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Parikh vectors
1SZS_1 6FZH_1 3MTR_1 Letter Amino acid
42 33 15 G Glycine
15 15 21 S Serine
28 35 17 V Valine
13 19 10 N Asparagine
8 2 2 C Cysteine
14 12 6 F Phenylalanine
20 10 14 P Proline
18 26 14 T Threonine
21 23 7 D Aspartic acid
12 7 2 H Histidine
28 21 11 I Isoleucine
12 7 12 Y Tyrosine
18 8 10 Q Glutamine
29 19 18 E Glutamic acid
41 22 12 L Leucine
21 20 9 K Lycine
11 8 2 M Methionine
1 2 5 W Tryptophan
54 34 23 A Alanine
20 13 5 R Arginine 

1SZS_1|Chains A, B, C, D|4-aminobutyrate aminotransferase|Escherichia coli (562)
>6FZH_1|Chains A, B|Glyceraldehyde-3-phosphate dehydrogenase|Streptococcus pyogenes MGAS8232 (186103)
>3MTR_1|Chains A, B|Neural cell adhesion molecule 1|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 \(w=f(c)\)
1SZS , Knot 175 426 0.83 40 225 408 
MNSNKELMQRRSQAIPRGVGQIHPIFADRAENCRVWDVEGREYLDFAGGQAVLNTGHLHPKVVAAVEAQLKKLSHTCFQVLAYEPYLELCEIMNQKVPGDFAKKTLLVTTGSEAVENAVKIARAATKRSGTIAFSGAYHGRTHYTLALTGKVNPYSAGMGLMPGHVYRALYPCPLHGISEDDAIASIHRIFKNDAAPEDIAAIVIEPVQGEGGFYASSPAFMQRLRALCDEHGIMLIADEVQSGAGRTGTLFAMEQMGVAPDLTTFAKSIAGGFPLAGVTGRAEVMDAVAPGGLGGTYAGNPIACVAALEVLKVFEQENLLQKANDLGQKLKDGLLAIAEKHPEIGDVRGLGAMIAIELFEDGDHNKPDAKLTAEIVARARDKGLILLSCGPYYNVLRILVPLTIEDAQIRQGLEIISQCFDEAKQ
6FZH , Knot 142 336 0.82 40 194 320 
MVVKVGINGFGRIGRLAFRRIQNIEGVEVTRINDLTDPNMLAHLLKYDTTQGRFDGTVEVKEGGFEVNGNFIKVSAERDPENIDWATDGVEIVLEATGFFAKKEAAEKHLHANGAKKVVITAPGGNDVKTVVFNTNHDILDGTETVISGASCTTNCLAPMAKALHDAFGIQKGLMTTIHAYTGDQMILDGPHRGGDLRRARAGAANIVPNSTGAAKAIGLVIPELNGKLDGAAQRVPVPTGSVTELVVTLDKNVSVDEINAAMKAASNDSFGYTEDPIVSSDIVGVSYGSLFDATQTKVMEVDGSQLVKVVSWYDNEMSYTAQLVRTLEYFAKIAK
3MTR , Knot 102 215 0.85 40 151 208 
ATYAAASSALVPRGSHMYAPKLQGPVAVYTWEGNQVNITCEVFAYPSATISWFRDGQLLPSSNYSNIKIYNTPSASYLEVTPDSENDFGNYNCTAVNRIGQESLEFILVQADTPSSPSIDQVEPYSSTAQVQFDEPEATGGVPILKYKAEWRAVGEEVWHSKWYDAKEASMEGIVTIVGLKPETTYAVRLAALNGKGLGEISAASEFKTQPVQGE

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 \(c\).

\(|P_{f(1SZS_1)}(2) \setminus P_{f(6FZH_1)}(2)|=95\), \(|P_{f(6FZH_1)}(2) \setminus P_{f(1SZS_1)}(2)|=64\). 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 1:100000110000011101110101111001000011010100010111101110010101011111010100100001011100101010011000111011000111001001100110110110000101110110010000011101010100111111110100110101101100001110100110001110011111101101011101001111001011000011111100100111001011110011111010011001111111111010101101111111110011011101111011011000011001001100100111111000101101011111111011001000010101010111010001111100110001101111101001010011011000100100
Pair \(Z_2\) Length of longest common subsequence
1SZS_1,6FZH_1 159 4
1SZS_1,3MTR_1 164 5
6FZH_1,3MTR_1 159 4

Newick tree

 
[
	3MTR_1:81.17,
	[
		1SZS_1:79.5,6FZH_1:79.5
	]:1.67
]

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 \((0.85)(0.8)(\frac{762 }{\log_{20} 762}-\frac{336}{\log_{20}336})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 1SZS_1 6FZH_1 148 132
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]