1.1.

Intracellular Transport

Throughout its cycle, a living cell is subject to highly dynamic processes, coordinated in both time and space.^1,2 Despite great progress on the subject, intracellular transport remains insufficiently understood.^3–5 Trafficking at intracellular level results from both thermal diffusion and activity of molecular motors.⁶ This process has been examined through techniques that generally rely on fluorescence imaging,⁷ e.g., fluorescent speckle microscopy,⁸ fluorescence correlation spectroscopy,⁹ fluorescence resonance energy transfer,¹⁰ fluorescence recovery after photobleaching,¹¹ and fluorescence lifetime imaging microscopy.¹² Recently, there has been significant progress in superlocalization microscopy, such as stochastic optical reconstruction microscopy,¹³ photoactivated localization microscopy,^14,15 and their precursor, fluorescence imaging with one nanometer accuracy,¹⁶ which are closing the gap between cellular and molecular scale biology. In combination with fluorescence microscopy, particle tracking is a widely used approach for studying the transport of discrete objects inside the cell, such as embedded particles, organelles, and vesicles.^17–20 Analyzing the trajectories of these particles in terms of their mean squared displacements provides valuable information about the nature of transport and even the rheology of the surrounding medium.²¹ However, this approach has limited applicability when the system of interest is continuous, such as actin and microtubule cytoskeleton at spatial scales larger than the mesh size. Furthermore, particle tracking does not contain explicit spatial information.

In this paper, we study the spatiotemporal dynamics of mass transport in unlabeled cells, quantifying differences between nuclear and cytoplasmic mass traffic. We are able to access numeric values associated with both active and passive transport simultaneously and faster than particle tracking-based approaches.

We performed “dispersion-relation” analysis to quantify spatiotemporal transport, which can be used for both continuous and discrete mass distributions.^22,23 Due to the label-free method employed, our measurements are not limited by photobleaching or photoxicity, and thus can monitor cell dynamics over broad time scales and multiple cell cycles. We used spatial light interference microscopy (SLIM), which is a highly sensitive form of quantitative phase imaging (QPI), implemented as an add-on to an existing phase contrast microscope.

1.2.

Quantitative Phase Imaging

QPI has received tremendous scientific interest, especially in the past decade.^24,25 Since the 1950s, scientists have recognized the potential of optical phase to provide “quantitative” information about biological specimens, including cell dry mass.^26,27 QPI, in which the optical path length map across a specimen is measured, has become a rapidly emerging field with important applications in biomedicine.²⁸ QPI utilizes the fact that the phase of a field is much more sensitive to the specimen structure than its amplitude. As fields from the source interact with the specimen, phase shifts are induced in the scattered field with respect to the incident light. This phase shift contains the desired structural information about the sample under investigation. However, cameras and detectors can only measure intensity. To obtain the phase information, interference methods must be employed. Recently, a number of such methods have been developed from various groups around the world.^29–42

2.1.

Spatial Light Interference Microscopy

We use SLIM^43,44 to quantitatively obtain the path length differences from the phase images without staining the samples. SLIM is based on the principles of phase contrast microscopy⁴⁵ and holography⁴⁶ and introduces additional phase delays between the scattered and incident light. As a result, SLIM enables retrieving quantitative phase information of unstained samples, including live cells, with high sensitivity due to the low level of spatial noise (0.3 nm) and temporal stability (0.03 nm).^43,44 Recently, SLIM has become a useful tool for research in several biomedical areas.^47–54 In particular, SLIM is a valuable tool for studying intracellular transport across a broad range of spatiotemporal scales, as detailed in the next Sec. 2.2.^22,23,55 For all the measurements presented here, we used the commercial SLIM module (Q100, Phi Optics, Inc.), which yields 4 MP images, at maximum acquisition rate of $12\text{frames}/\mathrm{s}$, and is fully programmable in terms of data acquisition. The module was coupled to a motorized inverted microscope (AxioObserver Z1, Zeiss), equipped with an incubator and temperature, humidity, and ${\mathrm{CO}}_2$ control for long-term live cell imaging.

2.2.

