The journal of the Symmetrion SYMMETRIES IN GENETIC INFORMATION AND ALGEBRAIC BIOLOGY CONTENTS

Symmetry: Founding editors: G. Darvas and D. Nagy The journal of the Symmetrion

Editor: György Darvas

Volume 23, Numbers 3-4,225-448, 2012

SYMMETRIES IN GENETIC INFORMATION AND ALGEBRAIC BIOLOGY

CONTENTS

ANNOUNCEMENT Symmetry Festival 2013, 2-7 August, Delft, The Netherlands 228

EDITORIAL, Sergey Petoukhov 229

SYMMETRY IN SCIENCE AND ART Genome symmetries, Paul Dan Cristea 233 Symmetry of mitochondrial DNA. The case of COXn genes in primates

and carnivores, Teodora Popovici and Paul Dan Cristea 255 Symmetries of the genetic code, hypercomplex numbers and

genetic matrices with internal complementarities, Sergey V. Petoukhov 275 Fractal genetic nets and symmetry principles in long nucleotide

sequences, S.V. Petoukhov, V.I. Svirin 303 A Markov information source for the syntactic characterization of amino

acid substitutions in protein evolution, Miguel A. Jiménez-Montaño 323 Symmetries in molecular-genetic systems and musical harmony,

G. Darvas, A.A. Koblyakov, S.V.Petoukhov, I.V.Stepanian 343 Modeling “cognition” with nonlinear dynamic systems, Yuri V. Andreyev,

Alexander S. Dmitriev 377 The irregular (integer) tetrahedron as a warehouse of biological

information, Tidjani Négadi 403 Theory of topological coding of proteins and nature of antisymmetry

of the amino acids canonical set, Vladimir A. Karasev 427

SYMMETRY: CULTURE AND SCIENCE is the journal of and is published by the Symmetrion, http://symmetry.hu/. Edition is backed by the Executive Board and the Advisory Board (http://symmetry.hu/isa_leadership.html) of the International Symmetry Association. The views expressed are those of individual authors, and not necessarily shared by the boards and the editor.

Editor: György Darvas

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Cover layout: Günter Schmitz; Image on the front cover: Matjuska Teja Krasek: Star(s) for Donald, 2000, (tribute to H.S.M. Coxeter);

Images on the back cover: Matjuska Teja Krasek: Twinstar and Octapent; Ambigram on the back cover: Douglas R. Hofstadter.

Symmetry: Culture and Science Vol. 23, Nos. 3-4, 323-342, 2012

A MARKOV INFORMATION SOURCE FOR THE SYNTACTIC CHARACTERIZATION OF AMINO ACID

SUBSTITUTIONS IN PROTEIN EVOLUTION

Miguel A. Jiménez-Montaño

BioPhysicist, (b. México, D.F., MEXICO, 1941).

Address: Faculty of Physics and Artificial Intelligence, University of Veracruz, Sebastián Camacho # 5, Col. Centro, C.P. 91000, Xalapa, Ver., México. E-mail: [email protected].

Fields of interest: The structure of the genetic code, technological evolution and informational measures and algorithmic complexity of sequences of symbols and nerve signals.

Awards: Research Award, 1989; Dean Award, 2004; both from Universidad Veracruzana. Fulbright Fellow, 1982. First Prize National Contest on Scientific Non-technical Essay, 1990.

Publications: Ebeling W., Jiménez-Montaño M. A. (1980)*. On Grammars, Complexity, and Information Measures of Biological Macromolecules. Mathematical Biosciences Vol. 52:53-71. Jiménez-Montaño M. A. (1984). On the Syntactic Structure of Protein Sequences, and Concept of Grammar Complexity. Bulletin of Mathematical Biology, Vol.46:641-660. Jiménez-Montaño M.A., de la Mora-Basáñez R., Pöschel T. (1996)*. The Hypercube Structure of the Genetic Code Explains Conservative and Non-Conservartive Aminoacid Substitutions in Vivo and in Vitro. BioSystems Vol. 39: 117-125. Jiménez-Montaño M.A (1999)* Protein Evolution Drives the Evolution of the Genetic Code and Vice Versa. BioSystems Vol. 54: 47-64. Weiss O., Jiménez-Montaño M.A, Herzel H. (2000) Information content of protein sequences. J. Theor. Biol., Vol 206: 379-386. . Jiménez-Montaño M. A. (2004). Applications of Hyper Genetic Code to Bioinformatics. Journal of Biological Systems. Vol. 12: 5-20. .Jiménez-Montaño M. A. (2009)*. The fourfold way of the genetic code. BioSystems, Vol. 98 (2), 105-114.

Abstract: We introduce a theoretical model, which consists of a Markov Information

Source that generates codon sequences, and from them amino acid sequences, that

maintain the same or very similar functions and structures, as a direct consequence of

the structure of the genetic code, and general physical chemical constraints. With the

help of the model, we propose a codon dendrogram to describe a hierarchy of codon

categorizations, which explain the pattern of frequent amino acid substitutions in short-

term evolution.

