Selective constraints on amino acids estimated by a mechanistic codon substitution model with multiple nucleotide changes

Background

Empirical substitution matrices represent the average tendencies of substitutions over various protein families by sacrificing gene-level resolution. We develop a codon-based model, in which mutational tendencies of codon, a genetic code, and the strength of selective constraints against amino acid replacements can be tailored to a given gene. First, selective constraints averaged over proteins are estimated by maximizing the likelihood of each 1-PAM matrix of empirical amino acid (JTT, WAG, and LG) and codon (KHG) substitution matrices. Then, selective constraints specific to given proteins are approximated as a linear function of those estimated from the empirical substitution matrices.

Results

Akaike information criterion (AIC) values indicate that a model allowing multiple nucleotide changes fits the empirical substitution matrices significantly better. Also, the ML estimates of transition-transversion bias obtained from these empirical matrices are not so large as previously estimated. The selective constraints are characteristic of proteins rather than species. However, their relative strengths among amino acid pairs can be approximated not to depend very much on protein families but amino acid pairs, because the present model, in which selective constraints are approximated to be a linear function of those estimated from the JTT/WAG/LG/KHG matrices, can provide a good fit to other empirical substitution matrices including cpREV for chloroplast proteins and mtREV for vertebrate mitochondrial proteins.

Conclusions/Significance

The present codon-based model with the ML estimates of selective constraints and with adjustable mutation rates of nucleotide would be useful as a simple substitution model in ML and Bayesian inferences of molecular phylogenetic trees, and enables us to obtain biologically meaningful information at both nucleotide and amino acid levels from codon and protein sequences.     

Modeling Markovian biological systems via optimization operators

A general, computer-oriented method permitting to derive Markovian models with required (desired) properties is suggested and illustrated by examples. The method is based on the concept of a transition matrices generating (tmg) optimization operator, which is defined as a pair involving a (linear) transformation T and the associate optimization problem L _T . When the latter one is solved a set of transition matrices with required properties (ergodicity, regularity etc.) is get by starting from a sequence of probability vectors {P _k } which expresses the test data. Since the corresponding measurements are inevitably subjected to errors, it is not required that {P _k } be reached in the step-wise evolution of the process. Instead, it is required to minimize the so-called v-distance with respect to the probability vectors {P _k }. The optimization is performed by taking into account some constraints expressing the prior-known properties of the chain. This enables to solve the following problem: Given a sequence of (measured) probability vectors {P _k }, find a sequence of transition matrices {P _k } leading to the smallest v-distance with respect to {P _k } subject to given constraints. Some fundamental properties of the resulting Markov chains are emphasized, which are useful in modeling concrete biological systems. Thus, more realistic Markovian models are obtained starting from test data, as compared with the methods using conventional means.     

On the eigenvalue distribution of genetic and phenotypic dispersion matrices: Evidence for a nonrandom organization of quantitative character variation

A quantitative genetic model of random pleiotropy is introduced as reference model for detecting the kind and degree of organization in quantitative genetic variation. In this model the genetic dispersion matrix takes the form of G = BB ^T, where B is a general, real, Gaussian random matrix. The eigenvalue density of the corresponding ensemble of random matrices (_G) is considered. The first two moments are derived for variance-covariance matrices G as well as for correlation matrices R, and an approximate expression of the density function is given. The eigenvalue distribution of all empirical correlation matrices deviates from that of a random pleiotropy model by a very large leading eigenvalue associated with a size factor. However the frequency-distribution of the remaining eigenvalues shows only minor deviations in mammalian skeletal data. A prevalence of intermediate eigenvalues in insect data may be caused by the inclusion of many functionally unrelated characters. Hence two kinds of deviations from random organization have been found: a mammal like and an insect like organization. It is concluded that functionally related characters are on the average more tightly correlated than by chance (= mammal like organization), while functionally unrelated characters appear to be less correlated than by random pleiotropy (insect like organization).     

