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Parikh vectors
4EJE_1 8TMC_1 3MJH_1 Letter Amino acid
8 21 8 D Aspartic acid
9 13 2 H Histidine
15 37 15 L Leucine
13 15 3 P Proline
12 18 12 S Serine
3 8 9 Q Glutamine
5 35 11 E Glutamic acid
9 28 9 I Isoleucine
11 24 13 K Lycine
7 11 4 M Methionine
2 5 2 W Tryptophan
10 18 5 Y Tyrosine
6 23 6 R Arginine
7 12 9 N Asparagine
2 1 2 C Cysteine
9 16 7 G Glycine
10 21 9 T Threonine
3 8 20 A Alanine
4 20 10 F Phenylalanine
14 39 12 V Valine 

4EJE_1|Chains A, B|Tumor susceptibility gene 101 protein|Homo sapiens (9606)
>8TMC_1|Chains A, B, C, D, E|Cobalt/magnesium transport protein CorA|Thermotoga maritima (2336)
>3MJH_1|Chains A, C|Ras-related protein Rab-5A|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 \(w=f(c)\)
4EJE , Knot 73 159 0.77 40 118 153 
MRGSHHHHHHGMASMAVSESQLKKMVSKYKYRDLTVRETVNVITLYKDLKPVLDSYVFNDGSSRELMNLTGTIPVPYRGNTYNIPICLWLLDTYPYNPPICFVKPTSSMTIKTGKHVDANGKIYLPYLHEWKHPQSDLLGLIQVMIVVFGDEPPVFSRP
8TMC , Knot 155 373 0.82 40 202 348 
MGSSHHHHHHSSGRENLYFQGHMEEKRLSAKKGLPPGTLVYTGKYREDFEIEVMNYSIEEFREFKTTDVESVLPFRDSSTPTWINITGIHRTDVVQRVGEFFGIHPLVLEDILNVHQRPKVEFFENYVFIVLKMFTYDKNLHELESEQVSLILTKNCVLMFQEKIGDVFDPVRERIRYNRGIIRKKRADYLLYSLIDALVDDYFVLLEKIDDEIDVLEEEVLERPEKETVQRTHQLKRNLVELRKTIWPLREVLSSLYRDVPPLIEKETVPYFRDVYDHTIQIADTVETFRDIVSGLLDVYLSSVSNKTNEVMKVLTIIATIFMPLTFIAGIYGMNFEYMPELRWKWGYPVVLAVMGVIAVIMVVYFKKKKWL
3MJH , Knot 82 168 0.83 40 131 164 
NKICQFKLVLLGESAVGKSSLVLRFVKGQFHEFQESTIGAAFLTQTVCLDDTTVKFEIWDTAGLERYHSLAPMYYRGAQAAIVVYDITNEESFARAKNWVKELQRQASPNIVIALSGNKADLANKRAVDFQEAQSYADDNSLLFMETSAKTSMNVNEIFMAIAKKLPK

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 \(c\).

\(|P_{f(4EJE_1)}(2) \setminus P_{f(8TMC_1)}(2)|=42\), \(|P_{f(8TMC_1)}(2) \setminus P_{f(4EJE_1)}(2)|=126\). 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 1:101000000011101110000100110000000101000101101000101110001100100001101010111100100001110111100010011101101000101001001010101011010010010001111101111111001111001
Pair \(Z_2\) Length of longest common subsequence
4EJE_1,8TMC_1 168 7
4EJE_1,3MJH_1 161 3
8TMC_1,3MJH_1 179 3

Newick tree

 
[
	8TMC_1:88.79,
	[
		4EJE_1:80.5,3MJH_1:80.5
	]:8.29
]

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 \((0.85)(0.8)(\frac{532 }{\log_{20} 532}-\frac{159}{\log_{20}159})=108.\)
Status Protein1 Protein2 d d1/2
Query variables 4EJE_1 8TMC_1 136 95.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]