Data-Driven Method to Estimate Nonlinear Chemical Equivalence

There is great need to express the impacts of chemicals found in the environment in terms of effects from alternative chemicals of interest. Methods currently employed in fields such as life-cycle assessment, risk assessment, mixtures toxicology, and pharmacology rely mostly on heuristic arguments to justify the use of linear relationships in the construction of “equivalency factors,” which aim to model these concentration-concentration correlations. However, the use of linear models, even at low concentrations, oversimplifies the nonlinear nature of the concentration-response curve, therefore introducing error into calculations involving these factors. We address this problem by reporting a method to determine a concentration-concentration relationship between two chemicals based on the full extent of experimentally derived concentration-response curves. Although this method can be easily generalized, we develop and illustrate it from the perspective of toxicology, in which we provide equations relating the sigmoid and non-monotone, or “biphasic,” responses typical of the field. The resulting concentration-concentration relationships are manifestly nonlinear for nearly any chemical level, even at the very low concentrations common to environmental measurements. We demonstrate the method using real-world examples of toxicological data which may exhibit sigmoid and biphasic mortality curves. Finally, we use our models to calculate equivalency factors, and show that traditional results are recovered only when the concentration-response curves are “parallel,” which has been noted before, but we make formal here by providing mathematical conditions on the validity of this approach.     

Connectivity, Probabilities and Persistence: Comparing Reserve Selection Strategies

Reserve selection methods are often based on information on species’ occurrence. This can be presence–absence data, or probabilities of occurrence estimated with species distribution models. However, the effect of the choice of distribution model on the outcome of a reserve selection method has been ignored. Here we test a range of species distribution models with three different reserve selection methods. The distribution models had different combinations of variables related to habitat quality and connectivity (which incorporates the effect of spatial habitat configuration on species occurrence). The reserve selection methods included (i) a minimum set approach without spatial considerations; (ii) a clustering reserve selection method; and (iii) a dynamic approach where probabilities of occurrence are re-evaluated according to the spatial pattern of selected sites. The sets of selected reserves were assessed by re-computing species probability of occurrence in reserves using the best probability model and assuming loss of non-selected habitat. The results show that particular choices of distribution model and selection method may lead to reserves that overestimate the achieved target; in other words, species may seem to be represented but the reserve network may actually not be able to support them in the long-term. Instead, the use of models that incorporated connectivity as a variable resulted in the selection of aggregated reserves with higher potential for species long-term persistence. As reserve design aims at the long-term protection of species, it is important to be aware of the uncertainties related to model and method choice and their implications.     

Endogenous agonists may change the concentration-response curves of exogenous agonists: Source of quantitative information about the endogenous tone

The pharmacological dogma that competitive antagonists cause parallel shifts to the right with sustained maximum effect of semi-logarithmic concentration-response curves of exogenous agonists may not be true if an endogenous agonist is present in the preparation. In this case, the antagonist and the exogenous agonist interfere in a complex way with an existing circuit of regulation between the response and the endogenous agonist. In consequence, it is difficult to determine the true shift in the concentration-response curves as induced by the antagonist, since a deviation of the curves in a non-parallel manner can be observed. The extent of this deviation may be used to learn more about the variables involved.The present paper discusses this phenomenon: The regulatory circuit of the (auto)receptor modulated release of neutrotransmitters is used as an example. Paired samples of data are analysed in this example. Since the extent of the non-parallel deviation also depends on the manner in which the paired samples are mathematically linked, two different ways of data evaluation have been used. A theoretical model of the relation between receptor activation and response is proposed which allows to evaluate experimental concentration-response curves by means of non-linear regression analysis. This evaluation yields quantitative information on the parameters of the regulatory circuit: the concentration of the endogenous agonist, itsK _D value and the true shift of the concentration-response curve caused by the applied antagonist.     

Effects of cellular pharmacology on drug distribution in tissues.