Cross entropy minimization in uninvadable states of complex populations

Selection is often. viewed as a process that maximizes the average fitness of a population. However, there are often constraints even on the phenotypic level which may prevent fitness optimization. Consequently, in evolutionary game theory, models of frequency dependent selection are investigated, which focus on equilibrium states that are characterized by stability (or uninvadability) rather than by optimality. The aim of this article is to show that nevertheless there is a biologically meaningful quantity, namely cross (fitness) entropy, which is optimized during the course of evolution: a dynamical model adapted to evolutionary games is presented which has the property that relative entropy decreases monotonically, if the state of a (complex) population is close to an uninvadable state. This result may be interpreted as if evolution has an order stabilizing effect.     

What Is the Optimal Value of the g-Ratio for Myelinated Fibers in the Rat CNS? A Theoretical Approach

Background

The biological process underlying axonal myelination is complex and often prone to injury and disease. The ratio of the inner axonal diameter to the total outer diameter or g-ratio is widely utilized as a functional and structural index of optimal axonal myelination. Based on the speed of fiber conduction, Rushton was the first to derive a theoretical estimate of the optimal g-ratio of 0.6 [1]. This theoretical limit nicely explains the experimental data for myelinated axons obtained for some peripheral fibers but appears significantly lower than that found for CNS fibers. This is, however, hardly surprising given that in the CNS, axonal myelination must achieve multiple goals including reducing conduction delays, promoting conduction fidelity, lowering energy costs, and saving space.

Methodology/Principal Findings

In this study we explore the notion that a balanced set-point can be achieved at a functional level as the micro-structure of individual axons becomes optimized, particularly for the central system where axons tend to be smaller and their myelin sheath thinner. We used an intuitive yet novel theoretical approach based on the fundamental biophysical properties describing axonal structure and function to show that an optimal g-ratio can be defined for the central nervous system (≈0.77). Furthermore, by reducing the influence of volume constraints on structural design by about 40%, this approach can also predict the g-ratio observed in some peripheral fibers (≈0.6).

Conclusions/Significance

These results support the notion of optimization theory in nervous system design and construction and may also help explain why the central and peripheral systems have evolved different g-ratios as a result of volume constraints.     

A New Approach for Inversion of Large Random Matrices in Massive MIMO Systems

We report a novel approach for inversion of large random matrices in massive Multiple-Input Multiple Output (MIMO) systems. It is based on the concept of inverse vectors in which an inverse vector is defined for each column of the principal matrix. Such an inverse vector has to satisfy two constraints. Firstly, it has to be in the null-space of all the remaining columns. We call it the null-space problem. Secondly, it has to form a projection of value equal to one in the direction of selected column. We term it as the normalization problem. The process essentially decomposes the inversion problem and distributes it over columns. Each column can be thought of as a node in the network or a particle in a swarm seeking its own solution, the inverse vector, which lightens the computational load on it. Another benefit of this approach is its applicability to all three cases pertaining to a linear system: the fully-determined, the over-determined, and the under-determined case. It eliminates the need of forming the generalized inverse for the last two cases by providing a new way to solve the least squares problem and the Moore and Penrose''s pseudoinverse problem. The approach makes no assumption regarding the size, structure or sparsity of the matrix. This makes it fully applicable to much in vogue large random matrices arising in massive MIMO systems. Also, the null-space problem opens the door for a plethora of methods available in literature for null-space computation to enter the realm of matrix inversion. There is even a flexibility of finding an exact or approximate inverse depending on the null-space method employed. We employ the Householder''s null-space method for exact solution and present a complete exposition of the new approach. A detailed comparison with well-established matrix inversion methods in literature is also given.     

The structure of assembled communities

A set of rules is formulated which expresses the random assembly of ecological communities by sequentially arriving species, subject to energetic constraints. It is shown that these “assembled communities” provide a reasonable model for 35 out of the 40 real food webs recently compiled by Briand (1981), on the basis of the statistics: species richness, proportion of herbivores, ratio of prey to predators, proportion of dietary specialists, number of trophic links, number of potential competitive links, connectance, and average maximal food chain length. However, the observed frequency of intervality among Briand's food webs deviates significantly from the value expected on the basis of random sampling from the mathematical universe of assembled webs. Finally, there are indications in this work that the process of community genesis may be fundamentally different in fluctuating and in constant environments.     

