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Parikh vectors
4KJP_1 5LXD_1 3GWI_1 Letter Amino acid
6 4 2 W Tryptophan
10 19 12 D Aspartic acid
3 9 2 C Cysteine
12 32 7 K Lycine
25 15 9 T Threonine
64 43 19 L Leucine
18 15 3 M Methionine
15 23 14 E Glutamic acid
57 32 8 G Glycine
6 16 12 H Histidine
37 19 10 I Isoleucine
11 21 10 S Serine
30 22 14 V Valine
27 16 3 F Phenylalanine
19 22 3 P Proline
9 17 4 Y Tyrosine
52 20 11 A Alanine
21 28 13 R Arginine
11 15 7 N Asparagine
13 20 7 Q Glutamine 

4KJP_1|Chains A, B|H(+)/Cl(-) exchange transporter ClcA|Escherichia coli (83333)
>5LXD_1|Chains A, B|Dual specificity tyrosine-phosphorylation-regulated kinase 2|Homo sapiens (9606)
>3GWI_1|Chain A|Magnesium-transporting ATPase, P-type 1|Escherichia coli (83333)
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 \(w=f(c)\)
4KJP , Knot 173 446 0.78 40 206 400 
MRRRQLIRQLLERDKTPLAILFMAAVVGTLVGLAAVAFDKGVAWLQNQRMGALVHTADNYPLLLTVAFLCSAVLAMFGYFLVRKYAPEAGGSGIPEIEGALEDQRPVRWWRVLPVKFFGGLGTLGGGMVLGREGPTVQIGGNIGRMVLDIFRLKGDEARHTLLATGAAAGLAAAFNAPLAGILFIIEEMRPQFRYTLISIKAVFIGVIMSTIMYRIFNHEVALIDVGKLSDAPLNTLWLYLILGIIFGIFGPIFNKWVLGMQDLLHRVHGGNITKWVLMGGAIGGLCGLLGFVAPATSGGGFNLIPIATAGNFSMGMLVFIFVARVITTLLCFSSGAPGGIFAPMLALGTVLGTAFGMVAVELFPQYHLEAGTFAIAGMGALLAASIRAPLTGIILVLEMTDNYQLILPMIITGLGATLLAQFTGGKPLYSAILARTLAKQEAEQK
5LXD , Knot 178 408 0.87 40 239 391 
SMGKVKATPMTPEQAMKQYMQKLTAFEHHEIFSYPEIYFLGLNAKKRQGMTGGPNNGGYDDDQGSYVQVPHDHVAYRYEVLKVIGKGSFGQVVKAYDHKVHQHVALKMVRNEKRFHRQAAEEIRILEHLRKQDKDNTMNVIHMLENFTFRNHICMTFELLSMNLYELIKKNKFQGFSLPLVRKFAHSILQCLDALHKNRIIHCDLKPENILLKQQGRSGIKVIDFGSSCYEHQRVYTYIQSRFYRAPEVILGARYGMPIDMWSLGCILAELLTGYPLLPGEDEGDQLACMIELLGMPSQKLLDASKRAKNFVSSKGYPRYCTVTTLSDGSVVLNGGRSRRGKLRGPPESREWGNALKGCDDPLFLDFLKQCLEWDPAVRMTPGQALRHPWLRRRLPKPPTGEKTSVKR
3GWI , Knot 83 170 0.83 40 126 162 
MIVLENHTDISGKTSERVLHSAWLNSHYQTGLKNLLDTAVLEGTDEESARSLASRWQKIDEIPFDFERRRMSVVVAENTEHHQLVCKGALQEILNVCSQVRHNGEIVPLDDIMLRKIKRVTDTLNRQGLRVVAVATKYLPAREGDYQRADESDLILEGYIAFLDHHHHHH

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 \(c\).

\(|P_{f(4KJP_1)}(2) \setminus P_{f(5LXD_1)}(2)|=66\), \(|P_{f(5LXD_1)}(2) \setminus P_{f(4KJP_1)}(2)|=99\). 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 1:10000110011000001111111111110111111111001111100001111100100011110111100111111101110001101110111010111000011011011110111111011111111001101011101101110110101001000111011111111110111111111100101010001101011111111001100110001111011010011100111011111111111111001111100110010110100111111111110111111111001111011111011010111111111101100110100111111111111110111011111110111000101101111111111110101110111111010000011111110111101110101101100111100110001000
Pair \(Z_2\) Length of longest common subsequence
4KJP_1,5LXD_1 165 4
4KJP_1,3GWI_1 164 3
5LXD_1,3GWI_1 185 3

Newick tree

 
[
	5LXD_1:89.44,
	[
		4KJP_1:82,3GWI_1:82
	]:7.44
]

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 \((0.85)(0.8)(\frac{854 }{\log_{20} 854}-\frac{408}{\log_{20}408})=119.\)
Status Protein1 Protein2 d d1/2
Query variables 4KJP_1 5LXD_1 148 146.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]