The efficacy of targeted therapeutics such as immunotoxins is directly related to both the extent of distribution achievable and the degree of drug internalization by individual cells in the tissue of interest. The factors that influence the tissue distribution of such drugs include drug transport; receptor/drug binding; and cellular pharmacology, the processing and routing of the drug within cells. To examine the importance of cellular pharmacology, previously treated only superficially, we have developed a mathematical model for drug transport in tissues that includes drug and receptor internalization, recycling, and degradation, as well as drug diffusion in the extracellular space and binding to cell surface receptors. We have applied this "cellular pharmacology model" to a model drug/cell system, specifically, transferrin and the well-defined transferrin cycle in CHO cells. We compare simulation results to models with extracellular diffusion only or diffusion with binding to cell surface receptors and present a parameter sensitivity analysis. The comparison of models illustrates that inclusion of intracellular trafficking significantly increases the total transferrin concentration throughout much of the tissue while decreasing the penetration depth. Increasing receptor affinity or tissue receptor density reduces permeation of extracellular drug while increasing the peak value of the intracellular drug concentration, resulting in "internal trapping" of transferrin near the source; this could account for heterogeneity of drug distributions observed in experimental systems. Other results indicate that the degree of drug internalization is not predicted by the total drug profile. Hence, when intracellular drug is required for a therapeutic effect, the optimal treatment may not result from conditions that produce the maximal total drug distribution. Examination of models that include cellular pharmacology may help guide rational drug design and provide useful information for whole body pharmacokinetic studies.     

两种自然保护区设计方法——数学建模和计算机模拟

传统的自然保护区设计方法是打分法和Gap分析法,这两种方法简单易行但可靠性不高;地理信息系统(GIS)在保护区设计领域的应用也为人熟悉.关注近年来快速发展而国内使用不多的两种方法——数学建模和计算机模拟.数学建模主要用来从一组备选地块中选择一部分组成自然保护区,包括线性和非线性模型,用启发式算法或最优化算法求解.启发式算法具有速度快、灵活等优点,但解通常不是最优的,不能保证稀缺资源的最优化利用.最优化算法运算效率低,变量较多比如数百时就可能遇到计算困难,但解是最优的.预计两种算法都将继续发展.计算机模拟主要用于保护区评价、功能区划分、预测特定环境比如空间特征和气候变化对物种的影响等,多用启发式算法,与其它软件结合把结果以图画显示出来.两种方法特别是计算机模拟均要求保护区设计者有较强的专业知识.讨论了两种方法面临的问题和新的研究方向,至少包括:1)基础数据依然需要完善;2)一些新的因素比如动态性和不确定性如何在模型中考虑并与其它因素结合;3)气候变化预景下模拟参数如何评估和调整;4)如何协调保护与发展的关系;5)方法的实际应用需要研究者与决策者之间建立交流机制;6)多领域专家和相关利益方应有机会参与保护区设计.     

Stochastic automaton models for interaction of excitatory and inhibitory impulse sequences in neurons

The mechanisms by which excitatory and inhibitory input impulse sequences interact in changing the spike probability in neurons are examined in the two mathematical neuron models; one is a real-time neuron model which is close to physiological reality, and the other a stochastic automaton model for the temporal pattern discrimination proposed in the previous paper (Tsukada et al., 1976), which is developed in this paper as neuron models for interaction of excitatory and inhibitory input impulse sequences. The interval distributions of the output spike train from these models tend to be multimodal and are compared with those used for experimental data, reported by Bishop et al. (1964) for geniculate neuron activity and Poisson process deleting model analyzed by Ten Hoopen et al. (1966). Special attention, moreover, should be paid to how different forms of inhibitory input are transformed into the output interval distributions through these neuron models. These results exhibit a clear correlation between inhibitory input form and output interval distribution. More detailed information on this mechanism is obtained from the computations of recurrence-time under the stationary condition to go from active state to itself for the first time, each of which is influenced by the inhibitory input forms. In addition to these facts, some resultant characteristics on interval histogram and serial correlation are discussed in relation to physiological data from the literature.     

