Erratic Cell Motility Generated By Deterministic Actin Polymerization Waves

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(1)Thesis. Erratic Cell Motility Generated By Deterministic Actin Polymerization Waves. ECKER, Nicolas. Abstract A cell's ability to move allows it to efficiently follow nutrient gradients and enables complex processes in tissues. Migration is mostly driven by the actin cytoskeleton. Spontaneous actin waves have been suggested to provide a physical mechanism for its organization during migration. We study theoretically the motion of cells driven by spontaneous acts polymerization waves. To this end, we introduce a mean-field description of actin waves. The actin field is confined to an evolving cellular domain by means of a phase-field. First, we show the emergence of spontaneous, traveling waves akin to excitable systems and quantify their dynamics. Next, the phase diagram of possible movement states is determined. In particular, we find erratic motion due to the deterministic emergence of spiral waves and compare these findings to the searching behaviour of immature dendritic cells. Lastly, we simulate cells in confinements and identify conditions of wave synchronization in neighboring cells.. Reference ECKER, Nicolas. Erratic Cell Motility Generated By Deterministic Actin Polymerization Waves. Thèse de doctorat : Univ. Genève, 2020, no. Sc. 5440. DOI : 10.13097/archive-ouverte/unige:141598 URN : urn:nbn:ch:unige-1415984. Available at: http://archive-ouverte.unige.ch/unige:141598 Disclaimer: layout of this document may differ from the published version..

(2) UNIVERSITÉ DE GENÈVE Départements de Biochimie et Physique. FACULTÉ DES SCIENCES Professeur K. Kruse. Erratic Cell Motility Generated By Deterministic Actin Polymerization Waves. THÈSE présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention physique. par. Nicolas ECKER de Neunkirchen (Sarre, Allemagne). Thèse n°5440 GENÈVE Atelier d’impression repromail 2020.

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(4) Declaration I declare that this thesis was composed by myself, that the work contained herein is my own except where explicitly stated otherwise in the text, and that this work has not been submitted for any other degree or processional qualification except as specified.. Parts of this work have been published in L. Stankevicins, N. Ecker, F. Lautenschläger et al: Deterministic actin waves as generators of cell polarization cues. PNAS 117 (2) 826-835 (2020). i.

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(6) Résumé en français La capacité d’une cellule à se déplacer est l’une de ses caractéristiques les plus fascinantes. Elle permet à la cellule de suivre efficacement les gradients de nutriments et de piloter des processus complexes dans les tissus, comme la recherche d’agents pathogènes par les cellules du système immunitaire. La motilité des cellules animales est souvent controlée par le cytosquelette d’actine. Bien que de nombreux facteurs importants impliqués dans le déplacement des cellules controlées par l’actine aient été identifiés et caractérisés avec précision, on comprend encore mal comment le réseau de filaments d’actine dans son ensemble est organisé dans ce processus. Des ondes spontanées d’actine ont été observées dans un grand nombre de différents types de cellules et présentent un angle intéressant pour comprendre l’organisation du réseau d’actine pendant le déplacement. Nous introduisons une déscription de l’assemblage d’actine en utilisant l’approximation des champs moyens. On prend en compte des facteurs favorisant la nucléation et une rétroaction négative des filaments d’actine sur l’activité des nucléateurs. Le système peut générer des ondes spontanées et progressives similaires à celles des systèmes excitables. Nous étudions le confinement de ce système dans un domaine cellulaire au moyen d’un champ de phase et déterminons le diagramme de phase correspondant. En particulier, nous trouvons un mouvement erratique dû à la formation déterministe d’ondes spirales. En outre, nous caractérisons le comportement de recherche des cellules dendritiques immatures, qui font partie du système immunitaire des animaux. Pour ce faire, nous analysons les trajectoires et les micrographies vidéo mesurées lors des expériences. Contrairement aux modèles stochastiques, comme les marches aléatoires, qui sont généralement utilisés pour décrire ce type de problèmes, nos simulations reproduisent des caractéristiques importantes de ce comportement migratoire. Les expériences et les simulations suggèrent toutes deux que les ondes de polymérisation de l’actine sont utilisées par les cellules pour se polariser, ce qui conduit à une migration dirigée. Enfin, nous étendons notre système pour simuler des cellules dans des confinements et des groupes de cellules. Notre approche nous permet d’étudier ces processus en fonction de la dynamique interne des cellules.. iii.

