Occupancy Modeling of Coverage Distribution for Whole Genome Shotgun Dna Sequencing

Expected-value models have long provided a rudimentary theoretical foundation for random DNA sequencing. Here, we are interested in improving characterization of genome coverage in terms of its underlying probability distributions. We find that the mathematical notion of occupancy serves as a good model for evolution of the coverage distribution function and reveals new insights related to sequence redundancy. Established concepts, such as “full shotgun depth,” have been assumed invariant, but actually depend on project size and decrease over time. For most microbial projects, the full shotgun milestone should be revised downward by about 30%. Accordingly, many already-completed genomes appear to have been over-sequenced. Results also suggest that read lengths for emerging high-throughput sequencing methods must be increased substantially before they can be considered as possible successors to the standard Sanger method. In particular, gains in throughput and sequence depth cannot be made to compensate for diminished read length. Limits are well approximated by a simple logarithmic equation, which should be useful in estimating maximum coverage-based redundancy for future projects.     

Validation of a mathematical approach to estimate dynamic scapular orientation

The goal of this study was to develop and validate a non-invasive approach to estimate scapular kinematics in individual patients. We hypothesized that individualized mathematical algorithms can be developed using motion capture data to accurately estimate dynamic scapula orientation based on measured humeral orientations and acromion process positions. The accuracy of the mathematical algorithms was evaluated against a gold standard of biplane fluoroscopy using a 2D to 3D fluoroscopy/model matching process. Individualized linear models were developed for nine healthy adult shoulders. These models were used to predict scapulothoracic kinematics, and the predicted kinematics were compared to kinematics obtained using biplane fluoroscopy to determine the accuracy of the algorithms. Results showed strong correlations between mathematically predicted kinematics and validation kinematics. Estimated kinematics were within 8° of validation kinematics. We concluded that individualized linear models show promise for providing accurate, non-invasive measurements of scapulothoracic kinematics in a clinical environment.     

A versatile ODE approximation to a network model for the spread of sexually transmitted diseases

 We develop a moment closure approximation (MCA) to a network model of sexually transmitted disease (STD) spread through a steady/casual partnership network. MCA has been used previously to approximate static, regular lattices, whereas application to dynamic, irregular networks is a new endeavour, and application to sociologically-motivated network models has not been attempted. Our goals are 1) to investigate issues relating to the application of moment closure approximations to dynamic and irregular networks, and 2) to understand the impact of concurrent casual partnerships on STD transmission through a population of predominantly steady monogamous partnerships. We are able to derive a moment closure approximation for a dynamic irregular network representing sexual partnership dynamics, however, we are forced to use a triple approximation due to the large error of the standard pair approximation. This example underscores the importance of doing error analysis for moment closure approximations. We also find that a small number of casual partnerships drastically increases the prevalence and rate of spread of the epidemic. Finally, although the approximation is derived for a specific network model, we can recover approximations to a broad range of network models simply by varying model parameters which control the structure of the dynamic network. Thus our moment closure approximation is very flexible in the kinds of network models it can approximate. Received: 26 August 2001 / Revised version: 15 March 2002 / Published online: 23 August 2002 C.T.B. was supported by the NSF. Key words or phrases: Moment closure approximation – Network model – Pair approximation – Sexually transmitted diseases – Steady/casual partnership network     

The simplest formal argument for fitness optimization

The Formal Darwinism Project aims to provide a formal argument linking population genetics to fitness optimization, which of necessity includes defining fitness. This bridges the gulf between those biologists who assume that natural selection leads to something close to fitness optimization and those biologists who believe on theoretical grounds that there is no sense of fitness that can usefully be said to be optimized. The current paper’s main objective is to provide a careful mathematical introduction to the project, and it also reflects on the project’s scope and limitations. The central argument is the proof of close ties between the mathematics of motion, as embodied in the Price equation, and the mathematics of optimization, as represented by optimization programmes. To make these links, a general and abstract model linking genotype, phenotype and number of successful gametes is assumed. The project has begun with simple dynamic models and simple linking models, and its progress will involve more realistic versions of them. The versions given here are fully mathematically rigorous, but elementary enough to serve as an introduction.     

