Planar morphometrics using Teichmüller maps

Inspired by the question of quantifying wing shape,
we propose a computational approach for analysing
planar shapes. We first establish a correspondence
between the boundaries of two planar shapes with
boundary landmarks using geometric functional data
analysis and then compute a landmark-matching
curvature-guided Teichmüller mapping with uniform
quasi-conformal distortion in the bulk. This allows
us to analyse the pair-wise difference between the
planar shapes and construct a similarity matrix on
which we deploy methods from network analysis
to cluster shapes. We deploy our method to study
a variety of Drosophila wings across species to
highlight the phenotypic variation between them,
and Lepidoptera wings over time to study the
developmental progression of wings. Our approach
of combining complex analysis, computation and
statistics to quantify, compare and classify planar
shapes may be usefully deployed in other biological
and physical systems.

Rigidity percolation and geometric information in floppy origami

Origami structures with a large number of excess folds are capable of storing distinguishable geometric states that are energetically equivalent. As the number of excess folds is reduced, the system has fewer equivalent states and can eventually become rigid. We quantify this transition from a floppy to a rigid state as a function of the presence of folding constraints in a classic origami tessellation, Miura-ori. We show that in a fully triangulated Miura-ori that is maximally floppy, adding constraints via the elimination of diagonal folds in the quads decreases the number of degrees of freedom in the system, first linearly and then nonlinearly. In the nonlinear regime, mechanical cooperativity sets in via a redundancy in the assignment of constraints, and the degrees of freedom depend on constraint density in a scale-invariant manner. A percolation transition in the redundancy in the constraints as a function of constraint density suggests how excess folds in an origami structure can be used to store geometric information in a scale-invariant way.

Spatial control of irreversible protein aggregation

Liquid cellular compartments form in the cyto- or nucleoplasm and can regulate aberrant protein aggregation. Yet, the mechanisms by which these compartments affect protein aggregation remain unknown. Here, we combine kinetic theory of protein aggregation and liquid-liquid phase separation to study the spatial control of irreversible protein aggregation in the presence of liquid compartments. We find that even for weak interactions aggregates strongly partition into the liquid compartment. Aggregate partitioning is caused by a positive feedback mechanism of aggregate nucleation and growth driven by a flux maintaining the phase equilibrium between the compartment and its surrounding. Our model establishes a link between specific aggregating systems and the physical conditions maximizing aggregate partitioning into the compartment. The underlying mechanism of aggregate partitioning could be used to confine cytotoxic protein aggregates inside droplet-like compartments but may also represent a common mechanism to spatially control irreversible chemical reactions in general.

Generalized Erdös numbers for network analysis

The identification of relationships in complex networks is critical
in a variety of scientific contexts. This includes the identification of
globally central nodes and analysing the importance of pairwise
relationships between nodes. In this paper, we consider the
concept of topological proximity (or ‘closeness’) between nodes
in a weighted network using the generalized Erdo´´s numbers
(GENs). This measure satisfies a number of desirable properties
for networks with nodes that share a finite resource. These
include: (i) real-valuedness, (ii) non-locality and (iii) asymmetry.
We show that they can be used to define a personalized
measure of the importance of nodes in a network with a natural
interpretation that leads to new methods to measure centrality.
We show that the square of the leading eigenvector of an
importance matrix defined using the GENs is strongly correlated
with well-known measures such as PageRank, and define a
personalized measure of centrality that is also well correlated
with other existing measures. The utility of this measure
of topological proximity is demonstrated by showing the
asymmetries in both the dynamics of random walks and the
mean infection time in epidemic spreading are better
predicted by the topological definition of closeness provided
by the GENs than they are by other measures.

Shape-shifting of things to come

Complex lattices that change in response to stimuli open a range of applications in electronics, robotics, medicin

Protein Clumping Best Blocked Using Specific Compounds at Distinct Disease Stages, Model Suggests

A mathematical model created to more effectively prevent protein clumping, widely thought to underlie diseases like Parkinson’s, found that different potential treatments work best at different disease stages

How early-stage embryos maintain their size

Mechanical cues play critical role in shaping form and function

Cutting into shape

Mathematical framework turns any sheet of material into any shape using kirigami cuts

Using math to help treat Alzheimer’s, Parkinson’s and other diseases

Insights into treatments for protein aggregation diseases from control theory and chemical kinetics

Using origami memory to encode geometric information in floppy structures

Researchers develop method to control the rigidity of structures through origami folds