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Parikh vectors
3QZR_1 4GXP_1 1NSA_1 Letter Amino acid
3 23 24 Y Tyrosine
17 34 22 V Valine
9 33 28 A Alanine
2 4 7 C Cysteine
9 8 13 Q Glutamine
8 23 20 K Lycine
10 25 16 R Arginine
11 26 20 I Isoleucine
18 29 29 L Leucine
9 27 19 F Phenylalanine
1 15 9 W Tryptophan
9 25 25 N Asparagine
9 33 23 D Aspartic acid
19 43 23 G Glycine
6 7 5 M Methionine
14 24 30 T Threonine
7 25 30 E Glutamic acid
7 16 7 H Histidine
8 23 13 P Proline
11 24 32 S Serine 

3QZR_1|Chains A, B|3C protein|Human enterovirus 71 (39054)
>4GXP_1|Chains A, B, C|Beta-glucosidase Chimeric protein|Thermotoga maritima (243274)
>1NSA_1|Chain A|PROCARBOXYPEPTIDASE B|Sus scrofa (9823)
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 \(w=f(c)\)
3QZR , Knot 91 187 0.84 40 143 185 
GSHMGPSLDFALSLLRRNVRQVQTDQGHFTMLGVRDRLAVLPRHSQPGKTIWIEHKLVNVLDAVELVDEQGVNLALTLITLDTNEKFRDITKFIPENISTASDATLVINTEHMPSMFVPVGDVVQYGFLNLSGKPTHRTMMYNFPTKAGQCGGVVTSVGKIIGIHIGGNGRQGFCAGLKRSYFASEQ
4GXP , Knot 194 467 0.85 40 247 439 
MHHHHHHMNVKKFPEGFLWGVATASYQIEGSPLADGAGMSIWHTFSHTPGNVKNGDTGDVACDHYNRWKEDIEIIEKLGVKAYRFSISWPRILPEGTGRVNQKGLDFYNRIIDTLLEKGITPFVTIFHWDLPFALQLKGGLLNREIADWFAEYSRVLFENFGDRVKNWITFNEPLCSAIPGYGSGTFAPGRQSTSEPWTVGHNILVAHGRAVKVFRETVKDGKIGIVLNGDFTYPWDAADPADKEAAERRLEFFTAWFADPIYLGDYPASMRKQLGDRLPTFTPEERALVHGSNDFYGMNHYTSNYIRHRSSPASADDTVGNVDVLFTNKQGNCIGPETAMPWLRPCAAGFRDFLVWISKRYGYPPIYVTENGAAFDDVVSEDGRVHDQNRIDYLKAYIGAMVTAVELDGVNVKGYFVWSLLDNFEWAEGYSKRFGIVYVDYSTQKRIVKDSGYWYSNVVKNNGLED
1NSA , Knot 169 395 0.85 40 226 377 
FEGEKVFRVNVEDENDISELHELASTRQIDFWKPDSVTQIKPHSTVDFRVKAEDILAVEDFLEQNELQYEVLINNLRSVLEAQFDSVSRTTGHSYEKYNNWETIEAWTEQVTSKNPDLISRSAIGTTFDGDNIYLLKVGKPGSNKPAIFMDCGFHAREWISQAFCQWFVRDAVRTYGYEAHMTEFLDNLDFYVLPVLNIDGYIYTWTKNRMWRKTRSTNAGSSCTGTDPNRNFNAGWCTVGASVNPCNETYCGSAAESEKETKALADFIRNNLSSIKAYLTIHSYSQMILYPYSYDYKLPENDAELNSLAKGAVKELASLYGTSYSYGPGSTTIYPAAGGSDDWAYNQGIKYSFTFELRDKGRFGFVLPESQIQATCQETMLAVKYVTNYTLEHL

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 \(c\).

\(|P_{f(3QZR_1)}(2) \setminus P_{f(4GXP_1)}(2)|=45\), \(|P_{f(4GXP_1)}(2) \setminus P_{f(3QZR_1)}(2)|=149\). 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 1:1001110101110110001001000010101111000111110000110011100011011011011000110111011010000010010011100100100101110000110111111011001110101010000110011001100111100110111101110100110111000011000
Pair \(Z_2\) Length of longest common subsequence
3QZR_1,4GXP_1 194 4
3QZR_1,1NSA_1 183 3
4GXP_1,1NSA_1 179 4

Newick tree

 
[
	3QZR_1:95.83,
	[
		1NSA_1:89.5,4GXP_1:89.5
	]:6.33
]

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 \((0.85)(0.8)(\frac{654 }{\log_{20} 654}-\frac{187}{\log_{20}187})=132.\)
Status Protein1 Protein2 d d1/2
Query variables 3QZR_1 4GXP_1 168 117.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]