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Parikh vectors
2KTM_1 1GNU_1 5LCY_1 Letter Amino acid
9 3 3 T Threonine
3 6 9 A Alanine
8 3 4 Q Glutamine
9 4 4 H Histidine
8 5 5 S Serine
4 8 1 Y Tyrosine
8 9 6 V Valine
2 0 1 C Cysteine
1 9 11 L Leucine
4 2 4 M Methionine
2 9 0 F Phenylalanine
5 8 10 R Arginine
5 5 6 G Glycine
3 7 6 I Isoleucine
3 12 4 K Lycine
2 7 4 P Proline
0 0 0 W Tryptophan
5 2 1 N Asparagine
3 7 3 D Aspartic acid
5 11 9 E Glutamic acid 

2KTM_1|Chain A|Major prion protein|Ovis aries (9940)
>1GNU_1|Chain A|GABARAP|HOMO SAPIENS (9606)
>5LCY_1|Chains A, B, C, D|Frmr|Salmonella enterica serovar Typhimurium (90371)
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 \(w=f(c)\)
2KTM , Knot 45 89 0.75 38 72 82 
MGSSHHHHHHSSGLVPRGSHMRPVDQYSNQNNFVHDCVNITVKQHTVTTTTKGENFTETDIKAMERVVEQMCITQYQRESQAYYQRGAS
1GNU , Knot 56 117 0.76 36 92 113 
MKFVYKEEHPFEKRRSEGEKIRKKYPDRVPVIVEKAPKARIGDLDKKKYLVPSDLTVGQFYFLIRKRIHLRAEDALFFFVNNVIPPTSATMGQLYQEHHEEDFFLYIAYSDESVYGL
5LCY , Knot 47 91 0.77 36 73 86 
MPHSPEDKKRILTRVRRIRGQVEALERALESGEPCLAILQQIAAVRGASNGLMSEMVEIHLKDHLVSGETTPDQRAVRMAEIGHLLRAYLK

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 \(c\).

\(|P_{f(2KTM_1)}(2) \setminus P_{f(1GNU_1)}(2)|=54\), \(|P_{f(1GNU_1)}(2) \setminus P_{f(2KTM_1)}(2)|=74\). 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 1:11000000000011110100101100000000110001010100001000001001000010110011001010000000010000110
Pair \(Z_2\) Length of longest common subsequence
2KTM_1,1GNU_1 128 3
2KTM_1,5LCY_1 109 4
1GNU_1,5LCY_1 111 3

Newick tree

 
[
	1GNU_1:61.59,
	[
		2KTM_1:54.5,5LCY_1:54.5
	]:7.09
]

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 \((0.85)(0.8)(\frac{206 }{\log_{20} 206}-\frac{89}{\log_{20}89})=38.3\)
Status Protein1 Protein2 d d1/2
Query variables 2KTM_1 1GNU_1 50 42
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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