Viscoelastic properties of suspended cells measured with shear flow deformation cytometry
Abstract
Numerous cell functions are accompanied by phenotypic changes in viscoelastic properties, and measuring them can help elucidate higherlevel cellular functions in health and disease. We present a highthroughput, simple and lowcost microfluidic method for quantitatively measuring the elastic (storage) and viscous (loss) modulus of individual cells. Cells are suspended in a highviscosity fluid and are pumped with high pressure through a 5.8 cm long and 200 µm wide microfluidic channel. The fluid shear stress induces large, near ellipsoidal cell deformations. In addition, the flow profile in the channel causes the cells to rotate in a tanktreading manner. From the cell deformation and tank treading frequency, we extract the frequencydependent viscoelastic cell properties based on a theoretical framework developed by R. Roscoe ^{1} that describes the deformation of a viscoelastic sphere in a viscous fluid under steady laminar flow. We confirm the accuracy of the method using atomic force microscopycalibrated polyacrylamide beads and cells. Our measurements demonstrate that suspended cells exhibit powerlaw, soft glassy rheological behavior that is cell cycledependent and mediated by the physical interplay between the actin filament and intermediate filament networks.
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Author Response
Reviewer 2
In the manuscript, the cellular deformation that is due to the shear stress generated in a classical microfluidic channel is used to deform detached cells that are moving in the flow. A very elegant point of the paper is that the same cells are used in the provided software to determine the fluid flow, which is a key parameter of the method. This is particularly important, as an independent way to crosscheck the fluid flow with the expected values is important for the reliability of the method. Instead of complicated shape analysis that are required in other microfluidic methods, here the authors simply use the elongation of the cell and the orientation angle with respect to the fluid flow direction. The nice thing here is that a wellknown theory from R. Roscoe can be successfully used to relate these …
Author Response
Reviewer 2
In the manuscript, the cellular deformation that is due to the shear stress generated in a classical microfluidic channel is used to deform detached cells that are moving in the flow. A very elegant point of the paper is that the same cells are used in the provided software to determine the fluid flow, which is a key parameter of the method. This is particularly important, as an independent way to crosscheck the fluid flow with the expected values is important for the reliability of the method. Instead of complicated shape analysis that are required in other microfluidic methods, here the authors simply use the elongation of the cell and the orientation angle with respect to the fluid flow direction. The nice thing here is that a wellknown theory from R. Roscoe can be successfully used to relate these quantities to the viscoelastic shear modulus. Thanks to the knowledge of the fluid flow profile, the mechanical properties can be related to the tank treading frequency of the cells, which in turn depends on the position in the channel, and the flow speed. Hence, after knowing the flow profile, which can be determined with a sufficiently fast camera, and the actual static cell shape, it is possible to obtain frequency dependent information. Assuming then that cells do have a statistically accessible mean viscoelastic property, the massive and quick data acquisition can be used to get the shear modulus over a large span of frequencies.
The very impressive strength of the paper is that it opens the door for basically any, nonspecialized cell biology lab to perform measurements of the viscoelastic properties of typically used cell types in solution. This allows to include global mechanical properties in any future analysis and I am convinced that this method can become a main tool for a rapid viscoelastic characterization of cell types and cell treatment.
Although it is both elegant and versatile, there remain a couple of important questions open to be further studied before the method is as reliable as it is suggested by the authors. A main problem is that the model and the data simply don't really work together. This is most prominent in Figure 3a. This is explained by the authors as a result of nonlinear stress stiffening. Surely this is a possible explanation, but the fact that the question is not fully answered in the paper makes the whole method seems not sufficiently backed. I agree that the test with the elastic beads are beautiful, but also here the results obtained with the microfluidic method and the AFM seem not to match sufficiently to simply use the proposed model in conjecture with a single power law approach to fully translate the single frequency data into a frequency dependent plot. There are more and more hints that two power law models are more reasonable to describe cell mechanics. If true this would abolish the approach to exploit only a single image to get the mechanical power law exponent and the prefactor in a single image. Despite all the excitement about the method, I have the feeling that the used models are stretched to their extreme, and the fact that the only real crosscheck (figure 3a) does not work for the power law exponent undermines this impression.
