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Parikh vectors
3KSH_1 9FCM_1 5SHC_1 Letter Amino acid
11 11 16 V Valine
12 15 21 A Alanine
3 8 14 R Arginine
13 6 13 H Histidine
18 3 21 I Isoleucine
10 5 18 K Lycine
1 4 7 W Tryptophan
2 13 12 Y Tyrosine
9 3 14 N Asparagine
13 9 15 D Aspartic acid
7 9 17 Q Glutamine
3 3 12 M Methionine
6 4 14 P Proline
5 4 15 F Phenylalanine
7 10 22 T Threonine
3 2 12 C Cysteine
9 4 24 E Glutamic acid
8 15 16 G Glycine
14 6 36 L Leucine
6 15 24 S Serine 

3KSH_1|Chain A|Putative uncharacterized protein|Staphylococcus aureus (282458)
>9FCM_1|Chains A, B, C|Single-domain antibody R3DC23|Lama glama (9844)
>5SHC_1|Chains A, B, C, D|cAMP and cAMP-inhibited cGMP 3',5'-cyclic phosphodiesterase 10A|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 \(w=f(c)\)
3KSH , Knot 78 160 0.82 40 121 153 
HHHHHHMTTINPTNYTLLKKQAASLIEDEHHMIAILSNMSALLNDNLDQINWVGFYLLEQNELILGPFQGHPACVHIPIGKGVCGTAVSERRTQVVADVHQFKGHIACDANSKSEIVVPIFKDDKIIGVLDIDAPITDRFDDNDKEHLEAIVKIIEKQLA
9FCM , Knot 68 149 0.76 40 111 142 
QVQLQESGGGLVQAGDSLTLSCAVSGRIFSTYTMGWFRQAPGKEREFVAAVRWGAGTIYYADSMKGRFTISRDSAKNTVDLQMNSTKPEDTAVYYCGAAYVSKANYGSLWYQDSRRYDYWGQGTQVTVSSAAAYPYDVPDYGSHHHHHH
5SHC , Knot 152 343 0.86 40 221 334 
GSSICTSEEWQGLMQFTLPVRLCKEIELFHFDIGPFENMWPGIFVYMVHRSCGTSCFELEKLCRFIMSVKKNYRRVPYHNWKHAVTVAHCMYAILQNNHTLFTDLERKGLLIACLCHDLDHRGFSNSYLQKFDHPLAALYSTSTMEQHHFSQTVSILQLEGHNIFSTLSSSEYEQVLEIIRKAIIATDLALYFGNRKQLEEMYQTGSLNLNNQSHRDRVIGLMMTACDLCSVTKLWPVTKLTANDIYAEFWAEGDEMKKLGIQPIPMMDRDKKDEVPQGQLGFYNAVAIPCYTTLTQILPPTEPLLKACRDNLSQWEKVIRGEETATWISSPSVAQKAAASED

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 \(c\).

\(|P_{f(3KSH_1)}(2) \setminus P_{f(9FCM_1)}(2)|=86\), \(|P_{f(9FCM_1)}(2) \setminus P_{f(3KSH_1)}(2)|=76\). 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 1:0000001001010000110001101100000111110010111000100101111011000011111101011010111101101011000000111010010101100100000111111000011111010111000100000001011101100011
Pair \(Z_2\) Length of longest common subsequence
3KSH_1,9FCM_1 162 6
3KSH_1,5SHC_1 174 3
9FCM_1,5SHC_1 214 4

Newick tree

 
[
	5SHC_1:10.42,
	[
		3KSH_1:81,9FCM_1:81
	]:21.42
]

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 \((0.85)(0.8)(\frac{309 }{\log_{20} 309}-\frac{149}{\log_{20}149})=49.1\)
Status Protein1 Protein2 d d1/2
Query variables 3KSH_1 9FCM_1 62 60
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]