Sampled-data vectors: A new means for biological system identification

The concept of the sampled-data vector is defined. It is shown that, single-valued as well as double-valued nonlinear biological models, however complicated, may be expressed by means of the sampled-data nonlinearity vector. By using this vector-valued model, a direct correlation may be established between the non-linearity and its equivalent gain. This is obtained by means of linear transformations using numerically known (invariant) matrices, i.e. matrices which are independent of the nonlinear model. Likewise, for linear frequency-dependent biological models, the stepresponse sampled-data vector as well as the real frequency sampled-data vector are defined. By means of these vectors, a direct correlation may be established between the time and frequency domains. This is a linear transformation too, using invariant matrices. The matrices permitting the inverse transformation (i.e. the identification) are given and, it is shown that, these transformations may be associated with a linear for quadratic programming procedure in order to get linear as well as nonlinear biological models, subject to some constraints. This leads to the conditional identification concept, which allows the use of our prior knowledge about the biological system in the identification procedure. Two examples concerning the identification procedure (a linear and a nonlinear model) are given and, it is shown that since invariant matrix operators in connection with customary optimization algorithms are used, the identification procedure is a computer-oriented one. Thus, by starting directly from the test-data and by observing some accuracy or stability constraints, various biological models may be obtained in a simple and general manner.

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Zusammenfassung Der Begriff des Abtastvektors wird definiert und gezeigt, daß sowohl ein- als auch mehrdeutige, beliebig komplizierte nichtlineare biologische Modelle mit Hilfe des Abtastvektors der Nichtlinearität ausgedrückt werden können. Zur Anwendung dieses vektoriellen Modelles wird eine direkte Beziehung zwischen der Nichtlinearität und ihrer äquivalenten Verstärkung aufgestellt. Dies geschieht mittels einer linearen Transformation, unter Anwendung zahlenmäßig bekannter, folglich vom nichtlinearen Modell unabhängiger Matrizen. Ebenso werden für lineare, frequenzabhängige biologische Modelle der Übergangs- und Realtilvektor definiert. Mit Hilfe der letzteren kann man—durch eine lineare Transformation—eine direkte Beziehung zwischen dem Zeit- und Frequenzgebiet herstellen, indem man ebenfalls invariante Matrizen anwendet. Es werden die umgekehrte Transformation ermöglichende Matrizen gegeben und man sieht, daß diese Transformationen mit einem linearen (oder quadratischen) Programmierungsverfahren verknüpft werden können, um sowohl lineare als auch nichtlineare, verschiedenen Einschränkungen unterworfene biologische Modelle zu erhalten. Dies führt zu dem Konzept der bedingten Identifikation, welche die Anwendung vorheriger Kenntnisse über das biologische System im Identifikationsverfahren ermöglicht. Zwei dieses Verfahren betreffende Beispiele (ein lineares und ein nichtlineares Modell) werden gegeben und es wird gezeigt, daß die Anwendung invarianter Matrixoperatoren zu einem für Computer geeigneten Identifikationsverfahren führt. Wenn man also direkt von den Testdaten ausgeht und bestimmte, die Genauigkeit oder die Stabilität des Modelles betreffende Einschränkungen berücksichtigt, kann man verschiedenartige nichtlineare und lineare biologische Modelle auf einfache und allgemeingültige Art erzielen.

     

Modeling of chromosome intermingling by partially overlapping uniform random polygons