We had assumed that the probing frequency equals the tank treading frequency. This is incorrect. As the cell undergoes a full rotation, any given volume element inside the cell is compressed twice and elongated twice. Hence, the frequency with which the cell is probed is twice the tanktreading frequency. This correction shifts the G’ and G” versus frequency curves to the right (by a factor of two), and in addition, the G” data points are shifted (increased) by a factor of two (Eq. 17). This also increases the fluidity alpha (and hence the slope of the powerlaw relationship) roughly by a factor of two (Eq. 22), and since the actual slope of the G’ and G” versus frequency data “cloud” is unchanged by the correction, the single powerlaw description now describes the data much better (see new Fig. 3a).
Regarding the critique that models are stretched to their extreme: The Roscoe model assumes that cells behave as the viscoelastic continuummechanics equivalent of a KelvinVoigt body consisting of an elastic spring in parallel with a resistive (or viscous) dashpot element . This then gives rise to a complex shear modulus with storage modulus G’ and loss modulus G”, measured at twice the tank treading frequency 𝜔. Roscoe makes no assumptions whatsoever about how G’ and G” might change as a function of frequency. Hence, our “raw” G’ and G” data, e.g. in Fig. 3a, are obtained without any power law assumption.
One could leave it at that, as the reviewer suggests below, and only present the raw G’ and G” vs. frequency plots. However, this would also make it nearly impossible to compare our measurements to those obtained with other techniques that operate at different, nonoverlapping time or frequencyscales. For such a comparison to work, one needs a model to predict how G’ and G” scale with frequency.
A commonly used and very simple model to predict how G’ and G” scale with frequency, which is also the model used by Fregin et al. and many others, is that of a KelvinVoigt body consisting of an elastic spring in parallel with a resistive element (dashpot), both with a frequencyindependent stiffness and resistance (viscosity), respectively. However, our data show that G’ and G” of different cells, all measured at different tanktreading frequencies, exhibit a behavior that is very unlike that of a simple KelvinVoigt body with a constant, frequencyindependent stiffness and resistance. In this case, G’ would be flat (power law exponent zero), and G” would increase proportional with frequency (power law exponent of unity). This is clearly not what our data show.
Rather, we find that G’ and G” increase with increasing frequency according to a power law, with the same exponent 𝛼 for G’ and G”. At high frequencies (beyond the range of our microfluidic method, but in the range of our AFM measurements), G” increases more strongly with frequency, akin to a Newtonian viscosity (power law exponent of unity), which we take into account in the case of the AFM measurements. A large number of publications have shown that many types of cells, including cells in suspension, follow power law rheology, regardless of the measurement method. Also the AFM measurements that we include in this study support the validity of powerlaw rheology.
Power law rheology predicts a peculiar behavior: The ratio of G”/G’ in the lowfrequency regime (where the highfrequency viscous term is not yet dominating) must be equal to tan(𝛼𝜋/2), for mathematical reasons (Eq. 22). With our correction (that the probing frequency is twice the tanktreading frequency), we find that Eq. 22 correctly predicts the powerlaw exponent of the G’ and G” vs. frequency data.
Note that we actually do not fit a power law model (Eq. 1) to the population data of G’ and G” vs. frequency in Fig. 3a. The G’ and G” data are obtained by applying Roscoetheory, without any further assumptions such as powerlaw rheology. Only the lines shown in Fig. 3a that go nicely through the data are a prediction of how a typical cell (selected from the mode of the joint probability density of alpha and k, see Fig. 3b) would behave if we had measured it at different frequencies, under the assumption that this cell follows power law rheology, based on Eq. 22. With this assumption, we can directly convert the measured G’ and G” of any cell into a stiffness k and power law exponent 𝛼 using Eqs. 21 and 22  no fit is needed here.
