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Parikh vectors
5TQB_1 3LJT_1 2ZKJ_1 Letter Amino acid
17 3 24 P Proline
15 19 37 S Serine
21 10 18 R Arginine
8 18 24 D Aspartic acid
9 13 21 E Glutamic acid
25 12 19 K Lycine
26 11 28 V Valine
9 14 12 H Histidine
17 26 40 L Leucine
9 6 18 F Phenylalanine
14 13 17 T Threonine
9 6 19 Q Glutamine
6 3 12 M Methionine
10 11 25 I Isoleucine
4 2 2 W Tryptophan
4 4 17 Y Tyrosine
37 20 21 A Alanine
5 6 19 N Asparagine
2 8 6 C Cysteine
30 13 15 G Glycine 

5TQB_1|Chain A|60S ribosomal protein L4-like protein|Chaetomium thermophilum (759272)
>3LJT_1|Chain A|A disintegrin and metalloproteinase with thrombospondin motifs 5|Homo sapiens (9606)
>2ZKJ_1|Chains A, B|[Pyruvate dehydrogenase [lipoamide]] kinase isozyme 4|Homo sapiens (9606)
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 \(w=f(c)\)
5TQB , Knot 119 277 0.80 40 164 263 
MASRPTVTVFGADGKPTGATEVLPKVFSAPIRPDIVKHVHTGMAKNKRQPYAVSEKAGHQTSAESWGTGRAVARIPRVSGGGTHRAGQGAFGNMCRSGRMFAPTKIWRKWHVKINQGQKRFATASALAASAVAPLLMARGHQVSTVPEVPLVVDSAAVAGDAVAKTAAAYKLLKAIGAGPDVEKVKKSKKLRAGKGKMRGRRHRQRRGPLIVYSPEHDGKELVKGFRNIPGVETCPVDALNLLQLAPGGHLGRFIVWTSAAIKQLDAVYESKKGFFL
3LJT , Knot 101 218 0.83 40 150 212 
ASRARQVELLLVADASMARKYGRGLQHYLLTLASIANRLYSHASIENHIRLAVVKVVVLGDKDKSLEVSKNAATTLKNFCKWQHQHNQLGDDHEEHYDAAILFTREDLCGHHSCDTLGMADVGTICSPERSCAVIEDDGLHAAFTVAHEIGHLLGLSHDDSKFCEETFGSTEDKRLMSSILTSIDASKPWSKCTSATITEFLDDGHGNCLLDLPRKQI
2ZKJ , Knot 173 394 0.87 40 234 382 
ASVPREVEHFSRYSPSPLSMKQLLDFGSENACERTSFAFLRQELPVRLANILKEIDILPTQLVNTSSVQLVKSWYIQSLMDLVEFHEKSPDDQKALSDFVDTLIKVRNRHHNVVPTMAQGIIEYKDACTVDPVTNQNLQYFLDRFYMNRISTRMLMNQHILIFSDSQTGNPSHIGSIDPNCDVVAVVQDAFECSRMLCDQYYLSSPELKLTQVNGKFPDQPIHIVYVPSHLHHMLFELFKNAMRATVEHQENQPSLTPIEVIVVLGKEDLTIKISDRGGGVPLRIIDRLFSYTYSTAPTPVMDNSRNAPLAGFGYGLPISRLYAKYFQGDLNLYSLSGYGTDAIIYLKALSSESIEKLPVFNKSAFKHYQMSSEADDWCIPSREPKNLAKEVAM

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 \(c\).

\(|P_{f(5TQB_1)}(2) \setminus P_{f(3LJT_1)}(2)|=93\), \(|P_{f(3LJT_1)}(2) \setminus P_{f(5TQB_1)}(2)|=79\). 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 1:1100101011110101011001110110111010110010011100000101100011000010011010111011010111000110111101000101111001100101010010001101011110111111110100100110111110011111011100111001101111110100100000101101010100000001111100100010011011001111000110110110111110110111100111001011000001111
Pair \(Z_2\) Length of longest common subsequence
5TQB_1,3LJT_1 172 3
5TQB_1,2ZKJ_1 180 3
3LJT_1,2ZKJ_1 176 4

Newick tree

 
[
	2ZKJ_1:89.98,
	[
		5TQB_1:86,3LJT_1:86
	]:3.98
]

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 \((0.85)(0.8)(\frac{495 }{\log_{20} 495}-\frac{218}{\log_{20}218})=80.0\)
Status Protein1 Protein2 d d1/2
Query variables 5TQB_1 3LJT_1 100 90
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]