(7) Abstract A cell’s ability to move is one of its most fascinating characteristics. It enables the cell to efficiently follow nutrient gradients and drives complex processes in tissues, like the search for pathogens by cells of the immune system. The motility of animal cells is often driven by the actin cytoskeleton. Although many important factors involved in actin-driven cell crawling have been identified and characterized in amazing detail, it is still poorly understood how the actin filament network as a whole is organized in this process. Spontaneous actin waves have been observed in a large number of different cell types and present an attractive concept to understand actin-network organization during crawling. We introduce a mean-field description for actin assembly by nucleation-promoting factors with negative feedback of actin filaments on the nucleators’ activity. The system can generate spontaneous, traveling waves akin to those in excitable systems. We study confinement of this system to a cellular domain by means of a phase-field and determine the corresponding phase diagram. In particular, we find erratic motion due to the deterministic formation of spiral waves. Additionally, we characterize the searching behavior of immature dendritic cells, which are part of the immune system of animals. To achieve this, we analyze the trajectories and micrograph videos measured in experiments. Our simulations reproduce important features of this migratory behavior, unlike stochastic models like random walks, which are typically used to describe these types of problems. Experiments and simulations both suggest that actin polymerization waves are used by the cells to spontaneously polarize, which leads to directed migration. Lastly, we expand our system to simulate cells in confinements and groups of cells. Our approach enables us to study these processes as a function of the internal dynamics of the cells.. iv.

(8) Acknowledgments I want to thank everybody who has supported me during my work and while writing the thesis. First and foremost, this includes my supervisor Karsten Kruse, who gave me the opportunity to work on this fascinating topic, provided steady feedback and was always open for questions and discussions over the course of the years. Similarly, I want to thank all of the current and recent members of Karstens group, especially Denis, Philipp, Wasnik, Carles, Dan and Nicolas, for tons of interesting discussions, remarks and talks about issues - be it related to physics or profession - as well as creating a very pleasant atmosphere at work. In this context, I’d also like to thank my colleagues in the biochemistry department of Geneva. Discussions with them sparked some new ideas and thoughts that I might not have had in an environment of only physicists, and explaining my work in more general terms has provided me with new insights into the subject. Next, I’d like to thank Mira, who has continuously supported me over the years. This was very valuable especially at crucial times when other parts of life outside of work caused problems. Lastly, a special thanks goes to my parents Marion and Heinz for their unwavering support for me. Without them, my time and work abroad would have been exponentially more stressful.. v.

