Measuring biological forces mini-lab

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20.309: Biological Instrumentation and Measurement

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Overview

The ability of biological systems to generate, carry, and sense mechanical forces is fundamental to an incredibly wide range of processes — in obvious places like muscle contraction and cell motility, but also in more unexpected settings like long term memory, cancer metastasis, cell division, and gene regulation. Methods for precisely exerting and measuring forces at the cellular, sub cellular, and molecular scale are crucial for many biological research and engineering applications. Laser tweezers and atomic force microscopes are two of the most frequently used instruments in force experiments. In this lab, you will make quantitative measurements with both of these instruments and determine their ultimate limit of detection. If you have time, there are some really cool optional activities available, such as Characterizing the e. coli flagellar motor, dna tethers, measuring elastic modulus by micro indentation, or imaging with the AFM.

Measuring focce

Use a spring. Calibrate the spring. You can do a higher-order calibration, but linear is good for small displacements. Measurements that need to be accurate to more than a sig fig or two should pay special attention.

Air molecules constantly boing into you. You become acutely aware of this on a cold february day, when the average velocity of the molecules gets reduced from a comfortable ... to ... You probably don't notice when you get on your bathroom scale, but the air molecules preterm the measurement... Thermal noise not much of a factor in precision of your bathroom scale. Ends up being the limiting factor for OT and AFM

AFM and optical trap model

Thermally excited AFM cantilever model.png Optical trap and AFM models.png Thermally excited Optical Trap model.png
RLC circuit model of AFM excited by collisions with air molecules. The integrator converts velocity to position.

The circuit model of the AFM and optical trap shown on the right includes a white-noise force source driving a mass, spring, and damper in parallel. (The parallel connection indicates that all of these model components have the same velocity.) The transfer function of the circuit model is:

$ \frac{x_{out}}{f_{in}}=\frac{\frac{1}{m}}{s^2+\frac{b}{m}s+\frac{k}{m}} $

In the time domain, random force input $ f(t) $ drives a second-order system to produce position output $ x(t) $.

AFM and optical trap signal processing.png
Time- and frequency-domain signals in the AFM. Air molecules collide with the cantilever to impart a random driving force $ f(t) $ with a white spectrum $ S_{ff}(f)=\frac{2K_BT}{\pi^2\Beta} $. The cantilever is an underdamped, second-order (mass-spring-damper) system with impulse response $ h(t) $ and transfer function $ \hat{H}(f) $. The transfer function has units of position divided by force (the inverse of a spring constant). The output power spectrum is equal to $ S_{ff} $ times the magnitude squared of $ \hat{h}(f) $.

The driving force has a constant power spectrum $ P_{ff}=\frac{2K_BT}{\pi^2\Beta} $. The output power spectrum $ P_{xx}(f) $ is equal to the input spectrum times the magnitude of the transfer function squared.