Exact occupation probabilities for intermittent transport and application to image correlation spectroscopy

Intermittent transport is frequently observed in nature and has been proven to accelerate search processes at both the macroscopic (e.g., animals looking for food) and microscopic scale (e.g., protein-DNA interactions). In living cells, active transport of membrane proteins (e.g., membrane receptors) or intracellular vesicles (organelles) has been extensively studied as an example of intermittent behavior. The intermittent stochastic process is commonly analyzed in terms of first-passage probabilities. Here we derive exact occupation probabilities of intermittent active transport, making such analysis available for image correlation spectroscopy techniques. The power of this new theoretical framework is demonstrated on intracellular trafficking of lipid/DNA nanoparticles in living cells for which we were allowed to quantify switching time scales.


Introduction
Intermittent mobility and transport, the stop and go movement, is a ubiquitous process that occurs during cargo transport in inhomogeneous media, shuttling proteins in a cell, according to human traffic patterns [1,2]. It has been argued that in some circumstances, intermittent movement has a distinct advantage in target finding processes that span, on macroscopic scales, from animals and bacteria looking for food [3] to, on the microscopic scale, the transport of vesicles to specific sites [4,5]. A particular cellular process that lies at the heart of neurotransmitter transport in axons and the gene delivery by natural and synthetic viruses is that of DNA-filled vesicle transport. This transport is characterized by a frequent switching between a passive diffusive mode and an active transport driven by molecular motors [6]. Various motor proteins such as kinesins, dyneins, or myosins are able to convert the chemical fuel provided by adenosine triphosphate (ATP) into mechanical work by interacting with the semiflexible oriented filaments (mainly F-actin and microtubules) of the cytoskeleton [7]. Because many molecules or larger cellular organelles such as vesicles, lysosomes, or mitochondria can randomly bind and unbind to motors, the overall transport in the cell is faithfully described by alternating (intermittent) phases of standard diffusive transport and phases of active directed transport powered by motor proteins [4,6].
Various experimental strategies have been developed that allow one to study the transport of vesicles in live cells at high spatial and temporal resolution. These include the direct visualization by single particle tracking (SPT) [6,[8][9][10][11] and a variety of image correlation spectroscopy (ICS) techniques [12][13][14][15]. Typically, data obtained from these techniques are analyzed in terms of mean squared displacements [16][17][18] that result in mean diffusion constants and mean transport velocities for the objects. Due to the averaging procedure, information about multiple independent populations and typical switching times is lost, despite the exquisite resolution the experimental techniques generate.
Here we derive the theoretical framework that allows one to use the full information density on squared-displacement distribution functions that are contained in tracking and correlation data. We obtained the exact occupation probability distribution for particles undergoing an n-state intermittent behavior between diffusive and active transport modes in a ddimensional space. Our result can be applied to all ICS techniques, such as spatio-temporal ICS (STICS) [19] (or its k-and ν-space versions [20,21]) and to particle ICS (PICS) [22] that is based on SPT containing super-resolution information.
The theoretical framework is subsequently applied to and validated by analysis of the intermittent transport properties of cationic-lipid/DNA nanoparticles in live cells by means of PICS. We found that a major fraction ( α 〈 〉 = ± 0.63 0.05) of nanoparticles experienced transitions between active and passive transport, while a minor fraction was locally trapped in subcellular crowded regions (e.g., the perinuclear region). Further, we determined average values of diffusion coefficients ( ) and velocities (〈 〉 = ± − v 34 6 nm s 1 ) that corroborate our previous results by SPT and STICS [15,23,24]. In addition, the method allowed us to investigate the time scales of the intermittent behavior, i.e., the average time scales T 1 and T 2 of diffusion and active transport, respectively, which define the corresponding average steady-state probabilities We have previously demonstrated that a lower transfection efficiency is related to lysosomal degradation, which, in turn, is enhanced by microtubule-mediated active transport [23]. An active transport with a large time scale might therefore increase degradation by lysosomal compartments, lowering the overall efficiency of the nanoparticles. This method can therefore highlight the correlation between switching time scales and transfection efficiencies of non-viral nanoparticles made of different lipid formulations.

