Effects of Sampling Rate on the Interpretation of Cellular Transport

Anal. Chem. 2008, 80, 7684–7689

Effects of Sampling Rate on the Interpretation of Cellular Transport Measurements Sumitha P. Nair and Miklos Gratzl* Department of Biomedical Engineering, Case Western Reserve University, Cleveland, Ohio 44106 Electrochemical sensing techniques are increasingly used to study biological processes by monitoring concentration changes of the molecule of interest close to cells. The measured concentration is the result of cellular transport across the cell membrane and diffusion of the released molecules from the cells to the sensing electrode. The objective of such experiments is to understand the cellular processes underlying the observed changes in concentration. Thus, the influence of mass transport on the measured concentration trace has to be removed. This is done by deconvolution of the impulse response function of diffusion from the concentration data. We have recently observed that measuring concentration at a sampling rate that satisfies the Nyquist criterion for the observed concentration dynamics may not be sufficient to correctly reconstruct cellular flux. This is because the impulse response function of diffusion also has to be represented with sufficient temporal resolution. We discuss this problem here using the example of monitoring drug efflux from a monolayer of cancer cells with microvoltammetry, and chloride secretion from an epithelial cell monolayer monitored with an ion-selective electrode. Very small to microscopic sensors based on electrochemical principles are increasingly used to gain insight into biological transport processes. Many applications of voltammetry have been reported in both in vitro1-8 and in vivo1,2,9-12 contexts. Potentiometry with ion-selective electrodes has also been used in * To whom correspondence should be addressed. E-mail: [email protected]. Phone: (216)-368-6589. Fax: (216)-368-4969. (1) Michael, A. C., Borland, L. M., Eds. Electrochemical Methods for Neuroscience, 1st ed.; Frontiers in Neuroengineering; CRC Press: Boca Raton, FL, 2006. (2) Ponchon, J. L.; Cespuglio, R.; Gonon, F.; Jouvet, M.; Pujol, J. F. Anal. Chem. 1979, 51, 1483–1486. (3) Lantoine, F.; Brunet, A.; Bedioui, F.; Devynck, J.; Devynck, M. A. Biochem. Biophys. Res. Commun. 1995, 215, 842–848. (4) Kumar, G. K.; Overholt, J. L.; Bright, G. R.; Hui, K. Y.; Lu, H. W.; Gratzl, M.; Prabhakar, N. R. Am. J. Physiol-Cell. Physiol. 1998, 274, C1592–C1600. (5) Yi, C.; Gratzl, M. Biophys. J. 1998, 75, 2255–2261. (6) Lu, H. W.; Gratzl, M. Anal. Chem. 1999, 71, 2821–2830. (7) Cserey, A.; Gratzl, M. Anal. Chem. 2001, 73, 3965–3974. (8) Mauzeroll, J.; Bard, A. J. Proc. Natl. Acad. Sci. U. S. A. 2004, 101, 7862– 7867. (9) Gratzl, M.; Tarcali, J.; Pungor, E.; Juhasz, G. Neuroscience 1991, 41, 287– 293. (10) Griveau, S.; Dume´zy, C.; Se´guin, J.; Chabot, G. G.; Scherman, D.; Bedioui, F. Anal. Chem. 2007, 79, 1030–1033. (11) Xu, W.; Ma, W.; Li, K.; Hu, J.; Shen, L.; Li, H.; Cao, L. Sens. Actuators, B: Chem. 2002, 86, 174–179. (12) Fleming, B. D.; Johnson, D. L.; Bond, A. M.; Martin, L. L. Expert Opin. Drug Metabol. Toxicol. 2006, 2, 581–589.

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Analytical Chemistry, Vol. 80, No. 20, October 15, 2008

