Intracellular Traffic Observed in in silico simulations by Pair Correlation Analysis

In live cells the default mechanism of motion for many biological molecules is diffusion. Although unregulated diffusion produces a spatially isotropic distribution of molecules, it is has been shown that structural features of the cell create intracellular compartments that generate spatially heterogeneous molecule distributions. For example, insights into intra-cellular trafficking have been derived from measuring the effect of the cell nucleus on the diffusion of biologically active and inert molecules [7]. The effect of the nucleus on diffusion is readily apparent, if we scan across the axis of an NIH3T3 cell transiently transfected with the biologically inert fluorescent protein 5-EGFP (Fig 1A). In this case, the fluorescence intensity profile clearly shows the exclusion of 5-EGFP from the nucleus (Fig 1B). From this simple experiment we can deduce that the nuclear envelope behaves as an impenetrable barrier, around which 5-EGFP must diffuse. Given that intracellular trafficking of biologically active molecules is far more complex, the diffusive route traveled by a fluorescently labeled protein is not always evident from simple inspection of the fluorescence intensity distribution, and thus a more dynamic approach is required.

To gain insight into the diffusive motion active proteins, we employ an analysis method that is based on pairwise correlation functions. Using pairwise correlation analysis, it is possible to discern both diffusion rates and particle fluxes along a confocal line scan. These quantities are inferred by measuring temporal cross-correlations in fluorescence intensity between pairs of points a distance δr apart as a function of the time delay τ between measurements. To illustrate this idea consider the following in silico example. We simulate the diffusion of individual Rac1 molecules (D = 10μm²/s) inside a cell containing four diffusive traps (Fig 1C). The cell is 25.6μm long and 5μm wide; the circular traps are 2μm in diameter (see Methods). Rac1 molecules that diffuse into a trapping area can reversibly bind to the trap. When bound in a trap, Rac1 molecules are (1) spatially restricted to remain inside the trap, and (2) the diffusion constant reduces to 1μm²/s. During the simulation we repeatedly take “line scans”, which are measurements along a line that traverses the cell (Fig 1C). Whereas the line scans for the in vivo experiments measured fluorescence intensity in each pixel (0.1μm)² along the line, the line scans for our in silico simulations measure the number of molecules in square bins (0.1μm)² along the line. In both cases, we summarize the resulting data as an intensity carpet (Fig 1D); wherein, each row is a single line scan (Fig 1E and 1G) and each column gives an intensity profile (intensity vs time) for a single pixel (Fig 1D).

The impact on mobility can be measured by pairwise correlation analysis between an intensity profile and a neighboring profile, a distance δr to the right, as a function of the time delay τ. We choose δr such that it is large enough to measure mobility around each trap. For example, we can calculate the correlation between the intensity profile at pixel 64 and the intensity profile at pixel 69 (δr = 5 or 0.5μm) τ seconds later. The characteristic delay time, defined as the τ that generates the highest correlation value, is a measure of the time scale for a Rac1 molecule at pixel 64 to diffuse to pixel 69. We repeat this process for all pixels to map the molecular flow pattern of Rac1 along the simulated cell’s axis. We first set δr = 0 and thus derive an autocorrelation profile (pCF(0)) for each pixel (Fig 1E and 1F). For τ = 0, the value of the autocorrelation is equal to the mean squared number of particles in the pixel. Hence, the high value areas in the pCF(0) carpet (Fig 1F) indicate areas of high Rac1 concentration. Unsurprisingly, these areas are co-localized with the actin island traps.

We next introduce a spatial component to the cross correlation function by setting δr = 5 pixels (pCF(5)) and recalculating the pCF carpet (Fig 1G and 1H). Taking δr = 5 pixels, corresponds to a distance of 0.5μm. This distance allows us to cross correlate intensity fluctuations located outside the trap with intensity fluctuations located inside the trap, thus measuring the time taken to enter or exit this environment. To help interpret the pCF carpet, we use the SimFCS software developed at the Laboratory for Fluorescence Dynamics (www.lfd.uci.edu); more details can be found in S1 Text as well as the literature [3,5,7]. Briefly, we combine and average every 20 pixels (columns) of the pair correlation values. Each average is smoothed with a Gaussian filter, and we highlight the peak times. Each peak time is the delay time with the maximum pair correlation value (Fig 1H). The highlighted points, plotted every 20 pixels, are connected by interpolation and indicate the time scale for Rac1 to diffuse 0.5μm to the right. We find Rac1 takes longer to diffuse in and around the actin islands than elsewhere in the cell. It takes about 0.3s to diffuse 0.5μm to the right inside the islands, which is consistent with the diffusional time scale of the islands: . Outside the islands, the same trajectory takes much less than 0.1s, which is as expected: . Hence, pCF analysis is capable of revealing that actin islands act as barriers to mobility.

