Mapping general anesthesia states based on electro-encephalogram transition phases.

Cortical electro-encephalography (EEG) served as the clinical reference for monitoring unconsciousness during general anesthesia. The existing EEG-based monitors classified general anesthesia states as underdosed, adequate, or overdosed, lacking predictive power due to the absence of transition phases among these states. In response to this limitation, we undertook an analysis of the EEG signal during isoflurane-induced general anesthesia in mice. Adopting a data-driven approach, we applied signal processing techniques to track θ- and δ-band dynamics, along with iso-electric suppressions. Combining this approach with machine learning, we successfully developed an automated algorithm. The findings of our study revealed that the dampening of the δ-band occurred several minutes before the onset of significant iso-electric suppression episodes. Furthermore, a distinct γ-frequency oscillation was observed, persisting for several minutes during the recovery phase subsequent to isoflurane-induced overdose. As a result of our research, we generated a map summarizing multiple brain states and their transitions, offering a tool for predicting and preventing overdose during general anesthesia. The transition phases identified, along with the developed algorithm, have the potential to be generalized, enabling clinicians to prevent inadequate anesthesia and, consequently, tailor anesthetic regimens to individual patients.

Live-cell three-dimensional single-molecule tracking reveals modulation of enhancer dynamics by NuRD.

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.

How large the number of redundant copies should be to make a rare event probable.

The redundancy principle provides a framework to study how rare events are made possible with probability 1 in accelerated time, by making many copies of similar random searchers. However, what is a large n? To estimate large n with respect to the geometrical properties of a domain and the dynamics, we present here a criterion based on splitting probabilities between a small fraction of the exploration space associated with an activation process and other absorbing regions where trajectories can be terminated. We obtain explicit computations especially when there is a killing region located inside the domain that we compare with stochastic simulations. We also present examples of extreme trajectories with killing in dimension 2. For a large n, the optimal trajectories avoid penetrating inside the killing region. Finally, we discuss some applications to cell biology.

Modeling the voltage distribution in a non-locally but globally electroneutral confined electrolyte medium: applications for nanophysiology.

The distribution of voltage in sub-micron cellular domains remains poorly understood. In neurons, the voltage results from the difference in ionic concentrations which are continuously maintained by pumps and exchangers. However, it not clear how electro-neutrality could be maintained by an excess of fast moving positive ions that should be counter balanced by slow diffusing negatively charged proteins. Using the theory of electro-diffusion, we study here the voltage distribution in a generic domain, which consists of two concentric disks (resp. ball) in two (resp. three) dimensions, where a negative charge is fixed in the inner domain. When global but not local electro-neutrality is maintained, we solve the Poisson-Nernst-Planck equation both analytically and numerically in dimension 1 (flat) and 2 (cylindrical) and found that the voltage changes considerably on a spatial scale which is much larger than the Debye screening length, which assumes electro-neutrality. The present result suggests that long-range voltage drop changes are expected in neuronal microcompartments, probably relevant to explain the activation of far away voltage-gated channels located on the surface membrane.

Active flow network generates molecular transport by packets: case of the endoplasmic reticulum.

Biological networks are characterized by their connectivity and topology but also by their ability to transport materials. In the case of random transportation, the efficacy is measured by the time it takes to travel between two nodes of the network. We study here the consequences of a unidirectional transport mechanism occurring in the endoplasmic reticulum (ER) network, a structure present in the cell cytoplasm. This unidirectional transport mechanism is an active-waiting transportation, where molecules have to wait a random time before being transported from one node to the next one. We develop here a general theory of transport in an active network and find an unusual network transportation, where molecules group together in redundant packets instead of being disperse. Finally, the mean time to travel between two nodes of the ER is of the order of 20 min, but is reduced to 30 s when we consider the fastest particles because it uses optimal paths. To conclude, the present theory shows that unidirectional transport is an efficient and robust mechanism for fast molecular redistribution inside the ER.

