INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

A MAJOR LIMITATION IN STUDIES of muscle contraction is the absence of an ideal force transducer; i.e., one with high sensitivity, low noise, high speed, low compliance, and linearity that is also rugged, easy to repair, and resistant to humidity. The need for high-speed transducers for use with intact skeletal or smooth muscle and with high sensitivity for skinned fibers, single fibers, or myotubules has led to many different designs (1-3, 6). The device described here integrates the above features of an ideal transducer to a significant degree. It has a cantilever beam that can be projected into a muscle bath. Rotational movement of the cantilever is amplified by an "optical lever" derived from a laser diode beam reflected from a mirror on the cantilever to a pair of diodes. This arrangement places the light source and sensor away from the damp environment of the bath. The dynamic properties of the device can be altered over a small range by adjusting the length of a compliant "hinge" at the proximal end of the beam, and over a larger range by substituting different beams. The principal description here is of a device for studying single skeletal muscle fibers that develop up to 1 mN force. Also described are cantilevers for studying muscles that develop orders of magnitude more and less force.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Principle of the Design

View larger version (22K): [in this window] [in a new window]   Fig. 1.   Two types of cantilever design. A: sculpted aluminum element for rapid recording in single cells, with side, front, and perspective views; a: clamped portion, b: hinge, c: brace and mirror, d: distal extension. B: Folded titanium element for slow recording. Shape before folding, on left. Tabs x and y are folded toward reader by ~120° and then distal extension z is folded to right by 30° to align vertically. Side, front, and perspective views after folding along dashed lines are shown, respectively, in the 3 rightmost drawings. Mirror, shown in bold outline, is glued in place.

View larger version (24K): [in this window] [in a new window]   Fig. 2.   A: mechanical and optical components of force transducer. Light-beam path indicated by dark lines with arrows. B (inset): details of mirrors (ms, stationary mirror; mc, mirror attached to the proximal end of cantilever). C: circuit used to measure separate current outputs from the 2 diodes. D: circuit used in normal operation.

The cantilever principally described was made from sheet aluminum milled flat in the clamped portion and hinge (segments a and b in Fig. 1A) to ~0.8 mm thick × 0.8 mm wide. The thickness of the beam increases abruptly distal to the hinge (from 0.8 to 2.5 mm), to brace against flexion (segment c in Fig. 1A). Below this brace, the beam tapers both in thickness and width toward the distal tip. The back upper edge of the brace is angled at ~38° to the horizontal and serves to support a 2.5 × 4 × 0.2-mm front surface mirror. A more robust but slower cantilever for measuring isometric force in permeabilized whole muscle was made from titanium folded as indicated in Fig. 1B. The advantages of the titanium are that it can be folded and that it is more resistant to corrosion, but its greater density diminishes its speed.

The length of the free end of the cantilever is 6 mm, with the proximal 2 mm comprising the hinge, brace, and mirror support (segments b and c in Fig. 1A). This length is the minimum for our applications. It allows up to 2 mm to be projected into a solution bath and ~2 mm of protection against a rise in the level of the bath to reach the moving mirror (segment d in Fig. 1A). To attach muscles to the transducer, a hook made from an 8 mm × 100-µm stainless steel wire was cemented perpendicularly to the tip of the cantilever with a thin layer of water-resistant epoxy (see side views in Fig. 1). Muscle ends were held with aluminum T clips (5), and holes in the clips were passed over the hooks to connect the muscle to transducer and servomotor.

The body of the transducer is made in three sections (Fig. 2A) that can be adjusted relative to each other during alignment of the beam. The middle section is an aluminum block that holds the laser diode, serves as a heat sink, and is attached to the apparatus. A "front piece" mounted toward the muscle holds both the stationary mirror and the clamp that grips the cantilever beam (Fig. 2B). This section can be moved vertically, i.e., along the axis of the cantilever beam, to adjust the position of the laser beam on both mirrors. Small rotations of the front piece (about an axis perpendicular to the bearing surface) cause horizontal displacements of the laser beam reflected from the mirrors. Such rotation is used to center the beam horizontally on the photodiodes. A "backplate" holds the photodiode pair.

