 Nano Express
 Open Access
 Published:
FrequencyModulated Wave Dielectrophoresis of Vesicles And Cells: Periodic UTurns at the Crossover Frequency
Nanoscale Research Letters volume 13, Article number: 169 (2018)
Abstract
We have formulated the dielectrophoretic force exerted on micro/nanoparticles upon the application of frequencymodulated (FM) electric fields. By adjusting the frequency range of an FM wave to cover the crossover frequency f_{ X } in the real part of the ClausiusMossotti factor, our theory predicts the reversal of the dielectrophoretic force each time the instantaneous frequency periodically traverses f_{ X }. In fact, we observed periodic Uturns of vesicles, leukemia cells, and red blood cells that undergo FM wave dielectrophoresis (FMDEP). It is also suggested by our theory that the video tracking of the Uturns due to FMDEP is available for the agile and accurate measurement of f_{ X }. The FMDEP method requires a short duration, less than 30 s, while applying the FM wave to observe several Uturns, and the agility in measuring f_{ X } is of much use for not only salty cell suspensions but also nanoparticles because the electricfieldinduced solvent flow is suppressed as much as possible. The accuracy of f_{ X } has been verified using two types of experiment. First, we measured the attractive force exerted on a single vesicle experiencing alternatingcurrent dielectrophoresis (ACDEP) at various frequencies of sinusoidal electric fields. The frequency dependence of the dielectrophoretic force yields f_{ X } as a characteristic frequency at which the force vanishes. Comparing the ACDEP result of f_{ X } with that obtained from the FMDEP method, both results of f_{ X } were found to coincide with each other. Second, we investigated the conductivity dependencies of f_{ X } for three kinds of cell by changing the surrounding electrolytes. From the experimental results, we evaluated simultaneously both of the cytoplasmic conductivities and the membrane capacitances using an elaborate theory on the singleshell model of biological cells. While the cytoplasmic conductivities, similar for these cells, were slightly lower than the range of previous reports, the membrane capacitances obtained were in good agreement with those previously reported in the literature.
Background
The polarizability of an electrical phenotype is primarily due to the cell membrane and the cytoplasmic electrical properties that depend on the frequency of the applied electric field. Accordingly, individual cells can be identified by the differences in the dielectric spectra using noninvasive electrical techniques. The electrical techniques are currently competent for separating cells with useful phenotypes from unknown samples [1–15]. Compared with other separation methods, these offer the major advantage that cell modification by antibodies or adherence to foreign material is unnecessary, whereby the potential for cell damage or activation by these probes is avoided [1–16]. The characterization of the cellular dielectric properties has been performed mainly using either impedance spectroscopy [10, 12, 13] or alternatingcurrent (AC) electrokinetics such as dielectrophoresis (DEP), travelingwave DEP (twDEP), and electrorotation [1, 9, 15]. Among them, we focus on extending the ACDEP method to develop a new method for dielectric characterization using the frequencymodulated (FM) waves instead of AC fields.
In general, the DEP occurs in an electricfield gradient that creates an electrokinetic force exerted on any polarizable object, charged or neutral, in the direction determined not only by the gradient vector, but also by the real part of the ClausiusMossotti (CM) factor [1–15, 17–21]. For instance, we consider the DEP force induced by the AC electric field E_{AC}(r,t) whose spacetime dependence is expressed as E_{AC}(r,t)=A(r) cosθ_{AC}(r,t) using the amplitude vector A(r) and the phase θ_{AC}(r,t). The ACDEP force is generated by the spatial gradient of the amplitude (i.e., ∇A) multiplied by the real part of the CM factor, as mentioned above, whereas the spatial gradient of the phase (i.e., ∇θ_{AC}) multiplied by the imaginary part of the CM factor creates the force of either twDEP or electrorotation, which therefore provides complementary information to the ACDEP method in terms of the dielectric characterization [9, 15, 20, 21].
In this letter, we aim to formulate the DEP force induced by an FM field and compare the AC and FMDEP methods, so that neither the AC nor the FM field considers the spatial dependence of the phase; therefore, we will set θ_{AC}(t)=2πf_{AC}t in proportion to the applied frequency f_{AC}. A significant feature of the ACDEP is that the force direction as well as its strength depends on f_{AC}. Most notably, the force direction is reversed at crossover frequency f_{AC}=f_{ X } due to the change in sign of the real part of the CM factor, which has been found available for the dielectric characterization using ACDEP [1–15].
