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Parikh vectors
4GHH_1 3SYC_1 3SUJ_1 Letter Amino acid
22 25 8 T Threonine
26 17 6 R Arginine
15 14 6 N Asparagine
33 13 10 D Aspartic acid
3 6 5 C Cysteine
16 7 1 H Histidine
18 29 4 I Isoleucine
16 19 4 F Phenylalanine
18 14 8 Y Tyrosine
23 15 9 A Alanine
27 25 6 E Glutamic acid
29 19 10 G Glycine
5 10 2 M Methionine
17 10 6 P Proline
26 26 7 V Valine
10 10 2 Q Glutamine
28 33 8 L Leucine
14 16 7 K Lycine
14 25 18 S Serine
5 7 0 W Tryptophan 

4GHH_1|Chains A, B, C, D|Homoprotocatechuate 2,3-dioxygenase|Brevibacterium fuscum (47914)
>3SYC_1|Chain A|G protein-activated inward rectifier potassium channel 2|Mus musculus (10090)
>3SUJ_1|Chain A|Cerato-platanin 1|Moniliophthora perniciosa (153609)
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 \(w=f(c)\)
4GHH , Knot 156 365 0.84 40 208 350 
MSNEIPKPVAPAPDILRCAYAELVVTDLAKSRNFYVDVLGLHVSYEDENQIYLRSFEEFIHHNLVLTKGPVAALKAMAFRVRTPEDVDKAEAYYQELGCRTERRKDGFVKGIGDALRVEDPLGFPYEFFFETTHVERLHMRYDLYSAGELVRLDHFNQVTPDVPRGRKYLEDLGFRVTEDIQDDEGTTYAAWMHRKGTVHDTALTGGNGPRLHHVAFSTHEKHNIIQICDKMGALRISDRIERGPGRHGVSNAFYLYILDPDNHRIEIYTQDYYTGDPDNPTITWNVHDNQRRDWWGNPVVPSWYTEASKVLDLDGNVQEIIERTDDSELEVTIGADGFSFTRAGDEDGSYHGQASKGFKLGNQL
3SYC , Knot 148 340 0.84 40 206 328 
MAKRKIQRYVRKDGKCNVHHGNVRETYRYLTDIFTTLVDLKWRFNLLIFVMVYTVTWLFFGMIWWLIAYIRGDMDHIEDPSWTPCVTNLNGFVSAFLFSIETETTIGYGYRVITDKCPEGIILLLIQSVLGSIVNAFMVGCMFVKISQPKKRAETLVFSTHAVISMRDGKLCLMFRVGNLRNSHIVEASIRAKLIKSKQTSEGEFIPLNQTDINVGYYTGDDRLFLVSPLIISHEINQQSPFWEISKAQLPKEELEIVVILEGMVEATGMTCQARSSYITSEILWGYRFTPVLTLEDGFYEVDYNSFHETYETSTPSLSAKELAELANRAESNSLEVLFQ
3SUJ , Knot 66 127 0.84 38 109 125 
SAVKLSYDEAYDNPSSSLLSVTCSDGENGLYPKYRTFGDLPGFPCIGGSSDIAGYNSPNCGSCYQLTYSSAHTTPKSIYMVAIDRSAEGFTASKQAMDDLTNKRAEELGTVNVDVRKVDFSRCERKS

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 \(c\).

\(|P_{f(4GHH_1)}(2) \setminus P_{f(3SYC_1)}(2)|=89\), \(|P_{f(3SYC_1)}(2) \setminus P_{f(4GHH_1)}(2)|=87\). 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 1:10001101111110110010101110011000010101111010000000101001001100011100111111011110100100100101000011000000001110111011010011111001110000100101000100110110100100101011010001001110100010000100011110001010001101101101001110000000110100011110100010011100110011010110100001010000000101001010101000000011101111010001001101010100110000000101011101101001100010001010011011001
Pair \(Z_2\) Length of longest common subsequence
4GHH_1,3SYC_1 176 5
4GHH_1,3SUJ_1 179 3
3SYC_1,3SUJ_1 183 4

Newick tree

 
[
	3SUJ_1:91.32,
	[
		4GHH_1:88,3SYC_1:88
	]:3.32
]

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 \((0.85)(0.8)(\frac{705 }{\log_{20} 705}-\frac{340}{\log_{20}340})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 4GHH_1 3SYC_1 130 124
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]