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New Method for Modeling Connective-Tissue Cell Migration: Improved Accuracy on Motility Parameters

Matt J. Kipper ^* , Hynda K. Kleinman ^* and Francis W. Wang 

^* National Institute of Dental and Craniofacial Research, National Institutes of Health, Bethesda, Maryland; and National Institute of Standards and Technology, Gaithersburg, Maryland

Correspondence: Address reprint requests to Francis W. Wang, National Institute of Standards and Technology, 100 Bureau Dr., Gaithersburg, MD 20899. Tel.: 301-975-6726; Fax: 301-975-9143; E-mail: francis.wang{at}nist.gov.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
Mathematical models of cell migration based on persistent random walks have been successfully applied to describe the motility of several cell types. However, the migration of slowly moving connective-tissue cells, such as fibroblasts, is difficult to observe experimentally and difficult to describe theoretically. We identify two primary sources of this difficulty. First, cells such as fibroblasts tend to migrate slowly and change shape during migration. This makes accurate determination of cell position difficult. Second, the cell population is considerably heterogeneous with respect to cell speed. Here we develop a method for fitting connective-tissue cell migration data to persistent random walk models, which accounts for these two significant sources of error and enables accurate determination of the cell motility parameters. We demonstrate the usefulness of this method for modeling both isotropic cell motility and biased cell motility, where the migration of a population of cells is influenced by a gradient in a surface-bound adhesive peptide. This method can discern differences in the motility of populations of cells at different points along the peptide gradient and can therefore be used as a tool to quantify the effects of peptide concentration and gradient magnitude on cell migration.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
Cell migration is an important class of phenomena that play key roles in physiological processes such as wound healing, metastasis, and embryonic development. Cell migration is most commonly described as a persistent random walk,

(1)

where indicate the mean over a population of cells, n_d is the number of dimensions in which the cells are migrating, S² is a characteristic mean-squared speed of the population, P is a persistence time in direction, and t is time (1,2) (see Appendix for usage of the notation ). This model is arrived at by assuming that the orthogonal components of the velocity are described by two-parameter Langevin equations, subject to certain assumptions regarding velocity distributions (3,4) (see Appendix). Dunn and Brown have demonstrated that a discrete form of this two-parameter model is sufficient for describing fibroblast motility (5), and Stokes et al. have extended the analysis to describe continuous motion (6). Although Eq. 1 only accounts for isotropic motility, the model can be modified in a number of ways to account for anisotropies in the cell motility. The simplest of these modifications is to add an additional parameter that accounts for directional bias (6–9) (see Appendix). The reader is referred to previous works for a description of the development of the model (1–6,10,11).

This and other mathematical models of cell migration provide insights into the mechanisms by which cells migrate, and may predict what cellular responses will be manifested by a cell population in response to alternative sets of stimuli. Regardless of the model used, distinctions among mechanisms of cell migration cannot be unambiguously made without obtaining a significant amount of data describing the individual paths of a large population of cells. This is because the phenomena of interest (i.e., cell speed and changes in cell orientation) are always coincident with a multitude of cellular processes (e.g., the cell cycle) that are manifested in the cell population as statistical, rather than deterministic, parameters. Additionally, systematic and stochastic errors may be associated with experimental observation. These sources of uncertainty are amplified in the case of connective-tissue cells important for wound healing (e.g., fibroblasts, keratinocytes) because these cells migrate very slowly compared to cell types whose migration has been previously well characterized (e.g., bacteria, neutrophils, and macrophages).

In this work, we develop a method for analyzing cell migration data that improves the accuracy with which the persistent random walk model parameters can be determined. This is accomplished by accounting for two common sources of error: accurate determination of the cell position and heterogeneity in the cell population with respect to speed. We demonstrate the usefulness of this method by comparing it to nonlinear regression on a persistent random walk model. We also briefly discuss applying this model to anisotropic cell migration on peptide gradient surfaces, and provide an example where this method can be used to differentiate among cell populations whose response is different at different points along a peptide gradient.

