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Parikh vectors
6AFE_1 3VQT_1 5FHE_1 Letter Amino acid
3 12 18 Y Tyrosine
7 41 19 R Arginine
6 12 21 N Asparagine
9 34 29 D Aspartic acid
3 23 8 H Histidine
18 44 41 L Leucine
9 24 23 S Serine
19 41 27 V Valine
24 53 17 A Alanine
3 4 6 C Cysteine
4 20 13 Q Glutamine
18 42 26 G Glycine
9 25 14 P Proline
8 33 27 T Threonine
10 25 42 I Isoleucine
15 39 33 E Glutamic acid
16 24 31 K Lycine
5 20 11 M Methionine
3 28 24 F Phenylalanine
0 4 3 W Tryptophan 

6AFE_1|Chain A|Protein/nucleic acid deglycase DJ-1|Homo sapiens (9606)
>3VQT_1|Chains A, B, C, D|Peptide chain release factor 3|Desulfovibrio vulgaris (883)
>5FHE_1|Chain A|Uncharacterized protein|Bacteroides sp. 2_1_16 (817)
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 \(w=f(c)\)
6AFE , Knot 86 189 0.79 38 124 182 
MASKRALVILAKGAEEMETVIPVDVMRRAGIKVTVAGLAGKDPVQCSRDVVICPDASLEDAKKEGPYDVVVLPGGNLGAQNLSESAAVKEILKEQENRKGLIAAICAGPTALLAHEIGFGSKVTTHPLAKDKMMNGGHYTYSENRVEKDGLILTSRGPGTSFEFALAIVEALNGKEVAAQVKAPLVLKD
3VQT , Knot 216 548 0.82 40 250 512 
GSSHHHHHHSSGLVPRGSHMSSRLEREAARRRTFAIISHPDAGKTTLTEKLLLFGGAIQMAGSVKARKAARHATSDWMAMERERGISVTTSVMQFPYRDRVVNLLDTPGHQDFSEDTYRVLTAVDSALVVIDAAKGVEAQTRKLMDVCRMRATPVMTFVNKMDREALHPLDVMADIEQHLQIECAPMTWPIGMGSSFKGTYDLLHKQLHLFSATHGGRIQSGIVIHGADDPQLDEYLGDQAEQLRMDLALLEEAGTPFDEERYLKGELTPVFFGSAINNFGVREMLDMFVEFAPGPQPRPAATRVVEPGEEAFTGVVFKIQANMDKAHRDRMAFLRICSGTFTRGMRLKHHRTGKDVTVANATIFMAQDRTGVEEAFPGDIIGIPNHGTIKIGDTFTESKEVLKFVGIPNFAPEHFRRVRLKNPLKAKQLQKGLEQLAEEGAVQLFRPLVNNDYILGAVGVLQFDVIVARLADEYGVDAVYEGVSTHTARWVYCEDKKIFADFQDYHRGELAVDAEGALAYLAPNPWRLESAMERYPKVEFRTTREIS
5FHE , Knot 183 433 0.85 40 243 415 
MEDMILTEEMQKIMNLIQDDENNVFVTGKAGSGKTTFLKYLIEKSGKNCIVAAPTGIAAINAGGVTLHSLFGIPFGPITPYDRLENKFSEYKVELLLKMELLIIDEISMVRPDILDTIDRKLRWVYESDEPFGGVQVVMFGDLFQLPPVTKKQEREILSDFYDGFFFFNALVFKRTGFHIVELTKIFRQTEPEFINVLNNIRNYQVTSDELDLLSELKDRKISSSYDNEYIHICTHKADVEKINADKLGEQEIRNYDIVIKDKFPESSIPCDLHLKLRVGARVMSLVNDSLKGYYNGMLGIVTALEDNVITVRMDNGRTIKFERYTWSNTQYTLKDNEIVKEEIGSCTQFPLTLAWAITIHKSQGLTFDKIIIHVSHTFCPGQLYVALSRCRTLEGIVSDAFITKQMIIPEYALIDFERAYKSEGNYYGKRLD

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 \(c\).

\(|P_{f(6AFE_1)}(2) \setminus P_{f(3VQT_1)}(2)|=28\), \(|P_{f(3VQT_1)}(2) \setminus P_{f(6AFE_1)}(2)|=154\). 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 1:110001111110110010011110110011101011111100110000011101010100100011001111111011100100011100110000000111111011101111001111001000111000110110000000010001111000111001011111101101001110101111100
Pair \(Z_2\) Length of longest common subsequence
6AFE_1,3VQT_1 182 4
6AFE_1,5FHE_1 185 4
3VQT_1,5FHE_1 147 4

Newick tree

 
[
	6AFE_1:97.07,
	[
		3VQT_1:73.5,5FHE_1:73.5
	]:23.57
]

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 \((0.85)(0.8)(\frac{737 }{\log_{20} 737}-\frac{189}{\log_{20}189})=153.\)
Status Protein1 Protein2 d d1/2
Query variables 6AFE_1 3VQT_1 193 127.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]