Mass Volume Relationship Definition Essay


One of the most fundamental and important descriptions of an object is its weight. Unlike other measurements of an object’s size, weight is not influenced by the shape of the object. Consequently, for at least 4000 years weight has been the basis for trade of goods and materials. And for living organisms, weight measurements provide valuable insights into an organism’s biological state and health. We measure the weight of our children from birth and plot these measurements over time on growth charts; any deviation from expected growth can indicate a health problem. People, birds [1], fish [2], and plants [3] are just a few of the organisms that are weighed to study their health and growth. But while children, animals, and plants can be weighed on a scale, some biological samples are more challenging to weigh. Samples like single cells, microorganisms, and embryos are too small to weigh on conventional laboratory balances, and these samples often live in a liquid environment that is incompatible with conventional scales.

Weighing the smallest living samples in their native liquid environments became possible with the advent of the suspended microchannel resonator or SMR [4]. The SMR consists of a microfluidic channel embedded inside a silicon cantilever. The cantilever is vibrated at its resonance frequency, which is a function of the overall mass of the cantilever. When a cell flows through the embedded microfluidic channel, the cell’s mass momentarily increases the cantilever’s mass. This momentarily decreases the resonance frequency of the cantilever by an amount proportional to the weight of the cell. By weighing cells in this manner, the SMR can be used to distinguish healthy and diseased cells, monitor cell growth, and observe the effects of drugs on cells [5–7].

However, there are many important samples that are too large to be weighed in the microfluidic SMR and too small to be weighed on a conventional balance. For example, fish embryos are used in a variety of developmental biology and toxicology applications [8]. If the weight of an embryo could be monitored as it grows or reacts to stimuli, these measurements could offer new insights into developmental biology and toxicology. However, millimeter-sized zebrafish embryos are far too large for the micron-scale channels in the SMR, and the aquatic environment of the embryos complicates weighing them using a traditional balance. In another example, materials with controlled degradation rates are finding important applications in medical implants, where they remain in the body for a predetermined period of time and then effectively disappear [9]. But accurately predicting the degradation rates of these materials in the fluidic environment of the body is challenging. If the weight of a biomaterial sample could be monitored as it degrades in a realistic artificial bodily fluid, these measurements would provide an unparalleled insight into the degradation of the material within the body. However, with their liquid environments and small size, these biomaterial samples are difficult to weigh using existing techniques, and manually monitoring the weight of these samples over weeks or months of degradation is a laborious and time-consuming process.

In this work, we demonstrate a simple and inexpensive sensor capable of weighing microgram-sized objects in fluid. Like the SMR, this sensor uses a change in resonance frequency to weigh an object in fluid with high precision. But unlike the SMR, this sensor can weigh samples with a large range of sizes and is extremely simple to fabricate. Our sensor consists of a short length of glass tubing bent into a “U” shape and attached to an inexpensive speaker that vibrates the glass tubing at its resonance frequency. The resulting sensor shown in Fig 1 costs about US $12 in materials and can be made in under 10 minutes. Additionally, by weighing samples in fluids of different densities, we can also use our sensor to measure the volume and density of samples in fluid. In this proof-of-concept demonstration, we used vibrating glass tubes to monitor fish embryos reacting to toxicant exposure, plant seeds undergoing rehydration and sprouting, and biomaterials undergoing controlled degradation.

Fig 1. Using a vibrating glass tube to weigh objects in fluid.

(A) The tube is bent into a “U” shape and mounted on an inexpensive piezoelectric speaker. The resonance frequency of the tube is detected by a photointerrupter, amplified, and fed into the speaker; the resulting feedback circuit keeps the glass tube vibrating at its resonance frequency. A peristaltic pump and computer (not shown) are used to flow samples through the tube and record the resonance frequency of the tube. (B) Detail of the glass tube shows the path followed by a sample inside the tube (red line) as it flows into the sensor (“a”), passes the tip of the sensor (“b”), and exits the sensor (“c”). (C) Resonance frequency of the vibrating tube vs. time as a 770 μm diameter glass bead is passed back and forth through the tube eighteen times. Each passage of the bead through the tube results in a momentary decrease in the tube’s resonance frequency; this is recorded as a downward peak in the resonance frequency (D). Each point on the peak (baseline “a,” tip “b,” and baseline “c”) corresponds with the bead’s location in (B) above. The height of this peak (72 millihertz) is used to determine the buoyant mass of the bead (344 micrograms). (E) Histograms showing the buoyant mass of another bead weighed thousands of times in two different fluid densities. In deionized water (density 1.000 g/mL) the bead has an average buoyant mass of 3.69 ±0.12 μg, and in a sodium chloride solution (density 1.046 g/mL) the bead has an average buoyant mass of −4.42 ±0.16 μg. The widths of these distributions—120 and 160 nanograms—provide an estimate of the resolution of our mass measurements. (F) By plotting the average buoyant mass of the bead vs. the density of the fluid in which the bead was measured from (E) and drawing a line between the two points, we can determine the absolute mass (180 ±4.4 μg), volume (176.3 ±4.3 nL), and density (1.0210 ±0.0005 g/mL) of the bead from the y-intercept, slope, and x-intercept, respectively. The measured bead density is in good agreement with the manufacturer-provided value of 1.023 g/mL, and the bead diameter calculated from the measured volume of the bead (700 μm) is very close to the measured diameter of the bead obtained using a digital caliper (710 μm).


In vibrating mass sensors, the resonance frequency of the vibrating sensor is inversely proportional to the effective mass of the sensor. For a vibrating sensor with a cantilever or “diving board” shape, the resonance frequency f of the sensor is (1) where ms is the effective mass of the sensor and k is the spring constant of the sensor [4]. If an additional mass is added to the sensor, the increase in ms in Eq (1) causes a measurable decrease in the cantilever’s resonance frequency f. For vibrating mass sensors containing a fluid-filled channel (like the vibrating tubes presented here), the contents of the channel contribute to the sensor’s mass and therefore affect its resonance frequency. The resonance frequency of the sensor is inversely proportional to the density of the fluid filling the channel; this is the basis for the long-established technique of using vibrating tubes to measure fluid density [10], and when two immiscible fluids are used, vibrating capillaries can be used to measure the density and radius of droplets of fluid in the tube [11].

In this work, we show that any object (not just fluids) flowing through a vibrating tube can affect the resonance frequency of the tube. Since the object is flowing in fluid, the mass it contributes to the vibrating tube is actually the object’s buoyant mass, mbo, defined as (2) where mo is the absolute (in vacuo) mass of the object, ρo is the density of the object, and ρf is the density of the fluid filling the channel. Stated in words, an object’s buoyant mass is equal to its real mass minus the mass of an equivalent-volume amount of fluid. If the object’s density is greater than the fluid’s density, then the object has a positive buoyant mass and its passage through the vibrating tube will be recorded as a momentary decrease in the tube’s resonance frequency (the downward peaks shown in Fig 1C and 1D). If the object’s density is less than the fluid’s density, then the object has a negative buoyant mass and its passage through the tube will result in a momentary increase in the tube’s resonance frequency (upward peaks). Finally, if the object’s density equals the fluid’s density, then the object will have zero buoyant mass and its passage through the tube will have no effect on the resonance frequency of the tube (although this situation can be easily avoided by changing the fluid density). Note that while the vibrating tube sensor is sensitive to an object’s buoyant mass, it is not affected by buoyant forces (that cause an object to sink or float relative to gravity) because the object being measured is confined to the tube and cannot sink or float vertically. Thus, the orientation of the vibrating tube with respect to gravity has no effect on its measurements.