Kinetic models of C3H mouse mammary tumor growth: implications regarding tumor cell loss

Three models of tumor cell loss are described. The effects of cell loss on other cellular kinetic parameters are evaluated, and experiments which may distinguish among the models are discussed. Each model is based on a different cell-loss mechanism, and equations for the cell-cycle, cell-frequency distribution, the growth of both the proliferating and non-proliferating cell population, the growth fraction (GF), and the relative rate of volumetric growth, (dV/dt)/V, are derived. The following types of data are simulated for each model: the pulse labelling index, the mitotic index, and the labeling index as a function of time after a single or a series of 3H-TdR injections. The relative volumetric growth rate has the same mathematical form for each model. The PLM curves predicted by each model for the tumor lines studied (S102F and Slow) are not appreciably different. The predicted initial labeling index and mitotic index may differ significantly among the models depending upon the tumor line. The most striking difference among the models lies in the predictions regarding the labeling index as a function of time after a single or after a series of 3H-TdR injections. These types of labeling experiments should be valuable for distinguishing the different cell-loss mechanisms in solid tumors.     

A new, vapour-phase mechanism for stomatal responses to humidity and temperature

A new mechanism for stomatal responses to humidity and temperature is proposed. Unlike previously-proposed mechanisms, which rely on liquid water transport to create water potential gradients within the leaf, the new mechanism assumes that water transport to the guard cells is primarily through the vapour phase. Under steady-state conditions, guard cells are assumed to be in near-equilibrium with the water vapour in the air near the bottom of the stomatal pore. As the water potential of this air varies with changing air humidity and leaf temperature, the resultant changes in guard cell water potential produce stomatal movements. A simple, closed-form, mathematical model based on this idea is derived. The new model is parameterized for a previously published set of data and is shown to fit the data as well as or better than existing models. The model contains mathematical elements that are consistent with previously-proposed mechanistic models based on liquid flow as well as empirical models based on relative humidity. As such, it provides a mechanistic explanation for the realm of validity for each of these approaches.     

Systems pharmacology and enhanced pharmacodynamic models for understanding antibody-based drug action and toxicity

Pharmacokinetic (PK) and pharmacodynamic (PD) models seek to describe the temporal pattern of drug exposures and their associated pharmacological effects produced at micro- and macro-scales of organization. Antibody-based drugs have been developed for a large variety of diseases, with effects exhibited through a comprehensive range of mechanisms of action. Mechanism-based PK/PD and systems pharmacology models can play a major role in elucidating and integrating complex antibody pharmacological properties, such as nonlinear disposition and dynamical intracellular signaling pathways triggered by ligation to their cognate targets. Such complexities can be addressed through the use of robust computational modeling techniques that have proven powerful tools for pragmatic characterization of experimental data and for theoretical exploration of antibody efficacy and adverse effects. The primary objectives of such multi-scale mathematical models are to generate and test competing hypotheses and to predict clinical outcomes. In this review, relevant systems pharmacology and enhanced PD (ePD) models that are used as predictive tools for antibody-based drug action are reported. Their common conceptual features are highlighted, along with approaches used for modeling preclinical and clinically available data. Key examples illustrate how systems pharmacology and ePD models codify the interplay among complex biology, drug concentrations, and pharmacological effects. New hybrid modeling concepts that bridge cutting-edge systems pharmacology models with established PK/ePD models will be needed to anticipate antibody effects on disease in subpopulations and individual patients.     