Gradient of Molecular Flow Observed in in vivo experiments by Pair Correlation Analysis

When probing the spatiotemporal dynamics of signaling molecules like Rac1, it is necessary to measure changes in both position and activity. These measurements are most often achieved by use of a FRET biosensor; wherein, changes in donor emission provide a readout of protein activity. Thus to determine the diffusive route Rac1 adopts upon activation, we recently combined biosensor FRET detection with pair correlation analysis [3]. Here, we repeat the experiment from [3]. We concomitantly measured the fluorescence intensity (Fig 2A) and fluorescence lifetime (Fig 2B) of a Rac1 biosensor in the donor channel along a line that extended from the rear to the front of a migrating cell. Lifetime analysis of the Rac1 biosensor FRET signal along the line scan revealed that the front of the cell (red time trace) is activated before the back of the cell (green time trace) (Fig 2C). In addition to this spatially dependent Rac1 activation, we found Rac1 mobility was also spatially dependent. We acquired three intensity carpets during the course of the experiment: one before EGF stimulation (Fig 2D), one between 0–180s after stimulation and one between 180–360s after stimulation. From each intensity carpet data, we performed pair correlation analysis of intensity fluctuations separated by a distance of 800nm (δr = 8 pixels) along the line scan (Fig 2E–2G). As in Fig 1H, we combined the data of every 20 columns into an average pair correlation vs delay time plot; we smooth this plot using a Gaussian filter. The highlighted red data points are the maxima of these smoothed curves (Fig 2E). For the data following EGF stimulation (Fig 2F and 2G), some of the smoothed curves have two maxima (see S1 Text for more details). The highlighted red and yellow data points are these maxima (Fig 2F and 2G). Before EGF stimulation (Fig 2E), Rac1 diffuses 800nm towards the front of the cell with a characteristic time of 0.03s. Note that this pCF carpet is qualitatively similar to the pCF carpet calculated from a simulation with uniform actin islands (Fig 1H). Three minutes after stimulation (Fig 2F), the time taken to travel 800nm is no longer constant across the axis of the cell (red curve Fig 2F); instead, there is a gradient of molecular flow with slower speeds near the front of the cell. After six minutes (Fig 2G), this gradient of flow is steeper (red curve Fig 2G); the characteristic times range from 0.1 to 1s. Additionally, the second set of highlighted data (yellow curve Fig 2G) is significantly different than the first, which indicates the presence of a second population of Rac1 whose mobility is spatially regulated separately from the first population.

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Fig 2. Pair correlation analysis of a Rac1 FRET biosensor reveals Rac1 activity to be spatiotemporally regulated by a dynamic gradient of protein mobility.

(A) Intensity image of a NIH3T3 cell expressing the Rac1 dual chain FRET biosensor in the donor channel before and after stimulation with epidermal growth factor (EGF). The white traces outline the cell’s position(s) from the previous panel(s). (B) Same cell as in (A) pseudo-colored according to donor lifetime. The blue to red color range corresponds to a change in lifetime from 2 to 3ns and therefore low to high Rac1 activity. The tau phase (in ns) is derived from phasor analysis of the fluorescence decay, as in [3]. Shorter tau-phase times correspond to higher FRET activity. (C) Average lifetime analysis of the first 10 pixels (back of the cell, green time series) and the last 10 pixels (front of the cell, red time series). This comparison reveals that after EGF stimulation Rac1 is activated earlier at the front than the back of the cell. (D) The intensity carpet that is derived from line scans acquired across the axis of the cell in (A). (E) Pair correlation analysis of the intensity carpet acquired before EGF stimulation. The highlighted data shows Rac1 molecular flow is uniform: it takes about 0.03s to traverse 0.8μm (pCF(8)). (F) Pair correlation analysis of the intensity carpet acquired 180s after EGF stimulation. The highlighted shows Rac1 molecular flow is non-uniform. At the back of the cell, traversing 0.8μm takes about 0.03s; this time becomes gradually longer towards the front of the cell. The second set of highlighted data (yellow curve) is not significantly different from the first (red curve). (G) Pair correlation analysis of data acquired 360s after EGF stimulation. The mobility gradient is steeper (red curve); the delay time ranges from 0.05s at the back to 1.2s at the front. A second gradient emerges (yellow curve); the delay times range from 0.03s to 0.08s.