Since we only measure two parameters for any given cell at twice its tanktreading frequency, namely strain and alignment angle, we can only extract two parameters for each cell (i.e., G’ and G”, or k and alpha) but not a third parameter. In essence, the reviewer expresses concerns that the G' and G" behavior of a typical cell, when extrapolated to higher or lower frequencies, may not necessarily match the frequency behavior of the entire cell population (Fig. 3a). However, our data show that a single (typical) cell that was measured at a single midrange frequency comes remarkably close to describing the G’ and G” versus frequency behavior of all other cells.
The reviewer suggests that a power law model with two exponents may be able to even more accurately describe the mechanics of the cell population. This is certainly correct, and in particular when cell mechanics is measured over a larger range of frequencies or strain rates, as we have done here using AFM, we find that at higher frequencies, G” deviates from a weak power law and merges into a different power law with a larger slope (i.e., power law exponent) that approaches unity or a value close to unity, akin to a Newtonian viscous term. Therefore, the single power law expression (Eq. 1) is not sufficient for the AFM data, and we use Eq. 2 instead. However, in the case of our shear stress cytometry measurements, the tanktreading frequency remains below the range where this second power law behavior becomes prominent. Therefore, the Newtonian viscosity term of Eq. 2 cannot be fitted with reasonable fidelity to the data from a single measurement.
In the case of polyacrylamide beads, we start to see a hint of an upward trend in G” versus frequency at tanktreading frequencies of around 10 Hz, and therefore have performed a global fit with Eq. 2 to the shear flow data where we keep the Newtonian viscous term constant for all conditions (different shear stresses and bead stiffnesses).
The reviewer furthermore cautioned that mechanical nonlinearities such as strain stiffening may distort or otherwise bias the results. As the reviewer brings up this issue in more detail below, we have addressed it there.
Regarding the concern that “results obtained with the microfluidic method and the AFM seem not to match sufficiently to simply use the proposed model in conjecture with a single power law approach to fully translate the single frequency data into a frequency dependent plot.”:
First, we tend to agree more with the opinion of Reviewer #1 who found it remarkable that results obtained with the microfluidic method and the AFM method are actually fairly similar. Now that we have introduced the correction that the probing frequency is twice the tanktreading frequency, the cells in suspension turn out to be softer and more fluidlike compared to the cells measured with AFM. But there are many more commonalities between the AFM data and the shear flow data, which we list above in our reply to reviewer #1, the most relevant here is that cells show powerlaw behavior both when measured with AFM and with our new method.
Second, we did not use a single power law to fit the AFM data. Rather, we used Eq. 2, which contains two power law relationships (the second power law exponent of unity for the Newtonian viscosity therm is usually not explicitly written). However, the origin of the Newtonian viscosity therm arises mainly from the hydrodynamic drag of the cantilever with the surrounding liquid, and less so from the cells. This hydrodynamic drag is absent in our shear flow deformation cytometry method, and moreover the tank treading frequency of most cells remains far below 10 Hz where an additional Newtonian viscosity therm does not yet come into play.
Third, we disagree that Fig. 3a is “the only real crosscheck for the power law exponent”. The inverse relation that we see between the power law exponent and the stiffness of individual cells (Fig. 3b) has been previously reported for different cell types and methods. Moreover, we find a power law exponent close to zero for PAA beads at small strain values, which is to be expected for a predominantly elastic material such as PAA. We think that this last result is a particularly convincing experimental crosscheck.

Evaluation Summary:
This paper describes a microfluidic approach to determine the viscoelastic properties of living cells from their deformation in a fluid flow. Its implementation seems accessible and the method offers the possibility to perform measurements on a large number of cells. This technique could eventually be used in many laboratories, including those not specialized in cell mechanics.
(This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. Reviewer #1 and Reviewer #2 agreed to share their name with the authors.)

