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Parikh vectors
4DJN_1 7WWB_1 6OOF_1 Letter Amino acid
22 52 11 I Isoleucine
4 9 2 H Histidine
21 34 10 K Lycine
8 20 5 M Methionine
26 38 4 F Phenylalanine
11 25 13 P Proline
22 48 13 S Serine
11 18 18 Q Glutamine
31 23 11 E Glutamic acid
8 35 29 T Threonine
4 27 1 C Cysteine
16 25 15 R Arginine
13 24 11 N Asparagine
17 18 13 D Aspartic acid
13 36 22 G Glycine
27 83 27 L Leucine
6 13 3 W Tryptophan
9 23 5 Y Tyrosine
22 52 35 A Alanine
20 54 15 V Valine 

4DJN_1|Chains A, B|Ribonucleoside-diphosphate reductase subunit M2 B|Homo sapiens (9606)
>7WWB_1|Chains A, B|Choline transporter-like protein 1|Homo sapiens (9606)
>6OOF_1|Chains A, B|Beta-lactamase|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 \(w=f(c)\)
4DJN , Knot 137 311 0.84 40 195 299 
GDQDERSSSDTNESEIKSNEEPLLRKSSRRFVIFPIQYPDIWKMYKQAQASFWTAEEVDLSKDLPHWNKLKADEKYFISHILAFFAASDGIVNENLVERFSQEVQVPEARCFYGFQILIENVHSEMYSLLIDTYIRDPKKREFLFNAIETMPYVKKKADWALRWIADRKSTFGERVVAFAAVEGVFFSGSFAAIFWLKKRGLMPGLTFSNELISRDEGLHCDFACLMFQYLVNKPSEERVREIIVDAVKIEQEFLTEALPVGLIGMNCILMKQYIEFVADRLLVELGFSKVFQAENPFDFMENISLEGKTN
7WWB , Knot 254 657 0.83 40 284 607 
MGCCSSASSAAQSSKREWKPLEDRSCTDIPWLLLFILFCIGMGFICGFSIATGAAARLVSGYDSYGNICGQKNTKLEAIPNSGMDHTQRKYVFFLDPCNLDLINRKIKSVALCVAACPRQELKTLSDVQKFAEINGSALCSYNLKPSEYTTSPKSSVLCPKLPVPASAPIPFFHRCAPVNISCYAKFAEALITFVSDNSVLHRLISGVMTSKEIILGLCLLSLVLSMILMVIIRYISRVLVWILTILVILGSLGGTGVLWWLYAKQRRSPKETVTPEQLQIAEDNLRALLIYAISATVFTVILFLIMLVMRKRVALTIALFHVAGKVFIHLPLLVFQPFWTFFALVLFWVYWIMTLLFLGTTGSPVQNEQGFVEFKISGPLQYMWWYHVVGLIWISEFILACQQMTVAGAVVTYYFTRDKRNLPFTPILASVNRLIRYHLGTVAKGSFIITLVKIPRMILMYIHSQLKGKENACARCVLKSCICCLWCLEKCLNYLNQNAYTATAINSTNFCTSAKDAFVILVENALRVATINTVGDFMLFLGKVLIVCSTGLAGIMLLNYQQDYTVWVLPLIIVCLFAFLVAHCFLSIYEMVVDVLFLCFAIDTKYNDGSPGREFYMDKVLMEFVENSRKAMKEAGKGGVADSRELKPMASGASSA
6OOF , Knot 117 263 0.82 40 163 250 
QTSAVQQKLAALEKSSGGRLGVALIDTADNTQVLYRGDERFPMCSTSKVMAAAAVLKQSETQKQLLNQPVEIKPADLVNYNPIAEKHVNGTMTLAELSAAALQYSDNTAMNKLIAQLGGPGGVTAFARAIGDETFRLDRTEPTLNTAIPGDPRDTTTPRAMAQTLRQLTLGHALGETQRAQLVTWLKGNTTGAASIRAGLPTSWTVGDKTGSGDYGTTNDIAVIWPQGRAPLVLVTYFTQPQQNAESRRDVLASAARIIAEGL

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 \(c\).

\(|P_{f(4DJN_1)}(2) \setminus P_{f(7WWB_1)}(2)|=48\), \(|P_{f(7WWB_1)}(2) \setminus P_{f(4DJN_1)}(2)|=137\). 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 1:10000000000000010000011100000011111100101101000101011010010100011010010100001100111111100111000110010001011010010110111001000100111000100100001110110011010001011101110000011001111111011110101111111000111111010001100001100011011100110010000100111011010001100111111111001110001011100111011100110100110110010101000
Pair \(Z_2\) Length of longest common subsequence
4DJN_1,7WWB_1 185 4
4DJN_1,6OOF_1 170 4
7WWB_1,6OOF_1 181 4

Newick tree

 
[
	7WWB_1:93.57,
	[
		4DJN_1:85,6OOF_1:85
	]:8.57
]

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 \((0.85)(0.8)(\frac{968 }{\log_{20} 968}-\frac{311}{\log_{20}311})=176.\)
Status Protein1 Protein2 d d1/2
Query variables 4DJN_1 7WWB_1 224 164
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]