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Parikh vectors
1ZCJ_1 6UWY_1 1UOS_1 Letter Amino acid
17 23 7 D Aspartic acid
43 23 6 G Glycine
16 19 6 Y Tyrosine
21 23 7 I Isoleucine
41 17 13 S Serine
24 20 3 R Arginine
11 17 5 N Asparagine
10 11 4 H Histidine
9 12 2 M Methionine
30 15 4 P Proline
20 16 8 F Phenylalanine
20 13 5 T Threonine
36 17 7 A Alanine
5 6 8 C Cysteine
18 14 6 Q Glutamine
27 21 9 E Glutamic acid
44 39 8 L Leucine
33 33 13 K Lycine
7 4 6 W Tryptophan
31 18 8 V Valine 

1ZCJ_1|Chain A|Peroxisomal bifunctional enzyme|Rattus norvegicus (10116)
>6UWY_1|Chains A, B, C, D|Dual specificity tyrosine-phosphorylation-regulated kinase 1A|Homo sapiens (9606)
>1UOS_1|Chains A, C|CONVULXIN ALPHA|CROTALUS DURISSUS TERRIFICUS (8732)
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 \(w=f(c)\)
1ZCJ , Knot 187 463 0.82 40 240 436 
ASGQAKALQYAFFAEKSANKWSTPSGASWKTASAQPVSSVGVLGLGTMGRGIAISFARVGISVVAVESDPKQLDAAKKIITFTLEKEASRAHQNGQASAKPKLRFSSSTKELSTVDLVVEAVFEDMNLKKKVFAELSALCKPGAFLCTNTSALNVDDIASSTDRPQLVIGTHFFSPAHVMRLLEVIPSRYSSPTTIATVMSLSKKIGKIGVVVGNCYGFVGNRMLAPYYNQGFFLLEEGSKPEDVDGVLEEFGFKMGPFRVSDLAGLDVGWKIRKGQGLTGPSLPPGTPVRKRGNSRYSPLGDMLCEAGRFGQKTGKGWYQYDKPLGRIHKPDPWLSTFLSQYREVHHIEQRTISKEEILERCLYSLINEAFRILEEGMAARPEHIDVIYLHGYGWPRHKGGPMFYAASVGLPTVLEKLQKYYRQNPDIPQLEPSDYLRRLVAQGSPPLKEWQSLAGPHGSKL
6UWY , Knot 159 361 0.86 40 226 348 
SMSSHKKERKVYNDGYDDDNYDYIVKNGEKWMDRYEIDSLIGKGSFGQVVKAYDRVEQEWVAIKIIKNKKAFLNQAQIEVRLLELMNKHDTEMKYYIVHLKRHFMFRNHLCLVFEMLSYNLYDLLRNTNFRGVSLNLTRKFAQQMCTALLFLATPELSIIHCDLKPENILLCNPKRSAIKIVDFGSSCQLGQRIYQYIQSRFYRSPEVLLGMPYDLAIDMWSLGCILVEMHTGEPLFSGANEVDQMNKIVEVLGIPPAHILDQAPKARKFFEKLPDGTWNLKKTKDGKREYKPPGTRKLHNILGVETGGPGGRRAGESGHTVADYLKFKDLILRMLDYDPKTRIQPYYALQHSFFKKTADE
1UOS , Knot 71 135 0.86 40 109 131 
GLHCPSDWYYYDQHCYRIFNEEMNWEDAEWFCTKQAKGAHLVSIKSAKEADFVAWMVTQNIEESFSHVSIGLRVQNKEKQCSTKWSDGSSVSYDNLLDLYITKCSLLKKETGFRKWFVASCIGKIPFVCKFPPQC

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 \(c\).

\(|P_{f(1ZCJ_1)}(2) \setminus P_{f(6UWY_1)}(2)|=84\), \(|P_{f(6UWY_1)}(2) \setminus P_{f(1ZCJ_1)}(2)|=70\). 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 1:1010101100111100010010010110100101011001111111011011110110111011110001001011001101010001001000101010101010000001001011101110010100011101011001111100000110100110000010111100110110110110111000001001101101000110111111000111100111100001111100100100101110011101111010011110111010010110110111101100010000011101100110110001011000001110100101110011000001001000010000110001001100110110011110100101101010111000111110110111101100100000001011010100010011101011100100111101001
Pair \(Z_2\) Length of longest common subsequence
1ZCJ_1,6UWY_1 154 4
1ZCJ_1,1UOS_1 205 4
6UWY_1,1UOS_1 201 3

Newick tree

 
[
	1UOS_1:10.44,
	[
		1ZCJ_1:77,6UWY_1:77
	]:31.44
]

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 \((0.85)(0.8)(\frac{824 }{\log_{20} 824}-\frac{361}{\log_{20}361})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 1ZCJ_1 6UWY_1 156 139
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]