arXiv subjects

Miranda Holmes-Cerfon

Publications and source records attributed to Miranda Holmes-Cerfon.

At least 19 recordsLinked to original sources

arXiv2026

Higher Order Rigidity and Energy

In this paper, we revisit the notion of higher-order rigidity of a bar-and-joint framework. In particular, we provide a link between the rigidity properties of a framework, and the growth order of an energy function defined on that framework. Using our approach, we propose a general definition for the rigidity order of a framework, and we show that this definition does not depend on the details of the chosen energy function. Then we show how this order can be studied using higher order derivative tests. Doing so, we obtain a new proof that the lack of a second order flex implies rigidity. Our proof relies on our construction of a fourth derivative test, which may be applied to a critical point when the second derivative test fails. We also obtain a new proof that when the dimension of non-trivial first-order flex coefficients $\p'$ equals $1$, then the lack of a $k$th order flex for some $k$ implies a framework is rigid. The higher order derivative tests that we study here may have applications in more general optimization problems.

math.MG↗

arXiv2026

Optimal interactions for addressable self-assembly

Addressable self-assembly asks that each building block assemble into a particular location in a target structure. Although particles may all be distinct, achieving high yield is a challenge because of monomer depletion: more target structures can nucleate than there are building blocks for, so they form partial fragments which cannot complete growth. We ask how to design the interactions between building blocks to achieve the highest yield in a given time. Using reaction equations describing all the intermediate steps of assembly, combined with numerical optimization, we show that the optimal interactions are such that (i) all bonds are either very strong or very weak, and (ii) the strong bonds form a spanning tree of the target structure. We then prove that when interactions form a spanning tree, monomer depletion cannot occur: assembly can always proceed downhill in energy space, with no kinetic traps. This result is a combinatorial property of the underlying interaction graph, and does not depend on the particular model for the kinetics. It suggests a robust design principle: create a network of strong interactions that has no loops, and make all other interactions much weaker. We validate this principle in numerical simulations of larger structures, and we further show that spanning trees that are more compact have typically better yield. Our results suggest a new framework for understanding monomer depletion and addressable self-assembly, which may be applied to DNA nanotechnology and which may give insight into the assembly pathways of certain multiprotein complexes.

cond-mat.soft↗

arXiv2026

Brownian dynamics with soft constraints in soft matter systems

Stiff forces, which bind objects together or otherwise confine motion, are found widely in soft-matter systems -- colloids with short range attractions, ligand-receptor contacts, particles in optical traps, fibres that resist stretching, etc. To assess the long-term effect of these stiff forces on dynamics and structure, it is useful to consider the limit where they are treated as constraints, so the system evolves strictly within allowed configurations. Efforts to derive equations involving both constraints, and the stochastic motion appropriate at the scales of soft matter, began around 50 years ago, yet, we are still lacking a straightforward way to extract the projected equations and apply them in modern formulations of mesoscale dynamics. Here, we address this gap with two key contributions: (1) a practical summary of the constrained Brownian dynamics equations with ``soft'' constraints, i.e. constraints imposed by stiff forces, which is illustrated through several representative examples, taking care to highlight the nontrivial effects of the constraints; and (2) a novel derivation using singular perturbation theory, establishing the validity of these equations over timescales exceeding the relaxation of stiffly constrained degrees of freedom. We further extend our approach to ``soft soft'' constraints, where mobility varies on lengthscales comparable to the restraining forces -- a scenario typical for particles in fluids experiencing hydrodynamic interactions. We hope our results will be useful for soft matter research, as a robust toolkit for studying tethered or confined systems.

cond-mat.soft↗

arXiv2026

A modified Euler-Maruyama method to simulate a one-dimensional sticky diffusion

A sticky diffusion is a process that can stick to and detach from a lower-dimensional boundary. A challenge in simulating such a process is in capturing the change in dimension in a dynamically consistent way. We introduce a numerical algorithm to simulate a one-dimensional sticky diffusion, which sticks to and detaches from a point. Our method is a simple modification of the standard Euler-Maruyama scheme, which chooses with some probability between a reflected Euler-Maruyama update and a jump to the sticky point. We show how to choose this probability to be consistent with the generator of the desired dynamics, and we prove that our scheme converges weakly to a sticky diffusion with order 1.

