A computational model of selection by consequences: log survivor plots

[McDowell, J.J, 2004. A computational model of selection by consequences. J. Exp. Anal. Behav. 81, 297-317] instantiated the principle of selection by consequences in a virtual organism with an evolving repertoire of possible behaviors undergoing selection, reproduction, and mutation over many generations. The process is based on the computational approach, which is non-deterministic and rules-based. The model proposes a causal account for operant behavior. McDowell found that the virtual organism consistently showed a hyperbolic relationship between response and reinforcement rates according to the quantitative law of effect. To continue validation of the computational model, the present study examined its behavior on the molecular level by comparing the virtual organism's IRT distributions in the form of log survivor plots to findings from live organisms. Log survivor plots did not show the "broken-stick" feature indicative of distinct bouts and pauses in responding, although the bend in slope of the plots became more defined at low reinforcement rates. The shape of the virtual organism's log survivor plots was more consistent with the data on reinforced responding in pigeons. These results suggest that log survivor plot patterns of the virtual organism were generally consistent with the findings from live organisms providing further support for the computational model of selection by consequences as a viable account of operant behavior.     

Undermatching is an emergent property of selection by consequences

A computational theory of selection by consequences [McDowell, J.J, 2004. A computational model of selection by consequences. J. Exp. Anal. Behav. 81, 297-317] was tested by studying the responding of virtual organisms that were animated by the theory on random interval schedules of reinforcement. The theory generated responding by applying principles of selection, reproduction, and mutation to a population of potential behaviors that evolved in response to the selection pressure exerted by reinforcement. The organisms' equilibrium response rates were well described by the modern version of the Herrnstein hyperbola, which includes an exponent on reinforcement rate. Under strong selection pressure this exponent decreased with increasing mutation rate from a value near 1.0 at 1% mutation to an asymptotic value of 0.83 at mutation rates of 10% and greater. This asymptotic value is consistent with values obtained by fitting the equation to data from live organisms responding on single schedules, and with the value of about 0.80 that is expected on the basis of extensive research with live organisms responding on concurrent schedules. These results show that the computational theory is consistent with the modern theory of matching [McDowell, J.J, 2005. On the classic and modern theories of matching. J. Exp. Anal. Behav. 84, 111-127], and that it is a viable candidate for a mathematical dynamics of behavior.     

Binary analysis and optimization-based normalization of gene expression data

MOTIVATION: Most approaches to gene expression analysis use real-valued expression data, produced by high-throughput screening technologies, such as microarrays. Often, some measure of similarity must be computed in order to extract meaningful information from the observed data. The choice of this similarity measure frequently has a profound effect on the results of the analysis, yet no standards exist to guide the researcher. RESULTS: To address this issue, we propose to analyse gene expression data entirely in the binary domain. The natural measure of similarity becomes the Hamming distance and reflects the notion of similarity used by biologists. We also develop a novel data-dependent optimization-based method, based on Genetic Algorithms (GAs), for normalizing gene expression data. This is a necessary step before quantizing gene expression data into the binary domain and generally, for comparing data between different arrays. We then present an algorithm for binarizing gene expression data and illustrate the use of the above methods on two different sets of data. Using Multidimensional Scaling, we show that a reasonable degree of separation between different tumor types in each data set can be achieved by working solely in the binary domain. The binary approach offers several advantages, such as noise resilience and computational efficiency, making it a viable approach to extracting meaningful biological information from gene expression data.     

New structural variation in evolutionary searches of RNA neutral networks

RNA secondary structure is an important computational model to understand how genetic variation maps into phenotypic (structural) variation. Evolutionary innovation in RNA structures is facilitated by neutral networks, large connected sets of RNA sequences that fold into the same structure. Our work extends and deepens previous studies on neutral networks. First, we show that even the 1-mutant neighborhood of a given sequence (genotype) G0 with structure (phenotype) P contains many structural variants that are not close to P. This holds for biological and generic RNA sequences alike. Second, we analyze the relation between new structures in the 1-neighborhoods of genotypes Gk that are only a moderate Hamming distance k away from G0, and the structure of G0 itself, both for biological and for generic RNA structures. Third, we analyze the relation between mutational robustness of a sequence and the distances of structural variants near this sequence. Our findings underscore the role of neutral networks in evolutionary innovation, and the role that high robustness can play in diminishing the potential for such innovation.     

