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Parikh vectors
8JKU_1 6FCE_1 2MQI_1 Letter Amino acid
2 0 5 Q Glutamine
9 1 4 F Phenylalanine
18 0 2 P Proline
18 0 6 T Threonine
26 0 9 G Glycine
27 0 10 L Leucine
13 1 9 R Arginine
16 1 6 D Aspartic acid
1 0 3 C Cysteine
3 1 2 W Tryptophan
7 0 6 Y Tyrosine
60 0 5 A Alanine
5 0 3 N Asparagine
20 0 6 E Glutamic acid
13 1 10 H Histidine
13 0 4 I Isoleucine
3 0 9 K Lycine
9 0 2 M Methionine
13 0 5 S Serine
21 0 5 V Valine 

8JKU_1|Chains A, B|Ancestral imine reductase N559|synthetic construct (32630)
>6FCE_1|Chain A|ACP-HIS-DPHE-ARG-TRP-ASP-NH2|synthetic construct (32630)
>2MQI_1|Chain A|Tyrosine-protein kinase Fyn|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 \(w=f(c)\)
8JKU , Knot 123 297 0.78 40 162 268 
MGHHHHHHMSNNTPVTVIGLGPMGRALAGAFLAAGHPTTVWNRTPGKADALVARGAVRAATVAEAVAASPLVVVCVTDYDAMRAILEPAADALAGRVLVNLTSDTPERAREAAAWAAEHGADYLDGAIMVPPPMIGTPEALIFYSGPREVFEAHRATLAALGGSTYLGEDPGLAALYDVALLDLFWTTMSGYLHAAALVGSEGVPAAEFAPFATAIFDSLSFLEEMAEQIDAGEYPGDESTLAMNAAGIDHIIHASRAQGIDTALPEAVKALAERAIAAGHGEDSFTSVIEVLRKPA
6FCE , Knot 7 7 0.64 12 6 5 
XHFRWDX
2MQI , Knot 58 111 0.82 40 90 106 
NKVHHHHHHMEWYFGKLGRKDAERQLLSFGNPRGTFLIRESETTKGAYSLSIRDWDDMKGDHVKHYKIRKLDNGGYYITTRAQFETLQQLVQHYSERAAGLCCRLVVPCHK

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 \(c\).

\(|P_{f(8JKU_1)}(2) \setminus P_{f(6FCE_1)}(2)|=162\), \(|P_{f(6FCE_1)}(2) \setminus P_{f(8JKU_1)}(2)|=6\). 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 1:110000001000011011111111011111111110100110001101011110111011011011110111110100001101110111011110111010000100100111111001100101111111111101011110011001101001011111100011001111110011110111001010101111110011111011111011100101100110010110011000011101111001101001011001110110111001111101000100110110011
Pair \(Z_2\) Length of longest common subsequence
8JKU_1,6FCE_1 168 1
8JKU_1,2MQI_1 172 7
6FCE_1,2MQI_1 94 2

Newick tree

 
[
	8JKU_1:94.33,
	[
		6FCE_1:47,2MQI_1:47
	]:47.33
]

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 \((0.85)(0.8)(\frac{304 }{\log_{20} 304}-\frac{7}{\log_{20}7})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 8JKU_1 6FCE_1 122 62.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]