Keywords: Markov source, genetic code, codon, amino acid, protein evolution.

M. A. JIMÉNEZ-MONTAÑO 324

1. INTRODUCTION

Understanding protein evolution remains today a major challenge in molecular biology

as it was a decade ago (Dokholyan and Shakhnovich, 2001 and references therein),

despite the huge amount of data gathered from genes, protein sequences and structures

presently available. Our knowledge of the relation between the genotype (DNA coding

for a protein) and the phenotype (a protein’s structure and its pattern of specific traits

related to its biological function), which is central to the Theory of Evolution and all

biology, is still at a very primitive stage (Thorne and Goldman, 2001; Wagner, 2012).

While the mechanisms of mutations in DNA sequences that code for proteins are known

(Parkhomchuk et al., 2009 ; Skipper,et al., 2012) the contribution of the genetic code in

creating new information, against the part played by natural selection in its fixation in

the population, is not completely appreciated. According to Abel and Trevors, (2006),

“Genetic prescription of computation precedes and produces phenotypic realization.

And this prescription is “written in stone”. Only recently, it has been recognized in the

literature the full complexity of the genotype/phenotype map (Crutchfield and Schuster,

2003).

According to DePristo et al., (2005), “Taken as whole, recent findings from

biochemistry and evolutionary biology indicate that our understanding of protein

evolution is incomplete, if not fundamentally flawed”. They suggest joining the fields of

protein biophysics and molecular evolution by highlighting the shared questions. In the

same line of thought, Pàl et al., (2006) argue that an integrated view of this field should

embrace genomic, structural and population levels of description. In Fig.1 these

different levels are graphically displayed. However, the problem to achieve this aim is,

on the one hand, that these levels belong to fields of knowledge with radically different

conceptual frameworks; and, on the other, the degenerate relationship between physics

and biology. It is well known that many proteins with no apparent sequence similarity

display the same folds (Kleiger et al., 2000). Thus, the many-to-one map M (Ai / S),

depicted in Fig.1, gives the amino acid at position i of any of the sequences

corresponding to the given structure (fold) S. In the concise statement by Hietpas et al.,

(2011): “Biology is governed by physical interactions, but biological requirements can

have multiple physical solutions”.

AMINO ACID SUBSTITUTIONS IN PROTEIN EVOLUTION 325

Figure 1: The conceptual scheme for protein evolution. The connection between the domain of population genetics (where Darwinian selection operates and true evolution ocurrs) and the genome of an organism (where mutations, an essential ‘raw material’ of evolution, occur), is mediated by the physics and chemistry of proteins. The crucial point to appreciate the complexity implicit in this diagram comes from the degenerate relationship between physics and biology: Given a population, e.g. of virus quasi-species with frequencies {fi, i = 1, 2, …,n}and with corresponding phenotypes {ψi, i = 1, 2, …,n}, M (S/ ψi ) specifies the common structure (S) of a protein family (the set π of orthologous proteins, associated phenotype ψi ). Then, M (Ai/S) is the amino acid at position i of any of the sequences corresponding to the structure S. In the same way, M(Ci/Ai) gives one of the codons codifying the amino acid, according to the genetic code. Let Cm be the chosen codon, then M (C’m / Cm) represents the single-nucleotide mutation from codon Cm to codon C’m , obeying the structure of the genetic code. In turn, M (A’j / C’m) gives the mutated amino acid A’j , coded by C’m, and M (S’/A’j) maps A’j to the corresponding new structure S’ . This modified protein structure is mapped to a new phenotype ψ+ though M (ψ+ / S’), which through Darwinian selection starts the evolutionary cycle again. Notice that M (S’/A’j) maps A’j to the corresponding new structure S’, and M (ψ+ / S’), maps the new protein structure (fold) to the new phenotype ψ+. Certainly, there is no direct feedback from the old to the new structure. Proteins are concrete molecules that do not evolve.

The main purpose of the present work is to provide a theoretical model, built upon an

empirical codon substitution matrix (Schneider et al., 2005), which explains the pattern

of amino acid substitutions in proteins that maintain the same or very similar functions

and structures, as a direct consequence of the structure of the genetic code (Jiménez-

Montaño, 1994), which controls the possible amino acid changes from single

Environment

NaturalSelectionNeutralEvolution

VirusQuasi‐Speciesf1, f2…fn;ψ1, ψ2… ψn

(Concentrationspace)

DNAsequence space(Genotypes)

Translation apparatus

GeneticCode Space(Hypercube)MUTATIONS

X

M (f’/S’)

f

M(S/ψi)

M (S’/Aj’)

Cm

M(Cm’/Cm)

Population genetics(PHENOTYPE)

(Codonusage)

Molecular biology &Bioinformatics(GENOTYPE)

Protein structure(formSpace)

Protein sequence space(Coding of structure)

Aj’

M(Ai/S)M(Cj /Ai)

M(Ai’/C’m)

Biophysics&

Biochemistry(Thermodynamics)

S = 3‐d structure of the proteinAi = aa at site i of protein sequence Ci = codon at site i of protein fi= frequency of species iψi= phenotype of species i


nucleotide mutations, and general physical chemical constraints which are responsible

for the stability of the protein.