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(10) Contents 1. Introduction 2. The 2.1. 2.2. 2.3.. 2.4.. 2.5. 2.6.. 2.7.. 1. basic concepts of cell migration Eukaryotic cells are highly compartmented . . . . . . . . . . . . . . . . . The membrane is an impermeable lipid bilayer . . . . . . . . . . . . . . Cell shape and flexibility are controlled by the cytoskeleton . . . . . . . 2.3.1. Actin filaments are highly adaptable and dynamic . . . . . . . . 2.3.2. Microtubules create pathways inside of the cell . . . . . . . . . . 2.3.3. Intermediary filaments are stable providers of tensile strength . . Cytoskeletal proteins generate mechanical stresses . . . . . . . . . . . . 2.4.1. Polymerization causes extensile stresses . . . . . . . . . . . . . . 2.4.2. Motor proteins cause contractile stresses . . . . . . . . . . . . . . Actin can exhibit spontaneous polymerization waves . . . . . . . . . . . Current biological models for amoeboid migration . . . . . . . . . . . . 2.6.1. Persistent motion is created by steady actin polymerization in lamellipodia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2. The balance of Arp2/3 and formins determines the shape of the leading edge and cell persistence . . . . . . . . . . . . . . . . . . 2.6.3. Polymerization waves can spontaneously polarize the cell . . . . 2.6.4. Myosin motors enhance motility, but are not essential . . . . . . 2.6.5. The role of microtubules and intermediate filaments . . . . . . . Summary and migration of immature dendritic cells . . . . . . . . . . .. . . . . . . . . . . .. 4 4 5 6 7 10 11 11 12 13 14 16. . 17 . . . . .. 18 19 21 21 22. 3. Theoretical approaches to studying the cytoskeleton 24 3.1. Stochastic reaction kinetics and laws of motion . . . . . . . . . . . . . . . 24 3.2. Physical principles, symmetries and conservation laws . . . . . . . . . . . 25 3.3. Mean-field descriptions of microscopic processes . . . . . . . . . . . . . . . 28 4. A mean-field description of actin polymerization waves 4.1. Dynamic equations for the actin concentration . . . . . . . . 4.2. Nucleator dynamics . . . . . . . . . . . . . . . . . . . . . . . 4.3. Nondimensionalization and biological range of parameters . . 4.4. Analysis of the system in zero dimensions . . . . . . . . . . . 4.4.1. Dynamics near the bifurcation . . . . . . . . . . . . . 4.4.2. Comparison to a FitzHugh-Nagumo system . . . . . . 4.5. Introducing one spatial dimension . . . . . . . . . . . . . . . 4.5.1. Simulated waves are driven by nucleators, not by actin. vii. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 30 31 33 34 35 38 40 41 41.

(11) 4.5.2. Linear stability analysis of the full system 4.5.3. Wave velocity . . . . . . . . . . . . . . . . 4.6. Analysis of the waves in two dimensions . . . . . 4.6.1. Overview of the wave dynamics . . . . . . 4.6.2. Wave length comparison . . . . . . . . . . 4.7. Summary . . . . . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 43 45 47 48 50 52. 5. Wave dynamics in spatial confinement 5.1. Sharp boundary conditions . . . . . . . . . . . . . . . . . . . . . 5.2. The phase-field approach . . . . . . . . . . . . . . . . . . . . . . 5.3. Implementing membrane impermeability in phase-field methods . 5.4. Numerical implementation of the system . . . . . . . . . . . . . . 5.4.1. Spacial derivatives are calculated using Fourier transforms 5.4.2. The code is parallelized for the use on GPUs . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 54 54 56 58 60 62 65. 6. Spontaneous actin waves enable different migration patterns 6.1. Overview of the migration patterns . . . . . . . . . . . . . 6.1.1. Stationary states . . . . . . . . . . . . . . . . . . . 6.1.2. Diffusive migration . . . . . . . . . . . . . . . . . . 6.1.3. Persistent migration . . . . . . . . . . . . . . . . . 6.1.4. Erratic migration . . . . . . . . . . . . . . . . . . . 6.1.5. Phase diagram of the possible migration patterns . 6.2. Cell velocity and diffusive constant . . . . . . . . . . . . . 6.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 66 66 66 67 69 71 73 74 76. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 7. Migration patterns of dendritic cells ex vivo and comparison with simulations 78 7.1. Confined immature dendritic cells show multiple migration states . . . . . 79 7.2. Quantification of the movement shows circular paths . . . . . . . . . . . . 81 7.3. Interfering with actin dynamics changes movement characteristics . . . . . 85 7.4. Cell shape, actin distribution and displacement depend on the migration state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.5. The deterministic system reproduces features of dendritic cell migration . 94 7.6. The role of Arp2/3 and actin waves for dendritic cell migration . . . . . . 97 7.7. Stochastic simulations cannot replicate the persistent state . . . . . . . . 100 7.8. Migration characteristics comparison and summary . . . . . . . . . . . . . 104 8. Extensions of the system 8.1. Effects of motor proteins . . . . . . . . . . . . . . . . 8.1.1. Implementing motor proteins implicitly . . . 8.1.2. Comparison of waves to experiments . . . . . 8.1.3. Possible migration states . . . . . . . . . . . 8.2. Obstacles and multiple cells . . . . . . . . . . . . . . 8.2.1. Dynamics of multiple interacting phase-fields 8.2.2. Cells within confinement . . . . . . . . . . . .. viii. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 108 108 108 108 110 113 113 114.