math.NA↗

arXiv2025

A Constrained Saddle Search Approach for Constructing Singular and Flexible Bar Frameworks

Singularity analysis is essential in robot kinematics, as singular configurations cause loss of control and kinematic indeterminacy. This paper models singularities in bar frameworks as saddle points on constrained manifolds. Given an under-constrained, non-singular bar framework, by allowing one edge to vary its length while fixing lengths of others, we define the squared length of the free edge as an energy functional and show that its local saddle points correspond to singular and flexible frameworks. Using our constrained saddle search approach, we identify previously unknown singular and flexible bar frameworks, providing new insights into singular robotics design and analysis.

cs.RO↗

arXiv2025

A Constrained Optimization Approach for Constructing Rigid Bar Frameworks with Higher-order Rigidity

We present a systematic approach for constructing bar frameworks that are rigid but not first-order rigid, using constrained optimization. We show that prestress stable (but not first-order rigid) frameworks arise as the solution to a simple optimization problem, which asks to maximize or minimize the length of one edge while keep the other edge lengths fixed. By starting with a random first-order rigid framework, we can thus design a wide variety of prestress stable frameworks, which, unlike many examples known in the literature, have no special symmetries. We then show how to incorporate a bifurcation method to design frameworks that are third-order rigid. Our results highlight connections between concepts in rigidity theory and constrained optimization, offering new insights into the construction and analysis of bar frameworks with higher-order rigidity.

math.MG↗

arXiv2025

Hierarchical self-assembly for high-yield addressable complexity at fixed conditions

There is evidence that the self-assembly of complex molecular systems often proceeds hierarchically, by first building subunits that later assemble in larger entities, in a process that can repeat multiple times. Yet, our understanding of this phenomenon and its performance is limited. Here we introduce a simple model for hierarchical addressable self-assembly, where interactions between particles can be optimised to maximise the fraction of a well-formed target structure, or yield. We find that a hierarchical strategy leads to an impressive yield up to at least five generations of the hierarchy, and does not require a cycle of temperatures as used in previous methods. High yield is obtained when the microscopic interaction decreases with the scale of units considered, such that the total interaction between intermediate structures remains identical at all scales. We provide thermodynamic and dynamical arguments constraining the interaction strengths where this strategy is effective. Overall, our work characterizes an alternative strategy for addressable self-assembly at a fixed temperature, and provides insight into the mechanisms sustaining hierarchical assembly in biology.

cond-mat.soft↗

arXiv2023

Hopping and crawling DNA-coated colloids

Understanding the motion of particles with ligand-receptors is important for biomedical applications and material design. Yet, even among a single design, the prototypical DNA-coated colloids, seemingly similar micrometric particles hop or roll, depending on the study. We shed light on this problem by observing DNA-coated colloids diffusing near surfaces coated with complementary strands for a wide array of coating designs. We find colloids rapidly switch between 2 modes: they hop - with long and fast steps - and crawl - with short and slow steps. Both modes occur at all temperatures around the melting point and over a wide array of designs. The particles become increasingly subdiffusive as temperature decreases, in line with subsequent velocity steps becoming increasingly anti-correlated. Overall, crawling (or hopping) phases are more predominant at low (or high) temperatures; crawling is also more efficient at low temperatures than hopping to cover large distances. We rationalize this behavior within a simple model: at lower temperatures, the number of bound strands increases, and detachment of all bonds is unlikely, hence, hopping is prevented and crawling favored. We thus reveal the mechanism behind a common design rule relying on increased strand density for long-range self-assembly: dense strands on surfaces are required to enable crawling, possibly facilitating particle rearrangements.

cond-mat.soft↗

arXiv2023

Monte Carlo on manifolds in high dimensions

We introduce an efficient numerical implementation of a Markov Chain Monte Carlo method to sample a probability distribution on a manifold (introduced theoretically in Zappa, Holmes-Cerfon, Goodman (2018)), where the manifold is defined by the level set of constraint functions, and the probability distribution may involve the pseudodeterminant of the Jacobian of the constraints, as arises in physical sampling problems. The algorithm is easy to implement and scales well to problems with thousands of dimensions and with complex sets of constraints provided their Jacobian retains sparsity. The algorithm uses direct linear algebra and requires a single matrix factorization per proposal point, which enhances its efficiency over previously proposed methods but becomes the computational bottleneck of the algorithm in high dimensions. We test the algorithm on several examples inspired by soft-matter physics and materials science to study its complexity and properties.