The frequency dependence of the ACDEP force has also made the following manipulations possible [1–15, 22–31]: electrically controllable trapping, focusing, and translation of colloidal particles, as well as the fractionation and characterization of living and/or dead cells. Conventional systems for the dielectrophoretic assembly and/or manipulation of colloidal particles have often made use of microfabricated electrodes between which the AC electric field has been applied to colloidal suspensions, benefiting from recent rapid advances in the fabrication of integrated semiconductor devices [24–30]. This technology, offering noncontact manipulation, is currently being integrated with a variety of labonachip systems that provide the advantage of accurate and repeatable handling. Nevertheless, the onchip electrodes that create highintensity spots in AC fields are incapable of changing their positions independently of the sample holder, in contrast to the laser focus that can be freely positioned in the optical manipulation. It follows from the limitation of onchip systems that the previous DEP methods have presented some difficulty and complexity in performing the types of operations for which optical tweezers are suitable. A candidate method to overcome these difficulties is optical imagedriven DEP [32].
Here, we adopted, as a simpler alternative, one of the electronic tweezers techniques [22, 23, 33–38] for ondemand dielectrophoretic assembly and/or manipulation without an optical apparatus (see Fig. 1). As seen from Fig. 1, our plugin style system uses a pair of microelectrode needles that are controlled by micromanipulators for applying the external electric fields in a colloidal suspension. The electrode probes were not fixed but rather movable in colloidal suspensions owing to their plugin style. There remains, however, a significant requirement for the practical use of dielectric characterization: the duration for which an electric field is applied to the cells surrounded by salty electrolytes should be minimized. For instance, the ACDEP method involves the use of interdigitated comblike electrodes that are embedded in a microfluidic system so that the AC fields of various frequencies can be simultaneously applied in a cell suspension [24–30]. While such refined onchip systems have been found relevant for the dielectric characterization, the multielectrodepair technique is inapplicable to the singleelectrodepair system which has been often used in the electronic tweezers techniques [22, 23, 33–38].
To accomplish simultaneous multifrequency measurements using the singleelectrodepair system (Fig. 1), the change in the applied electric field should be investigated. In this letter, we address the availability of timevarying DEP due to an FM wave (FMDEP) of the following form:
where the phase θ(t) of the FM wave is related to the instantaneous frequency f(t) as 2πf(t)=dθ(t)/dt and
with f_{ m } denoting the modulation frequency. We use the wide band FM satisfying that Δf/f_{ m }≫1, so that the conditions of f_{ m }/f(t), f_{ m }/f_{ c }, f_{ m }/Δf≪1 will be referred to as the wide band limit (WBL) in the theoretical formulation given below.
In this letter, particular attention is paid to the relationship between the characteristic frequency of f_{ X } and the trajectory of FMDEP. In the next section, we describe both the materials used and details of the plugin system for inducing FMDEP. The third section provides the results and discussion that consists of four parts. First, we investigate the details of repeating Uturns of a single leukemia cell by quantifying the reciprocating trajectory, whose periodicity is explained by the modulation frequency f_{ m }, or the periodical oscillation of f(t) given by Eq. (2). Next, we explain the reciprocating trajectory theoretically by deriving the timevarying dielectrophoretic force that modulates according to the instantaneous frequency f(t) of the FM field satisfying the WBL condition. The obtained form of the dielectrophoretic force provides the equation that determines the f_{ X } from the observed Uturns. Third, we measure the magnitude of the dielectrophoretic force on a multilamellar vesicle (MLV) that was attached to an electrode needle due to the attraction of the ACDEP. The frequency dependence of force was fitted using the spectral equation that was determined from the real part of the CM factor, so that f_{ X } was determined as the characteristic frequency at which the attractive force due to ACDEP vanishes. Because the FMDEP method also gives f_{ X } by analyzing the reciprocating trajectory of an MLV, we assess the extent of coincidence between the crossover frequencies evaluated from AC and FMDEPs. Finally, both of the cytoplasmic conductivities and the membrane capacitances of three kinds of cell were evaluated from f_{ X } as an increasing function of the solution conductivity, and the obtained values were compared with those reported in the literature.
Methods
Materials
For preparing multilamellar vesicles (MLVs), we used 1,2dioleoylsnglycero3phosphatidylcholine (DOPC) as lipids, purchased from Avanti Polar Lipids. The MLVs were obtained by the following procedure. The DOPC (1 mL, 20 mM) dissolved in chloroform/methanol (2:1 v/v) was dried with N_{2} gas, and the solvent was completely removed under a vacuum for more than 12 h. The thin film deposited on the glass vial due to the evaporation was rehydrated using deionized water and incubated at 25 °C for several hours.