Extreme escape from a cusp: When does geometry matter for the fastest Brownian particles moving in crowded cellular environments?

We study here the extreme statistics of Brownian particles escaping from a cusp funnel: the fastest Brownian particles among n follow an ensemble of optimal trajectories located near the shortest path from the source to the target. For the time of such first arrivers, we derive an asymptotic formula that differs from the mean first passage times obtained for classical narrow escape and dire strait. When particles are initially distributed at a given distance from a cusp, the time of the fastest particles depends on the cusp geometry. Therefore, when many particles diffuse around impermeable obstacles, the geometry plays a role in the time it takes to reach a target. In the context of cellular transduction with signaling molecules, having to escape from such cusp-like domains slows down signaling pathways. Consequently, generating multiple copies of the same molecule enables molecular signals to be delivered through crowded environments in sufficient time.

Biophysics of high density nanometer regions extracted from super-resolution single particle trajectories: application to voltage-gated calcium channels and phospholipids.

The cellular membrane is very heterogenous and enriched with high-density regions forming microdomains, as revealed by single particle tracking experiments. However the organization of these regions remain unexplained. We determine here the biophysical properties of these regions, when described as a basin of attraction. We develop two methods to recover the dynamics and local potential wells (field of force and boundary). The first method is based on the local density of points distribution of trajectories, which differs inside and outside the wells. The second method focuses on recovering the drift field that is convergent inside wells and uses the transient field to determine the boundary. Finally, we apply these two methods to the distribution of trajectories recorded from voltage gated calcium channels and phospholipid anchored GFP in the cell membrane of hippocampal neurons and obtain the size and energy of high-density regions with a nanometer precision.

Advances Using Single-Particle Trajectories to Reconstruct Chromatin Organization and Dynamics.

Chromatin organization remains complex and far from understood. In this article, we review recent statistical methods of extracting biophysical parameters from in vivo single-particle trajectories of loci to reconstruct chromatin reorganization in response to cellular stress such as DNA damage. We look at methods for analyzing both single locus and multiple loci tracked simultaneously and explain how to quantify and describe chromatin motion using a combination of extractable parameters. These parameters can be converted into information about chromatin dynamics and function. Furthermore, we discuss how the timescale of recurrent encounter between loci can be extracted and interpreted. We also discuss the effect of sampling rate on the estimated parameters. Finally, we review a polymer method to reconstruct chromatin structure using crosslinkers between chromatin sites. We list and refer to some software packages that are now publicly available to simulate polymer motion. To conclude, chromatin organization and dynamics can be reconstructed from locus trajectories and predicted based on polymer models.

Statistics of chromatin organization during cell differentiation revealed by heterogeneous cross-linked polymers.

Chromatin of mammalian nucleus folds into discrete contact enriched regions such as Topologically Associating Domains (TADs). Folding hierarchy and internal organization of TADs is highly dynamic throughout cellular differentiation, and are correlated with gene activation and silencing. To account for multiple interacting TADs, we developed a parsimonious randomly cross-linked (RCL) polymer model that maps high frequency Hi-C encounters within and between TADs into direct loci interactions using cross-links at a given base-pair resolution. We reconstruct three TADs of the mammalian X chromosome for three stages of differentiation. We compute the radius of gyration of TADs and the encounter probability between genomic segments. We found 1) a synchronous compaction and decompaction of TADs throughout differentiation and 2) high order organization into meta-TADs resulting from weak inter-TAD interactions. Finally, the present framework allows to infer transient properties of the chromatin from steady-state statistics embedded in the Hi-C/5C data.

Steady-state voltage distribution in three-dimensional cusp-shaped funnels modeled by PNP.

