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PubMed1998

Analysis of error propagation from NMR-derived internuclear distances into molecular structure of cyclo-pro-gly.

Analytical expressions have been derived that translate uncertainties in distance constraints (obtained from NMR investigations) into uncertainties in atom positions in the maximum likelihood (ML) structure consistent with these inputs. As a test of this approach, a comparison was made between test structures reconstructed by the new ML approach, which yields a single structure and a covariance matrix for coordinates, and those reconstructed by metric matrix distance-geometry (MMDG), which yields a family of structures that sample uncertainty space. The test structures used were 560 polyhedra, with edges of arbitrary length containing up to 50 vertices, and one polyhedron, with 100 vertices; randomized distance constraints generated from these structures were used in reconstructing the polyhedra. The uncertainties derived from the two methods showed excellent agreement, and the correlation improved, as expected, with increasingly larger numbers of MMDG structures. This agreement supports the validity of the rapid analytical ML approach, which requires the calculation of only a single structure. As a second test of the ML method, the approach was applied to the determination of uncertainties in the structure of a cyclic dipeptide, cyclo(DL-Pro-Gly) (cPG), derived from NMR cross-relaxation data. The input data were interproton distances calculated from NOEs measured for a solution of the peptide in 2:1 DMSO:H2O at -40 degreesC (so as to yield large negative NOEs). In order to evaluate effects of the quality of the input spectral parameters on the precision of the resulting NMR structure, information from the covalent geometry of cPG was not used in the structure calculations. Results obtained from the analytical ML approach compared favorably with those from the much slower random-walk variant of the Monte Carlo method applied to the same input data. As a third test, the ML approach was used with synthetic structural constraints for a small protein; the results indicate that it will be feasible to use this rapid method to translate uncertainties associated with a given set of distance restraints into uncertainties in atom positions in larger molecules.

Dipeptides↗

PubMed1999

Assignment of coherence features in NMR q-space plots to particular diffusion modes in erythrocyte suspensions.

NMR q-space plots derived from water diffusing inside and around erythrocytes in a suspension display reproducible and characteristic coherence features. The aim of the present work was to determine which water population gives rise to the respective features. The central experimental strategy was to use choline and choline phosphate which are virtually membrane impermeant on the time scale of the experiment; the former was incorporated into erythrocytes by a lysis-resealing method and the latter was simply added to the suspensions. Dimethyl sulfoxide, which readily but more slowly exchanges across the cell membranes than water, also yielded q-space plots which were similar to those of water, but the differences were able to be accounted for on the basis of its slower transmembrane exchange rate. Random walk simulations using a Monte Carlo procedure, together with a model of an array of biconcave discocytes, helped verify the interpretations of the assignment of the features of the plots to molecules diffusing in the two regions. In addition, the simulations revealed how the presence or absence of transmembrane exchange affects the form of q-space plots.

Aquaporins↗

PubMed1999

What are the conditions for exponential time-cubed echo decays?

Diffusion of precessing spins through a constant field gradient is well-known to produce two distinctive features: an exp(-bt(3)) decay of the echo amplitude in response to two pulses and a much slower decay of the Carr-Purcell echo train. These features will appear whenever the spin frequency is described by a continuous random-walk. The present work shows that this may also occur in the presence of motions with long correlation times tau(c)-continuous Gaussian frequency noise with an exponential autocorrelation has the correct properties over time durations smaller than tau(c). Thus, time-cubed echo decays will occur in situations other than physical diffusion. The decay rate of the Carr-Purcell echo train is shown to vary with the pulse spacing tau whenever the correlation time tau(c) is long; the slower Carr-Purcell decay compared to the two-pulse echo decay is not unique to diffusion. Simulations are presented that display time-cubed decays. The simulations confirm two important criteria: the echo time must be less than tau(c) and the frequency noise must consist of nearly continuous variations, as opposed to step-like changes. These criteria define the range of physical parameters for which time-cubed decays will be observable.

Computer Simulation↗

PubMed2001

Fat emulsification measured using NMR transverse relaxation.

