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Parikh vectors
8WGC_1 5MUJ_1 1LAR_1 Letter Amino acid
26 10 34 Q Glutamine
64 32 36 G Glycine
38 18 30 I Isoleucine
81 24 40 L Leucine
47 15 24 F Phenylalanine
14 13 9 W Tryptophan
92 29 39 A Alanine
26 12 20 N Asparagine
33 26 28 D Aspartic acid
31 4 12 C Cysteine
23 32 25 K Lycine
11 11 21 M Methionine
42 20 40 E Glutamic acid
62 16 39 R Arginine
16 11 15 H Histidine
43 16 26 P Proline
67 19 26 S Serine
47 26 42 T Threonine
28 25 28 Y Tyrosine
63 22 41 V Valine 

8WGC_1|Chain A[auth D]|Metabotropic glutamate receptor 2|Homo sapiens (9606)
>5MUJ_1|Chain A|Beta-galactosidase|Bacteroides thetaiotaomicron (strain ATCC 29148 / DSM 2079 / NCTC 10582 / E50 / VPI-5482) (226186)
>1LAR_1|Chains A, B|PROTEIN (LAR)|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 \(w=f(c)\)
8WGC , Knot 323 854 0.85 40 308 770 
EGPAKKVLTLEGDLVLGGLFPVHQKGGPAEDCGPVNEHRGIQRLEAMLFALDRINRDPHLLPGVRLGAHILDSCSKDTHALEQALDFVRASLSRGADGSRHICPDGSYATHGDAPTAITGVIGGSYSDVSIQVANLLRLFQIPQISYASTSAKLSDKSRYDYFARTVPPDFFQAKAMAEILRFFNWTYVSTVASEGDYGETGIEAFELEARARNICVATSEKVGRAMSRAAFEGVVRALLQKPSARVAVLFTRSEDARELLAASQRLNASFTWVASDGWGALESVVAGSEGAAEGAITIELASYPISDFASYFQSLDPWNNSRNPWFREFWEQRFRCSFRQRDCAAHSLRAVPFEQESKIMFVVNAVYAMAHALHNMHRALCPNTTRLCDAMRPVNGRRLYKDFVLNVKFDAPFRPADTHNEVRFDRFGDGIGRYNIFTYLRAGSGRYRYQKVGYWAEGLTLDTSLIPWASPSAGPLPASRCSEPCLQNEVKSVQPGEVCCWLCIPCQPYEYRLDEFTCADCGLGYWPNASLTGCFELPQEYIRWGDAWAVGPVTIACLGALATLFVLGVFVRHNATPVVKASGRELCYILLGGVFLCYCMTFIFIAKPSTAVCTLRRLGLGTAFSVCYSALLTKTNRIARIFGGAREGAQRPRFISPASQVAICLALISGQLLIVVAWLVVEAPGTGKETAPERREVVTLRCNHRDASMLGSLAYNVLLIALCTLYAFKTRKCPENFNEAKFIGFTMYTTCIIWLAFLPIFYVTSSDYRVQTTTMCVSVSLSGSVVLGCLFAPKLHIILFQPQKNVVSHRAPTSRFGSAAARASSSLGQGSGSQFVPTVCNGREVVDSTTSSL
5MUJ , Knot 162 381 0.84 40 220 364 
MIDKSIMKKLLLTFLSIAAVCSLYAQRKVTQERMEQIYEEVKTPYKYGLAVAPTDNYHKIDCPTVFRQGDKWLMTYVVYNGKTGTDGRGYETWIAESDNLLDWRTLGRVLSYRDGKWDCNQRGGFPALPDMEWGGSYELQTYKGRHWMTYIGGEGTGYEAVKAPLFVGLASTKGDISTAHEWESLDKPILSIHDKDAQWWEKLTQYKSTVYWDKDKTLGAPFVMFYNAGGRHPETDLKGERVGIALSKDMKTWKRYPGNPVFAHEADGTITGDAHIQKMGDVYVMFYFSAFEPSRKYKAFNTFAASYDLVNWTDWHGADLIIPSKNYDELFAHKSYVIKHDGVVYHFYCAVNNAEQRGIAIATSKPMGRSAVRFPKPETKN
1LAR , Knot 224 575 0.82 40 265 516 
MITDLADNIERLKANDGLKFSQEYESIDPGQQFTWENSNLEVNKPKNRYANVIAYDHSRVILTSIDGVPGSDYINANYIDGYRKQNAYIATQGPLPETMGDFWRMVWEQRTATVVMMTRLEEKSRVKCDQYWPARGTETCGLIQVTLLDTVELATYTVRTFALHKSGSSEKRELRQFQFMAWPDHGVPEYPTPILAFLRRVKACNPLDAGPMVVHCSAGVGRTGCFIVIDAMLERMKHEKTVDIYGHVTCMRSQRNYMVQTEDQYVFIHEALLEAATCGHTEVPARNLYAHIQKLGQVPPGESVTAMELEFKLLASSKAHTSRFISANLPCNKFKNRLVNIMPYELTRVCLQPIRGVEGSDYINASFLDGYRQQKAYIATQGPLAESTEDFWRMLWEHNSTIIVMLTKLREMGREKCHQYWPAERSARYQYFVVDPMAEYNMPQYILREFKVTDARDGQSRTIRQFQFTDWPEQGVPKTGEGFIDFIGQVHKTKEQFGQDGPITVHCSAGVGRTGVFITLSIVLERMRYEGVVDMFQTVKTLRTQRPAMVQTEDQYQLCYRAALEYLGSFDHYAT

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 \(c\).

\(|P_{f(8WGC_1)}(2) \setminus P_{f(5MUJ_1)}(2)|=133\), \(|P_{f(5MUJ_1)}(2) \setminus P_{f(8WGC_1)}(2)|=45\). 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 1:01110011010101111111110001111000111000011001011111100100010111110111011000000001100110110101001101000101010010010110110111110000101011011011011010010001010000000011001110110101110110110100100110010010011011010101001011000011011001110111011100101011111000001001111000101010111001111100111100111011101011001100110010010110000011100110001000100000110010111100000111110110111011001001101000010011011010010001110101011101100000101001101110001100101101000000110110110100011111010111111000001010001001011010011011001000010010010011101101010101011000101101111111011011111011111111000101110101001001111111100010111110100110010011110110100011100000110111110011001011011001110111101011111111110111010001100001101000000101110110011111100101100000100100101111010000111111111101000000100001010101010111101111010111101000110001100011011101000110101001110100100110000001
Pair \(Z_2\) Length of longest common subsequence
8WGC_1,5MUJ_1 178 4
8WGC_1,1LAR_1 139 4
5MUJ_1,1LAR_1 173 5

Newick tree

 
[
	5MUJ_1:93.05,
	[
		8WGC_1:69.5,1LAR_1:69.5
	]:23.55
]

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 \((0.85)(0.8)(\frac{1235 }{\log_{20} 1235}-\frac{381}{\log_{20}381})=222.\)
Status Protein1 Protein2 d d1/2
Query variables 8WGC_1 5MUJ_1 285 205.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]