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In the present work, we model single-cell movement as a random walk in an external potential observed within the extreme dumping limit, which we define herein as the extreme nonuniform behavior observed for cell responses and cell-to-cell communications. Starting from the Newton–Langevin equation of motion, we solve the corresponding Fokker–Planck equation to compute higher moments of the displacement of the cell, and then we build certain quantities that can be measurable experimentally. We show that, each time, the dynamics depend on the external force applied, leading to predictions distinct from the standard results of a free Brownian particle. Our findings demonstrate that cell migration viewed as a stochastic process is still compatible with biological and experimental observations without the need to rely on more complicated or sophisticated models proposed previously in the literature.

To understand many physiological processes in living organisms, such as embryogenesis and wound healing, among others, as well as their malfunctions, e.g., inflammatory diseases, tumor growth, and metastasis, it is fundamental and of great interest to comprehend the process of relocalization of cells, commonly known as cell migration. This term is used to refer to different processes that involve the movement of cells from one location to another. In living animals, embryonic development provides a clear example of importance of accuracy in cell migration, as errors in this process can result in birth defects. It is also known that proper cell migration is necessary for functional immune response and tissue repair in adults. Conversely, failure in cell migration or inappropriate migratory movements may result in life-threatening scenarios, such as autoimmune diseases, defective wound repair, inflammatory diseases, and tumor dissemination, promoting metastatic cancer progression [

The process of cell migration is very specific and depends on the cell type and the context of the migration process, thus several modes of cell migration have been described [

Cell migration is an important factor in physiological and pathological processes. (a) Inflammatory cells can migrate towards a site of interest via the sensing of chemokines and other inflammatory molecules. However, within the blood vessels, they are also subject to other relevant forces such as blood flow. (b) Osteoclast precursor cells are recruited into the tissues, where they can become activated by factors such as receptor activator of nuclear factor kappa-B ligand (RANKL) and cause bone resorption in health and disease. (c) Cell migration is also an important process that allows nonmotile bacteria to attach to surfaces, initiating biofilm formation. Migration of floating bacteria is also determined by important factors such as gravitational forces and flow. In all the above situations, migration is generated by a combination of stochastic (i.e., Brownian motion and random walk) and external forces (i.e., chemokines and flow).

At first sight, the migration of nonmotile cells is essentially a random walk, very similar to thermally driven Brownian particles. This is also true for bacteria in suspension. The observation that, when suspended in water, small pollen grains are found to be in a very animated and irregular state of motion was first systematically investigated by Scottish botanist Robert Brown in 1827, and the observed phenomenon took the name of Brownian motion. Albert Einstein in 1905 [

In modern times, Langevin’s approach is still used. In fact, during the last decades, physicist and mathematician modelers have viewed cell movement as a persistent random walk, which can be modeled using well-known stochastic differential equations. The most widely used model is the Ornstein–Uhlenbeck model (OU) [

Therefore, the aim of the present work is to revisit the idea that single-cell motility can be described as a random walk. We point out that previous statements are only valid in case of free Brownian particles. However, if we introduce an external applied force and work in extreme dumping limit, the dynamics change completely, and predictions of the model depend each time on the form of assumed external potential. We define the concept of extreme dumping limit as the extreme nonuniform behavior observed for cell responses and cell-to-cell communications in vivo, within a given biological context. Contrary to previous studies, where authors usually solve the Newton–Langevin equation, here we work with corresponding Fokker–Planck equation and explain why it admits an exact solution for cases we have considered, and we show in plots the kurtosis as well as the logarithmic derivative of mean square of displacement versus time for three different simple models. We show that the nonstandard behavior seen experimentally can be reproduced in the framework of random motion with an applied external force within the extreme dumping limit scenario. Therefore, the random motion paradigm minimally extended can still be used to describe cell motility successfully without the need of more complicated and sophisticated models introduced previously in the literature.

Let us consider a Brownian particle in one dimension in an external potential

The system eventually exhibits diffusive behavior at late times, while at early times the dynamics are dominated by the inertia of the particle, and the behavior is ballistic, which has been observed in [

As it is known that cells are able to feel and sense certain environmental cues such as the stiffness of their environment [

Next, we shall consider three cases in which we can find exact analytical solution of the FP equation.