Not shown in the diagrams are a framework and wrap made of hydrophobic strips (Parafilm, M-type, American National Can, Chicago, IL) that cover but do not touch the moving element. The strips offer additional protection against floods and slow thermal changes caused by air drafts.

Components

Laser diode. A 5-mW, 635-nm-wavelength laser diode with constant-current driver and focusable lenses was obtained from Merideth Instruments (Glendale, AZ; part no. VDM21). The assembly is mounted in a 9-mm-diameter tubular housing. The lens diameter is 5 mm, and its focal length is 4.8 mm. The light beam exiting the lens has an approximately oval cross section, with major and minor axes 5 × 3 mm. This reduces to ~3 × 1.8 mm at the moving mirror and to its smallest size, ~300 µm vertical diameter, at the diode pair, 135 mm from the lens and 90 mm from the moving mirror. Because the angular change of the light beam is twice the angular change of the mirror, the ratio 15:1 of rotating light beam length (90 mm) to cantilever length (6 mm) creates a 30:1 ratio of light spot movement to tip movement.

Photosensor. A photodiode pair with total area measuring 1 × 3 mm divided along the long axis, so that each diode measured 0.5 × 3 mm, was obtained from Hamamatsu (US representatives in Park Ridge, IL; part no. S2722). Figure 2, C and D, shows, respectively, the circuit used to measure the separate outputs of the diode when characterizing the apparatus and the circuit used in normal operation to obtain the difference between the outputs of the two diodes. The circuits are mounted on the transducer, near the diode pair, to reduce noise along the connecting cables.

Back-plate. The photodiode pair is mounted on a 6.3-mm-wide pedestal on a larger Plexiglas plate. This pedestal fits into a 6.3-mm-wide slot in the back of the center section of the back-plate, and the Plexiglas plate is mounted on rods that allow the plate and diodes to be moved vertically. Springs on the lower ends of the rods force the plate upward against a screw that adjusts the vertical position of the diodes so that the focused laser beam falls equally on both diodes.

Front piece. The front piece is a 19 × 19-mm-square plate with an angular slot milled partially through its center (Fig 2A). The base of the slot, at 60° to the horizontal, holds the stationary mirror (ms in Fig. 2B). The laser beam strikes this mirror at 30° to the horizontal and is then directed vertically downward to meet the moving mirror (mc in Fig. 2B). The apex of the angle at the end of the slot is cut off ~0.5 mm from its end, and the front piece is made of steel to provide a rigid clamp for the cantilever. A brass front piece having a sharper apex was found to be too flexible. The upper front face of the front piece is milled away to clear the line of sight from microscope to muscle. Slots for the attachment screws provide both vertical adjustment and small rotational adjustment of the front piece.

Cantilever beam. Cantilever beams were made of aluminum alloy, type 7075, hardened to T-6 temper (obtained from Pioneer Aluminum, Wichita, KS). The metal was first cut and milled to have the exact thickness of hinge and clamped region and the rough outlines of the distal extension. The final shape of the element (Fig. 1A) was achieved by hand-grinding and filing. With practice, a new element could be made from the milled sheet metal in about an hour.

Calculations

(1)

(2)

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Light Beam Width, Sensitivity to Movement, and Noise

View larger version (21K): [in this window] [in a new window]   Fig. 3.   Current outputs from 2-diode pair measured as light beam was swept across them. A: separate outputs of the 2 diodes plotted as solid curves and their sum plotted as dashed curve. B: output of 1 diode subtracted from the other, representing output obtained when using the circuit shown in Fig. 2D.

The data in Fig. 3 were obtained with the laser beam focused to as small a spot as possible at the diode pair. To assess whether the focus was critical, the maximum slope in the middle of the difference records (Fig. 3B) was measured when the laser diode was displaced along its axis in its holder. The slope increased by <5% when the diode was displaced up to 8 mm in either direction, indicating that the sensitivity of the device was not critically dependent on the focus of the laser beam.

