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Parikh vectors
8TDN_1 2ZJF_1 4QOT_1 Letter Amino acid
2 3 3 C Cysteine
11 42 6 G Glycine
12 13 5 T Threonine
3 5 2 N Asparagine
5 15 2 Y Tyrosine
8 23 2 P Proline
5 22 1 R Arginine
5 25 4 D Aspartic acid
0 15 0 Q Glutamine
5 18 2 H Histidine
8 6 8 K Lycine
6 14 1 F Phenylalanine
2 11 0 W Tryptophan
15 34 3 A Alanine
4 18 2 I Isoleucine
13 28 8 L Leucine
2 7 3 M Methionine
14 14 5 S Serine
14 29 6 V Valine
13 20 5 E Glutamic acid 

8TDN_1|Chains A, B, C, D, E|Transthyretin|Homo sapiens (9606)
>2ZJF_1|Chain A|Probable epoxide hydrolase ephB|Mycobacterium tuberculosis (83332)
>4QOT_1|Chains A, B|Copper transport protein ATOX1|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 \(w=f(c)\)
8TDN , Knot 73 147 0.82 38 112 144 
MASHRLLLLCLAGLVFVSEAGPTGTGESKCPLMVKVLDAVRGSPAINVAVHVFRKAADDTWEPFASGKTSESGELHGLTTEEEFVEGIYKVEIDTKSYWKALGSSPFHEHAEVVFTANDSGPRRYTIAALLSPYSYSTTAVVTNPKE
2ZJF , Knot 151 362 0.82 40 210 340 
HHHHHHVSQVHRILNCRGTRIHAVADSPPDQQGPLVVLLHGFPESWYSWRHQIPALAGAGYRVVAIDQRGYGRSSKYRVQKAYRIKELVGDVVGVLDSYGAEQAFVVGHDWGAPVAWTFAWLHPDRCAGVVGISVPFAGRGVIGLPGSPFGERRPSDYHLELAGPGRVWYQDYFAVQDGIITEIEEDLRGWLLGLTYTVSGEGMMAATKAAVDAGVDLESMDPIDVIRAGPLCMAEGARLKDAFVYPETMPAWFTEADLDFYTGEFERSGFGGPLSFYHNIDNDWHDLADQQGKPLTPPALFIGGQYDVGTIWGAQAIERAHEVMPNYRGTHMIADVGHWIQQEAPEETNRLLLDFLGGLRP
4QOT , Knot 40 68 0.82 36 60 66 
MPKHEFSVDMTCGGCAEAVSRVLNKLGGVKYDIDLPNKKVCIESEHSMDTLLATLKKTGKTVSYLGLE

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 \(c\).

\(|P_{f(8TDN_1)}(2) \setminus P_{f(2ZJF_1)}(2)|=42\), \(|P_{f(2ZJF_1)}(2) \setminus P_{f(8TDN_1)}(2)|=140\). 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 1:110001111011111110011101010000111101101101011101110110011000101110100000101011000001101100101000001011100110001011101000110000111110100000011100100
Pair \(Z_2\) Length of longest common subsequence
8TDN_1,2ZJF_1 182 4
8TDN_1,4QOT_1 126 3
2ZJF_1,4QOT_1 198 3

Newick tree

 
[
	2ZJF_1:10.59,
	[
		8TDN_1:63,4QOT_1:63
	]:40.59
]

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 \((0.85)(0.8)(\frac{509 }{\log_{20} 509}-\frac{147}{\log_{20}147})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 8TDN_1 2ZJF_1 133 93
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]