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Modeling the Lamellipodium of Motile Cells

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Modeling the Lamellipodium of Motile Cells Modeling the lamellipodium of motile cells Formation, stability and strength of membrane protrusions DISSERTATION zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) im Fach Biophysik eingereicht an der Mathematisch-Naturwissenschaftlichen Fakultät I Humboldt-Universität zu Berlin von Dipl.-Phys. Juliane Zimmermann Präsident der Humboldt-Universität zu Berlin: Prof. Dr. Jan-Hendrik Olbertz Dekan der Mathematisch-Naturwissenschaftlichen Fakultät I: Prof. Stefan Hecht, Ph.D. Gutachter: 1. Prof. Dr. Andreas Herrmann 2. Dr. habil. Martin Falcke 3. Prof. Dr. Josef Käs eingereicht am: 01.11.2012 Tag der mündlichen Prüfung: 24.04.2013 Abstract Many cells move over surfaces during embryonic development, immune response, wound healing or cancer metastasis by protruding flat lamellipodia into the direc- tion of migration. Although the phenomenon of cell motility has been fascinating scientists for many years, we are only beginning to gain a quantitative understand- ing. Many issues remain unresolved. Open questions are for example: Why and how do lamellipodia initially form? Why do some cells exhibit stable lamellipodia while others show protruding and retracting lamellipodia or form a lamellipodium that subsequently collapses? Which changes occur inside the lamellipodium during phases of protrusion and retraction? Is the activation of signaling molecules neces- sary? The force-velocity relation of keratocytes has been measured experimentally. What is the reason for its unexpected concave shape? Why do moving cells first react very sensitively to tiny forces but can then still move at forces that are orders of magnitude larger? How large are the forces that cells can exert on surround- ing cells in tissue and what determines the strength of lamellipodia? What are the mechanical properties of the lamellipodium and which structures bring them about? In this thesis, a mathematical model is developed that contributes substantially to answering those questions. It describes the formation of lamellipodia, their stability, strength and leading edge dynamics. Two regions of the lamellipodium are distin- guished in the model. The bulk of the lamellipodium contains a dense cross-linked actin network called actin gel. The newly polymerized tips of the actin filaments at the leading edge constitute a highly dynamic boundary layer called semiflexible region (SR). The forces that single treadmilling actin filaments in the SR exert on the leading edge membrane are calculated, and the balance of forces with viscous and external forces determines the velocity of the leading edge. Specifically, the previously published filament model is supplemented by including retrograde flow in the actin gel, and a feasible description of a variable filament density in the SR due to nucleation of new and capping and severing of existing filaments. With sev- eral simplifications, the model is represented by a system of ordinary differential equations for a one dimensional cross-section through the thin lamellipodium. A stability analysis of the model identifies three qualitatively different parameter regions: with stationary filament density and leading edge motion, oscillatory mo- tion, or a filament density of zero and no leading edge motion. That defines criteria for the existence of stable lamellipodia and allows for describing different cell types. Zero filament density means no stable lamellipodium can exist. However, due to excitability, lamellipodia can still form transiently. The measured subsequent pro- trusions and retractions of lamellipodia in epithelial cells are very well reproduced by assuming random nucleation of single short filaments from the actin cortex in the excitable regime. The modeling results support ideas on lamellipodium formation, the formation of actin arcs in the lamellum, and show that in principle no signaling is necessary for cycles of protrusion and retraction. Furthermore, the model results are fitted to the force-velocity relation of kera- tocytes, which has been measured by placing a flexible scanning force microscopy (SFM)-cantilever into the cell’s path of migration. Due to the good agreement between experiment and simulations, a mechanism leading to the characteristic fea- tures of the force-velocity relation is suggested. Moreover, properties of the struc- ture of the stable keratocyte lamellipodium, like the length of actin filaments at iii the leading edge and the branch point density, can be concluded. It is shown that the force-velocity relation measured with a cantilever is a dynamic phenomenon. A stationary force-velocity relation is predicted that should apply if cells experience a constant force, e.g. exerted by surrounding tissue. iv