Towards a full quantitative description of single-molecule reaction kinetics in biological cells

Abstract

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.

Full distribution of first exit times in the narrow escape problem

“…As the most probable time now belongs to this plateau region, it is very difficult to quantify. This is a radically new feature of the NEP, as compared to recent works [47,52].…”

“…Equations (2)-(5) completely determine the explicit form of H(r, θ; t). As we already mentioned above, this result is obtained by resorting to a self-consistent closure scheme, developed earlier for the calculation of the mean FRT in certain reaction-diffusion problems [4,46,47]. This approximation consists in replacing the actual mixed boundary condition (1) by an inhomogeneous Neumann condition and in the derivation of an appropriate closure relation, which ensures that the mixed boundary condition (1) holds on average.…”

Steady-state reaction rate of diffusion-controlled reactions in sheets

“…Even its numerical computation becomes time consuming because the truncation size of the infinite-dimensional matrix has to be large. To get a more suitable expression for the flux, we apply the self-consistent approximation (also known as constantflux approximation) originally devised by Shoup, Lipari and Szabo [48] and then extensively adapted to firstpassage time problems [46,47,49]. The accuracy of this approximation was investigated in [50].…”

Reversible reactions controlled by surface diffusion on a sphere

“…In both cases, the particle diffuses for long time until its binding to the target and thus can largely explore the whole bounded domain, resulting in such an exponential law. However, if the particle starts relatively close to the target, it can bind to the target very rapidly, on a time scale much shorter than that of the diffusive exploration of the domain [62][63][64]. As a consequence, the exponential function fails to describe the intricate short-time behavior of the survival probability.…”