arXiv subjects

Jean-Pierre Eckmann

Publications and source records attributed to Jean-Pierre Eckmann.

At least 19 recordsLinked to original sources

arXiv2026

Geometric-Chemical Distance Between Protein Surfaces

Proteins recognize, bind, and catalyze through molecular surfaces, where geometry and chemical patterning determine interaction. Comparing these surfaces requires both a geometric--chemical distance and a correspondence that relates one complete surface to another. Here we introduce IFACE (Intrinsic Field--Aligned Coupled Embedding). IFACE derives a symmetric geometric--chemical distance by optimizing a probabilistic coupling over intrinsic geometry, mean curvature, electrostatics, hydrophobicity, and hydrogen-bond propensity. The same coupling provides an explicit surface map. For molecular-dynamics conformers, IFACE distinguishes the same protein from distinct proteins more accurately than TM-distance and a Laplace--Beltrami spectral distance. A Jensen--Shannon distribution distance performs best in this binary identity test, because aggregate surface-feature distributions already identify each protein. A distance must also satisfy a global requirement: its pairwise values must place many distinct protein surfaces consistently in one space. We therefore tested IFACE across six protein families. It produces the strongest family classification and clustering among the distributional, spectral, MaSIF, and SurfaceID comparisons. The inferred maps preserve geodesic neighborhoods and transfer heme-centered pocket regions across cytochrome P450 proteins. IFACE therefore provides, from one construction, both a distance between complete protein surfaces and the local map that explains that distance.

q-bio.BM↗

arXiv2025

Walks in Rotation Spaces Return Home when Doubled and Scaled

The dynamics of numerous physical systems, such as spins and qubits, can be described as a series of rotation operations, i.e., walks in the manifold of the rotation group. A basic question with practical applications is how likely and under what conditions such walks return to the origin (the identity rotation), which means that the physical system returns to its initial state. In three dimensions, we show that almost every walk in SO(3) or SU(2), even a very complicated one, will preferentially return to the origin simply by traversing the walk twice in a row and uniformly scaling all rotation angles. We explain why traversing the walk only once almost never suffices to return, and comment on the problem in higher dimensions.

cond-mat.stat-mech↗

arXiv2024

Tumbling Downhill along a Given Curve

A cylinder will roll down an inclined plane in a straight line. A cone will roll around a circle on that plane and then will stop rolling. We ask the inverse question: For which curves drawn on the inclined plane $\mathbb{R}^2$ can one carve a shape that will roll downhill following precisely this prescribed curve and its translationally repeated copies? This simple question has a solution essentially always, but it turns out that for most curves, the shape will return to its initial orientation only after crossing a few copies of the curve - most often two copies will suffice, but some curves require an arbitrarily large number of copies.

math-ph↗

arXiv2024

Active self-disassembly enhances the yield of self-assembled structures

We introduce a lattice model to probe the effect of active self-disassembly on equilibrium self-assembly. Surprisingly, we find conditions under which active self-disassembly enhances the yield of a target structure above that achieved by self-assembly alone when the latter is already favoured thermodynamically. We discuss biological implications of our findings.

cond-mat.soft↗

arXiv2023

The detection of relativistic corrections in cosmological N-body simulations

Cosmological N-body simulations are done on massively parallel computers. This necessitates the use of simple time integrators, and, additionally, of mesh-grid approximations of the potentials. Recently, Adamek et al. (2015); Barrera-Hinojosa et al. (2019) have developed general relativistic N-body simulations to capture relativistic effects mainly for cosmological purposes. We therefore ask whether, with the available technology, relativistic effects like perihelion advance can be detected numerically to a relevant precision. We first study the spurious perihelion shift in the Kepler problem, as a function of the integration method used, and then as a function of an additional interpolation of forces on a 2-dimensional lattice. This is done for several choices of eccentricities and semi-major axes. Using these results, we can predict which precisions and lattice constants allow for a detection of the relativistic perihelion advance in N-body simulation. We find that there are only small windows of parameters -- such as eccentricity, distance from the central object and the Schwarzschild radius -- for which the corrections can be detected in the numerics.

math.NA↗

arXiv2023

An Introduction to the Unpublished Book "Reflections on a Tube" by Mitchell J. Feigenbaum

This paper is an adaptation of the introduction to a book project by the late Mitchell J. Feigenbaum (1944-2019). While Feigenbaum is certainly mostly known for his theory of period doubling cascades, he had a lifelong interest in optics. His book project is an extremely original discussion of the apparently very simple study of anamorphs, that is, the reflections of images on a cylindrical mirror. He observed that there are \emph{two images} to be seen in the tube, and discovered that the brain preferentially chooses one of them. I edited and wrote an introduction to this planned book. As the book is still not published, I have now adapted my introduction as a standalone version, so that some of Feigenbaum's remarkable work will be accessible to a larger audience.

math.HO↗

arXiv2023

Instabilities Appearing in Cosmological Effective Field theories: When and How?