In addition to buoyant mass, the vibrating tube sensor can measure other physical properties of an object. For example, by measuring the buoyant mass of an object in two fluids of different densities, we obtain two instances of Eq (2) that can be solved simultaneously to calculate the absolute (in vacuo) mass mo of the object, the density ρo of the object, and the volume Vo of the object (from the definition of density, ρo = mo/Vo) [7]. A graphical version of this calculation is shown in Fig 1E and 1F: by plotting two measurements of an object’s buoyant mass vs. the density of the fluid in which it was weighed and connecting the points with a line, the mass of the object is equal to the y-intercept of the line, the density of the object is equal to the x-intercept of the line, and the volume of the object is equal to the slope of the line.

Materials and methods

Sensor fabrication and oscillator circuit design

To fabricate vibrating glass tube sensors, glass tubing (1.50 mm inner diameter, 1.80 mm outer diameter; VitroCom, Mountain Lakes, NJ) was cut to length and bent into a “U” shape using a butane torch. The resulting sensors have a mass of around 1.4 g empty and 1.7 g when filled with water. Our technique is sensitive to the amount of change in the sensor’s mass when an object flows through the sensor, so it is generally advantageous to maximize this change by keeping the mass of the tube as small as possible.

The top of the “U”-shaped glass tube was attached to an inexpensive piezoelectric speaker (Jameco Electronics, Belmont, CA) using epoxy. The bottom or tip of the “U” was suspended in the slot of a photointerrupter (an inexpensive electronic component that uses a light-emitting diode and a light-sensing diode to detect the position of an object inside its slot). As the tube vibrates at its resonance frequency, it blocks the photointerrupter’s light beam once per oscillation; the resulting photointerrupter output is an AC signal with the same frequency as the tube’s vibration. When the output of the photointerrupter is amplified and fed into the piezoelectric speaker, the circuit will spontaneously and continuously vibrate the glass tube at its resonance frequency. Alternatively, for some experiments, vibrating glass tubes from commercial fluid density meters were used. Fluid density meters (e.g., DMA 35, Anton Paar, Graz, Austria) were obtained second-hand and modified to isolate the vibrating glass tube and supporting electronics.

Q-factor measurement

The quality factor or Q-factor of an oscillating system like our vibrating tube sensor is an important value for quantifying the sensor’s ability to measure mass. An oscillator with a higher Q-factor has a purer “tone,” and changes in this tone (when objects pass through the sensor) can be measured more precisely. We measured the Q-factor of our sensors using the ring-down method. A vibrating tube sensor with a known resonance frequency f was manually “pinged” by flicking the tube using a finger. Video of the vibrating tube was recorded from the side of the tube (to measure the decrease in vibrational amplitude over time) using an iPhone camera acquiring video at 240 frames per second. This video was imported into the software ImageJ [12] as an image stack and resliced vertically at the tip of the vibrating tube to create a bitmap that represents the tube’s vibrational amplitude on the Y axis and the frame number on the X axis (S5 Fig). We used ImageJ to measure the vibrational amplitude A0 of the tube at the start of the experiment (frame 0) and then located the frame number n at which the vibrational amplitude had decreased to a value of A0/e or about 37% of its initial amplitude. Dividing this number of frames n by the framerate of the camera yields is the exponential decay time τ of the sensor. Finally, the quality factor Q of the sensor was calculated using the relationship Q = πfτ [13].

Data acquisition and analysis

To record the tube’s resonance frequency over time, we connected the output of the photointerruptor circuit to a counter input on a multifunction data acquisition device (National Instruments, Austin, TX) and a computer running a custom LabVIEW program, although we have also used simpler and cheaper hardware like the open-source Arduino microcontroller to measure the tube’s resonance frequency.

When a sample passes through the vibrating tube sensor, its buoyant mass is recorded as a brief peak in the plot of resonance frequency vs. time (e.g., Fig 1C and 1D). To detect peaks in the data corresponding to object measurements, the raw frequency measurements were first filtered using a digital filter (either low-pass or Savitzky-Golay) and then subjected to a moving window average that identifies peaks based on their deviation from the baseline. Once a peak is located, the height of the peak can be measured and converted to a corresponding buoyant mass value using the sensor’s point mass calibration described below. Alternatively, a custom Python program can be used to fit the raw frequency measurements to an analytical equation of expected peak shape derived from Dohn et al. [14] The resulting buoyant mass measurements were recorded and processed using a moving window average filter with a window size of five data points to slightly reduce noise in the plots of buoyant mass vs. time. Additional details about signal processing (including sample frequency data before and after filtering) are provided in S6 Fig.

Sensor calibration

Before use, each glass tube sensor was calibrated in two different ways. In the first calibration, the fluid density calibration, the tube was filled with different fluids of known density and the resonance frequency of the tube was recorded. Sodium chloride solutions with precisely known densities were prepared using our software tool [15]. By plotting resonance frequency vs. fluid density and fitting the plot to a line (S1 Fig), we obtain from the slope of that line a constant c1 that can be used to determine the density ρf of any fluid inside the tube as a function of the tube’s measured resonance frequency f: (3) where c1 is the tube’s fluid density calibration constant in g mL−1 Hz−1. In the second calibration, the point mass calibration, the tube is filled with a fluid of known density ρf. A microbead of known mass mo and known density ρo is passed through the tube multiple times. The buoyant mass mbo of the bead is calculated using Eq (2). As the microbead passes through the tube, the microbead’s mass momentarily changes the resonance frequency of the tube by an amount Δf (the heights of the peaks in Fig 1C and 1D). We then solve for a constant c2 that can be used to determine the buoyant mass mbo of any object inside the tube as a function of the measured change in resonance frequency Δf as the object passes through the tube: (4) where c2 is the tube’s point mass calibration constant in g Hz−1.

Because of variation in the sizes of the vibrating glass tubes we used, the baseline resonance frequency f of the sensors used in this study ranged from below 100 Hz to around 500 Hz depending on the sensor size (larger sensors have a larger mass and therefore have lower resonance frequencies). Thus, each sensor has unique constants c1 and c2 which much be determined before use via the process described above. However, we have observed that a sensor’s calibration constants do not change significantly over long experiments (10 days of continuous operation), and some of our glass tube sensors have been in regular use for several years without a change in performance.

Fluid control

To flow samples into and out of the vibrating glass tube sensors, the tubes were connected via flexible tubing to either a peristaltic pump or a servomotor-driven reservoir lifter. The reservoir lifter consists of two fluid reservoirs suspended from a rotary servomotor. By using the servo to raise one reservoir and lower the other, the resulting gravity-driven head pressure pumps fluid (and the sample) through the sensor. Reversing the positions of the reservoirs with the servo reverses the direction of fluid flow through the sensor. A custom LabVIEW program was used to continuously alternate the reservoir positions and measure the sample every few seconds.

Measuring the buoyant mass of zebrafish embryos during toxicant exposure

This study was carried out in strict accordance with the recommendations in the Guide for the Care and Use of Laboratory Animals of the National Institutes of Health. Zebrafish (D. rerio) used in these experiments were purchased from Oregon State University and maintained following a protocol approved by the University of California, Riverside’s Animal Care and Use Committee (IACUC approval number 20130005). Wild-type AB-strain zebrafish were approximately 16 months old at the time of spawning and were kept in aerated aged tap water (dechlorinated) at a temperature of 27°C with a light/dark cycle of 14:10 hours. Males and females were kept separately and fed twice a day on Artemia sp. until the night before spawning, when they were transferred to breeding aquaria. Eggs were collected the next morning, examined, and separated based on the stage of the development. Measurements of embryo buoyant mass began at approximately 2 hours per fertilization. Using a peristaltic pump connected via tubing to the vibrating tube sensor, each embryo was passed back and forth through the sensor every 10 seconds for the duration of the experiment. The fluid surrounding the embryos was either water (Fig 2A and S3 Fig), a solution of silver nanoparticles (nanoComposix, San Diego, CA; B and C in Fig 2 and S4C–S4H Fig), or a solution of copper sulfate (Sigma Aldrich, St. Louis, MO; Fig 2D and S4A and S4B Fig). Embryo experiments were completed before hatching (and thus before the NIH Office of Laboratory Animal Welfare classifies the organisms as vertebrate animals). After each experiment, embryos were euthanized using either sodium hypochlorite solution or rapid freezing.

