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Parikh vectors
4PWG_1 1IDJ_1 7PJD_1 Letter Amino acid
5 20 5 Y Tyrosine
3 19 6 N Asparagine
0 7 10 Q Glutamine
13 18 13 E Glutamic acid
13 44 11 G Glycine
13 19 15 L Leucine
8 8 17 K Lycine
15 30 15 S Serine
15 29 16 A Alanine
2 5 7 W Tryptophan
13 33 12 V Valine
6 8 3 R Arginine
5 22 8 I Isoleucine
4 1 2 M Methionine
6 11 10 F Phenylalanine
8 8 5 P Proline
12 32 5 T Threonine
5 30 19 D Aspartic acid
2 9 2 C Cysteine
11 6 3 H Histidine 

4PWG_1|Chains A, B|Transthyretin|Homo sapiens (9606)
>1IDJ_1|Chains A, B|PECTIN LYASE A|Aspergillus niger (5061)
>7PJD_1|Chains A, B, C, D, E, F|Juvenile hormone diol kinase|Bombyx mori (7091)
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 \(w=f(c)\)
4PWG , Knot 76 159 0.80 38 116 152 
MRGSHHHHHHGSMASHRLLLLCLAGLVFVSEAGPTGTGESKCPLMVKVLDAVRGSPAINVAMHVFRKAADDTWEPFASGKTSESGELHGLTTEEEFVEGIYKVEIDTKSYWKALGISPFHEHAEVVFTANDSGPRRYTIAALLSPYSYSTTAVVTNPKE
1IDJ , Knot 150 359 0.82 40 191 336 
VGVSGSAEGFAEGVTGGGDATPVYPDTIDELVSYLGDDEARVIVLTKTFDFTDSEGTTTGTGCAPWGTASACQVAIDQDDWCENYEPDAPSVSVEYYNAGVLGITVTSNKSLIGEGSSGAIKGKGLRIVSGAENIIIQNIAVTDINPKYVWGGDAITLDDCDLVWIDHVTTARIGRQHYVLGTSADNRVSLTNNYIDGVSDYSATCDGYHYWGIYLDGDADLVTMKGNYIYHTSGRSPKVQDNTLLHCVNNYFYDISGHAFEIGEGGYVLAEGNVFQNVDTVLETYEGAAFTVPSTTAGEVCSTYLGRDCVINGFGCSGTFSEDSTSFLSDFEGKNIASASAYTSVASRVVANAGQGNL
7PJD , Knot 86 184 0.81 40 128 180 
HMVSEVRKKKLLHVFTVFFDSDKSGVVEKQDFELAAQNIAKLRGWAPGSPAYDILQESMIAIWLGLQKQADADGDGKVTQDEWLALWDEYAKDPAAAKDWQNLLCKSIFQIQDSSNDGSVDVNEYVTVHESFGLNKEESTEAFKKLAKGKDSISWADFQELWKEYFSSDDPDVPGNYIFGRLTC

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 \(c\).

\(|P_{f(4PWG_1)}(2) \setminus P_{f(1IDJ_1)}(2)|=47\), \(|P_{f(1IDJ_1)}(2) \setminus P_{f(4PWG_1)}(2)|=122\). 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 1:101000000010110001111011111110011101010000111101101101011101110110011000101110100000101011000001101100101000001011110110001011101000110000111110100000011100100
Pair \(Z_2\) Length of longest common subsequence
4PWG_1,1IDJ_1 169 4
4PWG_1,7PJD_1 148 4
1IDJ_1,7PJD_1 171 4

Newick tree

 
[
	1IDJ_1:88.36,
	[
		4PWG_1:74,7PJD_1:74
	]:14.36
]

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 \((0.85)(0.8)(\frac{518 }{\log_{20} 518}-\frac{159}{\log_{20}159})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 4PWG_1 1IDJ_1 130 91.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]