The method presented here is a general technique for obtaining persistent random walk model parameters. However, our work is motivated by our interest in developing assays for fibroblast migration that can be used to quantify the haptotactic and haptokinetic activity of peptides derived from the basement membrane protein, laminin-1. Laminin-1 contains at least 20 peptide sequences that have been associated with various biological activities, including promotion of cell adhesion, migration, and angiogenesis (12–14). We have recently developed a technique for fabricating covalent peptide gradients that is suitable for studies aimed at quantifying the biological activity of the peptides and studying cellular mechanisms of migration, and that can be applied to surfaces and hydrogels (15). The method presented here will allow more accurate quantitation of how these potentially haptotactic or haptokinetic peptides affect fibroblast migration.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
The details of the substrate preparation and the cell culture are given in Kipper et al. (15). Briefly, glass coverslips (18 mm x 18 mm, No. 1, Corning, Corning, NY) were cleaned and oxidized by oxygen plasma etch and then coated with poly(L-lysine) hydrobromide (PLL, 70,000 g mol^–1 to 150,000 g mol^–1, Sigma-Aldrich, Saint Louis, MO) by incubating in a PLL solution (600 µL, 0.1 mg mL^–1 in Dulbecco's phosphate-buffered saline) for 30 min, followed by rinsing in deionized water. The PLL provides an amine-functionalized surface onto which we can couple peptides with reasonably accurate control over the surface peptide concentration. For the biased cell motility, surfaces were prepared with a gradient in covalently bound laminin B160 peptide, as reported previously (15). B160 comes from the ß1 chain of laminin-1. B160 has the sequence ¹⁶⁰⁷VILLQQSAADAIR¹⁶¹⁸ (12). B160 has been shown to promote the integrin-mediated adhesion of human umbilical vein endothelial cells, angiogenesis, and the promotion of human umbilical vein endothelial cell migration when presented in solution (12). Its capacity to promote fibroblast migration has not been studied or quantified to our knowledge.

Primary human foreskin fibroblasts (HFF) were cultured in Dulbecco's modified Eagle's media (DMEM, Gibco, Carlsbad, CA), with 10% fetal bovine serum (10%), L-glutamine (2 mM), penicillin and streptomycin (100 U mL^–1), and 0.1 mg mL^–1 gentamicin. Medium was replaced every three days and cells were passed 3:1 or 4:1 at confluence. Nonsynchronized cells were used between the eleventh and eighteenth passage. The PLL-coated coverslips were fixed to the bottom of wells of Nunclon 4-well plates (Nalge Nunc International, Rochester, NY) using a small amount of silicone grease. Cells were plated at low confluence (35–70 cells mm^–2), with 7 mL of serum-free DMEM in each well, and allowed to attach to the surfaces for 1 h. Medium was then replaced with 5 mL of serum-free DMEM containing 1 x 10^–5 µmol L^–1 5-(and 6)-(((4-chloromethyl)benzoyl)amino)tetramethylrhodamine (CellTracker Orange CMTMR, Molecular Probes, Eugene, OR). Cells were incubated with the dye for 30 min, and then the medium was replaced with 7 mL of serum-free DMEM.

The stained cells were imaged using in situ time-lapse video microscopy on an Axiovert 100 inverted fluorescent microscope (Carl Zeiss International, Oberkochen, Germany) equipped with an incubation chamber kept at 37°C, humidified, with a 5% CO₂ atmosphere, and a LEP Bioprecision motorized stage (Ludl Electronic Products, Hawthorne, NY). A shutter was used to ensure cells were only exposed during image capture, and an excitation filter was used to minimize exposure to potentially toxic UV radiation. Images were captured every 10 min for 20 h, and analyzed using the Metamorph 5.0 software package. An automated image capture routine cycled the robotic stage through multiple fields of view at each time point so that multiple time-lapse videos could be captured in a single experiment. To obtain data on a sufficiently large number of individual cells, the cells were imaged at relatively low magnification. A 5x objective was used, resulting in images that were 1.8 mm across. Between 30 and 100 cells were typically present in each field of view. Cell migration on PLL-coated surfaces was studied in four replicate experiments. Cell migration on the B160 peptide gradient surfaces was studied in two replicates. Time-lapse videos were collected from nine adjacent fields of view across the surface and all of the cells within each field of view were pooled as a single population.

Cell tracks were reconstructed from the positions of individual cells at each time point using an automated image analysis algorithm. The algorithm attempts to match a fraction of the pixel intensities in each image to a template created from the previous image for each cell. Once this threshold is obtained, the template is updated for the next image, and the cell position is defined as the centroid of the cell shape. This algorithm is very efficient at not "losing" cells from one image to the next, and is able to account for changes in the cell shape and position with time.

