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Parikh vectors
4KOO_1 4RFL_1 5IAS_1 Letter Amino acid
5 16 11 Y Tyrosine
15 14 12 V Valine
1 3 8 C Cysteine
10 14 8 H Histidine
8 42 19 I Isoleucine
4 19 16 T Threonine
6 14 14 R Arginine
8 22 20 D Aspartic acid
10 26 20 E Glutamic acid
15 8 15 F Phenylalanine
17 26 12 A Alanine
5 16 15 N Asparagine
13 27 16 G Glycine
1 11 10 M Methionine
15 29 26 S Serine
2 2 2 W Tryptophan
4 4 4 Q Glutamine
18 33 21 L Leucine
13 34 22 K Lycine
10 11 7 P Proline 

4KOO_1|Chains A, B, C, D|Single-stranded DNA-binding protein WHY1, chloroplastic|Arabidopsis thaliana (3702)
>4RFL_1|Chains A, B, C, D|Glycerol-1-phosphate dehydrogenase [NAD(P)+]|Methanocaldococcus jannaschii DSM 2661 (243232)
>5IAS_1|Chain A|Caspase-3|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 \(w=f(c)\)
4KOO , Knot 83 180 0.79 40 124 170 
MAEGLPARFYVGHSIYKGKAALTVDPRAPEFVALDSGAFKLSKDGFLLLQFAPSAGVRQYDWSKKQVFSLSVTEIGTLVSLGPRESCEFFHDPFKGKSDEGKVRKVLKVEPLPDGSGHFFNLSVQNKLVNVDESIYIPITRAEFAVLISAFNFVLPYLIGWHAFANSIKAAALEHHHHHH
4RFL , Knot 155 371 0.82 40 197 340 
MRGSHHHHHHGMASMTGGQQMGRDLYDDDDKDHPFTMIIVTPRYTIIEDGAINKIEEILKKLNLKNPLVITGKNTKKYCRFFYDIVYYDEILNNLEIELKKYTAYDCVIGIGGGRSIDTGKYLAYKLGIPFISVPTTASNDGIASPIVSIRQPSFMVDAPIAIIADTEIIKKSPRRLLSAGMGDIVSNITAVLDWKLAYKEKGEKYSESSAIFSKTIAKELISYVLNSDLSEYHNKLVKALVGSGIAIAIANSSRPASGSEHLFSHALDKLKEEYNLNINSLHGEQCGIGTIMMSYLHEKENKKLSGLHEKIKMSLKKVDAPTTAKELGFDEDIIIEALTMAHKIRNRWTILRDGLSREEARKLAEETGVI
5IAS , Knot 123 278 0.83 40 183 273 
MENTENSVDSKSIKNLEPKIIHGSESMDSGISLDNSYKMDYPEMGLCIIINNKNFHKSTGMTSRSGTDVDAANLRETFRNLKYEVRNKNDLTREEIVELMRDVSKEDHSKRSSFVCVLLSHGEEGIIFGTNGPVDLKKITNFFRGDRCRSLTGKPKLFIIQACRGTELDCGIETDSGVDDDMACHKIPVEADFLYAYSTAPGYYSWRNSKDGSWFIQSLCAMLKQYADKLEFMHILTRVNRKVATEFESFSFDATFHAKKQIPCIYSMLTKELYFYHL

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 \(c\).

\(|P_{f(4KOO_1)}(2) \setminus P_{f(4RFL_1)}(2)|=47\), \(|P_{f(4RFL_1)}(2) \setminus P_{f(4KOO_1)}(2)|=120\). 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 1:110111101011001001011101010110111100111010001111101110111000010000110101001101101110000011001101000010100110101110101011010100011010001011100101111101101111011110111001011110000000
Pair \(Z_2\) Length of longest common subsequence
4KOO_1,4RFL_1 167 6
4KOO_1,5IAS_1 177 4
4RFL_1,5IAS_1 158 3

Newick tree

 
[
	4KOO_1:88.25,
	[
		4RFL_1:79,5IAS_1:79
	]:9.25
]

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 \((0.85)(0.8)(\frac{551 }{\log_{20} 551}-\frac{180}{\log_{20}180})=107.\)
Status Protein1 Protein2 d d1/2
Query variables 4KOO_1 4RFL_1 135 97.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]