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Parikh vectors
5KPR_1 7ZRU_1 3MMJ_1 Letter Amino acid
21 3 19 D Aspartic acid
7 4 0 C Cysteine
16 1 11 I Isoleucine
21 1 21 K Lycine
3 0 8 W Tryptophan
22 3 32 A Alanine
17 2 7 N Asparagine
38 0 20 G Glycine
12 3 13 H Histidine
21 0 13 F Phenylalanine
11 3 17 P Proline
15 1 16 Y Tyrosine
24 2 19 V Valine
8 0 13 Q Glutamine
17 0 23 R Arginine
24 2 23 E Glutamic acid
45 0 22 L Leucine
8 2 6 M Methionine
30 1 15 S Serine
20 0 16 T Threonine 

5KPR_1|Chain A|Pantothenate kinase 3|Homo sapiens (9606)
>7ZRU_1|Chain A|Potassium channel toxin kappa-KTx 2.9|Pandinus imperator (55084)
>3MMJ_1|Chains A, B|Myo-inositol hexaphosphate phosphohydrolase|Selenomonas ruminantium (971)
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 \(w=f(c)\)
5KPR , Knot 161 380 0.84 40 205 362 
MGSSHHHHHHSSGLVPRGSPWFGMDIGGTLVKLSYFEPIDITAEEEQEEVESLKSIRKYLTSNVAYGSTGIRDVHLELKDLTLFGRRGNLHFIRFPTQDLPTFIQMGRDKNFSTLQTVLCATGGGAYKFEKDFRTIGNLHLHKLDELDCLVKGLLYIDSVSFNGQAECYYFANASEPERCQKMPFNLDDPYPLLVVNIGSGVSILAVHSKDNYKRVTGTSLGGGTFLGLCSLLTGCESFEEALEMASKGDSTQADKLVRDIYGGDYERFGLPGWAVASSFGNMIYKEKRESVSKEDLARATLVTITNNIGSVARMCAVNEKINRVVFVGNFLRVNTLSMKLLAYALDYWSKGQLKALFLEHEGYFGAVGALLGLPNFSDD
7ZRU , Knot 17 28 0.67 26 23 25 
VDACYEACMHHHMNSDDCIEACKNPVPP
3MMJ , Knot 137 314 0.83 38 193 301 
QTVTEPVGSYARAERPQDFEGFVWRLDNDGKEALPRNFRTSADALRAPEKKFHLDAAYVPSREGMDALHISGSSAFTPAQLKNVAAKLREKTAGPIYDVDLRQESHGYLDGIPVSWYGERDWANLGKSQHEALADERHRLHAALHKTVYIAPLGKHKLPEGGEVRRVQKVQTEQEVAEAAGMRYFRIAATDHVWPTPENIDRFLAFYRTLPQDAWLHFHSEAGVGRTTAFMVMTDMLKNPSVSLKDILYRQHEIGGFYYGEFPIKTKDKDSWKTKYYREKIVMIEQFYRYVQENRADGYQTPWSVWLKSHPAKA

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 \(c\).

\(|P_{f(5KPR_1)}(2) \setminus P_{f(7ZRU_1)}(2)|=196\), \(|P_{f(7ZRU_1)}(2) \setminus P_{f(5KPR_1)}(2)|=14\). 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 1:11000000000011110101111101110110100101101010000001001001000100011010011001010100101110010101101100011011011000010010011010111100100010011010100100100110111010010101010000110100100000111010010111110110110111100000000101001111011110011010001001101100100001001100101100001111111110011011000000010000110101101000110110101100010011111011010010101110110010010101111000101111111111101000
Pair \(Z_2\) Length of longest common subsequence
5KPR_1,7ZRU_1 210 3
5KPR_1,3MMJ_1 176 3
7ZRU_1,3MMJ_1 200 3

Newick tree

 
[
	7ZRU_1:10.93,
	[
		5KPR_1:88,3MMJ_1:88
	]:18.93
]

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 \((0.85)(0.8)(\frac{408 }{\log_{20} 408}-\frac{28}{\log_{20}28})=121.\)
Status Protein1 Protein2 d d1/2
Query variables 5KPR_1 7ZRU_1 155 82.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]