Abstract
Scope of the present work is to infer the migratory ability of leukocytes by stochastic processes in order to distinguish the spontaneous organization of immune cells against an insult (namely cancer). For this purpose, spleen cells from immunodeficient mice, selectively lacking the transcription factor IRF8 (IRF8 knockout; IRF8 KO), or from immunocompetent animals (wildtype; WT), were allowed to interact, alternatively, with murine B16.F10 melanoma cells in an ad hoc microfluidic environment developed on a LabOnChip technology. In this setting, only WT spleen cells were able to establish physical interactions with melanoma cells. Conversely, IRF8 KO immune cells exhibited poor dynamical reactivity towards the neoplastic cells. In the present study, we collected data on the motility of these two types of spleen cells and built a complete set of observables that recapitulate the biological complexity of the system in these experiments. With remarkable accuracy, we concluded that the IRF8 KO cells performed pure uncorrelated random walks, while WT splenocytes were able to make singular drifted random walks that collapsed on a straight ballistic motion for the system as a whole, hence giving rise to a highly coordinate response. These results may provide a useful system to quantitatively analyse the real time cellcell interactions and to foresee the behavior of immune cells with tumor cells at the tissue level.
Introduction
The development of theoretical frames able to describe complex biological systems has been a leitmotif in the work of many physicists and mathematicians and it is deeply linked with the possibility to obtain measurements of the parameters and variables describing the systems under observation.
The immune system is a striking example of an integrated information system, engaged in coordinated hostprotective activities. Quoting Kim and coworkers, the immune system operates according to a diverse, interconnected network of interactions and the complexity of the network makes it difficult to understand experimentally. On one hand, in vitro experiments that examine a few or several cell types at a time often provide useful information about isolated immune interactions. However, these experiments also separate immune cells from the natural context of a larger biological network, potentially leading to nonphysiological behavior. On the other hand, in vivo experiments observe phenomena in a physiological context, but are usually incapable of resolving the contributions of individual regulatory components^{1}.
In this regard, the reconstitution of cellular microenvironments exploiting microfluidic chips and cell cocultures has emerged as valid and innovative approach to address limitations of in vitro static endpoint measurements and in vivo experiments. Highly controllable and engineered microenvironments have been realized to mimic in vivolike situations at different scales while maintaining compatibility with advanced modern imaging techniques and molecular biology methods^{2,3}. This has provided a new window to finely investigate on the realtime imaging of processes involving motility and cellular interactions in highly relevant context (normal or disease).
From the theoretical counterpart, models coming from mathematics or theoretical physics, such as maximum entropy principle^{4}, disordered statistical mechanics^{5,6}, complex optimization^{7}, graph theory^{8}, stochastic processes^{9} and dynamical systems^{10} are being adapted to biological systems, allowing a more complete picture of cellular behavior from advanced imaging studies. In parallel, the development of methods aimed at considering the system as a whole, hence with all its constituents mutually interacting, have become a major topic, involved as a necessary step beyond reductionism limitations^{6,11,12,13}. For example, cancer progression involves multiple events and is the result of the interactions with cells of the immune system within the tumor microenvironment^{14,15,16,17,18,19,20,21}.
Previous studies on the tumor microenvironment, performed mainly by using animal models^{22,23}, have suggested that immune cells, infiltrating the growing tumor mass, play a crucial role in the control of cancer expansion and of the spread of metastasis. Indeed, immune cells enter the cancer site by means of angiogenesisinduced blood vessels and by the generation of a large network of chemokines and cytokines^{24}. After their entry, immune cells actively interact with cancer cells and other components of the extracellular matrix, thus dictating the fate of the cancer tissue, namely expansion or eradication. In this complex scenario, the migratory capability of both immune cells and cancer cells plays an important role for tumor eradication or expansion, both inside the tumor microenvironment and at systemic level for metastatic spread. In addition, chemokines and their receptors finely orchestrate the destination of cancer cells and immune cells, thus meaning that the generation of multiple chemotactic gradients is pivotal for the initiation and sustainment of the interactions between cancer and immune cells. Despite the growing knowledge in this field, the mechanisms driving the mutual crosstalk between cancer and immune cells are poorly defined due to the difficulty in visualizing the physical interactions at the tumor site.