(12) 8.2.3. Cooperative effects of multiple cells . . . . . . . . . . . . . . . . . . 118 8.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 9. Discussion and outlook. 125. 10.Bibliography. 129. A. Solutions for the linear partial differential equations A.1. Reducing the complexity to 1D . . . . . . . . . . A.2. Solution for the actin dynamics . . . . . . . . . . A.3. Solution for the total nucleator dynamics . . . . A.4. Resulting equation for the active nucleators . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 142 142 142 146 148. B. Incorporating impermeable boundary conditions in the diffusion terms of proteins 150 C. Parallel programming on graphics cards C.1. CPU and GPU architecture . . . . . . . . . . . . . . . . . . . . . . . . . . C.2. Programming on GPUs with CUDA . . . . . . . . . . . . . . . . . . . . . C.3. General strategies when using GPUs . . . . . . . . . . . . . . . . . . . . .. ix. 153 153 155 157.

(13) 1. Introduction The self-driven migration of living cells allows them to search for nutrients or escape hazardous environments [1, 2], leading to a significant advantage over particles only subject to brownian motion and flows of the surrounding material. Out of the plethora of possible means of migration, the crawling motion of cells is of most interest for this work, as it is wide-spread around cells in complex tissues and environments [3, 4, 6, 5, 7]. Crawling cells depend on a substrate or surface to move on, either by anchoring themselves to it through the formation of chemical bonds or by exerting forces and using the resulting friction to propel themselves [2, 8]. Even among crawling cells, there still exist sub-types. Due to this, we focus on amoeboid crawling motion [9, 10], where the cell moves solely by active reorganization of its cytoskeleton [9, 10], a polymer network which grants it mechanical stability, but can also introduce stresses [2, 11]. Existing studies focus on the actin network, a highly dynamic part of the cytoskeleton [1, 2, 12]. The propulsion of the cell is attributed to steady, directed polymerization of the network against the cell membrane towards the direction of motion [1, 5, 6]. This has been well studied in fish keratocytes, where a broad front of polymerizing actin, the lamellipodium, creates a very persistent type of motion [6, 13, 14]. Other cells, like neutrophils, which are part of the immune system, undergo more erratic motion on the search for pathogens [15, 16]. ”Erratic” in this case describes a type of motion that can, randomly or chaotically, switch between different states and does not necessarily follow straight paths. This is a result of changes of the polarization of the cell, which can be given by chemotaxis, the evaluation of external chemical signals [1, 17]. In the case of missing external inputs, fluctuations regarding the direction of motion or the polarization of the actin network could be associated with molecular noise. In this case, polarity would be lost due to random microscopic fluctuations of the constituents, although it is unclear whether this has a significant contribution when integrated over the scale of an entire cell. Typical protein concentrations are on the order of 10−6 mol/l [5, 16], which coupled with a typical cell volume of 100 − 1.000µm3 [18] gives protein counts of 105 − 106 . Apart from stochastic effects, there have been studies describing spontaneous cell polarization through positive feedback loops inside of the cell involving the actin network [17, 19, 20]. In summary, it can be said that, while different aspects of amoeboid crawling motion have already been studied in detail, a key understanding of how those aspects are connected to each other is still missing. In order to develop a deeper understanding, knowledge of the underlying fundamental processes is needed. The finding of actin polymerization waves [15, 21, 22, 23, 24, 25, 26, 27], where actin-related proteins perform organized polymerization and degradation of. 1.