2. MODELS OF PROTEIN EVOLUTION

2.1 Amino acid models of protein evolution

Nonetheless the complexity of protein evolution, for a wide range of applications such

as database search, sequence alignment, protein family classification and phylogenetic

inference, among many others, the phenomenological approach to amino acid

substitutions in protein families, started with the empirical work of Margaret Dayhoff

and her colleagues (1978), is still widely used. Following Dayhoff’s footsteps, with the

help of large data bases available in subsequent years, various authors built several

amino acid substitution matrices based on observed mutation counts in protein

alignments (e.g. the updated Dayhoff matrices by Gonnet et al., 1992 or Jones et al.,

1992). This formalism operates in protein space (see below), thus completely ignores

the underlying mutational process that occurs at the DNA level. Dayhoff’s PAM

matrices describe the probabilities of amino acid substitutions, for a given period of

evolution. They are derived from a model in which amino acids mutate randomly and

independent of one another. Each substitution probability during some time interval

depends only on the identities of the initial and replacement residues. Mathematically

speaking, the dynamics of amino acid substitution resembles a time-homogenous first

order reversible Markov chain (Dayhoff et al., 1972, 1978; Gonnet et al., 1992; Jones et

al., 1992; Müller and Vingron, 2000).Of course, the above assumptions are not strictly

true, and various authors have pointed out that the dynamics of amino acid substitutions

is not Markovian, stationary, nor homogeneous (Crooks and Brenner, 2005).

Sequence space, the abstract space of all sequences drawn from an alphabet of k letters

and of length n, was first introduced in coding theory by Hamming (1950). It is a

metric space with respect to the Hamming distance, dH, (Hamming, 1950), which

represents the minimum number of changes that are required to convert one sequence

into another. Maynard Smith (1970) applied this concept to amino acid sequences

defining the concept of protein space (see also Kauffman, 1989 and references therein).

As recently described in a delightful paper by Frances Arnold (2011), in protein space

each sequence is surrounded by its one-mutant neighbors, that is, by all the proteins that

differ from it by a change in a single amino acid letter. As described in (Kauffman

1989), “The concept of protein space is a high-dimensional space in which each point


represents one protein, and is next to 19 N points representing all the 1-mutant

neighbors of that protein. The protein space therefore simultaneously represents the

entire ensemble of 20N proteins and keeps track of which proteins are 1-mutant

neighbors of each other”.

However, the autonomous description of protein evolution in protein (sequence) space

is misleading, because it violates the Central Dogma of molecular biology. In nature

amino acids do not interchange among themselves during evolution. The space of

possibilities is at the DNA level, where the meaningful unit is a base triplet or codon.

Therefore, the relevant space to describe codon substitutions is a genetic code space

(Fig. 1), where codon mutations occur (Swanson, 1984; Jiménez-Montaño et al., 1996;

Petoukhov, 1999; Stambuk, 2000; Jiménez-Montaño, 2004; Jiménez-Montaño and He,

2009). Thus, by single-nucleotide mutations many of the 19 amino acids are out of

reach from the original amino acid, and thus they have null probability of appearance.

This is the first place where the symmetry associated to the concept of random

mutations is broken. The single-nucleotide mutations among the four bases are indeed

random (although not necessarily equally probable), but the corresponding amino acid

substitution probabilities cannot be equal, due to the structure of the genetic code. The

dynamics of amino acid substitutions refers to an aggregate level (see below).

2.2 Codon models of protein evolution

Goldman and Yang (1994) and, independently, Muse and Gaut (1994) introduced the

first models of a Markovian dynamics at the DNA (codon) level. In these models all

substitution rates are derived from parameters. We will not discuss parametric codon

models here; instead, we are going to employ an adaptation for short-term evolution of

the empirical codon substitution model proposed by Gaston Gonnet and his group

(Schneider et al., 2005). In this case, all substitution rates were estimated from a large

data set of aligned vertebrate coding sequences and then fixed.

Assuming a Markovian dynamics at the DNA (codon) level, the dynamics of amino acid

substitutions is defined by an aggregation (grouping) of codon states. However,

Görnerup and Jacobi (2010) pointed out that in general the dynamics on the aggregated

level is not closed, since the partition of the original space introduces memory on the

aggregated level. Only in the special case when the aggregated dynamics indeed is

closed, the stochastic process over the partitions constitutes a Markov chain with the

same order as the original process. Employing the same empirical codon substitution

matrix (Schneider et al., 2005) as we do, they showed that the substitution process


hierarchically operates on multiple levels, from nucleotides to codons, to groups of

codons, associated with amino acids, and to amino acid groups which form “reduced

alphabets”. Since each level approximately has its own closed dynamics, the original

dynamics and the partition of the state space then define a new stochastic process on the

coarser level. These theoretical aspects of molecular evolution were corroborated by our

computer simulations.

