Persistence-Driven Durotaxis: Generic, Directed Motility in Rigidity Gradients

Elizaveta A. Novikova, Matthew Raab, Dennis E. Discher, and Cornelis Storm

Phys. Rev. Lett. 118, 078103 – Published 16 February 2017; Erratum Phys. Rev. Lett. 121, 159901 (2018)

Abstract

Cells move differently on substrates with different rigidities: the persistence time of their motion is higher on stiffer substrates. We show that this behavior—in and of itself—results in a net flux of cells directed up a soft-to-stiff gradient. Using simple random walk models with varying persistence and stochastic simulations, we characterize the propensity to move in terms of the durotactic index also measured in experiments. A one-dimensional model captures the essential features and highlights the competition between diffusive spreading and linear, wavelike propagation. Persistence-driven durokinesis is generic and may be of use in the design of instructive environments for cells and other motile, mechanosensitive objects.

Received 4 January 2016

DOI:https://doi.org/10.1103/PhysRevLett.118.078103

Physics Subject Headings (PhySH)

Chemotaxis

Cells

Diffusion & random walks

Physics of Living Systems

Erratum

Erratum: Persistence-Driven Durotaxis: Generic, Directed Motility in Rigidity Gradients [Phys. Rev. Lett. 118, 078103 (2017)]

Elizaveta A. Novikova, Matthew Raab, Dennis E. Discher, and Cornelis Storm

Phys. Rev. Lett. 121, 159901 (2018)

Authors & Affiliations

Elizaveta A. Novikova^1,4, Matthew Raab², Dennis E. Discher³, and Cornelis Storm^4,5

¹Institute for Integrative Biology of the Cell(I2BC), Institut de Biologie et de Technologies de Saclay(iBiTec-S), CEA, CNRS, Universite Paris Sud, F-91191 Gif-sur-Yvette cedex, France
²CNRS UMR144, Institut Curie, 12 rue Lhomond, 75005 Paris, France
³Molecular & Cell Biophysics and Graduate Group in Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
⁴Department of Applied Physics, Eindhoven University of Technology, P. O. Box 513, NL-5600 MB Eindhoven, The Netherlands
⁵Institute for Complex Molecular Systems, Eindhoven University of Technology, P. O. Box 513, NL-5600 MB Eindhoven, The Netherlands

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Vol. 118, Iss. 7 — 17 February 2017

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Images

Figure 1
Persistence-dependent motility. Simulated trajectories (2D model) of 25 cells, departing from the origin at $t = 0$ , with a linear velocity of $50 μ m / h$ . Total time is 12 h; cellular positions are recorded at 6-minute intervals. A black dot marks the end of each cell trajectory. (a) Cells on a soft substrate, with a low persistence time $τ_{p} = 0.2 h$ . (b) Stiff substrate: persistence time $τ_{p} = 2 h$ . (c) Gradient substrate, with persistence time increasing linearly from 0.2 to 2 h over the $x$ range $[- 0.1, 0.1] mm$ (i.e., $Δ τ_{p} / Δ x = 9 h / mm$ ). (d) Averaged $x$ displacement in the gradient region for different widths of the gradient regions and hence, the gradient steepnesses (top to bottom: $Δ τ_{p} / Δ x = 90 h / mm$ , $18 h / mm$ , $9 h / mm$ , $4.5 h / mm$ , $1.8 h / mm$ ).
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Figure 2
Evolution of probability with time. Simulated trajectories (2D model) of 50 cells, departing from the origin at $t = 0$ , with a linear velocity of $50 μ m / h$ on a persistence gradient, increasing linearly from 0.2 to 2 h over the $x$ range $[- 0.1, 0.1] mm$ (i.e., $Δ τ_{p} / Δ x = 9 h / mm$ ). The cells were tracked for 12 h, their positions recorded at 6-minute intervals. A black dot marks the end of each cell trajectory. (a)–(d) As time progresses, the asymmetry becomes increasingly clear. (e) The probability distribution $P (x, y)$ (rescaled such that its maximal value is 1) at $t = 4 h$ clearly shows a double-peaked structure: a diffusive peak on the soft side and a wave front further out on the rigid side.
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Figure 3
Durotactic index as a function of time. Main figure: $x$ component of the durotactic index vs time for cells moving in a rigidity gradient, with $τ_{p}$ increasing linearly from 0.2 to 2 h over the $x$ range $[- 0.1, 0.1] mm$ . Averages computed over $5 \times 1 0^{4}$ trajectories (2D model). Black line, black dots: stiffness-independent velocity $v_{c} = 50 μ m / h$ everywhere. Red dashed line: the same system, but with a velocity that rises with persistence; $v_{c} = 20 - 80 μ m / h$ across the gradient region. Blue dashed line: velocity decreases with persistence; $v_{c} = 80 - 20 μ m / h$ across the gradient region. Inset: the average velocity over the 12 h window as a function of the gradient strength. All gradients had $τ_{p}$ varying from 0.2 to 2 h, but over different spatial ranges.
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Figure 4
Evolution of 1D inhomogeneous telegraph probability. Probability distributions $P (x, t)$ determined by direct integration of Eq. (5) (1D model). The turning frequency $λ (x)$ decreased linearly from 0.4 to 0.02 over the $x$ interval $[- 5, 5]$ . From left to right, we plot distributions for $t = 10 \dots 100$ with 10 unit time intervals. Clearly visible is the diffusive spreading on the left, vs the wavelike propagation to the right. The inset shows the long-time $t^{- 1 / 2}$ behavior of $D I (t)$ .
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