Tools and methods to study muscle contraction

[Produced by Tiberius Brastaviceanu, 2010]

Introduction 




This page is about methods for studying the dynamics of muscle contraction.

When activated, muscles perform work. In order to lift an apple from the ground energy must be transferred to that apple, in the mechanical form. We say that work is performed on the apple, which gains potential energy. That energy is released within our muscles. When pulling on an elastic we transfer energy from our muscles into elastic energy.

Physiologists like to say that muscles produce force, which is not exactly the language a physicist would use. Let's just say for now that our muscles are capable of producing mechanical work. But what is the fundamental mechanism behind this phenomena?

See the video to get a general idea and move the the next section to dive deeper, at the molecular level.

 


The cross bridge or sliding filament model  



Although the above video might give the impression that there are no mysteries about muscle contraction, this is actually far from the truth. The video is just a cartoon, based on our actual understanding. No one can actually see the mechanism unfolding at the molecular level, and we are far from understanding this system in all its complexity. The research continues...

Thus the role of this section is to present cutting edge experimental techniques that help physiologists today to broaden our understanding about active biological processes, like mass transport, motion/displacement and force generation.




NOTE: in this video it is shown that for one stroke of a myosin head 2 ATP molecules are needed. Experimental data suggests that only one ATP is needed. Apart from this, the video illustrates very well the mechanisms at play during muscle contraction
 




Measuring dynamic mechanical properties


Force transducers are used to study the dynamics of muscle contraction at different levels of organization, from the actin-myosin molecular interaction to macroscopic muscle fibers. 

Vocabulary

    • Measurand: A particular property or phenomenon subject to measurement by the sensor. 
    • Measurement: An instance of a procedure to estimate of the value of a natural phenomenon, typically involving an instrument or sensor. This is implemented as a dynamic feature type, which has a property containing the result of the measurement. The measurement feature also has a location, time, and reference to the method used to determine the value. A measurement feature effectively binds a value to a location and to a method or instrument. 
    • Observed Value: A value describing a natural phenomenon, which may use one of a variety of scales including nominal, ordinal, ratio and interval. The term is used regardless of whether the value is due to an instrumental observation, a subjective assignment or some other method of estimation or assignment. 
    • Phenomenon: An event or physical property that can be observed and measured, such as temperature, gravity, chemical concentration, orientation, number-of-individuals. 
    • Process Model: A process that takes one or more inputs, and based on parameters and methodologies, generates one or more outputs. 
    • Process (Process Chain): A composite process that takes one or more inputs, and based several internal processes with linkage between them, generates one or more outputs. 
    • Response Model: A type of Process Model, typically associated with a Sensor that takes one or more Samples as input, and outputs one or more new Products, based on characteristic response characteristics. 
    • Sample: A subset of the physical entity on which an observation is made. 
    • Sensor: An entity capable of observing a phenomenon and returning an observed value. A sensor can be an instrument or a living organism (e.g. a person), but herein we concern ourselves primarily with modelling instruments, not people. 
    • Sensor Model: In line with traditional definitions of the remote sensing community, a sensor model is a type of Location Model that allows one to georegister observations from a sensor (particularly remote sensors). 
    • (Sensor) Platform: An entity to which can be attached sensors or other platforms. A platform has an associated local coordinate frame that can be referenced to an external coordinate reference frame and to which the frames of attached sensors and platforms can be referenced. 
    • Transducer: An entity that receives a signal as input and outputs a modified signal as output. Includes detectors, actuators, and filters.


Force transducers, important considerations, classification


In vitro contraction of a bundle of myofibrils in pCa=4.5 activation solution. The bundle is attached to an AFM cantilever. Visualization under magnification 60X. After activation the tension is released twice from the other end of the fiber (not in the field of view). Experiment by Tiberius Brastaviceanu and Ivan Pavlov, in collaboration with Dilson Rassier.  


The classic Newtonian notion of force subsumes to the notion of space. An actuator that exerts a force on something, a force transducer for instance, normally induces a displacement of this thing in space, or at least of some part of it. But the notion of force doesn't necessary imply displacement. For example, the tension in a cord of a static/fixed pendulum, or the gravitational force between the Earth and a geostationary satellite. Based on this observation we can form 3 classes of force transducers:

    1. mobile flexible, usually elastic material object. A portion of object also serves as point of contact between the transducer and the sample, or as anchor. Example: a cantilever, a spring... 
    2. system composed of a) a mobile material object serving as point of contact or anchor, within b) an immaterial potential pit created with a electric or magnetic field, or c) a focused optical beam for example. Example: optical tweezers
    3. fixed force-sensitive material object also serving as point of contact or anchor.

The measurands in displacement/force sensors are displacement and force.

