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Parikh vectors
2EBU_1 3PKI_1 7QZJ_1 Letter Amino acid
13 18 2 K Lycine
6 23 40 V Valine
7 29 70 A Alanine
4 8 7 N Asparagine
5 18 23 D Aspartic acid
1 9 9 Q Glutamine
10 37 55 L Leucine
2 6 7 M Methionine
12 21 17 S Serine
6 21 26 T Threonine
0 5 3 W Tryptophan
3 4 6 Y Tyrosine
1 6 6 C Cysteine
11 25 27 E Glutamic acid
15 29 43 G Glycine
9 11 9 I Isoleucine
1 8 13 F Phenylalanine
4 32 38 R Arginine
0 11 22 H Histidine
2 34 27 P Proline 

2EBU_1|Chain A|Replication factor C subunit 1|Homo sapiens (9606)
>3PKI_1|Chains A, B, C, D, E, F|NAD-dependent deacetylase sirtuin-6|Homo sapiens (9606)
>7QZJ_1|Chains A, B|Aspartate aminotransferase family protein|Streptomyces sp. HPH0547 (1203592)
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 \(w=f(c)\)
2EBU , Knot 56 112 0.78 36 84 106 
GSSGSSGKALGSKEIPKGAENCLEGLIFVITGVLESIERDEAKSLIERYGGKVTGNVSKKTNYLVMGRDSGQSKSDKAAALGTKIIDEDGLLNLIRTMPGKKSKYEIAVETE
3PKI , Knot 151 355 0.83 40 197 340 
GSVNYAAGLSPYADKGKCGLPEIFDPPEELERKVWELARLVWQSSSVVFHTGAGISTASGIPDFRGPHGVWTMEERGLAPKFDTTFESARPTQTHMALVQLERVGLLRFLVSQNVDGLHVRSGFPRDKLAELHGNMFVEECAKCKTQYVRDTVVGTMGLKATGRLCTVAKARGLRACRGELRDTILDWEDSLPDRDLALADEASRNADLSITLGTSLQIRPSGNLPLATKRRGGRLVIVNLQPTKHDRHADLRIHGYVDEVMTRLMEHLGLEIPAWDGPRVLERALPPLPRPPTPKLEPKEESPTRINGSIPAGPKQEPCAQHNGSEPASPKRERPTSPAPHRPPKRVKAKAVPS
7QZJ , Knot 172 450 0.77 40 189 392 
MAHHHHHHHRSSLTNVGNDRHAAGTDTDRARERDRRHLWHPWSSVRQPADDRPILVAGEGCRVRDVTGAGYLDAMASAMNSSCGYAHPALLEAARRQLELLPHFDLSAASHLPAGLAAERIAGLLPAGLERTFFVNSGSEATEAAVRIAHDHWTNRGEPRDRLVTFAAGYHGTTLVAQHLSGLPTNAIHGTAPFPVTRVELPLEPAALRTPEALTLLADAFERAVLDGPPPAAVMVEPLLNVGGGVVLPDGFLRALRALCDRTGALLIVDEVFCGFGRTGRMFGFQHDGVTPDLVTMSKGISGGYVPFAALTTTDEVYRSFAADPLLGGLRYGHTTGGHAVACAVALAVLDVIEERGLVGSAARLGAELLAGLAPLAEHPEVTDVRGLGLVATVECRQPESAAALVAEAKRQGVLLRQQGRAVMAIPPLVIDAAEVAELVRAVEQAVARL

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 \(c\).

\(|P_{f(2EBU_1)}(2) \setminus P_{f(3PKI_1)}(2)|=28\), \(|P_{f(3PKI_1)}(2) \setminus P_{f(2EBU_1)}(2)|=141\). 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 1:1001001011100011011000101111110111001000010011000110101010000001111000100000011111001100011101100111000000111000
Pair \(Z_2\) Length of longest common subsequence
2EBU_1,3PKI_1 169 3
2EBU_1,7QZJ_1 173 4
3PKI_1,7QZJ_1 130 5

Newick tree

 
[
	2EBU_1:91.32,
	[
		3PKI_1:65,7QZJ_1:65
	]:26.32
]

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 \((0.85)(0.8)(\frac{467 }{\log_{20} 467}-\frac{112}{\log_{20}112})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 2EBU_1 3PKI_1 132 85.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]