Stationary mutant distributions and evolutionary optimization

Molecular evolution is modelled by erroneous replication of binary sequences. We show how the selection of two species of equal or almost equal selective value is influenced by its nearest neighbours in sequence space. In the case of perfect neutrality and sufficiently small error rates we find that the Hamming distance between the species determines selection. As the error rate increases the fitness parameters of neighbouring species become more and more important. In the case of almost neutral sequences we observe a critical replication accuracy at which a drastic change in the “quasispecies”, in the stationary mutant distribution occurs. Thus, in frequently mutating populations fitness turns out to be an ensemble property rather than an attribute of the individual. In addition we investigate the time dependence of the mean excess production as a function of initial conditions. Although it is optimized under most conditions, cases can be found which are characterized by decrease or non-monotonous change in mean excess productions.     

Effects of limited input distance constraints upon the distance geometry algorithm.

In this paper we examine the distance geometry (DG) algorithm in the form used to determine the structure of proteins. We focus on three aspects of the algorithm: bound smoothing with the triangle inequality, the random selection of distances within the bounds, and the number of distances needed to specify a structure. Computational experiments are performed using simulated and real data for basic pancreatic trypsin inhibitor (BPTI) from nmr and crystallographic measurements. We find that the upper bounds determined by bound smoothing to be a linear function of the true crystal distance. A simple model that describes the results obtained with randomly selected trial distances is proposed. Using this representation of the trial distances, we show that BPTI DG structures are more compact than the true crystal structure. We also show that the DG-generated structures no longer resemble test structures when the number of these interresidue distance constraints is less than the number of degrees of freedom of the protein backbone. While the actual model will be sensitive the way distances are chosen, our conclusions are likely to apply to other versions of the DG algorithm.     

Hamming distance geometry of a protein conformational space: application to the clustering of a 4-ns molecular dynamics trajectory of the HIV-1 integrase catalytic core

Protein structures can be encoded into binary sequences (Gabarro-Arpa et al., Comput Chem 2000;24:693-698) these are used to define a Hamming distance in conformational space: the distance between two different molecular conformations is the number of different bits in their sequences. Each bit in the sequence arises from a partition of conformational space in two halves. Thus, the information encoded in the binary sequences is also used to characterize the regions of conformational space visited by the system. We apply this distance and their associated geometric structures to the clustering and analysis of conformations sampled during a 4-ns molecular dynamics simulation of the HIV-1 integrase catalytic core. The cluster analysis of the simulation shows a division of the trajectory into two segments of 2.6 and 1.4 ns length, which are qualitatively different: the data points to the fact that equilibration is only reached at the end of the first segment. The Hamming distance is compared also to the r.m.s. deviation measure. The analysis of the cases studied so far shows that under the same conditions the two measures behave quite differently, and that the Hamming distance appears to be more robust than the r.m.s. deviation.     

Plankton motility patterns and encounter rates

Many planktonic organisms have motility patterns with correlation run lengths (distances traversed before direction changes) of the same order as their reaction distances regarding prey, mates and predators (distances at which these organisms are remotely detected). At these scales, the relative measure of run length to reaction distance determines whether the underlying encounter is ballistic or diffusive. Since ballistic interactions are intrinsically more efficient than diffusive, we predict that organisms will display motility with long correlation run lengths compared to their reaction distances to their prey, but short compared to the reaction distances of their predators. We show motility data for planktonic organisms ranging from bacteria to copepods that support this prediction. We also present simple ballistic and diffusive motility models for estimating encounter rates, which lead to radically different predictions, and we present a simple criterion to determine which model is the more appropriate in a given case.     

SynPAM-a distance measure based on synonymous codon substitutions.

Measuring evolutionary distances between DNA or protein sequences forms the basis of many applications in computational biology and evolutionary studies. Of particular interest are distances based on synonymous substitutions, since these substitutions are considered to be under very little selection pressure and therefore assumed to accumulate in an almost clock-like manner. SynPAM, the method presented here, allows the estimation of distances between coding DNA sequences based on synonymous codon substitutions. The problem of estimating an accurate distance from the observed substitution pattern is solved by maximum-likelihood with empirical codon substitution matrices employed for the underlying Markov model. Comparisons with established measures of synonymous distance indicate that SynPAM has less variance and yields useful results over a longer time range.     