This paper presents a novel method of measuring the droplet size in oil-in-water emulsions. It is based on changes in the NMR transverse relaxation rate due to the effect of microscopic magnetic susceptibility differences between fat droplets and the surrounding water. The longitudinal and transverse relaxation rates of a series of emulsions with constant oil volume fraction and five different mean droplet sizes, in the range 0.4-20.9 microm, were measured in vitro at 37 degrees C using EPI. While the longitudinal relaxation rate 1/T(1) did not change significantly, 1/T(2) was observed to increase with mean droplet size. The measured changes in 1/T(2) were found to be in good agreement with results predicted from proton random walk simulations, and were also consistent with analytical solutions based on an outer sphere relaxation model. Measurements of 1/T(2) on emulsions with a higher oil volume fraction, and on emulsions of a fixed size where the water phase was doped with gadolinium to modulate the susceptibility difference between the phases, also showed the predicted behavior. As part of this study the susceptibility difference between olive oil and water was measured to be 1.55 ppm.

Algorithms↗

PubMed1993

Why some fitness landscapes are fractal.

Many biological and biochemical measurements, for example the "fitness" of a particular genome, or the binding affinity to a particular substrate, can be treated as a "fitness landscape", an assignment of numerical values to points in sequence space (or some other configuration space). As an alternative to the enormous amount of data required to completely describe such a landscape, we propose a statistical characterization, based on the properties of a random walk through the landscape and, more specifically, its autocorrelation function. Under assumptions roughly satisfied by two classes of simple model landscapes (the N-k model and the p-spin model) and by the landscape of estimated free energies of RNA secondary structures, this autocorrelation function, along with the mean and variance of individual points and the size of the landscape, completely characterize it. Having noted that these and other landscapes of estimated replication and degradation rates all have a well-defined correlation length, we propose a classification of landscapes depending on how the correlation length scales with the diameter of the landscape. The landscapes of some of the kinetic parameters of RNA molecules scale similarly to the model landscapes introduced into evolutionary studies from other fields, such as quadratic spin glasses and the traveling salesman problem, but the correlation length of RNA landscapes are considerably smaller. Nevertheless, both the model and some of the RNA landscapes satisfy a test of self-similarity proposed by Sorkin (1988).

Animals↗

PubMed1999

The evolutionary ecology of dominance-recessivity.

An "adaptive dynamics" modelling approach to the evolution of dominance-recessivity is presented. In this approach, fitness derives from an explicit ecological scenario, and both evolutionary attractivity and invasibility of resident populations are examined. The ecology consists of a within-individual part representing a locus with regulated activity and a between-individual part that is a two-patch soft selection model. Evolutionary freedom is allowed at a single locus. The evolutionary analysis considers directed random walks on trait space, generated by repeated invasions of mutants. The phenotype of an individual is determined by allelic parameters. Mutations can have two effects: they either affect the affinity of the promoter sequence for transcription factors, or they affect the gene product. The dominance interaction between alleles derives from their promoter affinities. Additive genetics is evolutionarily unstable when selection and evolution maintain two alleles in the population. In such a situation, dominance interactions can become stationary and close to additive genetics or they continue to evolve at a very slow pace towards dominance-recessivity. The probability that a specific dominance interaction will evolve depends on the relative mutation rate of promoter compared to gene product and the distribution of mutational effect sizes. Either allele in the dimorphism can become dominant, and dominance-recessivity is always most likely to evolve. Evolution then approaches a population state where every phenotype has maximum viability in one of the two patches. When the within-individual part is replaced by a housekeeping locus that codes for a metabolic enzyme, evolution favours a population of two alleles under the same conditions as for a regulated locus. In the case of a housekeeping gene, however, the evolutionary dynamical system approaches a population state where the heterozygote and only one homozygote phenotype are equivalent to the optimum phenotypes in the two patches.

Alleles↗

PubMed2002

Light-induced mass turnover in a mono-species community of mixotrophs.

We formulate a simple model for growth of a facultative photoautotroph with chemoheterotrophic capabilities. The organism is described by zero, one or three reserve components, and one structural component, all taken to be generalized compounds. The rules of synthesizing units are used for interactions among the uptake processes of the various nutrients and light (parallel processing), and for the merging of autotrophic and heterotrophic activities (sequential processing). For simplicity, we focus on the assimilation of inorganic carbon, inorganic nitrogen and light, and of two organic compounds (dead reserves and dead structure) that originate from aging. The process of resource recycling in a closed environment, as driven by light, and its links with community's structure (amount of biomass) is analysed in this simplest of all communities. Explicit analytical expressions for the steady states show how structure and function depend on the system parameters light, total carbon and total nitrogen. The behaviour resembles the Monod model for the Canonical Community, a three-species ecosystem consisting of producers, consumers and decomposers. If trophic preferences of a mixotroph are allowed to follow a random walk across generations, a trophic structure evolves where mixotrophs coexist with auto- and heterotrophs. Depth profiles are presented for the implied steady-state concentrations of dissolved inorganic carbon and nitrogen.

