López-Paleta Natalia, Moreno-Barbosa Eduardo, Velázquez-Castro Jorge
Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, 72570, Puebla, México.
J Biol Phys. 2025 May 19;51(1):16. doi: 10.1007/s10867-025-09681-x.
Since Waddington proposed the concept of the "epigenetic landscape" in 1957, researchers have developed various methodologies to represent it in diverse processes. Studying the epigenetic landscape provides valuable qualitative information regarding cell development and the stability of phenotypic and morphogenetic patterns. Although Waddington's original idea was a visual metaphor, a contemporary perspective relates it to the landscape formed by the basins of attraction of a dynamical system describing the temporal evolution of protein concentrations driven by a gene regulatory network. Transitions among these attractors can be driven by stochastic perturbations, with the cell state more likely to transition to the nearest attractor or to the one that presents the path of least resistance. In this study, we define the epigenetic landscape using the free energy potential obtained from the solution of the Fokker-Planck equation on the regulatory network. Specifically, we obtained a numerical approximate solution of the Fokker-Planck equation describing the Arabidopsis thaliana flower morphogenesis process. We observed good agreement between the coexpression matrix obtained from the Fokker-Planck equation and the experimental coexpression matrix. This paper proposes a method for obtaining this landscape by solving the Fokker-Planck equation (FPE) associated with a dynamical system describing the temporal evolution of protein concentrations involved in the process of interest. As these systems are high-dimensional and analytical solutions are often unfeasible, we propose a gamma mixture model to solve the FPE, transforming this problem into an optimization problem. This methodology can enhance the analysis of gene regulatory networks by directly relating theoretical mathematical models with experimental observations of coexpression matrices, thus providing a discriminating technique for competing models.
自1957年沃丁顿提出“表观遗传景观”概念以来，研究人员已开发出各种方法来在不同过程中对其进行表征。研究表观遗传景观能提供有关细胞发育以及表型和形态发生模式稳定性的有价值的定性信息。尽管沃丁顿最初的想法是一种视觉隐喻，但从当代角度来看，它与由描述基因调控网络驱动的蛋白质浓度随时间演变的动力系统的吸引盆所形成的景观相关。这些吸引子之间的转变可由随机扰动驱动，细胞状态更有可能转变为最近的吸引子或阻力最小的路径所对应的吸引子。在本研究中，我们使用从调控网络上的福克 - 普朗克方程解中获得的自由能势来定义表观遗传景观。具体而言，我们获得了描述拟南芥花形态发生过程的福克 - 普朗克方程的数值近似解。我们观察到从福克 - 普朗克方程获得的共表达矩阵与实验共表达矩阵之间具有良好的一致性。本文提出了一种通过求解与描述感兴趣过程中蛋白质浓度随时间演变的动力系统相关的福克 - 普朗克方程（FPE）来获得这种景观的方法。由于这些系统是高维的且解析解通常不可行，我们提出了一种伽马混合模型来求解FPE，将此问题转化为一个优化问题。这种方法可以通过直接将理论数学模型与共表达矩阵的实验观察联系起来，增强对基因调控网络的分析，从而为竞争模型提供一种区分技术。
