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Parikh vectors
4GVU_1 3UZC_1 8RWL_1 Letter Amino acid
8 14 3 C Cysteine
6 17 5 H Histidine
3 7 15 K Lycine
3 20 5 F Phenylalanine
7 14 12 P Proline
7 6 2 W Tryptophan
4 9 27 E Glutamic acid
25 18 25 G Glycine
18 38 25 L Leucine
2 8 8 M Methionine
11 10 4 Y Tyrosine
12 14 25 R Arginine
17 14 7 N Asparagine
7 4 21 D Aspartic acid
15 12 7 Q Glutamine
10 29 26 I Isoleucine
17 41 29 A Alanine
22 16 14 S Serine
19 11 19 T Threonine
27 27 38 V Valine 

4GVU_1|Chain A|Chymotrypsin-like elastase family member 1|Sus scrofa (9823)
>3UZC_1|Chain A|Adenosine A2A Receptor|Homo sapiens (9606)
>8RWL_1|Chains A, B|Malate dehydrogenase|Methanopyrus kandleri (2320)
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 \(w=f(c)\)
4GVU , Knot 108 240 0.82 40 162 230 
VVGGTEAQRNSWPSQISLQYRSGSSWAHTCGGTLIRQNWVMTAAHCVDRELTFRVVVGEHNLNQNNGTEQYVGVQKIVVHPYWNTDDVAAGYDIALLRLAQSVTLNSYVQLGVLPRAGTILANNSPCYITGWGLTRTNGQLAQTLQQAYLPTVDYAICSSSSYWGSTVKNSMVCAGGDGVRSGCQGDSGGPLHCLVNGQYAVHGVTSFVSRLGCNVTRKPTVFTRVSAYISWINNVIASN
3UZC , Knot 137 329 0.80 40 186 298 
MPIMGSSVYITVELAIAVLAILGNVLVCWAVWLNSNLQNVTNYFVVSLAAADILVGVLAIPFAITISTGFCAACHGCLFIACFVLVLAQSSIFSLLAIAIDRYIAIAIPLRYNGLVTGTRAAGIIAICWVLSFAIGLTPMLGWNNCGQPKEGKNHSQGCGEGQVACLFEDVVPMNYMVYFNFFACVLVPLLLMLGVYLRIFAAARRQLKQMESQPLPGERARSTLQKEVHAAKSAAIIAGLFALCWLPLHIINCFTFFCPDCSHAPLWLMYLAIVLAHTNSVVNPFIYAYRIREFRQTFRKIIRSHVLRQQEPFKAAAAHHHHHHHHHH
8RWL , Knot 135 317 0.81 40 182 303 
MSKVAVIGATGRVGSTAAARLALLDCVNEVTLIARPKSVDKLRGLRRDILDSLAAAQKDAEITIGCERDDYVDADVIVMTAGIPQKPGQTRLDLTKDNAAIIKKYLEGVAEENPEAIVLVVTNPVDVLTYVALKVSGLPKNRVIGLGTHLDSMRFKVLIAKHFNVHMSEVHTRIIGEHGDTMVPVISSTSVGGIPVTRMPGWEDFDVEEAVREVKEAGQRIIETWGGSQFGPAQAITNLVRTILQDERRVLTVSAYLDGEIDGIRDVCIGVPARLGREGVLEIVPIELEEDEMRAFRRSVKVVKEATREAMEAISER

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 \(c\).

\(|P_{f(4GVU_1)}(2) \setminus P_{f(3UZC_1)}(2)|=62\), \(|P_{f(3UZC_1)}(2) \setminus P_{f(4GVU_1)}(2)|=86\). 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 1:111100100001100101000010011000110110001110110010001010111100010000100001110011101010000111100111101100101000101111101101110001001011110000101100100101101001100000011001000110111011001001001111001101001101100110011001000101100101010110011100
Pair \(Z_2\) Length of longest common subsequence
4GVU_1,3UZC_1 148 4
4GVU_1,8RWL_1 154 4
3UZC_1,8RWL_1 178 5

Newick tree

 
[
	8RWL_1:86.06,
	[
		4GVU_1:74,3UZC_1:74
	]:12.06
]

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 \((0.85)(0.8)(\frac{569 }{\log_{20} 569}-\frac{240}{\log_{20}240})=93.5\)
Status Protein1 Protein2 d d1/2
Query variables 4GVU_1 3UZC_1 115 99.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]