Aging↗

PubMed2002

Futile cycles revisited: a markov chain model of simultaneous glycolysis and gluconeogenesis.

We have used a random walk model of glycolysis and gluconeogenesis to investigate the bioenergetic implications of considering the cell cytoplasm to be a uniform well-mixed compartment. Radiotracer studies conducted on hepatocytes harvested from fasted rats and incubated with 40 mM glucose and 10 mM lactate demonstrated simultaneous glycolysis and gluconeogenesis, with net glycolysis. Tracer introduced as glycerol was incorporated both into glucose (via gluconeogenesis) and into pyruvate (via glycolysis). The data allow us to place a lower bound on the energetic cost of futile cycles involving adenosine triphosphate (ATP) hydrolysis in the early phosphorylation steps of glycolysis. Applying the Markov Chain model for glucose undergoing metabolism to pyruvate, the expected number of ATP molecules hydrolysed is not less than 15 ATP molecules per glucose molecule. The data suggest that, in hepatocytes under the circumstances of this experiment, either glycolysis is a net consumer of ATP, or glycolysis and gluconeogenesis are compartmentalized to a greater extent than is generally supposed.

Animals↗

PubMed2003

Discrete and continuous mathematical models of DNA branch migration.

DNA junctions, known as Holliday junctions, are intermediates in genetic recombination between DNAs. In this structure, two double-stranded DNA helices with similar sequence are joined at a branch point. The branch point can move along these helices when strands with the same sequence are exchanged. Such branch migration is modeled as a random walk. First, we model this process discretely, such that the motion of the branch is represented as transfer between discrete compartments. This is useful in analysing the results of DNA branch migration on junction comprised of synthetic oligonucleotides. The limit in which larger numbers of smaller steps go to continuous motion of the branch is also considered. We show that the behavior of the continuous system is very similar to that of the discrete system when there are more than just a few compartments. Thus, even branch migration on oligonucleotides can be viewed as a continuous process. One consequence of this is that a step size must be assumed when determining rate constants of branch migration. We compare migration where forward and backward movements of the branch are equally probable to biased migration where one direction is favored over the other. In the latter case larger differences between the discrete and continuous cases are predicted, but the differences are still small relative to the experimental error associated with experiments to measure branch migration in oligonucleotides.

DNA↗

PubMed2003

Quasi-independence, homology and the unity of type: a topological theory of characters.

In this paper Lewontin's notion of "quasi-independence" of characters is formalized as the assumption that a region of the phenotype space can be represented by a product space of orthogonal factors. In this picture each character corresponds to a factor of a region of the phenotype space. We consider any region of the phenotype space that has a given factorization as a "type", i.e. as a set of phenotypes that share the same set of phenotypic characters. Using the notion of local factorizations we develop a theory of character identity based on the continuation of common factors among different regions of the phenotype space. We also consider the topological constraints on evolutionary transitions among regions with different regional factorizations, i.e. for the evolution of new types or body plans. It is shown that direct transition between different "types" is only possible if the transitional forms have all the characters that the ancestral and the derived types have and are thus compatible with the factorization of both types. Transitional forms thus have to go over a "complexity hump" where they have more quasi-independent characters than either the ancestral as well as the derived type. The only logical, but biologically unlikely, alternative is a "hopeful monster" that transforms in a single step from the ancestral type to the derived type. Topological considerations also suggest a new factor that may contribute to the evolutionary stability of "types". It is shown that if the type is decomposable into factors which are vertex irregular (i.e. have states that are more or less preferred in a random walk), the region of phenotypes representing the type contains islands of strongly preferred states. In other words types have a statistical tendency of retaining evolutionary trajectories within their interior and thus add to the evolutionary persistence of types.

Animals↗

PubMed1996

Distribution of genome shared IBD by half-sibs: approximation by the Poisson clumping heuristic.

The distribution of the proportion of genome shared identical by descent (IBD) by c half-sibs is calculated for a general c. With data availability the consideration of the chromosome as a whole rather than discrete loci becomes of increasing practical importance. Assuming the genomic continuum model, which allows for recombination, the crossover process in a chromosome pedigree is viewed as a continuous-time Markov random walk on the vertices of a hypercube with time parameter map distance along the chromosome. The desired distribution corresponds to the distribution of sojourn times of the process in a small set of vertices. The sojourn times are well approximated via the Poisson clumping heuristic. It is shown explicitly that there is no "equivalent" number of independently segregating loci that will yield the same results as with the genomic continuum model. Results are given for the human genome.