studying a number of biological processes.13-16 The common goal of these studies is to obtain information on the cellular processes that manifest themselves in changes in the measured concentration. Using the acquired concentration data directly for characterizing the underlying biology is, however, rarely satisfactory.17,18 Therefore, in most cases it is desirable to derive cellular flux from concentrations measured at some distance from the cells. This can be achieved by deconvolution of the impulse response function of diffusion from the concentration data. This can be thought of as undoing the process that created the detected concentrations from flux at the cells, i.e., diffusion from cells to sensor. It is well-known that to represent dynamics of the studied process properly the data need to be sampled at a frequency that is significantly higher than the highest frequency content of interest (Nyquist criterion19,20). We have recently observed that measuring concentration at a sampling interval that satisfies this criterion for the concentration dynamics may not be sufficient to correctly reconstruct cellular transport from the measured concentration trace. This is because the dynamics of the diffusion process that has translated transport at the cells’ surface to concentration seen by the electrode at some distance from the cells also has to be properly represented. Interestingly, in cases when a sampling rate that would be high enough to achieve this cannot be realized, such as when adsorptive preconcentration at the electrode surface is required before every measurement, interpolation between measured data points can improve accuracy of the obtained cellular flux. We explain and illustrate this problem using earlier experiments where we studied the process of passive and active efflux of a cancer drug, doxorubicin (DOX), from a monolayer of cancer cells with microvoltammetry where adsorptive preconcentration was necessary.5 We then use data obtained with an ion-selective electrode that was used to monitor concentration changes corresponding to stimulated chloride secretion from an epithelial cell monolayer16 where data acquisition was fast enough to properly represent both cellular transport and subsequent diffusion. This example made it possible to study the (13) Land, S. C.; Collett, A. J. Exp. Biol. 2001, 204, 785–795. (14) Smith, P. J. S.; Trimarchi, J. Am. J. Physiol. Cell. Physiol. 2001, 280, C1– C11. (15) Pretsch, E. Trends Anal. Chem. 2007, 26, 46–51. (16) Nair, S.; Kashyap, R.; Laboisse, C. L.; Hopfer, U.; Gratzl, M. Eur. Biophys. J. 2008, 37, 411–419. (17) Nair, S.; Gratzl, M. Trends Anal. Chem. 2004, 23, 459–467. (18) Nair, S.; Gratzl, M. Anal. Chem. 2005, 77, 2875–2881. (19) Lathi, B. P. Signal Processing and Linear Systems; Oxford Press: New York, 1998. (20) Proakis, J. G.; Monolakis, D. G. Digital Signal Processing Principles, Algorithm and Applications; Prentice Hall: Upper Saddle River, NJ, 1996. 10.1021/ac800842m CCC: $40.75  2008 American Chemical Society Published on Web 09/10/2008

effects of hypothetically decreasing the sampling rate on the accuracy of back-calculated flux. EXPERIMENTAL SECTION Drug Efflux Experiments. In vitro drug efflux data5 from a monolayer of a MDR subline (CHRC5) of an auxotrophic mutant of Chinese hamster ovary cells is used to elucidate the effect of sampling rate on the back-calculated flux. The cell density was 6.2 × 104 cells cm-2. The cells were incubated in 6.4 µM DOX containing medium at 37 °C for 1 h. Prior to the efflux measurements, the DOX-containing medium was quickly replaced with drug-free medium. A carbon fiber (CF) electrode (7.5 µm in diameter and 5 mm in length) placed horizontally ∼10-100 µm above the cell monolayer was used to monitor efflux of DOX from the cell monolayer. Cathodic current of DOX electroreduction was monitored in differential pulse amperometry mode every 6 min. Each measurement was preceded by adsorptive preconcentration of DOX on the CF surface for 5 min. Electrochemical cleaning of the CF surface was performed after each measurement by applying a negative potential (-1.0 V) for 30 s. Adsorption is complete at the end of the preconcentration time;5 thus, the electrode surface at the end of preconcentration time is nearly in equilibrium with the then actual drug concentration. This justifies plotting the concentration data at the end of the measurement time. Further details about electrode fabrication, cell culture, and monitoring of DOX efflux can be found in ref 5. Chloride Secretion from Epithelial Cells. Chloride secretion from a monolayer of HT29.Cl.16E cells in response to apical stimulation with 1 mM adenosine triphosphate (ATP) + 50 nM phorbolmyristate acetate was studied using a chloride ion-selective electrode (ISE). The experiments were carried out in a Ussing chamber arrangement under voltage clamp conditions. The apical chamber included four electrodes: a Cl- ISE and ISE reference in addition to a current and voltage electrode. The basolateral chamber incorporated only a current and voltage electrode. Further details of the experimental arrangement and ISE fabrication can be found in ref 16. The use of an ISE under voltage clamp conditions enabled the simultaneous monitoring of net charge transport across the epithelial monolayer in the form of short circuit current (Isc)21 and also chemically resolved secretion of chloride. Data Processing. DOX efflux rate and chloride secretion from the epithelial cell monolayer as functions of time were backcalculated from measured concentration data using shape error optimization that is described in ref 18. This technique is based on the premise that the time course of a continuous function sampled at N discrete time points can be represented as a vector in an N dimensional space. The angle between two such vectors is an indicator of how similar or dissimilar the sampled time courses are in “shape”. An angle of 0° implies identical shapes. The optimization converges to a flux time course that will replicate the measured concentration trace. Shape optimization has been successfully used to reconstruct dynamics of anticancer drug efflux from cancer cells18 and ion transport from epithelial cell monolayers.16 In the drug efflux studies, electrode distance from the cell monolayer was ∼20 µm.5,17,18 We note that DOX concentration in the medium is not sensitive to distance from the (21) Schultz, S. G.; Leaf, A. J. Membr. Biol. 2001, 184, 199–202.