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The simulation in Fig 1 (Fig 1H), wherein each barrier has equivalent affinity for Rac1, qualitatively matches the mobility of Rac1 in an unstimulated cell (Fig 2E); that is, Rac1 mobility is uniform across the cell. However, in a stimulated cell, Rac1 shows variable mobility across the cell (Fig 2F and 2G), possibly due to variable density of actin and hence variable Rac1 binding affinity. It may be that Rac1 interacts with different substrates with varying affinities in the membrane, and therefore, Rac1 mobility depends on cell polarization in response to external cues. We next perform simulations guided by this hypothesis.

Gradient of Rac1 Molecular Flow Produced by Actin Island Simulations

To test our hypothesis that actin-islands can create the spatially dependent mobility observed from in vivo experiments, we returned to in silico simulations. Instead of assuming all the actin islands bind Rac1 with equal probability, we first tested a scenario in which the affinity of the rear island was taken to be three-times higher than the others (Fig 3A). The resulting intensity carpet showed four regions with higher intensity than the background, corresponding to the islands (Fig 3B). The region with highest intensity corresponds to the island with highest affinity for Rac1. Calculating the pCF carpet (Fig 3C) reveals that Rac1 molecular flow is slowest in the island with highest affinity, slightly faster in the other three islands and fastest outside the islands. Hence the islands’ affinity inversely correlates with local Rac1 mobility.

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Fig 3. Simulations with islands of varying binding affinity.

(A) A simulation set-up showing the rear (leftmost) island binds on average 18.75% of all Rac1, and each of the other three bind 6.25% on average. Unbound Rac1 diffuses with D = 10μm²/s. If bound, Rac1 diffuses with D = 1μm²/s. (B) The resulting intensity carpet shows the highest accumulation at the rear island. (C) The pCF carpet (yellow curve) reveals four arc features, which indicate regions of slow molecular flow, across each island. The time needed to flow 0.5μm to the right (pCF(5)) is longer near the island with the highest affinity (0.6s) than the other islands (0.2s). (D) A simulation wherein the islands form a gradient of binding affinities. The affinities range from 37.5–6.25% from the rear to the front of the cell. (E) The resulting intensity carpet shows a gradient of accumulation of Rac1. (F) The pCF carpet reveals a gradient of arc features whose position corresponds to the position of the islands and whose length correlates with the affinity of the islands. The time scale for molecular flow near the islands ranges from 1s, 0.8s, 0.5s and 0.2s (back to front). (G) Simulation of a cell with actin islands forming a gradient of binding affinities. The affinities range from 6.25–37.5% from the back to the front of the cell. (H) The resulting intensity carpet shows a gradient of accumulation of Rac1. (I) The pCF carpet reveals a gradient of arc features whose position corresponds to the position of the islands and whose length correlates with the affinity of the islands. The pair correlation pattern is opposite of the previous simulation (F). The time scale for molecular flow near the islands ranges from 0.2s, 0.5s, 0.8s and 1s (back to front). This gradient is analogous to the gradient calculated for 3min after EGF stimulation (Fig 2F).

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Next, we extend our model by varying the affinity of each actin island. Specifically, islands closer to the leading edge have lower affinity than islands near the trailing edge (Fig 3D). The intensity carpet shows four regions with different intensities (Fig 3E). As before, the average intensity in each of the four regions is directly proportional to the binding affinity. The pCF carpet (Fig 3F) shows a gradient of molecular flow; wherein, Rac1 mobility is faster towards the leading edge. We can reverse this gradient by reversing the island affinities (Fig 3G). This model produces a pCF carpet (Fig 3I) qualitatively similar to the one observed from in vivo data (Fig 2F). That is, there is a diffusive gradient such that Rac1 mobility is slower towards the leading edge. Consistent throughout all our simulations, we find the characteristic time to diffuse around or through an actin island is proportional to the island’s binding affinity. We find that this mechanism to spatially regulate molecular flow is quite robust; the affinity of the actin island dictates the mobility at that location.

Finally, we aimed to reproduce the molecular flow observed 6 minutes after EGF stimulation (Fig 2G), in which there are two populations of Rac1 whose mobility is regulated separately. It is known that active Rac1 moves slower than inactive Rac1 [8]. In our model, diffusion is regulated by Rac1’s affinity for actin (Rac1’s primary effector). We postulated that inactive Rac1 also binds actin, but with a significantly reduced affinity. Therefore, keeping all other parameters equal, we consider two populations of Rac1 molecules (inactive and active) that possess different affinities for the actin islands (Fig 4F). In particular, we assumed the active population binds actin with twice the affinity as the inactive population. For computational simplicity, we simulated each population separately (Figs 3G and 4D) and combined the resulting intensity carpets. This approach does not affect the results of our simulations, because the two subpopulations are assumed to diffuse and react independently of one another. After combining the intensity carpets, we perform pCF analysis. The resulting pCF carpet shows two gradients of molecular flow: one for each Rac1 population (Fig 4G). Based on our previous results, we know the red curve corresponds to the high affinity active Rac1 population, and the yellow curve corresponds to the low affinity active Rac1 population. Our results indicate actin islands, which reversibly bind Rac1 and slow its diffusion, are sufficient to produce the spatially dependent molecular flow observed in vivo. Additionally, we find that if active and inactive Rac1 have different affinities for actin, then these two subpopulations would show different diffusive behaviors, similar to what has been observed experimentally.

