Splinelike interpolation in particle tracking microrheology

Converting time dependent creep compliance to frequency dependent complex shear modulus is an important step in analyzing the results of particle tracking microrheology. Fitting a function to the whole time range and transforming it to calculate the shear modulus is one way of solving this problem. However, the creep compliance of many samples, such as gels of biopolymers, shows different trends under different time regimes. Fitting in these regimes segmentwise results in a function which usually cannot be transformed in a closed analytical form. In general, unlike for beta and cubic splines, also the continuity of the first derivative cannot be ensured. In this paper, we present a method for using segmentwise fitting and numerical conversion, discussing interpolation for improving the transition between the fitted ranges, and propose a dynamic sampling technique to control the accuracy of the resultant complex shear modulus.

Figure 1

A schematic demonstrating how linear interpolation (4) describes the experimental data. The data after the ${(k-1)}^{\mathrm{th}}$ data point is described by a linear equation $A_{k}t+J_{k-1}$. A step function $H(t-t_{k-1})$ cuts the line to zero for lower $t$ values (dashed lines). To correct the slope of the data after the next data point $t_k$, the current slope $A_k$ has to be subtracted, and the next $A_{k+1}$ one added. The constant $J_{k-1}$ is automatically formed by the sum of the previous segments.

Figure 2

Calculating the complex shear modulus of a Maxwell fluid from the creep compliance using the methods of Mason (lower frequency range on the left; $G^{\prime }$ is denoted by [blue] $*$ symbols, $G^{\prime \prime }$ by [pink] $\square$) and Evans ($G^{\prime }$ by [red] $+$, $G^{\prime \prime }$ by [green] $\times$). The storage modulus ($G^{\prime }$) values are the lower and steeper data, the loss modulus ($G^{\prime \prime }$) are on the top part of the figure. The theoretical values predicted by Eq. (9) are plotted with a solid black line. Though in the log-log representation it is not apparent, the Mason method shows a relative error up to $40\%$, while the Evans method has a relative error of about ${10}^{-8}$.

Figure 3

Testing the conversion methods using a Kelvin-Voigt creep compliance [Eq. (10), $E=100$ Pa and $\eta =2.5$ Pa s]. The ($+$) symbols (red) depict the storage modulus, and the ($\times$) symbols (blue) the loss modulus. The solid and dashed lines are the theoretical values. Results from the “direct conversion” method are presented in the top panel, showing excellent match at low frequencies, but the storage modulus deviates strongly (up to $100\%$ relative error) above about 30 s${}^{-1}$. In the bottom panel the results from the Mason method are plotted. It shows a relative error for the storage modulus up to $50\%$ at high frequencies, and up to $100\%$ for the loss modulus at low frequencies (below about 10 s${}^{-1}$).

Figure 4

Comparison of the storage modulus calculated with the method of Evans, using the same settings as in Fig. 3 ($+$ [red]) and $10\times$ oversampled ($\times$ [blue]). The relative error of the oversampled data set is $<1\%$.

Figure 5

Mean squared displacement measured with a tracer bead (diameter of about $1.9\mu$m) embedded into a semiflexible biopolymer (actin) network ($+$ symbols [red]). The double-logarithmic presentation clearly shows two different power law domains. The dashed straight lines show local power law fits in separate time ranges, while the beginning was better described using a Kelvin-Voigt model. The full line (black) is the interpolated and resampled data, indicating the extended dynamic range in the 0.001–0.01 s interval.

Figure 6

Frequency dependent storage modulus $G^{\prime }\left(\omega \right)$ and loss modulus $G^{\prime \prime }\left(\omega \right)$ obtained using Eq. (5) on the experimental data ($+$ and $\times$ symbols, respectively) and the fitted, interpolated, and smoothed data ([blue and green] solid and dashed lines between the symbols). The experimental data are presented as values averaged in logarithmically equidistant bins. The high noise of the experimental data is indicated by the error bars, which represent an estimated $\pm$ standard deviation in the bin. The data also show a breakdown of the storage modulus at high frequencies, which is an artifact related to the finite sampling bandwidth. A slight remaining of the oscillatory noise caused by the transitions between the consecutive fit regimes is also visible in the figure.