Quantifying the roles of random motility and directed motility using advection-diffusion theory for a 3T3 fibroblast cell migration assay stimulated with an electric field
© The Author(s) 2017
Received: 19 January 2017
Accepted: 22 February 2017
Published: 17 March 2017
Directed cell migration can be driven by a range of external stimuli, such as spatial gradients of: chemical signals (chemotaxis); adhesion sites (haptotaxis); or temperature (thermotaxis). Continuum models of cell migration typically include a diffusion term to capture the undirected component of cell motility and an advection term to capture the directed component of cell motility. However, there is no consensus in the literature about the form that the advection term takes. Some theoretical studies suggest that the advection term ought to include receptor saturation effects. However, others adopt a much simpler constant coefficient. One of the limitations of including receptor saturation effects is that it introduces several additional unknown parameters into the model. Therefore, a relevant research question is to investigate whether directed cell migration is best described by a simple constant tactic coefficient or a more complicated model incorporating saturation effects.
We study directed cell migration using an experimental device in which the directed component of the cell motility is driven by a spatial gradient of electric potential, which is known as electrotaxis. The electric field (EF) is proportional to the spatial gradient of the electric potential. The spatial variation of electric potential across the experimental device varies in such a way that there are several subregions on the device in which the EF takes on different values that are approximately constant within those subregions. We use cell trajectory data to quantify the motion of 3T3 fibroblast cells at different locations on the device to examine how different values of the EF influences cell motility. The undirected (random) motility of the cells is quantified in terms of the cell diffusivity, D, and the directed motility is quantified in terms of a cell drift velocity, v. Estimates D and v are obtained under a range of four different EF conditions, which correspond to normal physiological conditions. Our results suggest that there is no anisotropy in D, and that D appears to be approximately independent of the EF and the electric potential. The drift velocity increases approximately linearly with the EF, suggesting that the simplest linear advection term, with no additional saturation parameters, provides a good explanation of these physiologically relevant data.
We find that the simplest linear advection term in a continuum model of directed cell motility is sufficient to describe a range of different electrotaxis experiments for 3T3 fibroblast cells subject to normal physiological values of the electric field. This is useful information because alternative models that include saturation effects involve additional parameters that need to be estimated before a partial differential equation model can be applied to interpret or predict a cell migration experiment.
KeywordsCell migration Random motility Directed motility Electrotaxis Partial differential equation Keller-Segal model
where D(S)>0 is the cell diffusivity, and χ(S) is the tactic sensitivity function. In this Keller-Segel  type model, the tactic flux is proportional to the gradient of some signal, S(x,y,t), and the strength of the tactic response is governed by the tactic sensitivity function, χ(S) [6, 7]. Setting χ(S)>0 represents attraction, since the directed component of the cell flux is in the direction of increasing S. Alternatively, setting χ(S)<0 represents repulsion. To maintain generality, the cell diffusivity D(S)>0 is also written as a function of the signal, S [1, 8, 9]. If D(S) is increasing, this model represents an increase in undirected motility with the signal, as in the case of chemokinesis . Since there is no source/sink term in Eq. (1) we are focusing on cell migration processes on short time scales so that cell proliferation and cell death have a negligible impact on the cell density.
Directed cell migration can occur in response to various types of external spatial gradients. In Eq. (1) we have not specified the physical interpretation of S. In a model of chemotaxis S would represent the concentration of a chemical signal, whereas in a model of thermotaxis S would represent the temperature. In a model of electrotaxis S represents the electric potential. In this work we focus on stimulating directed cell migration in an electric field.
