arXiv subjects

Michael A. Lomholt

Publications and source records attributed to Michael A. Lomholt.

At least 19 recordsLinked to original sources

arXiv2025

Objective comparison of methods to decode anomalous diffusion

Deviations from Brownian motion leading to anomalous diffusion are found in transport dynamics from quantum physics to life sciences. The characterization of anomalous diffusion from the measurement of an individual trajectory is a challenging task, which traditionally relies on calculating the trajectory mean squared displacement. However, this approach breaks down for cases of practical interest, e.g., short or noisy trajectories, heterogeneous behaviour, or non-ergodic processes. Recently, several new approaches have been proposed, mostly building on the ongoing machine-learning revolution. To perform an objective comparison of methods, we gathered the community and organized an open competition, the Anomalous Diffusion challenge (AnDi). Participating teams applied their algorithms to a commonly-defined dataset including diverse conditions. Although no single method performed best across all scenarios, machine-learning-based approaches achieved superior performance for all tasks. The discussion of the challenge results provides practical advice for users and a benchmark for developers.

physics.data-an↗

arXiv2024

Target search on DNA -- effect of coiling

Some proteins can find their targets on DNA faster than by pure diffusion in the three-dimensional cytoplasm, through the process of facilitated diffusion: They can loosely bind to DNA and temporarily slide along it, thus being guided by the DNA molecule itself to the target. This chapter examines this process in mathematical detail with a focus on including the effect of DNA coiling on the search process.

q-bio.BM↗

arXiv2022

Bayesian inference of scaled versus fractional Brownian motion

We present a Bayesian inference scheme for scaled Brownian motion, and investigate its performance on synthetic data for parameter estimation and model selection in a combined inference with fractional Brownian motion. We include the possibility of measurement noise in both models. We find that for trajectories of a few hundred time points the procedure is able to resolve well the true model and parameters. Using the prior of the synthetic data generation process also for the inference, the approach is optimal based on decision theory. We include a comparison with inference using a prior different from the data generating one.

stat.ME↗

arXiv2021

Bayesian inference of Lévy walks via hidden Markov models

The Lévy walk is a non-Brownian random walk model that has been found to describe anomalous dynamic phenomena in diverse fields ranging from biology over quantum physics to ecology. Recurrently occurring problems are to examine whether observed data are successfully quantified by a model classified as Lévy walks or not and extract the best model parameters in accordance with the data. Motivated by such needs, we propose a hidden Markov model for Lévy walks and computationally realize and test the corresponding Bayesian inference method. We introduce a Markovian decomposition scheme to approximate a renewal process governed by a power-law waiting time distribution. Using this, we construct the likelihood function of Lévy walks based on a hidden Markov model and the forward algorithm. With the Lévy walk trajectories simulated at various conditions, we perform the Bayesian inference for parameter estimation and model classification. We show that the power-law exponent of the flight-time distribution can be successfully extracted even at the condition that the mean-squared displacement does not display the expected scaling exponent due to the noise or insufficient trajectory length. It is also demonstrated that the Bayesian method performs remarkably inferring the Lévy walk trajectories from given unclassified trajectory data set if the noise level is moderate.

physics.bio-ph↗

arXiv2020

Stochastic unfolding of nanoconfined DNA: experiments, model and Bayesian analysis

Nanochannels provide means for detailed experiments on the effect of confinement on biomacromolecules, such as DNA. We here introduce a model for the complete unfolding of DNA from the circular to linear configuration. Two main ingredients are the entropic unfolding force as well as the friction coefficient for the unfolding process, and we describe the associated dynamics by a non-linear Langevin equation. By analyzing experimental data where DNA molecules are photo-cut and unfolded inside a nanochannel, our model allows us to extract values for the unfolding force as well as the friction coefficient for the first time. In order to extract numerical values for these physical quantities, we employ a recently introduced Bayesian inference framework. We find that the determined unfolding force is in agreement with estimates from a simple Flory type argument. The estimated friction coefficient is in agreement with theoretical estimates for motion of a cylinder in a channel. We further validate the estimated friction constant by extracting this parameter from DNA's center-of-mass motion before and after unfolding, yielding decent agreement.

physics.bio-ph↗

arXiv2018

Fitting a function to time-dependent ensemble averaged data

Time-dependent ensemble averages, i.e., trajectory-based averages of some observable, are of importance in many fields of science. A crucial objective when interpreting such data is to fit these averages (for instance, squared displacements) with a function and extract parameters (such as diffusion constants). A commonly overlooked challenge in such function fitting procedures is that fluctuations around mean values, by construction, exhibit temporal correlations. We show that the only available general purpose function fitting methods, correlated chi-square method and the weighted least squares method (which neglects correlation), fail at either robust parameter estimation or accurate error estimation. We remedy this by deriving a new closed-form error estimation formula for weighted least square fitting. The new formula uses the full covariance matrix, i.e., rigorously includes temporal correlations, but is free of the robustness issues, inherent to the correlated chi-square method. We demonstrate its accuracy in four examples of importance in many fields: Brownian motion, damped harmonic oscillation, fractional Brownian motion and continuous time random walks. We also successfully apply our method, weighted least squares including correlation in error estimation (WLS-ICE), to particle tracking data. The WLS-ICE method is applicable to arbitrary fit functions, and we provide a publically available WLS-ICE software.

