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Parikh vectors
8BUG_1 6DRS_1 5MLT_1 Letter Amino acid
69 24 31 V Valine
37 16 6 R Arginine
40 10 30 D Aspartic acid
71 28 34 E Glutamic acid
53 14 32 I Isoleucine
21 8 7 M Methionine
5 4 5 W Tryptophan
48 19 25 T Threonine
41 7 41 N Asparagine
15 1 0 C Cysteine
34 1 16 Q Glutamine
69 25 32 G Glycine
19 6 3 H Histidine
27 15 20 P Proline
84 17 34 L Leucine
35 7 17 F Phenylalanine
43 23 32 A Alanine
44 18 48 K Lycine
57 10 20 S Serine
28 3 22 Y Tyrosine 

8BUG_1|Chains A, D, G|DNA damage-binding protein 1|Homo sapiens (9606)
>6DRS_1|Chain A|Dihydrofolate reductase, putative|Aspergillus flavus (332952)
>5MLT_1|Chain A|ABC transporter, substrate-binding protein|Streptococcus pneumoniae serotype 4 (strain ATCC BAA-334 / TIGR4) (170187)
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 \(w=f(c)\)
8BUG , Knot 316 840 0.84 40 305 758 
GGGRMSYNYVVTAQKPTAVNGCVTGHFTSAEDLNLLIAKNTRLEIYVVTAEGLRPVKEVGMYGKIAVMELFRPKGESKDLLFILTAKYNACILEYKQSGESIDIITRAHGNVQDRIGRPSETGIIGIIDPECRMIGLRLYDGLFKVIPLDRDNKELKAFNIRLEELHVIDVKFLYGCQAPTICFVYQDPQGRHVKTYEVSLREKEFNKGPWKQENVEAEASMVIAVPEPFGGAIIIGQESITYHNGDKYLAIAPPIIKQSTIVCHNRVDPNGSRYLLGDMEGRLFMLLLEKEEQMDGTVTLKDLRVELLGETSIAECLTYLDNGVVFVGSRLGDSQLVKLNVDSNEQGSYVVAMETFTNLGPIVDMCVVDLERQGQGQLVTCSGAFKEGSLRIIRNGIGGNGNSGEIQKLHIRTVPLYESPRKICYQEVSQCFGVLSSRIEVQDTSGGTTALRPSASTQALSSSVSSSKLFSSSTAPHETSFGEEVEVHNLLIIDQHTFEVLHAHQFLQNEYALSLVSCKLGKDPNTYFIVGTAMVYPEEAEPKQGRIVVFQYSDGKLQTVAEKEVKGAVYSMVEFNGKLLASINSTVRLYEWTTEKELRTECNHYNNIMALYLKTKGDFILVGDLMRSVLLLAYKPMEGNFEEIARDFNPNWMSAVEILDDDNFLGAENAFNLFVCQKDSAATTDEERQHLQEVGLFHLGEFVNVFCHGSLVMQNLGETSTPTQGSVLFGTVNGMIGLVTSLSESWYNLLLDMQNRLNKVIKSVGKIEHSFWRSFHTERKTEPATGFIDGDLIESFLDISRPKMQEVVANLQYDDGSGMKREATADDLIKVVEELTRIH
6DRS , Knot 113 256 0.81 40 154 242 
MAPTNPLTLIVATTPIPTREKTLLGIGLNGTLPWPRIKADMSFFARVTTRPPRPGTTNAMIMGRKTYDSVPKSLRPLGKRINVIVTRDVEGVSKRVAEELKEKRAKMAAAAAAATSAGENKEEGPITDAIVSSGLEAALEDVEEKFKGGLGSVFVIGGAEIYATALGLGGDRPVRIVMTNVEKKGVDGEKAVFECDTFFPIDEELLMEKGWRKVSAEEVTEWVGEPVSGEWKDEGEVRIQMVGYERVNLEHHHHHH
5MLT , Knot 181 455 0.81 38 209 425 
GPDKPVIKMYQIGDKPDNLDELLANANKIIEEKVGAKLDIQYLGWGDYGKKMSVITSSGENYDIAFADNYIVNAQKGAYADLTELYKKEGKDLYKALDPAYIKGNTVNGKIYAVPVAANVASSQNFAFNGTLLAKYGIDISGVTSYETLEPVLKQIKEKAPDVVPFAIGKVFIPSDNFDYPVANGLPFVIDLEGDTTKVVNRYEVPRFKEHLKTLHKFYEAGYIPKDVATSDTSFDLQQDTWFVREETVGPADYGNSLLSRVANKDIQIKPITNFIKKNQTTQVANFVISNNSKNKEKSMEILNLLNTNPELLNGLVYGPEGKNWEKIEGKENRVRVLDGYKGNTHMGGWNTGNNWILYINENVTDQQIENSKKELAEAKESPALGFIFNTDNVKSEISAIANTMQQFDTAINTGTVDPDKAIPELMEKLKSEGAYEKVLNEMQKQYDEFLKNKK

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 \(c\).

\(|P_{f(8BUG_1)}(2) \setminus P_{f(6DRS_1)}(2)|=167\), \(|P_{f(6DRS_1)}(2) \setminus P_{f(8BUG_1)}(2)|=16\). 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 1:111010000110100101101010101001001011110000101011010110110011101011110110101000011111010001011000001001011001010100011010001111110100011110100111011110000001011010100101101011010011010110001010010000101000010011100001010101111110111111111000100001000111111110000110000101010001110101011111100000101010100101011100011001001001111110011000110101000001001111001001111101011010001010110001110010101100111101001010010100111000100100001000111100010100001100110101000110001000011000011000011001010011110000101101001100001101100011001000111101110100101001011110000101001100010111001101010111010001010010000010000000001111010001011111011001111100110101001100101011011011000011110011011100000110000000010011110110110110010111001100001001011110101111110010001001110100010011001101000110010000000110111010110011010010100111010000101100010100110110010010
Pair \(Z_2\) Length of longest common subsequence
8BUG_1,6DRS_1 183 3
8BUG_1,5MLT_1 154 4
6DRS_1,5MLT_1 149 4

Newick tree

 
[
	8BUG_1:87.65,
	[
		5MLT_1:74.5,6DRS_1:74.5
	]:13.15
]

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 \((0.85)(0.8)(\frac{1096 }{\log_{20} 1096}-\frac{256}{\log_{20}256})=224.\)
Status Protein1 Protein2 d d1/2
Query variables 8BUG_1 6DRS_1 284 183
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]