Nonlinear partial differential equations appear in many domains of physics, and we study here a typical equation which one finds in effective field theories (EFT) originated from cosmological studies. In particular, we are interested in the equation $\partial_t^2 u(x,t) = α(\partial_x u(x,t))^2 +β\partial_x^2 u(x,t)$ in $1+1$ dimensions. It has been known for quite some time that solutions to this equation diverge in finite time, when $α>0$. We study the nature of this divergence as a function of the parameters $α>0 $ and $β\ge0$. The divergence does not disappear even when $β$ is very large contrary to what one might believe (note that since we consider fixed initial data, $α$ and $β$ cannot be scaled away). But it will take longer to appear as $β$ increases when $α$ is fixed. We note that there are two types of divergence and we discuss the transition between these two as a function of parameter choices. The blowup is unavoidable unless the corresponding equations are modified. Our results extend to $3+1$ dimensions.

math-ph↗

arXiv2023

Abelian Sandpiles on Cylinders

We study here a variant of the Abelian Sandpile Model, where the playground is a cylinder of width $w$ and of circumference c. When c << w, we describe a phenomenon which has not been observed in other geometries: the probability distribution of avalanche sizes has a ladder structure, with the first step consisting of avalanches of size up to w c/2 that are essentially equiprobable, except for a small exponential tail of order about 10c. We explain this phenomenon and describe subsequent steps.

cond-mat.stat-mech↗

arXiv2022

General theory of specific binding: insights from a genetic-mechano-chemical protein model

Proteins need to selectively interact with specific targets among a multitude of similar molecules in the cell. But despite a firm physical understanding of binding interactions, we lack a general theory of how proteins evolve high specificity. Here, we present such a model that combines chemistry, mechanics and genetics, and explains how their interplay governs the evolution of specific protein-ligand interactions. The model shows that there are many routes to achieving molecular discrimination - by varying degrees of flexibility and shape/chemistry complementarity - but the key ingredient is precision. Harder discrimination tasks require more collective and precise coaction of structure, forces and movements. Proteins can achieve this through correlated mutations extending far from a binding site, which fine-tune the localized interaction with the ligand. Thus, the solution of more complicated tasks is enabled by increasing the protein size, and proteins become more evolvable and robust when they are larger than the bare minimum required for discrimination. The model makes testable, specific predictions about the role of flexibility and shape mismatch in discrimination, and how evolution can independently tune affinity and specificity. Thus, the proposed theory of specific binding addresses the natural question of "why are proteins so big?". A possible answer is that molecular discrimination is often a hard task best performed by adding more layers to the protein.

q-bio.BM↗

arXiv2022

Revisiting the Monge problem in the Landauer limit

We discuss the Monge problem of mass transportation in the framework of stochastic thermodynamics and revisit the problem of the Landauer limit for finite-time thermodynamics, a problem that got the interest of Krzysztof Gawedzki in the last years. We show that restricted to one dimension, optimal transportation is efficiently solved numerically by well known methods from differential equations. We add a brief discussion about the relevance this has on optimising the processing in modern computers.

cond-mat.stat-mech↗

arXiv2022

Imaging in Reflecting Spheres

We study the formation of images in a reflective sphere in three configurations using caustics of the field of light rays. The optical wavefront emerging from a source point reaching a subject following passage through the optical system is, in general, a Gaussian surface with partial focus along the two principal directions of the Gaussian surface; i.e. there are two images of the source point, each with partial focus. As the source point moves, the images move on two surfaces, referred to as \emph{viewable surfaces}. In our systems, one viewable surface consists of points with radial focus and the other consists of points with azimuthal focus. The problems we study are (1) imaging of a parallel beam of light, (2) imaging of the infinite viewed from a location outside the sphere, and (3) imaging of a planar object viewed through the point of its intersection with the radial line normal to the plane. We verify the existence of two images experimentally and show that the distance between them agrees with the computations.

physics.optics↗

arXiv2021

Dimensional Reduction in Complex Living Systems: Where, Why, and How

The unprecedented prowess of measurement techniques provides a detailed, multi-scale look into the depths of living systems. Understanding these avalanches of high-dimensional data -- by distilling underlying principles and mechanisms -- necessitates dimensional reduction. We propose that living systems achieve exquisite dimensional reduction, originating from their capacity to learn, through evolution and phenotypic plasticity, the relevant aspects of a non-random, smooth physical reality. We explain how geometric insights by mathematicians allow one to identify these genuine hallmarks of life and distinguish them from universal properties of generic data sets. We illustrate these principles in a concrete example of protein evolution, suggesting a simple general recipe that can be applied to understand other biological systems.