physics.data-an↗

arXiv2018

Bayesian model selection with fractional Brownian motion

We implement Bayesian model selection and parameter estimation for the case of fractional Brownian motion with measurement noise and a constant drift. The approach is tested on artificial trajectories and shown to make estimates that match well with the underlying true parameters, while for model selection the approach has a preference for simple models when the trajectories are finite. The approach is applied to observed trajectories of vesicles diffusing in Chinese hamster ovary cells. Here it is supplemented with a goodness-of-fit test, which is able to reveal statistical discrepancies between the observed trajectories and model predictions.

physics.data-an↗

arXiv2017

Bayesian inference with information content model check for Langevin equations

The Bayesian data analysis framework has been proven to be a systematic and effective method of parameter inference and model selection for stochastic processes. In this work we introduce an information content model check which may serve as a goodness-of-fit, like the chi-square procedure, to complement conventional Bayesian analysis. We demonstrate this extended Bayesian framework on a system of Langevin equations, where coordinate dependent mobilities and measurement noise hinder the normal mean squared displacement approach.

cond-mat.stat-mech↗

arXiv2016

Tracer Particles in Two-Dimensional Elastic Networks Diffuse Logarithmically Slow

Several experiments on tagged molecules or particles in living systems suggest that they move anomalously slow - their mean squared displacement (MSD) increase slower than linearly with time. Leading models aimed at understanding these experiments predict that the MSD grows as a power law with a growth exponent that is smaller than unity. However, in some experiments the exponent is so small $(\sim 0.1-0.2)$ that they hint towards other mechanisms at play. In this paper, we theoretically demonstrate how in-plane collective modes excited by thermal fluctuations in a two dimensional membrane lead to logarithmic time dependence for the mean square displacement of a tracer particle.

cond-mat.stat-mech↗

arXiv2015

Real sequence effects on the search dynamics of transcription factors on DNA

Recent experiments show that transcription factors (TFs) indeed use the facilitated diffusion mechanism to locate their target sequences on DNA in living bacteria cells: TFs alternate between sliding motion along DNA and relocation events through the cytoplasm. From simulations and theoretical analysis we study the TF-sliding motion for a large section of the DNA-sequence of a common E. coli strain, based on the two-state TF-model with a fast-sliding search state and a recognition state enabling target detection. For the probability to detect the target before dissociating from DNA the TF-search times self-consistently depend heavily on whether or not an auxiliary operator (an accessible sequence similar to the main operator) is present in the genome section. Importantly, within our model the extent to which the interconversion rates between search and recognition states depend on the underlying nucleotide sequence is varied. A moderate dependence maximises the capability to distinguish between the main operator and similar sequences. Moreover, these auxiliary operators serve as starting points for DNA looping with the main operator, yielding a spectrum of target detection times spanning several orders of magnitude. Auxiliary operators are shown to act as funnels facilitating target detection by TFs.

q-bio.QM↗

arXiv2015

Stochastic thermodynamics of a tagged particle within a harmonic chain

We study the stochastic thermodynamics of an overdamped harmonic chain, which can be viewed equivalently as a 1D Rouse chain or as an approximate model of single file diffusion. We discuss mainly two levels of description of this system: the Markovian level for which the trajectories of all the particles of the chain are known and the non-Markovian level in which only the motion of a tagged particle is available. For each case, we analyze the energy dissipation and its dependence on initial conditions. Surprisingly, we find that the average coarse-grained entropy production rate can become transiently negative when an oscillating force is applied to the tagged particle. This occurs due to memory effects as shown in a framework based on path integrals or on a generalized Langevin equation.

cond-mat.stat-mech↗

arXiv2014

Universality and non-universality of mobility in heterogeneous single-file systems and Rouse chains

We study analytically the tracer particle mobility in single-file systems with distributed friction constants. Our system serves as a prototype for non-equilibrium, heterogeneous, strongly interacting Brownian systems. The long time dynamics for such a single-file setup belongs to the same universality class as the Rouse model with dissimilar beads. The friction constants are drawn from a density $\varrho(ξ)$ and we derive an asymptotically exact solution for the mobility distribution $P[μ_0(s)]$, where $μ_0(s)$ is the Laplace-space mobility. If $\varrho$ is light-tailed (first moment exists) we find a self-averaging behaviour: $P[μ_0(s)]=δ[μ_0(s)-μ(s)]$ with $μ(s)\propto s^{1/2}$. When $\varrho(ξ)$ is heavy-tailed, $\varrho(ξ)\simeq ξ^{-1-α} \ (0<α<1)$ for large $ξ$ we obtain moments $\langle [μ_s(0)]^n\rangle \propto s^{βn}$ where $β=1/(1+α)$ and no self-averaging. The results are corroborated by simulations.

