Rigidity and Shape Transitions in Living and Nonliving Matter

NONTECHNICAL SUMMARY

This award supports theoretical and computational research and education to study materials undergoing transformations from floppy states to rigid states, much like a phase transition from liquid water to ice. Consider a silo of grain initially supporting a person's weight that suddenly collapses beneath them; the grain has transformed from a rigid state to a floppy state.

Consider a network of cytoskeletal fibers inside a biological cell becoming less dense to rapidly transform from a rigid state, that provides the cell with mechanical integrity, to a floppy state. These are examples of a rigidity transition. Understanding how rigidity transitions occur in both living and nonliving materials may enable their control - perhaps the grain can be made to keep supporting the person’s weight.

Another fundamental property of a material is its shape. Materials can transform from simple to more complex shapes. For example, a developing cerebellum, the “little brain”, begins as a smooth structure and becomes a foliated, or branched, structure in time.

A healthy biological cell contains a roughly spherical cell nucleus, while the cell nucleus of a diseased cell may contain bulges and be far from spherical. Abrupt shape transformations may be shape transitions which are prevalent in nature and help determine a system’s functionality. For example, the compartmentalization of DNA in the bulge of a cell nucleus may influence which genes are being transcribed. Understanding shape transitions and shape changes can illuminate how biology works.

The PI and her group will explore rigidity and shape transitions in different materials, including biomaterials, to arrive at an overarching, quantitative framework for rigidity and shape. Materials that will be studied include networks of cytoskeletal fibers, grain packings, developing brains, and cell nuclei. The team will investigate connections between properties of loops in the network of cytoskeletal fibers and rigidity.

They will construct a theory for the rigidity transition in granular packings that contains rigid clusters and floppy holes to pin down how friction between grains affects the rigidity transition. To study shape transitions, a cellular-based computational model will be developed for predicting how the cerebellum acquires its shape, and to determine how complex shapes arise in brain organoids.

A brain organoid is an artificially grown tissue model that mimics key features of a brain and can give insight into its workings. The PI will study a computational and simplified model for the shape and rigidity of a cell nucleus.

This project has a component aimed at recruiting more women and marginalized groups into physics. The PI will help lead a new bridge program as well as mentor undergraduates from underrepresented groups during the summer months and address ways to decouple the biological clock and the tenure clock. The PI will also continue to introduce farm physics to school children visiting a u-pick apple orchard/farm in Ithaca, NY.

The experience combines physics with orchardry/farming to make physics fun and helps create future physicists. TECHNICAL SUMMARY

This award supports theoretical and computational research and education to address open questions regarding emergent rigidity and shape in living and nonliving matter in which disorder is prevalent. The types of materials to be studied include a network of cytoskeletal fibers, granular packings, the developing brain, and a cell nucleus. Such materials toggle between fibrous materials and deformable particulate materials to enable progress towards an overarching, predictive framework for the rigidity and shape of disordered materials. The project is organized into four specific objectives.

1. Rigidity and convexity in underconstrained, disordered fiber networks. Underconstrained, disordered fiber networks can rigidify with sufficient external strain.

The PI and her group will explore how geometrical shape attributes of the loops in the network, such as convexity, correlates with rigidity. A focus on convexity may enable direct connections with spin models, which are well-studied, to better understand the nature of the transition. The PI and her group will also determine how the addition of inclusions in the fiber network, which provide additional constraints, influence the emergence of rigidity.

They will study the onset of nonlinear rigidity, or compression stiffening/softening, to better understand rigidity as a function of the packing fraction of inclusions, in addition to external strain and spring network connectivity.

2. The universality class search for the frictional jamming transition. The team can readily decompose a two-dimensional, frictional particle packing into rigid clusters interspersed with floppy holes to demonstrate that emergent rigidity, or frictional jamming, is defined as the onset of a spanning rigid cluster.

They will now extend the continuum theory for the elasticity of disordered materials further away from the frictional jamming transition to include deformable inclusions to mimic the presence of the floppy holes. Floppy holes, or floppy clusters, arise from the particle mechanics of the packings and are not fixed in size and shape. They grow as the rigidity transition is approached from the rigid phase.

The team will formulate a continuum theory for the floppy holes with rules that allow for exchanges between the two cluster types to arrive at a continuum description with two coupled fields. Using this theory, the team aims to investigate the underlying nature of the frictional jamming transition.

3. Developing brain organoids as a shape-changing material. The brain is a living material that changes shape as it develops.

Brain organoids also exhibit shape changes as they mature such as containing a core with globular-shaped cells and a boundary layer with extended-shape cells, or cortex. The team will use a 3D vertex model with cells as deformable polyhedrons to study cortex-core formation and cortex-lumen formation. They will also build a cellular-based version of the buckling without bending model for the developing cerebellum.

The aim is a predictive model for brain organoid and cerebellar shape at the tissue scale and the cell scale.

4. Activity driven-shape transitions in deformable cell nuclei. The mechanics of a cell nucleus can be minimally modeled as an active polymer(s) representing chromatin confined within, and tethered to, a deformable, thin elastic shell representing a lamina shell.

Nuclear blebs are extreme local deformations of the lamina shell. The team will study the initiation of nuclear bleb formation using the minimal mechanical model as a result of external stresses applied to the cell nucleus due to the surrounding cytoplasm and understand how such localized deformations affect chromatin rheology. This avenue of exploration will inspire new types of problems within classical elastic shell theory.

These objectives move toward building predictive models for the complex interplay between rigidity and shape that go well beyond mean-field constraint counting and discriminate when shape is synonymous with rigidity and when the shape of one object affects the rigidity of another, as is the case with an active polymer contained within an elastic shell.

This project has a component aimed at recruiting more women and marginalized groups into. The PI will help lead a new bridge program as well as mentor undergraduates from underrepresented groups during the summer months and address ways to decouple the biological clock and the tenure clock. The PI will also continue to introduce farm physics to school children visiting a u-pick apple orchard/farm in Ithaca, NY.

The experience combines physics with orchardry/farming to make physics fun and helps create future physicists.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.