Studies of folding and misfolding using simplified models

Computer simulations are as vital to our studies of biological systems as experiments. They bridge and rationalize experimental observations, extend the experimental "field of view", which is often limited to a specific time or length scale, and, most importantly, provide novel insights into biological systems, offering hypotheses about yet-to-be uncovered phenomena. These hypotheses spur further experimental discoveries. Simplified molecular models have a special place in the field of computational biology. Branded as less accurate than all-atom protein models, they have offered what all-atom molecular dynamics simulations could not--the resolution of the length and time scales of biological phenomena. Not only have simplified models proven to be accurate in explaining or reproducing several biological phenomena, they have also offered a novel multiscale computational strategy for accessing a broad range of time and length scales upon integration with traditional all-atom simulations. Recent computer simulations of simplified models have shaken or advanced the established understanding of biological phenomena. It was demonstrated that simplified models can be as accurate as traditional molecular dynamics approaches in identifying native conformations of proteins. Their application to protein structure prediction yielded phenomenal accuracy in recapitulating native protein conformations. New studies that utilize the synergy of simplified protein models with all-atom models and experiments yielded novel insights into complex biological processes, such as protein folding, aggregation and the formation of large protein complexes.     

Effect of nonuniform interstitial space properties on impulse propagation: a discrete multidomain model

This work presents a discrete multidomain model that describes ionic diffusion pathways between connected cells and within the interstitium. Unlike classical models of impulse propagation, the intracellular and extracellular spaces are represented as spatially distinct volumes with dynamic/static boundary conditions that electrically couple neighboring spaces. The model is used to investigate the impact of nonuniform geometrical and electrical properties of the interstitial space surrounding a fiber on conduction velocity and action potential waveshape. Comparison of the multidomain and bidomain models shows that although the conduction velocity is relatively insensitive to cases that confine 50% of the membrane surface by narrow extracellular depths (≥2 nm), the action potential morphology varies greatly around the fiber perimeter, resulting in changes in the magnitude of extracellular potential in the tight spaces. Results also show that when the conductivity of the tight spaces is sufficiently reduced, the membrane adjacent to the tight space is eliminated from participating in propagation, and the conduction velocity increases. Owing to its ability to describe the spatial discontinuity of cardiac microstructure, the discrete multidomain can be used to determine appropriate tissue properties for use in classical macroscopic models such as the bidomain during normal and pathophysiological conditions.     

Development of a 3D finite element model of lens microcirculation

Background

It has been proposed that in the absence of a blood supply, the ocular lens operates an internal microcirculation system. This system delivers nutrients, removes waste products and maintains ionic homeostasis in the lens. The microcirculation is generated by spatial differences in membrane transport properties; and previously has been modelled by an equivalent electrical circuit and solved analytically. While effective, this approach did not fully account for all the anatomical and functional complexities of the lens. To encapsulate these complexities we have created a 3D finite element computer model of the lens.

Methods

Initially, we created an anatomically-correct representative mesh of the lens. We then implemented the Stokes and advective Nernst-Plank equations, in order to model the water and ion fluxes respectively. Next we complemented the model with experimentally-measured surface ionic concentrations as boundary conditions and solved it.

Results

Our model calculated the standing ionic concentrations and electrical potential gradients in the lens. Furthermore, it generated vector maps of intra- and extracellular space ion and water fluxes that are proposed to circulate throughout the lens. These fields have only been measured on the surface of the lens and our calculations are the first 3D representation of their direction and magnitude in the lens.

Conclusion

Values for steady state standing fields for concentration and electrical potential plus ionic and fluid fluxes calculated by our model exhibited broad agreement with observed experimental values. Our model of lens function represents a platform to integrate new experimental data as they emerge and assist us to understand how the integrated structure and function of the lens contributes to the maintenance of its transparency.     