In this context, we recently exploited microfluidic devices to investigate in real time on the mutual interactions between cancer and immune cells^{25}. We cocultured mouse melanoma cells with competent or deficient immune cells and compared several mutual migratory parameters: this CellOnChip approach has the great advantage of creating realistic models of in vivo microenvironments onto substrate perfectly compatible to modern microscopy tools and molecular biology methods and it may constitute the lacking bridge between biology and theoretical models.
Results
Making up the experiment for datadrive
The data analyzed in this paper were gathered from an experiment whose biologic characteristics have been described in detail elsewhere^{25}. The results of such experiment are summarized hereafter for reader convenience. The basic idea was to reproduce on chip the interactions between cells of the immune system with the tumor, mimicking as much as possible those occurring in vivo. To do so, we realized a microfluidic chip, shown in Figs. 1, 2, which was basically divided in two different zones, one for the culture of immune cells (marked in red) and one for the tumor (green chamber), which consisted of the murine metastatic melanoma cell line B16.F10.
The two zones were connected by an array of microchannels (section 10 × 10 mm^{2}) which allowed the migration of cells. We used for the immune system a mouse spleen homogenate, which contains the whole pool of mature immune cell populations, ranging from T and B lymphocytes to phagocytes. The experiments were carried out taking advantage of two different sets of splenocytes, the first was from a wild type (WT) mouse, namely a healthy immune system, the second was from a mouse deficient for the transcription factor IRF8, essential for the induction of competent immune responses^{26,27,30,31,32,33}.
In this experimental setting, both melanoma and immune cells could mutually migrate through the whole microfluidic system. The two systems were monitored by means of fluorescence microscopy up to 144 h and timelapse recordings of the first 48 h of culture, chosen as the most representative ones (as we observed the major cell motility in that temporal gap). For the B16WT system we observed a clear migration of the immune cells towards the melanoma and the formation of immune cell clusters around the B16 cells (see Fig. 3). In the case of the B16KO system, on the contrary, nor the KO cells show any response to the melanoma neither their trajectories were focused toward the insult (see Fig. 4).
Furthermore, the findings illustrated in Fig. 5 suggest that WT splenocytes seem to be endowed with a certain ability in limiting the B16 liveliness in comparison to KO splenocytes cells loaded with melanoma cells. Indeed, the perturbation factor is significantly increased for B16 cells when loaded with KO rather than WT immune cells. The perturbation factor, measuring the B16 cell area fluctuation at each timepoint, may presumably be correlated to the vitality and thus aggressiveness of melanoma cells. This hypothesis is in agreement with previous findings showing how shape factor may be helpful for investigations in cell viability^{28,29}.
The experimental data resulting from microscopy analisys give information about the cell dynamics, so the starting data set of our mathematical analysis consists in the measurements of cell positions during time. We followed the trajectory of N distinct migrating splenocytes and for each of them we collected time series of their position. Data were acquired at a constant rate τ = 4 min (due to the experimental setting) and measured with precision ~0.1 μm (corresponding to 1 pixel). A total of T = 350 measurements were performed, in such a way that, for each splenocyte, we have a walk labelled as j, with j ∈ (1, …, N) and characterized by the set of consecutive positions (, ), with i ∈ (1, …, T). As a result, being Δx_{i} = x_{i}_{+ 1} − x_{i} and Δy_{i} = y_{i}_{+ 1} − y_{i}, we can also derive the series of steps lengt hs Δx_{i}_{(i = 1,…,T)} and Δy_{i}_{(i = 1,…,T)} taken by cells along the time intervals of length τ.
A vocabulary for a quantitative treatment
In what follows, we introduce the parameters (observables) that we candidate to suitably describe splenocyte behavior into a quantitative frame. We give here a simplified definition of them, in order to allow the full understanding of the analysis described in the following. Rigorous mathematical definitions and descriptions can be found in the appendix (Suppl.Mat.).
