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Parikh vectors
5KHL_1 7DKD_1 5PWL_1 Letter Amino acid
26 103 3 A Alanine
0 4 12 C Cysteine
0 10 3 W Tryptophan
20 49 8 V Valine
22 36 13 S Serine
10 33 11 N Asparagine
10 46 5 D Aspartic acid
22 60 6 G Glycine
17 19 10 I Isoleucine
10 49 13 K Lycine
14 32 19 E Glutamic acid
14 8 6 H Histidine
35 70 11 L Leucine
1 29 13 F Phenylalanine
3 26 5 Y Tyrosine
10 32 11 R Arginine
19 34 12 Q Glutamine
7 16 5 M Methionine
16 33 8 P Proline
15 31 6 T Threonine 

5KHL_1|Chain A[auth B]|Hemin ABC transporter, periplasmic hemin-binding protein HutB|Vibrio cholerae (345073)
>7DKD_1|Chains A, B|Dipeptidyl-peptidase|Stenotrophomonas maltophilia (strain R551-3) (391008)
>5PWL_1|Chains A, B|Nuclear autoantigen Sp-100|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 \(w=f(c)\)
5KHL , Knot 117 271 0.80 36 163 256 
HHHHHHSSGLVPRGSHMRIVSAGSAVTELILALGAEQQLVAVDVTSEVPSSLNLPTVGYHRRLAAEGLLTLEPTHLIGSDEMGPDTALQQLRSSGIQVNVINSDSTPQGLLTRIDQIAQITHTEQHAQKLKENVQQQINALQAKRPEKPKKVLFLLLHEGRAANVAGSDTVPDTIIGLIGAHNPASPSITSYKPLSMESMIEMQPDMVLVSGRSLEKLGGADAVLNAVPMLAATPAGQNKNIVAIDGHALVGGLGLKSLQEAQRIQTLLYP
7DKD , Knot 273 720 0.83 40 263 646 
MRSNLLAFSIVASLGLAQVAHAAEGMWVPQQLPEIAGPLQKAGLKLSPEQLANLTGDPMGAVVALGGCTASFVSPQGLVVTNHHCAYGAIQLNSTAQKNLIKDGFNAPTLKDELSAGPNARVFVLDQITDVTAQAKAAIAGAGNDPLARSRALDAFDKAQVAACEADAGFRCRLYSFSGGNTYRLFRNMEIKDVRLVYAPPGSVGKFGGDVDNWMWPRHTGDFSFYRAYVGKDGKPAAFAADNVPYQPKHFLKFADQPLGADDFVMVAGYPGRTNRYALAGEFNETASFTYPTIAKHYNAVLKMIADAGKADADVKVKYAATAASMNNVAKNYLGQLEGFKRIDAAGQKQAEEAAVLAWLKKQGAAGKPALAAHAQLLKHLDTSKSTRERDLFVGQFNNTSAVGAAITLYRLSIERSKPDAEREAGYQERDLTTIEGGLKQMDRRYVAKMDQQLQTYWLDQYVALPAAQRDNEVLNKWLAGSDAAAVKSLVNKLGGTELGSLDTRLKWFKADRAAFEASNDPAIQYAVAVMPALLKQEEQKKIREGESLTARPLYLQAVADYKKSQGEFVYPDANLSLRITFGNVMGYGKDGVKYTPFTTLEGVAAKETGEDPFDSPKALLDAVKAKRYGGLEDKRLGSVPVNFLSNLDITGGNSGSPVLDANGKLVGLAFDGNWESVSSNWVFDPVMTRMIAVDSRYMQWIMQEVAPAPQLLKELNLAK
5PWL , Knot 87 180 0.83 40 137 176 
ENSNICEVCNKWGRLFCCDTCPRSFHEHCHIPSVEANKNPWSCIFCRIKTIQERCPESQSGHQESEVLMRQMLPEEQLKCEFLLLKVYCDSKSCFFASEPYYNREGSQGPQKPMWLNKVKTSLNEQMYTRVEGFVQDMRLIFHNHKEFYREDKFTRLGIQVQDIFEKNFRNIFAIQETSK

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 \(c\).

\(|P_{f(5KHL_1)}(2) \setminus P_{f(7DKD_1)}(2)|=32\), \(|P_{f(7DKD_1)}(2) \setminus P_{f(5KHL_1)}(2)|=132\). 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 1:0000000011110100101101101100111111100011110100011001011011000011101110101001110001110011001000110101100000101110010011010000001001000100010110100100100111111001011011100011001111111001101010000110100110101011110100100111101110111111101110000111101011111111001001001001101
Pair \(Z_2\) Length of longest common subsequence
5KHL_1,7DKD_1 164 4
5KHL_1,5PWL_1 176 4
7DKD_1,5PWL_1 220 3

Newick tree

 
[
	5PWL_1:10.82,
	[
		5KHL_1:82,7DKD_1:82
	]:22.82
]

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 \((0.85)(0.8)(\frac{991 }{\log_{20} 991}-\frac{271}{\log_{20}271})=194.\)
Status Protein1 Protein2 d d1/2
Query variables 5KHL_1 7DKD_1 242 164
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]