Two cell lines used in the experiments were JKTbetadel of human T cell leukemia (TL) line and CCRFSB of human B cell leukemia (BL) line. Both kinds of TL and BL cells were used after 1 week incubation in a humidified incubator that contains 5 % CO_{2} at 37, so that we had the cell concentrations within the range of 0.5×10^{6} to 1×10^{6} cells/mL. The RPMI 1640 medium for the cell culture was supplemented with 10% fetal brovine serum and 100 mM sodium pyruvate. The cells were sedimented by centrifugation at 370g for 3 min twice so that the cells could be purely resuspended in 1 ml of the RPMI 1640 medium prior to the pipetting. The obtained cell suspensions were further diluted using the isotonic 200 mM sucrose solution in order to prepare for the solvent having a required conductivity.
We also used human red blood (RB) cells dispersed in the following suspensions. Freshly drawn whole blood samples were obtained from healthy volunteers in their early twenties. The cells, suspended in a mixture of the RPMI 1640 medium and hematocrit of 3.1%, were diluted using the isotonic 200 mM sucrose solution in order to prepare for the solvent having a required conductivity as well as the above leukemia cells. All of the dielectrophoretic experiments using the human RB cells were finished within 10 mins after drawing the whole blood samples.
Experimental Setup
The conductivities of the cell suspensions were measured using a conductivity meter (SevenMulti, MettlerToledo, Columbus, OH, USA). A schematic of the plugin system used is shown in Fig. 1. An external electric field with an AC or FM wave was applied via an arbitrary waveform generator (Agilent 33220A, Agilent Technologies, Santa Clara, CA, USA) with a current amplifier (F30PV, FLC Electronics, Partille, Sweden) to which plugintype microelectrodes were connected. The microelectrodes comprised tungsten needles with a tip diameter of 0.5 μm that were independently controlled by two sets of patchclamp micromanipulators (NMN21, Narishige, Setagayaku, Tokyo, Japan). In all of the experiments that follow, we maintained the tip separation at 100 μm when applying the external fields to the above suspensions, and the maximum magnitude was set to be 0.5 kV/cm. The needle pair was inserted into a sample drop mounted on the inverted optical microscope (TE2000U, Nikon, Minatoku, Tokyo, Japan), and the optical micrographs were obtained using a CCD camera (Retiga Exi, QImaging, Surrey, British Columbia, Canada) with a frame rate of 25 fps; incidentally, it was confirmed that the frequency resolution of FM waves due to the frame rate was always within the error bars for each data. A 50 μl drop of suspension was mounted on the sample stage of the inverted optical microscope, whose temperature was maintained at 25 °C using a heat controller.
The plugin technique allows the simple system to perform various noncontact manipulations of a single cell, such as pushing it into a narrow channel without any contact and orienting it toward the desired direction. Although it is often necessary to treat the cells in an isotonic solution with salt, it is easiest to implement the above DEP manipulations of cells surrounded by deionized water. In Additional file 1: Movies S1 to S3, the plugin system induced the ACDEP of diatom cells suspended in deionized water. We can see from Additional file 1: Movies S1 to S3 that an anisotropic diatom cell dispersed in saltfree water was manipulated like a postit tag by a pair of microelectrodes between which the AC electric field (1 kV/cm) was applied. The noncontact operations consist of three steps: (i) a target cell was first rotated in parallel to a glass wall positively charged by the combination of dipole alignment at a frequency of 30 kHz and positional change of each microelectrode (Additional file 1: Movie S1), (ii) we subsequently changed the frequency to 100 kHz for pushing it toward the wall to fix the request cell with negative charges on the glass surface electrostatically (Additional file 1: Movie S2), and (iii) the AC frequency was adjusted to 20 MHz for inducing the ACDEP in the opposite direction, so that the electrostatically attached cell could be pulled out (Additional file 1: Movie S3).
Results and Discussion
Experimental Observation of a Leukemia Cell Experiencing FMDEP
Our plugin microelectrodes (see Fig. 1) allow the electric field to be applied to the particles floating far above the sample substrate, which is of practical use for selecting appropriate cells. For instance, Additional file 1: Movie S4 shows that the microelectrode pair was controlled to approach a floating triangular diatom cell to which we applied the AC electric field with its frequency jumping between 100 and 500 kHz at intervals of 0.5 s. In Additional file 1: Movie S4, we see the triangular cell bouncing on a microelectrode due to the frequency jumping as a preliminary result in advance of the following manipulation using the FMDEP.