We study here the bulk electro-diffusion properties of micro- and nanodomains containing a cusp-shaped structure in three-dimensions when the cation concentration dominates over the anions. To determine the consequences on the voltage distribution, we use the steady-state Poisson-Nernst-Planck equation with an integral constraint for the number of charges. A non-homogeneous Neumann boundary condition is imposed on the boundary. We construct an asymptotic approximation for certain surface charge distribution that agree with numerical simulations. Finally, we analyze the consequences of several piecewise constant non-homogeneous surface charge densities, motivated by designing new nanopipettes. To conclude, when electro-neutrality is broken at the scale of hundreds of nanometers, the geometry of cusp-shaped domains influences the voltage profile, specifically inside the cusp structure. The main results are summarized in the form of new three-dimensional electrostatic laws for non-electroneutral electrolytes. These formula provide a refined characterization of voltage distribution at steady-state in neuronal microdomains such as dendritic spines, but can also be used to design nanometric patch-pipettes.

Redundancy principle and the role of extreme statistics in molecular and cellular biology.

The paradigm of chemical activation rates in cellular biology has been shifted from the mean arrival time of a single particle to the mean of the first among many particles to arrive at a small activation site. The activation rate is set by extremely rare events, which have drastically different time scales from the mean times between activations, and depends on different structural parameters. This shift calls for reconsideration of physical processes used in deterministic and stochastic modeling of chemical reactions that are based on the traditional forward rate, especially for fast activation processes in living cells. Consequently, the biological activation time is not necessarily exponentially distributed. We review here the physical models, the mathematical analysis and the new paradigm of setting the scale to be the shortest time for activation that clarifies the role of population redundancy in selecting and accelerating transient cellular search processes. We provide examples in cellular transduction, gene activation, cell senescence activation or spermatozoa selection during fertilization, where the rate depends on numbers. We conclude that the statistics of the minimal time to activation set kinetic laws in biology, which can be very different from the ones associated to average times.

Electrical transient laws in neuronal microdomains based on electro-diffusion.

The current-voltage (I-V) conversion characterizes the physiology of cellular microdomains and reflects cellular communication, excitability, and electrical transduction. Yet deriving such I-V laws remains a major challenge in most cellular microdomains due to their small sizes and the difficulty in assessing voltage with high nanometer precision. We present here novel analytical relations derived for different numbers of ionic species inside neuronal micro/nano-domains, such as dendritic spines. When a steady-state current is injected, we find a large deviation from the classical Ohm's law, showing that the spine neck resistance is insufficient to characterize electrical properties. For a constricted spine neck, modeled by a hyperboloid, we obtain a new I-V law that illustrates the consequences of narrow passages on electrical conduction. Finally, during a fast current transient, the local voltage is modulated by the distance between activated voltage-gated channels. To conclude, electro-diffusion laws can now be used to interpret voltage distribution in neuronal microdomains.

Encounter times of chromatin loci influenced by polymer decondensation.

The time for a DNA sequence to find its homologous counterpart depends on a long random search inside the cell nucleus. Using polymer models, we compute here the mean first encounter time (MFET) between two sites located on two different polymer chains and confined locally by potential wells. We find that reducing tethering forces acting on the polymers results in local decondensation, and numerical simulations of the polymer model show that these changes are associated with a reduction of the MFET by several orders of magnitude. We derive here new asymptotic formula for the MFET, confirmed by Brownian simulations. We conclude from the present modeling approach that the fast search for homology is mediated by a local chromatin decondensation due to the release of multiple chromatin tethering forces. The present scenario could explain how the homologous recombination pathway for double-stranded DNA repair is controlled by its random search step.

Do cells sense time by number of divisions?

Can the cell's perception of time be expressed through the length of the shortest telomere? To address this question, we analyze an asymmetric random walk that models telomere length for each division that can decrease by a fixed length a or, if recognized by a polymerase, it increases by a fixed length b ≫ a. Our analysis of the model reveals two phases, first, a determinist drift of the length toward a quasi-equilibrium state, and second, persistence of the length near an attracting state for the majority of divisions. The measure of stability of the latter phase is the expected number of divisions at the attractor ("lifetime") prior to crossing a threshold T that model senescence. Using numerical simulations, we further study the distribution of times for the shortest telomere to reach the threshold T. We conclude that the telomerase regulates telomere stability by creating an effective potential barrier that separates statistically the arrival time of the shortest from the next shortest to T. The present model explains how random telomere dynamics underlies the extension of cell survival time.

