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Parikh vectors
4IEM_1 7CKP_1 6URX_1 Letter Amino acid
11 10 17 N Asparagine
8 24 23 I Isoleucine
11 26 17 T Threonine
21 18 15 P Proline
7 3 1 W Tryptophan
27 62 24 A Alanine
17 23 21 R Arginine
7 1 2 C Cysteine
10 11 14 Q Glutamine
24 31 15 E Glutamic acid
29 10 2 K Lycine
12 9 14 Y Tyrosine
37 41 22 L Leucine
2 8 4 M Methionine
9 12 14 F Phenylalanine
20 18 19 S Serine
15 34 15 V Valine
19 33 8 D Aspartic acid
25 48 17 G Glycine
6 13 4 H Histidine 

4IEM_1|Chains A, E[auth B], I[auth C], M[auth D]|DNA-(apurinic or apyrimidinic site) lyase|Homo sapiens (9606)
>7CKP_1|Chain A|Enolase|Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv) (83332)
>6URX_1|Chain A|Ricin|Ricinus communis (3988)
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 \(w=f(c)\)
4IEM , Knot 138 317 0.83 40 194 300 
PKRGKKGAVAEDGDELRTEPEAKKSKTAAKKNDKEAAGEGPALYEDPPDQKTSPSGKPATLKICSWNVDGLRAWIKKKGLDWVKEEAPDILCLQETKCSENKLPAELQELPGLSHQYWSAPSDKEGYSGVGLLSRQCPLKVSYGIGDEEHDQEGRVIVAEFDSFVLVTAYVPNAGRGLVRLEYRQRWDEAFRKFLKGLASRKPLVLCGDLNVAHEEIDLRNPKGNKKNAGFTPQERQGFGELLQAVPLADSFRHLYPNTPYAYTFWTYMMNARSKNVGWRLDYFLLSHSLLPALCDSKIRSKALGSDHCPITLYLAL
7CKP , Knot 170 435 0.79 40 202 405 
MHHHHHHMPIIEQVRAREILDSRGNPTVEVEVALIDGTFARAAVPSGASTGEHEAVELRDGGDRYGGKGVQKAVQAVLDEIGPAVIGLNADDQRLVDQALVDLDGTPDKSRLGGNAILGVSLAVAKAAADSAELPLFRYVGGPNAHILPVPMMNILNGGAHADTAVDIQEFMVAPIGAPSFVEALRWGAEVYHALKSVLKKEGLSTGLGDEGGFAPDVAGTTAALDLISRAIESAGLRPGADVALALDAAATEFFTDGTGYVFEGTTRTADQMTEFYAGLLGAYPLVSIEDPLSEDDWDGWAALTASIGDRVQIVGDDIFVTNPERLEEGIERGVANALLVKVNQIGTLTETLDAVTLAHHGGYRTMISHRSGETEDTMIADLAVAIGSGQIKTGAPARSERVAKYNQLLRIEEALGDAARYAGDLAFPRFACET
6URX , Knot 116 268 0.80 40 171 260 
MIFPKQYPIINFTTAGATVQSYTNFIRAVRGRLTTGADVRHEIPVLPNRVGLPINQRFILVELSNHAELSVTLALDVTNAYVVGYRAGNSAYFFHPDNQEDAEAITHLFTDVQNRYTFAFGGNYDRLEQLAGNLRENIELGNGPLEEAISALYYYSTGGTQLPTLARSFIICIQMISEAARFQYIEGEMRTRIRYNRRSAPDPSVITLENSWGRLSTAIQESNQGAFASPIQLQRRNGSKFSVYDVSILIPIIALMVYRCAPPPSSQF

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 \(c\).

\(|P_{f(4IEM_1)}(2) \setminus P_{f(7CKP_1)}(2)|=70\), \(|P_{f(7CKP_1)}(2) \setminus P_{f(4IEM_1)}(2)|=78\). 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 1:10010011110010010001010000011000000111011110001100000101011010100101011011100011011000110110100000000011101001111000010110000100111110000110100111000000010111101001111010110110111010000010011001101110001111010101100010100101000011101000011101101111100100101001010011001101000011101001110001111100001000111000011010111
Pair \(Z_2\) Length of longest common subsequence
4IEM_1,7CKP_1 148 3
4IEM_1,6URX_1 187 3
7CKP_1,6URX_1 145 4

Newick tree

 
[
	4IEM_1:87.90,
	[
		7CKP_1:72.5,6URX_1:72.5
	]:15.40
]

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 \((0.85)(0.8)(\frac{752 }{\log_{20} 752}-\frac{317}{\log_{20}317})=119.\)
Status Protein1 Protein2 d d1/2
Query variables 4IEM_1 7CKP_1 147 128
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]