Stochastic kinetics of ribosomes: Single motor properties and collective behavior

Syntheses of protein molecules in a cell are carried out by ribosomes. A ribosome can be regarded as a molecular motor which utilizes the input chemical energy to move on a messenger RNA (mRNA) track that also serves as a template for the polymerization of the corresponding protein. The forward movement, however, is characterized by an alternating sequence of translocation and pause. Using a quantitative model, which captures the mechanochemical cycle of an individual ribosome, we derive an exact analytical expression for the distribution of its dwell times at the successive positions on the mRNA track. Inverse of the average dwell time satisfies a “Michaelis-Menten-type” equation and is consistent with the general formula for the average velocity of a molecular motor with an unbranched mechanochemical cycle. Extending this formula appropriately, we also derive the exact force-velocity relation for a ribosome. Often many ribosomes simultaneously move on the same mRNA track, while each synthesizes a copy of the same protein. We extend the model of a single ribosome by incorporating steric exclusion of different individuals on the same track. We draw the phase diagram of this model of ribosome traffic in three-dimensional spaces spanned by experimentally controllable parameters. We suggest new experimental tests of our theoretical predictions.

Figure 1

(Color online) (a) A cartoon for pictorial depiction of the mechanochemical cycle of an individual ribosome in our model. Some of the symbols are explained in (b).

Figure 2

Mechanochemical cycle of an individual ribosome shown in Fig. 1 is redrawn for the convenience of formulation of the master equations.

Figure 3

(Color online) Probability density $f\left(t\right)$ of the dwell times of a single ribosome in the most general case of our model for a few different values of (a) the parameter ${\omega }_a$ (which is proportional to the concentration of tRNA-bound amino-acid subunits) and (b) the parameter ${\omega }_{h2}$ (which determines the rate of “stepping”). The continuous curve corresponds to the analytically derived expression (11) whereas the discrete data points have been obtained from computer simulation of the same model.

Figure 4

(a) Catalytic cycle of an enzyme in the Michaelis-Menten theory. $E$ denotes the enzyme while $S$ and $P$ denote the substrate and product, respectively. The symbol $I_1$ represents the intermediate state of molecular complex of which the enzyme is a component. (b) Generalization of the cycle shown in (a) to $n$ number of intermediate states $I_1,\dots ,I_n$.

Figure 5

The mechanochemical cycle of a ribosome, shown in the Fig. 2, is redrawn for the convenience of comparison to the MM enzymatic reaction scheme shown in Fig. 4b. The symbols $I_1,I_2,I_3,I_4$ denote the five intermediate states which we labeled in Fig. 2 by the integers 1, 2, 3, 4, respectively.

Figure 6

The effective mechanochemical cycle of a ribosome, where the effective rate constants ${\omega }_2^{eff}$ and ${\omega }_{-1}^{eff}$ are given by the Eqs. (25, 27), respectively.

Figure 7

The mechanochemical cycle of the molecular motor in the Fisher-Kolomeisky model for $m=4$. The horizontal dashed line shows the lattice which represents the track; $j$ and $j+1$ represent two successive binding sites of the motor. The circles labeled by integers denote different “chemical” states in between $j$ and $j+1$.

Figure 8

(Color online) Force-velocity relation for a ribosome in our model for a few different values of the parameter (a) ${\omega }_a$ (which is proportional to the concentration of tRNA-bound amino-acid subunits) and (b) ${\omega }_g$ (which can be controlled by varying GTP concentration). The continuous curve has been obtained from formula (34) whereas the discrete symbols denote the numerical data points obtained from computer simulations of the model.

Figure 9

(Color online) The randomness parameter $r$, defined by Eq. (39), is plotted against the concentration of aatRNA for a few different values of ${\omega }_h$ [in (a)] and ${\omega }_p$ [in (b)].

Figure 10

(Color online) The randomness parameter $r$, defined by Eq. (39), is plotted against the concentration of aatRNA for a set of parameter values where all the magnitudes of all the rate constants, other than ${\omega }_a$, are quite high.

Figure 11

(Color online) Distribution of the run times of a ribosome in our model. The continuous curve is the Gaussian distribution predicted by our theory while the discrete data points have been obtained from computer simulations. Inset shows the dwell time distribution for the same set of parameter values.

Figure 12

(Color online) A 3D phase diagram of our ribosome traffic model. The LD and HD phases coexist on the surface I (red). Surfaces II (purple) and III (blue) separate the MC phase from the HD and LD phases, respectively. Phase diagram is shown in (a) and (b) from two different orientations. The parameters used are ${\omega }_p=0.0028{\text{s}}^{-1},{\omega }_a=25.0,k_2=2.4{\text{s}}^{-1},{\omega }_g=25.0{\text{s}}^{-1}$

Figure 13

(Color online) Same as in Fig. 12 except that ${\omega }_h$ has been used instead of ${\omega }_a$.

Figure 14

(Color online) 2D cross sections of the 3D phase diagrams of the ribosome traffic model parallel to the $\alpha -\beta$ plane for different values of (a) $P_{{\omega }_a}$ and (b) $P_{{\omega }_h}$, all projected onto the same figure. The numbers on the phase boundary lines represent different values of (a) $P_{{\omega }_a}$ and (b) $P_{{\omega }_h}$. The HD and LD phases coexist on the curved lines whereas the straight lines separate the MC phase from the HD and LD phases, as shown. The parameters used for this figure are ${\omega }_p=0.0028{\text{s}}^{-1},{\omega }_a=25.0,k_2=2.4{\text{s}}^{-1},{\omega }_g=25.0{\text{s}}^{-1}$.

Figure 15

(Color online) The probabilities $P_{\mu }\left(t\right)$ $\left(\mu =1,2,\dots ,5\right)$ corresponding to the initial conditions $P_1\left(0\right)=1$ and $P_2\left(0\right)=P_3\left(0\right)=P_4\left(0\right)=P_5\left(0\right)=0$ [see Eqs. (79)–(83)] are plotted for the parameter set ${\omega }_a=23.0$, ${\omega }_{h1}=24.0$, $k_2=25.0$, ${\omega }_g=26.0$, ${\omega }_{h2}=27.0$, and ${\omega }_p=0.5$.