A Mathematical Model of the Genetic Code, the Origin of Protein Coding, and the Ribosome as a Dynamical Molecular Machine Diego L. Gonzalez CNR-IMM Dipar)mento Is)tuto per la di Stas)ca Microele4ronica Università di e i Microsistemi Bologna Workshop on Dynamical Systems and Applicaons BCAM 10-11 December 2013 1 The Origin of the Genetic code and Protein Coding • “The origin of the code is perhaps the most perplexing problem in evolutionary biology. The existing translation machinery is at the same time so complex, so universal and so essential that is hard to see how it could have come into existence, or how life could have existed without it” Smith and Szathmary, “The Major Transitions in Evolution” Oxford university Press, New York (1995). • Is it possible to obtain new insight on this problem on a first principles basis? • “The genetic code is not immediately dictated by any known physics or chemistry…it evokes the scary spectra of irreducible complexity” Wolf and Kooning, “On the origin of the translation system and the genetic code…”, Biol Direct 2, 14 (2007). • We present here a mathematical model of the genetic code that furnishes new first principles insight on the origin of the genetic code and protein coding • The main motivation of our model is the search for error detection/correction mechanisms that should use the redundancy of genetic information Workshop on Dynamical Systems and Applicaons BCAM 10-11 December 2013 2 Fundamental Dogma of Molecular Biology DNA molecules GENETIC CODE PROTEIN Messenger RNA Thymine (T) Uracil (U) Protein Synthesis Adenine, Thymine, Guanine, Cytosine REPLICATION TRANSCRIPTION TRANSLATION Workshop on Dynamical Systems and Applicaons BCAM 10-11 December 2013 3 Symbolic Coding of Genetic Information DNA (T,C,A,G) TACGGCATGGTGATAGCCCTTA ATGCCGTACCACTATCGGGAAT Thymine » Uracil TRANSCRIPTION (mRNA Synthesis) mRNA (U,C,A,G) AUGCCGUACCACUAUCGGGAAU Three-letter groups Codons AUG CCG UAC CAC UAU CGG GAA U Amino acids Met Pro Tyr Gln Tyr Arg Stop TRANSLATION (Protein Synthesis) Workshop on Dynamical Systems and Applicaons BCAM 10-11 December 2013 4 Protein Synthesis Workshop on Dynamical Systems and Applicaons BCAM 10-11 December 2013 5 The Nuclear Genetic Code Workshop on Dynamical Systems and Applicaons BCAM 10-11 December 2013 6 The Genetic Code as a Mapping 64 codons; 4x4x4 = 64 (Words of three letters from an alphabet of four, i.e., A, T, C, G) 20 amino acids + Stop codon (Methyonine represents also the synthesis start signal) Redundancy and Degeneracy follow Workshop on Dynamical Systems and Applicaons BCAM 10-11 December 2013 7 Degeneracy Distribution of the Genetic Code Degeneracy distribu)on inside quartets Cys Euplotes nuclear gene)c code Workshop on Dynamical Systems and Applicaons BCAM 10-11 December 2013 8 Outline 1. Description of the degeneracy distribution of the Euplotes nuclear genetic code with a redundant integer number representation system 2. Isomorphisms and models, the role of symmetries in the construction of a non-power model of the Euplotes nuclear genetic code 3. Modelling of the vertebrate mitochondrial genetic code and the origin of degeneracy in amino acid’s coding Workshop on Dynamical Systems and Applicaons BCAM 10-11 December 2013 9 Outline 1. Description of the degeneracy distribution of the Euplotes nuclear genetic code with a redundant integer number representation system 2. Isomorphisms and models, the role of symmetries in the construction of a non-power model of the Euplotes nuclear genetic code 3. Modelling of the vertebrate mitochondrial genetic code and the origin of degeneracy in amino acid’s coding Workshop on Dynamical Systems and Applicaons BCAM 10-11 December 2013 10 Outline 1. Description of the degeneracy distribution of the Euplotes nuclear genetic code with a redundant integer number representation system 2. Isomorphisms and models, the role of symmetries in the construction of a non-power model of the Euplotes nuclear genetic code 3. Modelling of the vertebrate mitochondrial genetic code and the origin of degeneracy in amino acid’s coding Workshop on Dynamical Systems and Applicaons BCAM 10-11 December 2013 11 Outline 1. Description of the degeneracy distribution of the Euplotes nuclear genetic code with a redundant integer number representation system 2. Isomorphisms and models, the role of symmetries in the construction of a non-power model of the Euplotes nuclear genetic code 