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Parikh vectors
2MQP_1 4MSF_1 3NKO_1 Letter Amino acid
7 29 20 Q Glutamine
6 28 49 K Lycine
1 11 17 M Methionine
11 34 60 S Serine
5 15 44 Y Tyrosine
6 42 59 P Proline
8 25 50 T Threonine
6 38 25 A Alanine
6 38 49 R Arginine
4 15 31 C Cysteine
7 40 50 G Glycine
9 24 34 I Isoleucine
9 34 41 N Asparagine
2 32 53 E Glutamic acid
0 15 28 H Histidine
9 33 51 D Aspartic acid
8 63 70 L Leucine
4 30 38 F Phenylalanine
1 14 13 W Tryptophan
9 35 49 V Valine 

2MQP_1|Chain A|Protein Hnrnpl|Rattus norvegicus (10116)
>4MSF_1|Chain A|Lactoperoxidase|Capra hircus (9925)
>3NKO_1|Chain A|Ectonucleotide pyrophosphatase/phosphodiesterase family member 2|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 \(w=f(c)\)
2MQP , Knot 60 118 0.80 38 95 115 
QKISRPGDSDDSRSVNSVLLFTILNPIYSITTDVLYTICNPCGPVQRIVIFRKNGVQAMVEFDSVQSAQRAKASLNGADIYSGCCTLKIEYAKPTRLNVFKNDQDTWDYTNPNLSGQG
4MSF , Knot 238 595 0.85 40 282 562 
SWEVGCGAPVPLVTCDEQSPYRTITGDCNNRRSPALGAANRALARWLPAEYEDGLAVPFGWTQRKTRNGFRVPLAREVSNKIVGYLDEEGVLDQNRSLLFMQWGQIVDHDLDFAPETELGSSEHSKVQCEEYCVQGDECFPIMFPKNDPKLKTQGKCMPFFRAGFVCPTPPYQSLARDQINAVTSFLDASLVYGSEPSLASRLRNLSSPLGLMAVNQEAWDHGLAYPPFNNVKPSPCEFINTTAHVPCFQAGDSRASEQILLATVHTLLLREHNRLARELKRLNPHWDGEMLYQEARKILGAFIQIITFRDYLPIVLGSEMQKWIPPYQGYNNSVDPRISNVFTFAFRFGHMEVPSTVSRLDENYQPWGPEAELPLHTLFFNTWRIIKDGGIDPLVRGLLAKNSKLMNQNKMVTSELRNKLFQPTHKVHGFDLAAINLQRCRDHGMPGYNSWRGFCGLSQPKTLKGLQAVLKNKVLAKKLLDLYKTPDNIDIWIGGNAEPMVERGRVGPLLACLLGRQFQQIRDGDRFWWENPGVFTEKQRDSLQKVSFSRLICDNTHITKVPLHAFQANNYPHDFVDCSAVDKLDLSPWASREN
3NKO , Knot 321 831 0.86 40 316 764 
AEWDEGPPTVLSDSPWTNTSGSCKGRCFELQEVGPPDCRCDNLCKSYSSCCHDFDELCLKTARGWECTKDRCGEVRNEENACHCSEDCLSRGDCCTNYQVVCKGESHWVDDDCEEIRVPECPAGFVRPPLIIFSVDGFRASYMKKGSKVMPNIEKLRSCGTHAPYMRPVYPTKTFPNLYTLATGLYPESHGIVGNSMYDPVFDATFHLRGREKFNHRWWGGQPLWITATKQGVRAGTFFWSVSIPHERRILTILQWLSLPDNERPSVYAFYSEQPDFSGHKYGPFGPEMTNPLREIDKTVGQLMDGLKQLKLHRCVNVIFVGDHGMEDVTCDRTEFLSNYLTNVDDITLVPGTLGRIRPKIPNNLKYDPKAIIANLTCKKPDQHFKPYMKQHLPKRLHYANNRRIEDLHLLVERRWHVARKPLDVYKKPSGKCFFQGDHGFDNKVNSMQTVFVGYGPTFKYRTKVPPFENIELYNVMCDLLGLKPAPNNGTHGSLNHLLRTNTFRPTLPEEVSRPNYPGIMYLQSDFDLGCTCDDKNKLEELNKRLHTKGSTEERHLLYGRPAVLYRTSYDILYHTDFESGYSEIFLMPLWTSYTISKQAEVSSIPEHLTNCVRPDVRVSPGFSQNCLAYKNDKQMSYGFLFPPYLSSSPEAKYDAFLVTNMVPMYPAFKRVWTYFQRVLVKKYASERNGVNVISGPIFDYNYNGLRDIEDEIKQYVEGSSIPVPTHYYSIITSCLDFTQPADKCDGPLSVSSFILPHRPDNDESCNSSEDESKWVEELMKMHTARVRDIEHLTGLDFYRKTSRSYSEILTLKTYLHTYESEISRENLYFQ

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 \(c\).

\(|P_{f(2MQP_1)}(2) \setminus P_{f(4MSF_1)}(2)|=25\), \(|P_{f(4MSF_1)}(2) \setminus P_{f(2MQP_1)}(2)|=212\). 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 1:0010011000000010011110110110010001100100101110011110001101110100100100101010110100100010100101001011000000100001010101
Pair \(Z_2\) Length of longest common subsequence
2MQP_1,4MSF_1 237 3
2MQP_1,3NKO_1 251 3
4MSF_1,3NKO_1 126 4

Newick tree

 
[
	2MQP_1:13.15,
	[
		4MSF_1:63,3NKO_1:63
	]:73.15
]

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 \((0.85)(0.8)(\frac{713 }{\log_{20} 713}-\frac{118}{\log_{20}118})=170.\)
Status Protein1 Protein2 d d1/2
Query variables 2MQP_1 4MSF_1 223 132
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]