Dispersion-Relation Phase Spectroscopy

Using SLIM, we can measure the changes in the optical path length through live specimens. These changes in the optical path length are proportional to the values of the dry mass in the cell.⁵² Intracellular transport measurements are performed in the focal plane. In this way, our analysis provides information about two-dimensional (2-D) transport. While a live cell is a three-dimensional (3-D) system, mass movement has, to the same approximation, the same statistical behavior in 2-D and 3-D.

The temporal autocorrelation associated with the dry mass density, $\eta$ (in $\mathrm{pg}/\mu {\mathrm{m}}^2$), is defined as

In Eq. (2), $D$ is the diffusion coefficient of the Brownian (diffusion) component and $\mathit{v}$ is the velocity of the active (deterministic) component. In order to solve for $g$, we use Fourier transform [Eq. (2)] with respect to $\mathit{r}$ and use the differentiation properties of the Fourier transform to obtain

Equation (4) indicates that a mass drift at constant velocity $\mathit{v}$ introduces a sinusoidal modulation to the autocorrelation function. Clearly, in a living cell we expect a distribution of velocities, with various magnitudes and orientations, say $P(\mathit{v}-{\mathit{v}}_0)$, where ${\mathit{v}}_0$ is the mean velocity. Averaging the autocorrelation function over the ensemble of velocity distribution yields

Note that the integral in Eq. (5) amounts to a Fourier transform with respect to velocity $\mathit{v}$. The conjugated variable is $\tau \mathit{q}$. Thus, Eq. (5) can be rewritten as

If we consider the second-order Taylor expansion of $\tilde{P}$ about the origin, we can obtain an analytic expression for Eq. (6) that is not dependent on the specific shape of $\tilde{P}$. We start with the expansion of $\tilde{P}$ and assuming isotropy, i.e., $P(\mathit{q})=P(q)$, we find

Next, we use the “central ordinate theorem” (see, e.g., Chapter 2 in Ref. 24) to identify each term in the expansion of $\tilde{P}$ with the moments of $P$, namely

In deriving Eq. (8), we used the fact that $P(\mathit{v})$ is a probability density, such that it is normalized to unit area. For Eq. (9), in addition to the central ordinate theorem, we also used the differentiation theorem, $d/d(\tau q)\leftrightarrow i\mathit{v}$, where $\leftrightarrow$ indicates Fourier transformation. Deriving Eq. (10) requires the use of the differentiation theorem twice, $d^2/d{(\tau q)}^2\leftrightarrow -{\mathit{v}}^2$. Note that Eq. (9) amounts to the first-order moment of $P$, which is zero (already shifted the origin of the velocity distribution at $v_0$). Finally, Eq. (10) gives the variance of the velocity distribution, $\mathrm{\Delta }{v}^2$.

Combining Eqs. (7) and (8), we obtain

Therefore, the velocity averaged autocorrelation function, which is the main quantity of interest, is computed from our data as

In all the measurements presented here, we did not observe a dominant velocity, ${\mathit{v}}_0$. This can be readily understood by an equal probability for the mass to be transported in opposite directions. Thus, with ${\mathit{v}}_0\cong 0$, Eq. (12) can be expressed in the frequency domain as

Equation (14) relates the spatial frequency (mode) $q$, with a temporal frequency quantity, $\mathrm{\Gamma }$, the diffusion coefficient, $D$, and the standard deviation of the velocity distribution, $\mathrm{\Delta }v$. The expression shown in Eq. (14) combines the spatiotemporal frequencies associated with mass transport, where the active transport is expressed by the linear term and the passive transport with the quadratic term in $q$. Hence, through QPI and dispersion-relation analysis, it is possible to retrieve information about and distinguish between both active and passive intracellular transport.

2.3.

Cell Culture Preparation

The sample under study consists of living HeLa cells from human cervix. The cells were grown in adherent growth mode in Dulbecco’s modified Eagle medium containing 10% fetal calf serum and antibiotics. The cells were imaged under controlled physiological conditions, 37°C and 5% ${\mathrm{CO}}_2$, using a petri dish. HeLa cells were imaged with the SLIM setup (see Sec. 2.1), using a $40\times /0.75\mathrm{NA}$ objective, over a 30 h time period with an acquisition rate of $0.1\text{frame}\mathrm{s}/\text{s}$, across eight clusters of cells.