Effects of heterogeneity within the population on cell motility parameter estimation
Equation 1 implies that the mean-squared displacement should be a smooth, monotonic function of time. Measuring the d² of a population of cells over time should therefore provide us with data on which we can perform a nonlinear regression to obtain the parameters S² and P. In fact, the experimentally observed d² data are dominated by features that are not described by Eq. 1, even for a relatively large number of cells (i.e., n > 50). This is illustrated in Fig. 1. In Fig. 1 a, d² (solid line) ± SE (dashed lines) is plotted over 20 h for a population of 73 HFF on a PLL-coated surface (see Appendix for the definition of standard error). The deviations from the expected smooth, monotonic behavior may be due to some error in the cell position measurements or some true behavior of the cell population that the model represented by Eq. 1 fails to describe, or they may be due to both effects. The latter contribution represents a true component of the cell population behavior and should therefore be included in our description of the cell migration. First, we briefly discuss the heterogeneity within the population. The effects of cell position uncertainty are discussed in the following section.

View larger version (5K): [in this window] [in a new window]   FIGURE 1  Mean-squared displacement (a) and (b) as a function of time for a population of 73 HFF on a PLL-coated glass surface. Broken lines represent the standard error of the mean. Normalizing the mean-squared displacement by the cumulative mean-squared speed significantly reduces the standard error. (c) Cell tracks for four of the cells shown as the discrete cell positions measured in the experiment. The + indicates the initial position of each cell, and the scale bar applies to all four cell tracks.

We propose an alternative approach to dealing with the heterogeneity in cell speed. According to Eq. 1, the expected value of d² for times t >> P increases at a rate proportional to S². Therefore, normalizing d² for each cell by that cell's cumulative mean-squared speed should eliminate some undesired noise arising from speed fluctuations. The cumulative mean-squared speed of cell i, which we denote (t) to distinguish it from the fit parameter in Eq. 1, is computed from the displacements of each cell at each previous time point. That is, for each cell, i, we define a time-dependent, normalized migration parameter _i(t)

(2)

(We use the notation f(t) and f(t) to indicate that f is a function of time or time interval, not that time is a multiplicative factor.) The displacement of the ith cell over the jth time interval is denoted d_i,j, where the underscore distinguishes this displacement from the total displacement of the i^th cell from its starting position at t = 0, d_i(t). is the number of time intervals at time t, given by = t/(t). (t) is approximated by the expression in parentheses in the denominator of the righthand side of Eq. 2. Equation 2 assumes that we can accurately estimate the cell speed from the positions of a cell at successive time points. In doing so, we make the approximation

(3)

where is the path length of the i^th cell over the j^th time interval. Assuming that the mean of the ratio is a valid estimate for the ratio of the means, we can estimate the population mean, (t):

(4)

The righthand side of Eq. 4 comes from Eq. 1. The mean normalized migration parameter, (t), for the data in Fig. 1 a is plotted in Fig. 1 b. This normalization reduces the standard error by 50% over the entire range of t. That is, the standard error on d²(t) is about twice the standard error on (t) as a percentage of the respective means. This reduction in the standard error occurs because cells that have relatively large typically also have a relatively large (t), while cells with relatively small typically also have a relatively small (t).

Effects of cell position uncertainty on cell motility parameter determination
To more fully investigate the effects of cell position uncertainty, suppose the position of cell i is measured at two successive time points, r_i,j–1 = (x_i,j–1, y_i,j–1) and r_i,j = (x_i,j, y_i,j). Its corresponding squared displacement over the j^th time interval is

(5)

Further, assume that the error on each of the measured coordinates is an independent random variable with a normal distribution, and standard deviation, _x, and that the errors on the two coordinates are independent and identically distributed. The uncertainty (type B standard uncertainty) on the measurement of arising from the uncertainty on the cell position is

(6)

(See Appendix for notes about uncertainties.) To consider how this uncertainty influences the squared speed, recognize that the true squared speed of a cell would be given by the lefthand side of Eq. 3 if it were possible to measure the path length, If the persistent random walk model (Eq. 1) is accurate, then d²(t) is related to the mean-squared path length, ²(t) by

(7)

For computing the squared speed over a short time interval, t, we can replace d²(t) with d²(t). This ratio approaches unity as t goes to zero (for n_d = 2, ). For nonzero t, the path length approximation, introduces an approximation on the cumulative mean-squared speed of cell i.

