Some further comments on the DNA-protein coding problem

A notion of quasi-ergodicity is defined in free monoids, generalizing a notion introduced in a previous paper (Rosen, 1959,Bull. Math. Biophysics,21, 71–95). It is shown that under certain conditions the algebraic properties of quasi-ergodicity are very similar to those derived for the more specialized concept inloc. cit. If it is assumed that the DNA-protein coding processes in nature are of a quasi-ergodic nature, then a condition is specified under which only a finite number of different DNA-protein codes are possible, and an upper bound for the total number of different quasi-ergodic codes is obtained.     

Identification of DNA-binding protein target sequences by physical effective energy functions: free energy analysis of lambda repressor-DNA complexes.

Background  

Specific binding of proteins to DNA is one of the most common ways gene expression is controlled. Although general rules for the DNA-protein recognition can be derived, the ambiguous and complex nature of this mechanism precludes a simple recognition code, therefore the prediction of DNA target sequences is not straightforward. DNA-protein interactions can be studied using computational methods which can complement the current experimental methods and offer some advantages. In the present work we use physical effective potentials to evaluate the DNA-protein binding affinities for the λ repressor-DNA complex for which structural and thermodynamic experimental data are available.     

An hypothesis of freese and the DNA-protein coding problem

Freese’s Hypothesis states that a single specific alteration in the sequence of nucleotides of an information-bearing DNA molecule results in a specific mutational effect. Within the framework of the DNA-protein coding problem developed elsewhere, and assuming the quasi-ergodicity of the general coding process, it is shown that Freese’s Hypothesis allows us to derive expressions for the length of the smallest mutable DNA molecule and to obtain a bound for the maximal number of allelic molecules of fixed length. To illustrate these ideas, calculations are carried out on appropriate data from bacternophage and man, and the results are shown to differ by a factor of 10 (modulo the rather crude approximations used). It is further shown that, if ρ(N) and ϱ(N) are respectively the number of information-bearing words of lengthN in a given code and the number of words of lengthN, then the number lim ρ(N)/ϱ(N) depends sensitively on the parameter ∈ which specifiesN→∞ the given code. The implications of this result for the spontaneous aggregation of a sufficient number of information-bearing words to characterize an organism are discussed. This research was supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under Contract No. AF 49(638)-917.     

An Analysis of the Metabolic Theory of the Origin of the Genetic Code

A computer program was used to test Wong's coevolution theory of the genetic code. The codon correlations between the codons of biosynthetically related amino acids in the universal genetic code and in randomly generated genetic codes were compared. It was determined that many codon correlations are also present within random genetic codes and that among the random codes there are always several which have many more correlations than that found in the universal code. Although the number of correlations depends on the choice of biosynthetically related amino acids, the probability of choosing a random genetic code with the same or greater number of codon correlations as the universal genetic code was found to vary from 0.1% to 34% (with respect to a fairly complete listing of related amino acids). Thus, Wong's theory that the genetic code arose by coevolution with the biosynthetic pathways of amino acids, based on codon correlations between biosynthetically related amino acids, is statistical in nature. Received: 8 August 1996 / Accepted: 26 December 1996     

An information theory of the genetic code

An information theory of the genetic code is given, which deals with the process by which template codes (nucleotides or codons) choose substrate codes (nucleotides or anti-codons) in accordance with the base-pairing rules in the chain elongation phase of polynucleotide or polypeptide synthesis. A definite period of recognition time (τ) required for a template code to discriminate a substrate code is proposed, and an experimental method for determining the time is suggested. A substrate word is defined to be the sequence of substrate codes which have appeared at a recognition site in turn before a substrate code complementary to a template code first appears, and the mean length of substrate words (F) is derived from the mole fractions of template codes and substrate codes. The chain elongation rate is greatest when the mole fractions of template codes is proportional to the square of those of substrate codes to minimize the mean recognition time per word (Fτ). The uncertainty of a template (G) and the uncertainty of a medium (M) respectively are derived from the minimum of the function F. The amount of genetic information contained in a template is measured by the function G. The unit of the amount of genetic information is termed “cit”. The function M, the ratio of the number of all binary collisions to the number of homogeneous binary collisions in a mixture of different molecules, may be the new other “entropy” which represents informational properties of the mixture not represented by thermodynamic entropy of mixing. Both functions (G and M) have maxima when all random variables are equal and they are multiplicative in nature in contrast to entropy which is additive. The multiplicativity of the function G may contribute to the enormous informational capacity of genes.     

