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Parikh vectors
1UVU_1 3MVY_1 1JEC_1 Letter Amino acid
2 21 32 R Arginine
4 25 34 G Glycine
2 38 30 P Proline
3 29 24 T Threonine
1 19 19 Y Tyrosine
2 52 31 A Alanine
1 23 21 N Asparagine
3 47 52 L Leucine
0 10 4 M Methionine
2 26 32 S Serine
3 28 36 D Aspartic acid
4 27 23 Q Glutamine
8 32 32 E Glutamic acid
1 10 14 H Histidine
1 32 39 I Isoleucine
1 25 36 V Valine
1 0 1 C Cysteine
4 28 22 K Lycine
6 21 22 F Phenylalanine
0 9 6 W Tryptophan 

1UVU_1|Chain A[auth L]|THROMBIN|Bos taurus (9913)
>3MVY_1|Chains A, B|Nickel-binding periplasmic protein|Escherichia coli (83333)
>1JEC_1|Chain A|SULFATE ADENYLYLTRANSFERASE|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
1UVU , Knot 31 49 0.82 36 44 46 
TSEDHFQPFFNEKTFGAGEADCGLRPLFEKKQVQDQTEKELFESYIEGR
3MVY , Knot 202 502 0.83 38 246 472 
AAPDEITTAWPVNVGPLNPHLYTPNQMFAQSMVYEPLVKYQADGSVIPWLAKSWTHSEDGKTWTFTLRDDVKFSNGEPFDAEAAAENFRAVLDNRQRHAWLELANQIVDVKALSKTELQITLKSAYYPFLQELALPRPFRFIAPSQFKNHETMNGIKAPIGTGPWILQESKLNQYDVFVRNENYWGEKPAIKKITFNVIPDPTTRAVAFETGDIDLLYGNEGLLPLDTFARFSQNPAYHTQLSQPIETVMLALNTAKAPTNELAVREALNYAVNKKSLIDNALYGTQQVADTLFAPSVPYANLGLKPSQYDPQKAKALLEKAGWTLPAGKDIREKNGQPLRIELSFIGTDALSKSMAEIIQADMRQIGADVSLIGEEESSIYARQRDGRFGMIFHRTWGAPYDPHAFLSSMRVPSHADFQAQQGLADKPLIDKEIGEVLATHDETQRQALYRDILTRLHDEAVYLPISYISMMVVSKPELGNIPYAPIATEIPFEQIKPVKP
1JEC , Knot 208 510 0.84 40 246 487 
PAPHGGILQDLIARDALKKNELLSEAQSSDILVWNLTPRQLCDIELILNGGFSPLTGFLNENDYSSVVTDSRLADGTLWTIPITLDVDEAFANQIKPDTRIALFQDDEIPIAILTVQDVYKPNKTIEAEKVFRGDPEHPAISYLFNVAGDYYVGGSLEAIQLPQHYDYPGLRKTPAQLRLEFQSRQWDRVVAFQTRNPMHRAHRELTVRAAREANAKVLIHPVVGLTKPGDIDHHTRVRVYQEIIKRYPNGIAFLSLLPLAMRMSGDREAVWHAIIRKNYGASHFIVGRDHAGPGKNSKGVDFYGPYDAQELVESYKHELDIEVVPFRMVTYLPDEDRYAPIDQIDTTKTRTLNISGTELRRRLRVGGEIPEWFSYPEVVKILRESNPPRPKQGFSIVLGNSLTVSREQLSIALLSTFLQFGGGRYYKIFEHNNKTELLSLIQDFIGSGSGLIIPNQWEDDKDSVVGKQNVYLLDTSSSADIQLESADEPISHIVQKVVLFLEDNGFFVF

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 \(c\).

\(|P_{f(1UVU_1)}(2) \setminus P_{f(3MVY_1)}(2)|=11\), \(|P_{f(3MVY_1)}(2) \setminus P_{f(1UVU_1)}(2)|=213\). 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 1:0000010111000011110100110111000010000000110001010
Pair \(Z_2\) Length of longest common subsequence
1UVU_1,3MVY_1 224 3
1UVU_1,1JEC_1 216 3
3MVY_1,1JEC_1 142 4

Newick tree

 
[
	1UVU_1:12.24,
	[
		1JEC_1:71,3MVY_1:71
	]:49.24
]

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 \((0.85)(0.8)(\frac{551 }{\log_{20} 551}-\frac{49}{\log_{20}49})=152.\)
Status Protein1 Protein2 d d1/2
Query variables 1UVU_1 3MVY_1 192 105
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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