cells within the ∼0-100-µm range for slow efflux processes such as the one discussed here.17 The diffusion constant of DOX used was 1.5 × 10-6 cm2 s-1 as in ref 5. For the chloride secretion studies, the ion-selective electrode was placed ∼50 µm from the cell monolayer.16 The diffusion coefficient of chloride in water (2.03 × 10-5 cm2 s-1)22 was used in the results presented in this work. The optimization was performed using Matlab (The Mathworks, Natick, MA) 7.0.4, Release 14.0. Linear interpolation of fluxes and concentrations was done using the “interp1” function available in Matlab. RESULTS AND DISCUSSION Reconstructing Cellular Flux from Measured Concentration. Concentration in the vicinity of cells that are releasing material is the result of the efflux process at the cells and diffusion of the released molecules away from the cells. Concentration at a given distance from a cell monolayer can therefore be estimated for a known cellular flux by convolution of the flux time course with the characteristic function that describes the dynamics of diffusion from the cells to the site of measurement. This function is the impulse response of one-dimensional diffusion:23 δz,t )

1

√πDt

( )

exp

-z2 4Dt

(1)

where t is time, z is the distance from cell(s) surface, and D is diffusion coefficient of released material in the extracellular solution. Concentration at a distance z from the cell layer is then expressed as a convolution: Cz,t ) Et o δ )

∫ E(τ)δ (t - τ) dτ t

0

z

(2)

where the symbol o denotes convolution, E is the flux at the cells surface, and τ is a running variable in time. Calculating concentration that results from material release by cells and diffusion of the released material from the cells to the sensor is called the “forward problem”. In most cases, flux across the cell membrane is not known, but rather, it is the variable of interest. Concentration is measured to obtain information on cellular flux. This problem is called “inverse problem” and can be solved with deconvolution: the inverse of the operation in eq 2. Effect of Sampling Rate on Back-Calculated Fluxes. Drug Efflux from Cancer Cells. Efflux of DOX from cells is mainly passive, driven by a concentration gradient across the cell membrane in the drug-sensitive cells. Efflux from the drugresistant cell phenotype has an ATP-driven active component also via membrane p-glycoprotein pumps (pgp).24 Both passive and active transport are continuous and limited by slow extracellular diffusion of the effluxed molecules away from the cell monolayer.5,6 As a result, drug concentration changes slowly and continuously in the medium. DOX concentration measured every 6 min close to the preloaded drug-resistant cell layer is shown in Figure 1 (red circles). The characteristic time scale of change is in the (22) Lide, D. R., Ed. Handbook of Chemistry and Physics; CRC Press: FL, 2002. (23) Crank, J. The mathematics of diffusion; Oxford Press: New York, 1975. (24) Gottesman, M. Annu. Rev. Med. 2002, 53, 615–627.


