Freire Tomas, Hu Zhijian, Wood Kevin B, Gjini Erida
Center for Computational and Stochastic Mathematics, Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal.
Departments of Biophysics and Physics, University of Michigan, USA.
bioRxiv. 2023 Nov 17:2023.11.16.567447. doi: 10.1101/2023.11.16.567447.
Multi-drug combinations to treat bacterial populations are at the forefront of approaches for infection control and prevention of antibiotic resistance. Although the evolution of antibiotic resistance has been theoretically studied with mathematical population dynamics models, extensions to spatial dynamics remain rare in the literature, including in particular spatial evolution of multi-drug resistance. In this study, we propose a reaction-diffusion system that describes the multi-drug evolution of bacteria, based on a rescaling approach (Gjini and Wood, 2021). We show how the resistance to drugs in space, and the consequent adaptation of growth rate is governed by a Price equation with diffusion. The covariance terms in this equation integrate features of drug interactions and collateral resistances or sensitivities to the drugs. We study spatial versions of the model where the distribution of drugs is homogeneous across space, and where the drugs vary environmentally in a piecewise-constant, linear and nonlinear manner. Applying concepts from perturbation theory and reaction-diffusion equations, we propose an analytical characterization of in the spatial system based on the principal eigenvalue of our linear problem. This enables an accurate translation from drug spatial gradients and mutant antibiotic susceptibility traits, to the relative advantage of each mutant across the environment. Such a mathematical understanding allows to predict the precise outcomes of selection over space, ultimately from the fundamental balance between growth and movement traits, and their diversity in a population.
用于治疗细菌群体的多药联合疗法处于感染控制和预防抗生素耐药性方法的前沿。尽管抗生素耐药性的演变已通过数学种群动力学模型进行了理论研究，但文献中空间动力学的扩展仍然很少，特别是多药耐药性的空间演变。在本研究中，我们基于一种重新缩放方法（吉尼和伍德，2021年）提出了一个反应扩散系统，该系统描述了细菌的多药演变。我们展示了空间中对药物的耐药性以及随之而来的生长速率适应性是如何由一个带有扩散的普莱斯方程所控制的。该方程中的协方差项整合了药物相互作用以及对药物的附带耐药性或敏感性的特征。我们研究了模型的空间版本，其中药物在空间上的分布是均匀的，以及药物在环境中以分段常数、线性和非线性方式变化的情况。应用微扰理论和反应扩散方程的概念，我们基于线性问题的主特征值对空间系统中的 提出了一种解析表征。这使得能够从药物空间梯度和突变体抗生素敏感性特征准确转换到每个突变体在整个环境中的相对优势。这种数学理解有助于预测空间选择的精确结果，最终源于生长和移动特征之间的基本平衡以及它们在种群中的多样性。
