Nonidentifiability of the Source of Intrinsic Noise in Gene Expression from Single-Burst Data

Over the last few years, experimental data on the fluctuations in gene activity between individual cells and within the same cell over time have confirmed that gene expression is a “noisy” process. This variation is in part due to the small number of molecules taking part in some of the key reactions that are involved in gene expression. One of the consequences of this is that protein production often occurs in bursts, each due to a single promoter or transcription factor binding event. Recently, the distribution of the number of proteins produced in such bursts has been experimentally measured, offering a unique opportunity to study the relative importance of different sources of noise in gene expression. Here, we provide a derivation of the theoretical probability distribution of these bursts for a wide variety of different models of gene expression. We show that there is a good fit between our theoretical distribution and that obtained from two different published experimental datasets. We then prove that, irrespective of the details of the model, the burst size distribution is always geometric and hence determined by a single parameter. Many different combinations of the biochemical rates for the constituent reactions of both transcription and translation will therefore lead to the same experimentally observed burst size distribution. It is thus impossible to identify different sources of fluctuations purely from protein burst size data or to use such data to estimate all of the model parameters. We explore methods of inferring these values when additional types of experimental data are available. Author Summary: Recent experimental data showing fluctuations in gene activity between individual cells and within the same cell over time confirm that gene expression is a “noisy” process. This variation is partly due to the small number of molecules involved in gene expression. One consequence is that protein production often occurs in bursts, each due to the binding of a single transcription factor. Recently, the distribution of the number of proteins produced in such bursts has been experimentally measured, offering a unique opportunity to study the relative importance of different sources of noise in gene expression. We derive the theoretical probability distribution of these bursts for a wide variety of gene expression models. We show a good fit between our theoretical distribution and experimental data and prove that, irrespective of the model details, the burst size distribution always has the same shape, determined by a single parameter. As different combinations of the reaction rates lead to the same observed distribution, it is impossible to estimate all kinetic parameters from protein burst size data. When additional data, such as protein equilibrium distributions, are available, these can be used to infer additional parameters. We present one approach to this, demonstrating its application to published data.