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Parikh vectors
8FOK_1 6VBQ_1 8XOA_1 Letter Amino acid
66 13 51 A Alanine
86 7 20 R Arginine
94 8 19 N Asparagine
60 9 19 Q Glutamine
87 5 46 I Isoleucine
111 14 16 K Lycine
35 4 25 M Methionine
46 7 30 F Phenylalanine
68 13 30 P Proline
112 31 51 S Serine
91 24 45 V Valine
29 7 11 C Cysteine
78 24 28 T Threonine
7 4 7 W Tryptophan
119 10 20 D Aspartic acid
95 5 16 E Glutamic acid
68 19 51 G Glycine
22 3 6 H Histidine
145 13 60 L Leucine
49 10 22 Y Tyrosine 

8FOK_1|Chain A[auth 1]|DNA polymerase|Saccharomyces cerevisiae (4932)
>6VBQ_1|Chains A, F[auth C], G[auth E], H|DH822 heavy chain|Homo sapiens (9606)
>8XOA_1|Chain A|Synaptic vesicular amine transporter,Synaptic vesicular amine transporter,transporterA|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 \(w=f(c)\)
8FOK , Knot 511 1468 0.84 40 343 1248 
MSSKSEKLEKLRKLQAARNGTSIDDYEGDESDGDRIYDEIDEKEYRARKRQELLHDDFVVDDDGVGYVDRGVEEDWREVDNSSSDEDTGNLASKDSKRKKNIKREKDHQITDMLRTQHSKSTLLAHAKKSQKKSIPIDNFDDILGEFESGEVEKPNILLPSKLRENLNSSPTSEFKSSIKRVNGNDESSHDAGISKKVKIDPDSSTDKYLEIESSPLKLQSRKLRYANDVQDLLDDVENSPVVATKRQNVLQDTLLANPPSAQSLADEEDDEDSDEDIILKRRTMRSVTTTRRVNIDSRSNPSTSPFVTAPGTPIGIKGLTPSKSLQSNTDVATLAVNVKKEDVVDPETDTFQMFWLDYCEVNNTLILFGKVKLKDDNCVSAMVQINGLCRELFFLPREGKTPTDIHEEIIPLLMDKYGLDNIRAKPQKMKYSFELPDIPSESDYLKVLLPYQTPKSSRDTIPSDLSSDTFYHVFGGNSNIFESFVIQNRIMGPCWLDIKGADFNSIRNASHCAVEVSVDKPQNITPTTTKTMPNLRCLSLSIQTLMNPKENKQEIVSITLSAYRNISLDSPIPENIKPDDLCTLVRPPQSTSFPLGLAALAKQKLPGRVRLFNNEKAMLSCFCAMLKVEDPDVIIGHRLQNVYLDVLAHRMHDLNIPTFSSIGRRLRRTWPEKFGRGNSNMNHFFISDICSGRLICDIANEMGQSLTPKCQSWDLSEMYQVTCEKEHKPLDIDYQNPQYQNDVNSMTMALQENITNCMISAEVSYRIQLLTLTKQLTNLAGNAWAQTLGGTRAGRNEYILLHEFSRNGFIVPDKEGNRSRAQKQRQNEENADAPVNSKKAKYQGGLVFEPEKGLHKNYVLVMDFNSLYPSIIQEFNICFTTVDRNKEDIDELPSVPPSEVDQGVLPRLLANLVDRRREVKKVMKTETDPHKRVQCDIRQQALKLTANSMYGCLGYVNSRFYAKPLAMLVTNKGREILMNTRQLAESMNLLVVYGDTDSVMIDTGCDNYADAIKIGLGFKRLVNERYRLLEIDIDNVFKKLLLHAKKKYAALTVNLDKNGNGTTVLEVKGLDMKRREFCPLSRDVSIHVLNTILSDKDPEEALQEVYDYLEDIRIKVETNNIRIDKYKINMKLSKDPKAYPGGKNMPAVQVALRMRKAGRVVKAGSVITFVITKQDEIDNAADTPALSVAERAHALNEVMIKSNNLIPDPQYYLEKQIFAPVERLLERIDSFNVVRLSEALGLDSKKYFRREGGNNNGEDINNLQPLETTITDVERFKDTVTLELSCPSCDKRFPFGGIVSSNYYRVSYNGLQCKHCEQLFTPLQLTSQIEHSIRAHISLYYAGWLQCDDSTCGIVTRQVSVFGKRCLNDGCTGVMRYKYSDKQLYNQLLYFDSLFDCEKNKKQELKPIYLPDDLDYPKEQLTESSIKALTEQNRELMETGRSVVQKYLNDCGRRYVDMTSIFDFMLN
6VBQ , Knot 101 230 0.79 40 150 221 
QVQLVQSGAEVKKPGASVKLSCKASGYTFTDYYIHWVRQAPGQGLEWMGIINPRRDTTTYAQKFQGRVTMTRDTSTNTVYMELSSLRSDDTAVYFCAKNVGFCNSDSCYVGMDVWGPGTTVTISSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKRVEPKSCDK
8XOA , Knot 225 573 0.83 40 254 531 
SRKLILFIVFLALLLDNMLLTVVVPIIPSYLYSIKHEKNATEIQTARPVHTASISDSFQSIFSYYDNSTMVTGNATRDLTLHQTATQHMVTNASAVPSDCPSEDKDLLNENVQVGLLFASKATVQLITNPFIGLLTNRIGYPIPIFAGFCIMFVSTIMFAFSSSYAFLLIARSLQGIGSSCSSVAGMGMLASVYTDDEERGNVMGIALGGLAMGVLVGPPFGSVLYEFVGKTAPFLVLAALVLLDGAIQLFVLQPSRVQPESQKGTPLTTLLKDPYILIAAGSICFANMGIAMLEPALPIWMMETMCSRKWQLGVAFLPASISYLIGTNIFGILAHKMGRWLCALLGMIIVGVSILCIPFAKNIYGLIAPNFGVGFAIGMVDSSMMPIMGYLVDLRHVSVYGSVYAIADVAFCMGYAIGPSAGGAIAKAIGFPWLMTIIGIIDILFAPLCFFLRSPPQVQLVESGGALVQPGGSLRLSCAASGFPVNRYSMRWYRQAPGKEREWVAGMSSAGDRSSYEDSVKGRFTISRDDARNTVYLQMNSLKPEDTAVYYCNVNVGFEYWGQGTQVTVSSK