During the early phase of the cell cycle the eukaryotic genome is organized into chromosome territories. The geometry of the interface between any two chromosomes remains a matter of debate and may have important functional consequences. The Interchromosomal Network model (introduced by Branco and Pombo) proposes that territories intermingle along their periphery. In order to partially quantify this concept we here investigate the probability that two chromosomes form an unsplittable link. We use the uniform random polygon as a crude model for chromosome territories and we model the interchromosomal network as the common spatial region of two overlapping uniform random polygons. This simple model allows us to derive some rigorous mathematical results as well as to perform computer simulations easily. We find that the probability that one uniform random polygon of length n that partially overlaps a fixed polygon is bounded below by \({1-O(\frac{1}{\sqrt n})}\). We use numerical simulations to estimate the dependence of the linking probability of two uniform random polygons (of lengths n and m, respectively) on the amount of overlapping. The degree of overlapping is parametrized by a parameter \({\epsilon\in [0,1]}\) such that \({\epsilon=0}\) indicates no overlapping and \({\epsilon=1}\) indicates total overlapping. We propose that this dependence relation may be modeled as \({f(\varepsilon, m, n) =1-{\frac{a(\epsilon)}{b(\epsilon)\sqrt{mn}+c(\epsilon)}}}\). Numerical evidence shows that this model works well when \({\epsilon}\) is relatively large \({(\varepsilon \ge 0.5)}\). We then use these results to model the data published by Branco and Pombo and observe that for the amount of overlapping observed experimentally the URPs have a non-zero probability of forming an unsplittable link.     

A vector-sum process produces curved aiming paths under rotated visual-motor mappings

Under a 90° rotation of motor space relative to visual space, human two-dimensional aiming movements frequently take the form of smooth arcs such as spirals and semi-circles. A time-independent differential equation explains this tendency in terms of a rotation-induced vector field made up, at each point in the two-dimensional space, of two input vectors. One vector represents a visual error signal and the other represents a motor error signal. A trajectory's instantaneous direction of movement at each point can be described as the resultant of the two vectors. This mathematical formulation incorporates plausible visual-motor mechanisms and, when expressed in polar coordinates, leads to a new method for analyzing the spatial properties of movements (i.e., movement paths). Plots of the angle between the resultant and the target vector () against distance from the target (r, in the polar representation) summarize the arc-shaped movement paths as a simple relation that can be analyzed statistically with respect to properties such as monotonicity. The polar representation is a plausible representation of visually-guided movements, with the visual error vector functioning as an objective function relative to which behavior is optimized. We extend the model and ther, movement path analysis to non-90° rotations, and we find that the model predicts an observed qualitative shift in behavior for rotations greater than 90°. It also predicts qualitatively different path shapes observed under visual-motor reflections.This work was performed while the first author was under the support of Grant IST-8511589 from the National Science Foundation and Grant NCC2-307 from the National Aeronautics and Space Administration     

Island-sharing by archipelago species

Summary Diamond (1975) formulated assembly rules for avian species on islands in an archipelago, which made a successful colonisation depend essentially on which other species were present. Critically examining these rules, Connor and Simberloff (1979) maintained that, in the Vanuatu (New Hebrides) archipelago, the field data on species distribution was quite compatible with a null hypothesis, in which species colonise at random with no species interaction. Their work was in turn criticised (Diamond and Gilpin (1982), Gilpin and Diamond (1982)) and a vigorous controversy has ensued.Here we contribute a method in which a simple but hitherto neglected statistic is used as a probe: the number of islands shared by a pair of species, with its first and second moments. The matrix of these sharing values is given as a simple product of the incidence matrix, and its properties are examined — first, for the field data, and then in the random ensemble used by Connor and Simberloff (1979). It is shown that their constraints hold constant the mean number shared, so that any fall in the number that one pair of species share, due to their excluding each other, must imply a rise in the number shared by some other species pair-i.e., an aggregation.Turning to the second moment of the numbers shared, it is shown that its value in the Vanuatu field data exceeds the largest value to be found in a sample of 1000 matrices, these latter being constructed so that they obey the Connor and Simberloff constraints but are otherwise random. This indicates that exclusion and/or aggregation effects are present in the actual distribution of species, which are not catered for by the null hypothesis.The observed distribution thus emerges as much more exceptional than found by Connor and Simberloff (1979), and even by Diamond and Gilpin (1982), when examining the same ensemble. The reason for this disagreement are sought, and some cautions are offered, supported by numerical evidence, concerning the use of the chi-square test when the data points involved are mutually dependent.     