Additional file 1: Movies S5 and S6 shows typical behaviors of several TL cells experiencing the FMDEP, which are similar to those of mammalian cells manipulated by an electronic tweezers using a single electrode ACDEP [36]. Figure 2 depicts one of the periodic trajectories using the 3D plot of (x,y) along the t axis, where a relative coordinate of (x,y) is assigned to the temporary cell position with the origin of (0, 0) located at a specific point on a microelectrode needle for extracting the cellelectrode configuration. While the x axis represents the tangent to the electrode surface at (0, 0), the y axis, perpendicular to the tangent, mainly reflects the projection of the periodic Uturns explained below. In Fig. 2, we selected a floating TL cell to which we applied the FM electric field with its modulation frequency f_{ m } set to be f_{ m }=0.25 Hz in the range of 200 kHz ≤f(t)≤ 3 MHz. Because we have that Δf/f_{ m }, f(t)/f_{ m }<10^{−5}, the WBL condition actually holds, as mentioned after Eq. (2).
It is found from Additional file 1: Movies S5 and S6 as well as Fig. 2 that the periodic trajectory is constituted by three parts of leaving, approaching, and staying on the microelectrode: (i) the cell leaves the microelectrode, (ii) it approaches the microelectrode after doing a Uturn, and (iii) it stays on the microelectrode surface. The cell is often unable to return to the same position on the microelectrode surface because of the solvent flow, which is not only observed in Additional file 1: Movie S6, but is also represented by the Uturns with the cell migrating in the x direction in Fig. 2. Despite the interference with the solvent flow, it is possible to distinguish the moments when the cell starts to leave the microelectrode surface and does the Uturn in the periodic trajectory, respectively. Accordingly, we can see from Fig. 2 that these Uturns are repeated at intervals of 4 s in coincidence with the modulation frequency of 0.25 Hz, or the 4s period of the instantaneous frequency f(t).
Theoretical Study on the FMDEP
To explain the experimental trajectories, including the periodic Uturns, we consider a spherical object as a simplified model of a single cell, to which an arbitrary timevarying electric field E(r,t) is applied. Figure 3 shows a schematic of the timedependent DEP force acting on a spherical object [9]. As shown in Fig. 3, the permittivity and conductivity inside a spherical object are represented by ε_{in} and σ_{in}, respectively, and the subscript “out,” such as ε_{out} and σ_{out}, denotes the outside. In general, F_{DEP}(r,t) is related to the induced dipole moment p(r,t) as [17–19]
where K_{ H } and Δτ are defined as follows: K_{ H }=(ε_{in}−ε_{out})/(ε_{in}+2ε_{out}), and \(\Delta \tau ^{1}=\tau _{0}^{1}\tau ^{1}\) using the radius R of the spherical object and two characteristic times of τ_{0}=(ε_{in}−ε_{out})/(σ_{in}−σ_{out}) and τ=(ε_{in}+2ε_{out})/(σ_{in}+2σ_{out}).
Substituting the AC electric field E_{AC}(r,t)=A(r) cos(2πf_{AC}t) into Eqs. (3) to (5), we obtain the mean DEP force <F_{DEP}(r,t)> that has been averaged over cycles of the AC field [9, 15, 20]:
where A_{RMS} denotes the root mean squared (RMS) vector satisfying that \(\boldsymbol {A}_{\text {RMS}}^{2}=\boldsymbol {A}^{2}/2\), and χ(f_{AC})≡2πR^{3}ε_{out}Re[K(f_{AC})] depends on the applied frequency f_{AC} due to Re[K(f_{AC})], the real part of the CM factor [9, 15, 20]:
where K_{ L }=(σ_{in}−σ_{out})/(σ_{in}+2σ_{out}) and K_{ H }, defined above, correspond to the real CM values in the low and highfrequency limits, respectively, and these limiting values, K_{ L } and K_{ H }, need to have opposite signs so that f_{ X } defined by χ(f_{ X })=0 may exist [9, 15, 20].
Equations (6) and (7) indicate that the AC electric field creates the DEP force whose direction depends on the applied frequency f_{AC} through χ(f_{AC}) given by Eq. (7), which explains the bouncing diatom cell in Additional file 1: Movie S4 as follows (see also Fig. 3). When the applied frequency provides the plus sign of the real part of the CM factor (i.e., χ(f_{AC})>0), we can observe cells attracted toward the electrode needle tips (the positive DEP) on which the strength of the AC field, applied via a pair of electrode needles, is the largest. The sign of the real CM factor can be reversed to the negative at f_{ X }, the vanishing frequency of the real CM factor (i.e., χ(f_{ X })=0), where we have the zero dielectrophoretic force as found from Eq. (6). In the negative sign of the CM factor (i.e., χ(f_{AC})<0), individual colloids are repelled from the electrode needle pair (the negative DEP). The triangular diatom cell in Additional file 1: Movie S2 bounced because of the opposite direction of the ACDEPs induced by the AC fields with their frequencies of 100 and 500 kHz; combining Eq. (6) and the dielectrophoretic directions observed, we find that χ(100 kHz)>0 and χ(500 kHz)<0.
