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Parikh vectors
5SNR_1 8XCT_1 7TWC_1 Letter Amino acid
51 40 46 A Alanine
22 35 26 E Glutamic acid
34 12 20 S Serine
24 39 24 V Valine
52 37 23 G Glycine
10 5 11 H Histidine
21 29 26 I Isoleucine
18 9 18 F Phenylalanine
19 23 16 T Threonine
7 10 29 Q Glutamine
36 24 28 L Leucine
8 34 16 K Lycine
2 12 9 W Tryptophan
7 16 11 Y Tyrosine
29 20 27 R Arginine
15 15 22 N Asparagine
24 24 29 D Aspartic acid
4 1 6 C Cysteine
14 14 7 M Methionine
17 20 24 P Proline 

5SNR_1|Chains A, B|3-oxoacyl-[acyl-carrier-protein] synthase 2|Pseudomonas aeruginosa (287)
>8XCT_1|Chain A|Glutamate dehydrogenase|Thermococcus profundus (49899)
>7TWC_1|Chain A|DUF1479 domain-containing protein|Yersinia pestis CO92 (214092)
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 \(w=f(c)\)
5SNR , Knot 171 414 0.83 40 212 384 
MSRRRVVITGMGMLSPLGLDVPSSWEGILAGRSGIAPIEHMDLSAYSTRFGGSVKGFNVEEYLSAKEARKLDLFIQYGLAASFQAVRDSGLEVTDANRERIGVSMGSGIGGLTNIENNCRSLFEQGPRRISPFFVPGSIINMVSGFLSIHLGLQGPNYALTTAQTTGTHSIGMAARNIAYGEADVMVAGGSEMAACGLGLGGFGAARALSTRNDEPTRASRPWDRDRDGFVLSDGSGALVLEELEHARARGARIYAELVGFGMSGDAFHMTAPPEDGAGAARCMKNALRDAGLDPRQVDYINAHGTSTPAGDIAEIAAVKSVFGEHAHALSMSSTKSMTGHLLGAAGAVEAIFSVLALRDQVAPPTINLDNPDEGCDLDLVAHEAKPRKIDVALSNSFGFGGTNGTLVFRRFAD
8XCT , Knot 174 419 0.83 40 213 401 
MVEIDPFEMAVKQLERAAQYMDISEEALEWLKKPMRIVEVSVPIEMDDGSVKVFTGFRVQHNWARGPTKGGIRWHPAETLSTVKALATWMTWKVAVVDLPYGGGKGGIIVNPKELSEREQERLARAYIRAVYDVIGPWTDIPAPDVYTNPKIMGWMMDEYETIMRRKGPAFGVITGKPLSIGGSLGRGTATAQGAIFTIREAAKALGIDLKGKKIAVQGYGNAGYYTAKLAKEQLGMTVVAVSDSRGGIYNPDGLDPDEVLKWKREHGSVKDFPGATNITNEELLELEVDVLAPAAIEEVITEKNADNIKAKIVAEVANGPVTPEADDILREKGILQIPDFLCNAGGVTVSYFEWVQNINGYYWTEEEVREKLDKKMTKAFWEVYNTHKDKNIHMRDAAYVVAVSRVYQAMKDRGWVKK
7TWC , Knot 176 418 0.84 40 235 393 
SNAMASLHIDDIPAAIKAVKQQLRQALPDYQQVFQAVEENIRQQVMEIRRNLAEGKNPVPQLHADDIINGKVTEEQKAQIKQRGCCAILGVFPQEKATAWNREIGDYLDRNNFVERLKNAAEDNYFGTLAASKPQIYGIYWSTPQVEARQDKRMQAVQIFLNNLWQTESNGKQHFDANRVVTYADRTRRRPPKSSSLGLSPHVDGGSIERWLDENFRHVYRHVFSGQWQKYDPFAAEGRPEVREFPSPAVCSMFRTFQGWTALTPQRTHAGTLNVIPIANAMAYILLRALQDDVADDDLCGAAPGRALSASEQWHPLLMEAISPIPDLEAGDTVFWHCDVIHSVENEHNGEFDSNVMYIAAAPWCEKNAAYLPRQLASFIDGRSPPDFAADDFEVDFIGRATIKNLTEIGKQQLGITD

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 \(c\).

\(|P_{f(5SNR_1)}(2) \setminus P_{f(8XCT_1)}(2)|=74\), \(|P_{f(8XCT_1)}(2) \setminus P_{f(5SNR_1)}(2)|=75\). 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 1:100001110111110111101100101111100111110010101000011101011010001010010010111001111010110001101001000011101101111100100000011001100101111110110110111010111011001100100010001111100110101011111100111011111111110110000001001001100000111100101111100100101011010101111110101101011100111110010011001110100100101010001110110111100111001011010000010101111111101110111100011110101001001001011100101001011100011111001011100110
Pair \(Z_2\) Length of longest common subsequence
5SNR_1,8XCT_1 149 4
5SNR_1,7TWC_1 147 4
8XCT_1,7TWC_1 152 4

Newick tree

 
[
	8XCT_1:75.82,
	[
		5SNR_1:73.5,7TWC_1:73.5
	]:2.32
]

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 \((0.85)(0.8)(\frac{833 }{\log_{20} 833}-\frac{414}{\log_{20}414})=112.\)
Status Protein1 Protein2 d d1/2
Query variables 5SNR_1 8XCT_1 144 141.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]