Fig 2. Monitoring the buoyant mass of single zebrafish embryos with a vibrating glass tube sensor.

Most zebrafish embryos that were not exposed to toxicants have unchanging buoyant masses over the first 15 hours of their development (A). However, embryos exposed to known toxicants (silver nanoparticles and copper sulfate) displayed clear decreases in their buoyant masses over the same time period (B-D). Additional results from healthy and toxicant-exposed embryos are provided in Supporting Information.

Measuring the buoyant mass and density of seeds during imbibition and germination

Seeds of iceland poppy (Papaver nudicaule), oregano (Origanum vulgare), and foxglove (Digitalis purpurea) were obtained from Ferry-Morse (Norton, MA) and measured every 10 seconds in water until active germination (the emergence of the embryo) was observed using the servomotor reservoir lifter described above. The resulting buoyant mass measurements are shown in Fig 3D–3F. To estimate the density of these seeds during imbibition and germination, the average single-seed mass of each seed type was measured by weighing 30 seeds using a laboratory balance and dividing the total mass by 30. This estimate of single-seed mass was used along with the seed buoyant mass measurements to calculate the estimated density of each seed during imbibition and germination (Fig 3G–3I).

Fig 3. Using a vibrating tube mass sensor to measure the buoyant mass and density of three different types of seeds during imbibition and germination.

Each individual seed was measured between 1,500 and 2,600 times over 35 hours. The largest seed, the Iceland poppy seed (A; Papaver nudicaule, ∼155 μg mass), has the largest buoyant mass (9 μg; D) but the smallest density (1.06 g/mL; G). The smallest seed, the foxglove (B; Digitalis purpurea, ∼73 μg) had the smallest buoyant mass (7 μg; E) but one of the largest densities (1.10 g/mL; H). Finally, the mid-sized oregano seed (C; Origanum vulgare, ∼90 μg) has a buoyant mass that increases from 6 to 9 μg at a rate of 12 ng/min during the first four hours of immersion (F); the density of the seed increases from 1.07 to 1.11 g/mL during the same time period (I).

Measuring the degradation of a biomaterial

Magnesium ribbon with a thickness of 250 μm (98% pure; MiniScience Inc., Clifton, NJ) was used as a model biomaterial in our degradation rate measurement studies. Roughly 1 mm sized pieces of magnesium were cut from the ribbon. The samples were polished before measurement using 600, 800, and 1200 grit silicon carbide abrasive papers to remove the native oxide layer. Each magnesium sample was then immersed in simulated bodily fluid (phosphate-buffered saline [PBS]) with adjusted pH (either 7 or 8) and passed back and forth through the vibrating tube sensor every 30 seconds until the piece had fully degraded. Flow through the sensor was controlled using the servomotor as described previously. The resulting buoyant mass measurements are shown in Fig 4.

Fig 4. Using a resonating glass tube to measure the degradation rate of a sample biomaterial in different physiologically relevant fluids.

Two millimiter-sized pieces of magnesium were measured in two different phosphate-buffered saline (PBS) solutions, one with a neutral pH (7.0; green) and one with a higher pH (8.0; red). The more acidic conditions at pH 7.0 caused the magnesium sample to degrade at a rate that is about six times faster than the sample in pH 8.0 buffer.


Before using a vibrating tube to weigh a sample, the tube must first be calibrated. We calibrated the sensor shown in Fig 1A using a glass bead of known size and density in a fluid of known density (ethanol). Fig 1C shows eighteen downward peaks in the resonance frequency of the tube as the bead is pumped back and forth eighteen times through the tube. The height of each peak (72 millihertz; Fig 1D) is proportional to the buoyant mass of the bead (known to be 344 micrograms). From this relationship we obtain the calibration constant c2 for this tube sensor in micrograms per millihertz (4.8 μg/mHz) that can subsequently be used to determine the buoyant mass of any object flowing through this tube. Our homemade vibrating tube mass sensors typically have quality factors around 500, which is comparable to that of tuning forks and high enough for precise measurement of the tube’s resonance frequency (S5 Fig).

In Fig 1C a slight downward drift is visible in the baseline resonance frequency of the vibrating tube over time; the frequency is decreasing at about 2 millihertz per minute. Slow baseline frequency drift like this is common in vibrating mass sensors and is mostly due to small fluctuations in the temperature of the sensor. However, the magnitude of this drift is small compared to the duration of a peak, which is why the baseline of an individual peak in Fig 1D is relatively flat. Additionally, since we use the difference in frequency between the baseline frequency and the tip of the peak (the peak height Δf) to determine the buoyant mass of an object, this baseline drift does not affect the accuracy of our technique.

As an object passes through the vibrating tube sensor, the shape of each resulting peak in the tube’s resonance frequency (Fig 1D) is a function of the vibrational mode and amplitude of the tube. In this work, the tubes are vibrating at their primary vibrational mode, meaning that the amplitude of vibration is highest at the tip (the bottom of the glass “U”) and lowest at the base (the top of the “U”). When an object enters the tube (point “a” in Fig 1B), it is in a region with relatively low vibrational amplitude, so the object’s buoyant mass has a relatively small effect on the tube’s resonance frequency (point “a” in Fig 1D). However, as the object rounds the tip of the tube (point “b” in Fig 1B), the vibrational amplitude at the tip is the highest, so the object has a maximum effect on the tube’s resonance frequency here (point “b” in Fig 1D). As the particle leaves the tube (point “c” in Fig 1B) the tube’s vibrational amplitude in this region decreases again, so the resonance frequency of the tube returns to baseline (point “c” in Fig 1D). General mathematical expressions for predicting this peak shape for any vibrational mode were derived by Dohn et al. [14]

To determine the resolution of our technique, we used another glass tube to make 4,483 measurements of the buoyant mass of another bead of known size and density in water. The width of the distribution of these measurements (120 nanograms in the right-hand distribution in Fig 1E) represents the smallest difference or change in mass that can be detected by this glass tube.

In addition to measuring the buoyant mass of samples, vibrating glass tubes can also be used to measure the absolute mass, volume, and density of a sample. By measuring the buoyant mass of the object in two fluids of different known densities, the mass, volume, and density of the object may be calculated [7]. Fig 1E shows the distribution of buoyant mass measurements for the bead in salt water (left-hand distribution) and fresh water (right-hand distribution). The bead’s average buoyant mass is 3.69 ± 0.12 μg in fresh water and −4.42 ± 0.16 μg in salt water. Fig 1F shows a plot of these two average buoyant mass measurements vs. the density of the fluid used in each measurement (1.00 g/mL for fresh water and 1.046 g/mL for salt water). The y-intercept of a line drawn through these two points is the absolute (in vacuo) mass of the bead, 180 ± 4.4 micrograms. The slope of the line is the volume of the bead, 176.3 ± 4.3 nanoliters. Finally, the x-intercept of the line is the density of the bead, 1.0210 ± 0.0005 g/mL. The measured density of the bead agrees well with the manufacturer-provided density of the bead (1.023 g/mL). Additionally, by using the measured volume of the spherical bead to calculate the diameter of the bead, we obtain a diameter of 700 μm which is in good agreement with a caliper-based measurement of the diameter of the bead (710 μm).

To demonstrate that vibrating tubes can be used to measure real biological samples, we used our sensors to measure the buoyant mass of single zebrafish (D. rerio) embryos. These millimeter-sized embryos reside in water and are too small and fragile to weigh on a conventional balance. However, using a vibrating glass tube, we weighed 474 different zebrafish embryos and obtained an average value for the buoyant mass of a zebrafish embryo, 7.59 μg (S2 Fig). The width of this distribution (standard deviation = 0.56 μg) provides an insight into the intrinsic variation in zebrafish embryo size and should be of value to biologists studying the mechanisms of size regulation in these organisms.