Moral and nonmoral innate constraints

Charles J. Lumsden and E.O. Wilson, in their writings together and individually, have proposed that human behaviors, whether moral or nonmoral, are governed by innate constraints (which they have termed epigenetic rules). I propose that if a genetic component of moral behavior is to be discovered, some sorting out of specifically moral from nonmoral innate constraints will be necessary. That some specifically moral innate constraits exist is evidenced by virtuous behaviors exhibited in nonhuman mammals, whose behavior is usually granted to be importantly governed by genetic factors. Propensities for such virtuous behaviors may have been passed to humans as highly conserved mammalian genes and continue to influence us. I propose that these constitute at least a rudimentary morality and may account in part for the moral intuitions. But other innate constraints which are nonmoral in nature interact with the specifically moral innate constraints and with culture to yield human moral decisions and actions. Any model which aims to identify the genetic component of moral behaviors or behaviors with moral import must provide not only a delineation of cultural causes but must also distinguish between those genetic causes which may have their origin in innate moral constraints from others which are fundamentally nonmoral because the critical faculty necessary to higher level human morality itself arises in part from innate constraints of a nonmoral type; i.e., the processes of inductive reasoning common to both ethics and science. Finally, humans who could bring the nonmoral evaluative capacities to bear upon whatever moral intuitions might be genetically conserved in mammalian heritage would have an advantage over similar beings who could not.     

A protein alignment scoring system sensitive at all evolutionary distances

Summary Protein sequence alignments generally are constructed with the aid of a substitution matrix that specifies a score for aligning each pair of amino acids. Assuming a simple random protein model, it can be shown that any such matrix, when used for evaluating variable-length local alignments, is implicitly a log-odds matrix, with a specific probability distribution for amino acid pairs to which it is uniquely tailored. Given a model of protein evolution from which such distributions may be derived, a substitution matrix adapted to detecting relationships at any chosen evolutionary distance can be constructed. Because in a database search it generally is not known a priori what evolutionary distances will characterize the similarities found, it is necessary to employ an appropriate range of matrices in order not to overlook potential homologies. This paper formalizes this concept by defining a scoring system that is sensitive at all detectable evolutionary distances. The statistical behavior of this scoring system is analyzed, and it is shown that for a typical protein database search, estimating the originally unknown evolutionary distance appropriate to each alignment costs slightly over two bits of information, or somewhat less than a factor of five in statistical significance. A much greater cost may be incurred, however, if only a single substitution matrix, corresponding to the wrong evolutionary distance, is employed.     

A computational proposal for designing structured RNA pools for in vitro selection of RNAs

Although in vitro selection technology is a versatile experimental tool for discovering novel synthetic RNA molecules, finding complex RNA molecules is difficult because most RNAs identified from random sequence pools are simple motifs, consistent with recent computational analysis of such sequence pools. Thus, enriching in vitro selection pools with complex structures could increase the probability of discovering novel RNAs. Here we develop an approach for engineering sequence pools that links RNA sequence space regions with corresponding structural distributions via a "mixing matrix" approach combined with a graph theory analysis. We define five classes of mixing matrices motivated by covariance mutations in RNA; these constructs define nucleotide transition rates and are applied to chosen starting sequences to yield specific nonrandom pools. We examine the coverage of sequence space as a function of the mixing matrix and starting sequence via clustering analysis. We show that, in contrast to random sequences, which are associated only with a local region of sequence space, our designed pools, including a structured pool for GTP aptamers, can target specific motifs. It follows that experimental synthesis of designed pools can benefit from using optimized starting sequences, mixing matrices, and pool fractions associated with each of our constructed pools as a guide. Automation of our approach could provide practical tools for pool design applications for in vitro selection of RNAs and related problems.     

Non-locality in biological systems results from hierarchy

The concept of non-locality is deduced from a new concept for biological systems, the functional interaction. It is shown that a biological system, which is expressed in terms of functional interactions, can be constructed as a hierarchical system, the dynamics of which are represented by a non-local field at each level of organization. The two following constraints: continuous representation of state variables and hierarchy of the system, result in non-locality, i.e., a space property according to which the system depends on mechanisms that are located elsewhere in the space. Concepts and theory are illustrated in the case of the nervous system, where two levels of organization are considered, the level of neurons and the level of synapses. Non-local versus local field operators are discussed, and an interpretation of the field equation terms is proposed. A general formulation of non-local operators for hierarchical systems is given.     

