An information theoretic approach to macromolecular modeling: II. Force fields

Information-theoretic approach in allometric scaling relations of DNA and proteins

Allometric scaling relations can be observed in between molecular parameters. Hence, we looked for presence of such relation among sizes (i.e., lengths) of proteins and genes. Protein lengths exist in the literature as the number of amino acids. They can also be derived from the mRNA lengths. Here, we looked for allometric scaling relation by using such data and simultaneously, the data was compared with the sizes of genes and proteins that were obtained from our modified information-theoretic approach. Results implied presence of scaling relation in the calculated results. This was expected due to the implemented modification in the information-theoretic calculation. Relation in the literature-based data was lacking high goodness of fit value. It could be due to physical factors and selective pressures, which ended up in deviations of the literature-sourced values from those in the model. Genome size is correlated with cell size. Intracellular volume, which is related to the DNA size, would require certain number of proteins, the sizes of which can therefore be correlated with the protein sizes. Cell sizes, genome sizes, and average protein and gene sizes, along with the number of proteins, namely the expression levels of the genes, are the physical factors, and the molecular factors influence those physical factors. The selective pressures on those can act through the connection between those physical factors and limit the dynamic ranges. Biological measures could be prone to such forces and are likely to deviate from expected models, regardless of the validity of assumptions, unless those are also implemented in the models. Yet, present discrepancies could be pointing at the need for model improvement, data imperfection, invalid assumptions, etc. Still, current work highlights possible use of information-theoretic approach in allometric scaling relations' studies.

Equalizing the information amounts of protein and mRNA by information theory

Based on the Shannon's information communication theory, information amount of the entire length of a polymeric macromolecule can be calculated in bits through adding the entropies of each building block. Proteins, DNA and RNA are such macromolecules. When only the building blocks' variation is considered as the source of entropy, there is seemingly lower information in case of the protein if this approach is applied directly on a protein of specific size and the coding sequence size of the mRNA corresponding to the particular length of the protein. This decrease in the information amount seems contradictory but this apparent conflict is resolved by considering the conformational variations in proteins as a new variable in the calculation and balancing the approximated entropy of the coding part of the mRNA and the protein. Probabilities can change therefore we also assigned hypothetical probabilities to the conformational states, which represent the uneven distribution as the time spent in one conformation, providing the probability of the presence in either or one of the possible conformations. Results that are obtained by using hypothetical probabilities are in line with the experimental values of variations in the conformational-state of protein populations. This equalization approach has further biological relevance that it compensates for the degeneracy in the codon usage during protein translation and it leads to the conclusion that the alphabet size for the protein is rather optimal for the proper protein functioning within the thermodynamic milieu of the cell. The findings were also discussed in relation to the codon bias and have implications in relation to the codon evolution concept. Eventually, this work brings the fields of protein structural studies and molecular protein translation processes together with a novel approach.

Distance-dependent classification of amino acids by information theory

Reduced amino acid alphabets are useful to understand molecular evolution as they reveal basal, shared properties of amino acids, which the structures and functions of proteins rely on. Several previous studies derived such reduced alphabets and linked them to the origin of life and biotechnological applications. However, all this previous work presupposes that only direct contacts of amino acids in native protein structures are relevant. We show in this work, using information-theoretical measures, that an appropriate alphabet reduction scheme is in fact a function of the maximum distance amino acids interact at. Although for small distances our results agree with previous ones, we show how long-range interactions change the overall picture and prompt for a revised understanding of the protein design process.

Statistical mechanics analysis of sparse data

Inferential structure determination uses Bayesian theory to combine experimental data with prior structural knowledge into a posterior probability distribution over protein conformational space. The posterior distribution encodes everything one can say objectively about the native structure in the light of the available data and additional prior assumptions and can be searched for structural representatives. Here an analogy is drawn between the posterior distribution and the canonical ensemble of statistical physics. A statistical mechanics analysis assesses the complexity of a structure calculation globally in terms of ensemble properties. Analogs of the free energy and density of states are introduced; partition functions evaluate the consistency of prior assumptions with data. Critical behavior is observed with dwindling restraint density, which impairs structure determination with too sparse data. However, prior distributions with improved realism ameliorate the situation by lowering the critical number of observations. An in-depth analysis of various experimentally accessible structural parameters and force field terms will facilitate a statistical approach to protein structure determination with sparse data that avoids bias as much as possible.

Evolution of biological sequences implies an extreme value distribution of type I for both global and local pairwise alignment scores

Background

Confidence in pairwise alignments of biological sequences, obtained by various methods such as Blast or Smith-Waterman, is critical for automatic analyses of genomic data. Two statistical models have been proposed. In the asymptotic limit of long sequences, the Karlin-Altschul model is based on the computation of a P-value, assuming that the number of high scoring matching regions above a threshold is Poisson distributed. Alternatively, the Lipman-Pearson model is based on the computation of a Z-value from a random score distribution obtained by a Monte-Carlo simulation. Z-values allow the deduction of an upper bound of the P-value (1/Z-value²) following the TULIP theorem. Simulations of Z-value distribution is known to fit with a Gumbel law. This remarkable property was not demonstrated and had no obvious biological support.

Results

We built a model of evolution of sequences based on aging, as meant in Reliability Theory, using the fact that the amount of information shared between an initial sequence and the sequences in its lineage (i.e., mutual information in Information Theory) is a decreasing function of time. This quantity is simply measured by a sequence alignment score. In systems aging, the failure rate is related to the systems longevity. The system can be a machine with structured components, or a living entity or population. "Reliability" refers to the ability to operate properly according to a standard. Here, the "reliability" of a sequence refers to the ability to conserve a sufficient functional level at the folded and maturated protein level (positive selection pressure). Homologous sequences were considered as systems 1) having a high redundancy of information reflected by the magnitude of their alignment scores, 2) which components are the amino acids that can independently be damaged by random DNA mutations. From these assumptions, we deduced that information shared at each amino acid position evolved with a constant rate, corresponding to the information hazard rate, and that pairwise sequence alignment scores should follow a Gumbel distribution, which parameters could find some theoretical rationale. In particular, one parameter corresponds to the information hazard rate.

Conclusion

Extreme value distribution of alignment scores, assessed from high scoring segments pairs following the Karlin-Altschul model, can also be deduced from the Reliability Theory applied to molecular sequences. It reflects the redundancy of information between homologous sequences, under functional conservative pressure. This model also provides a link between concepts of biological sequence analysis and of systems biology.

An information theoretic approach to macromolecular modeling: I. Sequence alignments

We are interested in applying the principles of information theory to structural biology calculations. In this article, we explore the information content of an important computational procedure: sequence alignment. Using a reference state developed from exhaustive sequences, we measure alignment statistics and evaluate gap penalties based on first-principle considerations and gap distributions. We show that there are different gap penalties for different alphabet sizes and that the gap penalties can depend on the length of the sequences being aligned. In a companion article, we examine the information content of molecular force fields.