Side-chain modeling with an optimized scoring function

Modeling side-chain conformations on a fixed protein backbone has a wide application in structure prediction and molecular design. Each effort in this field requires decisions about a rotamer set, scoring function, and search strategy. We have developed a new and simple scoring function, which operates on side-chain rotamers and consists of the following energy terms: contact surface, volume overlap, backbone dependency, electrostatic interactions, and desolvation energy. The weights of these energy terms were optimized to achieve the minimal average root mean square (rms) deviation between the lowest energy rotamer and real side-chain conformation on a training set of high-resolution protein structures. In the course of optimization, for every residue, its side chain was replaced by varying rotamers, whereas conformations for all other residues were kept as they appeared in the crystal structure. We obtained prediction accuracy of 90.4% for chi(1), 78.3% for chi(1 + 2), and 1.18 A overall rms deviation. Furthermore, the derived scoring function combined with a Monte Carlo search algorithm was used to place all side chains onto a protein backbone simultaneously. The average prediction accuracy was 87.9% for chi(1), 73.2% for chi(1 + 2), and 1.34 A rms deviation for 30 protein structures. Our approach was compared with available side-chain construction methods and showed improvement over the best among them: 4.4% for chi(1), 4.7% for chi(1 + 2), and 0.21 A for rms deviation. We hypothesize that the scoring function instead of the search strategy is the main obstacle in side-chain modeling. Additionally, we show that a more detailed rotamer library is expected to increase chi(1 + 2) prediction accuracy but may have little effect on chi(1) prediction accuracy.     

Parameter sensitivity of distributed transfer properties of neuronal dendrites: a passive cable approximation

Geometry and membrane properties of the dendrites crucially determine input–output relations in neurons. Unlike geometry often available in detail from computer reconstruction, the membrane resistivity is fragmentarily known if at all. Moreover, it varies during ongoing activity. In this study we address the question: what is the impact of the variation in membrane resistivity on the transfer properties of dendrites? Following a standard approach of the control system theory, we derive and explore the sensitivity functions complementary to the transfer functions of the passive dendrites with arbitrary geometrical parameters (length and diameter) and boundary conditions. We use the location-dependent somatopetal current transfer ratio (the reciprocal of the somatofugal voltage) as the transfer function, and its membrane resistivity derivatives, as the sensitivity functions. In the dendrites, at every path distance from the origin, the sensitivity function in a common form relates the transfer function, membrane resistivity, characteristic input conductance of semi-infinite cable and directional somatofugal input conductances at the given internal site and origin, and the length. Plotted in membrane resistivity versus path distance coordinates, the sensitivity functions display common features: along any coordinate there are low and high ranges, in which the sensitivity, respectively, increases and decreases. The ranges and corresponding rates depend on morphology and boundary conditions in a characteristic manner. These features predict existence of the geometry-dependent range of membrane resistivity (the earlier unattended mid-conductance state), such that the dendrites with a given metrical asymmetry are most distinguished in their transfer properties and electrical states if membrane resistivity is within the range and are not otherwise.     

A continuous optimization approach for inferring parameters in mathematical models of regulatory networks

Background

The advances of systems biology have raised a large number of sophisticated mathematical models for describing the dynamic property of complex biological systems. One of the major steps in developing mathematical models is to estimate unknown parameters of the model based on experimentally measured quantities. However, experimental conditions limit the amount of data that is available for mathematical modelling. The number of unknown parameters in mathematical models may be larger than the number of observation data. The imbalance between the number of experimental data and number of unknown parameters makes reverse-engineering problems particularly challenging.

Results

To address the issue of inadequate experimental data, we propose a continuous optimization approach for making reliable inference of model parameters. This approach first uses a spline interpolation to generate continuous functions of system dynamics as well as the first and second order derivatives of continuous functions. The expanded dataset is the basis to infer unknown model parameters using various continuous optimization criteria, including the error of simulation only, error of both simulation and the first derivative, or error of simulation as well as the first and second derivatives. We use three case studies to demonstrate the accuracy and reliability of the proposed new approach. Compared with the corresponding discrete criteria using experimental data at the measurement time points only, numerical results of the ERK kinase activation module show that the continuous absolute-error criteria using both function and high order derivatives generate estimates with better accuracy. This result is also supported by the second and third case studies for the G1/S transition network and the MAP kinase pathway, respectively. This suggests that the continuous absolute-error criteria lead to more accurate estimates than the corresponding discrete criteria. We also study the robustness property of these three models to examine the reliability of estimates. Simulation results show that the models with estimated parameters using continuous fitness functions have better robustness properties than those using the corresponding discrete fitness functions.