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Fig 4. Investigating the cellular substructure during three stages of EGF stimulation by comparing pCF carpets.

(A) Three snapshots of FLIM data before, 3 minutes after and 6 minutes after EGF stimulation. We aim to replicate features in the pCF carpets from the in vivo data (Fig 2E–2G) using our in silico simulations. (B—C) Uniform actin islands, as shown in Fig 1C and 1H. (D) Actin islands across the axis of the cell bind Rac1 with different affinities. These affinities range from 3.12% to 12.15% from the back to the front, which is less than the simulation in Fig 3G. This arrangement of islands is similar to what we expect is present in vivo. (E) pCF analysis reveals four arc features of differing lengths, co-localized with the actin islands. Rac1 mobility is slower towards the front of the cell. (F) Modeling two populations of Rac1 by the superposition of two actin island gradients: one with twice the affinity (Fig 3G) as the other (Fig 4D). We combine the intensity carpets of the two simulations and perform pCF analysis. Although the two sets of islands are superimposed, here, we separate them for illustrative purposes. (G) The resulting pCF carpet for a cell with two populations of Rac1. We find two distinct gradients as highlighted by the curves: one for each population. The red curve corresponds to the Rac1 population with the higher actin affinity.

https://doi.org/10.1371/journal.pone.0143753.g004

Simulation Algorithm

A cell migrating on a 2D substrate in the xy-plane is modeled as an isosceles triangle whose base represents the direction of migration (Fig 5). In this system, diffusion along the z-axis is largely inconsequential, because the molecular counts used to produce the intensity carpets in vivo are projected onto the xy-plane, thereby destroying any information from the third dimension. Additionally, the computational complexity is significantly reduced when only 2 dimensions are considered. Rac1 molecules are modeled as non-colliding point particles; their positions are stored in (x,y) Cartesian coordinates. Our hypothesized actin islands, regions of dense actin molecules, are capable of binding freely diffusing Rac1 molecules; hence, these actin islands behave like diffusive traps, since actin-bound Rac1 has a smaller diffusion constant. The actin islands are modeled as circular patches, which can reversibly bind any Rac1 molecules located inside these regions (Fig 5).

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Fig 5. Simulation Geometry.

The cell boundary is shown in green as an isosceles triangle 25.6μm wide and 5μm tall. The actin islands are shown as black circles with 1μm radii. As discussed in the text, unbound Rac1 molecules are free to diffuse throughout the cell and reflect off the cell boundary. Bound Rac1 molecules diffuse slower and are restricted to the inside of the island in which they are bound. We can individually set the K_D of each island thereby setting, on average, the percent of the total Rac1 population bound to each island. The 256 magenta boxes represent the (0.1μm)² bins used to generate the intensity carpet; these bins do not affect the behavior of the molecules.

https://doi.org/10.1371/journal.pone.0143753.g005

As suggested by [10], we use the Euler-Maruyama method [11] to perform stochastic simulations of diffusing particles. That is, knowing the current position of a Rac1 particle, (x(t), y(t)), we calculate its position at the next time step, (x(t+Δt),y(t+Δt)), using the following equations: (1) (2) Here, D is the diffusion constant, Δt is the time step and the W_n’s are random numbers drawn from a Gaussian distribution with a mean of 0 and a variance of 1. Unbound Rac1 particles diffuse freely inside the cell with D_unbound = 10μm²/s. The boundaries of the cell are reflective. Hence, if a molecule attempts to leave the cell (i.e. the diffusion calculation places the particle outside the cell boundaries), then the molecule is elastically reflected back inside the cell. Bound Rac1 particles diffuse freely inside the actin island to which they are bound with D_bound = 1μm²/s. The boundary of the island is reflective only to bound molecules. Hence, if a bound molecule attempts to leave the island (i.e. the diffusion calculation places the particle outside the island), then the molecule is elastically reflected back inside the island. Consequently, all bound Rac1 molecules are restricted to an island. The converse, however, is not true; not every Rac1 molecule positioned inside an actin island is bound. Unbound Rac1 molecules, which are positioned inside an island, are considered to be diffusing over/under/through that island without penalty. It is only in this condition that a binding reaction may occur.