SynPAM—A Distance Measure Based on Synonymous Codon Substitutions

Measuring evolutionary distances between DNA or protein sequences forms the basis of many applications in computational biology and evolutionary studies. Of particular interest are distances based on synonymous substitutions since these substitutions are considered to be under very little selection pressure and therefore assumed to accumulate in an almost clock-like manner. SynPAM, the method presented here, allows the estimation of distances between coding DNA sequences based on synonymous codon substitutions. The problem of estimating an accurate distance from the observed substitution pattern is solved by maximum likelihood with empirical codon substitution matrices employed for the underlying Markov model. Comparisons with established measures of synonymous distance indicate that SynPAM has less variance and yields useful results over a longer time range.     

A taxonomic approach to evaluation of the charge state model using twelve species of sea anemone

The charge-state model of electrophoretic variation was tested by comparing the distances between nearest electromorphs of five enzyme loci within polymorphic species and among pooled species of sea anemone. If the charge-state model is generally true, and in particular if it allows linear distance between electromorphs to be used as a measure of genetic distance, then electromorphs of different species should be on the same "mobility ladder". Therefore, distances between adjacent electromorphs should be approximately equal for the two sets of comparisons. It was found that distances between adjacent electromorphs for each locus were significantly smaller for the pooled comparisons than within polymorphic species. Thus, it was concluded that much of the variation detected among different species does not conform to the charge-state model, and therefore that distance between electromorphs per se would not be a good measure of genetic distance. However, the charge-state model does appear to adequately account for most of the variation existing as common polymorphisms within species, or between very closely related species. Possible reasons for this apparent difference in the nature of the variation seen within and among species are discussed.     

Phylogeny of Organisms Investigated by the Base-Pair Changes in the Stem Regions of Small and Large Ribosomal Subunit RNAs

In order to obtain the evolutionary distance data that are as purely additive as possible, we have developed a novel method for evaluating the evolutionary distances from the base-pair changes in stem regions of ribosomal RNAs (rRNAs). The application of this method to small-subunit (SSU) and large-subunit (LSU) rRNAs provides the distance data, with which both the unweighted pair group method of analysis and the neighbor-joining method give almost the same tree topology of most organisms except for some Protoctista, thermophilic bacteria, parasitic organisms, and endosymbionts. Although the evolutionary distances calculated with LSU rRNAs are somewhat longer than those with SSU rRNAs, the difference, probably due to a slight difference in functional constraint, is substantially decreased when the distances are converted into the divergence times of organisms by the measure of the time scale estimated in each type of rRNAs. The divergence times of main branches agree fairly well with the geological record of organisms, at least after the appearance of oxygen-releasing photosynthesis, although the divergence times of Eukaryota, Archaebacteria, and Eubacteria are somewhat overestimated in comparison with the geological record of Earth formation. This result is explained by considering that the mutation rate is determined by the accumulation of misrepairs for DNA damage caused by radiation and that the effect of radiation had been stronger before the oxygen molecules became abundant in the atmosphere of the Earth. Received: 23 October 1997 / Accepted: 12 August 1998     

Evolutionary distances for protein-coding sequences: modeling site- specific residue frequencies

Estimation of evolutionary distances from coding sequences must take into account protein-level selection to avoid relative underestimation of longer evolutionary distances. Current modeling of selection via site-to-site rate heterogeneity generally neglects another aspect of selection, namely position-specific amino acid frequencies. These frequencies determine the maximum dissimilarity expected for highly diverged but functionally and structurally conserved sequences, and hence are crucial for estimating long distances. We introduce a codon- level model of coding sequence evolution in which position-specific amino acid frequencies are free parameters. In our implementation, these are estimated from an alignment using methods described previously. We use simulations to demonstrate the importance and feasibility of modeling such behavior; our model produces linear distance estimates over a wide range of distances, while several alternative models underestimate long distances relative to short distances. Site-to-site differences in rates, as well as synonymous/nonsynonymous and first/second/third-codon-position differences, arise as a natural consequence of the site-to-site differences in amino acid frequencies.      

