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Parikh vectors
4LDE_1 6KGS_1 5ZSC_1 Letter Amino acid
41 57 24 A Alanine
23 22 70 N Asparagine
23 61 25 G Glycine
11 13 11 M Methionine
22 33 78 S Serine
10 3 19 C Cysteine
6 9 19 H Histidine
40 24 42 I Isoleucine
28 12 44 F Phenylalanine
39 51 38 V Valine
16 38 37 Q Glutamine
46 42 134 L Leucine
26 18 51 K Lycine
10 40 38 P Proline
11 5 8 W Tryptophan
24 32 37 R Arginine
23 38 39 D Aspartic acid
20 18 40 E Glutamic acid
31 31 42 T Threonine
19 15 27 Y Tyrosine 

4LDE_1|Chain A|Lysozyme, Beta-2 adrenergic receptor|Enterobacteria phage T4 (10665)
>6KGS_1|Chain A|Penicillin-binding protein PbpB|Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv) (83332)
>5ZSC_1|Chains A, B|Toll-like receptor 7|Macaca mulatta (9544)
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 \(w=f(c)\)
4LDE , Knot 193 469 0.84 40 233 440 
DYKDDDDAENLYFQGNIFEMLRIDEGLRLKIYKDTEGYYTIGIGHLLTKSPSLNAAKSELDKAIGRNTNGVITKDEAEKLFNQDVDAAVRGILRNAKLKPVYDSLDAVRRAALINMVFQMGETGVAGFTNSLRMLQQKRWDEAAVNLAKSRWYNQTPNRAKRVITTFRTGTWDAYAADEVWVVGMGIVMSLIVLAIVFGNVLVITAIAKFERLQTVTNYFITSLACADLVMGLAVVPFGAAHILTKTWTFGNFWCEFWTSIDVLCVTASIETLCVIAVDRYFAITSPFKYQSLLTKNKARVIILMVWIVSGLTSFLPIQMHWYRATHQEAINCYAEETCCDFFTNQAYAIASSIVSFYVPLVIMVFVYSRVFQEAKRQLQKIDKFALKEHKALKTLGIIMGTFTLCWLPFFIVNIVHVIQDNLIRKEVYILLNWIGYVNSGFNPLIYCRSPDFRIAFQELLCLRRSSLK
6KGS , Knot 218 562 0.81 40 240 513 
GPLGSGQLKVTDVQPAARGSIVDRNNDRLAFTIEARALTFQPKRIRRQLEEARKKTSAAPDPQQRLRDIAQEVAGKLNNKPDAAAVLKKLQSDETFVYLARAVDPAVASAICAKYPEVGAERQDLRQYPGGSLAANVVGGIDWDGHGLLGLEDSLDAVLAGTDGSVTYDRGSDGVVIPGSYRNRHKAVHGSTVVLTLDNDIQFYVQQQVQQAKNLSGAHNVSAVVLDAKTGEVLAMANDNTFDPSQDIGRQGDKQLGNPAVSSPFEPGSVNKIVAASAVIEHGLSSPDEVLQVPGSIQMGGVTVHDAWEHGVMPYTTTGVFGKSSNVGTLMLSQRVGPERYYDMLRKFGLGQRTGVGLPGESAGLVPPIDQWSGSTFANLPIGQGLSMTLLQMTGMYQAIANDGVRVPPRIIKATVAPDGSRTEEPRPDDIRVVSAQTAQTVRQMLRAVVQRDPMGYQQGTGPTAGVPGYQMAGKTGTAQQINPGCGCYFDDVYWITFAGIATADNPRYVIGIMLDNPARNSDGAPGHSAAPLFHNIAGWLMQRENVPLSPDPGPPLVLQAT
5ZSC , Knot 299 823 0.81 40 278 687 
RSPWARWFPKTLPCDVTLDVSKNHVIVDCTDKHLTEIPGGIPTNTTNLTLTINHIPDISPASFHRLVHLVEIDFRCNCVPIRLGSKSNMCPRRLQIKPRSFSGLTYLKSLYLDGNQLLEIPQGLPPSLQLLSLEANNIFSIRKEQLTELANIEILYLGQNCYYRNPCYVSYSIEKDAFLNLTKLKVLSLKDNNVTTVPTVLPSTLTELYLYNNMIAEIQEDDFNNLNQLQILDLSGNCPRCYNAPFPCTPCKNNSPLQIPVNAFDALTELKVLRLHSNSLQHVPPRWFKNINNLQELDLSQNFLAKEIGDAKFLHFLPNLIQLDLSFNFELQVYRASMNLSQAFSSLKSLKILRIRGYVFKELKSFQLSPLHNLQNLEVLDLGTNFIKIANLSMFKQFKRLKVIDLSVNKISPSGDSLVPRGSSNARTSVESYEPQVLEQLYYFRYDKYARSCRFKNKEASFTSVQESCYKYGQTLDLSKNSIFFIKSSDFQHLSFLKCLNLSGNLISQTLNGSEFQPLAELRYLDFSNNRLDLLHSTAFEELRKLEVLDISSNSHYFQSEGITHMLNFTKNLKVLQKLMMNDNDISSSTSRTMESESLRTLEFRGNHLDVLWRDGDNRYLQLFKNLLKLEELDISKNSLSFLPSGVFDGMPPNLKNLSLAKNGLKSFIWEKLRYLKNLETLDLSHNQLTTVPERLSNCSRSLKNLILKNNQIRSLTKYFLQDAFQLRYLDLSSNKIQMIQKTSFPENVLNNLKMLLLHHNRFLCTCDAVWFVWWVQHTEVTIPYLATDVTCVGPGAHKGQSVISLDLYTCELDLTNEFLVPR

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 \(c\).

\(|P_{f(4LDE_1)}(2) \setminus P_{f(6KGS_1)}(2)|=74\), \(|P_{f(6KGS_1)}(2) \setminus P_{f(4LDE_1)}(2)|=81\). 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 1:0000000100101010110110100110101000001000111101100010101100010011100001110000100110001011101110010101100010110011110111011001111100010110000100111011000100001001001100100101010110011111111110111111111011110111010010010001100110101111111111111011000101101100110010110101010010111100011100110000110000101111111110110011110101001000011000100000011000101110011010111111111000110010001001001110000110011111101010111111101101100011000101110111010011011100001010111001101000010
Pair \(Z_2\) Length of longest common subsequence
4LDE_1,6KGS_1 155 6
4LDE_1,5ZSC_1 155 4
6KGS_1,5ZSC_1 160 5

Newick tree

 
[
	5ZSC_1:79.17,
	[
		4LDE_1:77.5,6KGS_1:77.5
	]:1.67
]

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 \((0.85)(0.8)(\frac{1031 }{\log_{20} 1031}-\frac{469}{\log_{20}469})=147.\)
Status Protein1 Protein2 d d1/2
Query variables 4LDE_1 6KGS_1 190 173.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]