Computational Study on Subdural Cortical Stimulation - The Influence of the Head Geometry,Anisotropic Conductivity,and Electrode Configuration

Subdural cortical stimulation (SuCS) is a method used to inject electrical current through electrodes beneath the dura mater, and is known to be useful in treating brain disorders. However, precisely how SuCS must be applied to yield the most effective results has rarely been investigated. For this purpose, we developed a three-dimensional computational model that represents an anatomically realistic brain model including an upper chest. With this computational model, we investigated the influence of stimulation amplitudes, electrode configurations (single or paddle-array), and white matter conductivities (isotropy or anisotropy). Further, the effects of stimulation were compared with two other computational models, including an anatomically realistic brain-only model and the simplified extruded slab model representing the precentral gyrus area. The results of voltage stimulation suggested that there was a synergistic effect with the paddle-array due to the use of multiple electrodes; however, a single electrode was more efficient with current stimulation. The conventional model (simplified extruded slab) far overestimated the effects of stimulation with both voltage and current by comparison to our proposed realistic upper body model. However, the realistic upper body and full brain-only models demonstrated similar stimulation effects. In our investigation of the influence of anisotropic conductivity, model with a fixed ratio (1∶10) anisotropic conductivity yielded deeper penetration depths and larger extents of stimulation than others. However, isotropic and anisotropic models with fixed ratios (1∶2, 1∶5) yielded similar stimulation effects. Lastly, whether the reference electrode was located on the right or left chest had no substantial effects on stimulation.     

Modeling Signal Transduction Leading to Synaptic Plasticity: Evaluation and Comparison of Five Models

An essential phenomenon of the functional brain is synaptic plasticity which is associated with changes in the strength of synapses between neurons. These changes are affected by both extracellular and intracellular mechanisms. For example, intracellular phosphorylation-dephosphorylation cycles have been shown to possess a special role in synaptic plasticity. We, here, provide the first computational comparison of models for synaptic plasticity by evaluating five models describing postsynaptic signal transduction networks. Our simulation results show that some of the models change their behavior completely due to varying total concentrations of protein kinase and phosphatase. Furthermore, the responses of the models vary when models are compared to each other. Based on our study, we conclude that there is a need for a general setup to objectively compare the models and an urgent demand for the minimum criteria that a computational model for synaptic plasticity needs to meet.     

Planar cell polarity in the mammalian eye lens

The major role of the eye lens is to transmit and focus images onto the retina. For this function, the lens needs to develop and maintain the correct shape, notably, the precise curvature and high-level order and organization of its elements. The lens is mainly comprised of highly elongated fiber cells with hexagonal cross-sectional profiles that facilitate regular packing. Collectively, they form concentrically arranged layers around the anterior-posterior polar axis, and their convex curvature contributes to the spheroidal shape of the lens. Although the lens has been a popular system for developmental studies, little is known about the mechanism(s) that underlies the development of its exquisite three-dimensional cellular architecture. In this review, we will describe our recent work, which shows how planar cell polarity (PCP) operates in lens and contributes to its morphogenesis. We believe that the lens will be a useful model system to study PCP in general and gain insights into mechanisms that generate high-level cellular order during development.     

Planar cell polarity in the mammalian eye lens

The major role of the eye lens is to transmit and focus images onto the retina. For this function, the lens needs to develop and maintain the correct shape, notably, the precise curvature and high-level order and organization of its elements. The lens is mainly comprised of highly elongated fiber cells with hexagonal cross-sectional profiles that facilitate regular packing. Collectively, they form concentrically arranged layers around the anterior-posterior polar axis, and their convex curvature contributes to the spheroidal shape of the lens. Although the lens has been a popular system for developmental studies, little is known about the mechanism(s) that underlies the development of its exquisite three-dimensional cellular architecture. In this review, we will describe our recent work, which shows how planar cell polarity (PCP) operates in lens and contributes to its morphogenesis. We believe that the lens will be a useful model system to study PCP in general and gain insights into mechanisms that generate high-level cellular order during development.     

Simulation of pressure drop and energy dissipation for blood flow in a human fetal bifurcation.

