Oscillatory motion of a droplet in an active poroelastic two-phase model

We investigate flow-driven amoeboid motility as exhibited by microplasmodia of Physarum polycephalum. A poroelastic two-phase model with rigid boundaries is extended to the case of free boundaries and substrate friction. The cytoskeleton is modeled as an active viscoelastic solid permeated by a fluid phase describing the cytosol. A feedback loop between a chemical regulator, active mechanical deformations, and induced flows gives rise to oscillatory and irregular motion accompanied by spatio-temporal contraction patterns. We cover extended parameter regimes of active tension and substrate friction by numerical simulations in one spatial dimension and reproduce experimentally observed oscillation periods and amplitudes. In line with experiments, the model predicts alternating forward and backward ectoplasmatic flow at the boundaries with reversed flow in the center. However, for all cases of periodic and irregular motion, we observe practically no net motion. A simple theoretical argument shows that directed motion is not possible with a spatially independent substrate friction

Active poroelastic two-phase model for the motion of physarum microplasmodia

The onset of self-organized motion is studied in a poroelastic two-phase model with free boundaries for Physarum microplasmodia (MP). In the model, an active gel phase is assumed to be interpenetrated by a passive fluid phase on small length scales. A feedback loop between calcium kinetics, mechanical deformations, and induced fluid flow gives rise to pattern formation and the establishment of an axis of polarity. Altogether, we find that the calcium kinetics that breaks the conservation of the total calcium concentration in the model and a nonlinear friction between MP and substrate are both necessary ingredients to obtain an oscillatory movement with net motion of the MP. By numerical simulations in one spatial dimension, we find two different types of oscillations with net motion as well as modes with time-periodic or irregular switching of the axis of polarity. The more frequent type of net motion is characterized by mechano-chemical waves traveling from the front towards the rear. The second type is characterized by mechano-chemical waves that appear alternating from the front and the back. While both types exhibit oscillatory forward and backward movement with net motion in each cycle, the trajectory and gel flow pattern of the second type are also similar to recent experimental measurements of peristaltic MP motion. We found moving MPs in extended regions of experimentally accessible parameters, such as length, period and substrate friction strength. Simulations of the model show that the net speed increases with the length, provided that MPs are longer than a critical length of ≈ 120 μm. Both predictions are in line with recent experimental observations.DFG, 163436311, SFB 910: Kontrolle selbstorganisierender nichtlinearer Systeme: Theoretische Methoden und AnwendungskonzepteDFG, 87159868, GRK 1558: Kollektive Dynamik im Nichtgleichgewicht: in kondensierter Materie und biologischen SystemenDFG, 414044773, Open Access Publizieren 2019 - 2020 / Technische Universität Berli