The slope of the difference trace in Fig. 3B is 0.33% of the full signal per micrometer. When used with the ×10 amplifier, shown in Fig. 2D, and a slightly brighter laser diode, this slope corresponded to ~800 mV/µm movement of the laser spot at the diode. An optical lever ratio of 30:1 makes the sensitivity of the device to tip movement 24 V/µm.

The root-mean-square (RMS) noise of the electrical signal with the laser on was 400 µV (measured with a Tektronix TDS-340A digital oscilloscope.) Most of the peak-to-peak noise is contained within a band ~4 times the RMS noise, equivalent to 1.6 mV. A goal of the design was to have sufficient sensitivity for this band to be <1% of the maximum expected signal. For single skeletal muscle fibers that develop 1 mN, the desired sensitivity was at least 160 mV/mN or 1.6 V/g.

Sensitivity and Resonant Frequency

View larger version (17K): [in this window] [in a new window]   Fig. 4.   Resonant frequency () and sensitivity () as a function of hinge length.

At each hinge length, the output of the transducer was found to be linear for loads up to 2 g. At the shortest hinge length, four times the RMS noise corresponded to 7.3 µN of force or ~0.7% of the expected maximum force. Thus the ratio of maximum signal-to-noise was ~550, well above the stated goal of 400.

The resonant responses of the device to sudden unloading at the three hinge lengths are shown in Fig. 5, A-C. The dominant resonance at the shortest hinge length was 22 kHz, with lesser resonances at higher frequencies. These other resonances were less obvious at longer hinge lengths, where the dominant resonance was of lower frequency and greater amplitude. The sources of these higher resonances were not identified. The dominant resonances seen after rapid unloading decayed with time constants between 2.5 ms for the short hinges and 8.9 ms for the longest. As described in Fig. 6, the steps typically used with a muscle attached did not excite these resonances appreciably. The dependence of resonant frequency on hinge length was greater than expected on the basis of the changes in sensitivity and compliance. This greater change was due to increased equivalent mass of the cantilever element as the hinge was lengthened.

View larger version (23K): [in this window] [in a new window]   Fig. 5.   Oscillatory response to sudden unloading of transducer element for 3 lengths of hinge (A-C) in Fig. 4.

View larger version (11K): [in this window] [in a new window]   Fig. 6.   A: force record (10 samples/s) during an isometric tetanus in tracheal smooth muscle. Record has been spliced to remove period of rapid recording associated with isotonic step shown in B. B: isotonic release to 50% of isometric force. Lowest record is force. Two records superimposed in top are of overall length (dotted trace) and central segment length (solid trace). Left-hand calibration bar is for segment length; right-hand bar is for overall muscle length. C: a 20-µm step, complete within 200 µs, was imposed on central segment (middle trace) while its length was servo-controlled. Changes in overall muscle length and force are shown in top and bottom traces, respectively. For these recordings, digitization rate was changed several times, as indicated by vertical cursors, and only a portion of the typical fast record is shown, from 5 ms before to 15 ms after the step. Initial isometric period is recorded at 1 sample in each channel per ms. First 2 ms associated with the step are recorded at 1 sample in each channel per 67 µs. Next 8 ms are recorded at 1 sample in each channel per 200 µs, and last period is shown at rate of 1 sample/ms.

Equivalent Mass

Similar calculations suggest that the equivalent mass of the cantilever was ~1.7 mg at the shortest hinge length.

Cantilever Designs for Other Force Ranges

At the opposite end of the sensitivity spectrum, a cantilever was made for measuring isometric force in permeabilized muscle preparations that develop 10-30 mN force. Because ruggedness is more important than speed in this application, the cantilever was made of titanium, and its width was not tapered (Fig. 1B). To increase sensitivity, as well as to provide greater clearance above the solutions, the cantilever beam length was increased to 10 mm. With a hinge length of ~0.1 mm, the sensitivity was 3.1 V/g and the resonant frequency was 7 kHz, several thousand times greater than needed for this application.