Human matching behavior in social networks: an algorithmic perspective

We argue that algorithmic modeling is a powerful approach to understanding the collective dynamics of human behavior. We consider the task of pairing up individuals connected over a network, according to the following model: each individual is able to propose to match with and accept a proposal from a neighbor in the network; if a matched individual proposes to another neighbor or accepts another proposal, the current match will be broken; individuals can only observe whether their neighbors are currently matched but have no knowledge of the network topology or the status of other individuals; and all individuals have the common goal of maximizing the total number of matches. By examining the experimental data, we identify a behavioral principle called prudence, develop an algorithmic model, analyze its properties mathematically and by simulations, and validate the model with human subject experiments for various network sizes and topologies. Our results include i) a [Formula: see text]-approximate maximum matching is obtained in logarithmic time in the network size for bounded degree networks; ii) for any constant [Formula: see text], a [Formula: see text]-approximate maximum matching is obtained in polynomial time, while obtaining a maximum matching can require an exponential time; and iii) convergence to a maximum matching is slower on preferential attachment networks than on small-world networks. These results allow us to predict that while humans can find a "good quality" matching quickly, they may be unable to find a maximum matching in feasible time. We show that the human subjects largely abide by prudence, and their collective behavior is closely tracked by the above predictions.     

Body size and food web structure: testing the equiprobability assumption of the cascade model

The cascade model successfuly predicts many patterns in reported food webs. A key assumption of this model is the existence of a predetermined trophic hierarchy; prey are always lower in the hierarchy than their predators. At least three studies have suggested that, in animal food webs, this hierarchy can be explained to a large extent by body size relationships. A second assumption of the standard cascade model is that trophic links not prohibited by the hierarchy occur with equal probability. Using nonparametric contingency table analyses, we tested this ”equiprobability hypothesis” in 16 published animal food webs for which the adult body masses of the species had been estimated. We found that when the hierarchy was based on body size, the equiprobability hypothesis was rejected in favor of an alternative, ”predator-dominance” hypothesis wherein the probability of a trophic link varies with the identity of the predator. Another alternative to equiprobabilty is that the probability of a trophic link depends upon the ratio of the body sizes of the two species. Using nonparametric regression and liklihood ratio tests, we show that a size-ratio based model represents a significant improvement over the cascade model. These results suggest that models with heterogeneous predation probabilities will fit food web data better than the homogeneous cascade model. They also suggest a new way to bridge the gap between static and dynamic food web models. Received: 3 February 1999 / Accepted: 26 October 1999     

Exact reductions of Markovian dynamics for ion channels with a single permissive state

We show that the cooperative model for the kinetics of a tetrameric potassium ion channel derived in Nekouzadeh et al. (Biophys J 95(7):3510–3520, 2008) is an invariant manifold reduction of the full master equation for the channel kinetics. We further establish the validity of this reduction for ion channel models consisting of multiple independent subunits with cooperative transitions from a single permissive state to a conducting state. Finally, we conclude that solutions of the reduced model are globally asymptotically stable solutions of the full master equation system.     

A problem with using derivatives to explain the matching law

Herrnstein (1979) recently claimed that the matching law could be derived from an ordinary differential equation. He failed, however, to analyze the dynamic properties of his proposed equation. I show that it implies behavior cannot stabilize in accordance with the matching law. Consequently, Herrnstein's equation cannot explain why matching behavior has been widely observed.     

Ideal Binocular Disparity Detectors Learned Using Independent Subspace Analysis on Binocular Natural Image Pairs

An influential theory of mammalian vision, known as the efficient coding hypothesis, holds that early stages in the visual cortex attempts to form an efficient coding of ecologically valid stimuli. Although numerous authors have successfully modelled some aspects of early vision mathematically, closer inspection has found substantial discrepancies between the predictions of some of these models and observations of neurons in the visual cortex. In particular analysis of linear-non-linear models of simple-cells using Independent Component Analysis has found a strong bias towards features on the horoptor. In order to investigate the link between the information content of binocular images, mathematical models of complex cells and physiological recordings, we applied Independent Subspace Analysis to binocular image patches in order to learn a set of complex-cell-like models. We found that these complex-cell-like models exhibited a wide range of binocular disparity-discriminability, although only a minority exhibited high binocular discrimination scores. However, in common with the linear-non-linear model case we found that feature detection was limited to the horoptor suggesting that current mathematical models are limited in their ability to explain the functionality of the visual cortex.     

