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Parikh vectors
5RYN_1 6TKR_1 7TFM_1 Letter Amino acid
22 17 10 E Glutamic acid
23 11 10 K Lycine
16 12 9 V Valine
16 11 3 F Phenylalanine
12 8 7 P Proline
4 1 1 W Tryptophan
8 5 0 C Cysteine
8 9 7 Q Glutamine
13 14 4 I Isoleucine
6 8 9 M Methionine
10 12 5 N Asparagine
18 7 12 D Aspartic acid
15 25 8 G Glycine
11 19 3 H Histidine
9 9 11 T Threonine
13 13 3 Y Tyrosine
14 11 23 A Alanine
18 14 11 R Arginine
31 15 15 L Leucine
19 19 7 S Serine 

5RYN_1|Chain A|Isoform 2 of Band 4.1-like protein 3|Homo sapiens (9606)
>6TKR_1|Chains A[auth AAA], B[auth BBB]|Tankyrase-2|Homo sapiens (9606)
>7TFM_1|Chain A|ATP-dependent Clp protease subunit, heat shock protein 100 (HSP100)|Leishmania mexicana MHOM/GT/2001/U1103 (929439)
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 \(w=f(c)\)
5RYN , Knot 130 286 0.85 40 190 277 
SMPKSMQCKVILLDGSEYTCDVEKRSRGQVLFDKVCEHLNLLEKDYFGLTYRDAENQKNWLDPAKEIKKQVRSGAWHFSFNVKFYPPDPAQLSEDITRYYLCLQLRDDIVSGRLPCSFVTLALLGSYTVQSELGDYDPDECGSDYISEFRFAPNHTKELEDKVIELHKSHRGMTPAEAEMHFLENAKKLSMYGVDLHHAKDSEGVEIMLGVCASGLLIYRDRLRINRFAWPKVLKISYKRNNFYIKIRPGEFEQFESTIGFKLPNHRAAKRLWKVCVEHHTFFRLL
6TKR , Knot 112 240 0.85 40 176 232 
MHHHHHHSSGVDLGTENLYFQSMLNTSGSGTILIDLSPDDKEFQSVEEEMQSTVREHRDGGHAGGIFNRYNILKIQKVCNKKLWERYTHRRKEVSEENHNHANERMLFHGSPFVNAIIHKGFDERHAYIGGMFGAGIYFAENSSKSNQYVYGIGGGTGCPVHKDRSCYICHRQLLFCRVTLGKSFLQFSAMKMAHSPPGHHSVTGRPSVNGLALAEYVIYRGEQAYPEYLITYQIMRPEG
7TFM , Knot 73 158 0.78 38 111 151 
MTTQQPEWTQAASDLMARTAALARKKANGYLDPVHLAYVMFEDENSLASRVVRKLGAASVKDGLEARVDAIPTQMPAPTQPRPNSDMMRVMNTAEQERVALGDTLMAADHFLLALHESKEVGRILDAAGAGKKAIRTTLLEMRKGKKITSDFQDDNYE

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 \(c\).

\(|P_{f(5RYN_1)}(2) \setminus P_{f(6TKR_1)}(2)|=91\), \(|P_{f(6TKR_1)}(2) \setminus P_{f(5RYN_1)}(2)|=77\). 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 1:0110010001111010000001000001011100100010110000111000010000011011001000100111010101010110110100010000101010001101011001101111100010001100010001000100101110000010001101000001101101010110010010101101001000011011111010111100001010011110110100000010101011010010001110110001100110101000011011
Pair \(Z_2\) Length of longest common subsequence
5RYN_1,6TKR_1 168 4
5RYN_1,7TFM_1 197 3
6TKR_1,7TFM_1 181 3

Newick tree

 
[
	7TFM_1:97.85,
	[
		5RYN_1:84,6TKR_1:84
	]:13.85
]

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 \((0.85)(0.8)(\frac{526 }{\log_{20} 526}-\frac{240}{\log_{20}240})=81.8\)
Status Protein1 Protein2 d d1/2
Query variables 5RYN_1 6TKR_1 107 97
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]