## Analysis of immune cell migration.

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This website is providing an overview of our current research results. The aim is to introduce different methods and tools to study immune cell migration in vivo. We recommend the interested reader to refer for full details and exact scientific descriptions to the provided references.

Analysis of in vivo cell migration data | Random walks in Biology | Inference of chemokine gradients | Cell migration on curved surfaces |

# Analysis of in vivo cell migration data

Developments in the field of live imaging of single-cell migration have enabled us to observe cellular processes and their temporal evolution at unprecedented detail. It is now possible to image the rich diversity of cellular dynamics inside living organisms. In this study, we examine the behaviour of leukocyte dynamics in zebrafish embryos in response to injury. The zebrafish, which has long been an important model system in developmental biology, has also become an attractive model in which to study inflammation and the immune system. Analyses in zebrafish allow in vivo imaging of immune processes to be combined with molecular studies that target signalling processes regulating leukocyte migration.

Automated leukocyte tracking system enables statistical analysis. pu.1:EGFP zebrafish embryos were injured via tail transection (blue dashed line). The blue-framed region was captured using time-lapse fluorescent microscopy resulting in image sequences with green fluorescent pu.1:EGFP-positive cells. Each cell was detected automatically using edge detection and was described as an object with the coordinates of its centre. In addition, bright field images were generated to normalise the data. In all generated movies the zebrafish was injured via tail transection, therefore the injury is located orthogonal to the notochord. To allow the analysis of data that are extracted from different movies it was necessary to normalise them. The image resolution was constant in all movies. Linear transformation of the trajectory data results into the new axes shown in red. The normalised data were used to track the cells over time resulting in cell trajectories (blue lines). Scale bars are 100mm (a). Time-lapse bright field images overlaid with images of single pu.1:EGFPpositive cell automatically detected (green cell) and tracked (blue trajectory line) produced by automated single-cell tracking system in a pu.1:EGFPþ transgenic zebrafish embryo injured by tail transection at 5 days post fertilisation (dpf) injured by tail transection. Time starts from 3-h post injury (h.p.w.) and the images shown are at 36s intervals (original time gap between images used for data analysis was 18s) (b). A trajectory (blue) of a pu.1:EGFPpositive cell (green) that was tested for random walk characteristics. Two motion vectors v1 and v2 (dark blue) with their projections onto the x axis and the y axis (x1, y1 and x2, y2) were used to test for isotropy, which is achieved by calculating the angle between v1 and v2. If the BM random walk model holds,the one-dimensional projections of the motion vectors onto the axes are Gaussian distributed (c).

After extracting the cell trajectory information from the imaging data, the next challenge is to understand migration patterns. A fast and easy way of analysis is the simple visualisation of the data. In the example below we plotted all macrophage and neutrophil data extracted from zebrafish after tail transection according to their position in the zebrafish embryo. Visual inspection already allows us to see some differences between the two cell types. However, the representation of the data in a different ways allows us to gain more information. We plot, for example, the cell tracks again, but this time shifting each trajectory, so that its origin starts in the point (0,0) of our coordinate system. In this way it becomes apparent that neutrophils spread out more in space and are moving more efficiently towards the wound compared to macrophages.

For more details please refer to:

p38 and JNK have opposing effects on persistence of in vivo leukocyte migration in zebrafish.

Taylor et al, Immunol Cell Biol. 2013 Jan;91(1):60-9. doi: 10.1038/icb.2012.57. Epub 2012 Nov 20.

# Random walks in Biology

There are different types of random walks that are commonly described in Biology. We can classify them into random walks that describe the step length distribution and random walks that describe the angular distributions. The definition of random walks via step length distribution is somewhat more frequently used. However, to investigate if a cell or a molecule is targeted in its movement, it is easier to look at angular distributions.

A random walk can be compared to a drunkard trying to go home after a long pub night.Random walks are not always completely ‘random’. In this example, our walker has a clear target: home. So he is biased towards a specific direction. However, the more drunk he is the less able he is to follow his target. In other terms, he performs a biased random walk, with a bias strength defined by his drunkenness.