Disparity estimation through Green’s functions of matching equations

Binocular disparities arise from positional differences of scene features projected in the two retinae, and constitute the primary sensory cue for stereo vision. Here we introduce a new computational model for disparity estimation, based on the Green’s function of an image matching equation. When filtering a Gabor-function-modulated signal, the considered Green’s function yields a similarly modulated but shifted version of the original signal. Since a Gabor function models the receptive field of a cortical simple cell, the Green’s kernel thus allows the simulation of relative shifts between the cell’s left and right binocular inputs. A measure of the local degree of matching of such shifted inputs can then be introduced which affords disparity estimation in a similar manner to the energy model of the complex cortical cells. We have therefore effectively reformulated, in physiologically plausible terms, an image matching approach to disparity estimation. Our experiments show that the Green’s function method allows the detection of disparities both from random-dot and real-world stereograms. Partially supported by CNPq-Brazil.     

A nonlinear treatment of the protocell model by a boundary layer approximation

The “protocell” is a mathematical model of a self-maintaining unity based on the dynamics of simple reaction-diffusion processes and a self-controlled dynamics of the surface. In this paper its spatio-temporal behaviour far from the stationary structure is investigated by means of a boundary layer approximation. It is shown in detail how a simplified and mathematically feasible equation can be derived from the original parabolic problem. It turns out that the known instability which is initiated in the linear region around the stationary structure is continued further in the direction to a division by nonlinear dynamics.     

Comparison and content of the Wright-Fisher model of random genetic drift, the diffusion approximation, and an intermediate model

We investigate the detailed connection between the Wright-Fisher model of random genetic drift and the diffusion approximation, under the assumption that selection and drift are weak and so cause small changes over a single generation. A representation of the mathematics underlying the Wright-Fisher model is introduced which allows the connection to be made with the corresponding mathematics underlying the diffusion approximation. Two ‘hybrid’ models are also introduced which lie ‘between’ the Wright-Fisher model and the diffusion approximation. In model 1 the relative allele frequency takes discrete values while time is continuous; in model 2 time is discrete and relative allele frequency is continuous. While both hybrid models appear to have a similar status and the same level of plausibility, the different nature of time and frequency in the two models leads to significant mathematical differences. Model 2 is mathematically inconsistent and has to be ruled out as being meaningful. Model 1 is used to clarify the content of Kimura's solution of the diffusion equation, which is shown to have the natural interpretation as describing only those populations where alleles are segregating. By contrast the Wright-Fisher model and the solution of the diffusion equation of McKane and Waxman cover populations of all categories, namely populations where alleles segregate, are lost, or fix.     

Converging evidence for a simplified biophysical model of synaptic plasticity

 Different mechanisms that could form the molecular basis for bi-directional synaptic plasticity have been identified experimentally and corresponding biophysical models can be constructed. However, such models are complex and therefore it is hard to deduce their consequences to compare them to existing abstract models of synaptic plasticity. In this paper we examine two such models: a phenomenological one inspired by the phenomena of AMPA receptor insertion, and a more complex biophysical model based on the phenomena of AMPA receptor phosphorylation. We show that under certain approximations both these models can be mapped on to an equivalent, calcium-dependent, differential equation. Intracellular calcium concentration varies locally in each postsynaptic compartment, thus the plasticity rule we extract is a single-synapse rule. We convert this single synapse plasticity equation to a multi-synapse rule by incorporating a model of the NMDA receptor. Finally we suggest a mathematical embodiment of metaplasticity, which is consistent with observations on NMDA receptor properties and dependence on cellular activity. These results, in combination with some of our previous results, produce converging evidence for the calcium control hypothesis including a dependence of synaptic plasticity on the level of intercellular calcium as well as on the temporal pattern of calcium transients. Received: 24 April 2002 / Accepted: 15 May 2002 Acknowledgements. LCY was supported by a Burroughs Wellcome fellowship, GCC by Murst 60%. Correspondence to: H. Z. Shouval (e-mail: Harel_Shouval@brown.edu)     

A mathematical model of the adaptive control of human arm motions

This paper discusses similarities between models of adaptive motor control suggested by recent experiments with human and animal subjects, and the structure of a new control law derived mathematically from nonlinear stability theory. In both models, the control actions required to track a specified trajectory are adaptively assembled from a large collection of simple computational elements. By adaptively recombining these elements, the controllers develop complex internal models which are used to compensate for the effects of externally imposed forces or changes in the physical properties of the system. On a motor learning task involving planar, multi-joint arm motions, the simulated performance of the mathematical model is shown to be qualitatively similar to observed human performance, suggesting that the model captures some of the interesting features of the dynamics of low-level motor adaptation. Received: 20 September 1994 / Accepted in revised form: 18 November 1998     

Reduced models for binocular rivalry

Binocular rivalry occurs when two very different images are presented to the two eyes, but a subject perceives only one image at a given time. A number of computational models for binocular rivalry have been proposed; most can be categorised as either “rate” models, containing a small number of variables, or as more biophysically-realistic “spiking neuron” models. However, a principled derivation of a reduced model from a spiking model is lacking. We present two such derivations, one heuristic and a second using recently-developed data-mining techniques to extract a small number of “macroscopic” variables from the results of a spiking neuron model simulation. We also consider bifurcations that can occur as parameters are varied, and the role of noise in such systems. Our methods are applicable to a number of other models of interest.     