The pressure drop from the umbilical vein to the heart plays a vital part in human fetal circulation. The bulk of the pressure drop is believed to take place at the inlet of the ductus venosus, a short narrow branch of the umbilical vein. In this study a generalized Bernoulli formulation was deduced to estimate this pressure drop. The model contains an energy dissipation term and flow-scaled velocities and pressures. The flow-scaled variables are related to their corresponding spatial mean velocities and pressures by certain shape factors. Further, based on physiological measurements, we established a simplified, rigid-walled, three-dimensional computational model of the umbilical vein and ductus venosus bifurcation for stationary flow conditions. Simulations were carried out for Reynolds numbers and umbilical vein curvature ratios in their respective physiological ranges. The shape factors in the Bernoulli formulation were then estimated for our computational models. They showed no significant Reynolds number or curvature ratio dependency. Further, the energy dissipation in our models was estimated to constitute 24 to 31 percent of the pressure drop, depending on the Reynolds number and the curvature ratio. The energy dissipation should therefore be taken into account in pressure drop estimates.     

The capabilities and limitations of conductance-based compartmental neuron models with reduced branched or unbranched morphologies and active dendrites

Conductance-based neuron models are frequently employed to study the dynamics of biological neural networks. For speed and ease of use, these models are often reduced in morphological complexity. Simplified dendritic branching structures may process inputs differently than full branching structures, however, and could thereby fail to reproduce important aspects of biological neural processing. It is not yet well understood which processing capabilities require detailed branching structures. Therefore, we analyzed the processing capabilities of full or partially branched reduced models. These models were created by collapsing the dendritic tree of a full morphological model of a globus pallidus (GP) neuron while preserving its total surface area and electrotonic length, as well as its passive and active parameters. Dendritic trees were either collapsed into single cables (unbranched models) or the full complement of branch points was preserved (branched models). Both reduction strategies allowed us to compare dynamics between all models using the same channel density settings. Full model responses to somatic inputs were generally preserved by both types of reduced model while dendritic input responses could be more closely preserved by branched than unbranched reduced models. However, features strongly influenced by local dendritic input resistance, such as active dendritic sodium spike generation and propagation, could not be accurately reproduced by any reduced model. Based on our analyses, we suggest that there are intrinsic differences in processing capabilities between unbranched and branched models. We also indicate suitable applications for different levels of reduction, including fast searches of full model parameter space.     

花背蟾蜍眼早期形态发生中其主要部分空间联系的研究

本文用扫描电镜研究了花背蟾蜍眼早期形态发生中视泡和预定晶状体、晶状体和预定角膜上皮间的紧密接触,此后在接触处出现间隙,其中存在呈网状的原纤维(fibril),这些原纤维的数量随两侧相连组织的分化,表现出增多、减少和逐渐消失的规律性变化,据此推测其成分属细胞外基质,对促进相连组织的分化起重要作用。     

The effects of simulating a realistic eye model on the eye dose of an adult male undergoing head computed tomography

In head computed tomography, radiation upon the eye lens (as an organ with high radiosensitivity) may cause lenticular opacity and cataracts. Therefore, quantitative dose assessment due to exposure of the eye lens and surrounding tissue is a matter of concern. For this purpose, an accurate eye model with realistic geometry and shape, in which different eye substructures are considered, is needed. To calculate the absorbed radiation dose of visual organs during head computed tomography scans, in this study, an existing sophisticated eye model was inserted at the related location in the head of the reference adult male phantom recommended by the International Commission on Radiological Protection (ICRP). Then absorbed doses and distributions of energy deposition in different parts of this eye model were calculated and compared with those based on a previous simple eye model. All calculations were done using the Monte Carlo code MCNP4C for tube voltages of 80, 100, 120 and 140 kVp. In spite of the similarity of total dose to the eye lens for both eye models, the dose delivered to the sensitive zone, which plays an important role in the induction of cataracts, was on average 3% higher for the sophisticated model as compared to the simple model. By increasing the tube voltage, differences between the total dose to the eye lens between the two phantoms decrease to 1%. Due to this level of agreement, use of the sophisticated eye model for patient dosimetry is not necessary. However, it still helps for an estimation of doses received by different eye substructures separately.     