Conclusions

The inference studies and robustness analysis suggest that the proposed continuous optimization criteria are effective and robust for estimating unknown parameters in mathematical models.

Electronic supplementary material

The online version of this article (doi:10.1186/1471-2105-15-256) contains supplementary material, which is available to authorized users.     

Parameter estimation in a special reaction-diffusion system modelling man-environment diseases

An approximation scheme for a reaction-diffusion system with distributed feedback through the boundary is developed. It is used to estimate the strength of the feedback mechanisms from measurements of the states. The results are illustrated by numerical examples.     

急诊PCI 治疗中应用血栓抽吸术及主动脉内球囊反搏术 与血浆脑钠肽水平、心功能参数的关系

目的:观察血栓抽吸术与主动脉内球囊反搏术(IABP)联用在急诊冠状动脉介入治疗(PCI)的疗效。方法:ST段抬高型急性心肌梗塞(AMI)行急诊冠状动脉造影提示大量血栓征象、并行血栓抽吸术患者98例,随机分为实验组和对照组,实验组术前行IABP后联合血栓抽吸;对照组仅进行血栓抽吸。观察两组患者的BNP及心功能参数。结果:术后24小时两组BNP有普遍升高趋势,对照组升高更明显(P<0.01),术后2周普遍回降,实验组下降更明显(P<0.01);2周后实验组的心脏指数(CI)、每搏指数(SI)、混合静脉血氧饱和度(SvO2)均高于对照组(P<0.01)。结论:对于行急诊冠状动脉介入治疗的患者联合使用主动脉内球囊反搏术和血栓抽吸术,可以明显改善患者的心肌缺血情况,增加冠脉灌注,有利于患者心功能的恢复。     

一种获得非线性生长函数参数初始值的新方法:杂种遗传算法

本研究旨在杂种遗传算法应用于非线性生长函数的参数估计.提出了杂种遗传算法估计非线性生长函数参数的数学模型.5种非线性生长函数Gompertz、Logistic、von Bertalanffy、Richards、Brody分别拟合一个较大型的、群体类型差异大的番鸭体重生长资料,利用杂种遗传算法获得了有效初始值,在lsqcurvefit与proc nlin中获得了一致最优解的结果.表明杂种遗传算法估计非线性函数参数的实际可行性.     

ECOBAS — A tool to develop ecosystem models exemplified by the shallow lake model EMMO

The modelling and simulation tool ECOBAS was extended in order to include special features supporting the development of ecological models. The «Graphical Model Editor» allows the connection of at least 2 modules in order to build a whole model to run simulations. With the ECOBAS simulation system the model can be tested extensively in order to find appropriate parameter sets («Parameter analysis» and «Parameter estimation») and to identify critical parameters («Sensitivity analysis»). The «Interaction Analysis» shows the internal dependencies of a model. ECOBAS integrates the steps of ecological modelling and creates well readable and complete documentations within one working step, supports modularization of models and the user is rid of the technical and numerical aspects of modelling. Hence ECOBAS is setting up complete, consistent and syntactical correct models.All new features of the ECOBAS-system will be introduced by applying it on the existing ecosystem model EMMO.     

Parameter optimization and sensitivity analysis for large kinetic models using a real-coded genetic algorithm

Dynamic modeling is a powerful tool for predicting changes in metabolic regulation. However, a large number of input parameters, including kinetic constants and initial metabolite concentrations, are required to construct a kinetic model. Therefore, it is important not only to optimize the kinetic parameters, but also to investigate the effects of their perturbations on the overall system. We investigated the efficiency of the use of a real-coded genetic algorithm (RCGA) for parameter optimization and sensitivity analysis in the case of a large kinetic model involving glycolysis and the pentose phosphate pathway in Escherichia coli K-12. Sensitivity analysis of the kinetic model using an RCGA demonstrated that the input parameter values had different effects on model outputs. The results showed highly influential parameters in the model and their allowable ranges for maintaining metabolite-level stability. Furthermore, it was revealed that changes in these influential parameters may complement one another. This study presents an efficient approach based on the use of an RCGA for optimizing and analyzing parameters in large kinetic models.     