Recently, Kosiol and Goldman (2011) proposed a closely related approach in terms of

aggregated Markov processes (AMPs), to model protein evolution as time-

homogeneous Markovian at the DNA (codon) level but observed (via the genetic code)

only at the amino acid level. They showed that this approach leads to time-dependent

and non-Markovian observations of amino acid sequence evolution. The main

difference between their work and the paper by Görnerup and Jacobi (2010) and our

model is that Kosiol and Goldman employed a parametric codon substitution matrix.

Nonetheless, our model is consistent with their assertion that the genetic code and

amino acids' physiochemical properties “influence the average substitution patterns

observed over collections of proteins at all evolutionary distances in the same way”.

That is, we assert that is not exact that the influence of the genetic dominates in the

short-term, and physiochemical properties in the long-term, as supposed by Benner et

al. (1994).

3. THE MARKOVIAN CODON-SUBSTITUTION MODEL

Markov processes/ chains/ models were first developed by Andrei A. Markov. Their

first use was for a linguistic purpose, modeling the letter sequences in works of Russian

literature (Markov, 1913). Later on, Markov models were developed as a general

statistical tool and applied to problems in the study of natural language processing

(Christopher and Schutze, 2003) and in computational biology (Nielsen, 2005; Ewens

et al., 2001; Yang, 2006), among many other applications.

3.1 The Markov Information Source

Probabilistic finite state automata, PFA, as hidden Markov models, HMM, are widely

used in computational linguistics, machine learning, time series analysis, computational

biology, and speech recognition among other fields of research. Their definition, given

in (Vidal et al., 2005), is equivalent to the definition of a stochastic regular grammar.

PFA are built to deal with the problem of probabilizing a structured space by adding


probabilities to structure. This is precisely what we want to do: To bring in codon

transition probabilities into the structure of the genetic code.

This is necessary because in our model, as in the parametric models by Goldman and

Yang (1994), and by Halpern and Bruno (1998), the state-space for the Markov process

corresponds to the standard genetic code (or its variants). In the model of Goldman and

Yang, “The states of the Markov process are the 61 sense codons. The three nonsense

(stop) codons are not considered in the model, as mutations to or from stop codons can

be assumed to affect drastically the structure and function of the protein and therefore

will rarely survive”. But, except for sharing the same abstract space, our approach is not

related with the mentioned models. Rather, its formulation and interpretation is closer to

that of informational and linguistic models. Therefore, we interpret the genetic code as a

Markov information source, exactly as this expression is understood in information

theory (Ash, 1965, p 172). That is, a finite Markov chain, together with a function f

whose domain is the set of states S and whose range is a finite set Γ called the alphabet

of the source. In our case, Γ = {A, G, S, T,…, Y, W} is the amino acid alphabet. The

PFA can be displayed graphically as a six-dimensional Boolean hypercube (Jiménez-

Montaño et al., 1996; Petoukhov, 1999; Stambuk, 2000; Jiménez-Montaño, 2004;

Sánchez et al., 2004; Karasev and Soronkin, 1997). In a forthcoming paper (Jiménez-

Montaño and Ramos-Fernández, 2013) we describe an implementation of the PFA with

the help of software tool GSEQUENCE that we developed specially to simulate the

generation of codon sequences, and from them amino acid sequences, in protein

evolution.

As Shannon (1948) did not mean that his statistical description of human language, with

a Markov information source, is the actual manner in which human discourse is

generated, it is clear that we are not suggesting that Nature really produces proteins with

the help of a Markov information source at the codon level. This is only a mathematical

device to describe the correlations among the amino acid substitutions along the

evolutionary process.

As mentioned above, for our model we have adapted for short term evolution (i.e., for

single-nucleotide changes) the 61 x 61 codon matrix introduced in (Schneider et al.,

2005), for which all substitution rates have been estimated from a set of 17,502

alignments of orthologous genetic sequences from five vertebrate genomes. The codon

transition probabilities are fixed, and correspond to protein divergence between 25 and

60 accepted point mutations per 100 amino acids (PAMs). Besides the influence of the

genetic code, this as any other empirical codon substitution matrix includes variable


factors such as codon usage, transition/transversion bias and selective pressures.

Inversions and duplications are not considered in this paper.

Out of the 190 possible interchanges among the 20 amino acids, we consider only 75

that can be obtained by single-base substitutions. Therefore, we employ a reduced

empirical matrix (REM), making zero all entries corresponding to more than one

nucleotide change in the original matrix and normalizing the resulting matrix; see

(Jiménez-Montaño and He, 2009) for more details. In this way, we take into account the

local structure of the genetic code around each codon. In one step, a codon can change

in nine different ways and generally can have from zero to three synonymous changes

and from six to nine non-synonymous changes, except in the cases of six-fold

degeneracy such as serine, leucine and arginine. We consider all possible one-step

changes for all 61 codons disregarding the three stop codons.

The important contribution of the genetic code to protein evolution has recently been

underlined by Hietpas et al., (2011), who found that the genetic code is highly

optimized (+2.4σ) to favor single-base substitutions between codons with WT-like

fitness compared with randomly generated codes. Thus, the genetic code generally

permits single-base substitution pathways between codons with WT-like fitness.