The piezo force transducer measures force directly. It is a classe 3 transducer. A cantilever-baced sensor falls in the class 1, using an elastic material object to measure displacement. The force is deduced from displacement taking into consideration the stiffness of the elastic object and a simple mathematical model that relates force to displacement. 
In transducers of category 2,  
the moving part can be a conductor loop through which a current is running, interacting with a magnetic field, or a small object interacting with a focused laser beam, this is the case of optical traps.

There are different ways to quantify the displacement of the moving material object in the first two categories.

  • By image analysis, where fixed reference images of the moving material object are taken at different times, and the position of the object is tracked frame by frame. 
  • By optical means, where some property of a laser beam is modified in some manner by the moving material object, and the change of property is analysed to deduce displacement 
    • deflection: the moving object causes a laser beam deflection, like in the case of the cantilever-based force transducer. The detection usually relies on a dual or quadrant photodiode. Example: cantilever-based force transducer, or the AFM
    • transmission and shadow: the moving object causes a change in the transmission of a laser beam, either by partial/total obstruction or total/partial absorption. In this case the detection mechanism can be a simple photodiode, or even dual or quadrant photodiode. Example: needle, micro-pipette, or cantilever -based force transducers, some optical guide as well as Pollak’s arrangement 
    • diffraction/interference pattern: the moving object causes a change in the diffraction pattern of one or two coherent laser beams interacting with the moving part of the force transducer. The detection mechanism in this case is composed of dual or quadrant photodiode, or even a CCD camera. Example: optical guide-based force transducer based on multi-core fibers. 
    • intensity modulation: the moving object causes a change in the intensity of a laser beam interacting with the moving part of the force transducer. The detection mechanism in this case is composed of a single photodiode. Example: optical guide-based force transducer based on leaky fiber. 
    • spectral modulation: the moving object causes a change in the spectral composition of a laser beam interacting with the moving part of the force transducer. The detection mechanism in this case is composed of a spectrometer coupled to a photodiode. Example: optical guide-based force transducer based on Bragg fiber. 
  • By electromagnetic means 
    • Voltage variation 
      • from capacitance variance 
    • Current variation 

Generally, sensors can also be classified by the spatio-temporal characteristics of their interaction with the sample/phenomena. In time, the interaction can be continuous for an interval ∆T, or discrete, happening only at precise moments. It can also be quasi continuous, or practically continuous, with interaction time dT very small (nano to microsecond), and with intervals between interactions on the same scale. In space, the interaction can be distributed over a given continuous domain in 1D, 2D or even 3D, or it can be discrete, in a single or multiple point(s). Direction and orientation can also characterize the interaction between the force transducer and the sample/phenomena, as well as the information obtained.

Temporal resolution

Temporal resolution for displacement/force measurement can be limited by the force transducer or by the acquisition system. In most cases they are limited by the force transducer.

Acquisition methods that use image analysis have a real handicap when it comes to temporal resolution because of the low frame rate of available cameras. The limit normally lies in the ms range.

Methods based on optical and electromagnetic effects measurements usually have acquisition rates well beyond the time-scale of most physiological processes. The limiting factor in these cases is the force transducer. The time response of elastic properties-based transducers is related to their resonance frequency.

Displacement and force measurement resolution and range

The resolution of systems based on acquisition methods that use image analysis are limited by the optical resolution of the electro-optical system, microscope+camera. The range is limited by the field of view. Spatial resolution is typically in the range of the micron, and the range is few hundreds of microns. The range can be limited to the range of motion of the elastic properties-based transducer.

In the case of methods based on optical and electromagnetic effects measurements with elastic properties-based transducers the range limitation is in part governed by the physical range of motion of these transducers. Furthermore, other range limitations can be imposed by the extent of the operating regime of the device, which can be of optical nature (for example in the case of systems based on an AFM cantilever the laser light path can only change only within a certain range before it becomes blocked/obstructed), or of electronic nature (in the case systems based on electromagnetic effects the range can be limited to the extent of the operating range of the relationship between measurand and the measured quantity - displacement vs voltage or current).

The resolution of most of these systems usually doesn't pose a limitation for myofibril level investigations. The sensitivity is related to the stiffness of the transducer. Capacitive force transducers are limited in the range of nano-microN. AFM cantilever-based transducers can, in principle, go down to the nN range. Optical traps are the least stiff, and can operate down to the pN, but they are not very suitable at the myofibril level due to their inherent low stiffness.
 



Examples of displacement/force measurement systems

Force transducers of class 1 (displacement, elastic)

This type of transducer gives direct access to the distance of shortening of the fiber during contraction. The sensor is essentially a one point contact displacement sensor. Knowing stiffness of the transducer one can calculate the instantaneous tension in the fiber, the work performed by the activating fiber on the elastic transducer, and the instantaneous power, or the energy transferred into elastic energy/unit of time. 

