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Parikh vectors
4IJW_1 3UCQ_1 2JFG_1 Letter Amino acid
10 17 21 N Asparagine
4 2 7 C Cysteine
16 16 14 I Isoleucine
9 26 13 F Phenylalanine
10 22 9 Y Tyrosine
27 72 52 A Alanine
10 23 16 H Histidine
17 9 15 K Lycine
5 37 17 P Proline
26 86 56 L Leucine
16 11 13 M Methionine
26 30 22 S Serine
25 44 34 V Valine
12 64 22 R Arginine
8 50 25 D Aspartic acid
9 17 15 Q Glutamine
18 41 27 E Glutamic acid
22 51 40 G Glycine
15 22 24 T Threonine
1 15 3 W Tryptophan 

4IJW_1|Chains A, B, C[auth D], D[auth E]|Corticosteroid 11-beta-dehydrogenase isozyme 1|Homo sapiens (9606)
>3UCQ_1|Chain A|Amylosucrase|Deinococcus geothermalis (319795)
>2JFG_1|Chain A|UDP-N-ACETYLMURAMOYLALANINE--D-GLUTAMATE LIGASE|ESCHERICHIA COLI (562)
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 \(w=f(c)\)
4IJW , Knot 129 286 0.85 40 183 279 
GSHMASMTGGQQMGRGSNEEFRPEMLQGKKVIVTGASKGIGREMAYHLAKMGAHVVVTARSKETLQKVVSHCLELGAASAHYIAGTMEDMTFAEQFVAQAGKLMGGLDMLILNHITNTSLNLFHDDIHHVRKSMEVNFLSYVVLTVAALPMLKQSNGSIVVVSSLAGKVAYPMVAAYSASKFALDGFFSSIRKEYSVSRVNVSITLCVLGLIDTETAMKAVSGIVHMQAAPKEECALEIIKGGALRQEEVYYDSSRWTTLLIRNPCRKILEELYSTSYNMDRFINK
3UCQ , Knot 251 655 0.82 40 256 578 
GPLGSMLKDVLTSELAAQVRDAFDDDRDAETFLLRLERYGEDLWESLRAVYGDQVRALPGRLLEVMLHAYHARPAELRRLDEARLLRPDWLQRPEMVGYVAYTDRFAGTLKGVEERLDYLEGLGVKYLHLMPLLRPREGENDGGYAVQDYRAVRPDLGTMDDLSALARALRGRGISLVLDLVLNHVAREHAWAQKARAGDPKYRAYFHLFPDRRGPDAFEATLPEIFPDFAPGNFSWDEEIGEGEGGWVWTTFNSYQWDLNWANPDVFLEFVDIILYLANRGVEVFRLDAIAFIWKRLGTDCQNQPEVHHLTRALRAAARIVAPAVAFKAEAIVAPADLIHYLGTRAHHGKVSDMAYHNSLMVQLWSSLASRNTRLFEEALRAFPPKPTSTTWGLYVRCHDDIGWAISDEDAARAGLNGAAHRHFLSDFYSGQFPGSFARGLVFQYNPVNGDRRISGSAASLAGLEAALETGDPGRIEDAVRRLLLLHTVILGFGGVPLLYMGDELALLNDYAFEDVPEHAPDNRWVHRPQMDWALAERVRQEPSSPAGRVNTGLRHLLRVRRDTPQLHASIESQVLPSPDSRALLLRRDHPLGGMVQVYNFSEETVMLPSHVLRDVLGDHVQDRLSGSAFRLDRPTVRLEGYRALWLTAGEAPA
2JFG , Knot 181 445 0.82 40 223 415 
MADYQGKNVVIIGLGLTGLSCVDFFLARGVTPRVMDTRMTPPGLDKLPEAVERHTGSLNDEWLMAADLIVASPGIALAHPSLSAAADAGIEIVGDIELFCREAQAPIVAITGSNGKSTVTTLVGEMAKAAGVNVGVGGNIGLPALMLLDDECELYVLELSSFQLETTSSLQAVAATILNVTEDHMDRYPFGLQQYRAAKLRIYENAKVCVVNADDALTMPIRGADERCVSFGVNMGDYHLNHQQGETWLRVKGEKVLNVKEMKLSGQHNYTNALAALALADAAGLPRASSLKALTTFTGLPHRFEVVLEHNGVRWINDSKATNVGSTEAALNGLHVDGTLHLLLGGDGKSADFSPLARYLNGDNVRLYCFGRDGAQLAALRPEVAEQTETMEQAMRLLAPRVQPGDMVLLSPACASLDQFKNFEQRGNEFARLAKELGSHHHHHH

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 \(c\).

\(|P_{f(4IJW_1)}(2) \setminus P_{f(3UCQ_1)}(2)|=55\), \(|P_{f(3UCQ_1)}(2) \setminus P_{f(4IJW_1)}(2)|=128\). 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 1:1001101011001101000010101101001110110011100110011011101110100000100110001011110100111010010110011101101111101111001000010110001001000101011001110111111100001011110011101101111100100111011100100000100101010101111100001101101110101110000110110111100001000000100111001000110010000001001100
Pair \(Z_2\) Length of longest common subsequence
4IJW_1,3UCQ_1 183 4
4IJW_1,2JFG_1 144 3
3UCQ_1,2JFG_1 149 4

Newick tree

 
[
	3UCQ_1:86.91,
	[
		4IJW_1:72,2JFG_1:72
	]:14.91
]

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 \((0.85)(0.8)(\frac{941 }{\log_{20} 941}-\frac{286}{\log_{20}286})=176.\)
Status Protein1 Protein2 d d1/2
Query variables 4IJW_1 3UCQ_1 215 156
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]