Displacement distributions in intermittent transport
Our theoretical description was inspired by Shlesinger & Klafter [16] who derived a probabilistic description of diffusive transport that is interlaced with random trapping events. For intermittent behavior between n mobility states (e.g., diffusion characterized by diffusion constant D i , transport characterized by velocity v i , trapped ) in a d-dimensional space, the occupation probability for a particle in mobility state i to be at position ⃗ r after a (relative) time t is given by: is the probability to be at ⃗ r given that the time spent in phase i is t. The subscript i indicates only that the probability densities refer to state i and it is not a random variable in the distribution, nor does it indicate if the probability is conditional on being in state i. ϕ t ( ) i is the probability density that the transport process lasts for a time t before the particle switches to a state ≠ j i with different transport characteristics (note The occupation probability ⃗ P r t ( , ) of being at position ⃗ r at time t is more complicated. It can be built from a memoryless Markovian scheme. The particle can reach position ⃗ r at time t while being in state i at position ⃗ − ′ ⃗ r r at an earlier time τ − t . The state i might be the result of a transition from state ≠ j i in ′ ⃗ − ″ ⃗ r r at an even earlier time τ τ − ′. Therefore, its general form is: ) satisfies the Markovian chain Greenʼs function equation and it is the propagator for this stochastic process [16].
are the asymmetric and symmetric transition probabilities from state i to state j, respectively, as given by: where p i SS is the steady-state probability of state i.
We seek the solution of (2), which is most easily obtained in Fourier-Laplace space leading to the following master equation: . Given a probability distribution, which describes a particular mobility state ⃗ p r t ( | ) i and a distribution for switching times between states ϕ t ( ) For the sake of concreteness, in what follows, we simply assume state i, i.e., Under these assumptions, (5) gives: We note that ⃗ P r t ( , ) is properly normalized (i.e.,  which is equivalent to the expression found by Shlesinger & Klafter [16]. The general solution we found in (6) has further interesting properties that are noteworthy to mention: (i) (6) is also valid for the limiting case n = 1 (i.e., no intermittent behavior but only one biased diffusion distribution, can be performed analytically since ⃗ P k s ( , ) is a rational function in s (see the appendix); (iii) (6) gives us an unexpected handle to detect and quantify the intermittent behavior also from the mean square displacement MSD(t), which can be directly calculated from ⃗ P k t ( , ) (see the appendix), but only in the presence of at least one biased diffusion state. Even for the simplest case (i.e., d = 1 and n = 2), MSD(t) not only contains a diffusion term linear in t and a transport term quadratic in t, but also athird non-linear term that dampens out on the time scale of the switching between the two states: being the respective weights. This is well indicated by the kink intercepting at = t 1 s the red solid line that represents the steadystate probability p 2

SS
. As t increases, the kink disappears (i.e., the two states become more coupled) and at large times ( ≫ t T 2 ), the CDF is quite comparable to the one of a single diffusive state with an effective diffusion coefficient , i.e., at large times, it is likely that the particles diffuse more slowly. Interestingly, this result is also confirmed by (8) if one sets = D 0 2 and = v 0 2 . The occupation probability in its Fourier space ⃗ P k t ( , ) can be directly exploited to evaluate the spatio temporal correlation function defined in the STICS master equation [21], which is proportional to the following spatial convolution: PP r is the spatial autocorrelation of the experimental point spread function (PSF). Alternatively, it might be used for calculating the cumulative probability C R t ( , ) necessary in the PICS analyses [22]. For this purpose, we obtained the analytical relation between C R t ( , ) and ⃗ P k t ( , ) (see the appendix): where J 1 is the Bessel function of the first kind (of order 1) and ω is the localization uncertainty of the particle.

