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Parikh vectors
4WVL_1 8JVL_1 4KPT_1 Letter Amino acid
22 10 1 R Arginine
11 3 1 H Histidine
3 2 0 C Cysteine
7 3 3 W Tryptophan
15 4 12 Y Tyrosine
29 16 17 L Leucine
13 5 12 F Phenylalanine
21 8 18 S Serine
17 7 17 T Threonine
20 9 17 D Aspartic acid
30 9 9 E Glutamic acid
19 11 14 I Isoleucine
17 7 21 G Glycine
24 11 26 K Lycine
10 5 12 M Methionine
18 17 7 P Proline
23 4 14 V Valine
20 11 29 A Alanine
16 6 11 N Asparagine
12 8 10 Q Glutamine 

4WVL_1|Chain A|Histone-lysine N-methyltransferase, H3 lysine-79 specific|Homo sapiens (9606)
>8JVL_1|Chain A|Ubiquitin-conjugating enzyme E2 T|Homo sapiens (9606)
>4KPT_1|Chains A, B|Glutamine ABC transporter permease and substrate binding protein protein|Lactococcus lactis subsp. lactis (272623)
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 \(w=f(c)\)
4WVL , Knot 157 347 0.88 40 224 339 
MGEKLELRLKSPVGAEPAVYPWPLPVYDKHHDAAHEIIETIRWVCEEIPDLKLAMENYVLIDYDTKSFESMQRLCDKYNRAIDSIHQLWKGTTQPMKLNTRPSTGLLRHILQQVYNHSVTDPEKLNNYEPFSPEVYGETSFDLVAQMIDEIKMTDDDLFVDLGSGVGQVVLQVAAATNCKHHYGVEKADIPAKYAETMDREFRKWMKWYGKKHAEYTLERGDFLSEEWRERIANTSVIFVNNFAFGPEVDHQLKERFANMKEGGRIVSSKPFAPLNFRINSRNLSDIGTIMRVVELSPLKGSVSWTGKPVSYYLHTIDRTILENYFSSLKNPKLREEQEAARRRQQR
8JVL , Knot 77 156 0.83 40 128 152 
GSMQRASRLKRELHMLATEPPPGITCWQDKDQMDDLRAQILGGANTPYEKGVFKLEVIIPERYPFEPPQIRFLTPIYHPNIDSAGRICLDVLKLPPKGAWRPSLNIATVLTSIQLLMSEPNPDDPLMADISSEFKYNKPAFLKNARQWTEKHARQK
4KPT , Knot 111 251 0.81 38 154 238 
MKKLFFALAMMLATVTAFLVAPSVKAETTVKIASDSSYAPFEFQNGQKKWVGIDVDIMQEVAKINDWKLEMSYPGFDAALQNLKAGQVDGIIAGMTITDERKETFDFSNPYYTSALTIATTKDSKLSDYSDLKGKAVGAKNGTAAQTWLQENQKKYGYTIKTYSDGVHMFAALSSGNIAGAMDEVPVISYAMKQGQDLAMNFPSISLPGGYGFAVMKGKNSTLVDGFNKALAEMKSNGDYDKILKKYGITA

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 \(c\).

\(|P_{f(4WVL_1)}(2) \setminus P_{f(8JVL_1)}(2)|=142\), \(|P_{f(8JVL_1)}(2) \setminus P_{f(4WVL_1)}(2)|=46\). 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 1:11001010100111101110111111000000110011001011000110101110001110000001001001000000110010011010001101000100111001100100001001001000011010101000101110110010100001110110111011101111000000011001011100100100010011010100010001001011000100011000111100111110100010001101001101100011111010100001001101101101011010101010110001001000110001001001010000011000000
Pair \(Z_2\) Length of longest common subsequence
4WVL_1,8JVL_1 188 4
4WVL_1,4KPT_1 178 3
8JVL_1,4KPT_1 168 4

Newick tree

 
[
	4WVL_1:93.91,
	[
		4KPT_1:84,8JVL_1:84
	]:9.91
]

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 \((0.85)(0.8)(\frac{503 }{\log_{20} 503}-\frac{156}{\log_{20}156})=101.\)
Status Protein1 Protein2 d d1/2
Query variables 4WVL_1 8JVL_1 131 92.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]