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Parikh vectors
5FQH_1 1KLM_1 8FYF_1 Letter Amino acid
41 58 7 K Lycine
34 40 30 T Threonine
27 26 52 A Alanine
36 25 19 D Aspartic acid
4 2 13 C Cysteine
25 13 34 F Phenylalanine
60 19 32 S Serine
18 19 5 W Tryptophan
55 19 3 N Asparagine
19 8 16 H Histidine
55 41 24 I Isoleucine
14 6 12 M Methionine
25 37 34 P Proline
15 40 41 V Valine
42 47 17 E Glutamic acid
12 37 22 Q Glutamine
34 34 26 G Glycine
45 49 77 L Leucine
30 22 10 Y Tyrosine
19 18 30 R Arginine 

5FQH_1|Chain A|BETA-N-ACETYLGALACTOSAMINIDASE|CLOSTRIDIUM PERFRINGENS (1502)
>1KLM_1|Chain A|HIV-1 REVERSE TRANSCRIPTASE|Human immunodeficiency virus 1 (11676)
>8FYF_1|Chains A, B|Endosomal/lysosomal potassium channel TMEM175|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 \(w=f(c)\)
5FQH , Knot 239 610 0.83 40 275 564 
MGSSHHHHHHSSGLVPRGSHMASMKKDTTLGASIGSTDFHYLQKDYDEIKKLNLNTWNEVAWIGDELNSKIVMWTNSSPVNNVTLSSSDFINENGDLISSNNIKISWLKETLANIGRSNPSAPLEPFPDIIHNSGSLNIEKNKIASAWINIKIPRNAKPGIYNGSIEVTADELEKSYTFDYSFEVLNLVQPLPSETNTQIEFWQHPYTIARYYKICKEDLFTEKHFKYLRGNLKEYRNMGGRGVIATIVHEAWNHQSYDSDPSMIKWRKNSYGTFEFDYSHFDKWIQLNIDLGILDPEKGFGQIKCYSIVPWNNRIQYFNEATNKEEAINPTPGSDLWINIWTQFLTSFMSHLEEKGWFNITYISMNQRSMDDLKACVDLIENITNNSYEHFKISSAMDYESGNDYSFLDRIDDISIGLSHINHNSDDMKNMATHRQELGLLTTIYTCTGDYPSSFTISDPSEGAFTIWYSLYQNTNGFLRWSWDGWVENPLENVSYKYWEPGDPFLIYPAEKDSIGKTFYSTPRLEKLKEGIRDINKAKYLMEKAPNLKNSIENLIYSLKRPNKGENAYGSAVAASKEDRDLTISEANRIKNGINNFAREFISLTMETL
1KLM , Knot 223 560 0.84 40 255 517 
PISPIETVPVKLKPGMDGPKVKQWPLTEEKIKALVEICTEMEKEGKISKIGPENPYNTPVFAIKKKDSTKWRKLVDFRELNKRTQDFWEVQLGIPHPAGLKKKKSVTVLDVGDAYFSVPLDEDFRKYTAFTIPSINNETPGIRYQYNVLPQGWKGSPAIFQSSMTKILEPFRKQNPDIVIYQYMDDLYVGSDLEIGQHRTKIEELRQHLLRWGLTTPDKKHQKEPPFLWMGYELHPDKWTVQPIVLPEKDSWTVNDIQKLVGKLNWASQIYPGIKVRQLCKLLRGTKALTEVIPLTEEAELELAENREILKEPVHGVYYDPSKDLIAEIQKQGQGQWTYQIYQEPFKNLKTGKYARMRGAHTNDVKQLTEAVQKITTESIVIWGKTPKFKLPIQKETWETWWTEYWQATWIPEWEFVNTPPLVKLWYQLEKEPIVGAETFYVDGAANRETKLGKAGYVTNRGRQKVVTLTDTTNQKTELQAIYLALQDSGLEVNIVTDSQYALGIIQAQPDQSESELVNQIIEQLIKKEKVYLAWVPAHKGIGGNEQVDKLVSAGIRKVL
8FYF , Knot 196 504 0.80 40 233 453 
MSQPRTPEQALDTPGDCPPGRRDEDAGEGIQCSQRMLSFSDALLSIIATVMILPVTHTEISPEQQFDRSVQRLLATRIAVYLMTFLIVTVAWAAHTRLFQVVGKTDDTLALLNLACMMTITFLPYTFSLMVTFPDVPLGIFLFCVCVIAIGVVQALIVGYAFHFPHLLSPQIQRSAHRALYRRHVLGIVLQGPALCFAAAIFSLFFVPLSYLLMVTVILLPYVSKVTGWCRDRLLGHREPSAHPVEVFSFDLHEPLSKERVEAFSDGVYAIVATLLILDICEDNVPDPKDVKERFSGSLVAALSATGPRFLAYFGSFATVGLLWFAHHSLFLHVRKATRAMGLLNTLSLAFVGGLPLAYQQTSAFARQPRDELERVRVSCTIIFLASIFQLAMWTTALLHQAETLQPSVWFGGREHVLMFAKLALYPCASLLAFASTCLLSRFSVGIFHLMQIAVPCAFLLLRLLVGLALATLRVLRGLARPEHPPPAPTGQDDPQSQLLPAPC

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 \(c\).

\(|P_{f(5FQH_1)}(2) \setminus P_{f(1KLM_1)}(2)|=80\), \(|P_{f(1KLM_1)}(2) \setminus P_{f(5FQH_1)}(2)|=60\). 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 1:1100000000001111010011010000011101100010010000001001010010011111001000111100001100101000011000101100001010110001101100010111011101100010101000011011101011001011100101010100100000100010110110111000000101100100110000100001100001001010100000111011110110011000000001011010000010101000010011010101111010011101000011110001001001000001101011001110110011001100100011101001010000100101010110010000000101001100001000011001001011100100000010011000001111001000010010010100100111011001000001110101011100110010000101101111011000011001000101001001100100100110011010001001100100100100101011110000001010010010011001100110101001
Pair \(Z_2\) Length of longest common subsequence
5FQH_1,1KLM_1 140 4
5FQH_1,8FYF_1 190 4
1KLM_1,8FYF_1 174 3

Newick tree

 
[
	8FYF_1:97.10,
	[
		5FQH_1:70,1KLM_1:70
	]:27.10
]

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 \((0.85)(0.8)(\frac{1170 }{\log_{20} 1170}-\frac{560}{\log_{20}560})=157.\)
Status Protein1 Protein2 d d1/2
Query variables 5FQH_1 1KLM_1 201 190
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]