Estimation of stage-specific developmental times and survivorship from stage frequency data

An algorithm for estimating mean stage durations, the standard deviations associated with the means, and stage-specific survivorships from stage frequency data is presented. The algorithm is based on an age-structured, distributed delay simulation model which usesErlang distributions to determine the probability of maturity for individuals in each stage. If the data set extends beyond the first generation, the algorithm requires a fecundity rate, as well as stage frequencies, as input. Goodness-of-fit was measured using a weighted least squares calculation summed over all observed stages and all sampling dates.     

数量性状主基因＋多基因混合遗传分析中鉴定多基因存在的IECM算法

为了进行２对主基因＋多基因混合遗传分析中的主基因存在的鉴定和多基因存在的鉴定以及多世代的联合遗传分析的分布参数估计,在ＥＣＭ算法和剖分成分分布方差为主基因变异组分、多基因变异组分和误差变异组分三部分基础上,提出了计算简便的迭代ＥＣＭ算法,简称ＩＥＣＭ算法,以利用 Ｐ_１、Ｆ_１、Ｐ_２和 Ｆ_（２：３）家系世代鉴定多基因存在为例阐明该算法．它的 ＣＭ步包含迭代ＣＭ_１、迭代ＣＭ_２和迭代ＣＭ_３步,在固定其它参数的情况下分别求分布平均数、多基因方差组分和误差方差的极大似然估计．通过１１３８－２ｘ邳县天鹅蛋杂交组合的Ｐ_１、Ｐ_２、Ｆ_１和Ｆ_（２：３）家系群体研究了大豆豆秆黑潜蝇的遗传规律．结果表明,它受 １对主基因的控制并有多基因的修饰．     

Optimized N-mineralization parameters of loess soils from incubation experiments

Summary Nitrogen mineralization from long term incubation experiments with dried loess soils from southern Lower Saxony has been split by an optimization procedure into three contributing fractions: one from the resistant organic material (index rpm), another from decomposable plant residues (index dpm), and a third one from autolyzing microbial biomass (index bom). The nitrogen contents at the end of February averages 650 kg nitrogen per hectare and the 30 cm arable layer for the rpm, 30 to 50 kg N/ha for the dpm and about 65 kg N/ha for microbial biomass. The corresponding mean reaction coefficients at 35°C are .0058, .16 and about .35 per day, respectively. A rough approximation of the temperature dependence of the reaction coefficients has also been estimated.This paper was presented in part at the 1981 Congress of the German Soil Science Society held in Berlin, and at a colloquium at the Institute of Soil Chemistry, Pisa, Italy, Oct. 19th, 1981; nothing of it has been published before, however.     

A model for hormonal control of the menstrual cycle: Structural consistency but sensitivity with regard to data

This study presents a 13-dimensional system of delayed differential equations which predicts serum concentrations of five hormones important for regulation of the menstrual cycle. Parameters for the system are fit to two different data sets for normally cycling women. For these best fit parameter sets, model simulations agree well with the two different data sets but one model also has an abnormal stable periodic solution, which may represent polycystic ovarian syndrome. This abnormal cycle occurs for the model in which the normal cycle has estradiol levels at the high end of the normal range. Differences in model behavior are explained by studying hysteresis curves in bifurcation diagrams with respect to sensitive model parameters. For instance, one sensitive parameter is indicative of the estradiol concentration that promotes pituitary synthesis of a large amount of luteinizing hormone, which is required for ovulation. Also, it is observed that models with greater early follicular growth rates may have a greater risk of cycling abnormally.     

