Traffic of interacting ribosomes: Effects of single-machine mechanochemistry on protein synthesis

Many ribosomes simultaneously move on the same messenger RNA (mRNA), each synthesizing separately a copy of the same protein. In contrast to the earlier models, here we develop a “unified” theoretical model that not only incorporates the mutual exclusions of the interacting ribosomes, but also describes explicitly the mechanochemistry of each of these macromolecular machines during protein synthesis. Using analytical and numerical techniques of nonequilibrium statistical mechanics, we analyze the rates of protein synthesis and the spatiotemporal organization of the ribosomes in this model. We also predict how these properties would change with the changes in the rates of the various chemomechanical processes in each ribosome. Finally, we illustrate the power of this model by making experimentally testable predictions on the rates of protein synthesis and the density profiles of the ribosomes on some mRNAs in E-coli.

Figure 1

A pictoral depiction of three major steps in the chemomechanical cycle of a single ribosome. The larger and smaller subunits have been depicted as two rectangles. The complementary shapes of the vertical tips and dips merely emphasize the codon-anticodon matching.

Figure 2

A schematic representation of the biochemical cycle of a single ribosome during the elongation stage of translation in our model. Each box represents a distinct state of the ribosome. The index below the box labels the codon on the mRNA with which the smaller subunit of the ribosome binds. The number above the box labels the biochemical state of the ribosome. Within each box, $1\left(0\right)$ represents presence (absence) of tRNA on binding sites E, P, A, respectively. $1^{*}$ is a elongation factor (EF)-Tu bound tRNA and G is a EF-G GTPase. The symbols accompanied by the arrows define the rate constants for the transitions from one biochemical state to another. As explained in Sec. 4, the dashed arrow represents the approximate pathway we have considered in the simplified dynamics of our model.

Figure 3

A schematic representation of the model. (a) A cartoon of a single ribosome that explicitly shows the three binding sites E, P, and A on the larger subunit which is represented by the ellipsoidal lobe. The rectangular lower part represents the smaller subunit of the ribosome. (b) The mRNA is represented as a one-dimensional lattice where each site corresponds to one single codon. The smaller subunit of each ribosome covers $\ell$ codons ($\ell =2$ in this figure) at a time.

Figure 4

Flux of ribosomes with $\ell =3$, 12, under periodic boundary conditions, plotted against density. The curves correspond to the analytical expressions obtained in the mean-field (MF) approximation whereas the discrete data points have been obtained by carrying out computer simulations (Sim). Values of all the parameters, except $\ell$, are the same as those listed in Table .

Figure 5

Flux of ribosomes with $\ell =12$, under periodic boundary conditions, plotted against density. The curves correspond to the analytical expressions obtained in the mean-field (MF) approximation whereas the discrete data points have been obtained by carrying out computer simulations (Sim). Values of all the parameters, which are not shown explicitly on the figure, are identical to those in Table .

Figure 6

Same as in Fig. 5 except that different curves correspond to different values of ${\omega }_a$. Values of all the parameters, which are not shown explicitly on the figure, are identical to those in Table .

Figure 7

Same as in Fig. 5 except that different curves correspond to different values of ${\omega }_g$. Values of all the parameters, which are not shown explicitly on the figure, are identical to those in Table .

Figure 8

Same as in Fig. 5 except that different curves correspond to different values of $k_2$. Values of all the parameters, which are not shown explicitly on the figure, are identical to those in Table .

Figure 9

Flux of ribosomes plotted against (a) ${\omega }_{\alpha }$ and (b) ${\omega }_h$ for the genes crr (170 codons) and cysK (324 codons) of Escherichia coli K-12 strain MG1655, as well as the corresponding curve for a homogenous mRNA strand of 300 codons. The insets show the average density profiles on a hypothetical homogeneous mRNA track for four different values of (a) ${\omega }_{\alpha }$ and (b) ${\omega }_h$, for fixed ${\omega }_a=25{\mathrm{s}}^{-1}$.

Figure 10

Incorporating $\alpha$ and $\beta$ through two reservoirs with appropriate densities.

Figure 11

Phase diagram in $\alpha -P_{{\omega }_a}$ plane. $P_{{\omega }_a}$ is the probability of attachment of a tRNA in time $\mathrm{\Delta }t=0.001\mathrm{s}$, and is related to ${\omega }_a$ by Eq. (A2). This diagram has been plotted for ${\omega }_{h1}={\omega }_{h2}={\omega }_h=10{\mathrm{s}}^{-1}$.

Figure 12

Phase diagram in $P_{{\omega }_h}-P_{{\omega }_a}$ plane. $P_{{\omega }_h}$ is the probability of hydrolysis in time $\mathrm{\Delta }t=0.001\mathrm{s}$, and is related to ${\omega }_h$ by Eq. (A2).