Electrotaxis plays an important role in guiding epithelial and corneal wound healing processes, and could potentially be used to design novel therapies [11–16]. While the precise molecular-level mechanisms behind electrotaxis remain unresolved, a common hypothesis is that exposing cells to an electric field leads to changes in plasma membrane potentials [11, 12] with the membrane facing the cathode becoming depolarized, and the membrane facing the anode becoming hyperpolarized [11, 12]. In a cell with negligible voltage-gated conductance, the hyperpolarized membrane attracts calcium ions, leading to a contraction of this side of the cell which propels the cell toward the cathode [11, 12]. In a cell with voltage-gated calcium channels, the channels near the depolarized side open to allow an influx of calcium ions leading to a rise in the intracellular calcium ion level throughout such a cell. The direction of cell movement in this situation will depend on the balance between the opposing contractile forces [11, 12].
The advantage of working with Eq. (2) compared to Eq. (1), is that there are just two unknown parameters in Eq. (2), χ and D. In contrast, the more complicated models involving receptor saturation effects can involve six or more unknown parameters [13, 18–23].
Making a distinction between choosing models where the tactic sensitivity incorporates receptor saturation effect (Eq. (1)) and a simpler model where the tactic sensitivity coefficient is constant (Eq. (2)) is not obvious unless we are guided by a reasonable quantity of experimental data. From a theoretical point of view, it might be attractive to incorporate receptor saturation dynamics into a mathematical model, but this comes with the trade off that this is typically achieved by introducing a complicated relationship between the tactic sensitivity coefficient and the attractant concentration, which can introduce several unknown parameters into the mathematical model thereby over complicating the process of model calibration . To provide some insight into this question, here we analyze a suite of cell migration data. The data we analyze comes from an electrotaxis experiment where the strength of the attraction gradient is carefully varied so that we can analyze both the random component of the cell migration as well as the directed component over a range of applied gradients.
Qualitative assessment of trajectory data
Quantitative assessment of trajectory data
Given our estimates of 〈v x 〉 and 〈v y 〉 (Fig. 3 a-h), we now estimate the diffusivity coefficients, D x and D y , for each experiment. Results showing estimates of D x and D y under the application of no gradient are summarised in Fig. 3 i-j. Averaging our estimates across the 80 trajectories we obtain 〈D x 〉=59 μm2/h and 〈D y 〉=50μm2/h for the experiments in which there is no gradient. The magnitude of these estimates of cell diffusivity are consistent with previous estimates 3T3 fibroblast cells obtained using single cell trajectory data [36, 37]. Additional estimates of D x and D y , and 〈D x 〉 and 〈D y 〉 are shown in Fig. 3 k-p for cell migration under the influence of gradients of 100, 200 and 400 mV/mm, respectively. For each of these data sets we have 〈D x 〉≈〈D y 〉, indicating that the random motility coefficient is isotropic. Furthermore, unlike our estimates of 〈v x 〉, our estimates of 〈D x 〉 and 〈D y 〉 appear not to depend on the electric field.
Relationship between the applied gradient, cell diffusivities and drift velocities
Results in Fig. 4 a-b show 〈v x 〉 and 〈v y 〉 as a function of the EF. As we anticipate, 〈v x 〉 increases with EF whereas 〈v y 〉≈0 for all EF considered. To examine the putative relationship between 〈v x 〉 and EF, and between 〈v y 〉 and EF, we perform an unconstrained linear regression. The coefficient of determination for the 〈v x 〉 data is very high, r 2=0.98, suggesting that the linear relationship between 〈v x 〉 and EF provides a good explanation of the variability. In contrast, the coefficient of determination for 〈v y 〉 is very low, r 2=0.00, suggesting that the null hypothesis is valid and there is no relationship between 〈v y 〉 and EF. In summary, these results imply that a linear relationship between 〈v x 〉 and EF is consistent with the observed data. To match the drift term in Eq. (1) with the advection-diffusion (Eq. (6)) we require that v x =χ(S)∂ S/∂ x. Since our data is consistent with a linear relationship between v x and the applied gradient, ∂ S/∂ x, it appears that a constant tactic sensitivity function, χ(S)=χ, provides the simplest explanation of our experimental results.
