Fluctuations in protein synthesis from a single RNA template: Stochastic kinetics of ribosomes

Proteins are polymerized by cyclic machines called ribosomes, which use their messenger RNA (mRNA) track also as the corresponding template, and the process is called translation. We explore, in depth and detail, the stochastic nature of the translation. We compute various distributions associated with the translation process; one of them—namely, the dwell time distribution—has been measured in recent single-ribosome experiments. The form of the distribution, which fits best with our simulation data, is consistent with that extracted from the experimental data. For our computations, we use a model that captures both the mechanochemistry of each individual ribosome and their steric interactions. We also demonstrate the effects of the sequence inhomogeneities of real genes on the fluctuations and noise in translation. Finally, inspired by recent advances in the experimental techniques of manipulating single ribosomes, we make theoretical predictions on the force-velocity relation for individual ribosomes. In principle, all our predictions can be tested by carrying out in vitro experiments.

Figure 1

A schematic representation of the biochemical cycle of a single ribosome during the elongation stage of translation in our model [13]. Each circle labeled by an integer index represents a distinct state in the mechanochemical state of a ribosome. The index below the box labels the codon on the mRNA with which the ribosome binds. The symbols accompanied by the arrows define the rate constants for the corresponding transitions from one state to another.

Figure 2

(Color online) Probability distribution of the dwell times of the ribosomes for a hypothetical homogeneous mRNA template for $\ell =12$. The set of data fits well with the difference of two exponentials. The parameters are ${\omega }_a={\omega }_g=25{\mathrm{s}}^{-1}$ and $\alpha =0.0001$.

Figure 3

(Color online) Typical time series of the translation events for (a) crr gene of Escherichia coli K-12 strain MG1655 and (b) the corresponding hypothetical homogeneous mRNA template, both corresponding to ${\omega }_a=2.5{\mathrm{s}}^{-1}$, ${\omega }_g=2.5{\mathrm{s}}^{-1}$, ${\omega }_h=10{\mathrm{s}}^{-1}$, and $\alpha =0.1$.

Figure 4

(Color online) Probability distribution of the times taken to complete the synthesis of a single polypeptide (which is identical to the probability distribution of the run times of ribosomes) for (a) crr gene of Escherichia coli K-12 strain MG1655 and (b) the corresponding hypothetical homogeneous mRNA template. In both (a) and (b), different curves correspond to different values of ${\omega }_a$, all for $\ell =12$. The discrete data points have been obtained from our computer simulations of the model whereas the lines denote the Gaussian best fits to these data. The insets show the exponential decrease of the corresponding noise strengths with ${\omega }_a$. In both (a) and (b), ${\omega }_g=2.5{\mathrm{s}}^{-1}$ and $\alpha =0.1$.

Figure 5

(Color online) Probability distribution of the time gaps between the completions of the synthesis of a successive polypeptides (which is identical to the probability distribution of the time headways in the ribosome traffic) for (a) crr gene of Escherichia coli K-12 strain MG1655 and (b) the corresponding hypothetical homogeneous mRNA template. In both (a) and (b) different curves correspond to different values of ${\omega }_a$, all for $\ell =12$. The discrete data points have been obtained from our computer simulations of the model, whereas the lines denote the $\Gamma$ distributions fitted to these data. The insets show the exponential decrease of the corresponding noise strengths with ${\omega }_a$. In both (a) and (b), ${\omega }_g=2.5{\mathrm{s}}^{-1}$ and $\alpha =0.1$.

Figure 6

(Color online) The average velocity of individual ribosomes as a function of the externally applied load force plotted for three different values of the rate constant ${\omega }_a$. The solid curves have been plotted using the analytical expression (3), together with (6), for an extremely small number density of the ribosomes. The discrete data points have been obtained by computer simulations of the same model.