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Figure 1. Effect of sampling rate on deconvolution. Panel A: Doxorubicin concentration measured by a carbon fiber electrode (red circle) is interpolated every 30 (green circle) and 1 s (blue dot). Panel B: Corresponding fluxes back-calculated using shape optimization is shown with the respective symbols. For the error optimization, electrode distance from the cells and diffusion constant of doxorubicin in the extracellular medium were assumed to be 20 µm and 1.5-10-6 cm2 s-1, respectively. As the sampling interval decreases, the amplitude of the back-calculated flux decreases especially in the initial period and the flux dynamics also changes.

order of 1 h or longer. This is also indicated by the Fourier transform of the data set that has negligible amplitude above ∼0.0003 Hz, corresponding to 50 min (not shown). Therefore, the 6-min sampling rate is sufficient to reproduce the frequency content of the concentration trace and it in fact represents a significant oversampling. We describe here that despite the fact that the sampling rate is more than sufficient to map the dynamics of concentration, very significant distortion in the back-calculated flux arises unless data are sampled at a much higher rate, i.e., at 1 s or faster. We show that as long as the Nyquist criterion for the measured concentration itself is satisfied, this error can be corrected for by interpolation between the measured data points even though interpolation does not add new independent information to the original data. In the results discussed below, linear interpolation is used only for demonstration purposes and using other functions for interpolation would not affect the conclusions of this work. Similar results were obtained with spline interpolation (not shown). Linear interpolation has, however, the advantage of being the simplest of possible interpolations and will not introduce implicit assumptions about dynamics between sampled points. Cellular flux of DOX from the loaded drug-resistant cells was back-calculated with shape-based deconvolution as described earlier.18 First we used the original data set containing concentra7686


Figure 2. Effect of sampling rate on convolution. Panel A: Flux backcalculated from measured concentration (red circle) is interpolated every 30 (green circle) and 1 s (blue dots). Panel B: The corresponding concentration calculated using eq 2 is shown with the respective symbols. As the sampling becomes sparser, the same amount of material translates into a lower concentration.

tions measured every 6 min,5,17 shown in Figure 1A (red circles). The result is an essentially linearly decreasing flux over the duration of the experiment as shown in Figure 1B, red circles. The flux obtained closely reproduces the originally measured concentration data points when the values in Figure 1B red circles are plugged into eq 2 to calculate concentration (not shown). We then repeated the same calculation procedure but using additional “data” points that were obtained by linear interpolation between the measured concentration data. First we used interpolated values 30 s apart that mimicked a sampling rate of 30 s (Figure 1A green circles). The back-calculated flux in this case was very different from the flux obtained without interpolation, especially for the first 20 min, in both dynamics and amplitude (Figure 1B red circles). Yet, by using these flux values to solve the forward problem (eq 2), we could reproduce again the originally measured concentration data very closely (not shown). Next we used interpolated concentration values at 1-s intervals (Figure 1A blue dots). The back-calculated flux was further modified (Figure 1B blue dots), but the concentration trace calculated from this flux was again correct (not shown). Further increase in the number of interpolated points produced negligible change in flux. To help interpret the above observation, we then performed the reverse procedure to test how the forward problem, convolution, is influenced by sampling rate. To do so, we first solved the

Figure 3. Impulse response function of planar diffusion calculated using eq 1 with z ) 20 µm and D ) 1.5 × 10-6 cm2 s-1 (shown as a black line). The impulse response sampled at 6 min, 30 s, and 1 s is shown in red, green, and blue, respectively. Panel A: Impulse response function for the entire measurement period of ∼80 min. Panels B-D show the impulse response shown in panel A on gradually expanded time scales. The effluxed material “lost” due to insufficient sampling of concentration at different sampling rates is proportional to the area under the function (black line) down to the labeled lines. Line a, 6-min sampling; line b, 30-s interpolation; lines c, d, 1-s interpolation.

forward problem using the flux values (Figure 2A red circles) that were obtained with back-calculation from the original data sampled at 6-min intervals. As expected, the procedure reproduced the concentration data very well (Figure 2B red circles). We then added interpolated values to the flux, every 30 s (Figure 2A green circles). The forward problem now resulted in higher concentration levels (Figure 2B green circles) than those that were measured. Further interpolation in flux at every 1 s (Figure 2A blue dots) produced even higher concentrations (Figure 2B blue dots). The results shown in Figures 1 and 2 summarize the effect of sampling rate on convolution and deconvolution. These results imply that more released material is “required” to reproduce the observed concentration values if sampling is sparse than when it is more frequent. The explanation involves the impulse response function of diffusion from a monolayer (eq 1). At 6-min sampling, a large part of the function is truncated between 0 and 6 min, as shown in Figure 3 (the individual panels of this figure show the initial period of the impulse response function on gradually expanded time scales). The physical meaning of this can be thought of as if material represented by the area under the impulse response function in Figure 3, black line, down to the labeled lines will never be sensed by the electrode. More frequent sampling decreases the extent of this material “loss”, and consequently, less efflux is sufficient to reproduce the measured concentrations. In the special experimental context discussed here, it is obvious that ∼1-s sampling is required to decrease the “material loss” to a negligible level. There are two conclusions that can be drawn from these results. First, in many cases, it may not be sufficient to use a