Estimating the Stoichiometry of Human Immunodeficiency Virus Entry

To enter target cells, human immunodeficiency virus (HIV) first attaches to the cells and fuses with the cell membrane. Attachment and fusion involve envelope glycoprotein trimers on the surface of the virion and the CD4 receptor and chemokine coreceptors on the surface of the target cell. The stoichiometry of entry, that is, the number of bonds between such trimers and CD4 that are required for infection, is unknown. Pseudotyped virions that express mixed trimers consisting of functional and nonfunctional envelope proteins have been used to study how many trimer-receptor interactions are required for virus entry. However, to extract information on the stoichiometry of entry from data generated in in vitro infectivity assays with such viruses, mathematical models are required. Here, we describe mathematical models that can be used to infer the stoichiometry of entry. By fitting our simplest model to previously published data (X. Yang, S. Kurteva, X. Ren, S. Lee, and J. Sodroski, J. Virol. 79: 12132-12147, 2005), we estimated that the number of trimer-receptor interactions required for HIV to infect a target cell is approximately eight, which is higher than previous estimates. We also consider model extensions that explain some systematic deviations of the data from the prediction of the simplest model. However, these extended models yield very different estimates of the stoichiometry of entry ranging from 2 to 19. These results strongly suggest that, based on our present knowledge of HIV entry, the stoichiometry of this process cannot be reliably estimated. Our study identifies parameters that need to be defined to render the estimation of the stoichiometry of HIV entry possible.     

最优化设计连续的自然保护区

生境破碎是导致生物多样性损失的重要原因之一,避免生境破碎的一个有效方式是建立连续的自然保护区使物种可在保护区内自由移动。不加选择地把大片土地都转为保护区是实现连续的一个途径,但资源是有限的,应当以最优的方式分配。如何最优化设计生态上和经济上都有效的保护区成为生物保护领域一个重要议题。从一组备选地块中选择一部分组成自然保护区,这样的问题主要有两种解法:启发式方法和最优化方法。启发式方法虽然灵活且运算速度快但不能保证最优解因而可能导致稀缺资源的浪费,最优化方法保证得到的解是最优的但建模和运算存在困难。建立一个线性整数规划模型用于设计一个最小的连续保护区,用Dantzig剪切法消除循环确保形成一个连续的树,对应一个连续的保护区,检验了模型的计算效率。结果显示,模型可在合理时间内解决一个包含100个备选地块和30个物种的连续保护区设计问题,计算效率显著优于同类目的的其它方法。以美国伊利诺伊州Cache河流域11种濒危鸟类的保护区设计为例说明了该方法的应用,设计了两种情况下连续的保护区。讨论了模型的局限和数据问题。     

Kinetic Models of C3h Mouse Mammary Tumor Growth: Implications Regarding Tumor Cell Loss

Three models of tumor cell loss are described. the effects of cell loss on other cellular kinetic parameters are evaluated, and experiments which may distinguish among the models are discussed. Each model is based on a different cell-loss mechanism, and equations for the cell-cycle, cell-frequency distribution, the growth of both the proliferating and non-proliferating cell population, the growth fraction (GF), and the relative rate of volumetric growth, (dV/dt)/V, are derived. The following types of data are simulated for each model: the pulse labelling index, the mitotic index, and the labeling index as a function of time after a single or a series of ³H-TdR injections. the relative volumetric growth rate has the same mathematical form for each model. the PLM curves predicted by each model for the tumor lines studied (S102F and Slow) are not appreciably different. the predicted initial labeling index and mitotic index may differ significantly among the models depending upon the tumor line. the most striking difference among the models lies in the predictions regarding the labeling index as a function of time after a single or after a series of ³H-TdR injections. These types of labeling experiments should be valuable for distinguishing the different cell-loss mechanisms in solid tumors.     

Evaluation of Multitype Mathematical Models for CFSE-Labeling Experiment Data

Carboxy-fluorescein diacetate succinimidyl ester (CFSE) labeling is an important experimental tool for measuring cell responses to extracellular signals in biomedical research. However, changes of the cell cycle (e.g., time to division) corresponding to different stimulations cannot be directly characterized from data collected in CFSE-labeling experiments. A number of independent studies have developed mathematical models as well as parameter estimation methods to better understand cell cycle kinetics based on CFSE data. However, when applying different models to the same data set, notable discrepancies in parameter estimates based on different models has become an issue of great concern. It is therefore important to compare existing models and make recommendations for practical use. For this purpose, we derived the analytic form of an age-dependent multitype branching process model. We then compared the performance of different models, namely branching process, cyton, Smith–Martin, and a linear birth–death ordinary differential equation (ODE) model via simulation studies. For fairness of model comparison, simulated data sets were generated using an agent-based simulation tool which is independent of the four models that are compared. The simulation study results suggest that the branching process model significantly outperforms the other three models over a wide range of parameter values. This model was then employed to understand the proliferation pattern of CD4+ and CD8+ T cells under polyclonal stimulation.     

