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Parikh vectors
8PTK_1 6TTX_1 5JWZ_1 Letter Amino acid
10 13 0 C Cysteine
31 38 23 E Glutamic acid
37 46 66 S Serine
26 43 58 D Aspartic acid
25 36 102 G Glycine
16 27 20 I Isoleucine
14 15 24 F Phenylalanine
27 35 70 T Threonine
12 8 31 W Tryptophan
36 32 47 V Valine
22 34 33 R Arginine
10 29 15 Q Glutamine
27 58 35 L Leucine
12 11 8 M Methionine
13 33 40 P Proline
10 9 29 Y Tyrosine
11 16 25 N Asparagine
16 19 7 H Histidine
34 36 18 K Lycine
21 42 73 A Alanine 

8PTK_1|Chains A[auth 1], B[auth 2], C[auth 3], D[auth 4]|Platelet-activating factor acetylhydrolase IB subunit beta|Homo sapiens (9606)
>6TTX_1|Chain A|N6-adenosine-methyltransferase catalytic subunit|Homo sapiens (9606)
>5JWZ_1|Chains A, B|Cellulose-binding family II|Streptomyces sp. (strain SirexAA-E / ActE) (862751)
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 \(w=f(c)\)
8PTK , Knot 170 410 0.83 40 222 374 
MVLSQRQRDELNRAIADYLRSNGYEEAYSVFKKEAELDVNEELDKKYAGLLEKKWTSVIRLQKKVMELESKLNEAKEEFTSGGPLGQKRDPKEWIPRPPEKYALSGHRSPVTRVIFHPVFSVMVSASEDATIKVWDYETGDFERTLKGHTDSVQDISFDHSGKLLASCSADMTIKLWDFQGFECIRTMHGHDHNVSSVAIMPNGDHIVSASRDKTIKMWEVQTGYCVKTFTGHREWVRMVRPNQDGTLIASCSNDQTVRVWVVATKECKAELREHEHVVECISWAPESSYSSISEATGSETKKSGKPGPFLLSGSRDKTIKMWDVSTGMCLMTLVGHDNWVRGVLFHSGGKFILSCADDKTLRVWDYKNKRCMKTLNAHEHFVTSLDFHKTAPYVVTGSVDQTVKVWECR
6TTX , Knot 231 580 0.84 40 260 535 
MSDTWSSIQAHKKQLDSLRERLQRRRKQDSGHLDLRNPEAALSPTFRSDSPVPTAPTSGGPKPSTASAVPELATDPELEKKLLHHLSDLALTLPTDAVSICLAISTPDAPATQDGVESLLQKFAAQELIEVKRGLLQDDAHPTLVTYADHSKLSAMMGAVAEKKGPGEVAGTVTGQKRRAEQDSTTVAAFASSLVSGLNSSASEPAKEPAKKSRKHAASDVDLEIESLLNQQSTKEQQSKKVSQEILELLNTTTAKEQSIVEKFRSRGRAQVQEFCDYGTKEECMKASDADRPCRKLHFRRIINKHTDESLGDCSFLNTCFHMDTCKYVHYEIDACMDSEAPGSKDHTPSQELALTQSVGGDSSADRLFPPQWICCDIRYLDVSILGKFAVVMADPPWDIHMELPYGTLTDDEMRRLNIPVLQDDGFLFLWVTGRAMELGRECLNLWGYERVDEIIWVKTNQLQRIIRTGRTGHWLNHGKEHCLVGVKGNPQGFNQGLDCDVIVAEVRSTSHKPDEIYGMIERLSPGTRKIELFGRPHNVQPNWITLGNQLDGIHLLDPDVVARFKQRYPDGIISKPKNL
5JWZ , Knot 264 724 0.80 38 247 611 
DTYTWKNARIDGGGFVPGIVFNRSEKNLAYARTDIGGAYRWDQSGKQWKPLLDWVDWDRWGWTGVVSLASDTVDPDNVYAAVGTYTNSWDPTDGAVLRSSDRGASWKAATLPFKLGGNMPGRGMGERLAVDPNKNSVLYLGAPSGNGLWRSTDAGVSWSEVTAFPNPGNYAQDPSDTSGYGNDNQGIVWVTFDERSGSAGSATQDIYVGVADKENTVYRSTDGGATWSRIPGQPTGYLAHKGVLDSATGHLYLTLSDTGGPYDGGKGRIWRYDTASGAWQDVSPVAEADAYYGFSGLSVDRQKPGTLMATAYSSWWPDTQIFRSTDSGATWTQAWDYTGYPNRSNRYTLDVSSVPWLSWGASPAPPETAPKLGWMTEALEIDPFDSDRMMYGTGATVYGTEDLTSWDSGGTFRITPMVKGIEETAVNDLASPPSGAPLLSALGDIGGFRHTDLDAVPDLMYTSPNLDSTTSLDFAESSPGTVVRVGNSDAAPHIGFSTDNGANWFQGSEPSGVTGGGTVAAAADGSGFVWSPEGAGVHHTTGFGTSWTASTGIPAGATVESDRKNPEKFYGFEAGTFYVSTDGGATFTAEATGLPAEGNVRFQALPGTEGDIWLAGGSDTGAYGLWRSTDSGATFTKSAGVEQADSVGFGKAAPGASYRTVFVSAKIGGVRGIFRSTDAGASWTRINDDAHQWGWTGAAITGDPRVYGRVYVSTNGRGIQVGET

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 \(c\).

\(|P_{f(8PTK_1)}(2) \setminus P_{f(6TTX_1)}(2)|=66\), \(|P_{f(6TTX_1)}(2) \setminus P_{f(8PTK_1)}(2)|=104\). 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 1:11100000001001110010001000100110001010100010000111100010011010001101000100100010011111000010011101100011010001100111011101110100010101100001010001010000100101000101110001010101101011001001010000100111110100110100000101101001001001010001101101000101110000000101111100000101000001100101110000001001010000001011111101000001011010011011011100011011110011011100100001011000000010010100011001010001101101010001011000
Pair \(Z_2\) Length of longest common subsequence
8PTK_1,6TTX_1 170 5
8PTK_1,5JWZ_1 175 5
6TTX_1,5JWZ_1 145 4

Newick tree

 
[
	8PTK_1:90.38,
	[
		6TTX_1:72.5,5JWZ_1:72.5
	]:17.88
]

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 \((0.85)(0.8)(\frac{990 }{\log_{20} 990}-\frac{410}{\log_{20}410})=153.\)
Status Protein1 Protein2 d d1/2
Query variables 8PTK_1 6TTX_1 195 162
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]