]|Zf-Tim10_DDP domain-containing protein|Tetrahymena thermophila SB210 (312017)
>4DNA_1|Chains A, B|Probable glutathione reductase|Sinorhizobium meliloti (266834)
>9EOS_1|Chains A, B|Albumin|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 \(w=f(c)\)
8GYM , Knot 50 93 0.81 36 80 91 
MSTRKIFDSEEQSFIRLVDKFYLGLSLTKLCAQSCNLLRNDISGSALTQKEKDCLSICYNNIEKTQSAFYAKVKTTMNLPAVEDDGEEGGDDE
4DNA , Knot 187 463 0.82 40 225 437 
MSAFDYDLFVIGGGSGGVRSGRLAAALGKKVAIAEEFRYGGTCVIRGCVPKKLYVYASQFAEHFEDAAGFGWTVGESRFDWAKLVAAKEQEIARLEGLYRKGLANAGAEILDTRAELAGPNTVKLLASGKTVTAERIVIAVGGHPSPHDALPGHELCITSNEAFDLPALPESILIAGGGYIAVEFANIFHGLGVKTTLIYRGKEILSRFDQDMRRGLHAAMEEKGIRILCEDIIQSVSADADGRRVATTMKHGEIVADQVMLALGRMPNTNGLGLEAAGVRTNELGAIIVDAFSRTSTPGIYALGDVTDRVQLTPVAIHEAMCFIETEYKNNPTSPDHDLIATAVFSQPEIGTVGITEEEAARKFQEIEVYRAEFRPMKATLSGRKEKTIMKLVVNAADRKVVGAHILGHDAGEMAQLLGISLRAGCTKDDFDRTMAVHPTAAEELVTMYQPSYRVRNGERVG
9EOS , Knot 241 610 0.84 40 246 553 
MKWVTFISLLFLFSSAYSRGVFRRDAHKSEVAHRFKDLGEENFKALVLIAFAQYLQQCPFEDHVKLVNEVTEFAKTCVADESAENCDKSLHTLFGDKLCTVATLRETYGEMADCCAKQEPERNECFLQHKDDNPNLPRLVRPEVDVMCTAFHDNEETFLKKYLYEIARRHPYFYAPELLFFAKRYKAAFTECCQAADKAACLLPKLDELRDEGKASSAKQRLKCASLQKFGERAFKAWAVARLSQRFPKAEFAEVSKLVTDLTKVHTECCHGDLLECADDRADLAKYICENQDSISSKLKECCEKPLLEKSHCIAEVENDEMPADLPSLAADFVESKDVCKNYAEAKDVFLGMFLYEYARRHPDYSVVLLLRLAKTYETTLEKCCAAADPHECYAKVFDEFKPLVEEPQNLIKQNCELFEQLGEYKFQNALLVRYTKKVPQVSTPTLVEVSRNLGKVGSKCCKHPEAKRMPCAEDYLSVVLNQLCVLHEKTPVSDRVTKCCTESLVNRRPCFSALEVDETYVPKEFNAETFTFHADICTLSEKERQIKKQTALVELVKHKPKATKEQLKAVMDDFAAFVEKCCKADDKETCFAEEGKKLVAASQAALGLL

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 \(c\).

\(|P_{f(8GYM_1)}(2) \setminus P_{f(4DNA_1)}(2)|=21\), \(|P_{f(4DNA_1)}(2) \setminus P_{f(8GYM_1)}(2)|=166\). 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 1:100001100000011011001011101001010000110001010110000000101000010000011010100010111100010011000
Pair \(Z_2\) Length of longest common subsequence
8GYM_1,4DNA_1 187 4
8GYM_1,9EOS_1 206 4
4DNA_1,9EOS_1 161 5

Newick tree

 
[
	8GYM_1:10.63,
	[
		4DNA_1:80.5,9EOS_1:80.5
	]:23.13
]

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 \((0.85)(0.8)(\frac{556 }{\log_{20} 556}-\frac{93}{\log_{20}93})=137.\)
Status Protein1 Protein2 d d1/2
Query variables 8GYM_1 4DNA_1 168 99.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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