Exact Solutions for Kinetic Models of Macromolecular Dynamics

Simulation and analysis of single-ribosome translation

In the cell, proteins are synthesized by ribosomes in a multi-step process called translation. The ribosome translocates along the messenger RNA to read the codons that encode the amino acid sequence of a protein. Elongation factors, including EF-G and EF-Tu, are used to catalyze the process. Recently, we have shown that translation can be followed at the single-molecule level using optical tweezers; this technique allows us to study the kinetics of translation by measuring the lifetime the ribosome spends at each codon. Here, we analyze the data from single-molecule experiments and fit the data with simple kinetic models. We also simulate the translation kinetics based on a multi-step mechanism from ensemble kinetic measurements. The mean lifetimes from the simulation were consistent with our experimental single-molecule measurements. We found that the calculated lifetime distributions were fit in general by equations with up to five rate-determining steps. Two rate-determining steps were only obtained at low concentrations of elongation factors. These analyses can be used to design new single-molecule experiments to better understand the kinetics and mechanism of translation.

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.

Kinesin's backsteps under mechanical load

Kinesins move processively toward the plus end of microtubules by hydrolyzing ATP for each step. From an enzymatic perspective, the mechanism of mechanical motion coupled to the nucleotide chemistry is often well explained using a single-loop cyclic reaction. However, several difficulties arise in interpreting kinesin's backstepping within this framework, especially when external forces oppose the motion of kinesin. We review evidence, such as an ATP-independent stall force and a slower cycle time for backsteps, that has emerged to challenge the idea that kinesin backstepping is due to ATP synthesis, i.e., the reverse cycle of kinesin's forward-stepping chemomechanics. Supplementing the conventional single-loop chemomechanics with routes for ATP-hydrolyzing backward steps and nucleotide-free steps, especially under load, gives a better physical interpretation of the experimental data on backsteps.

Intersubunit coordination in a homomeric ring ATPase

Homomeric ring ATPases perform many vital and varied tasks in the cell, ranging from chromosome segregation to protein degradation. Here we report the direct observation of the intersubunit coordination and step size of such a ring ATPase, the double-stranded-DNA packaging motor in the bacteriophage phi29. Using high-resolution optical tweezers, we find that packaging occurs in increments of 10 base pairs (bp). Statistical analysis of the preceding dwell times reveals that multiple ATPs bind during each dwell, and application of high force reveals that these 10-bp increments are composed of four 2.5-bp steps. These results indicate that the hydrolysis cycles of the individual subunits are highly coordinated by means of a mechanism novel for ring ATPases. Furthermore, a step size that is a non-integer number of base pairs demands new models for motor-DNA interactions.