stat.CO↗

arXiv2022

Mass changes the diffusion coefficient of particles with ligand-receptor contacts in the overdamped limit

Inertia does not generally affect the long-time diffusion of passive overdamped particles in fluids. Yet a model starting from the Langevin equation predicts a surprising property of particles coated with ligands, that bind reversibly to surface receptors -- heavy particles diffuse more slowly than light ones of the same size. We show this by simulation and by deriving an analytic formula for the mass-dependent diffusion coefficient in the overdamped limit. We estimate the magnitude of this effect for a range of biophysical ligand-receptor systems, and find it is potentially observable for tailored micronscale DNA-coated colloids.

cond-mat.soft↗

arXiv2022

The Nanocaterpillar's Random Walk: Diffusion With Ligand-Receptor Contacts

Particles with ligand-receptor contacts bind and unbind fluctuating "legs" to surfaces, whose fluctuations cause the particle to diffuse. Quantifying the diffusion of such "nanoscale caterpillars" is a challenge, since binding events often occur on very short time and length scales. Here we derive an analytical formula, validated by simulations, for the long time translational diffusion coefficient of an overdamped nanocaterpillar, under a range of modeling assumptions. We demonstrate that the effective diffusion coefficient, which depends on the microscopic parameters governing the legs, can be orders of magnitude smaller than the background diffusion coefficient. Furthermore it varies rapidly with temperature, and reproduces the striking variations seen in existing data and our own measurements of the diffusion of DNA-coated colloids. Our model gives insight into the mechanism of motion, and allows us to ask: when does a nanocaterpillar prefer to move by sliding, where one leg is always linked to the surface, and when does it prefer to move by hopping, which requires all legs to unbind simultaneously? We compare a range of systems (viruses, molecular motors, white blood cells, protein cargos in the nuclear pore complex, bacteria such as Escherichia coli, and DNA-coated colloids) and present guidelines to control the mode of motion for materials design.

cond-mat.soft↗

arXiv2021

Comprehensive view of microscopic interactions between DNA-coated colloids

The self-assembly of DNA-coated colloids into highly-ordered structures offers great promise for advanced optical materials. However, control of disorder, defects, melting, and crystal growth is hindered by the lack of a microscopic understanding of DNA-mediated colloidal interactions. Here we use total internal reflection microscopy to measure in situ the interaction potential between DNA-coated colloids with nanometer resolution and the macroscopic melting behavior. The range and strength of the interaction are measured and linked to key material design parameters, including DNA sequence, polymer length, grafting density, and complementary fraction. We present a first-principles model that quantitatively reproduces our experimental data without fitting parameters over a wide range of DNA ligand designs. Our theory identifies a subtle competition between DNA binding and steric repulsion and accurately predicts adhesion and melting at a molecular level. Combining experimental and theoretical results, our work provides a quantitative and predictive approach for guiding material design with DNA-nanotechnology and can be further extended to a diversity of colloidal and biological systems.

cond-mat.soft↗

arXiv2021

Thermodynamic stability versus Kinetic Accessibility: Pareto Fronts for Programmable Self-Assembly

A challenge in designing self-assembling building blocks is to ensure the target state is both thermodynamically stable and kinetically accessible. These two objectives are known to be typically in competition, but it is not known how to simultaneously optimize them. We consider this problem through the lens of multi-objective optimization theory: we develop a genetic algorithm to compute the Pareto fronts characterizing the tradeoff between equilibrium probability and folding rate, for a model system of small polymers of colloids with tunable short-ranged interaction energies. We use a coarse-grained model for the particles' dynamics that allows us to efficiently search over parameters, for systems small enough to be enumerated. For most target states there is a tradeoff when the number of types of particles is small, with medium-weak bonds favouring fast folding, and strong bonds favouring high equilibrium probability. The tradeoff disappears when the number of particle types reaches a value $m_*$, that is usually much less than the total number of particles. This general approach of computing Pareto fronts allows one to identify the minimum number of design parameters to avoid a thermodynamic-kinetic tradeoff. However, we argue, by contrasting our coarse-grained model's predictions with those of Brownian dynamics simulations, that particles with short-ranged isotropic interactions should generically have a tradeoff, and avoiding it in larger systems will require orientation-dependent interactions.