Mapping parameter spaces of biological switches

Since the seminal 1961 paper of Monod and Jacob, mathematical models of biomolecular circuits have guided our understanding of cell regulation. Model-based exploration of the functional capabilities of any given circuit requires systematic mapping of multidimensional spaces of model parameters. Despite significant advances in computational dynamical systems approaches, this analysis remains a nontrivial task. Here, we use a nonlinear system of ordinary differential equations to model oocyte selection in Drosophila, a robust symmetry-breaking event that relies on autoregulatory localization of oocyte-specification factors. By applying an algorithmic approach that implements symbolic computation and topological methods, we enumerate all phase portraits of stable steady states in the limit when nonlinear regulatory interactions become discrete switches. Leveraging this initial exact partitioning and further using numerical exploration, we locate parameter regions that are dense in purely asymmetric steady states when the nonlinearities are not infinitely sharp, enabling systematic identification of parameter regions that correspond to robust oocyte selection. This framework can be generalized to map the full parameter spaces in a broad class of models involving biological switches.     

The transparent lens and cornea in the mouse and zebra fish eye

The lens and cornea combine to form a single optical element in which transparency and refraction are the fundamental biophysical characteristics required for a functional visual system. Although lens and cornea have different cellular and extracellular specializations that contribute to transparency and refraction, their development is closely related. In the embryonic mouse, the developing cornea and lens separate early. In contrast, zebra fish lens and cornea remain connected during early development and the optical properties of the cornea and lens observed by slit lamp and quasielastic laser light scattering spectroscopy (QLS) are more similar in the zebra fish eye than in the mouse eye. Optical similarities between cornea and lens of zebra fish may be the result of similarities in the cellular development of the cornea and lens.     

Computational model of the in vivo development of a tissue engineered vein from an implanted polymeric construct

Advances in vascular tissue engineering have been tremendous over the past 15 years, yet there remains a need to optimize current constructs to achieve vessels having true growth potential. Toward this end, it has been suggested that computational models may help hasten this process by enabling time-efficient parametric studies that can reduce the experimental search space. In this paper, we present a first generation computational model for describing the in vivo development of a tissue engineered vein from an implanted polymeric scaffold. The model was motivated by our recent data on the evolution of mechanical properties and microstructural composition over 24 weeks in a mouse inferior vena cava interposition graft. It is shown that these data can be captured well by including both an early inflammatory-mediated and a subsequent mechano-mediated production of extracellular matrix. There remains a pressing need, however, for more data to inform the development of next generation models, particularly the precise transition from the inflammatory to the mechanobiological dominated production of matrix having functional capability.     

Power-law models of signal transduction pathways

The mathematical modelling of signal transduction pathways has become a valuable aid to understanding the complex interactions involved in intracellular signalling mechanisms. An important aspect of the mathematical modelling process is the selection of the model type and structure. Until recently, the convention has been to use a standard kinetic model, often with the Michaelis-Menten steady state assumption. However this model form, although valuable, is only one of a number of choices, and the aim of this article is to consider the mathematical structure and essential features of an alternative model form--the power-law model. Specifically, we analyse how power-law models can be applied to increase our understanding of signal transduction pathways when there may be limited prior information. We distinguish between two kinds of power law models: a) Detailed power-law models, as a tool for investigating pathways when the structure of protein-protein interactions is completely known, and; b) Simplified power-law models, for the analysis of systems with incomplete structural information or insufficient quantitative data for generating detailed models. If sufficient data of high quality are available, the advantage of detailed power-law models is that they are more realistic representations of non-homogenous or crowded cellular environments. The advantages of the simplified power-law model formulation are illustrated using some case studies in cell signalling. In particular, the investigation on the effects of signal inhibition and feedback loops and the validation of structural hypotheses are discussed.     

