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Parikh vectors
3OIC_1 4YIN_1 3QKS_1 Letter Amino acid
21 21 17 E Glutamic acid
9 2 9 H Histidine
17 16 20 I Isoleucine
17 21 22 K Lycine
14 23 12 S Serine
14 16 9 N Asparagine
2 2 0 C Cysteine
24 17 18 L Leucine
12 10 10 T Threonine
2 7 2 W Tryptophan
11 5 11 D Aspartic acid
8 10 3 Q Glutamine
19 19 13 G Glycine
8 4 3 M Methionine
5 5 8 Y Tyrosine
24 20 11 V Valine
28 19 10 A Alanine
12 8 14 R Arginine
6 13 8 F Phenylalanine
5 8 3 P Proline 

3OIC_1|Chains A, B[auth D]|Enoyl-[acyl-carrier-protein] reductase [NADPH]|Bacillus subtilis (1423)
>4YIN_1|Chains A, B|Beta-lactamase|Pseudomonas aeruginosa (287)
>3QKS_1|Chain A|DNA double-strand break repair rad50 ATPase|Pyrococcus furiosus (2261)
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 \(w=f(c)\)
3OIC , Knot 116 258 0.83 40 151 246 
MEQNKCALVTGSSRGVGKAAAIRLAENGYNIVINYARSKKAALETAEEIEKLGVKVLVVKANVGQPAKIKEMFQQIDETFGRLDVFVNNAASGVLRPVMELEETHWDWTMNINAKALLFCAQEAAKLMEKNGGGHIVSISSLGSIRYLENYTTVGVSKAALEALTRYLAVELSPKQIIVNAVSGGAIDTDALKHFPNREDLLEDARQNTPAGRMVEIKDMVDTVEFLVSSKADMIRGQTIIVDGGRSLLVLEHHHHHH
4YIN , Knot 112 246 0.83 40 165 236 
SSITENTSWNKEFSAEAVNGVFVLCKSSSKSCATNNLARASKEYLPASTFKIPNAIIGLETGVIKNEHQVFKWDGKPRAMKQWERDLSLRGAIQVSAVPVFQQIAREVGEVRMQKYLKKFSYGNQNISGGIDKFWEGQLRISAVNQVEFLESLFLNKLSASKENQLIVKEALVTEAAPEYLVHSKTGFSGVGTESNPGVAWWVGWVEKGTEVYFFAFNMDIDNENKLPLRKSIPTKIMASEGIIGG
3QKS , Knot 94 203 0.82 38 141 194 
MKLERVTVKNFRSHSDTVVEFKEGINLIIGQNGSGKSSLLDAILVGLYWPLRIKDIKKDEFTKVGARDTYIDLIFEKDGTKYRITRRFLKGYSSGEIHAMKRLVGNEWKHVTEPSSKAISAFMEKLIPYNIFLNAIYIRQGQIDAILESDEAREKVVREVLNLDKFETAYKKLSELKKTINNRIKEYRDILARTEGGHHHHHH

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 \(c\).

\(|P_{f(3OIC_1)}(2) \setminus P_{f(4YIN_1)}(2)|=76\), \(|P_{f(4YIN_1)}(2) \setminus P_{f(3OIC_1)}(2)|=90\). 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 1:100000111010001110111101100100111001000011100100100111011110101101101001100100011010111001101110111010000101010101011110100110110001110110100110100100000111001110110001110101001110110111100011001100001100100001110110100110010111000101101001110110011110000000
Pair \(Z_2\) Length of longest common subsequence
3OIC_1,4YIN_1 166 4
3OIC_1,3QKS_1 150 6
4YIN_1,3QKS_1 146 3

Newick tree

 
[
	3OIC_1:81.03,
	[
		3QKS_1:73,4YIN_1:73
	]:8.03
]

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 \((0.85)(0.8)(\frac{504 }{\log_{20} 504}-\frac{246}{\log_{20}246})=73.9\)
Status Protein1 Protein2 d d1/2
Query variables 3OIC_1 4YIN_1 93 91
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]