Physikalische minimale Modelle von amobioder Zellbewegung

Cell locomotion plays an important role in many biological processes such as the immune system, embryonic development, or cancer metastasis. In these examples, cells interact with their environments or coordinate their movements with other cells, creating collective behavior. In this thesis, we utilize minimal modeling to investigate single and collective cell motility in three different settings. The key question we strive to answer is whether cell locomotion is a physical process that can function without the biochemistry that controls it. First, we examine contact inhibition of locomotion (CIL), which is one of the ways that cells interact. In experiments, cell migration can be restricted to quasi-one-dimensional stripes. In these stripes, head-on collisions of two cells occur frequently with only a few outcomes, such as cells reversing their directions, sticking to one another, or walking past each other. By utilizing a phase field model that includes the mechanics of cell shape and a minimal chemical model for CIL, we are able to reproduce all cases seen in these collisions. In addition, we found qualitative agreements such as the occurrence of ``cell trains''. Next, we investigate cells migrating on substrates with heterogeneous rigidity. By utilizing substrate configurations where cells with varying propulsion strength and membrane stiffness behave differently, we demonstrate that heterogeneous substrates are able to sort and distinguish those cells. Further, we investigate collective interactions and reproduce collective phenomena such as persistent rotational motion. We then study flow-driven amoeboid motility that is exhibited by microplasmodia of physarum. This motion is caused by a feedback loop between a chemical regulator, active mechanical deformations, and induced flows that give rise to spatio-temporal contraction patterns. We develop a poroelastic model consisting of two phases: (1) an active viscoelastic gel representing the cytoskeleton, that is permeated by (2) a fluid depicting the cytosol. Our model incorporates active contractions of the gel that are controlled by calcium. In turn, the calcium is advected with the fluid. By using free boundary conditions, nonlinear substrate friction and a nonlinear reaction kinetic for the calcium regulator, we reproduce the oscillatory motion of these microplasmodia with a net motion in each cycle. We demonstrate in all three cases that we can reproduce experimental behavior with these minimal models. This substantiates our assumption that some aspects of cell motility can be thought of as a ``physical machine'' that is controlled by the cell's biochemistry but can operate without it.Zellbewegung ist die Grundlage vieler biologischer Vorgänge. Beispiele sind das Immunsystem, die Entwicklung von Embryonen und metastasierender Krebs. In diesen Beispielen wechselwirken Zellen mit ihrer Umgebung oder kooperieren untereinander, wodurch kollektive Bewegung entsteht. Mit Hilfe von Minimalmodellen versuchen wir zu beantworten, ob Zellbewegung durch physikalische Prozesse erklärbar ist, welche durch die Biochemie der Zelle gesteuert werden, aber auch ohne sie funktionieren können. Zuerst analysieren wir die Interaktion von Zellen über contact inhibition of locomotion (CIL). In Experimenten kann Zellbewegung auf quasi-eindimensionale Streifen beschränkt werden. Dadurch kommt es zu frontalen Zell-Zell-Kollisionen, bei denen es nur wenige mögliche Ergebnisse gibt: Beide Zellen drehen um oder die Zellen haften aneinander bzw. quetschen sich aneinander vorbei. Zur Modellierung dieser Kollisionen benutzen wir ein Phasenfeld-Modell, welches die Zellform berücksichtigt und einen minimalen Ansatz für CIL beinhaltet. Damit können wir alle Kollisionsergebnisse reproduzieren. Als nächstes untersuchen wir Zellbewegung auf Substraten mit heterogener Steifheit. Wir identifizieren Heterogenitäten, bei denen sich Zellen mit unterschiedlicher Membransteifigkeit oder Vortriebsstärke unterschiedlich verhalten und nutzen dies, um Zellen zu sortieren. Zusätzlich untersuchen wir kollektive Zellbewegung und reproduzieren einige kooperative Phänomene, z.B. persistente Rotation und stabile Zellpaare, deren Bewegung von einer der Zellen gesteuert wird. Im letzten Kapitel betrachten wir die Bewegung von Mikroplasmodien (MP). Deren Bewegung entsteht durch eine Rückkopplungsschleife zwischen einem chemischen Regulator und aktiver mechanischer Kontraktion, welche zur Ausbildung von raum-zeitlichen Mustern führt. Wir entwickeln ein poroelastisches Zweiphasen-Modell, dessen erste Phase das aktive viskoelastische Cytoskelett beschreibt. Dieses ist vom Cytosol, die zweite viskose Phase im Modell, durchdrungen. Unser Modell beinhaltet aktive Kontraktionen des Gels, welche von Kalzium reguliert werden. Kalzium wird wiederum mit dem Fluid advektiert. Mit freien Randbedingungen, nichtlinearer Substratreibung und Reaktionskinetik für Kalzium können wir die oszillatorische Bewegung von MP reproduzieren. Im Besonderen identifizieren wir die nötigen Voraussetzungen für gerichtete Netto-Bewegung. In allen behandelten Beispielen können wir experimentelles Verhalten mit minimalen Modellen reproduziert. Dies untermauert unsere Annahme, dass einige Aspekte von Zellbewegung durch physikalische Prozesse erklärbar sind, welche von der Biochemie der Zelle gesteuert werden, aber auch ohne sie funktionieren.DFG, GRK 1558, Nonequilibrium Collective Dynamics in Condensed Matter and Biological System

Modeling Contact Inhibition of Locomotion of Colliding Cells Migrating on Micropatterned Substrates

