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Conceived and designed the experiments: CB EB. Performed the experiments: MT. Analyzed the data: GA MT. Contributed reagents/materials/analysis tools: AB MT. Wrote the paper: GA CB.

The authors have declared that no competing interests exist.

Chemotaxis, the directed motion of a cell toward a chemical source, plays a key role in many essential biological processes. Here, we derive a statistical model that quantitatively describes the chemotactic motion of eukaryotic cells in a chemical gradient. Our model is based on observations of the chemotactic motion of the social ameba

Directional movement of cells in response to chemical cues is ubiquitous in nature. It is essential for many biological processes ranging from embryogenesis

The chemotactic performance of a cell line is commonly investigated using gradient methods like micropipette assays or diffusion chambers. To date, chemotaxis in such assays has been characterized based on averaged quantities taken over a population of cells as well as over time. Typical examples are the average velocity in gradient direction

Processes that exhibit both deterministic and stochastic components are commonly described by Langevin-type stochastic differential equations. This approach has a long-standing tradition in the study of random (non-chemotactic) cell motion. The first random walk models that describe the motion of microorganisms date back to the early 20th century

Here, we introduce an analogous statistical concept to describe the directional movement of chemotactic cells in a chemical gradient. Earlier Langevin-type chemotaxis models were based on the assumption that random cell motion can be described as an Ornstein-Uhlenbeck process

In summary, it is the overall objective of our work to advance our understanding of eukaryotic chemotaxis beyond a description in terms of averaged values. In particular, we will characterize the deterministic and stochastic components of chemotactic motion along with their dependence on external parameters. Our primary goal is thus to phrase a detailed statistical description of chemotactic motion that captures also the distribution functions of fluctuating quantities. At this stage, it remains a purely descriptive approach. In future studies, it will serve as a basis for the detailed comparison of different mutant cell lines. This will enable us to identify the molecular players that determine specific features in the motion patterns of eukaryotic cells and link our model parameters to the underlying signaling events. Ultimately, this will lead to a detailed understanding of how eukaryotic cells move in response to a chemical gradient, a long-term aim of quantitative biology.

We studied the chemotactic motion of starvation developed

(A) Definition of the coordinate system. (B) Microfluidic gradient mixer, adapted from

In our experiments, cAMP gradients were linearly extending over a distance of about 320

After considering the chemotactic index as a classical average measure of the chemotactic performance, we moved on to analyze the fluctuations in various motion parameters by extracting the probability distribution functions of these quantities from the data. The results are summarized in

Experimental histograms (gray boxes) and simulated histograms (red lines) of (A)

It is our aim to model chemotactic motion based on the generalized Langevin equation 1. To phrase a specific model equation of this type, we need the explicit functional dependencies of the deterministic and stochastic parts on the cell speed and direction. We determined these expressions from our experimental data by conditional averaging

Let us first consider the deterministic and stochastic parts in a fixed cAMP gradient of 1.5 nM/

Deterministic components of (A) the parallel and (B) the perpendicular acceleration for

For the stochastic part, we found multiplicative noise that can be approximated by a linear dependence on the cell speed. The noise parallel to the direction of motion could be fitted, at each angle, by a first-order polynomial

(A) Stochastic component of parallel acceleration. Black dots show the experimental data, the red line shows a linear fit

Thus, by conditional averaging, the following Langevin equation for the chemotactic movement of

The model incorporates quadratic damping and multiplicative noise.

In the previous section, we have derived a probabilistic model of chemotactic motion in one given gradient of

(A) Friction coefficient

We used an Euler-Maruyama scheme to simulate the model equations 2 and 3 based on the parameters that were retrieved from the experimental data. For details of the numerical scheme see

In

To illustrate how the model parameters reflect different types of cell motion within a population, we have divided the data set of

(A) Friction coefficient

We have recorded large data sets of

(A) Two examples of schematic trajectories are displayed that have the same chemotactic index and the same average speed, but a very different geometrical character. (B) Trajectories governed by the Langevin equation

Thus, when considering only the chemotactic index, many details of the cellular motion patterns are lost. For example, when comparing mutant cell lines with deficiencies in different cytoskeletal regulators, the character of the cell trajectories may change considerably without substantial changes in the chemotactic index. Such differences in the structure of the cell trajectories may yield interesting information about the role of the respective proteins in the regulatory network of the cytoskeleton and cannot be resolved by the chemotactic index alone.