4. SELECTIVE CONSTRAINTS

The functions and structure of individual proteins impose different constraints on their

evolution. Irrespective of their dispensability, most proteins require a suitable three

dimensional structure to function. Therefore, any polypeptide having a well defined

globular structure must be the subject of a strong selection and its sequence is, from this

point of view, nearly optimal in terms of stability (Sánchez et al., 2006). Therefore, a

majority of positions in a protein globular domain are selected for stability. The

fundamental role of selection for thermodynamic stability in shaping molecular

evolution has been demonstrated by studies that simulated sequence evolution under

structural constraints (Parisi and Echave, 2001). The amino acid substitution

probabilities derived from the REM matrix are highly anti-correlated with the values

taken from the amino acid substitution matrix suggested long ago by Miyata et al.,

(1979) (Table 2). Therefore, the selective constraints are approximately captured by the

amino acid properties of hydrophobicity and volume.


More than thirty years ago, while analyzing the globin fold, Lesk and Chothia (1980)

reached already a similar conclusion. So long as some basic physical-chemical

constraints are satisfied, there is considerable latitude in primary structure (Axe, 2004).

About the same time, a related conclusion was reached by Sander and Schulz (1979) in

their study of the degeneracy of information contents of amino acid sequences from

overlaid genes. From the families of homologous protein know at that time, they

concluded that the information contained in a sequence is degenerate with respect to

function. They quantified this degeneracy on the basis of viral overlays and found that

five amino acid groups is the largest number of groups for which the assumption is

tenable that there exists one group sequence per protein function. Ever since, a great

number of reduced alphabets have been proposed, based on different amino acid

properties or observed substitutions (Miyata et al., 1979; Jiménez-Montaño, 1984;

Murphy et al., 2000; Solis and Rackovsky, 2000; Cannata et al., 2002; Li et al., 2003),

among many others. Here, following (Görnerup and Jacobi, 2010), we interpret reduced

amino acid alphabets simply as a result of the various codon sub dynamics, among

different groups of codons, which are neighbors according to the topology of the genetic

code space.

Recently, Chothia (Sasidharan and Chothia, 2007) returned to the problem from a

different perspective. For the divergence process in proteins that maintain the same or

very similar functions and structures, Sasidharan and Chothia reported very similar

overall patterns of divergence by counting observed amino acid substitutions in three

very different groups of orthologs. They interpret this result to mean that individual

responses of most proteins are variations on a common set of selective constraints

which govern the types of frequent mutations that are acceptable. In RESULTS we

show that the frequencies of amino acid pair substitutions deduced from our computer

simulations are in very good agreement with the mutation profile obtained in their

paper.

5. PROTEIN SYNTAX

Paraphrasing Prince and Smolensky (1997) in their attempt to relate the sciences of the

brain with the sciences of the mind, we can say in the present context that: “It is evident

that statistical thermodynamics and molecular biology are separated by many gulfs, not

the least of which lies between the formal methods appropriate for continuous

dynamical systems and those for discrete symbol structures”.


In order that an amino acid substitution is acceptable is necessary, first of all, that the

alteration it produces in the protein structure be as small as possible. Therefore, the

general Darwinian principle of “gradual change” is interpreted in the sense that the

destabilization of the structure should be as small as possible. Thus, thermodynamics

requires minimization of the Gibbs free energy change. However, this continuous

optimization is hampered by the discrete nature of the amino acid change. A rough

estimation of the effect produced by the substitution consists, for example, in

calculating the Miyata et al., distance (1979) between the original and the new amino

acid. However, this distance should not calculated between the original and any of the

other 19 amino acids; only between the original amino acid and the accessible amino

acids after a single-nucleotide mutation (that is, at most nine amino acids). In this way

the genetic code modulates acceptable mutations.

Following the parallelism between linguistics and a formal protein language (Jiménez-

Montaño, 1984), we recall that an important challenge of the first discipline is to

discover an architecture for grammars that both allows variation and limits its range to

what is actually possible in human language (Prince and Smolensky, 1997).

Furthermore, these authors remark that “… a central element in the architecture of

grammar is a formal means for managing the pervasive conflict between grammatical

constraints”. “The key observation is this: In a variety of clear cases where there is a

strength asymmetry between two conflicting constraints, no amount of success on the

weaker constraint can compensate for failure on the stronger one”. Finally, “….a

grammar consists entirely of constraints arranged in a strict domination hierarchy, in

which each constraint is strictly more important than-takes absolute priority overall

constrains lower-ranked in the hierarchy. With this type of constraint interaction, it is

only the ranking of constraints in the hierarchy that matters for the determination of

optimally; no particular numerical strengths, for example, are necessary”.

Below we are going to show in which sense these concepts can be applied to the

characterization of amino acid substitutions. First, we need to say some words about

amino acid categorizations and the syntactic structure of proteins at the letter-unit level

which we discussed in detail in (Jiménez-Montaño, 1984).