In addition to diffusion, binding and unbinding events are also calculated at each time step. If a Rac1 molecule is, unbound and positioned inside an island at time t, then the state of that molecule can be switched to bound with probability: P_bind calculated for each island by: (3) Here, k_on is the rate constant for binding to the island; Δt is the time step; v is the volume of the island, and V is the volume of the entire cell (we assume the cell is 1μm thick).

Separately, we calculate all unbinding events between time steps. If a Rac1 molecule is bound at time t, then its state can switch to unbound with probability P_unbind: (4) Here, k_off is the dissociation rate constant. When changing the state of a Rac1 molecule, due to either a binding or unbinding reaction, we do not change the position of the molecule.

We calculate the reaction rates based on an average fraction of total Rac1 molecules that we wish to have bound in each island Eq 5. (5) Here, f is the average fraction of Rac1 molecules bound in the island, and 1–F is the average fraction of Rac1 molecules not bound in any island. Note k_on should be expressed in units of μm³/s using a conversion factor such as 0.6022/(nM μm³). Additionally, the fraction of molecules bound at each island also relates to the reaction probabilities Eq 6. (6)

For example, if we desire each of the four islands to bind, on average, 6.25% of all Rac1 molecules, then F = 0.25 and for each island f = 0.0625. We know the volumes (v and V) from Fig 5. Unfortunately, instead of a value for P_bind or P_unbind, we are only left with a ratio between these reaction probabilities. We proceed by assigning an off-rate within the Biological regime. We choose the off-rate such that the time-scale for unbinding is equal to the time-scale for a bound molecule to diffuse the length of the island: see Eq 7. (7)

This rate is substituted into Eq 4 to get P_unbind, which in turn, is substituted into Eq 6 to get P_bind. In this case, P_unbind = 0.5×10⁻⁶ and P_bind = 0.85×10⁻⁶ for a time-step Δt = 1μs. The reaction probabilities for each simulation are annotated in S1 Text.

Pair Correlation Carpet

A line-scan technique was used during in vivo experiments [3,5,6], which reports the fluorescence intensity of Rac1-GFP molecules. The intensity is sequentially measured from pixel 1 to, about, pixel 300 to complete one line scan. Many line scans are taken and these scans are compiled into an intensity carpet on which pair correlation analysis can be performed. To produce an in silico analog to the intensity carpet, we tally the particles by their position as they are simulated (see above). To establish the line along which we will scan, we arrange 256 square bins sized (0.1μm)² along the center of the cell (Fig 5). We cumulatively count the number of Rac1 molecules positioned inside a bin during a 25μs time window. Each bin is scanned sequentially; that is, we tally the counts from the first bin for the first 25μs; then we tally the counts from the second bin for the second 25μs etc. After 6.4ms (256*25μs) one full line scan is complete, and the next line scan begins immediately. Each intensity carpet has 47000 lines, corresponding to a simulation time of just over 5min. The bin size (0.1μm)² is roughly equivalent to the size of the laser used in the in vivo experiments, and the measuring time (25μs) is equivalent to the length of time a pixel was monitored in vivo.

To better understand the flow of particles within the cell, we calculate the pair correlation function (pCF) from the intensity carpet. First, we extract the intensity time trace (intensity versus time) for each pixel (e.g. cartooned white curves in Fig 1D). Second, we calculate the correlation coefficient between two of these traces while sweeping two parameters in space (δr) and time (τ). With regards to space, we consider the traces of two pixels a distance δr apart; note that when δr = 0, we are computing the autocorrelation value (Fig 1E and 1F). With regards to time, we shift the trace from the second pixel by τ seconds. Hence the correlation coefficient for a pixel given a pair of parameters (δr, τ) describes the likelihood a particle will diffuse from that pixel to a pixel δr away in τ time. A value of 0 indicates this diffusive trajectory is impossible; while, a value of 1 implies a complete directed flow. Note that there is directionality in our pCF. A left to right calculation (e.g. correlating pixel 0 with pixel 5) describes the diffusion landscape from the back of the cell to the front (line scans are taken as such). A right to left calculation (e.g. correlating pixel 5 with pixel 0) describes the diffusion landscape from the front of the cell to the back.

Our results are independent of the cell’s shape, because the cell boundary is sufficiently far from our region of interest. For example, the sharp, leftmost corner produces a slight force to the right; however, this force quickly dissipates and has little effect beyond x = 5μm. All the pCF carpets presented here show pCF analysis for x≥6.4μm.