We then hypothesized that mass measurements from vibrating tubes can give insights into the health of organisms. Previous work with the microfluidic SMR mass sensor showed that single cells undergo predictable changes in mass during normal development and reproduction [5, 7, 16]. Thus, any deviation from the normal growth trajectory can be used as a signal for abnormal development or the emergence of an illness in a cell. On the other end of the size spectrum from cells, measurements of human body weight provide some of the most fundamental insights in the health and growth of a person. We hypothesize that the same relationship between mass change and organism health applies for organisms throughout the size spectrum from cells to humans.

To test this hypothesis, we used vibrating glass tubes to continuously monitor the buoyant mass of zebrafish embryos during exposure to known toxicants. Zebrafish embryos are popular vertebrate model organisms for high throughput drug discovery and screening [17, 18] and human disease modeling [19, 20]. Additionally, as marine animals, they are used extensively in assessing the toxicity of substances in aquatic environments [21]. Current methods for assessing the health of zebrafish embryo are laborious, time-consuming, require a high degree of expertise, and can be subject to human error. Even the more automated techniques, such as newly developed optical instruments with automated tracking software [22], are expensive and might only detect changes in certain developmental stages such as hatchlings that are already exhibiting physical malformations and behavioral abnormalities. Vibrating tube mass sensors could offer an economical and high-throughput alternative to these existing techniques for assessing the health of an embryo. They can also provide information on organism mass, a primary metric in toxicology that is used as a normalizing factor for dosing of toxicants [23].

Using a vibrating glass tube, we monitored the buoyant mass of single zebrafish embryos during the first 15 hours of their development starting 2 hours post-fertilization (Fig 2). A zebrafish embryo in normal water conditions (Fig 2A) shows no significant change in its mass during the monitoring period. We have observed this flat line in many other embryos in normal water conditions (S3 Fig) and therefore associate this flat line with healthy embryos. However, zebrafish embryos exposed to known toxicants demonstrate significant changes in their mass. When exposed to solutions of silver nanoparticles (Fig 2B and 2C) and copper sulfate (Fig 2D), the embryos exhibited sometimes-abrupt decreases in their buoyant mass. Additional observations of this mass decrease accompanying toxicant exposure are provided in S4 Fig. By using our sensors to look for this mass decrease in embryos exposed to suspect toxicants, this technique could be the basis for a simple, inexpensive, and high-throughput toxicity screening tool.

To validate our technique with a diverse range of biological samples, we used vibrating tube sensors to monitor the mass of individual plant seeds during imbibition and germination. Seed imbibition, or water uptake, is used in agriculture as a metric of seed health and quality [24]. Seed germination is a change in seed metabolism when imbibition starts; germination culminates with the elongation of the embryonic axis that penetrates the seed coating. The imbibition of seeds is accompanied by a rapid leakage of cellular materials (such as sugars, amino acids, and inorganic ions) and the rate of this leakage is decreased as the tissues become hydrated [25]. If water uptake by the seed is too rapid, the seed tissue might experience injury, and if the seed enters an anaerobic state, the seed might experience accumulation of toxic chemicals such as ethanol. Both situations can encourage undesirable seed dormancy and delay germination [26]. In summary, seed imbibition and germination are important phenomena in plant research, and quantitative measurements of these phenomena would be valuable in a wide range of botanical and agricultural fields.

Fig 3A–3C show seeds from three different plants: iceland poppy (Papaver nudicaule), oregano (Origanum vulgare), and foxglove (Digitalis purpurea). The different sizes, shapes, and surface characteristics of these seeds would complicate optical measurements of seed size. However, by continuously weighing these seeds during imbibition and sprouting, we can obtain precise measurements of seed biomass regardless of the morphology of the seed. Using a resonating glass tube mass sensor, we found that the buoyant mass of a single Iceland poppy seed remains mostly unchanged at 9 μg during the first 35 hours of exposure to water and sprouting (Fig 3D). The smaller foxglove seed remains mostly unchanged at 7 μg during the same time period (Fig 3E). However, the buoyant mass of the oregano seed undergoes a clear increase in buoyant mass from 6 to 9 μg during the first four hours of exposure to water and germination, after which it remains constant (Fig 3F). By fitting the first four hours of the oregano seed buoyant mass to the equation of a line, we determined that the oregano seed’s buoyant mass is increasing during this period at a rate of 12 ± 0.6 nanograms per minute. Since the density of the water surrounding the oregano seed is unchanging, Eq (1) indicates that this increase in buoyant mass can be attributed to an increase in the absolute mass of the seed (due to e.g. biomass synthesis as the seed sprouts) or a decrease in the volume of the seed (due to e.g. water displacing air inside the seed as it rehydrates).

If water displacing air was responsible for the increasing buoyant mass of the oregano seed, then one would expect the density of the seed to change during imbibition. By measuring the average mass of each seed type using a population of seeds and a laboratory balance, we can use our buoyant mass measurements to estimate the density of each seed during imbibition and sprouting. This estimate assumes that the absolute mass of the seed remains unchanged during this process. Interestingly, the Iceland poppy seed (the largest of the seeds, with an average mass of 155 μg) has the lowest density of the seeds: 1.06 g/mL in Fig 3G). In contrast, the foxglove seed (the smallest of the seeds, with an average mass of only 73 μg) has a relatively-high density of 1.10 g/mL. Finally, the oregano seed (a medium-sized seed with an average mass of 90 μg) has a density that starts low at 1.07 g/mL but increases to 1.11 g/mL during the first four hours of exposure to water, after which it remains unchanged. Measurements like these can offer fundamental insights into both the composition of these seeds and the changes occurring within the seeds as they begin to grow into a plant.

Finally, to demonstrate our technique using a biologically-relevant sample other than an organism, we used vibrating glass tubes to precisely measure the degradation rates of biodegradable materials. For many applications in medical implants, it is desirable to have materials with known degradation rates. For example, a screw for repairing a broken bone might remain intact until the bone heals and then dissolve away. However, measuring the degradation rates of slow-degrading materials is a time-consuming and labor-intensive process. Centimeter-sized samples are usually immersed in physiologically relevant fluid for weeks or months, during which time the sample is periodically removed, weighed using a conventional balance, and returned to the fluid. This process slows the development of new biomaterials and introduces the potential for human error.

By using our resonating glass tube to automatically monitor the mass of a degrading material in fluid, we can use a much smaller sample of material than would normally be required. These small (millimeter-sized) samples have a much larger surface-area-to-volume ratio than the centimeter-sized samples required by current methods. This increases the relative degradation rate of the smaller samples in fluid, making our technique capable of measuring the degradation rate of a material in hours instead of weeks or months. We used our sensors to measure the degradation rate of a sample of magnesium, a biodegradable material that has been extensively studied for potential use in medical implants [27, 28].

