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Parikh vectors
6YRS_1 4CHU_1 3TLT_1 Letter Amino acid
5 6 12 N Asparagine
24 19 8 G Glycine
8 3 2 H Histidine
31 20 31 L Leucine
12 17 23 S Serine
3 1 5 W Tryptophan
22 13 18 R Arginine
18 12 19 D Aspartic acid
20 13 32 V Valine
39 13 28 A Alanine
13 6 28 I Isoleucine
4 2 25 F Phenylalanine
11 4 18 P Proline
16 9 19 T Threonine
1 0 1 C Cysteine
5 7 7 Q Glutamine
19 7 16 E Glutamic acid
8 6 10 K Lycine
6 2 4 M Methionine
4 3 15 Y Tyrosine 

6YRS_1|Chains A, B|ancestral beta-lactamase|synthetic construct (32630)
>4CHU_1|Chains A, B|HTH-TYPE TRANSCRIPTIONAL REGULATOR ISCR|ESCHERICHIA COLI K-12 (83333)
>3TLT_1|Chains A, B, C, D, E|Glr4197 protein|Gloeobacter violaceus (33072)
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 \(w=f(c)\)
6YRS , Knot 112 269 0.77 40 146 250 
MAAQLSEQLAELEKRSGGRVGVIVLDTATGRRIAYRGDERFPMMSTFKALLAAAVLARVDAGKERLGRRITYSKEDLVDYSPVTEKHVGDGMTVAELCEAAITLSDNTAANLLLEALGGPAALTAFLRSIGDEVTRLDRWEPELNEAAPGDERDTTMPAAMAATLRTLLLGDALSPASRQQLVDWLVANKTGGKLLRAGLPADWRIGDKSGAGEHGSRNIIAVIGPPGRAPIIVVIYLTESQVDADARDAVIAEVGRLVVEAFHHHHHH
4CHU , Knot 77 163 0.80 38 117 158 
MGRLTSKGRYAVTAMLDVALNSEAGPVPLADISERQGISLSYLEQLFSRLRKNGLVSSVRGPGGGYLLGKDASSIAVGEVISAVDESVDATRSQGKGGSQGGDKSLTHALWRDLSDRLTGFLNNITLGELVNNQEVLDVSGRQHTHDAPRTRTQDAIDVKLRA
3TLT , Knot 138 321 0.82 40 187 304 
GSAAAQDMVSPPPPIADEPLTVNTGIYLIECYSLDDKAETFKVNAFLSLSWKDRRLAFDPVRSGVRVKTYEPEAIWIPEIRFVNVENARDADVVDISVSPDGTVQYLERFSARVLSPLDFRRYPFDSQTLHIYLIVRSVDTRNIVLAVDLEKVGKNDDVFLTGWDIESFTAVVKPANFALEDRLESKLDYQLRISRQYFSYIPNIILPMLFILFISWTAFWSTSYEANVTLVVSTLIAFIAFNILVETNLPKTPYMTYTGAIIFMIYLFYFVAVIEVTVQHYLKVESQPARAASITRASRIAFPVVFLLANIILAFLFFGF

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 \(c\).

\(|P_{f(6YRS_1)}(2) \setminus P_{f(4CHU_1)}(2)|=73\), \(|P_{f(4CHU_1)}(2) \setminus P_{f(6YRS_1)}(2)|=44\). 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 1:11101000110100001101111110010100110010001111001011111111101011000110010000001100011000011011011010011101000011011101111111101110011001001001010100111100000011111110100111101101100001101111000110110111110101100011100100011111111101111111010000101010011110110111011000000
Pair \(Z_2\) Length of longest common subsequence
6YRS_1,4CHU_1 117 4
6YRS_1,3TLT_1 143 4
4CHU_1,3TLT_1 154 3

Newick tree

 
[
	3TLT_1:78.86,
	[
		6YRS_1:58.5,4CHU_1:58.5
	]:20.36
]

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 \((0.85)(0.8)(\frac{432 }{\log_{20} 432}-\frac{163}{\log_{20}163})=79.8\)
Status Protein1 Protein2 d d1/2
Query variables 6YRS_1 4CHU_1 96 77
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]