It is improper to say that this system measures the instantaneous force produced by the fiber, and the term force does not have the same meaning as in Newtonian force

The problem with this type of transducer is that while tension is developed within the fiber, the fiber shortens, because the transducer is compliant, and it bends under the tension generated. In order to achieve isometric contractions one needs to add an active feedback system, in order to pull the fiber as it contracts, in order to maintain its length constant. The Piezo transducer dose not present this problem, it can give direct access to the tension. 

Quantified by electromagnetic means: Capacitive force transducer

Consists a pair (or more pairs for multiple axes sensing) of parallel charged plates. The displacement can be measured from the gap change of the plate separation, which translated into a voltage change between the cathode and anode of the capacitor.
        • Advantage: can be easily microfabricated in different forms and dimensions. Some of them can measure force on 2 or even 3 axes. 
        • Disadvantage: what about plunging them in conductor liquids?

From Ref:Capacitive force sensor capable of resolving forces up to 490 μN with a resolution of 0.01 μN in x, and up to 900 μN with a resolution of 0.24 μN in y. (...) System testing demonstrates that the performance was not degraded in aqueous solutions, where only the probe of the force sensors was emersed in the solutions. The intended use of the sensor is to provide real-time force feedback for microrobotic cell manipulation, such as in cell positioning, embryo pronuclei DNA injection [26] and biomembrane mechanical property modeling for cell injury and recovery studies [27].



Several laboratories have used commercial capacitive force transducers from Cambridge Technology (Watertown, MA). The most sensitive model can resolve submicroNewton forces, but the resonance frequency is only ,100 Hz. Go to the source article:A bulk microfabricated multi-axis capacitive cellular force sensor using transverse comb drives

Quantified by image analysis

Consists of a passive elastic object with known stiffness, such as a spring or a cantilever. This object is attached to the sample. The displacement of the elastic object is measured on a video recorded during the experiment. This is also called time-lapse microscopy. Knowing the stiffness the force can be deduced from the displacement. The displacement/force can, in principle, be given live by using fast image analysis software with tracking capability, or after the experiment by post processing of a recorded video.

Advantage: the images of the sample in action are recorded, giving access to other type of information. Can operate in all directions within the focal plane, if imaged vertically. 
Disadvantage: slow acquisition rate, limited by the camera’s frame-rate (ms) time resolution. Slow image analysis and tracking processing, the recorded data takes a lot of memory.

Glass needles or micropipette

Glass needles can be made with a wide range of stiffnesses extending as low as 0.004 pN/nm [2], and are thus useful for molecular and supramolecular force measurements, as well as for other applications with similar force range. The needles can be calibrated to an accuracy of 5% to 15%. However, glass needles suffer from poor reproducibility and contrast, and require individual calibration. Go to the original text

Cantilevers

In some arrangements levers are constructed in pairs attached to one another at their base to allow for differential measurements.


2D marked membranes

During the last two decades, several methods have been developed to visualize and measure traction forces produced by single cells. Initial experiments were performed by growing fibroblasts on a thin silicon rubber substrate that was deformed by forces exerted by migrating cells.[5] Subsequently, elastic polyacrylamide substrates embedded with fluorescent latex beads were used to get an insight into the force vectors exerted by the cells.[6] These methods were further improved by producing regular arrays of traceable markers in the elastic substrates allowing a quantification of traction forces exerted by adhering cells.[7] A different approach has used arrays of elastic posts positioned on a flat 2D substrate, coated with extracellular-matrix (ECM) molecules.[8] Because each post effectively acts like a harmonic spring, local traction forces at multiple cell adhesion sites could then be measured by simply monitoring post deflections. A modification of this assay was recently applied to measure contraction forces of single cardiomyocytes.[9] Go to source of text

Reference

Quantified by optical means


Reflection - the cantilever, or AFM type transducer

The system I worked with belongs to this category. It consists of a moving part with known stiffness, which can be a passive elastic object (a cantilever with known stiffness). The moving part has a reflective surface that acts as a mirror for a narrow/focused laser beam. The sample is also attached to the moving part. Its displacement is measured from the deflection of the laser beam reflected by the mirror. This deflection is normally measured with a quadrant photodetector. Knowing the stiffness the force can be deduced from the displacement. 

Advantage: fast acquisition rate because the acquisition system is reduced to the difference between voltage between two photodiodes, The data generated doesn't take much memory space.
Disadvantage: other type of information about the experiment is lost, only total displacement is recorded and there is no access to individual sarcomere length; the beam propagates in free space and can be obstructed/refracted by debris or by transient interfaces between solutions with different indexes of refraction (activation and relaxation solution for example); needs optical alignment and repetitive calibration; the reflective surface can get dirty or damaged, distorting the beam, it works only in one direction. 