Next, we consider the FMDEP by plugging the phase given by Eqs. (1) and (2) into Eqs. (3) to (5). As proved in Additional file 2, the WBL condition of the FM wave validates the approximate form of the integration in Eq. (5), thereby providing
which becomes the same form as that of the ACDEP when the timedependent frequency f(t) is replaced by a constant frequency of f_{AC}. We thus obtain the limiting form of the mean DEP force <F_{DEP}(r,t)> that has been averaged over cycles of θ(t) in the FM field (see Eqs. (A1), (A13) and (A14) in Additional file 2):
being of a similar form to Eq. (6) for the ACDEP. The difference is whether or not the coefficient of χ{f(t)} depends on t through f(t), which changes cyclically according to the frequency modulation with the period of T_{ m }=1/f_{ m }.
Based on the simple expression (9) of the FMDEP, we illustrate with Fig. 4 the mechanism of the above Uturns due to the FM wave. Figure 4 shows a schematic of the DEP induced by the FM wave in the WBL when the range of f(t) covers the crossover frequency f_{ X } such that f_{ c }−Δf≤f_{ X }≤f_{ c }+Δf. It is supposed in Fig. 4 that the frequency dependence of the real part of the CM factor, or χ{f(t)}, provides the alternate sign changes as follows: minus sign (χ{f(t)}<0) for f(t)<f_{ X } and plus sign (χ{f(t)}>0) for f(t)>f_{ X }, which is the case with our experiments. The former period satisfying f(t)<f_{ X } has a duration time, whereas the latter f(t)>f_{ X } has been retained during the rest the period: one cycle is classified into two periods that are marked red and blue, respectively, in Fig. 4.
Similarly to the ACDEP, Eq. (9) implies that the minus sign (χ{f(t)}<0) creates a repulsive DEP force between the cell and the microelectrodes while satisfying that f(t)<f_{ X }. As a result, the cell leaves the area around the micfroelectrode needle tips between which the magnitude of the electric field is the largest: the cell experiences the negative DEP during the red period of Δt_{ n } in Fig. 4. At the instant t_{ X } as a solution of f(t_{ X })=f_{ X }, χ(f) vanishes, followed by the sign change into χ(f)>0 while f(t)>f_{ X }, and correspondingly the DEP force is switched to the attractive force at t_{ X }. After doing a Uturn at t_{ X } due to the reversal in direction of the DEP force, the targeted cell starts to approach the microelectrode migrating in the opposite direction and is eventually trapped between the tips of the electrode needles or attached on one of the electrodes: the cell experiences the positive DEP during the blue period of Δt_{ p } in Fig. 4. Figure 4 indicates that the cycle of leaving, approaching and staying on the microelectrode should be repeated with the modulation period of T_{ m }, in agreement with Fig. 2: Δt_{ n }+Δt_{ p }=T_{ m }. The dielectrophoretic mechanism depicted in Fig. 4 can thus explain the periodic Uturns observed in Additional file 1: Movies S5 and S6 as well as Fig. 2.
Let us consider the periodic solution of the equation, f(t_{ X })=f_{ X }. As seen from Fig. 4, t_{ X } is expressed as t_{ X }=nT_{ m }+0.5Δt_{ p }=nT_{ m }+0.5(T_{ m }−Δt_{ n }) using an integer of n=0, ±1, ±2,⋯, which further reads
Substituting Eq. (10) into Eq. (2), we have for n=0 that
clarifying that the FMDEP method determines the crossover frequency if the duration time Δt_{ n } from leaving the microelectrode to doing the Uturn can be measured precisely.
Comparing Crossover Frequencies of a Single MLV Determined from the FM and ACDEPs
We investigated the experimental accuracy of Eq. (11). Experimentally, it is often necessary for the biological cells to be dispersed in an electrolyte. For MLVs, however, the use of deionized water is allowed during the preparation process of rehydration and dilution. We thus used the saltfree MLV suspension for comparing the crossover frequencies determined from both AC and FMDEPs.
The dielectrophoretic Uturns of a targeted MLV were induced by the FM wave in the range of 10 kHz ≤f(t)≤ 50 kHz (i.e., f_{ c }=30 kHz and Δf=20 kHz) with a setting that f_{ m }=0.1 Hz, and correspondingly the FMDEP has a 10s period. In the experiments, it takes less than 30 s to observe a few Uturns of the targeted MLV from leaving to approaching microelectrodes. From the trajectory, we obtained the mean leaving time that \(\overline {\Delta t_{n}}=5.8\pm 0.2\) s. Because the WBL condition applies to the present experiment satisfying that f_{ m }/Δf/f_{ m }, f_{,}m/f(t)<10^{−5}, the crossover frequency was evaluated to be f_{ X }=35±1 kHz from substituting \(\overline {\Delta t_{n}}=5.8\pm 0.2\) s into Eq. (11).
