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Parikh vectors
4ZVN_1 4UZX_1 5LKJ_1 Letter Amino acid
11 5 18 D Aspartic acid
15 6 20 E Glutamic acid
11 1 18 P Proline
14 5 30 S Serine
15 0 23 T Threonine
12 4 20 Q Glutamine
8 0 17 Y Tyrosine
4 0 13 M Methionine
6 0 4 W Tryptophan
8 4 14 R Arginine
6 5 12 N Asparagine
4 0 3 C Cysteine
10 3 19 I Isoleucine
20 10 32 L Leucine
14 8 17 K Lycine
18 6 25 A Alanine
18 4 23 G Glycine
6 1 13 H Histidine
14 2 24 F Phenylalanine
16 3 26 V Valine 

4ZVN_1|Chains A, B|Ribosyldihydronicotinamide dehydrogenase [quinone]|Homo sapiens (9606)
>4UZX_1|Chain A|PROTEIN THO1|SACCHAROMYCES CEREVISIAE (4932)
>5LKJ_1|Chains A, B, C, D|Histone-arginine methyltransferase CARM1|Mus musculus (10090)
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 \(w=f(c)\)
4ZVN , Knot 108 230 0.85 40 169 224 
AGKKVLIVYAHQEPKSFNGSLKNVAVDELSRQGCTVTVSDLYAMNFEPRATDKDITGTLSNPEVFNYGVETHEAYKQRSLASDITDEQKKVREADLVIFQFPLYWFSVPAILKGWMDRVLCQGFAFDIPGFYDSGLLQGKLALLSVTTGGTAEMYTKTGVNGDSRYFLWPLQHGTLHFCGFKVLAPQISFAPEIASEEERKGMVAAWSQRLQTIWKEEPIPCTAHWHFGQ
4UZX , Knot 36 67 0.75 30 57 64 
GSALSPEEIKAKALDLLNKKLHRANKFGQDQADIDSLQRQINRVEKFGVDLNSKLAEELGLVSRKNE
5LKJ , Knot 161 371 0.85 40 217 366 
GHMGHTLERSVFSERTEESSAVQYFQFYGYLSQQQNMMQDYVRTGTYQRAILQNHTDFKDKIVLDVGCGSGILSFFAAQAGARKIYAVEASTMAQHAEVLVKSNNLTDRIVVIPGKVEEVSLPEQVDIIISEPMGYMLFNERMLESYLHAKKYLKPSGNMFPTIGDVHLAPFTDEQLYMEQFTKANFWYQPSFHGVDLSALRGAAVDEYFRQPVVDTFDIRILMAKSVKYTVNFLEAKEGDLHRIEIPFKFHMLHSGLVHGLAFWFDVAFIGSIMTVWLSTAPTEPLTHWYQVRCLFQSPLFAKAGDTLSGTCLLIANKRQSYDISIVAQVDQTGSKSSNLLDLKNPFFRYTGTTPSPPPGSHYTSPSENM

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 \(c\).

\(|P_{f(4ZVN_1)}(2) \setminus P_{f(4UZX_1)}(2)|=134\), \(|P_{f(4UZX_1)}(2) \setminus P_{f(4ZVN_1)}(2)|=22\). 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 1:11001111010001001010100111001000100101001011010101000010101001011001100001000001100100000010010111101110110111110111001100111101111000111010111101001101010000110100001111100101010110111101011101100000011111100010011000111001010110
Pair \(Z_2\) Length of longest common subsequence
4ZVN_1,4UZX_1 156 3
4ZVN_1,5LKJ_1 176 4
4UZX_1,5LKJ_1 194 4

Newick tree

 
[
	5LKJ_1:96.99,
	[
		4ZVN_1:78,4UZX_1:78
	]:18.99
]

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 \((0.85)(0.8)(\frac{297 }{\log_{20} 297}-\frac{67}{\log_{20}67})=73.8\)
Status Protein1 Protein2 d d1/2
Query variables 4ZVN_1 4UZX_1 93 59
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]