Here, our stochastic model of chemotactic cell motion will make a contribution. Using this more detailed description, it is possible to capture subtleties that go far beyond the information that is contained in the chemotactic index. We based our model on the assumption that chemotactic cell motion contains both deterministic and stochastic contributions. Such processes can be typically described by a Langevin-type equation. By applying angle-resolved conditional averaging to the experimental data, we obtained the deterministic and stochastic parts of the underlying Langevin equation and analyzed the dependence on the external gradient. To date, similar data-driven stochastic modeling has been only applied to non-directional, random cell motion in absence of external stimuli, see e.g. Refs.

The stochastic equation of motion showed quadratic damping and multiplicative noise, similar to non-directional random ameboid motion (see

The presence of a gradient introduced an additional effective force in the deterministic part of the equation of motion. It consisted of a component

With increasing gradient steepness, both

The parameters

Note that in general, chemotactic movement will depend on both the chemoattractant gradient and the average ambient chemoattractant concentration (the so-called midpoint concentration). In the data presented here, the cells are exposed to a constant gradient, while the midpoint concentration increases when the cells are moving up the gradient. Our data thus presents a global average over a range of midpoint concentrations for each gradient investigated. In order to also resolve the dependence on the midpoint concentration, the cell trajectories would need to be divided into small intervals along the gradient direction to perform the stochastic data analysis within each interval, i.e., for each midpoint concentration, separately. However, even though we have collected a substantial amount of data, much larger data sets would be required in order to obatin statistically meaningful results from this type of analysis. This is primarily because the stochastic data analysis requires an additional division of the data also according to angle and speed. In

In previous studies, it has been shown for human dermal fibroblasts that the damping parameter

Furthermore, the noise term was found to be independent of the external gradient. Together with our earlier observation that the stochastic components of non-directional motility are not affected by development or ambient cAMP

Our model can be considered as a description that captures the behavior of a representative, average individual from the chemotactic cell population. The mean values and fluctuations of various motion parameters are correctly captured for this average chemotactic cell. However, by subdividing the cells into subpopulations of different motility and chemotactic performance, we demonstrated that a considerable cell to cell variability exists and that the parameters of the Langevin equation are different for each subpopulation.

Also the form of the stochastic part is influenced by the heterogeneity of a typical cell population. While the slopes

We can also relate the multiplicative noise in our Langevin equation to the stochastic processes that occur in the cell during gradient sensing. To the best of our knowledge, the only purely stochastic model of gradient sensing was presented by Gamba et al.

In a recent study of

In future work, we will apply our analysis to mutant cell lines that carry deficiencies in various components of the chemotactic signaling pathway. The objective is to relate the specific parameters of our stochastic description to the individual molecular players in a chemotaxing cell. Such relations between microscopic molecular components and macroscopic dynamical observables are an essential building block for a comprehensive model of eukaryotic chemotaxis, the central aim of this field.

All experiments were performed with

The experiments were performed in a microfluidic gradient mixer, in which stable gradient profiles could be established over a region of

We have performed control experiments with cells migrating in the microfluidic device under identical flow conditions but in absence of a chemoattractant gradient. No effect of the fluid flow on the cell motion could be detected. In particular, the histograms of the x- and y-components of the velocity were symmetric and superposed almost perfectly. See

Cell tracks were recorded on a Deltavision RT microscope imaging system (Applied Precision, Inc.). Pictures were taken with a 10x plan apochromat (UPLSAPO, Olympus) objective every 40 seconds during 50 minutes using a Photometrics CoolSnap CCD camera (Princeton Instruments, Inc.) at a resolution of 1024x1024 pixels. Differential interference contrast (DIC) was used to enhance cell contour visualization. About 120 cell tracks were recorded for each experiment. The cell contours were automatically detected using a method inspired by Kam

For each cell track, the velocity and acceleration of the cell was calculated at each point by finite differences from the cell positions. The deterministic and the stochastic parts of motion were separated according to equation 1. We determined the functions

The cross-correlation of the acceleration components was found to be neglectible as compared to the autocorrelation of each individual component. We could therefore conclude that there were no mixed stochastic terms, so that the stochastic contributions in the parallel and perpendicular directions could be computed according to

Furthermore, because no angle dependence was found in either of the stochastic components, we re-evaluated them without angular binning and fitted the results by a first order polynomial

An Euler-Maruyama scheme was used to simulate the equations 2 and 3 with the parameters obtained from our experimental data

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We thank Katharina Schneider and Barbara Kasemann for assistance and Rainer Kree for discussions.