Fig 4 shows the buoyant masses of two millimeter-sized samples of magnesium metal as they degrade in different fluids. In neutral-pH physiologically relevant fluid (pH = 7.0; green in Fig 4) the sample of magnesium degrades relatively quickly and is effectively gone after 20 hours of continuous measurement in our sensor. The degradation products (Mg2+ ions) enter the fluid, where they have a negligible effect on the fluid density (as confirmed by monitoring the tube’s baseline resonance frequency over time) and therefore no effect on our measurements. We assume that the mass of magnesium m remaining in the sample at any given time t can be modeled as an exponential decay, m = miert (where mi is the starting mass of the magnesium sample at t = 0 and r is the degradation rate). By using a least-squares regression to fit the data in Fig 4, we obtained a degradation rate of 0.14 h−1 for the magnesium sample in pH 7.0 buffer. However, when this experiment was repeated for a similarly-sized magnesium sample in a more-alkaline buffer (pH = 8.0; red in Fig 4), the sample took nearly 120 hours to disappear completely—about six times longer than the sample in pH 7.0 buffer. Consequently, the measured degradation rate for the sample in pH 8.0 buffer, 0.021 h−1, is about six times lower than the rate measured in pH 7.0 buffer. This trend makes sense because magnesium reacts with acids to form magnesium ions and hydrogen gas; this reaction would be expected to accelerate the degradation of a magnesium sample at lower pH. Our measurements quantify the extent to which this pH change accelerates magnesium degradation. These results also highlight the fact that biomaterials can behave differently in different environments. By providing a fully automated and low-cost tool for measuring nanogram-scale degradation rates of biomaterials in virtually any fluid, vibrating tube sensors can accelerate the development and testing of new biomaterials for important biomedical applications.


The four proof-of-concept samples studied here—microbeads, embryos, seeds, and biomaterials—are representative of a wide range of samples that may be analyzed in fluid using vibrating glass tube sensors. Our technique is very versatile because all objects have fundamental physical properties like mass. Consequently, our mass sensor can be applied to problems as diverse as screening toxic substances, understanding the growth of plants, measuring the degradation of biomaterials, and many others. And unlike imaging-based measurements of size, our mass sensor is insensitive to the shape of the object. Finally, the automation, portability, and low cost of this technique make vibrating glass tubes particularly well suited for applications in the field or in resource-limited settings.

Supporting information

S1 Fig. Fluid density calibration for two different vibrating glass tube mass sensors, one obtained from a commercial fluid density meter (A; DMA 35, Anton Paar, Graz, Austria) and one homemade from glass tubing (B).

While the different sizes of the tubes lead to different resonance frequencies (∼400 Hz vs. 60 Hz) for the tubes, the resonance frequency of each tube is a linear function of the density of the fluid filling the tube. The slope of each line is the fluid density calibration constant c1 for the tube (described in Eq (3)).


S2 Fig. Distribution of buoyant mass measurements for 474 different zebrafish embryos at 2 hours post-fertilization, obtained using our vibrating glass tube sensor.

The average zebrafish embryo buoyant mass is 7.59 μg. The width of this distribution (standard deviation = 0.56 μg) provides an insight into the intrinsic variation in zebrafish embryo size.


S3 Fig. Additional measurements of the buoyant mass of single zebrafish embryos in water, obtained using our vibrating glass tube sensor.

For most of the zebrafish embryos in water, the buoyant masses of the embryos remained unchanged during measurement (typical results shown in A–E). A small fraction of these embryos exhibited changes in their buoyant masses even though they were not intentionally exposed to toxicants (typical results shown in F–H). We attribute these results to naturally nonviable embryos that are always present in these zebrafish populations.


S4 Fig. Additional measurements of the buoyant mass of single zebrafish embryos exposed to various known toxicants, obtained using our vibrating glass tube sensor.

These toxicant-exposed embryos exhibited clear decreases in their buoyant mass during exposure (typical results shown in A–H).


S5 Fig. Measuring the quality factor of a vibrating tube mass sensor.

The quality factor Q of one of our homemade vibrating tube mass sensors was measured using the ringdown method. The glass tube (with a resonance frequency f = 65.2 Hz) was manually “pinged” by flicking it with a finger, and an iPhone was used to record video of the vibrating glass tube from the side (to visualize the decreasing amplitude of the vibration over time). The video was imported into the software ImageJ [12] and resliced at the tip of the tube to create this image representing the vibrational amplitude of the tube on the Y axis and video frame number on the X axis. By counting the number of frames (639) between the start of the experiment (when the vibrational amplitude A0 = 101 pixels) and the point where the amplitude has dropped to A0/e = 37 pixels, then dividing this frame count by the frame rate of the camera (240 frames per second), we obtain an exponential decay time τ = 2.66 s for this sensor. A common expression for the quality factor Q of a cantilever-style mass sensor with resonance frequency f is Q = πfτ [13]; this results in a quality factor of 545 for this vibrating tube mass sensor.


S6 Fig. Processing typical data from our vibrating tube mass sensor.

(A) Raw resonance frequency data from repeated measurements of a single polyethylene microbead in water (from the right-hand distribution of measurements in Fig 1E). (B) The raw data is filtered using a digital lowpass filter with a cutoff frequency (1.5 Hz in this case) chosen that reduces the noise in the frequency data without decreasing (“rounding off”) the measured heights of peaks in the data. After zooming in to the filtered data (C), peaks corresponding to individual measurements of the microbead are visible. Zooming in further on one pair of peaks (D) shows the ∼2 millihertz height of these peaks (corresponding to a buoyant mass of ∼3.7 μg for this microbead). The peaks come in pairs because the particular vibrating tube sensor used for this measurement had a tuning-fork shape with two vibrating “U”-shaped lobes (and therefore two peaks measured per passage of the microbead through the tube).



This work was supported by the National Science Foundation’s Instrument Development for Biological Research program under award DBI-1353974.

Author Contributions

  1. Conceptualization: SMO HCB GD HL DS WHG.
  2. Formal analysis: SMO HCB GD HL DS WHG.
  3. Funding acquisition: WHG.
  4. Investigation: SMO HCB GD HL DS WHG.
  5. Methodology: SMO HCB GD HL DS WHG.
  6. Project administration: WHG.
  7. Resources: HL DS WHG.
  8. Software: SMO HCB WHG.
  9. Supervision: WHG.
  10. Validation: SMO HCB GD HL DS WHG.
  11. Visualization: SMO HCB GD HL DS WHG.
  12. Writing – original draft: SMO HCB GD HL DS WHG.
  13. Writing – review & editing: SMO HCB GD HL DS WHG.


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Accurate measurements of cell size are fundamental to understanding the cell cycle and to identifying cell type and cell state. During exponential growth, cells require coordination between growth and division to maintain the population’s size distribution, but it remains unclear how cells monitor and regulate cell cycle entry in response to cell size. The key to cell cycle control is the concentration of critical regulatory proteins, which is defined not only by expression levels, but the volume of the cell. Furthermore, cell volume is coupled to mass and energy requirements that control cell division and survival. Changes to rates of mass and volume accumulation may be correlated with cell cycle position and can be measured as variations in cell density. Yet even for model systems, such as the budding yeast Saccharomyces cerevisiae, these cellular-level parameters remain poorly investigated mainly because of a lack of tools for directly and accurately measuring cell mass or density.

Current methods for measuring cell density are limited to indirect measurements and density gradients, which can yield conflicting results. The density of fission yeast has been reported as cell cycle-dependent via interferometry (1) and cell cycle-independent via density gradient centrifugation (2). It is reported in ref. 2 that excessive sample manipulation may have influenced the interferometric results. Even so, measurements in density gradients can be complicated by possible interactions between the cells and the gradient medium (3, 4). Density measured by gradient centrifugation has been reported as cell cycle-independent during the CHO (5) and murine cell cycles (6), either independent or dependent on the Escherichia coli cell cycle (4, 7, 8), and cell cycle-dependent for budding yeast (9, 10). Studies using density gradient centrifugation have suggested that the changing density of budding yeast may correlate with bud formation (2, 11) or cyclic changes in water content and vacuole dynamics (10). Although more than one factor likely contributes to cell cycle-dependent density variation (or lack thereof) across cell types, mechanisms that give rise to density variation in budding yeast may be conserved in other eukaryotes.

