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Parikh vectors
1OBU_1 4EKJ_1 2VAO_1 Letter Amino acid
6 1 4 C Cysteine
7 17 18 Q Glutamine
9 36 47 G Glycine
12 42 51 L Leucine
0 6 21 M Methionine
12 46 31 A Alanine
15 9 28 N Asparagine
12 39 36 D Aspartic acid
12 24 21 F Phenylalanine
11 42 42 V Valine
2 11 14 W Tryptophan
14 23 21 Y Tyrosine
9 21 35 I Isoleucine
9 31 36 P Proline
14 27 33 S Serine
12 27 30 K Lycine
9 28 23 T Threonine
6 34 27 R Arginine
8 27 27 E Glutamic acid
2 9 15 H Histidine 

1OBU_1|Chains A, B|CRUSTACYANIN C1 SUBUNIT|HOMARUS GAMMARUS (6707)
>4EKJ_1|Chain A|Beta-xylosidase|Caulobacter vibrioides (155892)
>2VAO_1|Chains A, B|VANILLYL-ALCOHOL OXIDASE|Penicillium simplicissimum (69488)
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 \(w=f(c)\)
1OBU , Knot 88 181 0.84 38 140 178 
DKIPDFVVPGKCASVDRNKLWAEQTPNRNSYAGVWYQFALTNNPYQLIEKCVRNEYSFDGKQFVIESTGIAYDGNLLKRNGKLYPNPFGEPHLSIDYENSFAAPLVILETDYSNYACLYSCIDYNFGYHSDFSFIFSRSANLADQYVKKCEAAFKNINVDTTRFVKTVQGSSCPYDTQKTL
4EKJ , Knot 198 500 0.82 40 235 473 
MANAGPGARVIDLDLRRAAGPVDRFFDLSIGSDYPGTLIREDSQAQLKTTVDELGFRYIRFHAIFHDVLGTVKVQDGKIVYDWTKIDQLYDALLAKGIKPFIELGFTPEAMKTSDQTIFYWKGNTSHPKLGPWRDLIDAFVHHLRARYGVEEVRTWFFEVWNEPNLDGFWEKADQAAYFELYDVTARAIKAIDPSLRVGGPATAGAAWVPEFLAHVKKSGSAVDFVTTHTYGVDGGFLDEKGVQDTKLSPSPDAVVGDVRRVREQIEASAFPGLPLYFTEWSTSYTPRDSVHDSYVSAAYIVEKLRRVKGLVQAMSYWTYSDLFEEPGPPTAPFQGGFGLMNPQGIRKPSWFAYKYLNAIKGRELVCADDQVFAARDGDRVAIVAYAWRQPDQKVSNRPFYTKLHPASDVEPLKVRLTSLKPGRYKLRVRRVGYRRNDAYSAYIDMGSPTTLTESQLQSLQALTEDRPEIEKALKVSGETVVDLPMRANDVVLIELEPLA
2VAO , Knot 228 560 0.86 40 268 524 
MSKTQEFRPLTLPPKLSLSDFNEFIQDIIRIVGSENVEVISSKDQIVDGSYMKPTHTHDPHHVMDQDYFLASAIVAPRNVADVQSIVGLANKFSFPLWPISIGRNSGYGGAAPRVSGSVVLDMGKNMNRVLEVNVEGAYCVVEPGVTYHDLHNYLEANNLRDKLWLDVPDLGGGSVLGNAVERGVGYTPYGDHWMMHSGMEVVLANGELLRTGMGALPDPKRPETMGLKPEDQPWSKIAHLFPYGFGPYIDGLFSQSNMGIVTKIGIWLMPNPRGYQSYLITLPKDGDLKQAVDIIRPLRLGMALQNVPTIRHILLDAAVLGDKRSYSSRTEPLSDEELDKIAKQLNLGRWNFYGALYGPEPIRRVLWETIKDAFSAIPGVKFYFPEDTPENSVLRVRDKTMQGIPTYDELKWIDWLPNGAHLFFSPIAKVSGEDAMMQYAVTKKRCQEAGLDFIGTFTVGMREMHHIVCIVFNKKDLIQKRKVQWLMRTLIDDCAANGWGEYRTHLAFMDQIMETYNWNNSSFLRFNEVLKNAVDPNGIIAPGKSGVWPSQYSHVTWKL

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 \(c\).

\(|P_{f(1OBU_1)}(2) \setminus P_{f(4EKJ_1)}(2)|=50\), \(|P_{f(4EKJ_1)}(2) \setminus P_{f(1OBU_1)}(2)|=145\). 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 1:0011011111001010000111000100000111100111000100110001000001010011100011100101100010101011101010100000111111110000000101000100011000010111000101100010000111001010000110010100010000001
Pair \(Z_2\) Length of longest common subsequence
1OBU_1,4EKJ_1 195 3
1OBU_1,2VAO_1 194 3
4EKJ_1,2VAO_1 139 5

Newick tree

 
[
	1OBU_1:10.88,
	[
		2VAO_1:69.5,4EKJ_1:69.5
	]:35.38
]

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 \((0.85)(0.8)(\frac{681 }{\log_{20} 681}-\frac{181}{\log_{20}181})=141.\)
Status Protein1 Protein2 d d1/2
Query variables 1OBU_1 4EKJ_1 177 121.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]