Use of the Transducer in Muscle Experiments

The records in Fig. 6C illustrate a transient tension response to a rapid stretch complete within 200 µs. For this record, the servomotor was controlled from the central segment length sensor, and a stretch complete within 200 µs was applied to the otherwise isometric segment. Force rose in association with the stretch and then relaxed toward its isometric level with a complex time course. Again, even with this rapid step there is no suggestion of the inherent resonant frequency of the transducer.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

A major advantage of the present device is that cantilever beams with very different characteristics can be substituted in the same apparatus. Such substitutions allow the other components of the device to be used for a wide range of experiments. Although the apparatus described had characteristics for use in single, skinned skeletal muscle fibers, which produce up to 1 mN of force, the device can be used with preparations that develop greater and lesser forces. When similar dimensions of cantilever are used, the resonant frequency of the device will vary in inverse proportion to the square root of sensitivity.

Another advantage is that the elements are very rugged. A general rule of force transducers is that they should operate near their breaking point (7) to make them as light and therefore as fast as possible. This proved to be difficult in the present structures because such delicate elements frequently broke during the grinding and filing required for fabrication. The elements developed here were found to be much stronger than the minimum required by the muscle preparations. Although it seems likely that we could have developed the skill to make narrower and therefore lighter cantilevers, the observation that a substantial fraction of the equivalent mass of the element was in essential proximal structures (braces, mirror, etc.) suggested that the gains were likely to be small.

Comparison with Other Transducers

Recently, De Winkel et al. (4) made a transducer with a 2-mm extension arm with a 70-kHz resonant frequency. Although this arm would be impracticably short for the types of experiments anticipated here, this device illustrates the increases in speed that can be achieved when the length of the projection arm is reduced toward zero.

The method of sensing movement of the compliant element in the transducer can be divided broadly into those in which the sensor is incorporated into the moving element (strain gauges, piezoelectric elements, capacitance devices) and those in which the moving element is independent of the sensor (usually photoelectric devices). The main advantage of the latter category is that the characteristics and design of the moving element are independent of the material in the sensing element. A further advantage of the photoelectric devices is that they can be used in a damp environment where variations in moisture might alter the sensitivity of other sensing devices (e.g., capacitance transducers) or short circuit the output entirely [e.g., piezoelectric devices such as that used by Chiu et al. (3)].

Another advantage of the present device is ease of fabrication. The three main sections of the device, independent of the cantilever beams, can be made to relatively low tolerances in a few hours of shop time. With practice, the cantilever beams can be made within an hour. We have previously described a photoelectric force transducer that had a resonant frequency of 6 kHz and in which a lever was glued to a twisting strip hinge (2). An aluminum foil vane attached to the moving element interrupted a light beam (from an incandescent lamp) falling on a pair of phototransistors, the output of which was used to signal force. By substituting quieter photodiodes of the type used here, we have been able to increase the resonant frequency of that device to ~15 kHz (C. Y. Seow and L. E. Ford, unpublished observations). With a brighter light source or quieter photodiodes, some of the resulting higher sensitivity might be traded for higher resonant frequency by using a stiffer twisting hinge. The reason for developing the present instrument instead was the greater ease in making and aligning the moving elements. The older moving elements were built of carbon fiber struts glued to a twisting phosphor-bronze strip hinge. Making an element took several hours, and frequently less-than-satisfactory elements resulted when glue joints were found to have unacceptable levels of hysteresis. The absence of glue joints in the present device (except for those holding the hook and mirror) ensures an absence of hysteresis and greatly simplifies fabrication.

A final advantage of the present device over earlier photoelectric devices is that the position of the photodiode pair can be adjusted easily to bring the laser beam to the center of the two diodes. Because of the large ratio of light beam movement to cantilever tip movement, the device is relatively insensitive to small movements of the photodiodes on their adjustable holder that might result from thermal changes.

In conclusion, the transducer described has at least four significant advantages. First, the same instrument can be adapted for the study of very different sizes of muscle through substitution of different moving elements. Second, alignment of the focused laser beam on the diode pair is greatly facilitated by the division of the main transducer body into three sections that can be moved independently. Third, the electronic components are located at a safe distance from the damp environment of the muscle bath. Finally, with a resonant frequency of 22 kHz, it is the fastest transducer known to us, with a signal-to-noise ratio adequate for single-fiber study and with an arm that can be projected several millimeters into a muscle bath.