Results in Fig. 4 c-d show 〈D x 〉 and 〈D y 〉 as a function of EF. Visually, we see no discernible trend in the data for different values of EF. This visual interpretation is consistent with the fact that we obtain a small coefficient for each of the linear regressions in Fig. 4 c-d. Therefore, it is reasonable to assume that the cell diffusivities appear to be independent of the electric field. If we accept this assumption and further average the data in Fig. 3 i-p in each direction we obtain overall estimates of 〈D x 〉=48μm2/h and 〈D y 〉=46μm2/h. Again, this suggests that the diffusion of 3T3 fibroblast cells is approximately isotropic since we have D x ≈D y , across all the experimental conditions considered.
where v is the drift velocity, D is the diffusivity and L is a relevant length-scale, which here we will take to be the cell diameter of fibroblast cells, L≈25μm . The Peclet number is a measure of the time scale of advection to the time scale of diffusion . When Pe≪1, undirected diffusive transport dominates, when Pe≫1, directed transport dominates, and when Pe≈1 to two mechanisms are in balance. Comparing estimates of the drift velocity and the diffusivity in the x-direction suggests that our experiments deal with a range of Peclet numbers from Pe≈0 when EF=0 mV/mm to Pe≈10 when EF=400 mV/mm. Therefore, our experimental data covers a wide range of transport conditions ranging from purely undirected, diffusive transport to highly directed, advection-dominant conditions.
To summarise our findings, results in Fig. 4 suggest that 〈v x 〉 increases linearly with EF, whereas the data suggests that the other transport coefficients, 〈v y 〉, 〈D x 〉 and 〈D y 〉, appear to be independent of EF. Guided by these results, we assume that 〈v x 〉 increases linearly with EF, and that the other transport coefficients are independent of EF. Comparing the results in Fig. 1 c and d also allows us to also consider whether there is any possible relationship between the transport coefficients and the electric potential. Repeating the process of plotting our estimates of the four transport coefficients as a function of the electric potential (not shown) suggests that there is no obvious trends in the data. Furthermore, linear regressions between each transport coefficient and the associated value of the electric potential reveals a low coefficient of determination, r 2<1. Therefore, based on the data, we assume that the transport coefficients appear to be independent of the electric potential in these experiments.
Our results indicate that when we quantify the roles of directed and undirected migration of 3T3 fibroblast cells under the influence of an applied electric field, the undirected component of the migration appears to be independent of the EF, and the directed migration appears to increase linearly with EF. Furthermore, we observe no consistent differences in the cell diffusivity estimates in the x and y Cartesian directions, implying that the undirected migration is isotropic. The simplest way to explain these results in terms of a Keller-Segel-type continuum model (Eq. (1)) is that we have a constant diffusivity, D(S)=D, and a constant chemotactic sensitivity function, χ(S)=χ. While the assumption that the chemotactic sensitivity function can be treated as a constant is widely invoked [24–33], this assumption is infrequently tested using experimental data collected under a range of gradient conditions. The question of whether the tactic sensitivity function ought to be treated as a constant or a more complicated expression is of interest because many theoretical models incorporate these kinds of details, such as receptor saturation, without necessarily being guided by experimental observations [6, 7, 13, 18–23].
By examining trajectories of 3T3 fibroblast cells under a range of physiologically-relevant electric gradients [11, 14], we quantify the roles of directed and undirected migration. In summary we find that the undirected migration is isotropic and the cell diffusivity is approximately 50 μm2/h, and that the drift velocity increases approximately linearly with the applied electric field, suggesting that the tactic sensitivity function is a constant.
Although our results apply to 3T3 fibroblast cells, we anticipate that repeating the experiments and analysis outlined here for different cell lines would provide insight into the roles of directed and undirected motility for any cell line of interest. Although we have found that the drift of fibroblasts to increase approximately linearly with the electric field in the range of EF=0−400 mV/mm, it is possible that we may observe a different response for different cell lines, or we may observe a different response for the same cell line when we apply a stronger electric field. However, here we deal only with gradients in the range of 0-400 mV/mm because this is a physiologically relevant range [11, 14].