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 \(c\).

\(|P_{f(8FOK_1)}(2) \setminus P_{f(6VBQ_1)}(2)|=208\), \(|P_{f(6VBQ_1)}(2) \setminus P_{f(8FOK_1)}(2)|=15\). 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 1:1000000100100101100100100001000010010001000000100000110001110001110100110001001000000000101100000000010000000100110000000011101000000011100100111010010100101111001000100010001000100101000000011100010101000000010100011010000100100100110010001111000001100011101101001100000000000111000010010000010100000100011101110111101101000100000110111010000110100001011110000100011111010100000101110101100011111001001001000111111000110010101001000101101100000101111000100000011001000010011110001100111000111101101011010010010001101010010010100000110100101010011010000001101010100010100111001010010011011000011111111100011101011000011100101110100101111001001010111001001011010011001000110011010001001110010010110011001100101000010100100100000001101000010000010010111000100011010100010110100010011101110011100110000111001000111110001000010000000001011100001000111110100110000111101001010110010101001000000100110111001001111011101100000100110000010001000100011010100101011010001010111111000100111000011001011110100001110010000101101111100110000011010100110011101000011101010001010011010110100001011000101011001100001001100100010010101000010100001010100010101110011110111010011011011011011100000100110011101100101100111000011101000100011111001100100101101001111000001000110001001001011000100100100010101001000001111111000000100011000000011011010001000101010100111100000001110001011100010010011100000000100011010011000000000101101100100100010000101100000011001001100010001000101001101110
Pair \(Z_2\) Length of longest common subsequence
8FOK_1,6VBQ_1 223 4
8FOK_1,8XOA_1 133 4
6VBQ_1,8XOA_1 174 5

Newick tree

 
[
	6VBQ_1:10.90,
	[
		8FOK_1:66.5,8XOA_1:66.5
	]:42.40
]

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 \((0.85)(0.8)(\frac{1698 }{\log_{20} 1698}-\frac{230}{\log_{20}230})=378.\)
Status Protein1 Protein2 d d1/2
Query variables 8FOK_1 6VBQ_1 483 277.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]