(14) the network through self-organized dynamics [17, 19, 20, 28] provides a fascinating system to study these mechanisms. Since the biological system is complex, we have worked together with experimental collaborators in germany, Luiza Stankevicins and Franziska Lautenschläger at Saarland University to be able to test predictions and gain feedback on the functions of the proteins involved. In their lab, they study the movement of dendritic cells, another type of immune cell which, when immature, migrates through our tissue in search of potential pathogens [3, 4]. During this searching phase, they do not form adhesive bonds with the surrounding tissue, which makes it possible to extract them, put them in a quasi-2D confinement by squishing them between two glass plates and track them by labelling the nuclei without impairing their behaviour or movement capabilities [4]. On top of that, they show erratic migration [3, 4] and our collaborators found actin polymerization waves [29], making dendritic cells a fitting specimen. When describing these cells, simplifications are needed to break down the complexity. For our theory, the main questions that we ask are whether actin wave dynamics alone are already enough to cause motion and if yes, whether experimentally observed trajectories and behaviour can be reproduced. To focus on this, we neglect all dynamics that are not directly a part of the polymerization waves themselves, such as the actin cortex, a flexible structure that spans the cell just below its membrane. As the waves travel in this cortex [19, 20, 30], we can assume that they are not impaired by other structures inside of the cell, making us able to neglect those. As a result, to us, the bulk of the cell simply acts as a reservoir for actin, supplying a constant and homogeneous amount of monomers. The underlying dynamic equations build on protein interactions found previously when studying in vitro actin polymerization waves, where actin and its related proteins were simply put into a petri dish and the resulting dynamics captured [31]. This led to a first study of cell motility [32], which however could not be further explored due to the numerical complexity involved in solving the equations, with simulations for single cells taking between two to four weeks. A follow-up study [33] used an alternative description of the membrane to decrease the simulation time, but did not yet incorporate a robust way of limiting protein diffusion to the inside of the cell while also creating some difficulties in numerical accuracy. The aim of this work is making the numerical description efficient and accurate enough to explore the effect of parameter changes on cellular motility to get an idea on which are the most important processes that we describe. This raises further questions on how to quantify erratic motion as well as polymerization waves and the coupling between the two, which I have also done for the experimental data. Regarding the physical description, an implementation of the membrane as an actually impermeable boundary is needed. In the end, the advances on these aspects will make it possible to study the movement of cells with obstacles and cooperative effects in multi-cell systems. The thesis is set up as follows. Chapter 2 gives an introduction on the biological prop-. 2.

(15) erties of the system, focusing on the aspects important for cell motility and migration. Afterwards, chapter 3 presents a short overview of the possible physical approaches and descriptions for biological systems. In chapter 4, the theoretical description of actin waves will be introduced, explained and analyzed. Chapter 5 then adds the description of the membrane through means of a phase-field, making sure the impermeability condition is satisfied. The numerical description of the resulting system is presented and discussed in chapter 5.4. Chapter 6 presents and discusses possible movement states, showing the parameter dependencies of key quantities. The analysis of the experimental data is then performed in chapter 7 and compared to our simulations as well as classic random walks and active particles. This links the behaviour of cells with the internal actin wave dynamics. Lastly, the gained numerical improvements are utilized to add additional molecular processes in the cytoskeleton in chapter 8.1 and the interactions of cells with obstacles and themselves in multi-cell simulations, eventually leading to an analysis for cooperative effects in chapter 8.2.. 3.

(16) 2. The basic concepts of cell migration Before going into more detail regarding the theoretical description, this chapter will introduce all of the necessary biological aspects, emphasizing only those that contribute to cellular motility. After a brief introduction of the cell’s structure (2.1-2.3), the cytoskeletal dynamics (2.3-2.5) and their currently perceived role in cellular movement (2.6) will be presented.. 2.1. Eukaryotic cells are highly compartmented Eukaryotic cells are the family of cell types which make up all higher organisms [1]. Their basic features involve the plasma membrane, a lipid bilayer that functions as a barrier between the outside of the cell and the cytoplasm, which is the compartmented interior. It consists of functionally different and in some cases membrane-enveloped organelles and the liquid cytosol, which contains numerous macro-molecules and ions. Flexiblity and stability arise from different types of proteins which polymerize into filaments and permeate the entire cytoplasm, the cytoskeleton [1]. The basic structure of a eukaryotic cell is shown in figure 2.1. Eukaryotes differ from prokaryotic cells, e.g. bacteria, by having a distinct nucleus enclosed by a membrane to store their genome. In general, eukaryotes contain a standard set of the aforementioned membrane-enveloped organelles, whereas only specialized ones are found in some prokaryotes. An example of such a standard organelle are the Mitochondria, which, among other functions, provide the cell with energy stored in the form of adenosine triphosphate (ATP), which can be hydrolyzed to adenosine diphosphate (ADP) to fuel other reactions or conformations in macromolecules [1]. All in all, cells are active, meaning they steadily convert energy. This makes them systems far out of equilibrium [2]. Due to the plethora of molecules and organelles, they are visco-elastic [1] and the interior dynamics are overdamped [2]. Many of the complex processes happening inside of them are self-organized, which means that some sort of order, like spatial or temporal patterns, will be reached even from an initially disordered state simply through the internal dynamics given by the constituents of the system. Examples of such active, self-organized systems include the organization of the cytoskeleton [2] and the oscillations of proteins like the Min-system in Escherichia coli bacteria, which determines their center and is important in cell division [34]. The intricate structure of the outer membrane keeps all of the involved molecules inside of the cell and also helps with the localization of some of these processes.. 4.

