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Parikh vectors
5IGW_1 7JIJ_1 8IMA_1 Letter Amino acid
11 16 27 F Phenylalanine
0 0 13 C Cysteine
10 9 17 Q Glutamine
9 7 16 M Methionine
18 29 37 G Glycine
25 30 47 L Leucine
9 8 3 W Tryptophan
10 15 15 Y Tyrosine
20 20 37 V Valine
28 50 25 A Alanine
10 5 17 R Arginine
20 23 27 D Aspartic acid
10 21 21 P Proline
10 19 23 T Threonine
27 28 28 E Glutamic acid
10 3 12 H Histidine
25 22 18 I Isoleucine
17 22 21 N Asparagine
22 34 35 K Lycine
11 12 37 S Serine 

5IGW_1|Chain A|Macrolide 2'-phosphotransferase II|Escherichia coli (562)
>7JIJ_1|Chain A[auth M]|Maltose/maltodextrin ABC transporter substrate-binding protein MalE|Escherichia coli (562)
>8IMA_1|Chains A, B, C, D|Glutaminase kidney isoform, mitochondrial|Homo sapiens (9606)
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 \(w=f(c)\)
5IGW , Knot 132 302 0.83 38 196 290 
MSKDIKQVIEIAKKHNLFLKEETIQFNESGLDFQAVFAQDNNGIDWVLRLPRREDVMPRTKVEKQALDLVNKYAISFQAPNWIIYTEELIAYKKLDGVPAGTIDHNIGNYIWEIDINNVPELFHKSLGRVLAELHSIPSNKAAALDLVVHTPEEARMSMKQRMDAVRAKFGVGENLWNRWQAWLNDDDMWPKKTGLIHGDVHAGHTMIDKDANVTGLIDWTEAKVTDVSHDFIFNYRAFGEEGLEALILAYKEIGGYYWPKMKEHIIELNAAYPVSIAEFALVSGIEEYEQMAKEALEVQGS
7JIJ , Knot 153 373 0.81 38 190 343 
MAKIEEGKLVIWINGDKGYNGLAEVGKKFEKDTGIKVTVEHPDKLEEKFPQVAATGDGPDIIFWAHDRFGGYAQSGLLAEITPAAAFQDKLYPFTWDAVRYNGKLIAYPIAVEALSLIYNKDLLPNPPKTWEEIPALDKELKAKGKSALMFNLQEPYFTWPLIAADGGYAFKYENGKYDIKDVGVDNAGAKAGLTFLVDLIKNKHMNADTDYSIAEAAFNKGETAMTINGPWAWSNIDTSAVNYGVTVLPTFKGQPSKPFVGVLSAGINAASPNKELAKEFLENYLLTDEGLEAVNKDKPLGAVALKSYEEELAKDPRIAATMENAQKGEIMPNIPQMSAFWYAVRTAVINAASGRQTVDEALKDAQTNAAEF
8IMA , Knot 198 476 0.85 40 252 450 
LVASGENKIKQGLLPSLEDLLFYTIAEGQEKIPVHKFITALKSTGLRTSDPRLKECMDMLRLTLQTTSDGVMLDKDLFKKCVQSNIVLLTQAFRRKFVIPDFMSFTSHIDELYESAKKQSGGKVADYIPQLAKFSPDLWGVSVCTVDGQRHSTGDTKVPFCLQSCVKPLKYAIAVNDLGTEYVHRYVGKEPSGLRFNKLFLNEDDKPHNPMVNAGAIVVTSLIKQGVNNAEKFDYVMQFLNKMAGNEYVGFSNATFQSERESGDRNFAIGYYLKEKKCFPEGTDMVGILDFYFQLCSIEVTCESASVMAATLANGGFCPITGERVLSPEAVRNTLSLMHSCGMYDFSGQFAFHVGLPAKSGVAGGILLVVPNVMGMMCWSPPLDKMGNSVKGIHFCHDLVSLCNFHNYDNLRHFAKKLDPRREGGDQRHSFGPLDYESLQQELALKETVWKKVSPESNEDISTTVVYRMESLGEKS

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 \(c\).

\(|P_{f(5IGW_1)}(2) \setminus P_{f(7JIJ_1)}(2)|=85\), \(|P_{f(7JIJ_1)}(2) \setminus P_{f(5IGW_1)}(2)|=79\). 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 1:10001001101100001110000101000110101111000011011101100001110001000110110001101011011100001110001011111010001100110101001101100011011101001100011110111001001010100010110101111001100101110000111000111010101100110001010111010010100100011100011100110111110001110011010001101011011011011110110000011001101010
Pair \(Z_2\) Length of longest common subsequence
5IGW_1,7JIJ_1 164 5
5IGW_1,8IMA_1 174 4
7JIJ_1,8IMA_1 178 4

Newick tree

 
[
	8IMA_1:89.91,
	[
		5IGW_1:82,7JIJ_1:82
	]:7.91
]

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 \((0.85)(0.8)(\frac{675 }{\log_{20} 675}-\frac{302}{\log_{20}302})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 5IGW_1 7JIJ_1 124 115
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]