Nearly maximal information gain due to time integration in central dogma reactions

Abstract

Living cells process information about their environment through the central dogma processes of transcription and translation, which drive the cellular response to stimuli. Here, we study the transfer of information from environmental input to the transcript and protein expression levels. Evaluation of both experimental and analogous simulation data reveals that transcription and translation are not two simple information channels connected in series. Instead, we demonstrate that the central dogma reactions often create a time-integrating information channel, where the translation channel receives and integrates multiple outputs from the transcription channel. This information channel model of the central dogma provides new information-theoretic selection criteria for the central dogma rate constants. Using the data for four well-studied species we show that their central dogma rate constants achieve information gain because of time integration while also keeping the loss because of stochasticity in translation relatively low ($<0.5$ bits).

Keywords: Biophysics; Gene process; Information system model.

Graphical abstract

Figure 1

Information flow during transcription and translation (A) The experimental system where IPTG induces the expression of eYFP and we measure both the transcript and the protein expression. The central dogma rate constants are for transcription ($k_m$), translation ($k_g$), and transcript and protein decays ($k_{d,m},k_{d,g}$). (B) Sequential channel model of the central dogma process, which receives an input, X, and produces transcripts, m, and then proteins, g, as sequential outputs. (C) Experimental result for the transcript-level mutual information ($I\left(X;m\right)$, left), and the protein-level mutual information ($I\left(X;g\right)$, right) using data from. (D) $I\left(X;m\right)$ (left) and $I\left(X;g\right)$ (right) from simulated expression data using a lac operon-based reaction network. In (C) and (D), the green dots in the top panels are the average expression values. The shaded region bounds the 5%–95% percentiles; the 2D heat maps in the bottom panels show the mutual information values over the space of probability distributions of the input, $P\left(X\right)$. The white dots in the heatmap indicate the maximum mutual information. In (C) the transcript value m is in RNA counts/cell and the protein value g is in molecules of equivalent fluorescein. In (D) the transcript and protein values are molecules per cell from Gillespie simulations. The maximum mutual information, or the channel capacity, is associated with an optimal input distribution.^,, The mutual information is higher near the (mean(X),std(X)) coordinates for the optimal distribution and decreases for input distributions that are away from the optimal one.

Figure 2

Trends in channel capacity as a function of central dogma rate constants (A) Transcript-level channel capacity for increasing transcription rate constant with fixed transcript decay rate constant, $k_{d,m}=0.5{\min }^{-1}$. The transition in the growth rate of $c\left(X;m\right)$ is due to$P\left(m|X\right)$ becoming more over-dispersed with increasing $k_m/k_{d,m}$ (Figure S3 and method details). (B) Protein-level channel capacity for increasing translation rate constant with fixed protein decay rate constant, $k_{d,g}=0.2{\min }^{-1}$. The black lines highlight that $c\left(X;g\right)\approx c\left(X;m\right)$ when translation power is 1. (C) Protein-level channel capacity for increasing protein decay rate constant with fixed translation power ($k_g/k_{d,g}={10}^3$). The black lines highlight that $c\left(X;g\right)\approx c\left(X;m\right)$ when $k_{d,g}=k_{d,m}$. In (A) to (C) all other rate constants are the same as reported in.

Figure 3

Information gain during translation (A) Ideal information gain curves as a function of the dimensionless integration time, $T\equiv k_{d,m}/k_{d,g}$. $k_{d,m}=0.1{\min }^{-1}$ for all the curves. At $T=1$, the ideal channel capacity is a function of the transcription power, $k_m/k_{d,m}$. For $T>1$, $c_{\text{ideal}}\left(T\right)$, is generally greater than $c\left(X;m\right)$. (B) Protein-level information gain curves for different translation power values, $k_g/k_{d,g}$. For the results shown, $k_m=k_{d,m}=0.1{\min }^{-1}$, which results in relatively low transcript-level channel capacity, $c\left(X;m\right)=0.4$ bits. The information gain due to time integration is determined by the translation power, $k_g/k_{d,g}$, and T. (C) Distributions of T for four species from literature data. The black bars show the 5%–95% percentile range. The white dots show the median. (D) Ideal and protein-level information gain curves for the four species. The solid lines with filled circles is $c\left(X;g\right)$, covering the 5%–95% percentile range of T. The shaded regions highlight the translation loss $c_{\text{ideal}}\left(T\right)-c\left(X;g\right)$. The translation loss is shown both within and beyond the naturally occurring range of T. For each of the species, the transcription rate constant $k_m$ and the transcript decay rate constant $k_{d,m}$ used were near median values in the literature (SI:3B).