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Parikh vectors
8CGF_1 7JSP_1 7NZH_1 Letter Amino acid
5 4 6 H Histidine
12 11 9 K Lycine
1 4 3 M Methionine
6 5 15 F Phenylalanine
4 12 12 D Aspartic acid
7 7 3 Q Glutamine
3 1 7 G Glycine
1 1 4 W Tryptophan
2 7 4 Y Tyrosine
9 2 8 R Arginine
10 12 19 E Glutamic acid
6 2 2 C Cysteine
6 8 8 I Isoleucine
16 11 15 L Leucine
1 6 13 P Proline
8 9 6 S Serine
15 8 12 T Threonine
3 11 17 V Valine
8 8 8 A Alanine
6 8 9 N Asparagine 

8CGF_1|Chain A|Interleukin-4|Homo sapiens (9606)
>7JSP_1|Chain A|Transcription initiation factor TFIID subunit 1|Homo sapiens (9606)
>7NZH_1|Chains A[auth AAA], C[auth CCC]|HLA class II histocompatibility antigen, DR alpha chain|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 \(w=f(c)\)
8CGF , Knot 67 129 0.84 40 101 125 
HKCDITLQEIIKTLNSLTEQKTLCTELTVTDIFAASKNTTEKETFCRAATVLRQFYSHHEKDTRCLGATAQQFHRHKQLIRFLKRLDRNLWGLAGLNSCPVKEANQSTLENFLERLKTIMREKYSKCSS
7JSP , Knot 71 137 0.85 40 110 133 
SMDDDQVAFSFILDNIVTQKMMAVPDSWPFHHPVNKKFVPDYYKVIVNPMDLETIRKNISKHKYQSRESFLDDVNLILANSVKYNGPESQYTKTAQEIVNVCYQTLTEYDEHLTQLEKDICTAKEAALEEAELESLD
7NZH , Knot 85 180 0.81 40 139 176 
KEEHVIIQAEFYLNPDQSGEFMFDFDGDEIFHVDMAKKETVWRLEEFGRFASFEAQGALANIAVDKANLEIMTKRSNYTPITNVPPEVTVLTNSPVELREPNVLICFIDKFTPPVVNVTWLRNGKPVTTGVSETVFLPREDHLFRKFHYLPFLPSTEDVYDCRVEHWGLDEPLLKHWEFD

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 \(c\).

\(|P_{f(8CGF_1)}(2) \setminus P_{f(7JSP_1)}(2)|=60\), \(|P_{f(7JSP_1)}(2) \setminus P_{f(8CGF_1)}(2)|=69\). 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 1:000010100110010010000010001010011110000000001001101100100000000001110100100000110110010001111111000110010000100110010011000000000
Pair \(Z_2\) Length of longest common subsequence
8CGF_1,7JSP_1 129 3
8CGF_1,7NZH_1 154 3
7JSP_1,7NZH_1 145 3

Newick tree

 
[
	7NZH_1:77.91,
	[
		8CGF_1:64.5,7JSP_1:64.5
	]:13.41
]

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 \((0.85)(0.8)(\frac{266 }{\log_{20} 266}-\frac{129}{\log_{20}129})=42.9\)
Status Protein1 Protein2 d d1/2
Query variables 8CGF_1 7JSP_1 55 52
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]