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Parikh vectors
8QXA_1 8SGG_1 2CSS_1 Letter Amino acid
6 6 2 W Tryptophan
8 7 1 Y Tyrosine
6 5 0 C Cysteine
5 3 1 H Histidine
14 7 7 I Isoleucine
55 22 18 G Glycine
20 14 10 K Lycine
18 0 2 M Methionine
16 15 7 P Proline
15 20 6 T Threonine
26 10 5 A Alanine
22 8 5 D Aspartic acid
24 7 2 Q Glutamine
41 39 13 S Serine
25 24 11 V Valine
28 7 4 N Asparagine
22 5 8 E Glutamic acid
22 8 1 F Phenylalanine
20 7 6 R Arginine
21 18 12 L Leucine 

8QXA_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|TAR DNA-binding protein 43|Homo sapiens (9606)
>8SGG_1|Chains A, C|Cy137D09 Fab heavy chain|Macaca fascicularis (9541)
>2CSS_1|Chain A|Regulating synaptic membrane exocytosis protein 1|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 \(w=f(c)\)
8QXA , Knot 169 414 0.82 40 225 381 
MSEYIRVTEDENDEPIEIPSEDDGTVLLSTVTAQFPGACGLRYRNPVSQCMRGVRLVEGILHAPDAGWGNLVYVVNYPKDNKRKMDETDASSAVKVKRAVQKTSDLIVLGLPWKTTEQDLKEYFSTFGEVLMVQVKKDLKTGHSKGFGFVRFTEYETQVKVMSQRHMIDGRWCDCKLPNSKQSQDEPLRSRKVFVGRCTEDMTEDELREFFSQYGDVMDVFIPKPFRAFAFVTFADDQIAQSLCGEDLIIKGISVHISNAEPKHNSNRQLERSGRFGGNPGGFGNQGGFGNSRGGGAGLGNNQGSNMGGGMNFGAFSINPAMMAAAQAALQSSWGMMGMLASQQNQSGPSGNNQNQGNMQREPNQAFGSGNNSYSGSNSGAAIGWGSASNAGSGSGFNGGFGSSMDSKSSGWGM
8SGG , Knot 103 232 0.80 38 145 216 
QVQLQESGPGLVKPSETLSLTCDVSGGSFSGDFYWSWIRQPPGKGLDWIGNIHGSSAGTKYKPSLKSRVTISKDTSKNQFSLKLSSVTAADTAVYYCTRGPLSRIVAGFGRGINWFDVWGPGVLVTVSSASTKGPSVFPLAPSSRSTSESTAALGCLVKDYFPEPVTVSWNSGSLTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYVCNVNHKPSNTKVDKRVEIKTC
2CSS , Knot 60 121 0.79 38 94 115 
GSSGSSGVTWQPSKEGDRLIGRVILNKRTTMPKDSGALLGLKVVGGKMTDLGRLGAFITKVKKGSLADVVGHLRAGDEVLEWNGKPLPGATNEEVYNIILESKSEPQVEIIVSRPSGPSSG

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 \(c\).

\(|P_{f(8QXA_1)}(2) \setminus P_{f(8SGG_1)}(2)|=132\), \(|P_{f(8SGG_1)}(2) \setminus P_{f(8QXA_1)}(2)|=52\). 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 1:100010100000001101100001011100101011110110000110001011011011101101111011011001000000100001001101001100000111111110000001000100110111101000100100011111010000001011000011010100001100000000110000111100000100001001100010110111101101111101100011001010011101101010010100000001000101110111110011110001111111000100111110111101011111110111000111111110000001101000001010001001110100000100011111110100110101101111001000001111
Pair \(Z_2\) Length of longest common subsequence
8QXA_1,8SGG_1 184 4
8QXA_1,2CSS_1 179 4
8SGG_1,2CSS_1 143 3

Newick tree

 
[
	8QXA_1:96.32,
	[
		2CSS_1:71.5,8SGG_1:71.5
	]:24.82
]

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 \((0.85)(0.8)(\frac{646 }{\log_{20} 646}-\frac{232}{\log_{20}232})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 8QXA_1 8SGG_1 145 112
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]