A mathematical model and computer simulation study of insulin receptor regulation

A homeomorphic mathematical model of cell surface insulin receptor regulation is developed. The overall structure of the model is based on molecular mechanisms suggested by in vivo and in vitro experimental evidence from many different cell types. Model parameters correspond to cellular processes which are constrained by known boundry value conditions. As an example, computer simulation results are compared with published data from BC3H-1 myocytes in culture. With appropriate parameter choice, this model is able to simulate data from other cell types. Cellular processes which are explicitly represented in the model include: bound and unbound receptor endocytosis, receptor recycling, intracellular receptor degradation, and state-dependent receptor synthesis. Most of these processes are represented as first-order events. Using more complex representations of the model structure with higher order rate constants or saturable pathways does not qualitatively improve simulation results. Simulations are able to reproduce ligand-induced down and up regulation of receptors as well as the initial spontaneous display of surface insulin receptors. To demonstrate the behavior of our model and illustrate its utility for explaining insulin receptor regulation for a variety of conditions, simulations for which experimental data is unavailable for direct comparison are also shown. We believe the structure of our model is sufficient to explain insulin receptor regulation in a wide variety of cell types. In addition our model may aid in understanding the receptor component of insulin resistance (decreased sensitivity or responsiveness to insulin) seen in pathological states such as obesity and diabetes mellitus. Finally, this model may be applicable to the study of the regulation of other polypeptide hormone receptors.     

Cyberkelp: an integrative approach to the modelling of flexible organisms

Biomechanical models come in a variety of forms: conceptual models; physical models; and mathematical models (both of the sort written down on paper and the sort carried out on computers). There are model structures (such as insect flight muscle and the tendons of rats' tails), model organisms (such as the flying insect, Manduca sexta), even model systems of organisms (such as the communities that live on wave-swept rocky shores). These different types of models are typically employed separately, but their value often can be enhanced if their insights are integrated. In this brief report we explore a particular example of such integration among models, as applied to flexible marine algae. A conceptual model serves as a template for the construction of a mathematical model of a model species of giant kelp, and the validity of this numerical model is tested using physical models. The validated mathematical model is then used in conjunction with a computer-controlled tensile testing apparatus to simulate the loading regime placed on algal materials. The resulting information can be used to create a more precise mathematical model.     

Sensitivity and optimal performance in steroid receptor analysis

The application of various mathematical models for the identification of binding parameters in steroid receptor studies is discussed. In contrast to the usual statistical error considerations a deterministic error analysis based on the Euclidian norm and condition numbers is included. The predicted sensitivity behaviour with respect to data scattering is in good agreement with the experimental results.     

MIR: a versatile program for the statistical analysis of enzyme kinetic data

We have developed a package program for the estimation of Michaelis-Menten parameters for enzymes that conform to different kinetic mechanisms. Data from different experimental schemes can be fitted with appropriate weighing factors to any of 6 mathematical models, corresponding to 5 kinetic mechanisms: ordered bi-bi, Theorell-Chance, rapid equilibrium random bi-bi, rapid equilibrium ordered bi-bi and ping pong bi-bi. The program also performs a significance test to discriminate between different candidate models. To illustrate the performance of the program, real data from kinetic experiments with glucose 6-phosphate from Leuconostoc mesenteroides have been fitted to different mathematical models, and the results are discussed. The program can be easily implemented for the fitting of kinetic data to any other model.     