The splenocytes under investigation perform paths which can, in principle, exhibit some degree of bias (e.g. due to chemical gradients) and some degree of stochasticity (randomness, e.g. due to noise). Hence, we propose to model such paths by means of random walks characterized by (synchronized) discrete time steps and moving on a continuous twodimensional space (xy). This kind of random walks can be described in terms of a probability distribution, giving the probability that the walker has covered a distance r in a time t.
We will call ψ(r − r′, t) the probability that at time t a step from r′ to r is performed (see the appendix for details). This probability qualitatively controls the resulting random walk. Indeed, according to its mathematical expression, it possibly gives rise to deterministic walks (corresponding to a ballistic motion), to correlated walks (corresponding to a motion with a preferred direction), to completely stochastic walks (corresponding to an isotropic motion where steps have fixed length).
In Euclidean structures, like the twodimensional substrate considered here, we can decompose r into its normal coordinates, i.e. r = (x, y), and, analogously r − r′ = (x − x′, y − y′) ≡ (Δx, Δy). Moreover, assuming that Δx and Δy are independent, ψ(Δx, Δy) can be factorized as ψ(Δx, Δy) = ψ_{x}(Δx)ψ_{y}(Δy).
The knowledge of the specific distribution ψ(Δx, Δy) possibly allows to get important information about the resulting random walk. For instance, one can show that, when diffusion is isotropic, i.e. it is equal in the x and y directions, any distribution ψ(Δx, Δy) fulfilling the central limit theorem asymptotically leads to the wellknown diffusive limit characterized by the normal distribution^{34}.
Hence, asymptotically, whenever noise is prevailing, we expect to observe a Brownian motion, i.e. , while, whenever there is a real presence of a drift (signal), we expect a ballistic motion, i.e. r ∝ t.
Hereafter, we summarize the observables that we are going to analyze, stressing on the kind of information which can be conveyed via their investigation.
Observable one: Step length
Step lengths are measured in micrometers to quantify the distance covered by the cells during the timeinterval among two different (adiacent) frames, i.e. τ = 4 min. The distributions of step lengths immediately provide fundamental information:

Direction dependency. By means of specific correlation coefficient (Pearson's coefficient) ρ_{ΔxΔy} we can highlight possible correlations between the time series {Δx_{i}} and {Δy_{i}} for a given walk. This analysis also allows to check whether propagations along the two dimensions are independent, namely if ψ(Δx, Δy) can be (at least approximately) factorized into ψ_{x}(Δx)ψ_{y}(Δy).

Distribution of step lengths. If the distributions display a finite mean μ and a finite variance σ^{2}, one can apply the central limit theorem (CLT) to get that (where n is the number of considered steps) has average value converging to nμ with standard deviation scaling like , (and analogously along the y direction). In this case the walk displays a characteristic length scale given by μ.
If a distribution respects CLT, roughly speaking, no Levy flights for its variable are expected: this means that if, for instance, ψ_{x}(Δx) is the probability distribution of the jumps Δx performed by the cells among two adjacent frames and it respects CLT, then the probability to observe a huge jump (whose length is by far larger than the characteristic average amplitude of jumps μ) is for practical purposes negligible.
In turn, when CLT convergence is forbidden (as for Scale Free distributions) or strongly slowed (as in lognormal ones), a much more broad ensemble of step lengths is expected: as a result, we can infer the degree of homogeneity of a process (i.e. the way cells move toward a direction) directly by checking the CLT convergence of the probability distribution of its related variable(s). This is important because we will show that as far as KO spenocytes are concerned, their step length distributions strongly respect CLT (hence their movements are highly homogeneous, i.e. share a characteristic step length), while when WT are analyzed, their step length distributions are found to converge to lognormal shapes thus suggesting the presence of broaden steps which, in turn, possibly reveals a hidden heterogeneity at the cellular level.
However this observable alone is not able to distinguish among purely diffusive motion and ballistic one, thus further observables are required to properly frame the system's behavior.
