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Parikh vectors
1HRN_1 3VRW_1 1PJO_1 Letter Amino acid
23 15 0 V Valine
16 12 7 A Alanine
10 12 0 R Arginine
16 14 0 E Glutamic acid
6 11 0 H Histidine
17 8 0 Y Tyrosine
26 34 0 L Leucine
12 7 0 N Asparagine
34 12 2 G Glycine
22 17 0 I Isoleucine
15 16 0 K Lycine
8 10 0 M Methionine
33 28 0 S Serine
29 11 0 T Threonine
3 1 0 W Tryptophan
17 21 0 D Aspartic acid
6 3 1 C Cysteine
12 14 0 Q Glutamine
19 8 0 F Phenylalanine
13 17 0 P Proline 

1HRN_1|Chains A, B|RENIN|Homo sapiens (9606)
>3VRW_1|Chain A|Vitamin D3 receptor|Rattus norvegicus (10116)
>1PJO_1|Chain A|5'-R(*CP*AP*AP*AP*GP*AP*AP*AP*AP*G)-3'|
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 \(w=f(c)\)
1HRN , Knot 146 337 0.84 40 205 322 
GNTTSSVILTNYMDTQYYGEIGIGTPPQTFKVVFDTGSSNVWVPSSKCSRLYTACVYHKLFDASDSSSYKHNGTELTLRYSTGTVSGFLSQDIITVGGITVTQMFGEVTEMPALPFMLAEFDGVVGMGFIEQAIGRVTPIFDNIISQGVLKEDVFSFYYNRDSENSQSLGGQIVLGGSDPQHYEGNFHYINLIKTGVWQIQMKGVSVGSSTLLCEDGCLALVDTGASYISGSTSSIEKLMEALGAKKRLFDYVVKCNEGPTLPDISFHLGGKEYTLTSADYVFQESYSSKKLCTLAIHAMDIPPPTGPTWALGATFIRKFYTEFDRRNNRIGFALAR
3VRW , Knot 119 271 0.82 40 172 256 
GSHMGSPNSPLKDSLRPKLSEEQQHIIAILLDAHHKTYDPTYADFRDFRPPVRMDGSTGSVTLDLSPLSMLPHLADLVSYSIQKVIGFAKMIPGFRDLTSDDQIVLLKSSAIEVIMLRSNQSFTMDDMSWDCGSQDYKYDVTDVSKAGHTLELIEPLIKFQVGLKKLNLHEEEHVLLMAICIVSPDRPGVQDAKLVEAIQDRLSNTLQTYIRCRHPPPGSHQLYAKMIQKLADLRSLNEEHSKQYRSLSFQPENSMKLTPLVLEVFGNEIS
1PJO , Knot 5 10 0.38 6 4 5 
CAAAGAAAAG

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 \(c\).

\(|P_{f(1HRN_1)}(2) \setminus P_{f(3VRW_1)}(2)|=109\), \(|P_{f(3VRW_1)}(2) \setminus P_{f(1HRN_1)}(2)|=76\). 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 1:1000001110001000001011110110010111001000111100000010010100011010000000001001010000101011100011011110100111010011111111101011111111001110101110011001110001101000000000001110111110010000101001011001110101011011000110001011110011001010000100110111100011001100001101101010111000010010011000000001001110110111101101111101100100010000001111110
Pair \(Z_2\) Length of longest common subsequence
1HRN_1,3VRW_1 185 4
1HRN_1,1PJO_1 207 2
3VRW_1,1PJO_1 174 2

Newick tree

 
[
	1HRN_1:10.60,
	[
		3VRW_1:87,1PJO_1:87
	]:14.60
]

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 \((0.85)(0.8)(\frac{608 }{\log_{20} 608}-\frac{271}{\log_{20}271})=94.6\)
Status Protein1 Protein2 d d1/2
Query variables 1HRN_1 3VRW_1 121 108
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]