遗传密码格式的组合编码数分析

用N个密码子对m个编码对象进行编码的编码格式是m元N维空间中的一个顶点。64个密码子对20种氨基酸和终止密码子进行编码格式的组合编码数是一个十分巨大的数字。对多元高维编码空间的拓扑特性进行了分析和研究 ,并由此推导出m -N空间的特性三角的排列方式以及给出特性三角公式的数学证明。指出 ,目前的遗传密码的编码格式是21元64维编码空间的一个顶点。应用组合数学分析的方法 ,计算了遗传密码格式的最大组合编码数CM =4.19×1084 ,基因组遗传密码的组合编码数CG =1.13×1080 以及线粒体遗传密码的组合编码数CT =1.38×1079 等。分析结果表明 ,遗传密码的指定是一个小概率事件 ,可能来源于λ简并后的偶数三联密码配对的组合编码的对称破缺     

Recent Progress in First Principles O(N) Methods

Abstract

We describe recent progress in developing practical first principles methods for which the computer effort is proportional to the number of atoms: linear scaling or O(N) methods. It is shown that the locality property of the density matrix gives a general framework for constructing such methods. We then outline some of the main technical problems which must be solved in order to develop a practical O(N) method based on density functional theory and the pseudopotential method. Recent progress in solving these problems is presented, and we show that the spatial cut-off distances needed to achieve good accuracy are small enough to make the calculations feasible. Parallel implementation of the O(N) methods in the CONQUEST code is outlined, and it is shown that the code exhibits excellent linear-scaling behaviour on test systems of several thousand atoms. It is pointed out that the most important remaining problem concerns the optimal strategy for seeking the ground state. It is argued that there are three different mechanisms of ill-conditioning which cause present search methods to be inefficient, and some partial solutions are suggested.     

The Coevolution of Genes and Genetic Codes: Crick’s Frozen Accident Revisited

The standard genetic code is the nearly universal system for the translation of genes into proteins. The code exhibits two salient structural characteristics: it possesses a distinct organization that makes it extremely robust to errors in replication and translation, and it is highly redundant. The origin of these properties has intrigued researchers since the code was first discovered. One suggestion, which is the subject of this review, is that the code’s organization is the outcome of the coevolution of genes and genetic codes. In 1968, Francis Crick explored the possible implications of coevolution at different stages of code evolution. Although he argues that coevolution was likely to influence the evolution of the code, he concludes that it falls short of explaining the organization of the code we see today. The recent application of mathematical modeling to study the effects of errors on the course of coevolution, suggests a different conclusion. It shows that coevolution readily generates genetic codes that are highly redundant and similar in their error-correcting organization to the standard code. We review this recent work and suggest that further affirmation of the role of coevolution can be attained by investigating the extent to which the outcome of coevolution is robust to other influences that were present during the evolution of the code. Electronic Supplementary Material Electronic Supplementary material is available for this article at and accessible for authorised users. [Reviewing Editor: Dr. Martin Kreitman]     

The Duplexing of the Genetic Code and Sequence-Dependent DNA Geometry

It is well known that sequences of bases in DNA are translated into sequences of amino acids in cells via the genetic code. More recently, it has been discovered that the sequence of DNA bases also influences the geometry and deformability of the DNA. These two correspondences represent a naturally arising example of duplexed codes, providing two different ways of interpreting the same DNA sequence. This paper will set up the notation and basic results necessary to mathematically investigate the relationship between these two natural DNA codes. It then undertakes two very different such investigations: one graphical approach based only on expected values and another analytic approach incorporating the deformability of the DNA molecule and approximating the mutual information of the two codes. Special emphasis is paid to whether there is evidence that pressure to maximize the duplexing efficiency influenced the evolution of the genetic code. Disappointingly, the results fail to support the hypothesis that the genetic code was influenced in this way. In fact, applying both methods to samples of realistic alternative genetic codes shows that the duplexing of the genetic code found in nature is just slightly less efficient than average. The implications of this negative result are considered in the final section of the paper.     