Observable Two: Time Correlations
To this task, another key observable is the distribution of time correlations: the latter indeed has the capability to detect the presence of a bias (and to discover the presence of ballistic motion as a consequence of the bias itself): the underlying idea is that if a drift is present (e.g. driven by a chemical gradient), then step by step the jumps of the cells will no longer be uncorrelated as, on average, cells will point to the source of the gradient. Thus, analyzing the correlation among two (temporallyadjacent) jumps would reveal the presence of the drift itself. In absence of any drift, clearly, no angular correlation is expected among the direction of motions of cells collected between two closer frames.
It may be worth stressing that, beyond the scope of the present analysis, its knowledge may however contribute to the experimental detection of chemical gradients and of chemotactic phenomena.
We distinguish among

Angular correlation. Due to the presence of a forcing field (i.e. a chemical gradient), a diffusive particle may exhibit a preferred direction, which can be measured in terms of the angle θ between two consequent steps; in particular, to highlight the existence of a shortterm memory we consider the temporal angular correlations C(τ) defined as
where the average is performed over t and the average 〈·〉 is performed over the set of random walks. Hence, C ~ 0 implies isotropy, which, in turn, implies that migrating white cells are not pointing to any specific target; conversely, C ≠ 0 is a necessary requisite in order to keep a coordinate motion toward the target (melanoma cells in the present case).

Acceleration phenomena. In order to figure out the possible existence of slowing down and/or speeding up phenomena one can consider the step length of a single random walk at each time step k and calculate
which is the average of the steps taken up to the nth time step by the single cell. A (at least approximately) constant behavior of the average ensures that, independently of the instant of time and of the place where the cell is currently located, the length of the step taken tends to remain the same. On the contrary, in the case of an acceleration/deceleration, an increasing/decreasing behavior of is expected.
The quantities described so far focus on “microscopic” features of a walk as they imply a fine zoom on the walk itself. From such a description one is usually able to derive the “macroscopic” behavior, typically measured in terms of the mean displacement 〈r(t)〉 as a function of time, whose following observables are due to.
Observable Three: Mean displacement
In a nutshell we now deal with the stochastic version of the equation of motion in classical dynamics (that is the trajectory). While, at the microscopic level, temporal determinism coded by the Newton's law is lost and each single cell has its own unpredictability regarding its dynamics (thus focusing on single particle trajectory becomes meaningless), the ensemble average of the whole cells, however, still obeys a deterministic evolution such that inferring the parameter ruling such an averaged dynamics may allow quantitative prediction for future movements.
To this task one can consider both the displacements x(t) and y(t) along the x and y axes and the overall (absolute) displacement
Notice that x(t) and y(t) are calculated with respect to the initial point in order to get the effective displacement (the rigorous notation for r(t) should be in order to account for the effective displacement with respect to the original position; however, the notation has been lightened throughout the paper). All these quantities can be computed for every cell at each time and, then, they are averaged over the ensemble of cells, obtaining 〈x(t)〉, 〈y(t)〉 and 〈r(t)〉. In fact, the latter quantities represent the mean displacements of the system as a whole.
Now, the scaling of the mean squared displacement 〈r^{2}(t)〉 with respect to t is often used to qualitatively define the kind of diffusion. For instance 〈r^{2}(t)〉 ~ t^{2} (i.e. ) is typical of simple diffusion, 〈r^{2}(t)〉 ~ t^{2} (i.e. 〈r(t)〉 ~ t) is typical of drifted motion, while a power law 〈r^{2}(t)〉 ~ t^{α}, with 0 < α < 1 or 1 < α < 2, is referred to as anomalous diffusion emerging, for instance, in the presence of crowded environment and/or fractal substrates^{34}.
Observable Four: Tortuosity
When dealing with the movement of a biological particle one is often interested in the tortuosity of its path, namely in how twisted the path is in a given space or time^{36}. Clearly, this is related to the mean displacement: highly tortuous paths will spread out in space slowly, while straight paths will spread out in space quickly. Hence, it can be useful to measure and study the tortuosity of observed paths in order to understand the processes involved, estimate the area spanned by a cell and predict spatial dispersal.