The ideal density measurement would directly monitor the mass and volume of each cell within a population with minimal sample perturbation and allow cells to be collected for subsequent measurements. Toward this aim, we utilized the suspended microchannel resonator (SMR) to measure single-cell buoyant mass with high precision. The SMR is a silicon cantilever with an embedded microfluidic channel that resonates at a frequency proportional to its total mass, which changes as individual cells flow through the channel (Fig. 1) (12). We measured density in two ways with the SMR. The first provides an alternative to density gradient centrifugation and population-based approaches described previously (13) by measuring buoyant mass with the SMR and, on the same sample, volume with a commercial Coulter counter. Unlike density gradient centrifugation, this technique provides buoyant mass and volume information, measures growth-arrested cells in almost any medium, and does not require density gradient chemicals. With this method we found that cell density increases prior to bud formation at the G1/S transition. To investigate the origin of this density increase, we used the SMR in a second method where relative density changes of growing yeast cells are measured as the cells are sampled by the microchannel. We found that the density increase requires energy, passage through START, function of the protein synthesis regulator target of rapamycin (TOR), and an intact actin cytoskeleton. In conjunction with these density measurements, FACS analysis and bud emergence data suggest that this density change is independent of DNA replication and may result from several START-dependent events.

Results and Discussion

Density, Volume, and Buoyant Mass of Growth-Arrested Cells.

Cell buoyant mass, volume, and density were measured by combining the SMR’s buoyant mass readout with a commercial Coulter counter’s volume measurements. Buoyant mass is the change in cantilever mass as a cell transits the channel or the difference between the mass of the cell and the mass of the displaced fluid. The relationship between a cell’s buoyant mass and the fluid density is linear, and the slope of this line is related to the cell volume (Fig. 2). Cell density can be extrapolated from this line as the point at which buoyant mass is zero or at which fluid density exactly matches cell density. For populations, the flow-through configuration of the SMR allows cells to be collected during the buoyant mass measurement and used in downstream measurements for paired data analysis, such as volume, but this yields relatively few cells (n < 3000). Volume measurements (n > 20,000) recorded before the sample was loaded in the SMR were used in the density calculation. In this way buoyant mass and volume measurements are made with the same sample, the data are pooled into buoyant mass and volume histograms, and each histogram is fit to a log-normal function. To calculate the density of a cell population, the means of the fitted functions are substituted into ρ = ρf + mB/V, where ρ is the cell density, ρf is the fluid density, mB is the buoyant mass, and V is the cell volume. For comparisons between samples, the cell buoyant mass was adjusted for the fluid density of water at 20 °C (0.9982 g/mL) by mB = mB,measured + (ρf - ρwater,20 °CV.

Fig. 2.

Cell density, mass, and volume measurements. Cell density is calculated by the linear relationship between buoyant mass and fluid density (red line, equation in text). The slope of this line is determined by the cell volume (Inset A) and is measured by a commercial Coulter counter on the same sample loaded in the SMR. The buoyant mass (Inset B) is determined by a distribution of > 1500 SMR measurements. An example of four of these measurements made in phosphate buffer is in Fig. 1B. The means of the fitted log-normal functions (X on Insets) are used to calculate average cell density.

This technique was verified with 3.67 ± 0.1 μm diameter National Institute of Standards and Technology particle size standards. The beads have a mean 1.38 ± 0.010 pg buoyant mass in water (T ∼ 20 °C, fluid density = 0.9980 g/mL) and a 27.0 ± 0.10 μm3 mean volume. We calculated the density of the particles to be 1.0486 ± 0.0010 g/mL, which is in accordance with the accepted polystyrene density of 1.05 g/mL. Errors in both the volume and mass parameters contribute to this density calculation. The Coulter counter’s error results from inherent instrument noise. The SMR’s main source of error is variation in particle position at the cantilever tip. Position variation contributes ∼6% to the measurement’s uncertainty for a 3.67-μm diameter particle, slightly less than the inherent size deviation of the population. This error decreases with increased particle diameter and is generally not significant for populations with larger size variation, such as cells. The SMR also has a relatively low throughput compared to the Coulter counter. For National Institute of Standards and Technology particle standards we recorded > 300 SMR measurements in < 30 min, and for cell samples we recorded > 1,500 SMR measurements in < 1 hr. For both particle standards and cells, > 20,000 Coulter counter measurements were recorded in < 1 min.

Yeast Density and Growth Rates are Coordinated with the Cell Cycle.

The yeast cell cycle is a highly regulated series of events controlled by the activation of cyclin-dependent kinases (CDKs) by cyclins. CDK activation drives cell cycle progression, and it is suggested that the expression level of a certain cyclin, Cln3, serves to coordinate size, mass, and volume, with entry into the cell cycle (14). The point at which cells commit to cell cycle entry is called the START or restriction point (15). To determine how cell density correlates with the cell cycle, we measured the distributions of buoyant mass and volume in budding yeast populations. The density of asynchronous budding yeast in our strain background, W303, was 1.1029 ± 0.0026 g/mL (Fig. 3A), slightly lower than the density of 1.1126 g/mL reported by density gradient centrifugation in the Y55 strain (10). Because cells are not uniformly distributed through the cell cycle in an asynchronous population (∼80% of W303 cells are in the S or M phase, estimated by budded cell counts), this density is expected to be weighted by the time cells spend in each phase of the cell cycle. We next measured the buoyant mass, density, and volume of cells arrested in various stages of the cell cycle. The density of cells arrested in G1 by treatment with the pheromone alpha factor decreased 1.7% to 1.0846 ± 0.0043 g/mL. The decrease in G1 density was also observed in cells arrested by using the analog-sensitive CDK mutant cdc28-as1. This allele responds to the ATP analog 1-NM-PP1, which produces a G1 arrest by selectively occupying the modified ATP binding pocket of cdc28-as1 and inhibiting CDK activity (16). Cells arrested in G1 via cdc28 inactivation (pp1) had a density of 1.0812 ± 0.0100 g/mL, a 2% decrease from the 1.1034 ± 0.0022 g/mL density of asynchronous cdc28-as1 populations. Despite the longer arrest (3 hr) and larger cell volume (Fig. 3B), arrested cdc28-as1 cells had a density similar to that of a 2-hr alpha-factor arrest, which may indicate well-matched mass and volume growth rates between these G1 arrests. The density of cells arrested in S phase by the replication inhibitor hydroxyurea was 1.1049 ± 0.0024 g/mL, approximately the same as the asynchronous population. S-phase arrested cells could be expected to have a higher density on the basis of previous findings (9, 10), and our result may be an effect of the drug or indicate differences between S-phase arrested cells and those in an asynchronous population. The density of cells arrested in metaphase by the microtubule inhibitor nocodazole (NOC) was 1.0998 ± 0.0049 g/mL, or ∼1.5% decrease from the mock-treated (DMSO) asynchronous culture density of 1.1173 ± 0.0176 g/mL. These results suggest that density for arrested populations is cell cycle-dependent, which further supports the findings by others (9, 10) that density is regulated by cell cycle progression.

Fig. 3.

Density, buoyant mass, and volume of cells synchronized by cell cycle blocks. (A) WT (A2587) and cdc28-as1 (A4370) densities (mean ± SD) in arrested G1 and metaphase are lower than asynchronous mock-treated (untreated, DMSO) populations. The density of the asynchronous population is approximately the same as S-phase arrested cells and reflects the unequal distribution of cells throughout the cell cycle. Cells were synchronized with alpha factor (AF) for 2 hr (5 μM), 1-NM-PP1 (pp1) CDK inhibitor for 3 hr (5 μM), hydroxyurea (HU) for 2 hr (10 mg/mL), and nocodazole (NOC) for 2 hr (15 μg/mL), as indicated. Fixed cells were fixed in 3.7% formaldehyde for various amounts of time. (B) Buoyant mass (ρf = ρwater,20 °C) and volume measurements for the same cell populations (mean ± SD).