Reviewer #1 (Public Review):
This article presents an assay to measure the viscoelastic properties of living cells based on their deformation and tank treading motion in a viscous flow. This experimental technique could indeed be easy to implement and thus be adapted in other labs. The analysis of the experiment, measuring G' and G' and the derived quantities, for instance, is not so straightforward but the authors publicly share their analysis tools. They validate their approach by comparing their results to similar measurements made by AFM on elastic PAAm beads and THP1 cells, the two techniques give very close values of the mechanical parameters. Once the robustness of the technique is established, they apply it to three case studies: the doseresponse of cells to Latrunculin, the effect of cytochalasin on intermediate …
Reviewer #1 (Public Review):
This article presents an assay to measure the viscoelastic properties of living cells based on their deformation and tank treading motion in a viscous flow. This experimental technique could indeed be easy to implement and thus be adapted in other labs. The analysis of the experiment, measuring G' and G' and the derived quantities, for instance, is not so straightforward but the authors publicly share their analysis tools. They validate their approach by comparing their results to similar measurements made by AFM on elastic PAAm beads and THP1 cells, the two techniques give very close values of the mechanical parameters. Once the robustness of the technique is established, they apply it to three case studies: the doseresponse of cells to Latrunculin, the effect of cytochalasin on intermediate filamentdeficient cells and the effect of the cell cycle on cell mechanics.
I have no criticism on the experiments and their analysis that are very convincing.

Reviewer #2 (Public Review):
In the manuscript, the cellular deformation that is due to the shear stress generated in a classical microfluidic channel is used to deform detached cells that are moving in the flow. A very elegant point of the paper is that the same cells are used in the provided software to determine the fluid flow, which is a key parameter of the method. This is particularly important, as an independent way to crosscheck the fluid flow with the expected values is important for the reliability of the method. Instead of complicated shape analysis that are required in other microfluidic methods, here the authors simply use the elongation of the cell and the orientation angle with respect to the fluid flow direction. The nice thing here is that a wellknown theory from R. Roscoe can be successfully used to relate these …
Reviewer #2 (Public Review):
In the manuscript, the cellular deformation that is due to the shear stress generated in a classical microfluidic channel is used to deform detached cells that are moving in the flow. A very elegant point of the paper is that the same cells are used in the provided software to determine the fluid flow, which is a key parameter of the method. This is particularly important, as an independent way to crosscheck the fluid flow with the expected values is important for the reliability of the method. Instead of complicated shape analysis that are required in other microfluidic methods, here the authors simply use the elongation of the cell and the orientation angle with respect to the fluid flow direction. The nice thing here is that a wellknown theory from R. Roscoe can be successfully used to relate these quantities to the viscoelastic shear modulus. Thanks to the knowledge of the fluid flow profile, the mechanical properties can be related to the tank treading frequency of the cells, which in turn depends on the position in the channel, and the flow speed. Hence, after knowing the flow profile, which can be determined with a sufficiently fast camera, and the actual static cell shape, it is possible to obtain frequency dependent information. Assuming then that cells do have a statistically accessible mean viscoelastic property, the massive and quick data acquisition can be used to get the shear modulus over a large span of frequencies.
The very impressive strength of the paper is that it opens the door for basically any, nonspecialized cell biology lab to perform measurements of the viscoelastic properties of typically used cell types in solution. This allows to include global mechanical properties in any future analysis and I am convinced that this method can become a main tool for a rapid viscoelastic characterization of cell types and cell treatment.
Although it is both elegant and versatile, there remain a couple of important questions open to be further studied before the method is as reliable as it is suggested by the authors. A main problem is that the model and the data simply don't really work together. This is most prominent in Figure 3a. This is explained by the authors as a result of nonlinear stress stiffening. Surely this is a possible explanation, but the fact that the question is not fully answered in the paper makes the whole method seems not sufficiently backed. I agree that the test with the elastic beads are beautiful, but also here the results obtained with the microfluidic method and the AFM seem not to match sufficiently to simply use the proposed model in conjecture with a single powerlaw approach to fully translate the single frequency data into a frequency dependent plot. There are more and more hints that two powerlaw models are more reasonable to describe cell mechanics. If true this would abolish the approach to exploit only a single image to get the mechanical powerlaw exponent and the prefactor in a single image. Despite all the excitement about the method, I have the feeling that the used models are stretched to their extreme, and the fact that the only real crosscheck (figure 3a) does not work for the powerlaw exponent undermines this impression.