sampling rate that satisfies the Nyquist criterion for the concentration trace alone if the data are to be used to reconstruct cellular flux. Rather, it is necessary to sample at a rate that is high enough to “catch” all the material released in a hypothetical impulse efflux and is therefore representative of the impulse response function of the diffusion problem. Interestingly, however, actual finer sampling if difficult to achieve could be substituted by interpolation between measured data points. The flux obtained with the interpolated concentrations could be interpreted in terms of the way the drug efflux experiment was conducted.5 The cells were incubated in a medium containing 6.4 µM DOX at 37 °C. After incubation, the cells were washed in ice-cold phosphate-buffered saline solution 5 times. The cells were transferred to the efflux measurement container, and the measurements began immediately thereafter. It is reasonable to assume that it took some time for the temperature at the cells to reach body temperature and activate the pgp efflux pumps. This effect could not be reconstructed from the concentration data previously where no interpolation was used. Electrolyte Secretion from a Confluent Monolayer of Epithelial Cells. Mucin is one of the most common exocrine secretions in the human body. Mucin is a polyanionic glycoprotein stored inside secretory granules in a highly condensed state.25,26 Release of mucin granules from epithelial cells is accompanied by electrolyte secretion. The amount of electrolyte secreted relative to mucin is thought to be essential for the formation of well-hydrated (25) Moniaux, N.; Escande, F.; Porchet, N.; Aubert, J. P.; Batra, S. K. Front. Biosci. 2001, 6, D1192-D1206. (26) Verdugo, P. Annu. Rev. Physiol. 1990, 52, 157–176.


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Figure 4. Stimulated chloride secretion from a confluent epithelial monolayer. Chloride concentration measured ∼50 µm from an epithelial cell monolayer with a 1-s sampling rate is shown in panels A and D, black line. Flux corresponding to the measured concentration, back-calculated using the shape approach is shown in panels B, C, E, and F, black line. Panel A, blue dot; panel D, red dot: concentration corresponding to a sampling rate of 10 and 30 s, respectively, obtained by removing data points from the measured concentration shown as a black line. Panel B, blue dot; panel E, red dot: flux that corresponds to the concentration of lower sampling rate of 10 and 30 s, respectively, together with the flux that corresponds to the measured concentration shown as a black line. Panel C, blue line; panel F, red line: concentration that represents 10and 30-s sampling interval (panel A, blue dot, and panel B, red dot, respectively) was linearly interpolated every 1 s, and the corresponding flux back-calculated using the shape approach is shown together with the flux that corresponds to the measured concentration shown as a black line.

mucus.27,28 Decreased electrolyte secretion, as in cystic fibrosis, leads to abnormally thick mucus causing pulmonary infection and eventual death.26-28 H29.Cl.16E is a human colonic cancer cell line that secretes electrolyte and mucin in response to stimulation with ATP,29 and we used it as a model cell line to study ion secretion that accompanies mucin exocytosis. Isc as induced by pharmacological stimulation in a voltage clamp setup was used to characterize net transepithelial ionic flux that accompanies mucin secretion in HT29.Cl.16E cells.29 The Isc, although a direct measure of cellular transport, is only an indication of net charge movement as opposed to chemically resolved ionic flux. In order to obtain information on ionic secretion with chemical selectivity at high temporal resolution, a Cl- ISE and reference electrode were placed in the apical side (in addition to the electrodes for the voltage clamp) ∼50 µm from the cell monolayer. The measured concentration was translated to flux at the cells using shape optimization. The use of the ISE (27) Wine, J. J. J. Clin. Invest. 1999, 103, 309–312. (28) Verkman, A. S.; Song, Y.; Thiagarajah, J. R. Am. J. Physiol. Cell. Physiol. 2003, 284, C2–C15.