Because the differences in density could be artifacts of the different treatments or the cell cycle arrests, we sought to determine if cell density varied during an unperturbed cell cycle. For this purpose we measured the buoyant mass, volume, and density of G1 cells isolated by centrifugal elutriation, resuspended in media for synchronous cell cycle progression, and formaldehyde-fixed overnight. Although fixation artificially increases cell density (Fig. 3A), the relative density throughout the time course (Fig. 4A) agrees with our previous observations and others’ (9, 10). Cell density was correlated with cell cycle position by comparing the cell density with the percent of budded cells (Fig. 4A, numbers in brackets). The density is relatively low in elutriated G1 (unbudded) cells, increases to near maximum during late G1 and S-phase entry (bud formation), and then decreases through mitosis. The recorded cell density does not return to its minimum at the end of the first cell cycle because of the time course’s resolution, loss of synchrony, or a possible elutriation effect on the initial selection. Buoyant mass and volume continue to increase throughout the time course as the cells grow and the population adjusts from the elutriation’s size selection (Fig. 4B).

Fig. 4.

Density, buoyant mass, and volume of cells synchronized by elutriation. (A) Density of formaldehyde-fixed-cell populations (A11311) grown overnight in YEP + 2% raffinose and a synchronized sample was selected by centrifugal elutriation. Bud counts are reported as percent budded in brackets next to each measurement. Cells begin to enter S phase between 60 and 120 min. Error bars are the SEM measured with a single fixed sample of elutriated WT cells (A11311, n = 3). (B) Buoyant mass (ρf = ρwater,20 °C) and volume increase throughout the time course. Changes in cell density at the population level are the result of differences in the relative rates of mass and volume increase through the cell cycle.

Although the density of arrested cells can be calculated with high precision because buoyant mass and volume are measured over an extended period, this limits the temporal resolution and makes it difficult to observe growth dynamics. In order to address this limitation and to measure an uninterrupted cell cycle, we continuously sampled from split cultures of elutriated G1 yeast cells that remained in media for both measurements. Buoyant mass and volume growth rates increase during the cell cycle, and the coefficient of variation for volume is greater than that observed in buoyant mass (Figs. 5 and S2–S4). The distributions widen as the culture loses synchrony and cell size variation emerges. Although the smaller sample size prevented accurate density calculations, this approach offers a powerful means to observe growth dynamics in synchronized cell populations.

Fig. 5.

Buoyant mass and volume growth rates are cell cycle-dependent. Color designates the fraction of the population with the indicated buoyant mass and volume (color bar at right). Small unbudded cells (A2587) were isolated by elutriation for synchronous culture after overnight growth in YEPD, which partially reduced synchrony. (A) The SMR steadily sampled from the culture and measurements were broken into 10-min divisions (n = 7,839 cells). (B) Volume measurements on a split culture were made from aliquots drawn at 10-min intervals and recorded in < 1 min (n = 67,607 cells). Additional budding data is available in Fig. S1.

In summary, our data show budding yeast increase density prior to bud formation and increase volume and buoyant mass growth rates throughout the cell cycle. The density increase may result from several START-dependent events, and to investigate this we have developed a method that enables cell density to be measured in real time for unperturbed cells.

Relative Density Measurement of Single Cells During Growth.

The SMR reads out real-time changes in relative cell density via the direction of change in buoyant mass as synchronized and growing cells flow through the cantilever. A cell with a density greater than that of the fluid appears as a positive buoyant mass, and a cell with a density less than that of the fluid appears as a negative buoyant mass (Fig. 6). If the media density is adjusted to be between the initial and final cell densities, then the ratio of the positive to total number of measurements in a short time (1 min) approximates the fraction of cells in the higher density state at that time. This single-cell technique enables the timing and detection of density changes in biological samples as they occur in growth media. It has a higher time resolution than current gradient-based approaches and does not involve a subsequent gradient fractionation to define the density distribution (7). This method may also be applied to cells transitioning to lower density by decreasing the fluid density to below that of the cells’ initial density.

Fig. 6.

Real-time relative cell density measurement. Cell state, distinguished by cell density, is determined by the cell’s direction of frequency shift in media with a density slightly above that of G1 cells. G1-synchronized cells have a negative buoyant mass (positive frequency shift), and cells entering S phase at a later time point have a positive buoyant mass (negative frequency shift). The proportion of cells in each state is directly correlated to the percent of cells below or above fluid density and changes as cells synchronously progress through the cell cycle.

The precision of this technique relies on the method for modifying the fluid density, the SMR’s resolution, and the degree of overlap in the density distributions of the two cell states. To modify the density of YEPD (yeast extract/peptone supplemented with 2% glucose) we selected Percoll (Sigma-Aldrich), a colloidal silica suspension, for its low osmolality, low viscosity, and general impermeability of biological membranes (17). Experiments with Histodenz (Sigma-Aldrich) showed density changes for arrested yeast, likely through an osmotic response, such as is observed in bacteria (18). The resolution of the 8-μm-tall SMR used for these experiments is ∼3 fg (1-Hz bandwidth). This can detect a 0.1% yeast density change in a high-density solution, and we measured ∼2% change through the cell cycle. By measuring trends in synchronized populations, sample number increases and the method’s statistical precision and reproducibility further improves.

The G1/S Density Change in Yeast.

We used this method to further investigate cell density changes in yeast and how density is coupled to the G1/S transition. Changes in density are indicative of differences between rates of total mass and volume increase, which may be correlated to specific cell cycle events. Cells were first synchronized via alpha factor and released into a YEPD:Percoll media (fluid density ∼1.086 g/mL), and as the cells progressed through the cell cycle they were continuously flowed through the SMR for single-cell buoyant mass measurements over a time course. As cells neared S-phase entry and became denser, the proportion of cells with a density above that of the fluid increased (Fig. 7, mock-treated release). This shift demonstrates that alpha-factor treatment arrests cells before the density increase observed in the unperturbed cell cycle of G1 elutriated cells. The shift is consistently observed following alpha-factor release, but the kinetics are not always identical. The timing and rate of this density shift depends on the cell treatment, efficiency of release, and the difference between cell and media density. If the difference between the cell and media densities varies between experiments, then the timing of the density change, the point at which the cell density is greater than that of the media, will also vary. These variances are responsible for the differences in mock-treated WT behavior observed during the experiments of Fig. 7. Although the technique is not suitable for quantifying density, it does enable the real-time detection of relative density changes that occur near the media density and, by the rate of transition, provides some information regarding population synchrony.

Fig. 7.

Changes in cell density require energy, TOR function, passage through START, and an intact cytoskeleton. WT cells (A2587) were arrested in alpha factor and released in YEPD:Percoll media with each treatment. Increases in the percent of cells with a density above that of the fluid signify an overall increase in cell density. (A) Azide (0.1% wt/vol) prevented the density change and demonstrates an energy requirement for the density increase. Mock-treatment: equal volume water. (B) Rapamycin (10 μM) prevented the density change and establishes a TOR function requirement. Mock-treatment: equal volume 70% ethanol. (C) Alpha factor (5 μM) prevented the density change and confirms a passage through START requirement. Mock-treatment: equal volume DMSO. (D) LatA (100 μM) prevented the density change and establishes a requirement for an intact actin cytoskeleton. Mock-treatment: equal volume DMSO. Error bars are the standard error of the proportion. The Bonferonni-corrected significance for each treatment was p ∼ 0.01 for (A), p ∼ 0.05 for (B), p ∼ 0.04 for (C), and p ∼ 0.06 for (D).

The Change in Yeast Density at G1/S Requires Energy, TOR function, START, and an Actin Cytoskeleton.

To characterize the yeast density shift at the G1/S transition as an energy-dependent process, we blocked ATP synthesis with sodium azide, an inhibitor of F1-ATPase, immediately following release from an alpha-factor arrest. The SMR’s buoyant mass measurements detected a cell density change in mock-treated control cells and no change in density for cells treated with azide (Fig. 7A). Thus, the density change we previously observed is an active process requiring ATP.