q-bio.OT↗

arXiv2021

Broken Pencils and Moving Rulers: After an unpublished book by Mitchell Feigenbaum

Mitchell Feigenbaum discovered an intriguing property of viewing images through cylindrical mirrors or looking into water. Because the eye is a lens with an opening of about 5mm, many different rays of reflected images reach the eye, and need to be interpreted by the visual system. This has the surprising effect that what one perceives depends on the orientation of the head, whether it is tilted or not. I explain and illustrate this phenomenon on the example of a human eye looking at a ruler immersed in water.

physics.class-ph↗

arXiv2019

Proteins: the physics of amorphous evolving matter

Proteins are a matter of dual nature. As a physical object, a protein molecule is a folded chain of amino acids with multifarious biochemistry. But it is also an instantiation along an evolutionary trajectory determined by the function performed by the protein within a hierarchy of interwoven interaction networks of the cell, the organism and the population. A physical theory of proteins therefore needs to unify both aspects, the biophysical and the evolutionary. Specifically, it should provide a model of how the DNA gene is mapped into the functional phenotype of the protein. We review several physical approaches to the protein problem, focusing on a mechanical framework which treats proteins as evolvable condensed matter: Mutations introduce localized perturbations in the gene, which are translated to localized perturbations in the protein matter. A natural tool to examine how mutations shape the phenotype are Green's functions. They map the evolutionary linkage among mutations in the gene (termed epistasis) to cooperative physical interactions among the amino acids in the protein. We discuss how the mechanistic view can be applied to examine basic questions of protein evolution and design.

q-bio.BM↗

arXiv2019

Decay of Hamiltonian Breathers under Dissipation

We study metastable behavior in a discrete nonlinear Schrödinger equation from the viewpoint of Hamiltonian systems theory. When there are $n < \infty$ sites in this equation, we consider initial conditions in which almost all the energy is concentrated in one end of the system. We are interested in understanding how energy flows through the system, so we add a dissipation of size $γ$ at the opposite end of the chain, and we show that the energy decreases extremely slowly. Furthermore, the motion is localized in the phase space near a family of breather solutions for the undamped system. We give rigorous, asymptotic estimates for the rate of evolution along the family of breathers and the width of the neighborhood within which the trajectory is confined.

math.DS↗

arXiv2018

Non-Equilibrium Steady States for Networks of Oscillators

Non-equilibrium steady states for chains of oscillators (masses) connected by harmonic and anharmonic springs and interacting with heat baths at different temperatures have been the subject of several studies. In this paper, we show how some of the results extend to more complicated networks. We establish the existence and uniqueness of the non-equilibrium steady state, and show that the system converges to it at an exponential rate. The arguments are based on controllability and conditions on the potentials at infinity.

math-ph↗

arXiv2018

Green function of correlated genes in a minimal mechanical model of protein evolution

The function of proteins arises from cooperative interactions and rearrangements of their amino acids, which exhibit large-scale dynamical modes. Long-range correlations have also been revealed in protein sequences, and this has motivated the search for physical links between the observed genetic and dynamic cooperativity. We outline here a simplified theory of protein, which relates sequence correlations to physical interactions and to the emergence of mechanical function. Our protein is modeled as a strongly-coupled amino acid network whose interactions and motions are captured by the mechanical propagator, the Green function. The propagator describes how the gene determines the connectivity of the amino acids, and thereby the transmission of forces. Mutations introduce localized perturbations to the propagator which scatter the force field. The emergence of function is manifested by a topological transition when a band of such perturbations divides the protein into subdomains. We find that epistasis -- the interaction among mutations in the gene -- is related to the nonlinearity of the Green function, which can be interpreted as a sum over multiple scattering paths. We apply this mechanical framework to simulations of protein evolution, and observe long-range epistasis which facilitates collective functional modes.

q-bio.QM↗

arXiv2018

On the geometry of chemical reaction networks: Lyapunov function and large deviations

In an earlier paper, we proved the validity of large deviations theory for the particle approximation of quite general chemical reaction networks (CRNs). In this paper, we extend its scope and present a more geometric insight into the mechanism of that proof, exploiting the notion of spherical image of the reaction polytope. This allows to view the asymptotic behavior of the vector field describing the mass-action dynamics of chemical reactions as the result of an interaction between the faces of this polytope in different dimensions. We also illustrate some local aspects of the problem in a discussion of Wentzell-Freidlin (WF) theory, together with some examples.

math.DS↗