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Parikh vectors
5BPX_1 3ODD_1 6WXR_1 Letter Amino acid
12 17 87 A Alanine
15 4 86 E Glutamic acid
4 22 124 S Serine
4 17 50 N Asparagine
3 15 76 Q Glutamine
11 25 101 G Glycine
9 6 49 H Histidine
8 10 72 I Isoleucine
8 3 101 F Phenylalanine
9 7 89 P Proline
9 12 121 R Arginine
8 7 68 D Aspartic acid
2 3 50 K Lycine
2 2 29 M Methionine
5 7 27 W Tryptophan
9 11 47 Y Tyrosine
12 27 95 V Valine
2 8 27 C Cysteine
12 18 169 L Leucine
9 19 68 T Threonine 

5BPX_1|Chain A|2,4'-dihydroxyacetophenone dioxygenase|Alcaligenes sp. (512)
>3ODD_1|Chain A|Chymotrypsin-like elastase family member 1|Sus scrofa (9823)
>6WXR_1|Chain A|Dual oxidase 1|Mus musculus (10090)
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 \(w=f(c)\)
5BPX , Knot 76 153 0.83 40 126 150 
LPEAYIPNAATEDERYYVPFTETVASRPLWISPQQNRWCDILLAREAGLVNRHYHPHEVFAYTISGKWGYLEHDWTATRGDFVYETPGEGHTLVAFEHEEPMRVFFIVQGPLIWLDEAGNSIGHFDVHDYIAMCREHYEKVGLGADLVVTLFR
3ODD , Knot 108 240 0.82 40 162 230 
VVGGTEAQRNSWPSQISLQYRSGSSWAHTCGGTLIRQNWVMTAAHCVDRELTFRVVVGEHNLNQNNGTEQYVGVQKIVVHPYWNTDDVAAGYDIALLRLAQSVTLNSYVQLGVLPRAGTILANNSPCYITGWGLTRTNGQLAQTLQQAYLPTVDYAICSSSSYWGSTVKNSMVCAGGDGVRSGCQGDSGGPLHCLVNGQYAVHGVTSFVSRLGCNVTRKPTVFTRVSAYISWINNVIASN
6WXR , Knot 531 1536 0.84 40 363 1299 
GPSRGAQNSISWEVQRFDGWYNNLMEHRWGSKGSRLQRLVPASYADGVYQPLKEPYLPNPRHLSNRVMRGSAGQPSLRNRTVLGVFFGYHVLSDLVSVETPGCPAEFLNIYIPHGDPVFDPDKRGNVVLPFQRSRWDRNTGQSPSNPRDQSNQVTGWLDGSAIYGSSHSWSDTLRSFSGGQLASGPDPAFPSDSQSSLLMWMAPDPSTGQGGPRGVYAFGAQRGNREPFLQALGLLWFRYHNLCARKLAQEHPHWGDEELFQHARKRVIATYQNIAMYEWLPSFLKQTPPEYPGYRPFLDPSISPEFVVASEQFLSTMVPSGVYMRNASCHFQGIPSHNSSVSGALRVCNSYWSREHPKLQRAEDVDALLLGMASQIAEREDHVVVEDMQDFWPGPLKFSRTDYLASCLQRGRDLGLPSYTKAREALGLSPISHWQDINPALSRSNGTVLEATAALYNQDLSRLELLPGGLLESHGDPGPLFSTIVLDQFVRLRDGDRYWFENTRNGLFSKEEIAEIRNTSLRDILVAVTNVDPSALQPNVFFWLAGDPCPQPSQLSAKGLPACAPLFIRDYFEGSGFGFGLTIGTLCCFPLVSLLSAWIVARLRKRNFKRLQRQDRQSIMSEKLVGGVEALEWQGRNEPCRPVLVHLQPGQIRVVDGRLTVLRTIQLRPPQQVNLILSSNRGRRTLLLKIPKEYDLVLLFNMEEERQALVENVRGALKENGLSFQEWELREQELMRAAVTRQQRGHLLETFFRHLFSQVLDINQADAGTLPLDSSTKVREALTCELSRAEFADSLGLKPQDMFVESMFSLADKDGNGYLSFREFLDILVVFMKGSPEEKSRLMFRMYDFDGNGLISKDEFIRMLRSFIEISNNCLSKAQLAEVVESMFRESGFQDKEELTWEDFHFMLRDHDSDLRFTQLCVKGVEVPEVIKNLCRRASYISQEKICPSPRMSAHCARNNMKTASSPQRLQCPMDTDPPQEIRRRFGKKVTSFQPLLFTEAHREKFQRSRRHQTVQQFKRFIENYRRHIGCVAVFYTITGALFLERAYYYAFAAHHSGITDTTRVGIILSRGTAASISFMFSYILLTMCRNLITFLRETFLNRYIPFDAAVDFHRLIASTAIILTVLHSAGHVVNVYLFSISPLSVLSCLFPGLFHDDGSEFPQKYYWWFFQTVPGLTGVLLLLALAIMYVFASHHFRRRSFRGFWLTHHLYIFLYILLIIHGSFALIQMPRFHIFFLVPAIIYVGDKLVSLSRKKVEISVVKAELLPSGVTHLRFQRPQGFEYKSGQWVRIACLALGTTEYHPFTLTSAPHEDTLSLHIRAAGPWTTRLREIYSPPTGDTCARYPKLYLDGPFGEGHQEWHKFEVSVLVGGGIGVTPFASILKDLVFKSSVSCQVFCKKIYFIWVTRTQRQFEWLADIIREVEENDRQDLVSVHIYITQLAEKFDLRTTMLYICERHFQKVLNRSLFTGLRSITHFGRPPFEPFFNSLQEVHPQVRKIGVFSCGPPGMTKNVEKACQLINRQDRTHFSHHYENF

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 \(c\).

\(|P_{f(5BPX_1)}(2) \setminus P_{f(3ODD_1)}(2)|=68\), \(|P_{f(3ODD_1)}(2) \setminus P_{f(5BPX_1)}(2)|=104\). 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 1:110101101100000001110001100111101000010011110011110000010011100101011010001010010110001101001111000011011111011111100110011010100011100000001111101110110
Pair \(Z_2\) Length of longest common subsequence
5BPX_1,3ODD_1 172 3
5BPX_1,6WXR_1 247 4
3ODD_1,6WXR_1 217 4

Newick tree

 
[
	6WXR_1:12.70,
	[
		5BPX_1:86,3ODD_1:86
	]:38.70
]

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 \((0.85)(0.8)(\frac{393 }{\log_{20} 393}-\frac{153}{\log_{20}153})=72.0\)
Status Protein1 Protein2 d d1/2
Query variables 5BPX_1 3ODD_1 89 74.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]