As shown in Fig. 1 a, we use a specifically designed and fabricated microfluidic chip to study the electrotaxis of NIH 3T3 fibroblasts. A CO2 laser scriber (ILS2, Laser Tools & Technics Corp, Taiwan) is used to ablate desired patterns on polymethylmethacrylate (PMMA) substrates [39–41]. Four layers of PMMA sheets are thermally bonded to form the fluidic channel, which is then attached to a cover glass to act as the cell culture area. The thickness of the fluidic channel is 1 mm, and the widths of the four culture areas (two copies) are 4.00, 8.28, 4.14 and 2.07 mm, respectively. By applying a direct current (dc) of 80 μA, the EF inside these areas are calculated to be 0, 100, 200, and 400 mV/mm, respectively, based on Ohm’s law . Numerical simulations of the EF and the potential inside the microfluidic chip is simulated using the commercial software package COMSOL Multiphysics (COMSOL, USA) to confirm these calculations (see Fig. 1 c and d).
The NIH 3T3 fibroblast cell line, purchased from Bioresource Collection and Research Center (BCRC, Taiwan), is cultured in a complete medium composed of Dulbecco’s modified Eagle medium (Gibco, USA) and 10% calf serum (Invitrogen, USA). 106 cells are injected into the chip and the temperature is maintained at 37±0.5oC using a customized temperature controller. Different EF strengths are introduced by connecting the Ag(anode)/AgCl(cathode) electrodes (see Fig. 1 a) to a dc power supply (GWInstek, Taiwan) set at the constant-current model . The microfluidic chip is mounted on a motorized, bright-field inverted microscope (CKX41, Olympus, USA) to observe cell migration. Figure 1 b shows an image of the cells in one culture area. For each culture area, corresponding to a different EF, images are taken over a period of 2 h. In each area, at least 80 cells were selected at random for data analysis.
where A is a plane region that is a subset of Ω.
Therefore, for each trajectory, (x(t),y(t)), we can obtain separate estimates of v x and v y . Fitting a series of straight lines constrained to pass through the origin gives us an estimate of v x and v y for each trajectory. Since we have 80 trajectories for each gradient condition, we obtain 80 estimates of v x and 80 estimates of v y . The variability amongst these estimates can be observed by plotting the results as a histogram. Furthermore, we can characterise the average coefficients by evaluating the sample mean and sample standard deviation of these 80 estimates. We will denote the sample mean as 〈v x 〉 and 〈v y 〉, respectively.
Therefore, given our previous estimates of the average drift velocity in each direction 〈v x 〉 and 〈v y 〉, for each trajectory we can obtain separate estimates of D x and D y . Fitting a series of straight lines constrained to pass through the origin give us estimates of D x and D y for each trajectory. Since we have 80 trajectories for each gradient condition, we obtain 80 estimates of D x and 80 estimates of D y . The variability amongst these estimates can be observed by plotting the results as a histogram. Furthermore, we can characterise the average coefficients by evaluating the sample mean and sample standard deviation of these 80 estimates. We will denote the sample mean as 〈D x 〉 and 〈D y 〉, respectively.
The authors would like to thank Dr. Ji-Yen Cheng for help in fabricating microfluidic chips, and the Center for Emerging Material and Advanced Devices, National Taiwan University, for the cell culture room facility. We appreciate the helpful comments from three referees and the BMC Systems Biology editor.
This work is supported by the Australian Research Council (DP140100249, DP170100474). This work is financially supported by the Ministry of Science and Technology of Taiwan under Contract No. MOST 104-2311-B-002-026 (K. Y. Lo), No. MOST 104-2112-M-030-002 (Y. S. Sun), and the National Taiwan University Career Development Project (103R7888) (K. Y. Lo).
Availability of data and raw materials
Data is available in the Additional file 1.
MJS, K-YL and Y-SS conceived the study and designed the experiments. K-YL and Y-SS performed the experiments. MJS analysed the data. MJS wrote the manuscript. All authors read and approved the final manuscript.
The authors declare that they have no competing interests.
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