It is worth noticing that this observable quantifies the degree of straightness shared by the motion under analysis and, coupled with the angular correlations (see Time Correlations), it may help quantifying the presence (or lacking) of chemical gradients.
Tortuosity can be quantified by comparing the overall net displacement of a path with the total path length. For example, if a random walk starts at location (0, 0) and, after n steps with lengths l_{j} (j = 1, …, n), ends at (x(n), y(n)), then we can measure the socalled straightness index S as^{35}
which ranges in 0 and 1, where 1 corresponds to movement in a straight line (the shortest distance between two points in the two dimensional Euclidean space the LabOnChip has built on) and 0 corresponds to a returning (thus tortuous) path.
Observable Five: Ergodicity
In the context of stochastic processes, we define ergodic a system where time and ensemble averages converge^{37}, namely the time average of a sequence of events is equivalent to the average over all the possible states of the system at a fixed time. So in an ergodic system the evolution of a single element is representative of the whole system as an average on all possible states of it. Simple Brownian motion owns this property. Hence, the possible non ergodicity of the system is a measure of how large is the deviation of the process examined from a normal diffusion. A convenient way to check ergodicity is the comparison between the mean squared displacement (MSD) of diffusing particles and the timeaveraged MSD defined as:
where T is the total measurement time and τ is the lag time which separates the trajectory points.
In the case of Brownian motion in two dimensions, we have that, over long measurement times,
This is essentially the same as the MSD averaged over a large ensemble of particles,
This equivalence quantitatively confirms that in the Brownian motion the ergodicity is preserved. In particular, in this process, a measurement of and, therefore, D in the time interval (0, t) will be identical to a measurement in the interval (t, 2t) for large t. Therefore, if a system shows the ergodicity property, it surely respects the timetranslational invariance, which, instead, is not applicable in many kinds of anomalous diffusion, such as subdiffusion processes^{38}.
From a practical perspective, the introduction of the concept of ergodic motion in biological dynamics may significantly contribute in the experiment's management because, still roughly speaking, ergodicity allows to exchange temporal averages with ensemble averages (i.e. average over the cells). If we image an experiment whose temporal length should be, for example, 10 days, but we are allowed to use the laboratory for 5 days only, then, if the motion under study is ergodic (to be verified separately), it is possible to use a (twice) larger system (keeping fixed the cell's density obviously) for 5 days only in order to collect the same data available in the 10days experiment with the original (smaller) system.
Analysis of differential immune cell dynamics
Knockout splenocytes: a simple random walk
Once defined the observables of interest, we analysed the experimental data to verify if it was possible to identify a clear trend in their distribution that would describe their dynamic behavior. In the following we will call “Spl compartment” (the central one in Fig. 2) the channel where the splenocytes were at the beginning of the experiment and “B16 compartment” the channel in which melanoma cell were cultivated. We started analysing the KO splenocytes behavior. During the experiment, they were poorly reactive to melanoma cells, almost no cell was able to get into the microchannels. Thus, the motion of these cells was studied only in the Spl compartment.
The available data were filtered with the compromise of obtaining the positions of splenocytes monitored from the same instant of time t_{0} (and not to include splenocytes initially too close to the channel wall, in order to avoid collisions with it, which could distort results) and, at the same time, of getting a reasonable statistics with the minimum number of analyzed cells. From this selection procedure we outlined N = 30 splenocytes for our analysis. It is remarkable that with such a small number of elements the statistics were already very significant (as shown below). Implication on collective capabilities of leukocytes will be discussed at the end of the paper.
As anticipated, our analysis begins with the determination of microscopic quantities.
First, we notice that there is no manifest spatial correlation between Δx and Δy along a single walk. Also, the histogram of the Pearson correlation coefficient ρ_{ΔxΔy} for each walk peaks at zero (not shown). As for the distribution of step lengths ψ_{x}(Δx) and ψ_{y}(Δy), we find that, at each time step, splenocytes perform a jump whose width is stochastic and exponentially distributed, as shown in Fig. 6. In particular, for both directions (displacements along x and along y) and for both branches (negative and positive displacements) the best fit is given by
best fit coefficients, summarized in Tab. I, are consistent with the experimental average values and highlight overall, within the error, a good symmetry. This suggests that KO cells are not pointing to any target.
