Biological membranes and vesicles play a vital role in critical physiological processes like endocytosis, cell division, and cell motility. Over decades, statistical and continuum mechanics studies have provided phenomenal insights into the mechanics of these membranes and their biophysical implications. However, most studies, until recently, have focused entirely on passive (or “dead”) membranes that exhibit only equilibrium thermal fluctuations. In this work, we acknowledge the growing consensus that the active nature of membranes is vital to their biophysical functioning. Active membranes contain proteins that are fueled by energy from external sources such as adinosine triphosphate hydrolysis or light. These proteins exert forces on the membrane during their activity causing the membrane to exhibit fluctuations that are non-thermal in origin, thus driving them away from equilibrium. In short, active membranes are “alive” with their own energy source capable of circumventing equilibrium considerations. In this paper, we present a theory for active membranes based on principles of continuum mechanics and a variational formulation that (we hope) provides a unifying framework to understand seemingly disparate but insightful prior works in this field. Using our developed continuum model, we study the statistical mechanics of active closed membranes or vesicles. A specific highlight of our work is that we incorporate nonlinear curvature elasticity in our continuum theory and obtain closed-form results for the fluctuation spectra for active vesicles. Our numerical results reveal a rather unanticipated interplay of active forces, surface tension as well as nonlinear elasticity for small vesicles. To gain insights into the biophysical implications of activity in vesicles, we revisit the problem of determining the vesicle size distribution by assuming that the active vesicles are close to thermal equilibrium. Our numerical calculations show that active forces may significantly impact vesicle size distribution. A tantalizing implication of this finding is that by tuning active forces, active vesicles may attain vesicle size distributions that are improbable for passive vesicles governed only by thermal fluctuations.

生物膜和囊泡在细胞内吞作用、细胞分裂和细胞运动等关键生理过程中起着至关重要的作用。几十年来，统计和连续介质力学研究为这些膜的力学及其生物物理学意义提供了惊人的见解。然而，直到最近，大多数研究都完全集中在仅表现出平衡热波动的被动（或“死”）膜上。在这项工作中，我们承认越来越多的共识认为膜的活性对其生物物理功能至关重要。活性膜含有蛋白质，这些蛋白质由三磷酸腺苷水解或光等外部来源的能量提供燃料。这些蛋白质在其活动期间对膜施加力，导致膜表现出非热源的波动，从而使他们远离平衡。简而言之，活性膜是“活的”，其自身的能量来源能够规避平衡考虑。在本文中，我们提出了一种基于连续介质力学原理的活性膜理论和一种变分公式，（我们希望）它提供了一个统一的框架来理解该领域看似不同但富有洞察力的先前工作。使用我们开发的连续体模型，我们研究了活性封闭膜或囊泡的统计力学。我们工作的一个具体亮点是，我们将非线性曲率弹性纳入连续统理论，并获得活性囊泡波动谱的封闭形式结果。我们的数值结果揭示了主动力之间相当出乎意料的相互作用，小囊泡的表面张力和非线性弹性。为了深入了解囊泡中活性的生物物理学意义，我们通过假设活性囊泡接近热平衡来重新审视确定囊泡大小分布的问题。我们的数值计算表明，主动力可能会显着影响囊泡大小分布。这一发现的一个诱人含义是，通过调整主动力，主动囊泡可能会获得囊泡大小分布，这对于仅受热波动控制的被动囊泡来说是不可能的。我们的数值计算表明，主动力可能会显着影响囊泡大小分布。这一发现的一个诱人含义是，通过调整主动力，主动囊泡可能会获得囊泡大小分布，这对于仅受热波动控制的被动囊泡来说是不可能的。我们的数值计算表明，主动力可能会显着影响囊泡大小分布。这一发现的一个诱人含义是，通过调整主动力，主动囊泡可能会获得囊泡大小分布，这对于仅受热波动控制的被动囊泡来说是不可能的。