Intellectual merit. This investigation will result in a new molecular dynamic model for nanosized motor proteins. This new model addresses an issue with the most commonly used models that omit the mass and acceleration terms that appear in Newton's second law. The most common approach assumes that mass and inertia have negligible effects on the dynamics of nano-sized objects moving through a fluid environment. This practice is paradoxical because the motion predicted by this massless model may not actually satisfy the theory from which it was derived, Newton's second law. The effort in this research is to develop a model which is consistent with Newton's second law by retaining the mass and acceleration terms. The method of multiple scales, when applied in this research, provides insights into how to obtain the desired model. Preliminary work with this approach has yielded motion of the motor protein myosin V, which appears more realistic. The goal here is the continued refinement of this model along with experimental validation of its predictions using advanced laser microscopic techniques. The use of this highly intense light source allows capture of the motor protein's movements at a rate fast enough, and at a length scale small enough, to discern whether the motor protein exhibits the behaviors predicted by the model. These measurements are difficult to perform using conventional microscopy techniques.
Broader impacts. Involvement in this research will facilitate the training of two graduate students in the dynamics of nano-scale systems and provide learning experiences for undergraduate students in this project. These students will be involved in developing the proposed theory, performing single molecule experiments, and disseminating the results through journal publications and presentations at conferences, which will provide training for future careers in academia or industry. Both investigators are currently involved in training minority students in order to increase the pool of individuals qualified for academic and industrial careers. Because the proposed research is far reaching in terms of its general applicability to the study of phenomena occurring at small scales, especially particle dynamics in low Reynolds number flow regimes, training in this area will equip the students with tools that they can use to launch their post-graduate careers in many directions beyond the study of motor proteins.
The key outcome of this award was the development and validation of a new model for nanoscale mechanics in environments characterized by a low Reynolds number. This type of environment exists within living cells in the body and in many other situations. A straightforward Newton-Euler model of small objects in a fluid environment suggests that they exhibit overdamped motion. Thus the objects appear to have negligible momentum, which means their motion relative to the fluid stops almost instantaneously in the absence of an external force. We have shown that this assumption is incorrect as the size of the object decreases to within the nanoscale range. Our model predicts underdamped motion of these objects, which was observed using an optical tweezer to trap, at its focal point, successively smaller spherical beads in water. Beads 990 nanometers in diameter and smaller, would overshoot the focal point and return to it, an example of underdamped motion; larger beads exhibited overdamped motion in that they moved directly to the focal point and stopped with no overshoot. This observation provides some validation of the proposed model, which predicts significantly different behavior than suggested by most researchers investigating the dynamic behavior of objects at the nanoscale as well as those attempting to use nano-sized objects in low Reynolds number environments in many applications involving cell mechanics, drug delivery, and optical trapping. A key reason why the overdamped motion assumption persists is because it takes a very long time to solve the standard models of these systems, on the order of days, weeks, or months. Most people will use shortcuts or other approaches to obtain a solution, rather than wait for one from the standard model. Unfortunately, they tailor these shortcuts and other approaches to predict what they already think should happen, thus most of them predict overdamped behavior. Our method is based on techniques from the method of multiple scales, an established approach for addressing systems with several time scales. The multiscale nature of the motor protein modeling stems from the fact that the mass of the object is very small but the forces acting on it are very large. This results in large accelerations whose effect can only be examined over a small amount of time, on the order of femtoseconds. Thus the problem is said to have fast dynamics, acting only for a short time before significant changes in the motion occur, and slow dynamics, where changes in the motion occur slowly over a longer period of time. We are interested in the protein’s motion over a longer period of time, so the goal is to isolate the slower dynamics. The solution of the standard model requires that we account for the fast dynamics, thus it takes much longer to obtain a solution, on the order of days, weeks, and months. The new model can be solved in a fraction of that time, on the order of seconds, minutes, or hours. We typically see about a 90% decrease in solution time for the proposed model, which means that we can investigate these small systems at a much faster pace than those using standard models. In addition, a straightforward application of the method of multiple scales for this type of dynamic system yields a model classified as a singular perturbation problem, which is difficult to solve. The proposed model is a new solution to this type of problem, which is significantly different from the proposed approaches. We applied the proposed modeling technique to motor proteins in order to investigate their dynamic behavior and function within the cell. Motor proteins are nature’s actuators that convert chemical energy into mechanical work. We are particularly interested in processive motor proteins, such as Kinesin and Myosin V that perform a biped walking motion as they travel through the cell along the cytoskeleton transporting vesicles and nutrient packets to where they are needed. The motor protein is much more complex than a spherical bead, but the modeling approach can still be applied and provides new insights into their behavior. These insights include the following: The heads perform an underdamped, oscillatory searching motion that helps them to dock at particular sites along actin filaments comprising the cytoskeleton. The heads must dock, form a strong bond between the head and the substrate, in order for the protein to process/move effectively. The magnitude of forces acting on the protein must balance each other in order to prevent physically infeasible behaviors. This suggests an efficiency for the amount of work that is converted from chemical energy. (We are close to determining a value to quantify this efficiency, but did not nail it down as yet.) The effect of Brownian motion on the motion of motor proteins is much less than suggested by many in the field. It has been suggested that the movement of motor proteins is completely determined by Brownian motion, a process referred to as diffusion. Our model suggests that it is the built in behavior, the coordination of a chemical reaction called ATP hydrolysis with binding and unbinding with the substrate, that drive the proteins movements; diffusion acts as a disturbance to this process.