Circular codes revisited: a statistical approach

In 1996 Arquès and Michel [1996. A complementary circular code in the protein coding genes. J. Theor. Biol. 182, 45-58] discovered the existence of a common circular code in eukaryote and prokaryote genomes. Since then, circular code theory has provoked great interest and underwent a rapid development. In this paper we discuss some theoretical issues related to the synchronization properties of coding sequences and circular codes with particular emphasis on the problem of retrieval and maintenance of the reading frame. Motivated by the theoretical discussion, we adopt a rigorous statistical approach in order to try to answer different questions. First, we investigate the covering capability of the whole class of 216 self-complementary, C³ maximal codes with respect to a large set of coding sequences. The results indicate that, on average, the code proposed by Arquès and Michel has the best covering capability but, still, there exists a great variability among sequences. Second, we focus on such code and explore the role played by the proportion of the bases by means of a hierarchy of permutation tests. The results show the existence of a sort of optimization mechanism such that coding sequences are tailored as to maximize or minimize the coverage of circular codes on specific reading frames. Such optimization clearly relates the function of circular codes with reading frame synchronization.     

Codon Usage Bias and Mutation Constraints Reduce the Level of ErrorMinimization of the Genetic Code

Studies on the origin of the genetic code compare measures of the degree of error minimization of the standard code with measures produced by random variant codes but do not take into account codon usage, which was probably highly biased during the origin of the code. Codon usage bias could play an important role in the minimization of the chemical distances between amino acids because the importance of errors depends also on the frequency of the different codons. Here I show that when codon usage is taken into account, the degree of error minimization of the standard code may be dramatically reduced, and shifting to alternative codes often increases the degree of error minimization. This is especially true with a high CG content, which was probably the case during the origin of the code. I also show that the frequency of codes that perform better than the standard code, in terms of relative efficiency, is much higher in the neighborhood of the standard code itself, even when not considering codon usage bias; therefore alternative codes that differ only slightly from the standard code are more likely to evolve than some previous analyses suggested. My conclusions are that the standard genetic code is far from being an optimum with respect to error minimization and must have arisen for reasons other than error minimization.[Reviewing Editor: Martin Kreitman]     

The coexistence of the nucleosome positioning code with the genetic code on eukaryotic genomes

It is known that there are several codes residing simultaneously on the DNA double helix. The two best-characterized codes are the genetic code—the code for protein production, and the code for DNA packaging into nucleosomes. Since these codes have to coexist simultaneously on the same DNA region, both must be degenerate to allow this coexistence. A-tracts are homopolymeric stretches of several adjacent deoxyadenosines on one strand of the double helix, having unusual structural properties, which were shown to exclude nucleosomes and as such are instrumental in setting the translational positioning of DNA within nucleosomes. We observe, cross-kingdoms, a strong codon bias toward the avoidance of long A-tracts in exon regions, which enables the formation of high density of nucleosomes in these regions. Moreover, long A-tract avoidance is restricted exclusively to nucleosome-occupied exon regions. We show that this bias in codon usage is sufficient for enabling DNA organization within nucleosomes without constraints on the actual code for proteins. Thus, there is inter-dependency of the two major codes within DNA to allow their coexistence. Furthermore, we show that modulation of A-tract occurrences in exon versus non-exon regions may result in a unique alternation of the diameter of the ‘30-nm’ fiber model.     

On the possibilities of error-detecting codes in nucleic acid molecules

Error-detecting codes have been known to mathematicians and to electrical engineers for over ten years. In general, such codes utilize an additional orparity bit for purposes of detecting errors by the addition of all positive binary bits or “1’s” occurring in any code word. However, since the process of addition is required for such code detection, it is not surprising that these codes have not been applied to the nucleic acid molecule. In 1962, P. I. Hershberg (Trans. I.R.E., CS-10, 280–4, 1962) outlined a categorical constraint which permitted the realization of a class of error-detecting codes which did not require parity bits. This class of codes is applied to the nucleic acid molecule in the present paper.     

On combinatorial DNA word design.

We consider the problem of designing DNA codes, namely sets of equi-length words over the alphabet [A, C, G, T] that satisfy certain combinatorial constraints. This problem is motivated by the task of reliably storing and retrieving information in synthetic DNA strands for use in DNA computing or as molecular bar codes in chemical libraries. The primary constraints that we consider, defined with respect to a parameter d, are as follows: for every pair of words w, x in a code, there are at least d mismatches between w and x if w not equal x and also between the reverse of w and the Watson-Crick complement of x. Extending classical results from coding theory, we present several upper and lower bounds on the maximum size of such DNA codes and give methods for constructing such codes. An additional constraint that is relevant to the design of DNA codes is that the free energies and enthalpies of the code words, and thus the melting temperatures, be similar. We describe dynamic programming algorithms that can (a) calculate the total number of words of length n whose free energy value, as approximated by a formula of Breslauer et al. (1986) falls in a given range, and (b) output a random such word. These algorithms are intended for use in heuristic algorithms for constructing DNA codes.     