Zusammenfassung Das Kriechen von Zellen über Oberflächen spielt eine entscheidende Rolle bei lebenswichtigen Prozessen wie der Embryonalentwicklung, der Immunantwort und der Wundheilung, aber auch bei der Metastasenbildung von Tumoren. Die Zellbe- wegung erfolgt über die Bildung einer flachen Ausstülpung der Zellmembran, des Lamellipodiums. Unterhalb der Zellmembran polymerisieren die Aktinfilamente des Zytoskeletts und schieben die Vorderkante des Lamellipodiums in Bewegungsrich- tung. Obwohl das Phänomen der Zellmigration Wissenschaftler seit Jahrzehnten fasziniert, sind viele Fragen noch immer unbeantwortet, wie zum Beispiel: Wie und weshalb bilden sich neue Lamellipodien? Wieso weisen manche Zelltypen stabile Lamellipodien auf, während das Lamellipodium anderer Zellen vorgeschoben und zurückgezogen wird, oder sich bildet und anschließend kollabiert? Wie verändert sich die Struktur des Lamellipodiums während der Phasen des Vorschiebens und Zurückziehens? Wird diese Dynamik durch Signalmoleküle gesteuert? Wie reagieren Zellen auf die Einwirkung äußerer Kräfte? Wie kann die Kraft-Geschwindigkeits- Kurve kriechender Zellen erklärt werden? Welche Strukturen bestimmen die mecha- nischen Eigenschaften des Lamellipodiums? Mathematische Modellierung kann einen wichtigen Beitrag zu einem quantitativen Verständnis der Zellmigration liefern. In dieser Arbeit wird ein Modell entwickelt, das die Bildung, Stabilität und Stärke des Lamellipodiums, sowie die Dynamik der Zell- vorderkante beschreibt. Dabei werden zwei Bereiche innerhalb des Lamellipodiums unterschieden. Im Hauptteil besteht das Lamellipodium aus einem dichten Netzwerk von Aktinfilamenten mit vielen Querverbindungen, dem sogenannten Aktingel. An der Vorderkante wachsen die Enden der Aktinfilamente durch Polymerisation und bilden einen dynamischen Grenzbereich, der semiflexible Region genannt wird. Die Kräfte, die die einzelnen Aktinfilamente in der semiflexiblen Region auf die Zellmem- bran an der Vorderkante ausüben, werden berechnet. Das Gleichgewicht zwischen den Filamentkräften und den viskosen sowie den äußeren Kräften bestimmt die Geschwindigkeit, mit der sich die Zellvorderkante bewegt. Das bereits publizierte Modell für die semiflexible Region wird in dieser Arbeit ergänzt. Eingefügt werden der retrograde Fluss im Aktingel und eine einfache Beschreibung einer variablen Fi- lamentdichte aufgrund der Verzweigung und Kappung und Zertrennung vorhandener Filamente. Mit einigen Vereinfachungen wird das Modell für einen eindimensionalen Querschnitt durch das flache Lamellipodium durch ein System gewöhnlicher Diffe- rentialgleichungen dargestellt. Mithilfe einer Stabilitätsanalyse des dynamischen Systems können drei qualitativ verschiedene Bereiche im Parameterraum identifiziert werden: (1) das Lamellipodi- um weist eine stationäre Filamentdichte und Bewegung der Vorderkante auf; (2) die Filamentdichte und Position der Zellvorderkante oszillieren; (3) die Filament- dichte fällt auf Null und die Zellvorderkante bewegt sich nicht. Dadurch werden Bedingungen für die Existenz stabiler Lamellipodien definiert und eine Beschrei- bung verschiedener Zelltypen wird ermöglicht. Im Bereich mit Filamentdichte Null kann kein stabiles Lamellipodium existieren, Lamellipodien können aber aufgrund von Anregbarkeit trotzdem vorrübergehend gebildet werden. Wenn man annimmt, dass sich einzelne kurze Filamente zufällig am Aktinkortex bilden, beschreibt das Modell im anregbaren Parameterbereich sehr gut das in Epithelzellen gemessene aufeinanderfolgende Vorschieben und Zurückziehen von Lamellipodien. Ideen zur Beschreibung der Neubildung von Lamellipodien und der Formation bogenförmiger Aktinbündel im Lamellum werden durch die Ergebnisse bestätigt. Das Modell zeigt, dass prinzipiell keine Änderung in der Konzentration von Signalmolekülen inner- halb der Zelle notwendig ist, um die Zyklen von Vorschieben und Zurückziehen des Lamellipodiums zu beschreiben. Das Modell wird auch auf die Kraft-Geschwindigkeits-Beziehung von Fischke- ratozyten angewandt. Diese wird gemessen, indem eine Zelle auf einen flexiblen Cantilever zukriecht. Die Modellparameter werden durch Fits der Ergebnisse an den Geschwindigkeitsverlauf der gesamten Messung bestimmt. Dies führt zu ex- perimentell bestätigten oder biologisch und physikalisch sinnvollen Parameterwer- ten. Aufgrund der guten Übereinstimmung zwischen Experiment und Simulatio- nen wird ein Mechanismus vorgeschlagen, der die charakteristischen Merkmale der Kraft-Geschwindigkeits-Kurve erklärt. Außerdem können Eigenschaften des stabilen Lamellipodiums von Keratozyten, wie die Längen der Filamente und die Dichte der Verzweigungspunkte, abgeleitet werden. Es wird gezeigt, dass die mit dem Cantilever gemessene Kraft-Geschwindigkeits-Beziehung ein dynamisches Phänomen ist. Eine stationäre Kraft-Geschwindigkeits-Beziehung, die unter
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