Refine
Language
- English (22)
Is part of the Bibliography
- yes (22)
Keywords
- diffusion (10)
- first passage time (5)
- random diffusivity (4)
- aspect ratio (3)
- cylindrical geometry (3)
- first-passage time (3)
- protein search (3)
- Brownian motion (2)
- approximate methods (2)
- extremal values (2)
Institute
- Institut für Physik und Astronomie (22) (remove)
The time instant-the first-passage time (FPT)-when a diffusive particle (e.g., a ligand such as oxygen or a signalling protein) for the first time reaches an immobile target located on the surface of a bounded three-dimensional domain (e.g., a hemoglobin molecule or the cellular nucleus) is a decisive characteristic time-scale in diverse biophysical and biochemical processes, as well as in intermediate stages of various inter- and intra-cellular signal transduction pathways. Adam and Delbruck put forth the reduction-of-dimensionality concept, according to which a ligand first binds non-specifically to any point of the surface on which the target is placed and then diffuses along this surface until it locates the target. In this work, we analyse the efficiency of such a scenario and confront it with the efficiency of a direct search process, in which the target is approached directly from the bulk and not aided by surface diffusion. We consider two situations: (i) a single ligand is launched from a fixed or a random position and searches for the target, and (ii) the case of 'amplified' signals when N ligands start either from the same point or from random positions, and the search terminates when the fastest of them arrives to the target. For such settings, we go beyond the conventional analyses, which compare only the mean values of the corresponding FPTs. Instead, we calculate the full probability density function of FPTs for both scenarios and study its integral characteristic-the 'survival' probability of a target up to time t. On this basis, we examine how the efficiencies of both scenarios are controlled by a variety of parameters and single out realistic conditions in which the reduction-of-dimensionality scenario outperforms the direct search.
We investigate a class of diffusion-controlled reactions that are initiated at the time instance when a prescribed number K among N particles independently diffusing in a solvent are simultaneously bound to a target region.
In the irreversible target-binding setting, the particles that bind to the target stay there forever, and the reaction time is the Kth fastest first-passage time to the target, whose distribution is well-known. In turn, reversible binding, which is common for most applications, renders theoretical analysis much more challenging and drastically changes the distribution of reaction times.
We develop a renewal-based approach to derive an approximate solution for the probability density of the reaction time.
This approximation turns out to be remarkably accurate for a broad range of parameters.
We also analyze the dependence of the mean reaction time or, equivalently, the inverse reaction rate, on the main parameters such as K, N, and binding/unbinding constants. Some biophysical applications and further perspectives are briefly discussed.
We develop an encounter-based approach for describing restricted diffusion with a gradient drift toward a partially reactive boundary. For this purpose, we introduce an extension of the Dirichlet-to-Neumann operator and use its eigenbasis to derive a spectral decomposition for the full propagator, i.e. the joint probability density function for the particle position and its boundary local time. This is the central quantity that determines various characteristics of diffusion-influenced reactions such as conventional propagators, survival probability, first-passage time distribution, boundary local time distribution, and reaction rate. As an illustration, we investigate the impact of a constant drift onto the boundary local time for restricted diffusion on an interval. More generally, this approach accesses how external forces may influence the statistics of encounters of a diffusing particle with the reactive boundary.
High diffusion-sensitizing magnetic field gradients have been more and more often applied nowadays to achieve a better characterization of the microstructure. As the resulting spin-echo signal significantly deviates from the conventional Gaussian form, various models have been employed to interpret these deviations and to relate them with the microstructural properties of a sample. In this paper, we argue that the non-Gaussian behavior of the signal is a generic universal feature of the Bloch-Torrey equation. We provide a simple yet rigorous description of the localization regime emerging at high extended gradients and identify its origin as a symmetry breaking at the reflecting boundary. We compare the consequent non-Gaussian signal decay to other diffusion NMR regimes such as slow-diffusion, motional-narrowing and diffusion-diffraction regimes. We emphasize limitations of conventional perturbative techniques and advocate for non-perturbative approaches which may pave a way to new imaging modalities in this field.
We study the probability density function (PDF) of the first-reaction times between a diffusive ligand and a membrane-bound, immobile imperfect target region in a restricted 'onion-shell' geometry bounded by two nested membranes of arbitrary shapes. For such a setting, encountered in diverse molecular signal transduction pathways or in the narrow escape problem with additional steric constraints, we derive an exact spectral form of the PDF, as well as present its approximate form calculated by help of the so-called self-consistent approximation. For a particular case when the nested domains are concentric spheres, we get a fully explicit form of the approximated PDF, assess the accuracy of this approximation, and discuss various facets of the obtained distributions. Our results can be straightforwardly applied to describe the PDF of the terminal reaction event in multi-stage signal transduction processes.