Crossing Over, Genetic↗

PubMed1999

Diffusion models for population dynamics incorporating individual behavior at boundaries: applications to refuge design.

We construct models for dispersal of a population which incorporate the response of individuals to interfaces between habitat types. The models are based on random walks where there may be a bias in the direction an individual moves when it encounters an interface. This sort of dispersal process is called skew Brownian motion. Our models take the form of diffusion equations with matching conditions across the interface between regions for population densities and fluxes. We combine the dispersal models with linear population growth models which assume that the population growth rate differs between regions of different habitat types. We use those models to study issues of refuge design. We specifically consider how the effectiveness of buffer zones depends on their size, quality, and the population's response to the interface between the buffer zone and the refuge.

Animals↗

PubMed2000

Carrying capacity and demographic stochasticity: scaling behavior of the stochastic logistic model.

The stochastic logistic model is the simplest model that combines individual-level demography with density dependence. It explicitly or implicitly underlies many models of biodiversity of competing species, as well as non-spatial or metapopulation models of persistence of individual species. The model has also been used to study persistence in simple disease models. The stochastic logistic model has direct relevance for questions of limiting similarity in ecological systems. This paper uses a biased random walk heuristic to derive a scaling relationship for the persistence of a population under this model, and discusses its implications for models of biodiversity and persistence. Time to extinction of a species under the stochastic logistic model is approximated by the exponential of the scaling quantity U=(R-1)(2) N/R(R+1), where N is the habitat size and R is the basic reproductive number.

Animals↗

PubMed1984

Mathematical analysis of transport and consumption of molecules in heterogeneous brain tissue (methodology).

A computer model of metabolite transport and consumption in heterogeneous brain tissue, using a combination of probabilistic and deterministic techniques is being developed. The metabolites are put into two separate classes: (I) those that have reached a membrane for the first time during a small time step, delta t, and (II) those that have not yet reached a cell membrane for the first time during that time step. The time dependent spatial distribution of class (I) molecules is determined using random walk theory, which takes into account the actual paths of the molecules. The variation of the spatial distribution of class (II) molecules with time is determined using the time dependent diffusion equation with a boundary condition of zero concentration on the enclosing membrane boundaries.

Biological Transport, Active↗

PubMed1990

An analysis of neural spike-train distributions: determinants of the response of visual cortex neurons to changes in orientation and spatial frequency.

A previously unexploited method of examining neural spike-trains was applied to data obtained from cells in the visual cortex. Distributions of interspike intervals recorded extracellularly from cat visual cortex under four conditions were analyzed. Stimuli were gratings differing in orientation and spatial frequency. The probability density function of first passage time for a random walk with drift process, which is defined by its barrier height and drift coefficient, was used to characterize the generating process of axonal discharge under resting and stimulus conditions. Drift coefficient and barrier height were derived from the sample mean and standard deviation of the measured inter-spike intervals. For cells with simple receptive fields, variations in the drift coefficient were produced by changes in orientation and spatial frequency. Variations in barrier height were produced only by changes in orientation of the stimulus.

Animals↗

PubMed1991

Frontal eye field lesions impair predictive and visually-guided pursuit eye movements.

The study initially explored the frontal eye field's (FEF) control of predictive eye movements, i.e., eye movements driven by previous rather than current sensory signals. Five monkeys were trained to pursue horizontal target motion, including sinusoidal targets and "random-walk" targets which sometimes deviated from a sine motion. Some subjects also tracked other target trajectories and optokinetic motion. FEF ablations or cold lesions impaired predictive pursuit, but also degraded visually guided foveal pursuit of all targets. Unilateral lesions impaired pursuit of targets moving in both horizontal orbital fields and in both directions of movement. Saccadic estimates of target motion were generally accurate. The slow-phase velocity of optokinetic pursuit (collected after 54 s of OKN) also appeared normal. Pursuit recovered over 1-3 weeks after surgery but the deficits were then reinstated by removal of FEF in the other hemisphere. Thereafter, a slight deficit persisted for up to 10 weeks of observation in two subjects. The pattern of symptoms suggests that FEF lies subsequent to parietal area MST and prior to the pontine nuclei in controlling pursuit eye movements.