Free energy gap and statistical thermodynamic fidelity of DNA codes.

DNA nanotechnology often requires collections of oligonucleotides called "DNA free energy gap codes" that do not produce erroneous crosshybridizations in a competitive muliplexing environment. This paper addresses the question of how to design these codes to accomplish a desired amount of work within an acceptable error rate. Using a statistical thermodynamic and probabilistic model of DNA code fidelity and mathematical random coding theory methods, theoretical lower bounds on the size of DNA codes are given. More importantly, DNA code design parameters (e.g., strand number, strand length and sequence composition) needed to achieve experimental goals are identified.     

Load minimization of the genetic code: history does not explain the pattern

The average effect of errors acting on a genetic code (the change in amino-acid meaning resulting from point mutation and mistranslation) may be quantified as its ''load''. The natural genetic code shows a clear property of minimizing this load when compared against randomly generated variant codes. Two hypotheses may be considered to explain this property. First, it is possible that the natural code is the result of selection to minimize this load. Second, it is possible that the property is an historical artefact. It has previously been reported that amino acids that have been assigned to codons starting with the same base come from the same biosynthetic pathway. This probably reflects the manner in which the code evolved from a simpler code, and says more about the physicochemical mechanisms of code assembly than about selection. The apparent load minimization of the code may therefore follow as a consequence of the fact that the code could not have evolved any other way than to allow biochemically related amino acids to have related codons. Here then, we ask whether this ''historical'' force alone can explain the efficiency of the natural code in minimizing the effects of error. We therefore compare the error-minimizing ability of the natural code with that of alternative codes which, rather than being a random selection, are restricted such that amino acids from the same biochemical pathway all share the same first base. We find that although on average the restricted set of codes show a slightly higher efficiency than random ones, the real code remains extremely efficient relative to this subset P = 0.0003. This indicates that for the most part historical features do not explain the load- minimization property of the natural code. The importance of selection is further supported by the finding that the natural code''s efficiency improves relative to that of historically related codes after allowance is made for realistic mutational and mistranslational biases. Once mistranslational biases have been considered, fewer than four per 100,000 alternative codes are better than the natural code.     

Immune networks modeled by replicator equations

In order to evaluate the role of idiotypic networks in the operation of the immune system a number of mathematical models have been formulated. Here we examine a class of B-cell models in which cell proliferation is governed by a non-negative, unimodal, symmetric response function f(h), where the field h summarizes the effect of the network on a single clone. We show that by transforming into relative concentrations, the B-cell network equations can be brought into a form that closely resembles the replicator equation. We then show that when the total number of clones in a network is conserved, the dynamics of the network can be represented by the dynamics of a replicator equation. The number of equilibria and their stability are then characterized using methods developed for the study of second-order replicator equations. Analogies with standard Lotka-Volterra equations are also indicated. A particularly interesting result of our analysis is the fact that even though the immune network equations are not second-order, the number and stability of their equilibria can be obtained by a superposition of second-order replicator systems. As a consequence, the problem of finding all of the equilibrium points of the nonlinear network equations can be reduced to solving linear equations.     

Selective advantages created by codon ambiguity allowed for the evolution of an alternative genetic code in Candida spp

Several species of the genus Candida decode the standard leucine CUG codon as serine. This and other deviations from the standard genetic code in both nuclear and mitochondrial genomes invalidate the notion that the genetic code is frozen and universal and prompt the questions ‘why alternative genetic codes evolved and, more importantly, how can an organism survive a genetic code change?’ To address these two questions, we have attempted to reconstruct the early stages of Candida albicans CUG reassignment in the closely related yeast Saccharomyces cerevisiae. These studies suggest that this genetic code change was driven by selection using a molecular mechanism that requires CUG ambiguity. Such codon ambiguity induced a significant decrease in fitness, indicating that CUG reassignment can only be selected if it introduces an evolutionary edge to counteract the negative impact of ambiguity. We have shown that CUG ambiguity induces the expression of a novel set of stress proteins and triggers the general stress response, which, in turn, creates a competitive edge under stress conditions. In addition, CUG ambiguity in S. cerevisiae induces the expression of a number of novel phenotypes that mimic the natural resistance to stress characteristic of C. albicans. The identification of an evolutionary advantage created by CUG ambiguity is the first experimental evidence for a genetic code change driven by selection and suggests a novel role for codon reassignment in the adaptation to new ecological niches.