(8)

The expected value is the value we would expect if Eq. 1 were an accurate model for the motility of the population. We use the notation _t to indicate the mean over time. In Eq. 8, we approximate that the mean-squared displacement and mean-squared path length of a single cell over many time intervals are equal to the mean-squared displacement and the mean-squared path length of many cells over a single time interval. The righthand side of Eq. 8 is from Eq. 7, where t is replaced with t, because the squared speed is computed over the interval t. The cell position uncertainty can then be propagated to Eq. 7 to determine the uncertainty on Eq. 8:

(9)

We continue to use (t) to remind ourselves that the last factor in Eq. 9 is a function of t. To understand the sources of this uncertainty, we replace ²(t) on the righthand side of Eq. 9 with S²t², where S² is the fit parameter from Eq. 1.

(10)

Here, we introduce the diffusion coefficient, which is also referred to as a random motility coefficient (17,18), for the cell population, D = 0.5S²P. Equation 10 represents the squared uncertainty on the mean of the path length approximation, not the uncertainty on a single measurement. For a single measurement of the squared cell speed, we consider the case of = 1. Taking the square root of Eq. 10 and the identity in Eq. 8 with = 1 gives the uncertainty of a squared speed measurement arising from the uncertainty in the cell position, and accounting for the path length approximation (Eq. 3):

(11)

(See Appendix for notes about the uncertainties in Eqs. 9–11.) Because Eq. 11 represents the special case of the squared speed of a single cell over a single time interval, we write as For simplicity, we call this ratio S*. The righthand side of Eq. 11 is written as a product of three important dimensionless groups. Each of these dimensionless groups has a physical significance. The left most group on the righthand side is the ratio of the uncertainty on the cell position to the expected distance the cells travel in one persistence time (equivalent to a mean free path). We will call this the normalized position uncertainty. The center group on the righthand side is a normalized observation time. The rightmost group on the righthand side represents the path length approximation. Note that the error arising from the path length approximation can be estimated by Eq. 8, and is also determined by the normalized observation time, t/P. As the observation time is reduced, the uncertainty on the squared speed measurement increases, but increasing the observation time too much reduces the accuracy of the path length approximation.

Combining Eqs. 8 and 11, with = 1, we find that the ratio and its uncertainty can be estimated by

(12)

where µ_S* is the value of S* predicted by Eq. 8 (see Appendix, Eq. A10). Equation 12 is only valid for S* > 0. Equation 12 is plotted in Fig. 2 a, with the solid line representing the first term on the righthand side, and the broken lines representing different values of _x(DP)^–1/2. Fig. 2 a shows that for t > P, the squared speed is underestimated, due to the path length approximation, but the error diverges as t approaches zero, due to the normalized position uncertainty.

View larger version (19K): [in this window] [in a new window]   FIGURE 2  Ratio of the observed squared speed to the expected squared speed (solid line) ± standard uncertainty (dashed lines) according to Eq. 12 (a), and the mean of the ratio of observed squared speed to expected squared speed (see Eq. 14) (b) for different values of the normalized position uncertainty. Ratio of the observed squared displacement to the expected squared displacement (solid line) ± standard uncertainty (dashed lines) according to Eq. 15 (c), and the error of the of the observed squared displacement to the expected squared displacement according to Eq. 16 (d) for different values of the normalized position uncertainty.

(13)

By integrating S*(S*) over the range 0 S* < , we can obtain the error that we would expect on the mean of many measurements of the squared speed

(14)

(see Eq. A11). Here, erf is the error function. The mean of the squared speed normalized by its expected value, is plotted in Fig. 2 b for different values of the normalized position uncertainty. Notice that the average of the observed squared speed is likely to significantly overestimate the squared speed for short observation times, due to the nonnormality of the probability distribution function (Eq. 13). Thus, accurate determination of the squared cell speed is difficult from individual measurements of cell position, particularly when the persistence time is not known. This divergence at short observation times has not been dealt with analytically by others who have analyzed the error on determination of cell speed (1,18). Dunn recognized that at short observation times the uncertainty in the cell displacement becomes large, but in his formulation, shorter observation times provide more nonoverlapping time intervals over which to average the cell speed. However he noted that accurate measurements of cell speed would be limited by the resolution of the images at short observation times.