3. Modelling of the vertebrate mitochondrial genetic code and the origin of degeneracy in amino acid’s coding Workshop on Dynamical Systems and Applicaons BCAM 10-11 December 2013 12 Mathematical model: integer number representation systems Usual positional notation – power representation systems – base k d5 d4 d3 d2 d1 d0 Digits: di (0, (k-1)) k5 k4 k3 k2 k1 k0 5 4 3 2 1 0 R (integer) = d5k + d4k + d3k + d2k + d1k + d0k m Univocity condition: (k −1)∑k n = k m+1 −1 n=0 Example 1: k=10; decimal system Digits: di (0, 9) 0 0 0 0 1 9 105 104 103 102 101 100 R = 1.101 + 9.100 = 19 Workshop on Dynamical Systems and Applicaons BCAM 10-11 December 2013 13 Redundant representation systems Example 2: Binary power system Digits: di (0, 1) 0 1 0 0 1 1 25 24 23 22 21 20 R = 0.25 + 1.24 + 0.23 + 0.22 + 1.21 + 1.20 = 19 Redundant Power Representations Using Out-of-Range Digits Example 3: Binary signed digits with three possible values: -1, 0, 1 1 -1 0 0 1 1 25 24 23 22 21 20 R = 1.25 + 0.24 - 1.23 + 0.22 + 1.21 - 1.20 = 19 Redundant Workshop on Dynamical Systems and Applicaons BCAM 10-11 December 2013 14 Non-power redundant representation systems posi)onal bases grow slowly than the powers of k Example 4: k=2; Fibonacci system 1 1 1 1 0 1 Digits: di (0, 1) 8 5 3 2 1 1 R = 1.8 + 1.5 + 1.3 + 1.2 + 0.1 + 1.1 = 19 Non-power representation systems are redundant, for example, the number 19 can be represented in the Fibonacci system in another alternative way: Maya 1 1 1 1 1 0 serpent numbers 8 5 3 2 1 1 R = 1.8 + 1.5 + 1.3 + 1.2 + 1.1 + 0.1 = 19 Workshop on Dynamical Systems and Applicaons BCAM 10-11 December 2013 15 Fibonacci’s non-power representation system Degeneracy Number Signed Binary 1 5 2 3 3 4 4 2 5 2 Fibonacci System 1 2 2 4 3 8 4 5 5 2 Genetic Code 1 2 2 12 3 2 4 8 Workshop on Dynamical Systems and Applicaons BCAM 10-11 December 2013 16 Non-power representation system 1,1,2,4,7,8 Represented) Length 6 binary strings Degeneracy distribution number! Non-power representation system ) ) 8) 7) 4) 2) 1) 1) ) 8) 7) 4) 2) 1) 1) ) 8) 7) 4) 2) 1) 1) ) 8) 7) 4) 2) 1) 1) 1,1,2,4,7,8 0) ) 0) 0) 0) 0) 0) 0) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) # of integer Degeneracy 1) ) 0) 0) 0) 0) 0) 1) ) 0) 0) 0) 0) 1) 0) ) ) ) ) ) ) ) ) ) ) ) ) ) ) numbers 2) ) 0) 0) 0) 0) 1) 1) ) 0) 0) 0) 1) 0) 0) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 2 1 3) ) 0) 0) 0) 1) 0) 1) ) 0) 0) 0) 1) 1) 0) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 4) ) 0) 0) 1) 0) 0) 0) ) 0) 0) 0) 1) 1) 1) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 12 2 5) ) 0) 0) 1) 0) 0) 1) ) 0) 0) 1) 0) 1) 0) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 2 3 6) ) 0) 0) 1) 1) 0) 0) ) 0) 0) 1) 0) 1) 1) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 8 4 7) ) 0) 0) 1) 1) 0) 1) ) 0) 0) 1) 1) 1) 0) ) 0) 1) 0) 0) 0 ) 0) ) ) ) ) ) ) ) 8) ) 0) 1) 0) 0) 0) 1) ) 0) 1) 0) 0) 1) 0) ) 1) 0) 0) 0) 0) 0) ) 0) 0) 1) 1) 1) 1) Unique solution 1,1,2,4,7,8 9) ) 1) 0) 0) 0) 0) 1) ) 1) 0) 0) 0) 1) 0) ) 0) 1) 0) 1) 0) 0) ) 0) 1) 0) 0) 1) 1) 10) ) 0) 1) 0) 1) 0) 1) ) 0) 1) 0) 1) 1) 0) ) 1) 0) 0) 1) 0) 0) ) 1) 0) 0) 0) 1) 1) 11) ) 1) 0) 0) 1) 0) 1) ) 1) 0) 0) 1) 1) 0) ) 0) 1) 1) 0) 0) 0) ) 0) 1) 0) 1) 1) 1) 12) ) 0) 1) 1) 0) 0) 1) ) 0) 1) 1) 0) 1) 0) ) 1) 0) 1) 0) 0) 0) ) 1) 0) 0) 1) 1) 1) 13) ) 1) 0) 1) 0) 0) 1) ) 1) 0) 1) 0) 1) 0) ) 0) 1) 1) 1) 0) 0) ) 0) 1) 1) 0) 1) 1) 14) ) 0) 1) 1) 1) 0) 1) ) 0) 1) 1) 1) 1) 0) ) 1) 0) 1) 1) 0) 0) ) 1) 0) 1) 0) 1) 1) 15) ) 1) 0) 1) 1) 0) 1) ) 1) 0) 1) 1) 1) 0) ) 1) 1) 0) 0) 0) 0) ) 0) 1) 1) 1) 1) 1) Degeneracy distribution 16) ) 1) 1) 0) 0) 0) 1) ) 1) 1) 0) 0) 1) 0) ) 1) 0) 1) 1) 1) 1) ) ) ) ) ) ) ) Euplotes nuclear genetic code 17) ) 1) 1) 0) 0) 1) 1) ) 1) 1) 0) 1) 0) 0) ) ) ) ) ) ) ) ) ) ) ) ) ) ) # of amino acids Degeneracy 18) ) 1) 1) 0) 1) 0) 1) ) 1) 1) 0) 1) 1) 0) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 19) ) 1) 1) 1) 0) 1) 1) ) 1) 1) 0) 1) 1) 1) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 2 1 20) ) 1) 1) 1) 0) 0) 1) ) 1) 1) 1) 0) 1) 0) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 12 2 21) ) 1) 1) 1) 1) 0) 0) ) 1) 1) 1) 0) 1) 1) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 2 3 22) ) 1) 1) 1) 1) 0) 1) ) 1) 1) 1) 1) 1) 0) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 23) ) 1) 1) 1) 1) 1) 1) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 8 4 ! Workshop on Dynamical Systems and Applicaons BCAM 10-11 December 2013 17 Outline 1.