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Parikh vectors
2ZBX_1 8IJW_1 1GHW_1 Letter Amino acid
29 72 4 G Glycine
17 13 0 H Histidine
5 46 3 K Lycine
8 29 1 Y Tyrosine
32 75 0 V Valine
30 53 3 R Arginine
3 30 1 C Cysteine
18 71 1 I Isoleucine
48 101 5 L Leucine
12 50 2 F Phenylalanine
10 34 0 N Asparagine
18 48 0 Q Glutamine
26 61 6 E Glutamic acid
30 60 2 T Threonine
2 11 0 W Tryptophan
44 92 1 A Alanine
27 52 3 D Aspartic acid
8 27 0 M Methionine
26 49 1 P Proline
19 59 3 S Serine 

2ZBX_1|Chain A|Cytochrome P450-SU1|Streptomyces griseolus (1909)
>8IJW_1|Chain A|Sodium/potassium-transporting ATPase subunit alpha|Sus scrofa (9823)
>1GHW_1|Chain A[auth L]|THROMBIN|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 \(w=f(c)\)
2ZBX , Knot 169 412 0.82 40 206 396 
MTDTATTPQTTDAPAFPSNRSCPYQLPDGYAQLRDTPGPLHRVTLYDGRQAWVVTKHEAARKLLGDPRLSSNRTDDNFPATSPRFEAVRESPQAFIGLDPPEHGTRRRMTISEFTVKRIKGMRPEVEEVVHGFLDEMLAAGPTADLVSQFALPVPSMVICRLLGVPYADHEFFQDASKRLVQSTDAQSALTARNDLAGYLDGLITQFQTEPGAGLVGALVADQLANGEIDREELISTAMLLLIAGHETTASMTSLSVITLLDHPEQYAALRADRSLVPGAVEELLRYLAIADIAGGRVATADIEVEGQLIRAGEGVIVVNSIANRDGTVYEDPDALDIHRSARHHLAFGFGVHQCLGQNLARLELEVILNALMDRVPTLRLAVPVEQLVLRPGTTIQGVNELPVTWHHHHHH
8IJW , Knot 377 1033 0.84 40 321 923 
GKAENYELYQVELGPGPSGDMAAKMSKKKAGRGGGKRKEKLENMKKEMEINDHQLSVAELEQKYQTSATKGLSASLAAELLLRDGPNALRPPRGTPEYVKFARQLAGGLQCLMWVAAAICLIAFAIQASEGDLTTDDNLYLALALIAVVVVTGCFGYYQEFKSTNIIASFKNLVPQQATVIRDGDKFQINADQLVVGDLVEMKGGDRVPADIRILQAQGCKVDNSSLTGESEPQTRSPECTHESPLETRNIAFFSTMCLEGTAQGLVVNTGDRTIIGRIASLASGVENEKTPIAIEIEHFVDIIAGLAILFGATFFIVAMCIGYTFLRAMVFFMAIVVAYVPEGLLATVTVCLSLTAKRLASKNCVVKNLEAVETLGSTSVICSDKTGTLTQNRMTVSHLWFDNHIHSADTTEDQSGQTFDQSSETWRALCRVLTLCNRAAFKSGQDAVPVPKRIVIGDASETALLKFSELTLGNAMGYRERFPKVCEIPFNSTNKFQLSIHTLEDPRDPRHVLVMKGAPERVLERCSSILIKGQELPLDEQWREAFQTAYLSLGGLGERVLGFCQLYLSEKDYPPGYAFDVEAMNFPTSGLCFAGLVSMIDPPRATVPDAVLKCRTAGIRVIMVTGDHPITAKAIAASVGIISEGSETVEDIAARLRVPVDQVNRKDARACVINGMQLKDMDPSELVEALRTHPEMVFARTSPQQKLVIVESCQRLGAIVAVTGDGVNDSPALKKADIGVAMGIAGSDAAKNAADMILLDDNFASIVTGVEQGRLIFDNLKKSIAYTLTKNIPELTPYLIYITVSVPLPLGCITILFIELCTDIFPSVSLAYEKAESDIMHLRPRNPKRDRLVNEPLAAYSYFQIGAIQSFAGFTDYFTAMAQEGWFPLLCVGLRPQWENHHLQDLQDSYGQEWTFGQRLYQQYTCYTVFFISIEMCQIADVLIRKTRRLSAFQQGFFRNRILVIAIVFQVCIGCFLCYCPGMPNIFNFMPIRFQWWLVPMPFSLLIFVYDEIRKLGVRCCPGSWWDQELYY
1GHW , Knot 24 36 0.79 28 34 34 
TFGSGEADCGLRPLFEKKSLEDKTERELLESYIDGR

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 \(c\).

\(|P_{f(2ZBX_1)}(2) \setminus P_{f(8IJW_1)}(2)|=20\), \(|P_{f(8IJW_1)}(2) \setminus P_{f(2ZBX_1)}(2)|=135\). 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 1:1000100100001111100000100110101010001111001010010011110000110011101010000000011100101011000101111101100100001010010100101101010011011100111111010110011111101110011111010001100100011000010011010001110101110010001111111111100110101000011001111111100001010010110110010001110100011111100110011110111101101010101011011011111001100010100010110100010001111111000110011010101110111001101011111001110110010110011101000000
Pair \(Z_2\) Length of longest common subsequence
2ZBX_1,8IJW_1 155 4
2ZBX_1,1GHW_1 200 3
8IJW_1,1GHW_1 293 4

Newick tree

 
[
	1GHW_1:13.74,
	[
		2ZBX_1:77.5,8IJW_1:77.5
	]:60.24
]

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 \((0.85)(0.8)(\frac{1445 }{\log_{20} 1445}-\frac{412}{\log_{20}412})=265.\)
Status Protein1 Protein2 d d1/2
Query variables 2ZBX_1 8IJW_1 337 231.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]