We study the probability density function (PDF) of the first-reaction times between a diffusive ligand and a membrane-bound, immobile imperfect target region in a restricted 'onion-shell' geometry bounded by two nested membranes of arbitrary shapes. For such a setting, encountered in diverse molecular signal transduction pathways or in the narrow escape problem with additional steric constraints, we derive an exact spectral form of the PDF, as well as present its approximate form calculated by help of the so-called self-consistent approximation. For a particular case when the nested domains are concentric spheres, we get a fully explicit form of the approximated PDF, assess the accuracy of this approximation, and discuss various facets of the obtained distributions. Our results can be straightforwardly applied to describe the PDF of the terminal reaction event in multi-stage signal transduction processes.
Towards a full quantitative description of single-molecule reaction kinetics in biological cells
(2018)
The first-passage time (FPT), i.e., the moment when a stochastic process reaches a given threshold value for the first time, is a fundamental mathematical concept with immediate applications. In particular, it quantifies the statistics of instances when biomolecules in a biological cell reach their specific binding sites and trigger cellular regulation. Typically, the first-passage properties are given in terms of mean first-passage times. However, modern experiments now monitor single-molecular binding-processes in living cells and thus provide access to the full statistics of the underlying first-passage events, in particular, inherent cell-to-cell fluctuations. We here present a robust explicit approach for obtaining the distribution of FPTs to a small partially reactive target in cylindrical-annulus domains, which represent typical bacterial and neuronal cell shapes. We investigate various asymptotic behaviours of this FPT distribution and show that it is typically very broad in many biological situations, thus, the mean FPT can differ from the most probable FPT by orders of magnitude. The most probable FPT is shown to strongly depend only on the starting position within the geometry and to be almost independent of the target size and reactivity. These findings demonstrate the dramatic relevance of knowing the full distribution of FPTs and thus open new perspectives for a more reliable description of many intracellular processes initiated by the arrival of one or few biomolecules to a small, spatially localised region inside the cell.
We consider a sequential cascade of molecular first-reaction events towards a terminal reaction centre in which each reaction step is controlled by diffusive motion of the particles. The model studied here represents a typical reaction setting encountered in diverse molecular biology systems, in which, e.g. a signal transduction proceeds via a series of consecutive 'messengers': the first messenger has to find its respective immobile target site triggering a launch of the second messenger, the second messenger seeks its own target site and provokes a launch of the third messenger and so on, resembling a relay race in human competitions. For such a molecular relay race taking place in infinite one-, two- and three-dimensional systems, we find exact expressions for the probability density function of the time instant of the terminal reaction event, conditioned on preceding successful reaction events on an ordered array of target sites. The obtained expressions pertain to the most general conditions: number of intermediate stages and the corresponding diffusion coefficients, the sizes of the target sites, the distances between them, as well as their reactivities are arbitrary.
We consider a sequential cascade of molecular first-reaction events towards a terminal reaction centre in which each reaction step is controlled by diffusive motion of the particles. The model studied here represents a typical reaction setting encountered in diverse molecular biology systems, in which, e.g. a signal transduction proceeds via a series of consecutive 'messengers': the first messenger has to find its respective immobile target site triggering a launch of the second messenger, the second messenger seeks its own target site and provokes a launch of the third messenger and so on, resembling a relay race in human competitions. For such a molecular relay race taking place in infinite one-, two- and three-dimensional systems, we find exact expressions for the probability density function of the time instant of the terminal reaction event, conditioned on preceding successful reaction events on an ordered array of target sites. The obtained expressions pertain to the most general conditions: number of intermediate stages and the corresponding diffusion coefficients, the sizes of the target sites, the distances between them, as well as their reactivities are arbitrary.
We study the extremal properties of a stochastic process xt defined by the Langevin equation ẋₜ =√2Dₜ ξₜ, in which ξt is a Gaussian white noise with zero mean and Dₜ is a stochastic‘diffusivity’, defined as a functional of independent Brownian motion Bₜ.We focus on threechoices for the random diffusivity Dₜ: cut-off Brownian motion, Dₜt ∼ Θ(Bₜ), where Θ(x) is the Heaviside step function; geometric Brownian motion, Dₜ ∼ exp(−Bₜ); and a superdiffusive process based on squared Brownian motion, Dₜ ∼ B²ₜ. For these cases we derive exact expressions for the probability density functions of the maximal positive displacement and of the range of the process xₜ on the time interval ₜ ∈ (0, T).We discuss the asymptotic behaviours of the associated probability density functions, compare these against the behaviour of the corresponding properties of standard Brownian motion with constant diffusivity (Dₜ = D0) and also analyse the typical behaviour of the probability density functions which is observed for a majority of realisations of the stochastic diffusivity process.