No abstract

Stochastic transcription and translation dynamics of protein accumulation up to some concentration threshold sets the timing of many cellular physiological processes. Here we obtain the exact distribution of first threshold-crossing times of protein concentration, in either Laplace or time domain, and its associated cumulants: mean, variance and skewness. The distribution is asymmetric and its skewness non-monotonically varies with the threshold. We study lysis times of E-coli cells for holin gene mutants of bacteriophage-λ and find a good match with theory. Mutants requiring higher holin thresholds show more skewed lysis time distributions as predicted.

Stochastic transcription and translation dynamics of protein accumulation up to some concentration threshold sets the timing of many cellular physiological processes. Here we obtain the exact distribution of first threshold-crossing times of protein concentration, in either Laplace or time domain, and its associated cumulants: mean, variance and skewness. The distribution is asymmetric and its skewness non-monotonically varies with the threshold. We study lysis times of E-coli cells for holin gene mutants of bacteriophage-λ and find a good match with theory. Mutants requiring higher holin thresholds show more skewed lysis time distributions as predicted.

Protein thresholds have been shown to act as an ancient timekeeping device, such as in the time to lysis of E. coli infected with bacteriophage lambda. The time taken for protein levels to reach a particular threshold for the first time is defined as the first passage time of the protein synthesis system, which is a stochastic quantity. It had been shown previously that it was possible to obtain the mean and higher moments of the distribution of first passage times, but an analytical expression for the full distribution was not available. In this work, we derive an analytical expression for the first passage times for a long-lived protein. This expression allows us to calculate the full distribution not only for cases of no self-regulation, but also for both positive and negative self-regulation of the threshold protein. We show that the shape of the distribution matches previous experimental data on lambda-phage lysis time distributions. We study the noise in the precision of the first passage times by calculating the coefficient of variation (CV) of the distribution under various conditions. In agreement with previous results, we show that the CV of a protein that is not self-regulated is less than that of a self-regulated protein for both positive and negative regulation. We show that under conditions of positive self-regulation, the CV declines sharply and then roughly plateaus with increasing protein threshold, while under conditions of negative self-regulation, the CV declines steeply and then increases with increasing protein threshold. In the latter case therefore there is an optimal protein threshold that minimizes the noise in the first passage times. We also provide analytical expressions for the FPT distribution with non-zero degradation in Laplace space. These analytical distributions will have applications in understanding the stochastic dynamics of threshold determined processes in molecular and cell biology.

The transfer of electro-chemical signals from the pre-synaptic to the post-synaptic terminal of a neuronal or neuro-muscular synapse is the basic building block of neuronal communication. When triggered by an action potential the pre-synaptic terminal releases neurotransmitters in the synaptic cleft through vesicle fusion. The number of vesicles that fuse, i.e., the burst size, is stochastic, and widely assumed to be binomially distributed. However, the burst size depends on the number of release-ready vesicles, a random variable that depends upon a stochastic replenishment process, as well as stochastic inter-spike intervals of action potentials. The burst size distribution suitably averaged over these two stochastic processes was not known in the literature. Here we analytically obtain the exact probability distribution of the number of vesicles released in the synaptic cleft, in the steady state reached during stimulation by a spike train of action potentials. We show that this distribution is binomial, with modified parameters, only when stimulated by constant frequency signals. Other forms of input, e.g. Poisson-distributed action potentials, lead to distributions that are non-binomial. The general formula valid for arbitrary distributions of the input inter-spike interval, may be employed to study neuronal transmission under diverse experimental conditions. We corroborate our theoretical predictions through comparison with the burst size distributions obtained from electrophysiological recordings from MNTB-LSO synapses of juvenile mice. We also confirm our theoretically predicted frequency dependence of mean burst size by comparing with experimental data from hippocampal and auditory neurons.