Intracellular trafficking of nanoparticles in living cells
We applied the intermittent transport model to the intracellular trafficking of cationic lipid/DNA complexes in living Chinese hamster ovarian (CHO) cells. Lipid/DNA nanoparticles are a selfassembled complex of plasmid DNA and a mixture of cationic liposomes. Depending on mixing ratio and the particular lipid used, the DNA is packed within the lipid bilayers, building up a multilamellar structure, or may be coated with a lipid monolayer arranged in a hexagonal lattice [25,26]. Several biomedical applications have exploited lipid/DNA complexes as nanoparticle-based carriers thanks to their low immunogenicity/toxicity and relative ease in synthesis and functionalization [27,28]. In previous studies by SPT and STICS [15,23,24], we demonstrated that lipid/DNA nanoparticles undergo both pure diffusion and biased diffusion, no matter which lipid formulation was prepared. We found that the active transport in the cell was mainly mediated by the microtubule network.
Hence, for data interpretation, we consider here a 3-state model in which transitions between a regular diffusion state '1' , switching timescale T 1 ) and a biased diffusion state '2' , switching timescale T 2 ) may occur. In addition, we considered a state '3' that characterizes immobile vesicles within the experimental timescale and that is uncoupled from states '1' and '2' ). This last fraction resembles nanoparticles that are trapped in subcellular crowded regions (e.g., the perinuclear region). The intermittent behavior of lipid/DNA nanoparticles is schematically described in figure 2.
Culture conditions for CHO cells, and lipid/DNA nanoparticle preparation and administration were carried out as described in [23]. The DNA inside the nanoparticles was Cy3-labeled plasmid DNA (λ = 550 nm exc and λ = 570 nm em ), purchased from Mirus Bio. Samples were mounted onto an inverted widefield microscope equipped with a × 100 oil objective (NA = 1.4). Cells were kept at°37 C and 5% of CO 2 during data acquisition. Measurements were done by illumination of the sample with a 514 nm laser for 5 ms at an intensity of − 0.3 kW cm 2 . Time series of images (200 frames @ 1 Hz) were acquired, setting a region of interest of 100 × 100 pixels (pixel size of ± 202 2 nm) on a liquid nitrogen-cooled black-illuminated charge coupled device (CCD)-camera. The image analysis was performed following the PICS method, discussed in detail in [22]. Briefly, the x y position of individual nanoparticles was determined frame by frame using a least-squares 2D Gaussian fit with high localization precision (ω = ± 27 12 nm exp ). The super-resolved images were spatio temporally correlated analogously to what is done by STICS [19]. For every time delay t, an integration over a circular spatial domain allowed us to obtain experimental CDFs that needed to be further corrected for random correlations.
Using an unweighted non-linear least-squares method, the experimental CDFs were therefore individually fitted according to the following model:   Figure 3 shows the experimental (blue circles) and fitted (red lines) cumulative distribution functions at = t 1, 5, 10, 20 s. It is noteworthy to observe that the kink introduced in figure 1 is here overshadowed for the presence of biased diffusion and a third independent (i.e., not switching) immobile fraction.
We were therefore able to determine the average fraction α 〈 〉 = ± 0.63 0.05 of nanoparticles undergoing transitions between passive and active transport with average steady-state probabilities  . The time-scales that characterize the intermittent behavior of lipid/DNA nanoparticles provide important insights into the time spent in the diffusive state, when the nanoparticles are much more likely to react and to release the DNA.
In conclusion, we presented here a robust analytical tool to investigate the intermittent transport of particles by either image correlation spectroscopy techniques or single particle tracking techniques.

Appendix.
To obtain ⃗ P k t ( , ), it is necessary to calculate the inverse Laplace transform of the following rational functions:  From the PICS theory [22], we know that C R t ( , ) is obtained by integrating the 2-dimensional spatial correlation function (substantially the same as the STICS master equation [21]) over a circular domain = + ⩽ C x y x y R {( , ): } 2 2 . Hence, we have to calculate: