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Parikh vectors
3VQF_1 5GCH_1 7NJG_1 Letter Amino acid
2 0 11 K Lycine
0 0 8 M Methionine
4 0 23 F Phenylalanine
4 2 19 P Proline
5 1 47 A Alanine
0 1 1 C Cysteine
7 0 28 E Glutamic acid
5 1 11 I Isoleucine
0 0 16 T Threonine
0 0 6 W Tryptophan
10 2 19 V Valine
7 0 37 D Aspartic acid
6 0 10 H Histidine
10 2 39 L Leucine
8 1 11 S Serine
8 0 35 R Arginine
9 2 37 G Glycine
0 0 9 Y Tyrosine
5 0 10 N Asparagine
4 1 11 Q Glutamine 

3VQF_1|Chain A|E3 ubiquitin-protein ligase LNX|Mus musculus (10090)
>5GCH_1|Chain A[auth E]|GAMMA-CHYMOTRYPSIN A|Bos taurus (9913)
>7NJG_1|Chain A|Xylose isomerase|Streptomyces rubiginosus (1929)
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 \(w=f(c)\)
3VQF , Knot 47 94 0.75 30 76 90 
GPLGSDHDDSFHVILNKSSPEEQLGIKLVRRVDEPGVFIFNVLNGGVADRHGQLEENDRVLAINGHDLRFGSPESAAHLIQASERRVHLVVSRQ
5GCH , Knot 11 13 0.72 18 12 11 
CGVPAIQPVLSGL
7NJG , Knot 159 388 0.81 40 206 357 
MNYQPTPEDRFTFGLWTVGWQGRDPFGDATRRALDPVESVRRLAELGAHGVTFHDDDLIPFGSSDSEREEHVKRFRQALDDTGMKVPMATTNLFTHPVFKDGGFTANDRDVRRYALRKTIRNIDLAVELGAETYVAWGGREGAESGGAKDVRDALDRMKEAFDLLGEYVTSQGYDIRFAIEPKPNEPRGDILLPTVGHALAFIERLERPELYGVNPEVGHEQMAGLNFPHGIAQALWAGKLFHIDLNGQNGIKYDQDLRFGAGDLRAAFWLVDLLESAGYSGPRHFDFKPPRTEDFDGVWASAAGCMRNYLILKERAAAFRADPEVQEALRASRLDELARPTAADGLQALLDDRSAFEEFDVDAAAARGMAFERLDQLAMDHLLGARG

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 \(c\).

\(|P_{f(3VQF_1)}(2) \setminus P_{f(5GCH_1)}(2)|=72\), \(|P_{f(5GCH_1)}(2) \setminus P_{f(3VQF_1)}(2)|=8\). 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 1:1111000000101110000100011101100100111111011011110001010000011110100101101001101101000010111000
Pair \(Z_2\) Length of longest common subsequence
3VQF_1,5GCH_1 80 2
3VQF_1,7NJG_1 166 5
5GCH_1,7NJG_1 204 2

Newick tree

 
[
	7NJG_1:10.85,
	[
		3VQF_1:40,5GCH_1:40
	]:64.85
]

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 \((0.85)(0.8)(\frac{107 }{\log_{20} 107}-\frac{13}{\log_{20}13})=36.3\)
Status Protein1 Protein2 d d1/2
Query variables 3VQF_1 5GCH_1 44 24.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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