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Parikh vectors
4XEN_1 6MLI_1 6ASA_1 Letter Amino acid
11 7 10 R Arginine
3 11 9 Q Glutamine
2 10 14 E Glutamic acid
3 11 6 F Phenylalanine
6 2 0 W Tryptophan
14 5 4 N Asparagine
8 2 3 C Cysteine
8 19 11 L Leucine
6 24 11 K Lycine
2 7 4 P Proline
6 12 15 V Valine
7 25 14 D Aspartic acid
12 19 11 G Glycine
6 12 11 I Isoleucine
12 29 9 A Alanine
1 1 4 H Histidine
2 3 4 M Methionine
10 15 8 S Serine
7 14 12 T Threonine
3 10 8 Y Tyrosine 

4XEN_1|Chain A|Lysozyme C|Gallus gallus (9031)
>6MLI_1|Chain A[auth E]|Lysine/arginine/ornithine-binding periplasmic protein|Salmonella typhimurium (strain LT2 / SGSC1412 / ATCC 700720) (99287)
>6ASA_1|Chain A|GTPase KRas|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 \(w=f(c)\)
4XEN , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
6MLI , Knot 108 238 0.82 40 150 225 
ALPQTVRIGTDTTYAPFSSKDAKGEFIGFDIDLGNEMCKRMQVKCTWVASDFDALIPSLKAKKIDAIISSLSITDKAQQEIAFSDKLYAADSRLIAAKGSPIQPTLESLKGKHVGVLQGSTQEAYANDNWRTKGVDVVAYANQDLIYSDLTAGRLDAALQDEVAASEGFLKQPAGKEYAFAGPSVKDKKYFGDGTGVGLRKDDTELKAAFDKALTELRQDGTYDKMAKKYFDFNVYGD
6ASA , Knot 81 168 0.82 38 132 166 
MTEYKLVVVGAGGVGKSALTIQLIQNHFVDEYEPTIEDSYRKQVVIDGETCLLDILDTAGQEEYSAMRDQYMRTGEGFLCVFAINNTKSFEDIHHYREQIKRVKDSEDVPMVLVGNKCDLPSRTVDTKQAQDLARSYGIPFIETSAKTRQGVDDAFYTLVREIRKHKE

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 \(c\).

\(|P_{f(4XEN_1)}(2) \setminus P_{f(6MLI_1)}(2)|=58\), \(|P_{f(6MLI_1)}(2) \setminus P_{f(4XEN_1)}(2)|=104\). 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 1:011100011111000110000100110110110100010001000000100001110100011000100110001001100111000101010010011001011011111000001001011101001
Pair \(Z_2\) Length of longest common subsequence
4XEN_1,6MLI_1 162 4
4XEN_1,6ASA_1 178 3
6MLI_1,6ASA_1 150 4

Newick tree

 
[
	4XEN_1:88.20,
	[
		6MLI_1:75,6ASA_1:75
	]:13.20
]

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 \((0.85)(0.8)(\frac{367 }{\log_{20} 367}-\frac{129}{\log_{20}129})=72.5\)
Status Protein1 Protein2 d d1/2
Query variables 4XEN_1 6MLI_1 91 70
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]