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Parikh vectors
3INF_1 1YXM_1 1CZK_1 Letter Amino acid
24 28 11 A Alanine
35 30 18 G Glycine
15 22 8 K Lycine
23 7 15 D Aspartic acid
13 6 0 H Histidine
31 27 12 S Serine
34 27 10 V Valine
29 25 12 L Leucine
8 4 1 M Methionine
22 13 3 P Proline
19 9 2 R Arginine
15 15 12 N Asparagine
6 5 1 C Cysteine
17 10 12 Q Glutamine
24 18 13 I Isoleucine
26 14 8 T Threonine
7 6 4 W Tryptophan
28 18 11 E Glutamic acid
20 11 8 F Phenylalanine
19 8 8 Y Tyrosine 

3INF_1|Chain A|Beta-secretase 1|Homo sapiens (9606)
>1YXM_1|Chains A, B, C, D|peroxisomal trans 2-enoyl CoA reductase|Homo sapiens (9606)
>1CZK_1|Chain A|FLAVODOXIN|Synechococcus elongatus (1140)
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 \(w=f(c)\)
3INF , Knot 175 415 0.84 40 232 393 
ETDEEPEEPGRRGSFVEMVDNLRGKSGQGYYVEMTVGSPPQTLNILVDTGSSNFAVGAAPHPFLHRYYQRQLSSTYRDLRKGVYVPYTQGKWEGELGTDLVSIPHGPNVTVRANIAAITESDKFFINGSNWEGILGLAYAEIARPDDSLEPFFDSLVKQTHVPNLFSLQLCGAGFPLNQSEVLASVGGSMIIGGIDHSLYTGSLWYTPIRREWYYEVIIVRVEINGQDLKMDCKEYNYDKSIVDSGTTNLRLPKKVFEAAVKSIKAASSTEKFPDGFWLGEQLVCWQAGTTPWNIFPVISLYLMGEVTNQSFRITILPQQYLRPVEDVATSQDDCYKFAISQSSTGTVMGAVIMEGFYVVFDRARKRIGFAVSACHVHDEFRTAAVEGPFVTLDMEDCGYNIPQTDESTHHHHHH
1YXM , Knot 132 303 0.83 40 181 289 
MASWAKGRSYLAPGLLQGQVAIVTGGATGIGKAIVKELLELGSNVVIASRKLERLKSAADELQANLPPTKQARVIPIQCNIRNEEEVNNLVKSTLDTFGKINFLVNNGGGQFLSPAEHISSKGWHAVLETNLTGTFYMCKAVYSSWMKEHGGSIVNIIVPTKAGFPLAVHSGAARAGVYNLTKSLALEWACSGIRINCVAPGVIYSQTAVENYGSWGQSFFEGSFQKIPAKRIGVPEEVSSVVCFLLSPAASFITGQSVDVDGGRSLYTHSYEVPDHDNWPKGAGDLSVVKKMKETFKEKAKL
1CZK , Knot 82 169 0.83 38 129 161 
AKIGLFYGTQTGVTQTIAESIQQEFGGESIVDLNDIANADASDLNAYDYLIIGCPTWNVGELQSDWEGIYDDLDSVNFQGKKVAYFGAGDQVGYSDNFQNAMGILEEKISSLGSQTVGYWPIEGYDFNESKAVRNNQFVGLAIDEDNQPDLTKNRIKTWVSQLKSEFGL

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 \(c\).

\(|P_{f(3INF_1)}(2) \setminus P_{f(1YXM_1)}(2)|=103\), \(|P_{f(1YXM_1)}(2) \setminus P_{f(3INF_1)}(2)|=52\). 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 1:0000010011001011011001010010100101011011001011100100011111110111000000010000001001101100010101011001101101101010101111000001110100101111110101101000101110011000011011010101111110000111011101111110001001011001100010001111010101001010000000000110010001011001101110010110000011011111001101011001101111101011101000010101110001011001100000000111000001011111110110111001000111110100100010011101111010100010011000000000000
Pair \(Z_2\) Length of longest common subsequence
3INF_1,1YXM_1 155 5
3INF_1,1CZK_1 193 4
1YXM_1,1CZK_1 158 4

Newick tree

 
[
	1CZK_1:91.46,
	[
		3INF_1:77.5,1YXM_1:77.5
	]:13.96
]

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 \((0.85)(0.8)(\frac{718 }{\log_{20} 718}-\frac{303}{\log_{20}303})=114.\)
Status Protein1 Protein2 d d1/2
Query variables 3INF_1 1YXM_1 144 123.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]