Mixed analytical-stochastic simulation method for the recovery of a Brownian gradient source from probability fluxes to small windows.

Is it possible to recover the position of a source from the steady-state fluxes of Brownian particles to small absorbing windows located on the boundary of a domain? To address this question, we develop a numerical procedure to avoid tracking Brownian trajectories in the entire infinite space. Instead, we generate particles near the absorbing windows, computed from the analytical expression of the exit probability. When the Brownian particles are generated by a steady-state gradient at a single point, we compute asymptotically the fluxes to small absorbing holes distributed on the boundary of half-space and on a disk in two dimensions, which agree with stochastic simulations. We also derive an expression for the splitting probability between small windows using the matched asymptotic method. Finally, when there are more than two small absorbing windows, we show how to reconstruct the position of the source from the diffusion fluxes. The present approach provides a computational first principle for the mechanism of sensing a gradient of diffusing particles, a ubiquitous problem in cell biology.

Electrostatics of non-neutral biological microdomains.

Voltage and charge distributions in cellular microdomains regulate communications, excitability, and signal transduction. We report here new electrical laws in a biological cell, which follow from a nonlinear electro-diffusion model. These newly discovered laws derive from the geometrical cell-membrane properties, such as membrane curvature, volume, and surface area. The electro-diffusion laws can now be used to predict and interpret voltage distribution in cellular microdomains such as synapses, dendritic spine, cilia and more.

Statistics of randomly cross-linked polymer models to interpret chromatin conformation capture data.

Polymer models are used to describe chromatin, which can be folded at different spatial scales by binding molecules. By folding, chromatin generates loops of various sizes. We present here a statistical analysis of the randomly cross-linked (RCL) polymer model, where monomer pairs are connected randomly, generating a heterogeneous ensemble of chromatin conformations. We obtain asymptotic formulas for the steady-state variance, encounter probability, the radius of gyration, instantaneous displacement, and the mean first encounter time between any two monomers. The analytical results are confirmed by Brownian simulations. Finally, the present results are used to extract the mean number of cross links in a chromatin region from conformation capture data.

Search for a small egg by spermatozoa in restricted geometries.

The search by swimmers for a small target in a bounded domain is ubiquitous in cellular biology, where a prominent case is that of the search by spermatozoa for an egg in the uterus. This is one of the severest selection processes in animal reproduction. We present here a mathematical model of the search, its analysis, and numerical simulations. In the proposed model the swimmers' trajectories are rectilinear and the speed is constant. When a trajectory hits an obstacle or the boundary, it is reflected at a random angle and continues the search with the same speed. Because hitting a small target by a trajectory is a rare event, asymptotic approximations and stochastic simulations are needed to estimate the mean search time in various geometries. We consider searches in a disk, in convex planar domains, and in domains with cusps. The exploration of the parameter space for spermatozoa motion in different uterus geometries leads to scaling laws for the search process.

Recovering a stochastic process from super-resolution noisy ensembles of single-particle trajectories.

Recovering a stochastic process from noisy ensembles of single-particle trajectories is resolved here using the coarse-grained Langevin equation as a model. The massive redundancy contained in single-particle tracking data allows recovering local parameters of the underlying physical model. We use several parametric and nonparametric estimators to compute the first and second moments of the process, to recover the local drift, its derivative, and the diffusion tensor, and to deconvolve the instrumental from the physical noise. We use numerical simulations to also explore the range of validity for these estimators. The present analysis allows defining what can exactly be recovered from statistics of super-resolution microscopy trajectories used for characterizing molecular trafficking underlying cellular functions.