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Parikh vectors
6CJB_1 9IUP_1 2BIA_1 Letter Amino acid
36 37 29 A Alanine
10 30 23 Q Glutamine
41 60 39 L Leucine
22 32 18 T Threonine
23 31 18 V Valine
27 32 22 G Glycine
20 16 8 H Histidine
16 27 14 P Proline
31 34 25 S Serine
10 27 14 Y Tyrosine
21 47 24 E Glutamic acid
11 21 10 M Methionine
3 20 2 W Tryptophan
15 18 10 R Arginine
17 39 29 N Asparagine
19 32 13 D Aspartic acid
5 8 2 C Cysteine
25 22 23 I Isoleucine
23 34 22 K Lycine
16 27 15 F Phenylalanine 

6CJB_1|Chains A, B, C, D|Cystathionine beta-lyase|Legionella pneumophila subsp. pneumophila (strain Philadelphia 1 / ATCC 33152 / DSM 7513) (272624)
>9IUP_1|Chain A|Processed angiotensin-converting enzyme 2|Homo sapiens (9606)
>2BIA_1|Chains A, B|PHOSPHOSERINE AMINOTRANSFERASE|BACILLUS ALCALOPHILUS (1445)
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 \(w=f(c)\)
6CJB , Knot 165 391 0.84 40 214 370 
MAHHHHHHMNKTHFDTRAIHAGQEPCKSTGAVMTPIYATSTYKQIAPGEHLGYEYSRTQNPTRKAYEDCIASLESGQKGFAFASGMAAINTVIDLLGSGDHVVAMDDLYGGTFRLFDKVKTRTSNLSFSFIDMSVPENIEAAITPKTKLLWLETPSNPMLKLANLRKIAAIAKKYNLITVADNTFATPWIQRPLELGFDIVLHSATKYLNGHSDVVSGVVVVGDNSVLSDKIAFLQNSCGAVAGPFDSFLVLRSLKTLSVRMQRHCENANHLANWLSSHPKIEKVIYPGLKSHPQYSLAKEQMNNFGGMISLVLKGSLEDAKRFLARCELFTLAESLGGVESLIEHPAIMTHASIPVEQRKALGIEDGFIRLSVGIEHIDDLRADLEHALG
9IUP , Knot 245 594 0.87 40 287 567 
STIEEQAKTFLDKFNHEAEDLFYQSSLASWNYNTNITEENVQNMNNAGDKWSAFLKEQSTLAQMYPLQEIQNLTVKLQLQALQQNGSSVLSEDKSKRLNTILNTMSTIYSTGKVCNPDNPQECLLLEPGLNEIMANSLDYNERLWAWESWRSEVGKQLRPLYEEYVVLKNEMARANHYEDYGDYWRGDYEVNGVDGYDYSRGQLIEDVEHTFEEIKPLYEHLHAYVRAKLMNAYPSYISPIGCLPAHLLGDMWGRFWTNLYSLTVPFGQKPNIDVTDAMVDQAWDAQRIFKEAEKFFVSVGLPNMTQGFWENSMLTDPGNVQKAVCHPTAWDLGKGDFRILMCTKVTMDDFLTAHHEMGHIQYDMAYAAQPFLLRNGANEGFHEAVGEIMSLSAATPKHLKSIGLLSPDFQEDNETEINFLLKQALTIVGTLPFTYMLEKWRWMVFKGEIPKDQWMKKWWEMKREIVGVVEPVPHDETYCDPASLFHVSNDYSFIRYYTRTLYQFQFQEALCQAAKHEGPLHKCDISNSTEAGQKLFNMLRLGKSEPWTLALENVVGAKNMNVRPLLNYFEPLFTWLKDQNKNSFVGWSTDWSP
2BIA , Knot 153 360 0.83 40 201 340 
VKQVFNFNAGPSALPKPALERAQKELLNFNDTQMSVMELSHRSQSYEEVHEQAQNLLRELLQIPNDYQILFLQGGASLQFTMLPMNLLTKGTIGNYVLTGSWSEKALKEAKLLGETHIAASTKANSYQSIPDFSEFQLNENDAYLHITSNNTIYGTQYQNFPEINHAPLIADMSSDILSRPLKVNQFGMIYAGAQKNLGPSGVTVVIVKKDLLNTKVEQVPTMLQYATHIKSDSLYNTPPTFSIYMLRNVLDWIKDLGGAEAIAKQNEEKAKIIYDTIDESNGFYVGHAEKGSRSLMNVTFNLRNEELNQQFLAKAKEQGFVGLNGHRSVGGCRASIYNAVPIDACIALRELMIQFKENA

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 \(c\).

\(|P_{f(6CJB_1)}(2) \setminus P_{f(9IUP_1)}(2)|=50\), \(|P_{f(9IUP_1)}(2) \setminus P_{f(6CJB_1)}(2)|=123\). 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 1:1100000010000100011011001000011110110100000011110011000000001000100001101001001111101111100110111010011110010110101100100000010101101011001011101000111100100111011010011111000011011000110111001101110111001000101000110111111000110001111000011111110011110010010101000000100110110001010011011100010001100010011111011101010010011100011011001111001100111100101110000111100111010111001001010100111
Pair \(Z_2\) Length of longest common subsequence
6CJB_1,9IUP_1 173 4
6CJB_1,2BIA_1 145 4
9IUP_1,2BIA_1 160 3

Newick tree

 
[
	9IUP_1:86.61,
	[
		6CJB_1:72.5,2BIA_1:72.5
	]:14.11
]

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 \((0.85)(0.8)(\frac{985 }{\log_{20} 985}-\frac{391}{\log_{20}391})=157.\)
Status Protein1 Protein2 d d1/2
Query variables 6CJB_1 9IUP_1 202 166
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]