Curved crease origami and topological singularities enable hyperextensibility of L. olor

Single-celled protists display a remarkable diversity of forms and accomplish a wide range of functions, including hunting live prey in dynamic environments. Lacrymaria olor, a predatory ciliate, hunts its prey by launching a necklike proboscis that can reversibly extend more than 30 times its original body length in <30 s and perform this task repeatably (more than 20,000 times in its life cycle). Such large-scale morphodynamics-an ability to shapeshift in real time-can be quantified by the large strain and strain rate seen in a single cell. Fundamental limits of morphodynamics and how geometry encodes behavior in single cells remain largely unknown.

Protists display remarkable strategies to thrive in almost all ecological niches in our planet, from the deep sea to our river streams. Although the relationship between form and function is the bedrock of biological studies, we still have a poor understanding when it comes to explaining the explosive morphological diversity of protists. By applying a lens of geometry, we explored the link between form and function in an iconic, shapeshifting protist, L. olor, which is known to capture prey through the dynamics of an ultralong necklike proboscis. With the advent of various cutting-edge imaging tools, we mapped subcellular components such as the cortical cytoskeleton and membrane architecture of this cell caught in various morphological states, including a contracted and an extended state. Because geometry is scale free, the essential features of the coupled cytoskeletal-membrane architecture can be captured in a scaled-up physical origami model. In this work, we demonstrate how topological singularities in this geometry can control physical transformation of a cell. The deployment of a necklike protrusion through cellular scale origami is one of the largest strain and extension rates observed in a single cell.

We compared largest known strain and strain rates in single-cell morphodynamics and identified L. olor as an outlier. With high-resolution imaging, we discovered that this linear extension is supported by a helical architecture of the cortical cytoskeleton composed of microtubule bands layered in multiple layers forming membrane pleats. This particular geometry stores both membrane and microtubule filaments necessary for rapid deployment of a long proboscis, forming a curved crease origami. The sharp transition between the folded and unfolded state in this curved crease cellular origami is controlled by the presence of two topological singularities: a "d-cone" (developable cone) and a "twist singularity" of the microtubule band. We also built a scaled-up model of this origami to reveal how the coupled dynamics of d-cone and twist traversal leads to the nonaffine nature (spooling) of this deployable origami. Our work reveals how topological singularities can be used by a cell to control deployment of subcellular components and unravels the embodied nature of control of behavior through geometry in this ciliate.

As recent studies continue to highlight important ecological roles of protists, it has become critical to understand the origins of complex behavior in these remarkable single cells. Much effort has been put on mapping the genetic diversity of these cells, but we still know very little about the morphological (geometrical) diversity and its function in protists at large. By mapping the subcellular geometry of the cytoskeleton of L. olor, we uncovered geometrical control of extreme morphing behavior in a single cell. As a living example of a microtubule-patterned curved crease origami, our deeper understanding of this structure opens new doors for synthesis of cytoskeleton-based bioengineered materials with transformable characteristics such as deployability. Our work also provides direct inspiration for deployable microrobotics and light-weight space architecture. The blueprints we have been looking for to bring agency and embedded control in microrobotics might be hidden in plain sight in the geometrical diversity of protists.