Rates of root and organism growth, soil conditions, and temporal and spatial development of the rhizosphere

BACKGROUND: Roots growing in soil encounter physical, chemical and biological environments that influence their rhizospheres and affect plant growth. Exudates from roots can stimulate or inhibit soil organisms that may release nutrients, infect the root, or modify plant growth via signals. These rhizosphere processes are poorly understood in field conditions. SCOPE AND AIMS: We characterize roots and their rhizospheres and rates of growth in units of distance and time so that interactions with soil organisms can be better understood in field conditions. We review: (1) distances between components of the soil, including dead roots remnant from previous plants, and the distances between new roots, their rhizospheres and soil components; (2) characteristic times (distance(2)/diffusivity) for solutes to travel distances between roots and responsive soil organisms; (3) rates of movement and growth of soil organisms; (4) rates of extension of roots, and how these relate to the rates of anatomical and biochemical ageing of root tissues and the development of the rhizosphere within the soil profile; and (5) numbers of micro-organisms in the rhizosphere and the dependence on the site of attachment to the growing tip. We consider temporal and spatial variation within the rhizosphere to understand the distribution of bacteria and fungi on roots in hard, unploughed soil, and the activities of organisms in the overlapping rhizospheres of living and dead roots clustered in gaps in most field soils. CONCLUSIONS: Rhizosphere distances, characteristic times for solute diffusion, and rates of root and organism growth must be considered to understand rhizosphere development. Many values used in our analysis were estimates. The paucity of reliable data underlines the rudimentary state of our knowledge of root-organism interactions in the field.     

First and second moments and the mean hamming distance in a stochastic replication-mutation model for biological macromolecules

In this work first and second moments for a many species Moran model are calculated. The model describes by means of a time-continuous birth- and death process the evolution of an ensemble of N macromolecules out of n possible species. The molecules may replicate (correct or erroneous, in the latter case producing mutants) and may undergo elimination. Replication and elimination will be coupled in order to keep population size constant. In the case of arbitrary replication rates an expansion of the moments in powers of 1/N is found. For equal replication rates exact calculation of the moments is possible. In the case of a v-cube model (binary macromolecules) the second moments may be used to find a simple expression for the mean Hamming distance in the system. This quantity provides a measure for the localization of the ensemble.Supported by Stiftung Volkswagenwerk     

Basic properties of populations generated in the frame of one-parameter discrete model of genetic diversity

Previously, when discussing the properties of one parameter discrete model of genetic diversity (M.Yu. Shchelkanov et al, J. Biomol. Struct. Dyn. 15, 887-894 (1998)), we took into account Hamming distance distribution only between precursor and arbitrary descendant sequences. However, really there are sets of sequence populations produced during amplification process. In the presented work we have investigated Hamming distance distributions between sequences from different descendant sets produced in the frame of one parameter discrete model. Two basic descendant generation operators (so called amplifiers) are introduced: 1) the last generation amplifier, L, which produces descendants with precursor elimination; 2) all generations amplifier, G, which produces descendants without precursor elimination. Generalization of one-parameter discrete model for the case when precursor sequences do not coincide are carried out. Using this generalization we investigate the distribution of Hamming distances between L- and G-generated sequences. Basic properties of L and G operators, L/G-choice alternative problem have been discussed. Obtained results have common theoretical significance, but they are more suitable for high level genetic diversity process (for example, HIV diversity).     

An in silico exploration of the neutral network in protein sequence space

Designating amino-acid sequences that fold into a common main-chain structure as "neutral sequences" for the structure, regardless of their function or stability, we investigated the distribution of neutral sequences in protein sequence space. For four distinct target structures (alpha, beta,alpha/beta and alpha+beta types) with the same chain length of 108, we generated the respective neutral sequences by using the inverse folding technique with a knowledge-based potential function. We assumed that neutral sequences for a protein structure have Z scores higher than or equal to fixed thresholds, where thresholds are defined as the Z score for the corresponding native sequence (case 1) or much greater Z score (case 2). An exploring walk simulation suggested that the neutral sequences mapped into the sequence space were connected with each other through straight neutral paths and formed an inherent neutral network over the sequence space. Through another exploring walk simulation, we investigated contiguous regions between or among the neutral networks for the distinct protein structures and obtained the following results. The closest approach distance between the two neutral networks ranged from 5 to 29 on the Hamming distance scale, showing a linear increase against the threshold values. The sequences located at the "interchange" regions between the two neutral networks have intermediate sequence-profile-scores for both corresponding structures. Introducing a "ball" in the sequence space that contains at least one neutral sequence for each of the four structures, we found that the minimal radius of the ball that is centered at an arbitrary position ranged from 35 to 50, while the minimal radius of the ball that is centered at a certain special position ranged from 20 to 30, in the Hamming distance scale. The relatively small Hamming distances (5-30) may support an evolution mechanism by transferring from a network for a structure to another network for a more beneficial structure via the interchange regions.     

