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Parikh vectors
3DDF_1 5KQO_1 5BSM_1 Letter Amino acid
11 7 10 C Cysteine
57 28 18 Q Glutamine
63 28 34 G Glycine
52 18 39 K Lycine
57 21 22 T Threonine
38 9 17 Y Tyrosine
53 26 31 E Glutamic acid
50 12 10 H Histidine
101 50 48 L Leucine
28 15 13 M Methionine
63 17 34 P Proline
76 28 33 S Serine
21 3 1 W Tryptophan
73 36 46 V Valine
53 52 45 A Alanine
59 14 17 R Arginine
65 19 34 D Aspartic acid
41 18 45 I Isoleucine
45 11 23 F Phenylalanine
39 15 22 N Asparagine 

3DDF_1|Chain A|alpha-mannosidase 2|Drosophila melanogaster (7227)
>5KQO_1|Chains A, B|1-deoxy-D-xylulose 5-phosphate reductoisomerase|Vibrio vulnificus (strain CMCP6) (216895)
>5BSM_1|Chains A, B|4-coumarate--CoA ligase 2|Nicotiana tabacum (4097)
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 \(w=f(c)\)
3DDF , Knot 386 1045 0.85 40 338 956 
RSSHHHHHHGEFDDPIRPPLKVARSPRPGQCQDVVQDVPNVDVQMLELYDRMSFKDIDGGVWKQGWNIKYDPLKYNAHHKLKVFVVPHSHNDPGWIQTFEEYYQHDTKHILSNALRHLHDNPEMKFIWAEISYFARFYHDLGENKKLQMKSIVKNGQLEFVTGGWVMPDEANSHWRNVLLQLTEGQTWLKQFMNVTPTASWAIDPFGHSPTMPYILQKSGFKNMLIQRTHYSVKKELAQQRQLEFLWRQIWDNKGDTALFTHMMPFYSYDIPHTCGPDPKVCCQFDFKRMGSFGLSCPWKVPPRTISDQNVAARSDLLVDQWKKKAELYRTNVLLIPLGDDFRFKQNTEWDVQRVNYERLFEHINSQAHFNVQAQFGTLQEYFDAVHQAERAGQAEFPTLSGDFFTYADRSDNYWSGYYTSRPYHKRMDRVLMHYVRAAEMLSAWHSWDGMARIEERLEQARRELSLFQHHDGITGTAKTHVVVDYEQRMQEALKACQMVMQQSVYRLLTKPSIYSPDFSFSYFTLDDSRWPGSGVEDSRTTIILGEDILPSKHVVMHNTLPHWREQLVDFYVSSPFVSVTDLANNPVEAQVSPVWSWHHDTLTKTIHPQGSTTKYRIIFKARVPPMGLATYVLTISDSKPEHTSYASNLLLRKNPTSLPLGQYPEDVKFGDPREISLRVGNGPTLAFSEQGLLKSIQLTQDSPHVPVHFKFLKYGVRSHGDRSGAYLFLPNGPASPVELGQPVVLVTKGKLESSVSVGLPSVVHQTIMRGGAPEIRNLVDIGSLDNTEIVMRLETHIDSGDIFYTDLNGLQFIKRRRLDKLPLQANYYPIPSGMFIEDANTRLTLLTGQPLGGSSLASGELEIMQDRRLASDDERGLGQGVLDNKPVLHIYRLVLEKVNNCVRPSKLHPAGYLTSAAHKASQSLLDPLDKFIFAENEWIGAQGQFGGDHPSAREDLDVSVMRRLTKSSAKTQRVGYVLHRTNLMQCGTPEEHTQKLDVCHLLPNVARCERTTLTFLQNLEHLDGMVAPEVCPMETAAYVSSHSS
5KQO , Knot 170 427 0.80 40 220 397 
MHHHHHHSSGVDLGTENLYFQSGSGMQKLTILGATGSIGASTLKVIEQNPDKFSVVALAADSNVEKMQQLCQRWQPEYAVMANKEAALRLKMALAVLAPNTQVLGGQEALCYVATLEQVDSVMAAIVGAAGLVPTMAAVKAGKRILLANKEALVMSGQLFIDEVEKSGAQLLPVDSEHNAIFQCLPQTVQGNLGRCDLASQGVSHILLTGSGGPFRYTDVAELEAVTPEQAIAHPNWSMGPKISVDSATMMNKGLEYIEAKWLFNASRDQLKVIIHPQSVIHSMVQYLDGSVLAQMGEPDMATPIALTLSYPERVKAGVKPLDFTQVGELTFLQPDFERYPCLALAIEACYLGQHATTTLNAANEVAVAAFLARQIKFTDIARVNDSVLNQVCKQSLASGLDSLESLLELDRMARTLADEVVRERAQ
5BSM , Knot 216 542 0.83 40 245 508 
MEKDTKQVDIIFRSKLPDIYIPNHLPLHSYCFENISEFSSRPCLINGANKQIYTYADVELNSRKVAAGLHKQGIQPKDTIMILLPNSPEFVFAFIGASYLGAISTMANPLFTPAEVVKQAKASSAKIIVTQACHVNKVKDYAFENDVKIICIDSAPEGCLHFSVLTQANEHDIPEVEIQPDDVVALPYSSGTTGLPKGVMLTHKGLVTSVAQQVDGENPNLYIHSEDVMLCVLPLFHIYSLNSVLLCGLRVGAAILIMQKFDIVSFLELIQRYKVTIGPFVPPIVLAIAKSPMVDDYDLSSVRTVMSGAAPLGKELEDTVRAKFPNAKLGQGYGMTEAGPVLAMCLAFAKEPFEIKSGACGTVVRNAEMKIVDPKTGNSLPRNQSGEICIRGDQIMKGYLNDPEATARTIDKEGWLYTGDIGYIDDDDELFIVDRLKELIKYKGFQVAPAELEALLLNHPNISDAAVVPMKDEQAGEVPVAFVVRSNGSTITEDEVKDFISKQVIFYKRIKRVFFVDAIPKSPSGKILRKDLRAKLAAGLPN