For comparison, we made use of the programmable manipulator in the ACDEP method that tries to evaluate the crossover frequency of the same targeted MLV to which the sinusoidal electric field with a frequency in the range of 30 to 100 kHz was applied via the electrode needle pair for inducing the ACDEP. Because the programmable manipulator carries the electrode needle pair at a constant speed in one direction, we can measure the dielectrophoretic force similarly to the lasertrapping experiments [39]. Attaching the MLV on an electrode tip that undergoes uniform linear motion, not only the ACDEP force but also the hydrodynamic force caused by the onedimensional motion are exerted on the MLV. With the gradual increase of electrode velocity, F_{DEP} eventually becomes smaller than the hydrodynamic force. As a result, the MLV initially attached to the moving electrode, owing to the DEP attraction, is desorbed by the hydrodynamic force. Defining the critical value, v_{ c }, by the maximum velocity value of the microelectrode pair prior to the desorption, the force balance equation between the DEP and hydrodynamic forces reads [39]
where F_{DEP}(f_{AC})e≡<F_{DEP}> with the unit vector e defined by \(\boldsymbol {e}=\nabla {\boldsymbol {A}}^{2}_{\text {RMS}}/\nabla {\boldsymbol {A}}^{2}_{\text {RMS}}\), η the water viscosity at 25 °C and 2R the diameter of the MLV.
Additional file 1: Movies S7 and S8 demonstrates the force measurement using the above ACDEP method at the applied frequency of f_{AC}= 60 kHz. In Additional file 1: Movie S7, the velocity of the electrode pair controlled by the programmed manipulator is 110 μm/s, which is lower than v_{ c }; therefore, the MLV remains attached to one part of the electrode pair owing to the dielectrophoretic attraction. Additional file 1: Movie S8, on the other hand, shows the higher electrode speed of 120 μm/s, under which the dielectrophoretic force becomes smaller than the hydrodynamic force that is exerted on the MLV, thereby desorbing the MLV from the electrode. Accordingly, v_{ c } is evaluated to be 110 μm/s ≤v_{ c }≤ 120 μm/s, and we can calculate F_{DEP}(60 kHz) using Eq. (12).
We can determine f_{ X } from the experimental results of F_{DEP} at various external frequencies. Figure 5 shows the frequency dependence of F_{DEP}, indicating that the DEP force experienced by the MLVs was reduced by lowering the applied frequency. It is found from Eqs. (6) and (7) that the fitting function of F_{DEP}(f_{AC}) can be expressed as
implying that
Equation (13) is depicted by the solid line in Fig. 5 that has been fitted to the experimental data using the bestfit results of three parameters: L=−21.02 pN, H=19.03 pN, and τ=4.9 μs. Substituting these results into Eq. (14), we evaluate that f_{ X }= 34.15 kHz, which coincides with the result of f_{ X }=35±1 kHz evaluated from the FMDEP method. The FMDEP method is thus validated in terms of the consistency with the direct force measurement using the ACDEP method.
Conductivity Dependencies of the Crossover Frequencies for Biological Cells
Let us return to the dielectrophoretic Uturns of biological cells mentioned in Fig. 2 to assess the practical reliability of the crossover frequencies when the FMDEP method is applied to cell suspensions. Recently, an elaborate theory [40] has investigated, in more detail than before, the relationship between the homogeneous sphere model (see Fig. 3) and the singleshell model where the inner structure of cell is represented by a smearedout cytoplasm surrounded by a membrane. As a result, the relation between f_{ X } and the suspension conductivity σ_{out} has been formulated using radius R of a cell, membrane capacitance C_{ m }, and cytoplasmic conductivity σ_{cyt} [40]:
where f_{X0} is the extrapolated value to the crossover frequency at σ=0 mS/m and will be treated as a fitting parameter herein. The elaborate treatment adds the squared term, the second term on the right hand side of Eq. (15), to the conventional linear relation which has mainly been used for evaluating C_{ m } from f_{ X } [40–45]. Theoretically, it has still been claimed [40] that Eq. (15) is valid within a lower range of σ_{out} such that σ_{out}<10 mS/m; however, it should be better to include the squared term in the evaluation of C_{ m }, considering that our range of σ_{out} is relatively high compared with previous results in the range of 10 mS/m ≤σ_{out}≤ 100 mS/m [40–45]. Hence, we determined σ_{cyt} as well as C_{ m } from fitting Eq. (15) to the experimental results of f_{ X } as an increasing function of σ_{out}.