The most prominent random walk is Brownian motion. The angular distribution is isotropic, meaning that at each step a cell or molecule has equal probability to move in any direction. If we measure the angles between a motion vector (cell step) and a reference direction, we will find that the resulting angular distribution is flat (uniformly distributed). If, on the contrary, a cell has a specific target direction, then the cell has higher probability to move towards that target direction compared to all remaining directions. In this case we speak about a biased random walk. The expected angular distribution will have a peak at the angle which points towards the target direction. The remaining characteristics of the angular distribution of such biased random walk depend on the details of the exhibited walk, which are usually unknown. However, a commonly used description of the angular distribution is a wrapped normal distribution (a normal distribution wrapped around a circle to describe circular variables such as angles). The mean of the wrapped normal distribution indicates the bias direction and the variance indicates the strength of the bias. The lower the variance, the narrower is the distribution and the stronger is the exhibited bias. A further type of random walk frequently used to describe animal movement and cell migration is a persistent random walk. A cell exhibiting this type of walk has higher probability of moving in the same direction as in the previous step compared to changing its direction. If we measure the angles between consecutive motion vectors (consecutive cell steps) we will observe a peak at 0, i.e. no change of direction. As for the biased random walk, the persistent random walk can also be described using a wrapped normal distribution with 0 mean and a variance which indicates the strength of the persistence (the lower the variance the stronger the persistence).

All three types of walks have been described for migration of immune and other cells, migration of animals, and movement of molecules inside the cell. Often a mix of these three types is observed.

Types of random walks and their analysis. Cell bias and persistence are computed from the angles βt (green, B) between the motion vector and the direction towards the source (red dot, B) and the angles αt (blue, B) between the current and proceeding motion vector, using an inference-based approach (D).

An other way to investigate cell migration behaviour is via so called transition matrices. A transition matrix can capture unexpected behaviour, because it is not assuming a specific random walk type. Below are shown several transition matrices typical for the above discussed random walks.

Transition matrices. A key to indicate the cell migration transitions captured by the transition matrix (a). Red arrows indicate the first step, followed by green arrows representing the consecutive step. The angles provide the absolute orientation in the fish, where the negative y axis (notochord) is used as a reference (a). Sample paths of the four described random walk models: BM; BRW, biased random walk; PRW; BPRW, biased persistence random walk. Initial conditions for numeric simulation: x=0; y=10 (b). Probability matrices for transitions of b for the four random walk models plotted as heat maps (blue, lowest probability; red, highest probability). Matrices are computed from 100 trajectories over 50 time units. The matrices show clearly distinctive patterns and can therefore be used to distinguish between the different random walk types (c).

In the following we computed transition matrices from macrophage and neutrophil trajectories observed in zebrafish embryos after tail transection. It can be clearly seen that the pattern in the matrices changes over time after wounding. Furthermore, the observed matrices differ when the zebrafish is treated with MAPK inhibitors.

Leukocyte dynamics change with time after injury. Transition matrices as heat maps for the four treatment groups are presented. Leukocyte trajectories detected at the injury site (distance from injury between 0 and 300mm) were divided into four time intervals post injury T1–T4 (see legend) and transition matrices plotted for each (a). We compute the average correlation time at each time interval (circles) with its bootstrap confidence interval of the mean (error bars). Note that the scales differ in between the treatment groups (b). To explain the unexpected dynamical patterns that appear in some of the transition matrix, we formulated two models, forward–backward random walk (c) and trafficking (d), to numerically simulate trajectories and compute their transition matrices for comparison. Initial conditions for numeric simulation: x=0; y=10.

The transition matrices can be seen solely as a tool to detect migration patterns. However, they can also be used to infer parameters of random walk models. Two of these parameters are for example the bias strength and the persistence strength. In the following we use a combination of Approximate Bayesian Computation and the transition matrices as inference framework in order to study differences between macrophages and neutrophils in response to wounding.