Note: our system integrates visual feedback and video recording with the displacement measurement. Other imaging techniques can be integrated with it such as fluorescent and confocal microscopy. 
Reference

Transmission and shadow

In this case laser light is used to record displacement of a moving elastic force transducer. The detection is similar to the reflexion case, the information is extracted from the difference in signal from two photodiodes. 

Example of transmission
 


Using one stiffness-calibrated optical fiber (the force transducer) which also acts as a light source, and two receiving fibers, which direct the light to two separate photodetectors. 

Advantage: fast acquisition rate, the data generated doesn't take much memory space.
Disadvantages: the beam propagates in free space and can be obstructed/refracted by debris or by transient interfaces between solutions with different indexes of refraction (activation and relaxation solution for example); needs optical alignment, operates in only one direction. 

Example of shadow

Two micropipettes were used to hold the sample, a muscle cell. One pipette was of a known stiffness, and was used as a force transducer. The signal carrying the information is light in transmission, intensity variation on a quadrant photodetector. The laser beam is partially obstructed by one of the pipettes, and the movement of the shadow is recorded with the quadrant photodetector.



Optical guide
 
Consists of a flexible optical guide, an optical fiber, which changes some property of the light passing through it while bending. We call this bend-sensitive optical fiber. 

Inventors: Tiberius Brastaviceanu and Ivan Pavlov

Advantage: the laser beam doesn't propagate in free space, there are no problems associated with the quality of the reflective surface of the cantilever; there is no need for recalibration before every set of experiments; depending on the operation principle, the acquisition system can be very fast, especially in the case where the information is encoded in intensity or spectral variation rather than spatial intensity distribution (diffraction pattern); systems based on intensity variation as the operation principle can be very low cost, this is the only type of device that operates well in all directions within a plane. 
Disadvantage: the fabrication process of the sensitive fiber can pose some problems.

Force transducers of class 2 (mobile object + potential pit)

Optical trap or tweezers

For measuring single-molecule mechanics optical traps are the instrument of choice. 



Advantages: low stiffness (as low as 0.01 pN/nm) and the high time and position resolution available when used in conjunction with a quadrant photodiode detector. 
Disadvantages: are not practical for measuring forces in excess of 100 pN; higher laser powers that might stiffen the trap may cause damage to the biological specimen. This severely limits the application of optical traps to measurement of small numbers of molecular interactions.

Force transducers of class 2c

Based on piezoelectric fiber, it doesn't exist yet

Force transducers of class 3

This transducer gives direct access to instantaneous force.

Piezo force transducer

Consists of a piezo crystal and some sort of lever to transfer and amplify/diminish motion.

Advantage: can measure isometric force production.
Disadvantage: not very sensitive?



Problems with variations of AFM 

I did extensive work with a AFM-type displacement/force measurement system. In this section I discuss some problems I encountered.

Inherent complexity

Before every experiment the user must go through different preparation and calibration procedures. The most critical procedure is the Displacement Calibration, during which the real displacement of the cantilever is plotted against the the voltage signal from the quadrant or dual photodetector (see the standard hardware architecture of this kind of device). The optical alignment of the cantilever and the laser beam is also important and sometimes very tedious. 

Ambient light influences sensor readings

If the photodetector of the device is sensitive to visible light, ambient light can change readings. Variations in ambient light can constitute serious problems. The Displacement Calibration procedure will give different results for different illumination conditions. The Displacement Calibration must be performed at the same conditions as the experiment. 

The solution would be to use a special wavelength and a narrow band pass filter in front of the detector. The filter let’s only a small fraction of the ambient light to reach the detector, and all the laser light. Even if the intensity in the ambient light varies, the total energy variation will be small, because the laser light represents the bulk of the energy. If there is no filter ambient light of all colors “seen” by the detector reaches the detector, and this might represent a large portion of the total light reaching it including the laser.

Optical artifact

In the case where the rapid solution switching technique is used to change the state of the fiber between activation and relaxation, because the activation and relaxation solutions have different refractive indices the laser beam refracts/reflects from the interface between these two streams during the motion of the pipette, or during the switch between these two fluids. The transition appears as a broad peak right before activation and relaxation of the myofibril. This effect was well described by some recent experiments in our lab.


 

Debris

Because the beam propagates in free space to the cantilever and back to the detector, any particle that gets in front of it gives a false signal. 

Moreover, the cantilever itself must be very clean to reflect the beam back without any imperfection. Any small imperfection on the surface of the cantilever will distort the beam by diffracting it, giving raise to a distorted signal. We encounter this problem very often and we always make sure to scan the surface of the cantilever to eliminate this artifact. The effects are seen during the Displacement Calibration scan.


[Produced by Tiberius Brastaviceanu, 2010]
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