cond-mat.soft↗

arXiv2021

Transverse Rigidity is Prestress Stability

Recently, V. Alexandrov proposed an intriguing sufficient condition for rigidity, which we will call "transverse rigidity". We show that transverse rigidity is actually equivalent to the known sufficient condition for rigidity called "prestress stability". Indeed this leads to a novel interpretation of the prestress condition.

math.MG↗

arXiv2021

Realizations of Isostatic Material Frameworks

This paper studies the set of equivalent realizations of isostatic frameworks in two dimensions, and algorithms for finding all such realizations. We show that an isostatic framework has an even number of equivalent realizations that preserve edge lengths and connectivity. We enumerate the complete set of equivalent realizations for a toy framework with pinned boundary in two dimensions and study the impact of boundary length on the emergence of these realizations. To ameliorate the computational complexity of finding a solution to a large multivariate quadratic system corresponding to the constraints; alternative methods - based on constraint reduction and distance-based covering map or Cayley parameterization of the search space - are presented. The application of these methods is studied on atomic clusters, a model two-dimensional glasses, and jamming.

cond-mat.dis-nn↗

arXiv2020

Almost-rigidity of frameworks

We extend the mathematical theory of rigidity of frameworks (graphs embedded in $d$-dimensional space) to consider nonlocal rigidity and flexibility properties. We provide conditions on a framework under which (I) as the framework flexes continuously it must remain inside a small ball, a property we call "almost-rigidity"; (II) any other framework with the same edge lengths must lie outside a much larger ball; (III) if the framework deforms by some given amount, its edge lengths change by a minimum amount; (IV) there is a nearby framework that is prestress stable, and thus rigid. The conditions can be tested efficiently using semidefinite programming. The test is a slight extension of the test for prestress stability of a framework, and gives analytic expressions for the radii of the balls and the edge length changes. Examples illustrate how the theory may be applied in practice, and we provide an algorithm to test for rigidity or almost-rigidity. We briefly discuss how the theory may be applied to tensegrities.

math.MG↗

arXiv2020

Simulating sticky particles: A Monte Carlo method to sample a stratification

Many problems in materials science and biology involve particles interacting with strong, short-ranged bonds, that can break and form on experimental timescales. Treating such bonds as constraints can significantly speed up sampling their equilibrium distribution, and there are several methods to sample probability distributions subject to fixed constraints. We introduce a Monte Carlo method to handle the case when constraints can break and form. More generally, the method samples a probability distribution on a stratification: a collection of manifolds of different dimensions, where the lower-dimensional manifolds lie on the boundaries of the higher-dimensional manifolds. We show several applications of the method in polymer physics, self-assembly of colloids, and volume calculation in high dimensions.

math.NA↗

arXiv2020

From canyons to valleys: Numerically continuing sticky hard sphere clusters to the landscapes of smoother potentials

We study the energy landscapes of particles with short-range attractive interactions as the range of the interactions increases. Starting with the set of local minima for $6\leq N\leq12$ hard spheres that are "sticky", i.e. they interact only when their surfaces are exactly in contact, we use numerical continuation to evolve the local minima (clusters) as the range of the potential increases, using both the Lennard-Jones and Morse families of interaction potentials. As the range increases, clusters merge, until at long ranges only one or two clusters are left. We compare clusters obtained by continuation with different potentials and find that for short and medium ranges, up to about 30\% of particle diameter, the continued clusters are nearly identical, both within and across families of potentials. For longer ranges the clusters vary significantly, with more variation between families of potentials than within a family. We analyze the mechanisms behind the merge events, and find that most rearrangements occur when a pair of non-bonded particles comes within the range of the potential. An exception occurs for nonharmonic clusters, those that have a zero eigenvalue in their Hessian, which undergo a more global rearrangement.

cond-mat.soft↗