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Parikh vectors
4QLZ_1 2PYB_1 4MWX_1 Letter Amino acid
19 7 29 N Asparagine
20 13 21 D Aspartic acid
5 3 9 Q Glutamine
9 5 7 H Histidine
17 1 21 P Proline
20 12 18 E Glutamic acid
17 5 31 G Glycine
12 18 18 L Leucine
18 3 22 V Valine
19 6 18 A Alanine
6 5 23 R Arginine
24 17 25 I Isoleucine
13 8 9 F Phenylalanine
11 6 18 Y Tyrosine
4 2 18 C Cysteine
27 13 16 K Lycine
9 7 5 M Methionine
16 12 38 S Serine
13 6 28 T Threonine
8 2 14 W Tryptophan 

4QLZ_1|Chains A, B|SJCHGC07024 protein|Schistosoma japonicum (6182)
>2PYB_1|Chains A, B, C, D|Neutrophil activating protein|Borrelia burgdorferi
>4MWX_1|Chain A|Neuraminidase|Influenza A virus (11320)
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 \(w=f(c)\)
4QLZ , Knot 129 287 0.84 40 194 282 
MSVERGTSNSASYKMFLTHGGSPISYFHDVPLFADATNNCYNMIVEIPRWTNAKMEICKEELMNPIKHDVKNNKLRYIYNVFPHKGYIWNYGALPQTWEDPSYVDEDTKAKGDNDPIDVCEIGSKIWPSGSVIPVKVLGILGMIDEGETDWKVIAINVADPMAEKLNDILDVDAHMPGFLKATRDWFKYYKVPAGKPENSFAFNGEFKNKEFAAKIISKTHEHWQKLISTKVEAGPIIRANVTVKGSPYMVSKEDFIDALQKHEDFKRGSEPTDQAIEQWHFCNTNI
2PYB , Knot 73 151 0.80 40 118 148 
DDLDAIQLKLQELLASLHIFYSNLRGIHWNIKDTNFFVIHKKTQKLYEYIEKIIDIVAERSRMLGYDSEFRYSEFMKKSFIKELDIESTSNFLPSMESIVCSLTEILKNIFGMRKLIDTAGDYGTANIMDDIMSDLEKHLWMHKALLENCD
4MWX , Knot 165 388 0.84 40 225 370 
RNFNNLTKGLCTINSWHIYGKDNAVRIGESSDVLVTREPYVSCDPDECRFYALSQGTTIRGKHSNGTIHDRSQYRALISWPLSSPPTVYNSRVECIGWSSTSCHDGKSRMSICISGPNNNASAVVWYNRRPVAEINTWARNILRTQESECVCHNGVCPVVFTDGSATGPADTRIYYFKEGKILKWESLTGTAKHIEECSCYGERTGITCTCKDNWQGSNRPVIQIDPVAMTHTSQYICSPVLTDNPRPNDPNIGKCNDPYPGNNNNGVKGFSYLDGANTWLGRTISTASRSGYEMLKVPNALTDDRSKPIQGQTIVLNADWSGYSGSFMDYWAEGDCYRACFYVELIRGRPKEDKVWWTSNSIVSMCSSTEFLGQWNWPDGAKIEYFL

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 \(c\).

\(|P_{f(4QLZ_1)}(2) \setminus P_{f(2PYB_1)}(2)|=129\), \(|P_{f(2PYB_1)}(2) \setminus P_{f(4QLZ_1)}(2)|=53\). 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 1:10100100001000111001101100100111110100000011101101001010100001101100010000100100111001011001111001001001000001010001101001100111010111101111111100100010111101101110010011010101111101000110000111101000111010100001110110000001001100010111110101010101011000011011000001001001000110010100001
Pair \(Z_2\) Length of longest common subsequence
4QLZ_1,2PYB_1 182 3
4QLZ_1,4MWX_1 179 4
2PYB_1,4MWX_1 201 3

Newick tree

 
[
	2PYB_1:97.89,
	[
		4QLZ_1:89.5,4MWX_1:89.5
	]:8.39
]

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 \((0.85)(0.8)(\frac{438 }{\log_{20} 438}-\frac{151}{\log_{20}151})=85.3\)
Status Protein1 Protein2 d d1/2
Query variables 4QLZ_1 2PYB_1 114 84.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]