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Parikh vectors
5TNS_1 8CRF_1 5YIF_1 Letter Amino acid
11 4 29 D Aspartic acid
11 6 40 K Lycine
9 1 9 M Methionine
5 0 14 W Tryptophan
29 5 27 A Alanine
12 6 12 Q Glutamine
13 3 34 Y Tyrosine
19 19 29 V Valine
8 1 24 N Asparagine
2 1 3 C Cysteine
23 12 39 E Glutamic acid
9 3 29 I Isoleucine
28 15 31 L Leucine
18 2 23 F Phenylalanine
19 6 19 P Proline
17 8 16 R Arginine
18 5 21 H Histidine
15 5 16 S Serine
11 4 22 T Threonine
24 11 38 G Glycine 

5TNS_1|Chains A, B, C, D|CFTR inhibitory factor|Pseudomonas aeruginosa (strain UCBPP-PA14) (208963)
>8CRF_1|Chain A|Host translation inhibitor nsp1|Severe acute respiratory syndrome coronavirus 2 (2697049)
>5YIF_1|Chains A, B|Pyruvylated beta-D-galactosidase|Bacillus sp. (1409)
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 \(w=f(c)\)
5TNS , Knot 136 301 0.86 40 202 292 
AEEFPVPNGFESAYREVDGVKLHYVKGGQGPLVMLVHGFGQTWYEWHQLMPELAKRFTVIAPDLPGLGQSEPPKTGYSGEQVAVYLHKLARQFSPDRPFDLVAHSIGIWNTYPMVVKNQADIARLVYMEAPIPDARIYRFPAFTAQGESLVWHFSFFAADDRLAETLIAGKERFFLEHFIKSHASNTEVFSERLLDLYARSYAKPHSLNASFEYYRALNESVRQNAELAKTRLQMPTMTLAGGGHGGMGTFQLEQMKAYAEDVEGHVLPGCGHWLPEECAAPMNRLVIDFLSRGRHHHHHH
8CRF , Knot 58 117 0.78 38 87 113 
EKTHVQLSLPVLQVRDVLVRGFGDSVEEVLSEARQHLKDGTCGLVEVEKGVLPQLEQPYVFIKRSDARTAPHGHVMVELVAELEGIQYGRSGETLGVLVPHVGEIPVAYRKVLLRKN
5YIF , Knot 193 475 0.83 40 246 451 
MKFPHDFLFGAASASYQVEGAWNEDGKGVTNWDEFSKIPGKTYNGTNGDIAVDHYHRYKEDVRLMAEMGLESYRFSISWARILPTGDGKVNEKGIEFYNNLIDECLKYGIVPFVTLYHWDLPLPLEKDGGWTNKRTAEAFVKYAETCFKAFGDRVKHWITFNATVMFCGLGYLKGAHPPGIQNDVPKYFQATHYVFYAHAKTVAVYKQLKQYGEIGITHVFLPAYSVDDQKENIQAANHANEYETYWYYDPILKGEYPSYVVQQLKEKGWTPNWTVEELEIIKQNAEENDFIGLNYYQPIRVERYDMDIKSEEHSRENSTLAPGNPSFDGFYRTVKMDDKTYTKWGWEISPEGFLEGLHMLKARYGDIKMYVTENGLGDEDPIIDGEIVDVPRIKFIEAHLKVMKRAIEEGINLKGYYAWSVIDLLSWLNGYKKQYGFIFVDHNDNLKRKKKLSFHWYKRVVETRGEELHHHHHH

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 \(c\).

\(|P_{f(5TNS_1)}(2) \setminus P_{f(8CRF_1)}(2)|=139\), \(|P_{f(8CRF_1)}(2) \setminus P_{f(5TNS_1)}(2)|=24\). 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 1:1001111011001000101101001011011111110111001001001110110010111101111100011001001001110100110010100110111001111000111100010110110101111010100111101010011101011110001100111100011100110001000011000110101000101001010100001100010001011000101101011111011110101001010100101011110101110001111001110110010000000
Pair \(Z_2\) Length of longest common subsequence
5TNS_1,8CRF_1 163 4
5TNS_1,5YIF_1 168 6
8CRF_1,5YIF_1 207 3

Newick tree

 
[
	5YIF_1:98.13,
	[
		5TNS_1:81.5,8CRF_1:81.5
	]:16.63
]

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 \((0.85)(0.8)(\frac{418 }{\log_{20} 418}-\frac{117}{\log_{20}117})=91.0\)
Status Protein1 Protein2 d d1/2
Query variables 5TNS_1 8CRF_1 114 76.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]