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Parikh vectors
1XED_1 2VAO_1 7ZTD_1 Letter Amino acid
3 36 23 D Aspartic acid
6 23 26 T Threonine
12 47 28 G Glycine
4 21 13 F Phenylalanine
0 21 17 M Methionine
9 42 22 V Valine
7 15 9 H Histidine
6 35 28 I Isoleucine
4 31 29 A Alanine
7 27 18 R Arginine
8 28 9 N Asparagine
4 4 6 C Cysteine
3 18 12 Q Glutamine
7 27 26 E Glutamic acid
7 51 27 L Leucine
4 30 19 K Lycine
5 36 19 P Proline
14 33 25 S Serine
1 14 4 W Tryptophan
6 21 15 Y Tyrosine

1XED_1|Chains A, B, C, D, E, F|Polymeric-immunoglobulin receptor|Homo sapiens (9606)
>2VAO_1|Chains A, B|VANILLYL-ALCOHOL OXIDASE|Penicillium simplicissimum (69488)
>7ZTD_1|Chains A, B, C, D, E, F, G, H|actin, cytoplasmic 1|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 \(w=f(c)\)
1XED , Knot 57 117 0.77 38 93 111
KSPIFGPEEVNSVEGNSVSITCYYPPTSVNRHTRKYWCRQGARGGCITLISSEGYVSSKYAGRANLTNFPENGTFVVNIAQLSQDDSGRYKCGLGINSRGLSFDVSLEVLEHHHHHH
2VAO , Knot 228 560 0.86 40 268 524
MSKTQEFRPLTLPPKLSLSDFNEFIQDIIRIVGSENVEVISSKDQIVDGSYMKPTHTHDPHHVMDQDYFLASAIVAPRNVADVQSIVGLANKFSFPLWPISIGRNSGYGGAAPRVSGSVVLDMGKNMNRVLEVNVEGAYCVVEPGVTYHDLHNYLEANNLRDKLWLDVPDLGGGSVLGNAVERGVGYTPYGDHWMMHSGMEVVLANGELLRTGMGALPDPKRPETMGLKPEDQPWSKIAHLFPYGFGPYIDGLFSQSNMGIVTKIGIWLMPNPRGYQSYLITLPKDGDLKQAVDIIRPLRLGMALQNVPTIRHILLDAAVLGDKRSYSSRTEPLSDEELDKIAKQLNLGRWNFYGALYGPEPIRRVLWETIKDAFSAIPGVKFYFPEDTPENSVLRVRDKTMQGIPTYDELKWIDWLPNGAHLFFSPIAKVSGEDAMMQYAVTKKRCQEAGLDFIGTFTVGMREMHHIVCIVFNKKDLIQKRKVQWLMRTLIDDCAANGWGEYRTHLAFMDQIMETYNWNNSSFLRFNEVLKNAVDPNGIIAPGKSGVWPSQYSHVTWKL
7ZTD , Knot 161 375 0.84 40 219 362
MDDDIAALVVDNGSGMCKAGFAGDDAPRAVFPSIVGRPRHQGVMVGMGQKDSYVGDEAQSKRGILTLKYPIEHGIVTNWDDMEKIWHHTFYNELRVAPEEHPVLLTEAPLNPKANREKMTQIMFETFNTPAMYVAIQAVLSLYASGRTTGIVMDSGDGVTHTVPIYEGYALPHAILRLDLAGRDLTDYLMKILTERGYSFTTTAEREIVRDIKEKLCYVALDFEQEMATAASSSSLEKSYELPDGQVITIGNERFRCPEALFQPSFLGMESCGIHETTFNSIMKCDVDIRKDLYANTVLSGGTTMYPGIADRMQKEITALAPSTMKIKIIAPPERKYSVWIGGSILASLSTFQQMWISKQEYDESGPSIVHRKCF

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 \(c\).

\(|P_{f(1XED_1)}(2) \setminus P_{f(2VAO_1)}(2)|=28\), \(|P_{f(2VAO_1)}(2) \setminus P_{f(1XED_1)}(2)|=203\). 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 1:001111100100101001010000110010000000100011011010110001010000110101001100101110110100000100001111000110101010110000000
Pair \(Z_2\) Length of longest common subsequence
1XED_1,2VAO_1 231 3
1XED_1,7ZTD_1 198 3
2VAO_1,7ZTD_1 175 4

Newick tree

 
[
	1XED_1:11.46,
	[
		7ZTD_1:87.5,2VAO_1:87.5
	]:25.96
]

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 \((0.85)(0.8)(\frac{677 }{\log_{20} 677}-\frac{117}{\log_{20}117})=161.\)
Status Protein1 Protein2 d d1/2
Query variables 1XED_1 2VAO_1 208 123
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]