Let us now consider the ratio of the observed squared displacement to the expected squared displacement for the i^th cell at time t. This ratio, which we abbreviate, d*, can be arrived at by extension of Eq. 5 to compute the uncertainty on a squared displacement measurement from the origin at t = 0 and Eq. 1 for the expected value of the squared displacement:

(15)

(see Eq. A10). Here, for simplicity, we drop the notation (t), which indicates a function of time. As with Eq. 12, Eq. 15 is only valid for Equation 15 is expressed in terms of the same important dimensionless groups that appear in Eq. 12, the normalized position uncertainty and the normalized observation time. The PDF of d* is similar to the PDF given by Eq. 13 for S*, with µ_S* replaced by µ_d* = 1, and _S* replaced by _d*, equal to the uncertainty indicated in Eq. 15 by the terms after the symbol ±. Thus, the mean error on the squared displacement has a similar form as in Eq. 14:

(16)

(see Eq. A11). Equations 15 and 16 are plotted in Fig. 2, c and d, respectively, for different values of the normalized position uncertainty. Just as for S* (Fig. 2 a), d* and its error diverge for small observation times due to the normalized position uncertainty. However, converges to for long observation times. Dickinson and Tranquillo predicted a different result for the effect of the position error on the mean-squared displacement (18). In their analysis, the position error results in a constant offset of 2, where is the variance of the cell position uncertainty. In developing Eqs. 15 and 16, we have assumed that uncertainty on the cell position is uncorrelated to the direction of the cell displacement.

Based on Fig. 2, c and d, one might assume that it would be possible to obtain S² and P from nonlinear regression of Eq. 1, provided we use sufficiently long observation times to reduce the error on d². However, as we have already noted from Fig. 1, these data may be too noisy to perform a reliable regression. Let us then consider the alternative mean normalized migration parameter, , which we proposed in Eq. 2. The uncertainty on a measurement of the normalized migration parameter for a single cell, _i

(17)

(see Eq. A9). If we fit the cell position data to Eq. 4 rather than to Eq. 1, then the ratio of the observed value of _i to the expected value of _i, is given by

(18)

(see Eq. A10). Again, we wish to emphasize that _i and are functions of time, though we drop the notation (t) for simplicity. Equation 18 is arrived at by using the definition in Eq. 2 for the observed value of _i. _i,Expected is the population mean, , and is computed according to the approximation in Eq. 4. Equation 18 has two normalized timescales, t/P = t/P (the time over which _i is measured) and t/P (the time step over which the squared speed is measured), which appear in _d* and _S*, respectively. The probability density function for * is also similar to Eq. 13, and results in a mean error on the observed value of the normalized migration parameter:

(19)

(see Eq. A11). * and ( + 1) are plotted in Fig. 3. In Fig. 3, a and b, the effect of varying the observation time is illustrated, with the total number of time steps held constant at = 25. The effect of varying is illustrated in Fig. 3, c and d, where t/P is held constant at 0.3. Comparing Fig. 2, c and d, to Fig. 3, a and b, we can see that normalizing by reduces the error on the measurement for small observation times. However, at longer observation times, * diverges due to divergence in whereas the error on converges. Thus, while _i may have significantly less error than does not converge to 0.

View larger version (18K): [in this window] [in a new window]   FIGURE 3  The ratio of the observed value of i to the expected value of i (solid line) ± standard uncertainty (dashed lines) for different values of the normalized position uncertainty according to Eq. 18 (a and c), and the error on computed from Eq. 19 for different values of the normalized position uncertainty (b and d). In a and b, the number of time steps, , is held constant at 25. In c and d, the normalized observation time is held constant at 0.30.

View larger version (10K): [in this window] [in a new window]   FIGURE 4  (a) Fit of the observed mean-squared speed as a function of observation time to Eq. 14 to obtain estimates for the normalized position uncertainty, persistence time, and expected mean-squared speed. Model parameters are reported as mean ± the standard deviation. The speed data were computed from the displacements of the 73 HFF for the first 900 min of migration using the overlapping time intervals with increasing observation time from 10 min to 150 min. Error bars represent the standard deviation of the speed. RMS speed as a function of time for the population of 73 HFF from Fig. 1 b. Error bars represent the standard error of the mean.