Modeling Imatinib-Treated Chronic Myelogenous Leukemia: Reducing the Complexity of Agent-Based Models

We develop a model for describing the dynamics of imatinib-treated chronic myelogenous leukemia. Our model is based on replacing the recent agent-based model of Roeder et al. (Nat. Med. 12(10):1181–1184, 2006) by a system of deterministic difference equations. These difference equations describe the time-evolution of clusters of individual agents that are grouped by discretizing the state space. Hence, unlike standard agent-base models, the complexity of our model is independent of the number of agents, which allows to conduct simulation studies with a realistic number of cells. This approach also allows to directly evaluate the expected steady states of the system. The results of our numerical simulations show that our model replicates the averaged behavior of the original Roeder model with a significantly reduced computational cost. Our general approach can be used to simplify other similar agent-based models. In particular, due to the reduced computational complexity of our technique, one can use it to conduct sensitivity studies of the parameters in large agent-based systems.     

From computational quantum chemistry to computational biology: experiments and computations are (full) partners

Computations are being integrated into biological research at an increasingly fast pace. This has not only changed the way in which biological information is managed; it has also changed the way in which experiments are planned in order to obtain information from nature. Can experiments and computations be full partners? Computational chemistry has expanded over the years, proceeding from computations of a hydrogen molecule toward the challenging goal of systems biology, which attempts to handle the entire living cell. Applying theories from ab initio quantum mechanics to simplified models, the virtual worlds explored by computations provide replicas of real-world phenomena. At the same time, the virtual worlds can affect our perception of the real world. Computational biology targets a world of complex organization, for which a unified theory is unlikely to exist. A computational biology model, even if it has a clear physical or chemical basis, may not reduce to physics and chemistry. At the molecular level, computational biology and experimental biology have already been partners, mutually benefiting from each other. For the perception to become reality, computation and experiment should be united as full partners in biological research.     

Bidirectional coupling between astrocytes and neurons mediates learning and dynamic coordination in the brain: a multiple modeling approach

In recent years research suggests that astrocyte networks, in addition to nutrient and waste processing functions, regulate both structural and synaptic plasticity. To understand the biological mechanisms that underpin such plasticity requires the development of cell level models that capture the mutual interaction between astrocytes and neurons. This paper presents a detailed model of bidirectional signaling between astrocytes and neurons (the astrocyte-neuron model or AN model) which yields new insights into the computational role of astrocyte-neuronal coupling. From a set of modeling studies we demonstrate two significant findings. Firstly, that spatial signaling via astrocytes can relay a "learning signal" to remote synaptic sites. Results show that slow inward currents cause synchronized postsynaptic activity in remote neurons and subsequently allow Spike-Timing-Dependent Plasticity based learning to occur at the associated synapses. Secondly, that bidirectional communication between neurons and astrocytes underpins dynamic coordination between neuron clusters. Although our composite AN model is presently applied to simplified neural structures and limited to coordination between localized neurons, the principle (which embodies structural, functional and dynamic complexity), and the modeling strategy may be extended to coordination among remote neuron clusters.     

Simplified gene synthesis: a one-step approach to PCR-based gene construction

PCR-based gene synthesis conventionally requires two steps: first, all overlapping oligonucleotides are assembled by self-priming; then an additional pair of primers is used to amplify the full-length gene product. Here we propose a simplified method of gene synthesis which combines these two steps into one. We have found that the efficiency of this one-step method, which we term "Simplified Gene Synthesis", is affected by multiple parameters of the PCR reactions. In particular, the choice of polymerase is critical for successful one-step assembly. Other important factors include the concentration of assembly oligonucleotides and amplification primers. Moreover, we offer a general method to estimate, given a known mutation rate, how many clones should be sequenced in order to be confident of obtaining at least one correct gene product. Having determined the accuracy of gene products synthesized under optimal conditions with Simplified Gene Synthesis, we show that our estimation works well. Overall, the simplified gene synthesis provides an easier and more efficient approach to gene synthesis, providing a further step towards the future goal of generalized automation for this process.