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Parikh vectors
4PVR_1 3GHC_1 3LDP_1 Letter Amino acid
11 9 17 N Asparagine
20 9 26 D Aspartic acid
18 16 29 E Glutamic acid
22 19 25 L Leucine
3 3 1 W Tryptophan
4 6 9 Y Tyrosine
6 1 1 C Cysteine
14 9 24 I Isoleucine
23 17 33 K Lycine
5 9 18 F Phenylalanine
19 7 27 T Threonine
15 14 17 S Serine
34 5 30 A Alanine
10 8 19 R Arginine
7 6 16 Q Glutamine
41 13 34 G Glycine
9 6 6 M Methionine
8 3 4 H Histidine
11 12 13 P Proline
30 14 35 V Valine 

4PVR_1|Chains A, B|Isoaspartyl peptidase/L-asparaginase|Homo sapiens (9606)
>3GHC_1|Chain A|Dihydrofolate reductase|Homo sapiens (9606)
>3LDP_1|Chains A, B|78 kDa glucose-regulated protein|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 \(w=f(c)\)
4PVR , Knot 133 310 0.82 40 183 292 
GHMNPIVVVHGGGAGPISKDRKERVHQGMVRAATVGYGILREGGSAVDAVEGAVVALEDDPEFNAGCGSVLNTNGEVEMDASIMDGKDLSAGAVSAVQCIANPIKLARLVMEKTPHCFLTDQGAAQFAAAMGVPEIPGEKLVTERNKKRLEKEKHEKGAQKTDCQKNLGTVGAVALDCKGNVAYATSTGGIVNKMVGRVGDSPCLGAGGYADNDIGAVSTTGHGESILKVNLARLTLFHIEQGKTVEEAADLSLGYMKSRVKGLGGLIVVSKTGDWVAKWTSTSMPWAAAKDGKLHFGIDPDDTTITDLP
3GHC , Knot 87 186 0.81 40 136 178 
VGSLNCIVAVSQNMGIGKNGDLPWPPLRNEFRYFSRMTTTSSVEGKQNLVIMGKKTWFSIPEKSRPLKGRINLVLSRELKEPPQGAHFLSRSLDDALKLTEQPELANKVDMVWIVGGSSVYKEAMNHPGHLKLFVTRIMQDFESDTFFPEIDLEKYKLLPEYPGVLSDVQEEKGIKYKFEVYEKND
3LDP , Knot 161 384 0.83 40 211 365 
GSDVGTVVGIDLGTTYSCVGVFKNGRVEIIANDQGNRITPSYVAFTPEGERLIGDAAKNQLTSNPENTVFDAKRLIGRTWNDPSVQQDIKFLPFKVVEKKTKPYIQVDIGGGQTKTFAPEEISAMVLTKMKETAEAYLGKKVTHAVVTVPAYFNDAQRQATKDAGTIAGLNVMRIINEPTAAAIAYGLDKREGEKNILVFDLGGGTFDVSLLTIDNGVFEVVATNGDTHLGGEDFDQRVMEHFIKLYKKKTGKDVRKDNRAVQKLRREVEKAKRALSSQHQARIEIESFYEGEDFSETLTRAKFEELNMDLFRSTMKPVQKVLEDSDLKKSDIDEIVLVGGSTRIPKIQQLVKEFFNGKEPSRGINPDEAVAYGAAVQAGVLSG

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 \(c\).

\(|P_{f(4PVR_1)}(2) \setminus P_{f(3GHC_1)}(2)|=101\), \(|P_{f(3GHC_1)}(2) \setminus P_{f(4PVR_1)}(2)|=54\). 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 1:1010111110111111100000001001110110110111001101101101111110001010110101100010101010110100101111011001101101101110001001100011101111111101110011000000010000000110000000011011111100010110100011110011101100101111101000111100010100110101101011010010010011010110100010111111110001011101000011111100101011101000010011
Pair \(Z_2\) Length of longest common subsequence
4PVR_1,3GHC_1 155 3
4PVR_1,3LDP_1 148 3
3GHC_1,3LDP_1 163 4

Newick tree

 
[
	3GHC_1:81.28,
	[
		4PVR_1:74,3LDP_1:74
	]:7.28
]

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 \((0.85)(0.8)(\frac{496 }{\log_{20} 496}-\frac{186}{\log_{20}186})=90.2\)
Status Protein1 Protein2 d d1/2
Query variables 4PVR_1 3GHC_1 108 86.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]