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Parikh vectors
2PTZ_1 4ZMN_1 7SVW_1 Letter Amino acid
10 5 0 M Methionine
24 16 0 S Serine
4 2 0 W Tryptophan
42 15 19 A Alanine
20 21 0 D Aspartic acid
33 12 0 K Lycine
10 0 16 C Cysteine
36 16 0 E Glutamic acid
19 17 0 P Proline
35 13 0 L Leucine
12 8 0 F Phenylalanine
27 20 0 V Valine
18 6 0 R Arginine
42 16 16 G Glycine
27 14 0 I Isoleucine
24 19 19 T Threonine
12 7 0 Y Tyrosine
16 10 0 N Asparagine
13 8 0 Q Glutamine
8 6 0 H Histidine 

2PTZ_1|Chain A|Enolase|Trypanosoma brucei (5691)
>4ZMN_1|Chain A|Cadherin-3|Homo sapiens (9606)
>7SVW_1|Chains A[auth 1], D[auth 4]|STC_LE_For|synthetic construct (32630)
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 \(w=f(c)\)
2PTZ , Knot 183 432 0.85 40 233 409 
GSHMTIQKVHGREVLDSRGNPTVEVEVTTEKGVFRSAVPSGASTGVYEACELRDGDKKRYVGKGCLQAVKNVNEVIGPALIGRDELKQEELDTLMLRLDGTPNKGKLGANAILGCSMAISKAAAAAKGVPLYRYLASLAGTKELRLPVPCFNVINGGKHAGNALPFQEFMIAPVKATSFSEALRMGSEVYHSLRGIIKKKYGQDAVNVGDEGGFAPPIKDINEPLPILMEAIEEAGHRGKFAICMDCAASETYDEKKQQYNLTFKSPEPTWVTAEQLRETYCKWAHDYPIVSIEDPYDQDDFAGFAGITEALKGKTQIVGDDLTVTNTERIKMAIEKKACNSLLLKINQIGTISEAIASSKLCMENGWSVMVSHRSGETEDTYIADLVVALGSGQIKTGAPCRGERTAKLNQLLRIEEELGAHAKFGFPGWS
4ZMN , Knot 106 231 0.83 38 155 219 
DWVVAPISVPENGKGPFPQRLNQLKSNKDRDTKIFYSITGPGADSPPEGVFAVEKETGWLLLNKPLDREEIAKYELFGHAVSENGASVEDPMNISIIVTDQNDHKPKFTQDTFRGSVLEGVLPGTSVMQVTATDEDDAIYTYNGVVAYSIHSQEPKDPHDLMFTIHRSTGTISVISSGLDREKVPEYTLTIQATDMDGDGSTTTAVAVVEILDANDNAPMFDPQKYEAHVP
7SVW , Knot 23 70 0.46 8 16 41 
ATGACATTAATCTGTCACCGACGACAGATAATTTGTCACTGTACAGGCCCTAGGTCTACGGTTAGAGGCT

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 \(c\).

\(|P_{f(2PTZ_1)}(2) \setminus P_{f(4ZMN_1)}(2)|=116\), \(|P_{f(4ZMN_1)}(2) \setminus P_{f(2PTZ_1)}(2)|=38\). 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 1:100101001010011000101010101000011100111011001100100100100000110101011001001111111100010000100111010101001011101111001110011111011110001101110001011110101101100110111100111111010010011011001000101110000100110110011111110010011111101100110010111010011000000000000101001010110100100000011000111010010000011111110011010001110010100000101110001000111010011010011100010100110111000010000001101111110101001110010001010011010001110101111110
Pair \(Z_2\) Length of longest common subsequence
2PTZ_1,4ZMN_1 154 4
2PTZ_1,7SVW_1 225 3
4ZMN_1,7SVW_1 159 3

Newick tree

 
[
	7SVW_1:10.31,
	[
		2PTZ_1:77,4ZMN_1:77
	]:26.31
]

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 \((0.85)(0.8)(\frac{663 }{\log_{20} 663}-\frac{231}{\log_{20}231})=121.\)
Status Protein1 Protein2 d d1/2
Query variables 2PTZ_1 4ZMN_1 151 114.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]