Exploring the conformational space of membrane protein folds matching distance constraints

Herein we present a computational technique for generating helix-membrane protein folds matching a predefined set of distance constraints, such as those obtained from NMR NOE, chemical cross-linking, dipolar EPR, and FRET experiments. The purpose of the technique is to provide initial structures for local conformational searches based on either energetic considerations or ad-hoc scoring criteria. In order to properly screen the conformational space, the technique generates an exhaustive list of conformations within a specified root-mean-square deviation (RMSD) where the helices are positioned in order to match the provided distances. Our results indicate that the number of structures decreases exponentially as the number of distances increases, and increases exponentially as the errors associated with the distances increases. We also found the number of solutions to be smaller when all the distances share one helix in common, compared to the case where the distances connect helices in a daisy-chain manner. We found that for 7 helices, at least 15 distances with errors up to 8 A are needed to produce a number of solutions that is not too large to be processed by local search refinement procedures. Finally, without energetic considerations, our enumeration technique retrieved the transmembrane domains of Bacteriorhodopsin (PDB entry1c3w), Halorhodopsin (1e12), Rhodopsin (1f88), Aquaporin-1 (1fqy), Glycerol uptake facilitator protein (1fx8), Sensory Rhodopsin (1jgj), and a subunit of Fumarate reductase flavoprotein (1qlaC) with Calpha level RMSDs of 3.0 A, 2.3 A, 3.2 A, 4.6 A, 6.0 A, 3.7 A, and 4.4 A, respectively.     

Cooperation under predation risk: a data-based ESS analysis

Two fish that jointly approach a predator in order to inspect it share the deadly risk of capture depending on the distance between them. Models are developed that seek ESS inspection distances of both single prey and pairs, based on experimental data of the risk that prey (sticklebacks) incur when they approach a predator (pike) to varying distances. Our analysis suggests that an optimal inspection distance can exist for a single fish, and for two equal fish behaving entirely cooperatively so as to maximize the fitness of the pair. Two equal fish inspecting cooperatively should inspect at an equal distance from the predator. The optimal distance is much closer to the predator for cooperative pairs than for single inspectors. However, optimal inspection for two equal fish behaving cooperatively operates across a rather narrow band of conditions relating to the benefits of cooperation. Evolutionarily stable inspection can also exist for two equal fish behaving non-cooperatively such that each acts to make a best reply (in terms of its personal fitness) to its opponent''s strategy. Non-cooperative pairs should also inspect at equal distance from the pike. Unlike the ''single fish'' and ''cooperative'' optima, which are unique inspection distances, there exists a range of ESS inspection distances. If either fish chooses to move to any point in this zone, the best reply of its opponent is to match it (move exactly alongside). Unilateral forward movement in the ''match zone'' may not be possible without some cooperation, but if the pair can ''agree'' to move forward synchronously, maintaining equal distance, inspection will occur at the nearest point in this zone to the predator. This ''near threshold'' is an ESS and is closer to the predator than the single fish optimum: pairs behaving almost selfishly can thus attain greater benefits from inspection by the protection gained from Hamilton''s dilution effect. That pairs should inspect more closely than single fish conforms with empirical findings. Phenotypic asymmetries in costs and benefits between the fish are not yet included in the model.     

Conceptual analysis of methods applied to assessment of diversity within and distance between populations with asexual or mixed mode of reproduction

Measures of diversity within populations, and distance between populations, are compared for organisms with an asexual or mixed mode of reproduction. Examples are drawn from studies of plant pathogenic fungi based on binary traits including presence/absence of DNA bands or virulence/avirulence to differential hosts. Commonly used measures of population diversity or genetic distance consider either genotype frequencies or allele frequencies. Kosman's diversity and distance measures are the most suitable for populations with an asexual or mixed mode of reproduction, because by considering genetic patterns of all individuals they take into account not just the genotype frequencies but also the genetic similarities between genotypes in the populations. The Kosman distance and diversity measures for populations can be calculated using different measures of dissimilarity between individuals (the simple mismatch, Jaccard and Dice coefficients of dissimilarity). Kosman's distances based on the simple mismatch and Jaccard dissimilarities are metrics. Comparisons of diversity indices for hypothetical examples as well as for actual data sets are presented to demonstrate that inferences from diversity analysis of populations can be driven by techniques of diversity and distance assessments and not only data driven.