Improved method for fitting cell migration data to the persistent random walk model
Now that we have an estimate for _x, we can develop a practical method for fitting the experimental data to the model. We use Eq. 19, and the approximation first introduced in Eq. 4 to model the mean normalized migration parameter as

(20)

Here, is an estimate for the true value and is also the expected value from the model, and we define the factor f(t) as the ratio of the observed value to the true value of . We drop the subscript "Expected" from to reflect the assumption that the model is accurate. The ratio f(t) can be computed from µ_* and _*, which can be computed from P and _x(DP)^–1/2 (using Eqs. 12, 15, and 19) . We use the values of P and _x(DP)^–1/2 that were obtained from the regression in Fig. 4 a as initial guesses to compute f(t). Once f(t) is known, _Observed can be transformed to using Eq. 20. An improved prediction for the persistence time can be obtained from a fit to Eq. 4. Finally, we use this improved value of P to compute the error on the observed mean-squared speed (Eq. 14) to find the expected value of the mean-squared speed. This procedure can be iterated using the improved prediction of P to reevaluate f(t) if suitable agreement between the values of P from the two fits is not obtained after the first iteration. Using this method, the correct values of S² and P will be obtained, regardless of the time interval, t, at which images are collected, provided that there are data at normalized time intervals below the convergence of the curves in Fig. 2 b.

Table 1 compares the results obtained from the procedure outlined above to results obtained from fitting the experimentally observed displacement data to Eq. 1. Data are fit to the models using the Levenberg-Markuardt nonlinear regression algorithm. The model fits to the experimental d² and data are shown in Fig. 5, a and b, respectively. For each technique, only the first 900 min of data were used; there appears to be a change in the persistence time after 900 min, which the persistent random walk model does not account for. We attribute this to some phenotypic change in the cell population that may arise from extended culture in serum-free media. Because the regressions are performed over the mean values, d² and , the standard uncertainties indicated for the model parameters (P and S² for Eq. 1 and P for Eq. 20) represent the accuracy with which the model is able to predict the parameters, and do not reflect the variance in the cell population. Therefore the uncertainties on these quantities shown in Table 1 provide us with a measure of the performance of each of the methods.

View larger version (9K): [in this window] [in a new window]   FIGURE 5  Model fits of the data from (a) Fig. 1 a to Eq. 1, and (b) Fig. 1 b to Eq. 4. Model parameters are reported as mean ± standard deviation.

Model development for anisotropic cell motility
For the case where cell motility is affected by some anisotropic property of the surface, such as a gradient in adhesion ligand concentration in one direction, the cell motility is biased in one direction. The anisotropy can be accounted for by adding a term to the Langevin equations for the velocity distributions resulting in a linear drift in the population, parallel to the gradient, which is characterized by a bias speed, S_Bias (6–9) (see Appendix). The movement of the cells in the direction orthogonal to the gradient is unaffected. That is, for a gradient in the x direction, y(t) 0, whereas x(t) S_Biast. S_Bias can therefore be estimated by fitting the average x position of the population as a function of time to a line. To correct for the anisotropic contribution to the total displacement of each cell, we compute the isotropic component of each cell's squared displacement as

(21)

The isotropic component of the cell motility, can then be further analyzed as above. It is important to note that there will be some variance within the population with respect to the degree to which each cell responds to the gradient. Thus, S_Bias cannot be computed for each cell. Rather, S_Bias is a statistical quantity associated with the cell population.

On the B160 peptide gradient surface, we observe an RMS bias speed of 1.6 µm h^–1 in the direction of increasing peptide concentration. More interestingly, there is a significant change in the isotropic component of the cell migration with changing peptide concentration. The isotropic cell motility parameters are shown for populations of cells at nine different adjacent fields of view across a B160 peptide gradient in Fig. 6. Statistical significance among populations was determined using Welch's t-test (p < 0.05). Because the speed data are log-normally distributed, statistics were computed on log-transformed speed and diffusivity data. We do not necessarily expect that the populations in adjacent fields of view should be statistically different since each population represents cells on a distribution of surface peptide concentrations. Fig. 6 c illustrates that the diffusivity of the cell population can be more than doubled by selecting an appropriate peptide concentration. However, if the surface is too adhesive, having too much peptide, the cell motility is reduced to about half that on unmodified PLL. The observation of a maximum in motility at an intermediate adhesion ligand concentration is consistent with our previous observations of cell motility on other laminin peptide gradients (15), as well as the observations of others who have studied the migration of anchorage-dependent cells (19,20) and the responses of neutrophils to soluble chemoattractants (21,22).