Habitat size and number in multi-habitat landscapes: a model approach based on species-area curves

This paper discusses species diversity in simple multi-habitat environments. Its main purpose is to present simple mathematical and graphical models on how landscape patterns affect species numbers. The idea is to build models of species diversity in multi-habitat landscapes by combining species-area curves for different habitats. Predictions are made about how variables such as species richness and species overlap between habitats influence the proportion of the total landscape each habitat should constitute, and how many habitats it should be divided into in order to be able to sustain the maximal number of species. Habitat size and numbers are the only factors discussed here, not habitat spatial patterns. Among the predictions are: 1) where there are differences in species diversity between habitats, optimal landscape patterns contain larger proportions of species rich habitats. 2) Species overlap between habitats shifts the optimum further towards larger proportions of species rich habitat types. 3) Species overlap also shifts the optimum towards fewer habitat types. 4) Species diversity in landscapes with large species overlap is more resistant to changes in landscape (or reserve) size. This type of model approach can produce theories useful to nature and landscape management in general, and the design of nature reserves and national parks in particular.     

Mechanistic Mathematical Modeling Tests Hypotheses of the Neurovascular Coupling in fMRI

Functional magnetic resonance imaging (fMRI) measures brain activity by detecting the blood-oxygen-level dependent (BOLD) response to neural activity. The BOLD response depends on the neurovascular coupling, which connects cerebral blood flow, cerebral blood volume, and deoxyhemoglobin level to neuronal activity. The exact mechanisms behind this neurovascular coupling are not yet fully investigated. There are at least three different ways in which these mechanisms are being discussed. Firstly, mathematical models involving the so-called Balloon model describes the relation between oxygen metabolism, cerebral blood volume, and cerebral blood flow. However, the Balloon model does not describe cellular and biochemical mechanisms. Secondly, the metabolic feedback hypothesis, which is based on experimental findings on metabolism associated with brain activation, and thirdly, the neurotransmitter feed-forward hypothesis which describes intracellular pathways leading to vasoactive substance release. Both the metabolic feedback and the neurotransmitter feed-forward hypotheses have been extensively studied, but only experimentally. These two hypotheses have never been implemented as mathematical models. Here we investigate these two hypotheses by mechanistic mathematical modeling using a systems biology approach; these methods have been used in biological research for many years but never been applied to the BOLD response in fMRI. In the current work, model structures describing the metabolic feedback and the neurotransmitter feed-forward hypotheses were applied to measured BOLD responses in the visual cortex of 12 healthy volunteers. Evaluating each hypothesis separately shows that neither hypothesis alone can describe the data in a biologically plausible way. However, by adding metabolism to the neurotransmitter feed-forward model structure, we obtained a new model structure which is able to fit the estimation data and successfully predict new, independent validation data. These results open the door to a new type of fMRI analysis that more accurately reflects the true neuronal activity.     

Optimization and Control of Agent-Based Models in Biology: A Perspective

Agent-based models (ABMs) have become an increasingly important mode of inquiry for the life sciences. They are particularly valuable for systems that are not understood well enough to build an equation-based model. These advantages, however, are counterbalanced by the difficulty of analyzing and using ABMs, due to the lack of the type of mathematical tools available for more traditional models, which leaves simulation as the primary approach. As models become large, simulation becomes challenging. This paper proposes a novel approach to two mathematical aspects of ABMs, optimization and control, and it presents a few first steps outlining how one might carry out this approach. Rather than viewing the ABM as a model, it is to be viewed as a surrogate for the actual system. For a given optimization or control problem (which may change over time), the surrogate system is modeled instead, using data from the ABM and a modeling framework for which ready-made mathematical tools exist, such as differential equations, or for which control strategies can explored more easily. Once the optimization problem is solved for the model of the surrogate, it is then lifted to the surrogate and tested. The final step is to lift the optimization solution from the surrogate system to the actual system. This program is illustrated with published work, using two relatively simple ABMs as a demonstration, Sugarscape and a consumer-resource ABM. Specific techniques discussed include dimension reduction and approximation of an ABM by difference equations as well systems of PDEs, related to certain specific control objectives. This demonstration illustrates the very challenging mathematical problems that need to be solved before this approach can be realistically applied to complex and large ABMs, current and future. The paper outlines a research program to address them.