Shown are the posterior distributions of the rescaled persistence, p′, and rescaled bias, b′, following parameter inference using the data shown in figure 5. Results from 1 to 6 h of the inferred persistence (A) and bias (B) as well as the results from 6 to 11 h of the inferred persistence (C) and bias (D) are shown.

For more details please refer to:

p38 and JNK have opposing effects on persistence of in vivo leukocyte migration in zebrafish.

Taylor et al, Immunol Cell Biol. 2013 Jan;91(1):60-9. doi: 10.1038/icb.2012.57. Epub 2012 Nov 20.

Calibrating spatio-temporal models of leukocyte dynamics against in vivo live-imaging data using approximate Bayesian computation.

Liepe et al,Integr Biol (Camb). 2012 Mar;4(3):335-45. doi: 10.1039/c2ib00175f. Epub 2012 Feb 10.

Inference of random walk models to describe leukocyte migration.

Jones et al, Phys Biol. 2015 Sep 25;12(6):066001. doi: 10.1088/1478-3975/12/6/066001.

# Inference of chemokine gradients

While it is already fascinating to understand the migration modes of immune cells, a further goal would be to understand the underlying signalling processes. We might for example ask: How does the chemokine gradient look like that guides macrophages and neutrophils to a site of injury?

This can be done in another inference framework.

To do so we need a mathematical description of cell migration and a set of potential chemokine gradient shapes. We then need to find out, which gradient shape can explain our observed experimental data. This is a common model selection problem.

Leukocyte trajectory data were extracted from time lapse microscopy experiments and used for model selection and parameter inference using the ABC SMC framework. A model for leukocyte migration was constructed, which involves the production of a cytokine gradient after wounding (red line), the sensing of the gradient using receptor binding kinetics and the translation of the signal into movement of the leukocyte. 3 different models for the stimulus gradients (M1–M3) were proposed. From the model that explains the experimental data best information about the migration mechanism can be obtained as well as information about the stimulus gradient.

For more details please refer to:

Calibrating spatio-temporal models of leukocyte dynamics against in vivo live-imaging data using approximate Bayesian computation.

Liepe et al,Integr Biol (Camb). 2012 Mar;4(3):335-45. doi: 10.1039/c2ib00175f. Epub 2012 Feb 10.

# Cell migration on curved surfaces

If the movement of the cell is restricted to a curved surface, then directly measuring turning angles based on the original (untransformed) data will provide us with artifacts, which in some cases could mimic a target bias where in reality there is none. In order to still be able to extract bias and persistence information from such data, we need to either transform the trajectory data in such way that they lie on a flat surface (and then apply the standard analysis tools), or use some methods to learn the exact surface (manifold) and compute the angles on such manifold. Either way, the aim is to remove any artifacts that appear through curved surfaces from the analysis. The first solution can be obtained via unwrapping; the second brings us to the field of manifold learning.

Artefacts that appear when random walks happen on curved surfaces but are analysed in the 2D projections.

Unwrapping trajectory data aims to transform data from a curved surface so that they lie on a flat surface. The basic idea behind this method is rather simple and intuitive: Imagine our cells are migrating on the peel of an orange, which is clearly a curved surface describing a sphere or an ellipsoid. The aim is now to peel the orange in such way that we can lay the peel on the flat table and still conserve the characteristics of the cell trajectories. We are here interested to conserve directional characteristics, more than distances.

We first look at the orange from one direction and cut the peel it into slices of equal widths. In this way we obtain for each slice data points in 2D that can be described by an ellipse. We now cut the ellipse at the top and unroll the peel towards the table. Once this is done for each slice, we turn the orange 90 degree and repeat the same from this perspective. We finally obtain all pieces of the peel that lie now on a flat surface.

The resulting transformed cell trajectories can now be analysed with the commonly used tools.

3D representation of neutrophil cell tracks extracted from laser wounded epidermis of the yolk syncytium of a zebrafish (blue) with the xy-, xz- and yz-projections (grey). Shown are example trajectories on a hemi-sphere and there transformation via Unwrapping.