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Parikh vectors
7NZW_1 1HBZ_1 5GUJ_1 Letter Amino acid
18 25 27 S Serine
19 37 21 D Aspartic acid
13 32 26 E Glutamic acid
8 17 18 I Isoleucine
2 7 8 M Methionine
17 26 6 P Proline
14 37 11 T Threonine
10 17 13 Y Tyrosine
12 37 22 A Alanine
8 25 8 N Asparagine
1 0 2 C Cysteine
12 23 13 Q Glutamine
22 44 21 G Glycine
11 20 9 H Histidine
7 11 1 W Tryptophan
17 30 18 V Valine
7 34 16 R Arginine
26 31 34 L Leucine
25 20 30 K Lycine
11 25 20 F Phenylalanine 

7NZW_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>1HBZ_1|Chain A|CATALASE|MICROCOCCUS LYSODEIKTICUS (1270)
>5GUJ_1|Chain A|DNA primase|Bacillus subtilis (strain 168) (224308)
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 \(w=f(c)\)
7NZW , Knot 111 260 0.79 40 173 248 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNTGYLFQVEFDDSQDKAVLKGGPLDGTYRLKQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDYGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLTECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
1HBZ , Knot 204 498 0.84 38 250 468 
TTPHATGSTRQNGAPAVSDRQSLTVGSEGPIVLHDTHLLETHQHFNRMNIPERRPHAKGSGAFGEFEVTEDVSKYTKALVFQPGTKTETLLRFSTVAGELGSPDTWRDVRGFALRFYTEEGNYDLVGNNTPIFFLRDPMKFTHFIRSQKRLPDSGLRDATMQWDFWTNNPESAHQVTYLMGPRGLPRTWREMNGYGSHTYLWVNAQGEKHWVKYHFISQQGVHNLSNDEATKIAGENADFHRQDLFESIAKGDHPKWDLYIQAIPYEEGKTYRFNPFDLTKTISQKDYPRIKVGTLTLNRNPKNHFAQIESAAFSPSNTVPGIGLSPDRMLLGRAFAYHDAQLYRVGAHVNQLPVNRPKNAVHNYAFEGQMWYDHTGDRSTYVPNSNGDSWSDETGPVDDGWEADGTLTREAQALRADDDDFGQAGTLVREVFSDQERDDFVETVAGALKGVRQDVQARAFEYWKNVDATIGQRIEDEVKRHEGDGIPGVEAGGEARI
5GUJ , Knot 145 324 0.86 40 202 311 
SGEQKMAEAHELLKKFYHHLLINTKEGQEALDYLLSRGFTKELINEFQIGYALDSWDFITKFLVKRGFSEAQMEKAGLLIRREDGSGYFDRFRNRVMFPIHDHHGAVVAFSGRALGSQQPKYMNSPETPLFHKSKLLYNFYKARLHIRKQERAVLFEGFADVISAVSSDVKESIATMGTSLTDDHVKILRRNVEEIILCYDSDKAGYEATLKASELLQKKGCKVRVAMIPDGLDPDDYIKKFGGEKFKNDIIDASVTVMAFKMQYFRKGKNLSDEGDRLAYIKDVLKEISTLSGSLEQEVYVKQLASEFSLSQESLTEQLSVFS

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 \(c\).

\(|P_{f(7NZW_1)}(2) \setminus P_{f(1HBZ_1)}(2)|=45\), \(|P_{f(1HBZ_1)}(2) \setminus P_{f(7NZW_1)}(2)|=122\). 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 1:10001101000110010001111010000110100001000101011010000100101100101101010000001110111101000100101011010101000010000011010110100001001011001011111111101101011100110110010001001010010101111001001001101001110001011110011010000110100101010101001110010110110000101010
Pair \(Z_2\) Length of longest common subsequence
7NZW_1,1HBZ_1 167 4
7NZW_1,5GUJ_1 163 4
1HBZ_1,5GUJ_1 160 4

Newick tree

 
[
	7NZW_1:83.32,
	[
		5GUJ_1:80,1HBZ_1:80
	]:3.32
]

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 \((0.85)(0.8)(\frac{758 }{\log_{20} 758}-\frac{260}{\log_{20}260})=137.\)
Status Protein1 Protein2 d d1/2
Query variables 7NZW_1 1HBZ_1 174 131.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]