There are three kinds of biological cell used: TL and BL cells of human leukemia and RB cells of three human volunteers. In all the experiments using any species of cell, the conductivities were within the range of 60 to 160 mS/m, and the modulation frequency was set to be 0.25 Hz. Regarding the instantaneous frequency, most of the experiments adopted the range from 100 to 1.5 MHz (i.e., f_{ c }= 800 kHz and Δf=700 kHz); exceptionally for leukemia cells, the frequency range was extended to 50 kHz ≤f(t)≤1550 kHz (i.e., f_{ c }=800 kHz and ΔfX=750 kHz) in the conductivity range of 60 mS/m≤σ≤80 mS/cm because f_{ X } in this σrange has been found to be lower than 100 kHz, and we were unable to observe the DEP Uturns in the range of 100 kHz ≤f(t)≤1500 kHz. Both of these frequency sets satisfy the WBL condition of Δf/f_{ m }, f(t)/f_{ m }<10^{−5} as before.
Each time we measured the leaving times of cells dispersed in a suspension, we looked for an appropriate spot at which a few cells having a similar size could simultaneously experience the FMDEP above the substrate, and the microelectrode tips were placed at the measurable position using the micromanipulator. We continued such scanning inside the cell suspensions until the FMDEP trajectories of 10 cells were collected in total at a couple of appropriate positions. For each kind of cell, the measurement of 10 cells was repeated twice using different drops of the same cell suspension. As mentioned, it is indispensable for the implementation of the FMDEP measurement at each spot to suppress the electrically induced solvent flows as much as possible. Hence, we traced only two cycles of the Uturn path so that the duration time of applying the electric field could be adjusted to be less than 10 s, and, correspondingly, the leaving time of each cell is given as the average of each trajectory, including the two Uturns. The mean leaving time \(\overline {\Delta t_{n}}\) of each cell suspension is thus obtained from averaging the leaving times of 20 cells. Particularly for human RB cells, we further averaged three sets of the mean crossover frequencies obtained for three RB cell suspensions of three human beings, supposing that cells of the same species are similar in C_{ m } and σ_{cyt} as well as in R. The twostep averaging of Δt_{ n } will be denoted by \(\left <\overline {\Delta t_{n}}\right >\). Substituting into Eq. (10) the experimental data of either \(\overline {\Delta t_{n}}\) or \(\left <\overline {\Delta t_{n}}\right >\), the mean crossover frequency <f_{ X }> was obtained.
Figure 6 shows the σ_{out}dependencies of <f_{ X }> measured for the above three kinds of biological cells using the FMDEP method. The solid lines in Fig. 6 depict the bestfit results of Eq. (15). We evaluated C_{ m } and σ_{cyt} from the best fitting of Eq. (15) into which the observed radii (R_{obs}) were inserted. Table 1 lists the fitting results of C_{ m } and σ_{cyt}, where we used the observed radii of 10 μm≤ 2R_{obs}≤ 15 μm for TL and BL cells, and 7.5 μm≤ 2R_{obs}≤ 10 μm for RB cells in evaluating C_{ m }. It is to be noted from Table 1 that different species have different membrane capacitances, which are in good agreement with those reported in the literature [40–47]; the C_{ m } values of RB cells with stationary whole blood samples from normal (healthy) donors are in excellent agreement with our value [46, 47], but are substantially higher than those of washed RB cells in isotonic buffered saline as noted in [47]. The bestfit results simultaneously provided cytoplasmic conductivities, which were consistently similar as seen from Table 1, but were slightly lower than the range of previous reports that 0.2 S/m ≤σ_{cyt}≤1 S/m [40, 45, 48–51]. These results support that the FMDEP method retains the practical reliability needed for the treatment of living cells.
Conclusions
Our theoretical treatment of the FMDEP has mainly focused on the WBL condition. In this limit, we have proved theoretically that the direction of the FMDEP force switches each time when the instantaneous frequency of the FM wave traverses the crossover frequency, thereby implying the periodic Uturns of micro/nanoparticles that undergo the FMDEP. Two kinds of experiment have demonstrated the accuracy and reliability of f_{ X } obtained from the observed trajectories of MLVs and cells using our formulation of the FMDEP (Eqs. (9) and (11)): While the f_{ X } evaluated from the FMDEP of a single MLV coincides with that obtained from the force measurement of the same MLV experiencing ACDEP, the conductivity dependencies of f_{ X } provide the membrane capacitances of various cells that are in close agreement with the literature values. In other words, it has been validated theoretically and experimentally that the FMDEP in the WBL limit can be mimicked by the timevarying ACDEP induced by the AC wave with its frequency changing continuously according to the periodic function of f(t). The simple view applies to other electrokinetics, including the twDEP and the electrorotation by applying the FM wave that has the spatial dependence of the phase as well as the magnitude. The AC and FMDEPs are associated with the real part of the dielectric spectra (or the CM factor), whereas the electrokinetics due to the spatial gradient of the phase reflect the imaginary part of the CM factor as mentioned before. Therefore, the application of the FM wave to either twDEP or electrorotation will be required for completing the dielectric characterization (the dielectric spectroscopy, in general) using the electrokinetics.