We will call any classification of the 20 amino acid types in r groups (under different

criteria), an amino acid categorization. The set of symbols denoting the group names

will be called a reduced alphabet. When the reduced alphabet corresponds to a pattern

of substitutions according to an empirical matrix (Dayhoff et al., 1972, 1978; Gonnet et

al., 1992; Jones et al., 1992; etc.), the pattern is called a pattern of substitution classes.


If the categorization is based on physical-chemical properties (Grantham; 1964; Miyata

et al., 1979; etc.), the resulting patterns are called amino acid property sequences. In the

paper mentioned above, we approached the question of how we can select a fixed

number of categories which best mirror amino acid replacements. The mutational

categories should reflect the most frequent amino acid substitutions observed. In the

same way, if the selected physical chemical properties truly determine the protein´s

architecture, both patterns will be consistent. In other words, the reduced alphabets

obtained under different criteria will be almost equivalent. It is on this basis that is

reasonable to assume that the constraints responsible for a given fold are somehow

encoded in the pattern of substitution classes.

A hierarchy (inverted tree) of amino acid categorizations represents a syntactic structure

of proteins at the letter-unit level. We shall employ for our discussion the hierarchy

introduced in Fig. 1 of (Jiménez-Montaño, 1984). In the same paper we discussed two

more hierarchies, one from Sneath (1966) and the other from Lim (1974). Twelve more

hierarchies (also called dendrograms), associated with the same number of popular

amino acid substitution matrices are displayed in Fig. 4 of (Johnson and Overington,

1993). See also Fig. 1 in (Fan and Wang, 2003), and (Venkatarajan and Braun, 2001),

among many other proposals in the literature.

With this background, the interpretation of the above quotations from Optimality

Theory (Prince and Smolensky (1997) in the context of the present article is

straightforward. A hierarchy of amino acid categorizations encodes physical chemical

constraints arranged in a strict domination hierarchy. Thus, the dominant partition in the

dendrogram in Fig. 1 of (Jiménez Montaño, 1984) separates amino acids into non-

hydrophobic, represented by the group symbol a, and hydrophobic, represented by the

group symbol b. This constraint dominates over the lower constrains (for example size).

Therefore, we expect that an amino acid of a given class will be substituted with another

amino acid of the same class. In this case, we say that the substitution is syntactically

correct, and that the new sequence belongs to the language generated by the grammar.

Let us illustrate with an example how the grammar generates average amino acid

substitutions obeying general physical chemical constraints that preserve the stability of

the protein. If, in a given site of a protein sequence, we have aspartic acid (D) we expect

that it will be replaced by glutamic acid (E) because the node n in Fig. 1 of (Jiménez-

Montaño, 1984) is the smallest class that includes both amino acids. Next, we have node

e which includes two more amino acids, Q and N, that is, e = {D, E, N, Q}. Thus, the

category represented by the symbol n corresponds to the most conservative substitution,


then follows the wider category e, and so on up to the category represented by the

symbol a, which embraces the non-hydrophobic amino acids.

The amino acid dendrograms we are considering were derived from a number of amino

acid substitution matrices, by several authors that employed different clustering

procedures which have a significant influence in the result. Besides, this approach in

protein space disregards the fact that two amino acids in the same group may be

separated by two or three nucleotide substitutions, thus unlikely to substitute one

another. As pointed out long ago by Miyata et al., (1979): “Amino acids separated by

two or three codon position differences are unlikely to interchange even if they are

chemically similar”. Recently, we discussed this problem (Jiménez-Montaño and He,

2009). For example, the category i = {F, Y, W} in Fig. 1 of (Jiménez-Montaño, 1984) ,

which includes the three large hydrophobic amino acids should be refined into two

groups:{F, Y} and {W}. This is so because to go from the codon of W to any of the

codons of the other two amino acids we need two nucleotide changes; thus, W

constitutes a separate group by itself. This splitting of W was already proposed, for

example, in (Murphy et al., 2000) but for a different reason (which is a consequence of

the above reason): The small number of substitutions observed between W and the other

two amino acids, as reflected in the empirical BLOSUM 50 matrix. Therefore, it is clear

that to improve over previous approaches it is necessary to have a syntactic structure at

the codon level.

6. RESULTS

The first result of this paper is the proposal of the codon dendrogram shown in Fig. 2. It

was obtained by applying the clustering algorithm UPGMA (unweighted pair-group

method using arithmetic averages) to the full codon substitution matrix introduced in

(Schneider et al., 2005). This classification of codons, inferred from an empirical

matrix, induces a corresponding arrangement for amino acids. We observe that the

codons for D and E share the same group, therefore, we expect these two amino acids

exchange frequently both because they are very similar and because they are neighbors

in codon space. They are ranked one in our simulations (Table 1) and in Human-

Chicken orthologs, and ranked two in Escherichia coli and Salmonella orthologs; both

from the observed data (Table 6 in supp. material from Sasidharan and Chothia, 2007).

However, the group e = {D, E, N, Q} does not occur in Fig.2; N and S2 (S with AGY

codons) form one category, and Q and H another. Therefore, the letters in category e are

not completely equivalent from the point of view of their substitutability. From the


results in Table 1, the substitutions DN, and EQ are in ranks between 12 and

14, while NQ and EN, are ranked 44 and 45, respectively, in (Table 6 in supp.