Bilateral matching of frequency tuning in neural cross-correlators of the owl

Sound localization requires comparison between the inputs to the left and right ears. One important aspect of this comparison is the differences in arrival time to each side, also called interaural time difference (ITD). A prevalent model of ITD detection, consisting of delay lines and coincidence-detector neurons, was proposed by Jeffress (J Comp Physiol Psychol 41:35–39, 1948). As an extension of the Jeffress model, the process of detecting and encoding ITD has been compared to an effective cross-correlation between the input signals to the two ears. Because the cochlea performs a spectrotemporal decomposition of the input signal, this cross-correlation takes place over narrow frequency bands. Since the cochlear tonotopy is arranged in series, sounds of different frequencies will trigger neural activity with different temporal delays. Thus, the matching of the frequency tuning of the left and right inputs to the cross-correlator units becomes a ‘timing’ issue. These properties of auditory transduction gave theoretical support to an alternative model of ITD-detection based on a bilateral mismatch in frequency tuning, called the ‘stereausis’ model. Here we first review the current literature on the owl’s nucleus laminaris, the equivalent to the medial superior olive of mammals, which is the site where ITD is detected. Subsequently, we use reverse correlation analysis and stimulation with uncorrelated sounds to extract the effective monaural inputs to the cross-correlator neurons. We show that when the left and right inputs to the cross-correlators are defined in this manner, the computation performed by coincidence-detector neurons satisfies conditions of cross-correlation theory. We also show that the spectra of left and right inputs are matched, which is consistent with predictions made by the classic model put forth by Jeffress. This article is part of a special issue on Neuronal Dynamics of Sensory Coding.     

Neural spike generator entrainment by regularly spaced synaptic input

 It has been known for 30 years that the output of a repetitively firing neuron or pacemaker can be synchronized (locked) to regularly spaced inhibitory or excitatory postsynaptic input potentials. Conditions for stable locking have been determined mathematically, demonstrated in computer simulation, and locking has been observed in vivo. We have developed a neural spike generator circuit model which exhibits stable locking to externally derived simulated inhibitory or excitatory post-synaptic inputs. Conditions for stable 1 : 1 lock, in which pacemaker output frequency matches that of the periodic input, are derived. These take the form of expressions for stable delay and convergence factor which incorporate known or measurable parameters of the circuit model. The expressions have been evaluated and shown to compare satisfactorily with experimental observations of locking by our circuit model. Received: 28 March 1996 / Accepted in revised form: 18 February 1997     

Interactions among biological systems: An analysis of asymptotic stability

An analysis of the interactions among asymptotically stable dynamical systems is formulated by making use of the dynamical system theory. Some results coming from previous mathematical analyses have been slightly modified to take into account some typical biological constraints as the boundedness properties of the solutions. In particular it has been shown that when the “coupling” among the subsystems is “loose” enough (in a sense that has to be made mathematically precise) the asymptotic behaviour of a complex system is the same of that of its individual components. The mathematical theory has been used to analyze two systems of biological significance: the coupling among chemical reactions and the stability properties of a 4-dimensional system describing the kinetics of a chemical transmitter.     

Using extracellular action potential recordings to constrain compartmental models

We investigate the use of extracellular action potential (EAP) recordings for biophysically faithful compartmental models. We ask whether constraining a model to fit the EAP is superior to matching the intracellular action potential (IAP). In agreement with previous studies, we find that the IAP method under-constrains the parameters. As a result, significantly different sets of parameters can have virtually identical IAP’s. In contrast, the EAP method results in a much tighter constraint. We find that the distinguishing characteristics of the waveform—but not its amplitude- resulting from the distribution of active conductances are fairly invariant to changes of electrode position and detailed cellular morphology. Based on these results, we conclude that EAP recordings are an excellent source of data for the purpose of constraining compartmental models. Action Editor: Alain Destexhe