View larger version (17K): [in this window] [in a new window]   FIGURE 6  Persistence time (a), root mean-squared speed (b) and diffusivity (c) for nine populations of HFF on a laminin B160 peptide gradient. Error bars represent the standard deviations over the mean values obtained at each time point. In b and c, the populations marked with symbols are statistically different from populations not marked with the same symbol, as determined using Welch's t-test (p < 0.05) on the log-transformed speed and diffusivity data. When applying the t-test, a reduced number of degrees of freedom was assumed to account for the fact that the cell speed is correlated over the persistence time. This correlation means that not all speed observations are necessarily independent measurements, requiring that the number of degrees of freedom be normalized by the factor P/t.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

(22)

Equation 22 is the sum of a linear component, with slope n_dS²P and intercept -n_dS²P², and an exponential component with a time constant of P. However, because our d² data in this case are nearly linear with zero intercept, the fit eliminates the exponential and the intercept by setting P close to zero, and essentially fits the slope to the product of two parameters. This contributes to the large uncertainties on S^21/2 and P indicated in Table 1. Therefore, we might expect to improve the fit by rejecting the first 100 min of migration data, setting the initial position of each cell to its position at t = 100 min. However, this results in P = (0.02 ± 0.31) h and S^21/2 = (60 ± 500) µm h^–1, which is little improvement over the original fit shown in Table 1.

Alternative methods of analyzing cell motility that eliminate immotile cells from the analysis (e.g., (23,24)) may intrinsically correct for the large error in cell speed at short observation times by only considering cells whose distance traveled is greater than the uncertainty on the cell position. Demou and McIntire suggest that a description of the cell migration is not complete without information on how the measured cell speeds are distributed (24). Fig. 7 is a probability distribution of the cell speed data shown in Fig. 4 b, taken only from 100 to 900 min (eliminating the relatively high speeds observed during the first 100 min). The curve in Fig. 7 is a log-normal fit to the data.

(23)

The log-transformed speed data have a mean of 1.95 and standard deviation of 0.75. From this distribution, the RMS speed is computed as S^21/2 = (e^2µ+2)^1/2 = 12.3 µm h^–1, and the standard deviation of the speed is computed as _S = (e^2µ+2 – e^2µ+)^1/2 = 8.1 µm h^–1. This observed RMS speed of (12.3 ± 8.1) µm h^–1 accounts for the log-normal distribution. This is probably the best estimate for the RMS speed of the population if we want to describe the population dynamics.

View larger version (8K): [in this window] [in a new window]   FIGURE 7  Probability distribution function for the speed data from Fig. 4 b over the range 100 min t 900 min. The curve represents a log-normal fit to the data (Eq. 23).

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
We have demonstrated an improved method for fitting cell motility data to a persistent random walk model, which is suitable for slow moving connective tissue cells, such as fibroblasts. This method permits the accurate determination of cell motility parameters, by reducing the sensitivity to two common sources of error: uncertainty on determination of the cell position, and large variance in cell speed. There are three benefits obtained by using the procedure outlined above: First, the noise in the mean-squared displacement data associated with heterogeneity in the population is reduced by 50% by normalizing each cell's squared displacement by its cumulative mean-squared speed. Second, as illustrated in Fig. 2, the accuracy of the observed cell speed and displacements are extremely sensitive to cell position uncertainty for short observation times when the cell position uncertainty is not close to zero. The method outlined above corrects for this. Third, the cell speed measurements are sensitive to the path length approximation at long observation times. Our method accounts for the path length approximation by adding the term f(t). The method proposed here will be used to analyze the migration of fibroblasts on laminin peptide gradients and will enable accurate determination of cell migration parameters.

	APPENDIX

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
Isotropic and anisotropic persistent random walks from Langevin equations
Equation 1 is derived from the Ornstein-Uhlenbeck process, which uses Langevin equations for the orthogonal components of the velocity (1,3,4,6):

(A1)

where W_x and W_y are continuous random "white noise" functions such that W(t) – W(t') for t t' has a normal distribution with mean zero and a spectrum of magnitude . The cell position as a function of time is then found by integrating these equations, for the x-component, for instance:

(A2)

Doob (3) (after Uhlenbeck and Ornstein (4)) showed that by integrating the stochastic differential equations for each of the velocity components, the expected value of the squared displacement components, E{[x(t) – x(0)]²} and E{[y(t) – y(0)]²} become

(A3)

The orthogonal components of the squared displacements are added by the Pythagorean theorem to obtain the expected squared displacement in n dimensions. Making the substitutions S² = /ß and P = 1/ß, one recovers for the persistent random walk (Eq. 1). Thus, the cell motility can be described by either pair of parameters (S² and P or and ß).