Material from Sasidharan and Chothia, 2007). In the first case the amino acids have

neighboring codons; in the second case the corresponding codons differ by two bases.

Figure 2: Dendrogram (Codon rooted tree) obtained from the full empirical codon-substitution matrix (Schneider et al., 2005), employing the UPGMA method.


E. COLI-S. ENTERICA MARKOV SOURCE RANDOM GENERATOR

RANK ORDER MUTATION RANK ORDER MUTATION RANK ORDER MUTATION 2 DE 1 DE/ED 10 DE/ED 1 IV 3 IV/VI 6 IV/VI 4 ST 4 ST/TS 3 ST/TS 5 AT 8 AT/TA 5 AT/TA 10 NS 5 SN/NS 6 SN/NS 3 AS 9 AS/SA 2 AS/SA 15 KR 2 RK/KR 6 RK/KR 18 GS 15 SG/GS 2 SG/GS 8 AV 13 AV/VA 5 AV/VA 6 IL 14 IL/LI 4 IL/LI 12 SP/PS 3 SP/PS

13 LV 6 VL/LV 3 VL/LV 7 LM 7 LM/ML 9 LM/ML 19 AP 20 AP/PA 5 AP/PA 16 AG 17 AG/GA 5 AG/GA 21 HQ 3 QH/HQ 10 QH/HQ 17 FY 10 FY/YF 10 FY/YF 23 NT 19 TN/NT 8 TN/NT 16 VM/MV 10 VM/MV

29 IM 10 IM/MI 11 IM/MI 23 PQ/QP 8 PQ/QP

25 FL 10 LF/FL 6 LF/FL 22 TP/PT 5 TP/PT 19 TI/IT 6 TI/IT

24 QR 18 RQ/QR 6 RQ/QR 14 DN 12 ND/DN 10 ND/DN 19 SC/CS 7 SC/CS

11 KQ 9 KQ/QK 10 KQ/QK 28 TM/MT 10 TM/MT

20 HN 17 NH/HN 10 NH/HN 9 AE 24 AE/EA 8 AE/EA 12 EQ 8 EQ/QE 10 EQ/QE 22 KN 11 KN/NK 10 KN/NK 26 AD 25 AD/DA 8 AD/DA 27 TV 5 TV/VT 28 HR 20 RH/HR 6 RH/HR 30 LQ 22 QL/LQ 6 QL/LQ 20 GE/EG 8 GE/EG 24 PL/LP 2 PL/LP

Table 1: Comparison of rank positions of the most frequent mutations types found in pairs of orthologs from

Escherichia coli and Salmonella enteric, for < 10 % divergence (Sasidharan and Chothia, 2007), with the ones obtained from simulations generated with our Markov information source, implemented with the help of

software tool GSEQUENCE (Jiménez-Montaño and Ramos-Fernández, 2013). In the third column we display results obtained with a random source

Despite the just explained discrepancies, there is a very good agreement between most

of the categories in Fig. 1 from (Jiménez-Montaño, 1984) and those in the codon


dendrogram (Fig. 2). For example, the important category h = {L, I, V, M} of aliphatic

amino acids in our hierarchy of amino acid categorizations, coincides with the category

d in Fig.2. The same is true for the small neutral amino acids in group c = {P, A, G, S,

T}, except for glycine (G), which in Fig. 2 belongs class b = {N, S2, G, D, E}. There are

other coincidences and differences that the reader can easily find. An important

difference worth mention is the following: while in the grouping based on amino acid

properties I and L are in the same category q, in the codon dendrogram I joins V to form

a group; even though there are proofreading process in protein synthesis to correct

translation errors, the substitutions VI are ranked three and the substitutions

LI are ranked fourteen in our simulations; and are ranked one and six, respectively,

in pairs of orthologs from Escherichia coli and Salmonella enterica (See our Table 1

and Table 6 in supp. Material from Sasidharan and Chothia, 2007).

The amino acid substitution pairs from our codon dendrogram, (D,E),(K,R),(I,V),

(Y,F),(M,L),(N, S2 ) (Q, H) and (A,S) are consistent to the ones reported in the

dendrogram displayed in Fig. 3 of (Görnerup and Jacobi, 2010), except for minor

differences. These are in the two last groups; in their paper (which employs a

completely different agglomeration procedure), Q and H form separate groups, and A

pairs with T instead of S. However, the higher categorizations are completely different

in both dendrograms. The separation of hydrophobic and hydrophilic amino acids in our

dendrogram (Fig. 2) is consistent with that in Fig. 1 of (Jiménez-Montaño, 1984). The

only difference comes from the small neutral amino acids, which are in the non-

hydrophobic group in our former publication, and are grouped with the hydrophobic

amino acids in the codon dendrogram.