We have treated microparticles such as MLVs and cells for the precise tracking of particle trajectories. In these experiments, sedimented particles as well as floating ones have been observed; we need to increase the magnitude of electric field for inducing the DEP of the sedimented particles which are likely to be aggregated. Accordingly, we have used the plugin system for applying the FM wave to a targeted particle floating above the substrate.
It is promising to further develop the FMDEP method for smaller particles with their sizes of submicron to nanoscale, such as dispersed carbon nanotubes, thereby opening up the possibility of realtime spectroscopy using the FMDEP as described below. When we apply the FM wave to the smaller colloids using the onchip systems whose electrode configuration is designed to create a constant gradient of the applied electric field, the timevarying velocity vector v(t) of the FMDEP caused by the time dependence of the FMDEP force is ascribed to the variation in χ(f) (or the real part of the CM factor): it is found from Eqs. (9) and (12) that
Hence, measuring the velocity vector v(t) of a submicron to nanoparticle could provide the frequency dependence of the real part of the CM factor directly, which would be nothing but the electrokinetic FM spectroscopy.
Abbreviations
 AC:

Alternating current
 BL:

B cell leukemia
 CM:

ClausiusMossotti
 DEP:

Dielectrophoresis
 DOPC:

1,2Dioleoylsnglycero3phosphatidylcholine
 FM:

Frequency modulated
 MLV:

Multilamellar vesicle
 RB:

Red blood
 RMS:

Root mean squared
 TL:

T cell leukemia
 twDEP:

Traveling wave dielectrophoresis
 WBL:

Wide band limit
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Acknowledgements
We are grateful to Mr. Yoshihiro Hatajiri, Mrs. Misuzu Matsuura, and Mrs. Yuki Yajima from Kochi University of Technology for the experimental help.
Funding
This research was partly supported by the Yamada Science Foundation.
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The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
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HF formulated the theory, designed and performed the experiments, analyzed the data, and wrote the manuscript. The author read and approved the final manuscript.
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HF is a professor at the School of Environmental Science & Engineering, Kochi University of Technology, TosaYmada, Japan.
Additional files
Additional file 1
Eight movie clips of an elongated diatom cell manipulated like a postit tag without contact to a microelectrode pair (Movies S1–3), a triangular diatom cell reciprocated on a microelectrode due to ACDEP (Movie S4), human T cell leukemia cells reciprocated on a microelectrode due to FMDEP (Movies S5 and S6), and a multilamellar vesicle (MLV) attracted to a moving microelectrode (Movies S7 and 8). Movie S1: Rotating a diatom cell by noncontact manipulation using the combination of dipole alignment and positional change of each microelectrode (alignment operation). Movie S2: Pushing a diatom cell toward a glass wall by noncontact manipulation to fix the request cell with negative charges on the glass surface electrostatically (paste operation). Movie S3: Pulling off a diatom cell from the glass wall by noncontact manipulation (peeloff operation). Movie S4: Periodic Uturns of a diatom cell due to ACDEPs induced by frequency jumps of applied sinusoidal electric fields. Movie S5: Periodic Uturns of leukemia cells due to FMDEP. Movie S6: Periodic Uturns of another leukemia cell due to FMDEP under solvent flow. Movie S7: Carrying an MLV attached to one part of the microelectrode pair moving with lower speed. Movie S8: Desorption of an MLV attached to one part of the microelectrode pair moving with higher speed. (ZIP 27000 kb)
Additional file 2
Derivation of Eqs. (8) and (9) in the wide band limit (WBL). We provide the details in obtaining the approximate expression of Eqs. (8) and (9). Our focus is on clarifying how the integral in Eq. (5) is reduced to a simple form in the WBL, as seen from Eqs. (A2), (A3) and (A10). (PDF 33 kb)
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Frusawa, H. FrequencyModulated Wave Dielectrophoresis of Vesicles And Cells: Periodic UTurns at the Crossover Frequency. Nanoscale Res Lett 13, 169 (2018). https://doi.org/10.1186/s1167101825835
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Keywords
 Dielectrophoresis
 Frequencymodulated wave
 The ClausiusMossotti factor
 Crossover frequency
 Cell
 Vesicle
 Spectroscopy