To describe effects of external fields, or, in our case, the anisotropic response to the gradient, the original Langevin equations can be modified to account for external forces by adding acceleration terms, A(t):

(A4)

Uhlenbeck and Ornstein considered the case where A(t) represents a harmonic oscillator (4). Stokes et al. allowed A(t) to be a function of the cell orientation, exerting a stronger influence over cells aligned with the gradient (6). Distasi et al. discussed the case where A(t) is a constant (7). Setting A_y = 0 and A_x as constant, the simplest solution is obtained:

(A5)

Assuming that the initial x and y components of the velocities are distributed with a mean of 0 within the population, we obtain

(A6)

Thus, x increases linearly with time, at the rate A_x/ß, whereas y remains near 0. In our analysis, we do not require that v_x(0) = v_y(0) = 0, we merely assume that y(t) 0, whereas x(t) S_Biast, where A_x/ß = S_Bias. If we subtract the cumulative bias, S_Biast, from the x position of each cell, we should obtain for the mean x position a value near 0. Furthermore the expected value for the corrected x displacements for each of the cells behaves as

(A7)

which is identical to the expected x displacement for the case A_x = 0. Therefore, correcting the x position of each cell with the mean bias yields an isotropic population that behaves identically to the original isotropic persistent random walk (Eq. 1).

Notation for the population mean
Throughout the text we use the notation to indicate the result of the mean operation over a population of size n (rather than the operation itself). Thus, we write the mean of f_i over the population as f (rather than f_i):

(A8)

Uncertainties
The standard deviation refers to the combined standard uncertainty, and the standard error of the mean refers to the combined standard uncertainty of the mean. Eq. 6, 9–11, and 17 are for the theoretical uncertainty on an observation of the cell displacement, displacement to path length ratio, squared speed, and normalized migration parameter. These uncertainties represent type B standard uncertainties, because they are estimated assuming that they can be related to the standard uncertainty on the cell position coordinates, _x. For each of these quantities, we use the general formula for estimating the uncertainty of g(x₁, x₂, ...x_n), _g, from the normally-distributed uncertainties on the independent variables x_i, _x,i

(A9)

The uncertainty on the x and y components of the cell position are assumed to be independent and identically distributed with normal distributions, and are therefore indicated by a single variable, _x.

Errors
In Eqs. 12, 15, and 18, we represent the observed value of a function, f, normalized by its expected value as

(A10)

The expected value is the value obtained assuming that the model given by Eq. 1 is an accurate description of the cell migration. µ_f* is the predicted value of the ratio, f*, accounting for the path length approximation, and _f* is the type B standard uncertainty of f*, computed from Eq. A9. For the squared speed (Eq. 12) and the squared displacement (Eq. 15), we use S* and d* (rather than S^2* and d^2*).In Eqs. 14, 16, and 19 the error, _f, is computed from

(A11)

	ACKNOWLEDGEMENTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
Funding for this work was provided by the National Research Council through a Research Associateship to M.J.K.

	FOOTNOTES

 
Matt J. Kipper's present address is Dept. of Chemical and Biological Engineering, Colorado State University, Fort Collins, CO 80523-1370.

Official contribution of the National Institute of Standards and Technology; not subject to copyright in the United States.

Disclaimer: Certain equipment, instruments or materials are identified in this article to adequately specify the experimental details. Such identification does not imply recommendation by the National Institute of Standards and Technology, nor does it imply that the materials are necessarily the best available for the purpose.

Editor: Elliot L. Elson.

Submitted on September 8, 2006; accepted for publication April 18, 2007.

	REFERENCES

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
1. Dunn, G. A. 1983. Characterising a kinesis response: time averaged measures of cell speed and directional persistence. Agents Actions Suppl. 12:14–33.[Medline]

2. Gail, M. H., and C. W. Boone. 1970. Locomotion of mouse fibroblasts in tissue culture. Biophys. J. 10:980–993.[Abstract/Free Full Text]

3. Doob, J. L. 1942. The Brownian movement and stochastic equations. Ann. Math. 43:351–369.[CrossRef]

4. Uhlenbeck, G. E., and L. S. Ornstein. 1930. On the theory of the Brownian motion. Phys. Rev. 36:0823–0841.