Estimation of additive genetic variance in commercial layer poultry and simulated populations under selection

Changes in genetic parameters over generations for a selected commercial population and simulated populations of poultry with different sizes were studied. The traits analyzed from the commercial population were rate of lay, age at first egg, egg weight, deformation, and body weight. In the simulated population, a trait measured on both sexes and a sex-limited trait, measured only on one sex, each with a heritability of 0.1 and 0.5, were analyzed. In the commercial and simulated populations, males and females were selected on the basis of family selection indexes and data was available only after many generations of selection. Parameters for each generation were estimated by fitting an animal model using derivative free maximum likelihood (DFREML) with different data structures. In structure 1, data included the given (base) generation for which the parameters were to be estimated, and all subsequent generations. In structure 2, only data on birds in the given generation and their progeny were included. In both structures, parents of base-generation birds were assumed unrelated and pedigrees traced back to these parents. With commercial data using structure 1, estimates of _a ² and h² decreased by 14 to 37% across five generations. With structure 2, no trends were observed, though estimates were lower than for structure 1. For simulated data, with a heritability of 0.1, both structures yielded apparently unbiased estimates of the observed additive genetic variances in the (selected) base generation, no matter how many generations of data were utilized, for both sex-limited and normal traits. However, with a heritability of 0.5 the estimated additive genetic variance for both types of trait decreased with a decrease in the number of generations used in the estimation. Estimates based on the first two generations underestimated, while estimates based on five generations of data overestimated, the observed genetic variances in the defined base. The combinations of conditions that lead to varying degrees of bias remain undefined.     

全退化型小参数问题周期解的存在性

运用拓扑度方法给出高维非自治系统全爱化型小参数问题周期解存在性的一些简明的判别方法,即在非线性项满足强制性条件时,无需计算便能判别周期解的存在。     

Bifurcation analysis of a model for hormonal regulation of the menstrual cycle

A model for hormonal control of the menstrual cycle with 13 ordinary differential equations and 41 parameters is presented. Important changes in model behavior result from variations in two of the most sensitive parameters. One parameter represents the level of estradiol sufficient for significant synthesis of luteinizing hormone, which causes ovulation. By studying bifurcation diagrams in this parameter, an interval of parameter values is observed for which a unique stable periodic solution exists and it represent an ovulatory cycle. The other parameter measures mass transfer between the first two stages of ovarian development and is indicative of healthy follicular growth. Changes in this parameter affect the uniqueness interval defined with respect to the first parameter. Hopf, saddle-node and transcritical bifurcations are examined. To attain a normal ovulatory menstrual cycle in this model, a balance must be maintained between healthy development of the follicles and flexibility in estradiol levels needed to produce the surge in luteinizing hormone.     

A methodology for parameter estimation in seaweed productivity modelling

This paper presents a combined approach for parameter estimation in models of primary production. The focus is on gross primary production and nutrient assimilation by seaweeds.A database of productivity determinations, biomass and mortality measurements and nutrient uptake rates obtained over one year for Gelidium sesquipedale in the Atlantic Ocean off Portugal has been used. Annual productivity was estimated by harvesting methods, and empirical relationships using mortality/wave energy and respiration rates have been derived to correct for losses and to convert the estimates to gross production. In situ determinations of productivity have been combined with data on the light climate (radiation periods, intensity, mean turbidity) to give daily and annual productivity estimates. The theoretical nutrient uptake calculated using a Redfield ratio approach and determinations of in situ N and P consumption by the algae during incubation periods have also been compared.The results of the biomass difference and incubation approaches are discussed in order to assess the utility of coefficients determined in situ for parameter estimation in seaweed production models.     

Logistic方程参数估计中的错误与修正

Logistic方程在种群生态学研究中被广泛应用,其积分式为N=k／（1＋e~（a-rt））,式中e~a=（k-N_0）／N_0。以往对该模型参数的各种估计方法,均将r,K和a作为三个相互独立的参数对待,而与e~a=（K-N_0）/N_0的假设相矛盾。由此估计出的参数K和a使实验初值N_0（t=0时的N值）发生偏离。笔者认为N_0是一个不带随机误差的常数,a值决定于K和N_0,而不是一个独立的参数,因而以往的参数估计方法是错误的,必须予以修正。本文提出了具体的修正方法,即用Marquardt方法或单纯形加速法求参数r和K,然后根据实验初值N_0求a,从而使Logistic微积分方程的共同参数r和K的估计一致。