Good afternoon everyone. Today I am going to talk about the design of an optical force clamp for molecular motor investigation.
In tis presentation, first I will give a brief introduction of motor proteins and then discuss about the optical tweezers, which are one of the primary probing devices for motor protein study. In this context, I would give the motivation for the development of a force clamp and I would point out the deficiencies of the existing force clamps . I would then show how system theoretic tools can help circumvent the problems and present a LMI based mixed objective controller design methodology. Finally, I would present the performance of the controller in an actual experimental test bed with simulated motor protein movements and compare the results with traditional force clamps.
Processive motor proteins are fascinating Nano machines that convert the chemical energy of ATP in cell into mechanical energy to move forward. They are enzymatic molecules that carry cellular cargo from one cell to another walking over polymeric lattice structured tracks called microtubules. The video explains a schematic of transport of a motor protein. The exact mechanism and energetics of their transport characteristic is an active field of research, as their malfunction leads to diseases like Alzheimer’s. One such motor protein of particular interest for in vitro studies is Kinesin. The two motor domains attaches and detaches to the microtubule lattice for the Kinesin to move forward, while it carries the cellular cargo at its tail end. Although Kinesin has an approximately 100 nm long stalk, it can carry cargos several times bigger, up to a few microns of size.
This capability of Kinesin of carrying larger cargoes is exploited by optical probing for their investigation. I will talk more about optical trapping in a moment, but here note that beads of 500 nm or larger are attached to the kinesin which in turn is probed by a laser trap to infer about the force and motion of the kinesin. Kinesin takes steps of size 8 nm and can respond to as small as fN order of forces, which makes optical tweezer an ideal candidate for their study. The video shows a kinesin bead assay under fluorescence microscopy, where you can see the cargo moving on the microtubule, but you can’t see the kinesin. The figure shows a schematic of an optically trapped bead by a Kinesin. Here note that the bead position is determined by the trapping force, thermal noise and the force exerted by the kinesin.
Ashkin in his seminal paper showed that it is possible to trap micron and smaller sized particles of suitable refractive index by focusing a collimated laser beam of appropriate intensity profile using a high NA objective. The laser creates a stable trap near the focus and thus for small deviations, the trap force can be modeled as a linear Hookean spring. By steering the laser beam with appropriate optics, it is possible to manipulate the position of the trapped bead in the x-y plane. The laser passed through the bead is collimated using a condenser and collected on a photo diode, whose position gives a measure of the bead position in the x-y plane. For more sophisticated application a separate laser is used for measuring the bead position.
The figure shows the schematic of the optical tweezer set up that we have developed in our lab. Here we use a separate detection laser for bead position measurement. The trapping laser is blocked by a ND filter from reaching the QPD. Although the spatial response of the QPD is non-linear, using the intensity we can extend the linear detection region by neural networks. An AOD is used to steer the trapping beam at a very high bandwidth. An conjugate lens pair arrangement ensures that the movement inflicted by the AOD is translated to the sample plane. The AOD receives its command from an FPGA, which in turn processes the QPD measurements. Using this high speed FPGA , loop closure rates up to 200 kHz is achievable for control applications.
This figure shows you a typical force-extension curve for a kinesin molecule. Generally, for mathematical analysis, the kinesin is modeled as a linear Hookean spring, which can obviously be not true the excursion in load force and hence its extension is too high. The second figure shows the force-extension curve for DNA, where unfolding of domains introduces hysteresis. Thus, the validity of the linearized analysis warrants for a constant force clamp. Also, one can infer about the motor stepping motion from the bead motion only if the extension and hence the load force remains constant. Although the huge thermal noise variance, which is comparable to the motor step size, makes the real time step estimation difficult and people has to rely on post-processing.
The existing force clamps can be categorized in active and passive clamps. The active clamps employ active feedback, whereas the passive clamps exploit physical property of the laser trap to maintain constant force. The figure shows the spring mass damper model of an optically probed kinsein where… The control law used in these clamps is ad hoc and do not use any formal design methodolgy. They just measure the change in bead position and move the trap by same amount. Here the aod dynamics is ignored or it is assumed the coomand u given to the aod is same as the trap position xt. Clearly, the operational bw of this scheme will be limited by the valdity of the assumption u=xt and the figure shows how their performance suffers at high bw. Here… the motor proteins are slowed down by an order of magnitude from their natural speed to meet this bw constraint. Passive clamps on the other hand uses a zero stiffness region of the trap force displacement curve and thus theoretically can have infinite bw. However, the extent of this region is too limited to use with processive motor proteins.
To understand the AOD dynamics, lets discuss the operating principle of the AOD briefly. In AOD, a sound wave is propagated through a crystal …thus the aod dynamics can not be ignored.
We used system identification tools to characterize the AOD dynamics precisely. We gave a chirp input to the AOD and measured the bead position to obtain the transfer function from the AOD to the bead position. Now, the tf of the trapped bead can be easily found out as … The phase plot clearly shows that … Also, the undershoot present in both the experimental and simulated step response confirm the presence of the rhp zeros. This rhp zeros poses a fundamental limitation on the performance that can be achieved.
We next cast our problem in the modern control framework. The figure summarizes all the signals and weighting funtions for the controller design. Red signals… the control also generates an estimate of stepping motion in addition to moving the trap position for force regulation. … To find the optimal controller, we take resort to a LMI …
I will not go into the detail of the solution methodolgy. To give you a flavor, we choose certain performace channels psi, psi2 and psi3 from the closed loop tm psi and impose constraints on them. … This has the adv that we do not impose artificial constraints on other channels like we would have if we were to minimize the hinfinity norm of the entire tm psi.
The problem turns out to be a non convex optimization from KYP lemma, but fortunately certain transformations can be made to map it to and euivalent convex optimization problem. Details of the method can be found in the paper … . Here we solve a set of LMIs that incorporates all the constartints and the solution can be transformed to get the optimal stabilizing controller. Details about how to form this LMIs and the the transformation techniues can be found in our paper. The main advantage of this method is that it allows for different objectives for different channels and we do not need to consider the channels whose performance we are not worried about.
Here we present the performance of the controller where force regulation is the only objective. The motor motion is simulated by multiplexing the trap and giving suare input pulses at xm. We can see for low velocity both of the traditional controller and our controller yields 5% regulation, which is similar to what is reported in literature. For High speeds, however, the regulation of the traditional controller deteriorates by 150% where as for the modern controller it only deteriorates by 30%. This shows that the modern controller can be used as force clamps for much highers speeds successfully, possibly near the natural speeds of the motor protein movements.
This slide shows the performance of the modern controller where step estimation is the main goal while still maintaining a good force regulation to validate the linear analysis. Typically, steps are estimated offline via post processing from the noisy bead trace, as the thermal noise variace is comparable to the step sizes. Sometimes real time step estimation is done by long time averaging, but that is limited to slowly moing motors. The figure shows that using the modern control approach we can obtain good estimates of motor stepping motion in real time for velocities above 1 micron/ s, while maintaining force regulation better than 7%.
I will not go into the detail of the solution methodolgy. To give you a flavor, we choose certain performace channels psi, psi2 and psi3 from the closed loop tm psi and impose constraints on them. … This has the adv that we do not impose artificial constraints on other channels like we would have if we were to minimize the hinfinity norm of the entire tm psi.