Moreover, the exponential distribution clearly satisfies the central limit theorem and this rules out the existence of Lévy flights among KO splenocytes. In other words, these splenocytes proceed smoothly and with rather regular steps.
Let us now consider the turning angle θ between two consecutive steps. The distribution of θ over the whole set of walks and the related time correlation C_{KO}(τ), (see Eq. 1) are shown in Fig. 7. The turn amplitude has zero mean, implying again that every choice of direction is not correlated with the previous one and the motion is isotropic.
Moreover, C_{KO}(τ) has zero average, confirming that there is no connection between the direction of a step with that of the following one so that one can exclude the presence of memory or collective organization in the process.
Finally, we do not find any significant temporal correlation among steps since, for each cell, the step (see Eq. 2) converges to a constant value; as shown in Fig. 8, no acceleration is observed in the process and the instantaneous speed 〈v(t)〉 is stable. More precisely, it fluctuates around 1.5 ± 0.1 μm/min in agreement with the results of other in vitro experiments, showing that, in the absence of external gradient guiding the KO splenocytes, these move with an average speed 1 ~ 4 μm/min^{39}.
Thus, from this microscopic analysis we can confidently derive that KO splenocytes move rather uniformly and isotropically, with no manifest bias, consistently with the expected lack of collective organization.
As for the macroscopic analysis, we show the time evolution of the mean displacement 〈r(t)〉 and of the meansquare displacement 〈r^{2}(t)〉 (see Fig. 9), which are proportional to and to t, respectively. This is the typical behavior of pure diffusion (see Eq. 3), in agreement with the results above.
From the whole set of results described so far we can consistently derive that KO cells perform a simple random motion (at least as for their free path) and that any anomalous diffusion can be excluded in this context. This can be further corroborated by the straightness index S (see Eq. 4), which, as shown in Fig. 10, decreases rapidly over time, approaching to zero. Indeed, in a normal diffusion process, the tortuosity of the path is high, because the particles do not move in a specific direction, but tend to explore the space rather compactly. For this reason, a simple Brownian motion spreads more slowly than a random walk with bias (as described in the next subsection).
Finally we consider the ergodicity problem. We measure for each trajectory the time average (see Eq. 5), which is then averaged over all trajectories to get .
More precisely, and 〈r^{2}(t)〉 have the same linear shape with comparable slope within the error (see Tab. II).
Thus, → 4D and there is equivalence of time and ensemble average, which is the hallmark of ergodicity. In particular, both procedures agree on the estimate of the diffusion coefficient, which turns out to be approximately D ≈ 9 μm^{2}/min.
In conclusion, the behavior of KO splenocytes can be characterized by a simple random walk, hence with a manifest lack of collective organization despite the presence of an insult (melanoma cells). In this respect, this result confirms the important role played by IRF8 as a central regulator of immune response and anticancer immunosurveillance^{22}.
Wild type splenocytes: a biased random walk
Since WT splenocytes express IRF8, they are expected to have a competent response to the tumor. Indeed, as we will see, WT splenocytes migrate towards B16 melanoma cells, in the attempt to contain their expansion^{22,23,25}.
Pointbypoint tracking, between 24 and 48 hours from the beginning of the experiment, showed that WT splenocytes are endowed with the potential ability to cross the microchannels connecting the Spl compartment with the B16 compartment (see Fig. 2). Thereafter, the performances of WT splenocytes in the Spl compartment, i.e., before passing the microchannels (WTPRE) and of WT splenocytes in the B16 compartment, i.e., after passing the microchannels (WTPOST), are treated separately.