Animals↗

PubMed1995

Age-related changes in open-loop and closed-loop postural control mechanisms.

In an earlier posturographic investigation (Collins and De Luca 1993) it was proposed that open-loop and closed-loop control mechanisms are involved in the regulation of undisturbed, upright stance. In this study, stabilogram-diffusion analysis was used to examine how the natural aging process affects the operational characteristics of these control mechanisms. Stabilogram-diffusion analysis leads to the extraction of repeatable center-of-pressure (COP) parameters that can be directly related to the steady-state behavior and functional interaction of the neuromuscular mechanisms underlying the maintenance of erect posture. Twenty-five healthy young males (aged 19-30 years) and twenty-five elderly males (aged 71-80 years) who were free of major gait and postural disorders were included in the study. An instrumented force platform was used to measure the time-varying displacements of the COP under each subject's feet during quiet standing. The COP trajectories were analyzed as one-dimensional and two-dimensional random walks, according to stabilogram-diffusion analysis. Using this technique, it was demonstrated cross-sectionally that healthy aging is associated with significant changes in the 'quasi-static' dynamics of the postural control system. (It was also shown that more traditional posturographic analyses, i.e., summary statistics, were not sensitive enough to detect these age-related differences.) It was found that the steady-state behavior of the open-loop postural control mechanisms in the elderly is more positively correlated and therefore perhaps more unstable, i.e., the output of the overall system has a greater tendency to move or drift away from a relative equilibrium point over the short term. In contrast with this result, it was also found that the steady-state behavior of the closed-loop postural control mechanisms in the elderly is more negatively correlated and therefore perhaps more stable, i.e., over the longer term, there is an increased probability that movements away from a relative equilibrium point will be offset by corrective adjustments back towards the equilibrium position. In addition, it was demonstrated that the elderly utilize open-loop control schemes for longer time intervals and correspondingly larger COP displacements during periods of undisturbed stance. This result suggests that in the elderly there is a greater delay, on average, before closed-loop feedback mechanisms are called into play. Finally, it was shown that there is an increased heterogeneity of postural control abilities in healthy older adults.

Adult↗

PubMed1995

The effects of visual input on open-loop and closed-loop postural control mechanisms.

In an earlier posturographic investigation (Collins and De Luca 1993) it was proposed that open-loop and closed-loop control mechanisms are involved in the regulation of undisturbed, upright stance. In this study, stabilogram-diffusion analysis was used to examine how visual input affects the operational characteristics of these control mechanisms. Stabilogram-diffusion analysis leads to the extraction of repeatable center-of-pressure (COP) parameters that can be directly related to the resultant steady-state behavior and functional interaction of the neuromuscular mechanisms underlying the maintenance of erect posture. Twenty-five healthy male subjects (aged 19-30 years) were included in the study. An instrumented force platform was used to measure the time-varying displacements of the COP under each subject's feet during quiet standing. The subjects were tested under eyes-open and eyes-closed conditions. The COP trajectories were analyzed as one-dimensional and two-dimensional random walks, according to stabilogram-diffusion analysis. Using this technique, it was found that visual input affects the performance of the postural control system in one of two different ways--either it significantly modifies the steady-state behavior of the open-loop postural control mechanisms, or it significantly alters the characteristics of the other closed-loop feedback mechanisms that are involved in balance control. This result is interpreted as an indication that the visual system is integrated into the postural control system in one of two different ways. The experimental population was roughly evenly divided between these two schemes. For the first group (13 of 25 subjects), visual input principally caused a decrease in the "effective" stochastic activity of the open-loop control mechanisms in both the mediolateral and anteroposterior directions. For the second group (12 of 25 subjects), visual input caused an increase in the effective stochastic activity and uncorrelated behavior of the closed-loop control mechanisms in the anteroposterior direction only. On the basis of these results, it is hypothesized that visual input, in both schemes, serves to decrease the stiffness of the musculoskeletal system. In the former case, this may be accomplished by decreasing the level of muscular activity across the joints of the lower limb, whereas, in the latter case, reduced stiffness may be achieved by reducing the gain(s) of the other postural feedback mechanisms, i.e., the proprioceptive and/or vestibular systems. Using stabilogram-diffusion analysis, it was also found that the two groups of subjects behaved similarly under eyes-closed conditions.(ABSTRACT TRUNCATED AT 400 WORDS)

Adult↗