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Parikh vectors
5VOS_1 4TQB_1 3RPZ_1 Letter Amino acid
1 10 33 A Alanine
0 4 5 C Cysteine
2 11 11 I Isoleucine
0 6 6 Y Tyrosine
0 11 19 E Glutamic acid
1 17 32 L Leucine
0 8 7 W Tryptophan
0 12 11 R Arginine
1 11 8 N Asparagine
0 6 8 Q Glutamine
1 15 13 K Lycine
0 3 6 M Methionine
0 8 5 F Phenylalanine
0 11 20 T Threonine
1 11 19 V Valine
0 14 14 D Aspartic acid
3 11 26 G Glycine
0 5 10 H Histidine
0 6 18 P Proline
1 11 8 S Serine 

5VOS_1|Chain A|Amyloid beta A4 protein|Homo sapiens (9606)
>4TQB_1|Chains A, B|Eukaryotic translation initiation factor 4E|Homo sapiens (9606)
>3RPZ_1|Chain A|ADP/ATP-dependent NAD(P)H-hydrate dehydratase|Bacillus subtilis (1423)
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 \(w=f(c)\)
5VOS , Knot 9 11 0.65 16 10 9 
VGSNKGAIIGL
4TQB , Knot 95 191 0.87 40 153 187 
MVANPEHYIKHPLQNRWALWFFKNDKSKTWQANLRLISKFDTVEDFWALYNHIQLSSNLMPGCDYSLFKDGIEPMWEDEKNKRGGRWLITLNKQQRRSDLDRFWLETLLCLIGESFDDYSDDVCGAVVNVRAKGDKIAIWTTECENREAVTHIGRVYKERLGLPPKIVIGYQSHADTATKSGSTTKNRFVV
3RPZ , Knot 120 279 0.80 40 168 262 
SNAMNVPFWTEEHVRATLPERDAESHKGTYGTALLLAGSDDMPGAALLAGLGAMRSGLGKLVIGTSENVIPLIVPVLPEATYWRDGWKKAADAQLEETYRAIAIGPGLPQTESVQQAVDHVLTADCPVILDAGALAKRTYPKREGPVILTPHPGEFFRMTGVPVNELQKKRAEYAKEWAAQLQTVIVLKGNQTVIAFPDGDCWLNPTGNGALAKGGTGDTLTGMILGMLCCHEDPKHAVLNAVYLHGACAELWTDEHSAHTLLAHELSDILPRVWKRFE

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 \(c\).

\(|P_{f(5VOS_1)}(2) \setminus P_{f(4TQB_1)}(2)|=2\), \(|P_{f(4TQB_1)}(2) \setminus P_{f(5VOS_1)}(2)|=145\). 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 1:11000111111
Pair \(Z_2\) Length of longest common subsequence
5VOS_1,4TQB_1 147 2
5VOS_1,3RPZ_1 164 2
4TQB_1,3RPZ_1 171 3

Newick tree

 
[
	3RPZ_1:86.92,
	[
		5VOS_1:73.5,4TQB_1:73.5
	]:13.42
]

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 \((0.85)(0.8)(\frac{202 }{\log_{20} 202}-\frac{11}{\log_{20}11})=68.1\)
Status Protein1 Protein2 d d1/2
Query variables 5VOS_1 4TQB_1 90 46.5
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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