A dynamical neural network model for motor cortical activity during movement: population coding of movement trajectories

As a dynamical model for motor cortical activity during hand movement we consider an artificial neural network that consists of extensively interconnected neuron-like units and performs the neuronal population vector operations. Local geometrical parameters of a desired curve are introduced into the network as an external input. The output of the model is a time-dependent direction and length of the neuronal population vector which is calculated as a sum of the activity of directionally tuned neurons in the ensemble. The main feature of the model is that dynamical behavior of the neuronal population vector is the result of connections between directionally tuned neurons rather than being imposed externally. The dynamics is governed by a system of coupled nonlinear differential equations. Connections between neurons are assigned in the simplest and most common way so as to fulfill basic requirements stemming from experimental findings concerning the directional tuning of individual neurons and the stabilization of the neuronal population vector, as well as from previous theoretical studies. The dynamical behavior of the model reveals a close similarity with the experimentally observed dynamics of the neuronal population vector. Specifically, in the framework of the model it is possible to describe a geometrical curve in terms of the time series of the population vector. A correlation between the dynamical behavior of the direction and the length of the population vector entails a dependence of the neural velocity on the curvature of the tracing trajectory that corresponds well to the experimentally measured covariation between tangential velocity and curvature in drawing tasks.On leave of absencefrom the Institute of Molecular Genetics, Russian Academy of Sciences, Moscow, Russia.     

A solvable selfreproductive hypercycle model for the selection of biological molecules

Summary By introducing a modified enzyme coupling in Eigen's hypercycle equations we have produced an exactly solvable selfreproductive hypercycle model. The model shows explicitly how collections of different macromolecular information carriers may coexist, through internal couplings, in the presence of constraints. We show how the selective value concept is manifested in the model and give simple criteria for selection among competing hypercycles.     

A general model of stochastic neural processing

A model of neural processing is proposed which is able to incorporate a great deal of neurophysiological detail, including effects associated with the mechanics of postsynaptic summation and cell surface geometry and is capable of hardware realisation as a probabilistic random access memory (pRAM). The model is an extension of earlier work by the authors, which by operating at much shorter time scales (of the order of the lifetime of a quantum of neurotransmitter in the synaptic cleft) allows a greater amount of information to be retrieved from the simulated spike train. The mathematical framework for the model appears to be that of an extended Markov process (involving the firing histories of the N neurons); simulation work has yielded results in excellent agreement with theoretical predictions. The extended neural model is expected to be particularly applicable in situations where timing constraints are of special importance (such as the auditory cortex) or where firing thresholds are high, as in the case for the granule and pyramidal cells of the hippocampus.     

Reliable RT-qPCR-based titration of retroviral and lentiviral vectors via quantification of residual vector plasmid DNA in samples

Objectives

To develop a method for reliable quantification of viral vectors, which is necessary for determining the optimal dose of vector particles in clinical trials to obtain the desired effects without severe unwanted immune responses.

Results

A significant level of vector plasmid remained in retroviral and lentiviral vector samples, which led to overestimation of viral titers when using the conventional RT-qPCR-based genomic titration method. To address this problem, we developed a new method in which the residual plasmid was quantified by an additional RT-qPCR step, and standard molecules and primer sets were optimized. The obtained counts were then used to correct the conventionally measured genomic titers of viral samples. While the conventional method produced significantly higher genomic titers for mutant retroviral vectors than for wild-type vectors, our method produced slightly higher or equivalent titers, corresponding with the general idea that mutation of viral components mostly results in reduced or, at best, retained titers.

Conclusion

Subtraction of the number of residual vector plasmid molecules from the conventionally measured genomic titer can yield reliable quantification of retroviral and lentiviral vector samples, a prerequisite to advancing the safety of gene therapy applications.