cond-mat.stat-mech↗

arXiv2013

Aging dynamics in interacting many-body systems

Low-dimensional, complex systems are often characterized by logarithmically slow dynamics. We study the generic motion of a labeled particle in an ensemble of identical diffusing particles with hardcore interactions in a strongly disordered, one-dimensional environment. Each particle in this single file is trapped for a random waiting time $τ$ with power law distribution $ψ(τ)\simeqτ^{-1- α}$, such that the $τ$ values are independent, local quantities for all particles. From scaling arguments and simulations, we find that for the scale-free waiting time case $0<α<1$, the tracer particle dynamics is ultra-slow with a logarithmic mean square displacement (MSD) $\langle x^2(t)\rangle\simeq(\log t)^{1/2}$. This extreme slowing down compared to regular single file motion $\langle x^2(t)\rangle\simeq t^{1/2}$ is due to the high likelihood that the labeled particle keeps encountering strongly immobilized neighbors. For the case $1<α<2$ we observe the MSD scaling $\langle x^2(t)\rangle\simeq t^γ$, where $γ<1/2$, while for $α>2$ we recover Harris law $\simeq t^{1/2}$.

physics.chem-ph↗

arXiv2013

Single-file diffusion with non-thermal initial conditions

Single-file diffusion is a theoretically challenging many-body problem where the calculation of even the simplest observables, e.g. mean square displacement, for a tracer particle requires a heavy mathematical machinery. There is therefore a need for simple approaches which predict qualitatively correct behaviours. Here we put forward one such method which we use to investigate the influence of non-thermal initial conditions on the dynamics of a tracer particle. With our new approach we reproduce, up to scaling, several known asymptotic results for the tracer particle mean square displacement.

cond-mat.stat-mech↗

arXiv2013

Transition dynamics in aging systems: microscopic origin of logarithmic time evolution

There exists compelling experimental evidence in numerous systems for logarithmically slow time evolution, yet its theoretical understanding remains elusive. We here introduce and study a generic transition process in complex systems, based on non-renewal, aging waiting times. Each state n of the system follows a local clock initiated at t=0. The random time τbetween clock ticks follows the waiting time density ψ(τ). Transitions between states occur only at local clock ticks and are hence triggered by the local forward waiting time, rather than by ψ(τ). For power-law forms ψ(τ) ~ τ^{-1-α} (0<α<1) we obtain a logarithmic time evolution of the state number ~ log(t/t_0), while for α>2 the process becomes normal in the sense that ~ t. In the intermediate range 1<α<2 we find the power-law growth ~ t^{α-1}. Our model provides a universal description for transition dynamics between aging and non-aging states.

cond-mat.stat-mech↗

arXiv2012

Bulk-mediated diffusion on a planar surface: full solution

We consider the effective surface motion of a particle that intermittently unbinds from a planar surface and performs bulk excursions. Based on a random walk approach we derive the diffusion equations for surface and bulk diffusion including the surface-bulk coupling. From these exact dynamic equations we analytically obtain the propagator of the effective surface motion. This approach allows us to deduce a superdiffusive, Cauchy-type behavior on the surface, together with exact cutoffs limiting the Cauchy form. Moreover we study the long-time dynamics for the surface motion.

cond-mat.stat-mech↗

arXiv2012

A solution to the subdiffusion-efficiency paradox: Inactive states enhance reaction efficiency at subdiffusion conditions in living cells

Macromolecular crowding in living biological cells effects subdiffusion of larger biomolecules such as proteins and enzymes. Mimicking this subdiffusion in terms of random walks on a critical percolation cluster, we here present a case study of EcoRV restriction enzymes involved in vital cellular defence. We show that due to its so far elusive propensity to an inactive state the enzyme avoids non-specific binding and remains well-distributed in the bulk cytoplasm of the cell. Despite the reduced volume exploration capability of subdiffusion processes, this mechanism guarantees a high efficiency of the enzyme. By variation of the non-specific binding constant and the bond occupation probability on the percolation network, we demonstrate that reduced non-specific binding are beneficial for efficient subdiffusive enzyme activity even in relatively small bacteria cells. Our results corroborate a more local picture of cellular regulation.

q-bio.SC↗

arXiv2011

Effective surface motion on a reactive cylinder of particles that perform intermittent bulk diffusion

In many biological and small scale technological applications particles may transiently bind to a cylindrical surface. In between two binding events the particles diffuse in the bulk, thus producing an effective translation on the cylinder surface. We here derive the effective motion on the surface, allowing for additional diffusion on the cylinder surface itself. We find explicit solutions for the number of adsorbed particles at one given instant, the effective surface displacement, as well as the surface propagator. In particular sub- and superdiffusive regimes are found, as well as an effective stalling of diffusion visible as a plateau in the mean squared displacement. We also investigate the corresponding first passage and first return problems.

cond-mat.stat-mech↗