The biophysics of an organism span multiple scales from subcellular to organismal and include processes characterized by spatial properties, such as the diffusion of molecules, cell migration, and flow of intravenous fluids. Mathematical biology seeks to explain biophysical processes in mathematical terms at, and across, all relevant spatial and temporal scales, through the generation of representative models. While non-spatial, ordinary differential equation (ODE) models are often used and readily calibrated to experimental data, they do not explicitly represent the spatial and stochastic features of a biological system, limiting their insights and applications. However, spatial models describing biological systems with spatial information are mathematically complex and computationally expensive, which limits the ability to calibrate and deploy them and highlights the need for simpler methods able to model the spatial features of biological systems. In this work, we develop a formal method for deriving cell-based, spatial, multicellular models from ODE models of population dynamics in biological systems, and vice versa. We provide examples of generating spatiotemporal, multicellular models from ODE models of viral infection and immune response. In these models, the determinants of agreement of spatial and non-spatial models are the degree of spatial heterogeneity in viral production and rates of extracellular viral diffusion and decay. We show how ODE model parameters can implicitly represent spatial parameters, and cell-based spatial models can generate uncertain predictions through sensitivity to stochastic cellular events, which is not a feature of ODE models. Using our method, we can test ODE models in a multicellular, spatial context and translate information to and from non-spatial and spatial models, which help to employ spatiotemporal multicellular models using calibrated ODE model parameters. We additionally investigate objects and processes implicitly represented by ODE model terms and parameters and improve the reproducibility of spatial, stochastic models. We developed and demonstrate a method for generating spatiotemporal, multicellular models from non-spatial population dynamics models of multicellular systems. We envision employing our method to generate new ODE model terms from spatiotemporal and multicellular models, recast popular ODE models on a cellular basis, and generate better models for critical applications where spatial and stochastic features affect outcomes.

中文翻译：

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从常微分方程生成群体动态的多细胞时空模型，并在病毒感染中应用


生物体的生物物理学跨越从亚细胞到生物体的多个尺度，包括以空间特性为特征的过程，例如分子扩散、细胞迁移和静脉内液体流动。数学生物学试图通过生成代表性模型，在所有相关的空间和时间尺度上用数学术语解释生物物理过程。虽然非空间常微分方程 (ODE) 模型经常被使用并且很容易根据实验数据进行校准，但它们没有明确表示生物系统的空间和随机特征，限制了它们的见解和应用。然而，用空间信息描述生物系统的空间模型在数学上很复杂，计算成本也很高，这限制了校准和部署它们的能力，并强调需要能够对生物系统的空间特征进行建模的更简单的方法。在这项工作中，我们开发了一种形式化方法，用于从生物系统中种群动态的 ODE 模型导出基于细胞的空间多细胞模型，反之亦然。我们提供了从病毒感染和免疫反应的 ODE 模型生成时空多细胞模型的示例。在这些模型中，空间和非空间模型一致性的决定因素是病毒产生的空间异质性程度以及细胞外病毒扩散和衰变的速率。我们展示了 ODE 模型参数如何隐式表示空间参数，并且基于细胞的空间模型可以通过对随机细胞事件的敏感性生成不确定的预测，这不是 ODE 模型的特征。 使用我们的方法，我们可以在多细胞、空间环境中测试 ODE 模型，并将信息转换为非空间和空间模型，这有助于使用使用校准的 ODE 模型参数来使用时空多细胞模型。我们还研究了 ODE 模型项和参数隐式表示的对象和过程，并提高了空间随机模型的再现性。我们开发并演示了一种从多细胞系统的非空间群体动力学模型生成时空多细胞模型的方法。我们设想采用我们的方法从时空和多细胞模型生成新的 ODE 模型项，在细胞基础上重新构建流行的 ODE 模型，并为空间和随机特征影响结果的关键应用生成更好的模型。