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 \(c\).

\(|P_{f(3DDF_1)}(2) \setminus P_{f(5KQO_1)}(2)|=140\), \(|P_{f(5KQO_1)}(2) \setminus P_{f(3DDF_1)}(2)|=22\). 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 1:0000000001010011011101100101100001100110101011010001010010111100110100011000100010111110000011110010000000000110011001000101011110100110100011000010100110010101101111110010001001110100100110011010101011101110010110110001100111000000100011000010111001100010011100111100001100011010100010100110111001101110010000111000111001000101000011111110010100000101001000011001000101010101101000101100100110101101010110010000001010000010000100111001011011011001011101000100100010110000110101000111000001001101001110001001100101001010100101000011101100000011110011100011100011010001101010011101001100110101011101000010001010100000011101011111110011010000100000100111000100111100100101101001010110110111000111001010000101110101100110001000110111101110110110111110010100010111101100011011110100110110100001110100010010110001011011000010011101000111011110010001011010111100110101011000011000001110111000111010011100100010100101110100110010001101100111100011110101110010100010101100100001000011011000011001010000001010011101100000010110010010111110101100110100000
Pair \(Z_2\) Length of longest common subsequence
3DDF_1,5KQO_1 162 6
3DDF_1,5BSM_1 137 4
5KQO_1,5BSM_1 141 5

Newick tree

 
[
	5KQO_1:78.25,
	[
		3DDF_1:68.5,5BSM_1:68.5
	]:9.75
]

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 \((0.85)(0.8)(\frac{1472 }{\log_{20} 1472}-\frac{427}{\log_{20}427})=267.\)
Status Protein1 Protein2 d d1/2
Query variables 3DDF_1 5KQO_1 345 239.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]