The TOR pathway controls translation initiation and stimulates protein synthesis in response to nutrients (19). To examine whether the density shift depends on protein synthesis, we asked whether the TOR pathway was required. We treated cells released from an alpha-factor-induced arrest with rapamycin, an inhibitor of TORC1 function (20), and compared the percent of cells with a change in density to the resulting percentages for a parallel-grown mock-treated control culture. Cells from the control culture changed in density, and cells with TOR inactivation did not change in density (Fig. 7B). Thus, the change requires TOR function and, likely, protein synthesis.

To determine whether START is required for the density change, we prolonged a G1 arrest via continuous treatment with alpha factor. Mock-treated cells released into fresh YEPD changed in density, and cells that were resuspended in alpha-factor-containing medium displayed no significant change in density during an 85-min time course (Fig. 7C). These results suggest that the density shift is START-dependent but do not indicate whether it results from bud formation, DNA replication, or some other START-dependent process.

Bud formation requires polarization of the actin cytoskeleton, and we investigated the possibility that the actin cytoskeleton has a function in the cell density increase. We treated cells released from an alpha-factor arrest with an inhibitor of actin polymerization, latrunculin A (LatA). We measured a change in density for mock-treated control cells and no change in density following LatA treatment (Fig. 7D). We also monitored bud appearance for similarly treated cells and confirmed that the LatA treatment severely inhibited bud formation (Fig. S5A). Thus, disruption to the actin cytoskeleton prevents the density change, which may be the direct effect of perturbing actin-dependent processes such as vesicular transport required for growth, or result from other LatA effects on cell growth (21).

To investigate if the density change also requires DNA replication, we examined DNA content by FACS analysis. Cellular DNA was fluorescently labeled and the distribution of relative DNA content (C, 2C) was measured for synchronized samples. The LatA and rapamycin treatments following release from an alpha-factor arrest reduced DNA replication by nearly the same amount (Fig. S5B). At 60 min, ∼10% of the mock-treated and ∼35% of the LatA- and rapamycin-treated cells exhibited 1C DNA, or had not yet replicated (Fig. S5C). The rapamycin treatment following release from an alpha-factor arrest also severely inhibited bud formation (Fig. S5A). Rapamycin is known to prevent entry into the cell cycle by inhibiting translation initiation (19); however, we have observed the effects of rapamycin to be more severe on budding than on replication. Rapamycin- and LatA-treated cells do not change in density and do not form buds but do replicate to a significant extent. Therefore, the change in density near the G1/S transition is independent of DNA replication, and inactivation of actin-related processes such as cell polarization, budding, and/or vesicular transport may inhibit pathways required for this density change.

Understanding the mechanism for this cell cycle-dependent density change is important in describing cell growth dynamics. A density change confirms that changes to total mass (protein synthesis, vacuolar dynamics) and volume increase (membrane growth) are not directly proportional, and the requirements for a density change at the G1/S transition of budding yeast demonstrate a coupling between these two processes. We have observed that perturbations to membrane growth early in the cell cycle, such as the inhibition by LatA of budding and growth, abolish changes in density, and previous studies have shown that protein synthesis is linked to membrane growth (22). A change in density, or differences between mass and volume rates of increase, may occur as a result of a transient increase to mass accumulation during polarized membrane growth at the G1/S transition, a slowdown in membrane expansion during coordination with bud formation, and/or changes to cell water content. One caveat in this hypothesis is that pheromone-treated cells, while polarized, remain low in density. However, this lower density may result from decreased protein synthesis and other effects of alpha-factor treatment (23). Another possibility is that changing vacuole size may alter density. Cln3 regulates vacuole size at the G1/S transition (24), and cells with larger vacuoles have decreased density (9). We have measured vps33Δ mutants (small, fragmented vacuoles) in the S288C background to have a density ∼0.8% greater than the WT density of 1.1174 ± 0.0039 g/mL, as in ref. 11. However, analysis of the vacuole’s role in cell cycle-dependent density changes is complicated by long cell cycle times in class I vacuole mutants, and thus little is known about how the vacuole affects cell density at the single-cell level and during a normal cell cycle.

Conclusions and Future Directions

We have presented mass, density, and volume measurements throughout the cell cycle and developed two unique measurement techniques to identify the density change in budding yeast at the G1/S transition as dependent on energy, TOR function, passage through START, and the actin cytoskeleton. One possible model for the density variation with these dependencies is one in which polarized growth enables processes during late G1 to increase cell density and changes in volume during bud formation at S phase decrease cell density. This linkage between mass and membrane growth is central to how the cell coordinates growth with division and may have an important role in the signal for cell cycle progression. The density measurement techniques may be generalized to other cells and subcellular particles, and future technology developments aim to acquire single-cell mass and density measurements at once and with only the SMR. This, combined with the ability to continuously monitor single cells, could detail the cell-to-cell variations that are otherwise obscured in population measurements and would provide the basis for a more complete understanding of cell growth, division, and response.

Materials and Methods

SMR Measurements.

Devices and fluidic controls are as described in ref. 12 with the exception of a larger (∼8 × 8 μm) cantilever channel cross-sectional area. Devices were further enlarged to ∼9 × 9 μm by a KOH etch at 40 °C postfabrication. Cell volume was measured twice for the density measurement in the first method—once before the mass measurement and once on the sample of cells collected from the SMR waste channel—to pair the measurements and ensure that the measured cells were representative of the original sample.

Strains, Growth Conditions, and Sample Preparation.

The yeast strains used in this study are listed in Table 1. Cells were grown in YEPD at room temperature (21 ºC). Cells were synchronized with 5 μg/mL alpha factor at 0 and 90 min, for a total arrest of 120 min, with the exception of cells synchronized by the 120-min hydroxyurea (10 mg/mL) or nocodazole (15 μg/mL) treatment. An equal volume DMSO mock-treated culture was a control for the nocodazole measurements. For cdc28-as1, cells were arrested with 1-NM-PP1 (5 μM) for 3 hr. For the density measurements on growth-arrested cells, cells were washed via vacuum filtration at 120 min and concentrated in phosphate buffer to ∼108 cells/mL. Volume was measured on a Multisizer 3 Coulter counter fitted with a 100-μm aperture tube, the sample was delivered to the SMR, and the cells were collected from the waste side of the SMR to be measured again with the Coulter counter. We observed no change in cell volume resulting from suspension in phosphate buffer. We compared the original sample volume distribution to that of cells collected from the SMR waste and directly observed the SMR’s preferential selection of small particles. This bias was mitigated with higher flow rates.

For real-time sample preparation, cells were grown and arrested as described above. Cells were washed via vacuum filtration and concentrated in a 65∶25∶10 Percoll (Sigma-Aldrich): 4xYEPD:H2O solution to ∼107 cells/mL. Depending on treatment, alpha factor (5 ug/mL), azide (0.1% wt/vol), rapamycin (10 μM), LatA (100 μM), DMSO, or ethanol control was added to the cell suspension. Cell volume was recorded by a Coulter counter and the sample was delivered to the SMR at room temperature.

Table 1.

Yeast strains used in this study


Cells were grown overnight in YEP +2% raffinose for density measurements (Fig. 4) and YEPD for continuous measurements (Fig. 5), synchronized by centrifugal elutriation (25), and resuspended in YEPD. For continuous measurements the sample was concentrated and delivered to the SMR for mass measurement in media. Aliquots for fixed-cell density measurements were collected into 3.7% formaldehyde at indicated time points. Fixation was required for a complete cell cycle time course because the time required for the density measurement is much longer than the cell cycle. After fixation, cells were washed and resuspended in phosphate buffer, the volume distribution was recorded on the Coulter counter, and the sample was delivered to the SMR for mass measurement.

Flow Cytometric Analysis.

Cells released from alpha-factor arrest were concentrated to ∼108

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