First, we consider the probability distributions of step lengths ψ_{x}(Δx) and ψ_{y}(Δy). Interestingly, a qualitative difference with respect to the case of KO splenocytes emerges: along the direction pointing to melanoma cells (i.e. along negative x and positive y directions) distributions are broadened and bestfits are now provided by lognormal distributions (see Fig. 11 and Fig. 12, lower panels), namely
where the parameters μ and σ are related to the expected value as .
On the other hand, along the opposite direction (i.e. along positive x and negative y directions) distributions are still exponential (see Fig. 11 and Fig. 12, upper panels), namely
This kind of distribution is compatible with the ability of the cells to perceive a chemotactic gradient along negative x and positive y directions which, on the cartesian xy plane, corresponds to a drift towards the second quadrant, where the source of melanoma cells resides. This kind of behavior is evidenced for both WTPRE and WTPOST; related fitting coefficient are reported in Tabs. III,IV,V,VI: notice that for WTPRE the effect is stronger.
This may be due to the fact that WTPOST splenocytes, after the migration to the B16 compartment, are at least partially surrounded by tumor cells in such a way that the resulting signaling is less focussed and, consequently, this drift becomes weaker.
The data described so far suggest that WT splenocytes can be modeled by biased random walks. This is indeed corroborated by the analysis of the turning angle θ and by the related angular correlation C_{WT}(τ) (see Fig. 13). Also in this case, the bias is especially stronger for WTPRE. Indeed, as mentioned above, in the B16 compartment, splenocytes tend to change direction slightly more frequently because of a broadened presence of melanoma cells.
However, no significant temporal correlation among steps is evidenced since, for each splenocyte, the mean step (see Eq. 2) turns out to be (approximately) constant in time for both WTPRE and WTPOST splenocytes. Thus, no acceleration is observed in the process and the instantaneous speed is stable (see Fig. 14 and 15). Of note, the speed of the splenocytes decreases, once they have crossed the microchannels: while in the Spl compartment it is 3.7 ± 0.1 μm/min, in the B16 compartment it decreases down to 2.4 ± 0.1 μm/min. Again, we can notice how the behavior of the splenocytes changes according to their proximity with the tumor.
Focusing on the analysis of the macroscopic process, the most remarkable point is that the mean displacement grows linearly with time (see Fig. 16), for both WTPRE and WTPOST splenocytes, as expected for a biased random walk, strongly supporting the evidence of a highly coordinate motion for the system as a whole. Notably, also in this case, it appears evident that WTPRE splenocytes are faster than the WTPOST, since they have a greater mean displacement over time (see angular coefficients of the linear fit of 〈r(t)〉 in Tab. VII).
Moreover, we checked that the linear behavior is also observed along both x and y directions of motion (of course, 〈x(t)〉 decreases and 〈y(t)〉 increases over time because the drift is directed along the negative x and the positive y axis).
In contrast to the isotropic unbiased random walk of KO splenocytes, here the mean squared displacement is proportional to t^{2} for large t (see Fig. 17), so the signal propagates as a wave, as expected for a ballistic motion.
This picture of random walk with bias is also confirmed by the straightness index (see Fig. 18) whose mean value, for WTPRE splenocytes, ranges between 0.9 and 1.0, while for WTPOST, ranges between 0.7 and 0.8; this means that for WTPOST the motion is less straight, consistently with results discussed above.
Finally, we successfully checked ergodicity also in WT splenocytes: as shown in Fig. 17, experimental data for and for 〈r^{2}(t)〉 are nicely overlapped and bestfitted by a power law with exponent approximately equal to 2. Fit coefficient, reported in Tab. VIII, provide us with further important information: the mean displacement of splenocytes (both in time and on the ensamble) shows a value that is one order of magnitude higher in case of WTPRE compared with WTPOST. This result expresses in a clearly measurable way the role of melanoma cells on the motion of splenocytes. This observation is further in line with the concept that spleen cells motion is driven by a chemotactic gradient: as they approach to the target, namely the tumor cell, these cells tend to slow down in order to establish a physical contact. Moreover, the importance of the performed data analysis consists in providing quantitative values describing the cell dynamics which is fundamental to compare data acquired in different experimental conditions.