The second but not less important result is that six of the ten more frequent amino acid

substitutions pairs, obtained from simulations generated with our Markov information

source, implemented with the help of software tool GSEQUENCE (Jiménez-Montaño

and Ramos-Fernández, 2013), agree with the pairs in the three sets of orthologs from E.

coli – S. enterica, Human-Mouse and Human-Chicken, respectively, from Table 6 in

supp. material from (Sasidharan and Chothia, 2007). These pairs are: DE, IV, ST, NS,

AT, AS. The pair KR agrees with two of the three sets of orthologs. Additionally, we

have in descending order the pairs VL, LM and FY which are ranked twelve, thirteen

and seventeen, respectively, in the same source. Seven of these amino acid substitution

pairs agree with the codon pairs in the Codon Dendrogram (Fig. 2), they are: DE, IV,

NS, AS, KR, LM, FY. These results are consistent with the most frequent amino acid

exchanges found in (Schmitt et al., 2007). In our simulations, the corresponding codons


outline approximately closed dynamics, as discussed in (Görnerup and Jacobi, 2010).

Therefore, these cycles of the codon dynamics produce amino acid substitutions which

are fixed in the population because of the similarity of the corresponding amino acids.

These outcomes take us to the third and last result of this paper.

AA/CODON CORR COEF AA/CODON CORR COEF AA/CODON CORR COEF

K AAA -0.8791 I ATA -0.5617 D GAC -0.7193 K AAG -0.9266 I ATC -0.5668 D GAT -0.7201 N AAC -0.6477 I ATT -0.5791 A GCA -0.3349 N AAT -0.7568 M ATG -0.6726 A GCC -0.3815 T ACA -0.6894 Q CAA -0.7166 A GCG 0.7656 T ACC -0.4522 Q CAG -0.5851 A GCT -0.4195 T ACG 0.4707 H CAC -0.6559 G GGA -0.4947 T ACT -0.4727 H CAT -0.6578 G GGC -0.8047 R AGA -0.8543 P CCA -0.8373 G GGG -0.2542 R AGG -0.8397 P CCC -0.8056 G GGT -0.7988 R CGA -0.8787 P CCG 0.3259 V GTA -0.4626 R CGC -0.9202 P CCT -0.7635 V GTC -0.6288 R CGG -0.9238 L CTA -0.6903 V GTG -0.8257 R CGT -0.9403 L CTC -0.964 V GTT -0.6141 S2 AGC -0.7744 L CTG -0.9467 Y TAC -0.9487 S2 AGT -0.6739 L CTT -0.9273 Y TAT -0.9621 S TCA -0.9765 L TTA -0.5547 C TGC -0.788 S TCC -0.8915 L TTG -0.7221 C TGT -0.7812 S TCG -0.7629 E GAA -0.6831 W TGG -0.7817 S TCT -0.9292 E GAG -0.8145 F TTC -0.4676

F TTT -0.4737

Table 2: Anti-correlation between the substitution probabilities from the reduced empirical matrix, REM (Jiménez-Montaño and He, 2009), and the physical-chemical dissimilarity index (distance) from (Miyata et

al., 1979). For a given codon, e.g. AAA (K), I calculated the correlation between the list of values of substitution probabilities with its neighbors (AGA (R), GAA (E), etc, and the list of values of the index of the

associated amino acids (in parenthesis)

In Table 2 we display the anti-correlation between the substitution probabilities from the

reduced empirical matrix, REM (Jiménez-Montaño and He, 2009), and the dissimilarity

physical-chemical index (distance) from (Miyata et al., 1979), which is based on

hydrophobicity and volume of amino acids. As expected, amino acid pairs which have

codons that substitute frequently have small values of the dissimilarity index and vice

versa. So, this well-known result from comparisons of amino acid substitution matrices,

is corroborated at the codon level.

7. CONCLUSIONS

After presenting a general conceptual framework for the analysis of protein evolution,

we introduced a theoretical model, which consists of a Markov Information Source that

generates codon sequences, and from them amino acid sequences, that maintain the

same or very similar functions and structures. This invariance is a consequence not only


of natural selection (that preserves the sequences that obey general physical chemical

constraints, which are responsible of the stability of the protein), but also of the

structure of the genetic code, which controls the possible amino acid changes, from

single nucleotide mutations. With the help of the model, we introduced a syntactic

formulation (codon dendrogram) to describe a hierarchy of codon categorizations which

explain the pattern of frequent amino acid substitutions in short-term evolution. From

our computer simulations (Jiménez-Montaño and Ramos-Fernández, 2013) we

interpreted the reduced amino acid alphabets simply as a result of the various codon

sub dynamics, among different clusters of codons, which are neighbors according to the

topology of the genetic code space.

Acknowledgements: I wrote this paper while commissioned at Dirección General de

Investigaciones de la Universidad Veracruzana. I want to thank director César I.

Beristain-Guevara for his support. I also thank Q.F.B. Antero Ramos-Fernández for his

help in doing some calculations and preparing the tables and figures. I thank David Abel

for suggestions to make clearer the manuscript and some references. I express thanks to

Sistema Nacional de Investigadores, México, for partial support. Finally, I thank my

wife, Ma. Eta. Castellanos G. for her patience and understanding.

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