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Parikh vectors
4QUJ_1 7QDV_1 5SOA_1 Letter Amino acid
17 2 52 G Glycine
10 2 10 H Histidine
19 8 21 I Isoleucine
2 0 2 W Tryptophan
15 3 18 F Phenylalanine
15 7 19 T Threonine
14 0 29 R Arginine
15 6 15 N Asparagine
8 0 4 C Cysteine
20 5 22 E Glutamic acid
20 11 36 L Leucine
22 16 8 K Lycine
10 6 7 Y Tyrosine
12 2 24 V Valine
4 5 7 Q Glutamine
12 6 51 A Alanine
20 3 24 D Aspartic acid
10 3 14 M Methionine
7 6 17 P Proline
26 9 34 S Serine 

4QUJ_1|Chain A|Caspase-3|Homo sapiens (9606)
>7QDV_1|Chain A|BBA14 (OrfD)|Borreliella burgdorferi B31 (224326)
>5SOA_1|Chains A, B|3-oxoacyl-[acyl-carrier-protein] synthase 2|Pseudomonas aeruginosa (287)
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 \(w=f(c)\)
4QUJ , Knot 123 278 0.83 40 183 273 
MENTENSVDSKSIKNLEPKIIHGSESMDSGISLDNSYKMDYPEMGLCIIINNKNFHKSTGMTSRSGTDVDAANLRETFRNLKYEVRNKNDLTREEIVELMRDVSKEDHSKRSSFVCVLLSHGEEGIIFGTNGPVDLKKIGNFFRGDRCRSLTGKPKLFIIQACRGTELDCGIETDSGVDDDMACHKIPVEADFLYAYSTAPGYYSWRNSKDGSWFIQSLCAMLKQYADKLEFMHILTRVNRKVATEFESFSFDATFHAKKQIPCIHSMLTKELYFYHH
7QDV , Knot 52 100 0.79 34 80 97 
GAMGLPEEPSSPQESTLKALSLYEAHLSSYIMYLQTFLVKTKQKVNNKNYPEFTLFDTSKLKKDQTLKSIKTNIAALKNHIDKIKPIAMQIYKKYSKNIP
5SOA , Knot 171 414 0.83 40 212 384 
MSRRRVVITGMGMLSPLGLDVPSSWEGILAGRSGIAPIEHMDLSAYSTRFGGSVKGFNVEEYLSAKEARKLDLFIQYGLAASFQAVRDSGLEVTDANRERIGVSMGSGIGGLTNIENNCRSLFEQGPRRISPFFVPGSIINMVSGFLSIHLGLQGPNYALTTAQTTGTHSIGMAARNIAYGEADVMVAGGSEMAACGLGLGGFGAARALSTRNDEPTRASRPWDRDRDGFVLSDGSGALVLEELEHARARGARIYAELVGFGMSGDAFHMTAPPEDGAGAARCMKNALRDAGLDPRQVDYINAHGTSTPAGDIAEIAAVKSVFGEHAHALSMSSTKSMTGHLLGAAGAVEAIFSVLALRDQVAPPTINLDNPDEGCDLDLVAHEAKPRKIDVALSNSFGFGGTNGTLVFRRFAD

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 \(c\).

\(|P_{f(4QUJ_1)}(2) \setminus P_{f(7QDV_1)}(2)|=135\), \(|P_{f(7QDV_1)}(2) \setminus P_{f(4QUJ_1)}(2)|=32\). 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 1:10000001000010010101101000100110100000100101110111000010000110000100101101000100100010000010000110110010000000000110111001001111100111010011011010000010101011110100100100110000110001100011101011010001110001000001011100101110001001011011001000110010010101010100011010011000101000
Pair \(Z_2\) Length of longest common subsequence
4QUJ_1,7QDV_1 167 4
4QUJ_1,5SOA_1 157 4
7QDV_1,5SOA_1 204 3

Newick tree

 
[
	7QDV_1:97.62,
	[
		4QUJ_1:78.5,5SOA_1:78.5
	]:19.12
]

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 \((0.85)(0.8)(\frac{378 }{\log_{20} 378}-\frac{100}{\log_{20}100})=85.5\)
Status Protein1 Protein2 d d1/2
Query variables 4QUJ_1 7QDV_1 112 74
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]