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enabled dynamic monitoring of chloride secretion that accompanies mucin exocytosis.16 The ISE measurements were carried out every 1 s. Interpolating between the measured concentrations at 0.1 s did not produce any observable change in the back-calculated flux (results not shown). This implies that the sampling rate of 1 s used in these studies was sufficient to correctly represent the frequency content of the biological process and also that of diffusion from the cells to the sensor. Interpolation can thus be used to test whether the actual sampling rate is sufficient to also represent the dynamics of extracellular mass trasnport. Here we demonstrate how eliminating some of the data points from the measured concentration (Figure 4A, black line), equivalent to lowering the sampling rate, affects the back-calculated flux. Data points were eliminated to simulate a sampling rate of 10 s (Figure 4A, blue dots). As seen in the figure, a sampling rate of 10 s satisfies the Nyquist criterion for the measured concentration for both its general trend and its finer details. The lower sampling (29) Merlin, D.; Augeron, C.; Tien, X. Y.; Guo, X.; Laboisse, C. L.; Hopfer, U. J. Membr. Physiol. 1994, 137, 137–149.

rate affected, however, the amplitude of the back-calculated flux and also produced a shift in time in the flux peak as shown in Figure 4B (blue dots). We then tested how linearly interpolating (1 s) between the sparsely sampled (10 s) concentration data would affect the backcalculated flux. The flux reconstructed from the interpolated concentrations is shown in Figure 4C with a blue line. Comparison with with flux back-calculated from the originally measured concentration (Figure 4C, black line) shows that interpolation has largely eliminated the change in amplitude and shift in time of the peak. In panels D-F of Figure 4, we also demonstrate the effect of an even lower sampling rate, which still correctly represents the overall trend but is not sufficient to reproduce some of the finer details of the concentration trace. Data points were eliminated from the measured concentration (Figure 4D, black line) to simulate a sampling rate of 30 s (Figure 4D, red dots). This produced a further increase in the amplitude of the back-calculated flux and also a shift in the peak (Figure 4E, red dots). Interpolating at every 1 s those concentration data that simulate a sampling rate of 30 s brought again the calculated flux close to that one corresponding to the actually measured concentration data (Figure 4F). However, interpolation could not reproduce the finer details that were lost because of undersampling. These results show that interpolation can be used to improve the accuracy of estimated cellular flux when it is not possible to realize a sampling rate that would be high enough to also properly represent the dynamics of diffusion, provided that the Nyquist criterion for the measured concentration itself is satisfied. CONCLUSIONS Reconstruction of cellular transport from measured concentration is strongly influenced by the sampling rate of the measurement. The sampling rate must not only be sufficient to represent the frequency content of the measured concentration time course but also that of mass transport from cells to the measurement site. Interpolation can be used in cases where this second condition cannot be experimentally satisfied. Though interpolation does not add any new information to the measured data, it significantly improves the accuracy of both the amplitude and dynamics of the back calculated flux.

The observations made in this work may lead to a more general conclusion that to the best of our knowledge has not been realized before. When the measured variable is the result of convolution, then it is not possible to reconstruct the underlying cause, which is often the actual aim of the measurement, unless two criteria are simultaneously satisfied: (1) the well-known Nyquist criterion for the measured signal, and (2) a similar criterion for the dynamics of the measurement system. In the examples discussed here, it is dynamics of extracellular diffusion that is convolved with cellular flux as the underlying cause of the signal. However, many other types of measurement can be modeled the same way: as convolution of the variable of interest and the dynamic response of the measurement system. Therefore, the conclusions of this work may have a broader impact than just in cellular transport studies. It is interesting that the second criterion can be satisfactorily addressed by interpolation, which will not produce new pieces of information. This observation can be explained as follows: independent pieces of information required to map the dynamics of the actually measured variable (concentration in this case) are already available when the Nyquist criterion for this variable is satisfied. Therefore, there is no need for further independent pieces of information and interpolation suffices to fulfill the second criterion. ACKNOWLEDGMENT We thank Drs. C. Yi and R. Kashyap for their contribution toward obtaining the experimental data used in this work for demonstration. The experimental work was supported by grant CA61860 from NIH-NCI. A NASA grant in Interdisciplinary Research in Support of Health-Related Countermeasures for Astronaut Crews, John Glenn Biomedical Engineering Consortium, provided support for developing the shape-based deconvolution approach. We also gratefully acknowledge Ohio Board of Regents Innovation Incentive Fellowship award to S.N.

Received for review April 25, 2008. Accepted August 7, 2008. AC800842M


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Effects of Sampling Rate on the Interpretation of Cellular Transport

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