Therefore, the solution finally is given by [

It is easy to check that when

Looking at the expression for MSD obtained, the first term is the contribution of the diffusion term, while the second term is the contribution from the applied external force.

Therefore, the system exhibits the diffusive behavior only in the beginning of the evolution, contrary to the OU model, and eventually the deterministic force takes over.

This can explain the observation that sometimes cell motion is more directed than random, such as in the case of cancer cell migration [

Now the kurtosis

Therefore, the solution finally is given by [

Therefore, the kurtosis in this case is always a constant in time

Finally, with help of Gaussian integrals and defining

The MSD in short-time regime is diffusive

In both the second and third models, MSD asymptotically in time goes to a constant value due to the harmonic trap, and their logarithmic derivatives of MSD exhibit similar behavior, namely, they both are a monotonically decreasing function of time in the interval 0 <

In the corresponding Figures, we show functions

Kurtosis versus time for three models considered in the present work. The constant function corresponds to the parabolic potential (second model, blue color), the one that eventually goes to 1 corresponds to the constant force (first model, red color), and the last one corresponds to the third model that combines the two (constant force in a harmonic trap, black color) for

Logarithmic derivative of MSD

We see that each model exhibits its own dynamics, and they behave differently at late times, although in the short-range regime (t ⟶ 0), they all exhibit diffusive behavior.

The simplest model with a constant applied force can explain (i) the scaling behavior with power

In addition, a more complicated model with a harmonic trap and constant force can explain asymptotic value of 2.3 seen previously [

It would be interesting to obtain more data that could verify (or falsify) the predictions of the models considered here in a quantitative manner.

Although it is widely accepted that cell migration is complex and multifactorial, our results show that this process can be described with a modification of the random walk model that also considers the application of external forces within a complex biological environment. Therefore, it remains possible to model cell migration in a minimalistic fashion without the need of using more complex calculations. Furthermore, this model is in line with several experimental observations in the laboratory. For example, despite the fact that fibroblasts can display random migration patterns in 2D tissue culture [

Furthermore, it has also been discussed that macrophages and neutrophils migrate using various modes of random walks (i.e., biased random walks) in response to acute injury, most of which include external factors that are guiding the cell towards an area of interest [

Studying properties of cell migration is of fundamental interest to understand many physiological processes in living organisms, as well as some pathological processes such as tumor metastasis or bacterial infection. Single-cell motility of nonmotile cells can be viewed as a random walk assuming that cells are thermally driven Brownian particles and can be modeled using well-known stochastic differential equations such as Langevin and Fokker–Planck equations. Recent experimental results have questioned the archetypical Ornstein–Uhlenbeck model, as they have shown some departures from standard predictions of persistent random motion paradigm based on a free Brownian particle driven by a drag Stokes's force, as well as by a random force due to molecule thermal motion. In the present work, we have revisited the issue of cell migration viewed as random walk by adding an applied external force and working in extreme dumping limit. We have studied three concrete cases for which the Fokker–Planck equation can be solved exactly, and we have provided analytical expressions for MSD and for certain quantities of interest that can be used to make contact observations. Our results show that predictions of the model and behavior of the system depend on form of the applied force (although all models exhibit diffusive behavior in short-time regime), and they all differ compared to the standard OU model. Our work shows that random motion paradigm minimally extended can still be used to describe cell motility successfully without introduction of sophisticated models. Overall, this model could be potentially beneficial to understand the migration behavior of cells during relevant biological processes such as wound healing, inflammation, and embryonic development from a minimalistic approach.

Mathematical calculations utilized to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

The authors acknowledge grants provided to BioMAT’X Research Group, (Laboratorio de Biomateriales, Farmacéuticos y Bioingeniería de Tejidos Cráneo Máxilo-Facial), member of CIIB (Centro de Investigación e Innovación Biomédica), Faculty of Dentistry, Universidad de los Andes, Santiago de Chile. The corresponding author acknowledges supplementary operating funding provided from CONICYT-FONDEF, Chile, under awarded project/grant (national) no. ID16I10366 (2016–2019) and Fondo de Ayuda a la Investigacion (FAI) Universidad de los Andes no. INV-IN-2015-101 (2015–2019).