Comparison with Numerical Simulations
In this Section we further check the robustness of our model, via numerical simulations: in particular we simulated random walks with step lengths drawn from Eq. 8 and Eqs. 9 and 10 and we measured the resulting displacement 〈r(t)〉. Such values are then compared with those obtained from real data of splenocyte displacement and previously shown in Figs. 9 and 16, respectively.
As shown in Fig. 19, the comparison is, in general, very good. In the simulations meant to recover the behavior of KO splenocytes we realized N = 30 random walks made of T = 350 steps; in the simulations meant to recover the behavior of WTPRE splenocytes we realized N = 31 random walks made of T = 20 steps, while in the simulations meant to recover the behavior of WTPOST splenocytes we realized N = 14 random walks made of T = 71 steps, consistently with raw data available (see Results).
Moreover, the parameters λ_{x} and λ_{y} for the KO case and of the parameters p_{1x}, p_{1y}, , σ_{x}, , σ_{y} for the WT case are those given in Tabs. I, III,IV,V,VI, according to the case considered. In addition, for the WT case, a bias has been introduced in such a way that the correlation among successive steps is consistent with the experimental data depicted in Fig. 13.
Conclusion
In this work, we report a data analysis on the motility of immune cells towards tumor cells by exploiting splenocytes deficient for the transcription factor IRF8, in comparison to WT cells during coculture with melanoma cells in a microfluidic device.
We framed their dynamics within the stochastic process theory and built a library of suitable observables to tackle their motion complexity. Through this analysis, we could show that every single IRF8 KO cell performs pure uncorrelated random walks without pointing to the target, represented by the melanoma cell. Conversely, WT splenocytes, singly, are able to perform drifted random walks, which, collectively, collapse onto a straight ballistic motion for the system as a whole, giving rise to an effectively high coordinate motion towards melanoma cells. At a more detailed level of investigation, IRF8 KO cells move rather uniformly since their step lengths are exponentially distributed with a characteristic step length λ in agreement with literature, e.g. λ ~ 4.5 μm. On the contrary, WT cells display a qualitatively broader motion, due to their step lengths along the direction of the melanoma lognormally distributed. The resulting dynamics are in good agreement with models of in vivo behavior of immune cells, showing poor ability of IRF8 KO immune cells to migrate towards the tumor site, in mice inoculated with B16 melanoma^{22,23}.
In conclusion, our analysis clearly evidences the value of an integrated approach of CellonChip devices and highdata driving theoretical scaffolds, such as those derived from the stochastic theory, for quantifying experiment's outcomes. In fact, Cellonchip devices allow to fully exploit the modern microscopy tools and, through image data extraction, give access to the exploitation of the suitable mathematical tools able to quantify information.
Future progress in combining nanoengineering, cell biology, live microscopy and computational sciences will promote a new level of insight to studies of host immune systempathogen interplay.
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Acknowledgements
Italian Minister of University and Research through the grant FIRBRBFR08EKEV and through INdAMGNFM (Progetto Giovani Agliari2014) support, Italian Ministry of Health through grant RF201102347120 to FM and through ”Programma Integrato Oncologia„ 2006, Italian Association for Cancer Research (AIRC) through projects no. 14297 to GS, no. 11610 to LG and no. 10720 to Dr. Filippo Belardelli are acknowledged. The authors are grateful to Giorgio Parisi for useful discussions.
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G.S., L.G. and F.M. handled all the purebiological procedures required to set up the experiment, while A.D.N., A.G. and L.B. built the LabOnChip where the experiment was hosted and all of them carried out the experiment and produced data. E.A., E.B. and A.B. analyzed them via theoretical physics glasses and framed them within the theory of stochastic processes. The general guidelines of investigations have been decided in constant mutual collaboration among all the researchers. L.B., F.M. and A.B. share equal credit for senior authorship.
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Agliari, E., Biselli, E., De Ninno, A. et al. Cancerdriven dynamics of immune cells in a microfluidic environment. Sci Rep 4, 6639 (2015). https://doi.org/10.1038/srep06639
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