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To understand how the nucleosome remodeling and deacetylase (NuRD) complex regulates enhancers and enhancer–promoter interactions, we have developed an approach to segment and extract key biophysical parameters from live-cell three-dimensional single-molecule trajectories. Unexpectedly, this has revealed that NuRD binds to chromatin for minutes, decompacts chromatin structure and increases enhancer dynamics. We also uncovered a rare fast-diffusing state of enhancers and found that NuRD restricts the time spent in this state. Hi-C and Cut&Run experiments revealed that NuRD modulates enhancer–promoter interactions in active chromatin, allowing them to contact each other over longer distances. Furthermore, NuRD leads to a marked redistribution of CTCF and, in particular, cohesin. We propose that NuRD promotes a decondensed chromatin environment, where enhancers and promoters can contact each other over longer distances, and where the resetting of enhancer–promoter interactions brought about by the fast decondensed chromatin motions is reduced, leading to more stable, long-lived enhancer–promoter relationships.

We report here recent progress in computing the search time for a stochastic particle to find a small target hidden in cusp-like pockets. The target is a small segment in dimension two, a small hole or a narrow ribbon in dimension three, placed at the end of a cusp. The asymptotic analysis of the diffusion equation reveals the role of the local geometry, and a mathematical difficulty comes from the boundary layer near the target. The methods are conformal mapping and matching asymptotic. We present applications in cell biology where cellular activation occurs when a diffusing particle finds a hidden site. This is the case during vesicular fusion initiated after a protein located between the vesicular and cell membranes binds to several diffusing calcium ions. Another example is a drug activation site located inside a deep molecular pocket. The analytical formulas clarify the role of small parameters.

A large number (tens of thousands) of single molecular trajectories on a cell membrane can now be collected by superresolution methods. The data contains information about the diffusive motion of molecule, proteins, or receptors and here we review methods for its recovery by statistical analysis of the data. The information includes the forces, organization of the membrane, the diffusion tensor, the long-time behavior of the trajectories, and more. To recover the long-time behavior and statistics of long trajectories, a stochastic model of their nonequilibrium motion is required. Modeling and data analysis serve extracting novel biophysical features at an unprecedented spatiotemporal resolution. The review presents data analysis, modeling, and stochastic simulations applied in particular on surface receptors evolving in neuronal cells.

We report here recent progress in computing the search time for a stochastic particle to find a small target hidden in cusp-like pockets. The target is a small segment in dimension two, a small hole or a narrow ribbon in dimension three, placed at the end of a cusp. The asymptotic analysis of the diffusion equation reveals the role of the local geometry, and a mathematical difficulty comes from the boundary layer near the target. The methods are conformal mapping and matching asymptotic. We present applications in cell biology where cellular activation occurs when a diffusing particle finds a hidden site. This is the case during vesicular fusion initiated after a protein located between the vesicular and cell membranes binds to several diffusing calcium ions. Another example is a drug activation site located inside a deep molecular pocket. The analytical formulas clarify the role of small parameters.

We present a rate model of the spontaneous activity in the auditory cortex, based on synaptic depression. A Stochastic integro-differential system of equations is derived and the analysis reveals two main regimes. The first regime corresponds to a normal activity. The second regime corresponds to epileptic spiking. A detailed analysis of each regime is presented and we prove in particular that synaptic depression stabilizes the global cortical dynamics. The transition between the two regimes is induced by a change in synaptic connectivity: when the overall connectivity is strong enough, an epileptic activity is spontaneously generated. Numerical simulations confirm the predictions of the theoretical analysis. In particular, our results explain the transition from normal to epileptic regime which can be induced in rats auditory cortex, following a specific pairing protocol. A change in the cortical maps reorganizes the synaptic connectivity and this transition between regimes is accounted for by our model. We have used data from recording experiments to fit synaptic weight distributions. Simulations with the fitted distributions are qualitatively similar to the real EEG recorded in vivo during the experiments.We present a rate model of the spontaneous activity in the auditory cortex, based on synaptic depression. A Stochastic integro-differential system of equations is derived and the analysis reveals two main regimes. The first regime corresponds to a normal activity. The second regime corresponds to epileptic spiking. A detailed analysis of each regime is presented and we prove in particular that synaptic depression stabilizes the global cortical dynamics. The transition between the two regimes is induced by a change in synaptic connectivity: when the overall connectivity is strong enough, an epileptic activity is spontaneously generated. Numerical simulations confirm the predictions of the theoretical analysis. In particular, our results explain the transition from normal to epileptic regime which can be induced in rats auditory cortex, following a specific pairing protocol. A change in the cortical maps reorganizes the synaptic connectivity and this transition between regimes is accounted for by our model. We have used data from recording experiments to fit synaptic weight distributions. Simulations with the fitted distributions are qualitatively similar to the real EEG recorded in vivo during the experiments.

A dendritic spine is an intracellular compartment in synapses of central neurons. The role of the fast twitching of spines, brought about by a transient rise of internal calcium concentration above that of the parent dendrite, has been hitherto unclear. We propose an explanation of the cause and effect of the twitching and its role in the functioning of the spine as a fast calcium compartment. Our molecular model postulates that rapid spine motility is due to the concerted contraction of calcium-binding proteins. The contraction induces a stream of cytoplasmic fluid in the direction of the dendritic shaft, thus speeding up the time course of spine calcium dynamics, relative to pure diffusion. Simulations indicate that chemical reaction rate theory at the molecular level can explain spine motility. They reveal two time periods in calcium dynamics, as measured recently by other researchers. It appears that rapid motility in dendritic spines increases the efficiency of calcium conduction to the dendrite and speeds up the emptying of the spine. This could play a major role in the induction of synaptic plasticity. A prediction of the model is that alteration of spine motility will modify the time course of calcium in the dendritic spine and could be tested experimentally.

Abstract.: Dans cet article, on considère le problème de Dirichlet pour une EDP non linéaire avec une donnée au bord qui change de signe. L'existence d'une famille de solutions minimisantes est obtenue sur les variétés riemaniennes compactes non localement conformément plates à bord.¶¶This paper deals with solutions of Dirichlet problem for a nonlinear PDE with boundary data changing of sign. A family of minimizing nodal solutions are found on nonlocally conformally flat manifolds with boundary.