<div><p>In cancer metastasis, embryonic development, and wound healing, cells can coordinate their motion, leading to collective motility. To characterize these cell-cell interactions, which include contact inhibition of locomotion (CIL), micropatterned substrates are often used to restrict cell migration to linear, quasi-one-dimensional paths. In these assays, collisions between polarized cells occur frequently with only a few possible outcomes, such as cells reversing direction, sticking to one another, or walking past one another. Using a computational phase field model of collective cell motility that includes the mechanics of cell shape and a minimal chemical model for CIL, we are able to reproduce all cases seen in two-cell collisions. A subtle balance between the internal cell polarization, CIL and cell-cell adhesion governs the collision outcome. We identify the parameters that control transitions between the different cases, including cell-cell adhesion, propulsion strength, and the rates of CIL. These parameters suggest hypotheses for why different cell types have different collision behavior and the effect of interventions that modulate collision outcomes. To reproduce the heterogeneity in cell-cell collision outcomes observed experimentally in neural crest cells, we must either carefully tune our parameters or assume that there is significant cell-to-cell variation in key parameters like cell-cell adhesion.</p></div

Reversal is robust, but chains require tuning.

<p>The percentage of collisions that result in chains is plotted; all other collisions create reversal, except in the marked region with <i>k</i>_<i>CR</i> ≤ 0.01<i>s</i>⁻¹ and <i>k</i>_<i>FR</i> ≤ 0.01<i>s</i>⁻¹ where mechanical interactions can dominate (discussed in the text). The parameters are <i>σ</i> = 2.25<i>σ</i>₀, <i>O</i>_<i>crit</i> = 0<i>μm</i>⁻² and <i>α</i> = 0.4<i>α</i>₀. We did 100 simulations for each point of the grid, which has a step size of 0.0025<i>s</i>⁻¹ for <i>k</i>_<i>CR</i> and 0.025<i>s</i>⁻¹ for <i>k</i>_<i>FR</i>. It should be noted that <i>k</i>_<i>FR</i> is an order of magnitude larger than <i>k</i>_<i>CR</i>.</p

Elements of our model.

<p>The cell shape is tracked by a phase field <i>ϕ</i>(<b>r</b>). The cell boundary (<i>ϕ</i> = 0.5 contour line) is plotted in black. On the left side the Rac concentration <i>ρ</i>(<b>r</b>) is shown, which defines the cell front. The inhibitor level <i>I</i>(<b>r</b>) is plotted on the right. To limit the internal fields to the inside of the cell, we plot <i>I</i>(<b>r</b>) × <i>ϕ</i>(<b>r</b>) (<i>ρ</i>(<b>r</b>) × <i>ϕ</i>(<b>r</b>), respectively). Throughout this work we use the shown color scales. To indicate the (static) micropattern, the contour line with <i>χ</i>(<b>r</b>) = 0.5 is displayed as a thick blue line.</p

Transition between sticking and reversal is sharp and depends on balance of adhesion and propulsion.

<p>We show the fraction of sticking events; all other events are reversals. The parameters are <i>k</i>_<i>CR</i> = 0.1<i>s</i>⁻¹, <i>k</i>_<i>FR</i> = 0<i>s</i>⁻¹ and <i>O</i>_<i>crit</i> = 0<i>μm</i>⁻². The simulations run for <i>T</i> = 2500<i>s</i>. For <i>α</i> the step size is 0.025<i>α</i>₀ and for <i>σ</i> it is 0.02<i>σ</i>₀ close to the transition and 0.25 further away. We did 100 simulations for points near the transition and otherwise 10.</p

Basic experimental observations.

<p>Basic experimental observations.</p

Snapshots of different outcomes.

<p>In each panel <i>ρ</i>(<b>r</b>) is on the left, <i>I</i>(<b>r</b>) on the right and the edges of the adhesive micropattern are indicated in blue. <i>α</i> = 0.4<i>α</i>₀ for all cases. The outcomes are i) reversals, ii) sticking, iii) walk-past, and iv) chaining. Next to each outcome are the parameters of the snapshots and the rate of the outcome for the given parameters based on 100 simulations. We chose the parameters that yield the maximum rate for each outcome. <i>α</i>₀ = 1<i>pN</i>/<i>μm</i>³ and <i>σ</i>₀ = 1<i>pN</i>/<i>μm</i>. Times are measured relative to the time of first contact.</p

Schematic picture of parameters controlling the different outcomes.

<p>Schematic picture of parameters controlling the different outcomes.</p