(17) Figure 2.1.: Structure of a eucaryotic cell. In addition to organelles, the plasma membrane, the nucleus and the cytoskeleton, it also shows a flagellum, which only some Eucaryots use to propel themselves. Taken from [36].. 2.2. The membrane is an impermeable lipid bilayer. Figure 2.2.: Schematic view of a lipid bilayer. Polar, hydrophilic tails of the phospholipids are shown as violet spheres. Unpolar, hydrophobic tails are the blue lines. A number of proteins with and without penetration of the membrane are embedded. From [37], coloured. As previously described, a cell’s plasma membrane is a phospholipid bilayer with embedded proteins. The latter can amount to about 60% of the membrane’s mass. It’s stabilized by hydrophobic interactions between the lipid’s unpolar tails as well as the. 5.

(18) hydrophilic interactions between the polar heads and the surrounding water molecules [35]. Due to this structure, the layer is essentially impermeable for smaller molecules and ions. Only certain ones, e.g. Ca2+ , water or glucose, are targeted by specific channel and shuttle proteins [1, 37]. However, the membrane is all but static. Shown in figure 2.2 is the so-called fluid-mosaic model originally introduced by Singer and Nicholson, where lipids and proteins can freely diffuse along the surface [37], giving rise to the terminology of a ”two-dimensional fluid” [1]. This model has been steadily expanded, because it has since been found that also certain membrane proteins like to conglomerate and form clusters. Additionally, there exist randomly forming and collapsing crystal-like domains of lipids and cholesterol, which are called lipid rafts[35]. These result in a heterogeneous structure of the membrane. The flexibility is dependent on the composition of structurally different lipids as well as the size and composition of rafts and protein clusters, which affects their interactions. Its actual shape is mostly determined by its interaction with the cytoskeletal cortex directly below it [1, 35], which will be described in the next section.. 2.3. Cell shape and flexibility are controlled by the cytoskeleton. Figure 2.3.: Fluorescence image of the cytoskeleton of a cell. Shown is the actin network (red), the microtubules (green) and the nucleus (blue). From [1]. The cytoskeleton is a polymer network consisting of different types of proteins, spanning the entire cell and forming bonds with the plasma membrane and organelles [1]. Its main components are the actin filaments, the microtubules and the intermediary filaments [1]. The former two are shown in figure 2.3. It is essential for a cell’s mechanical properties like elasticity, flexibility and motility, granting protection against external influences like shear forces, pressure and tension. On top of that, interplay with motorproteins, which can move along actin filaments and microtubules while carrying cargo. 6.

(19) ranging from proteins and vesicles up to organelles, makes the cytoskeleton important for internal organisation and cargo localization [2]. It orchestrates complex processes like cell division, migration and the generation of force dipoles in muscle cells [1, 2, 11, 35, 38]. In order to accomplish these feats, the cytoskeleton can be highly dynamic and enter a state of constant polymerization and degradation under consumption of chemical energy in the form of ATP, while being influenced and regulated by a vast amount of associated proteins. Because of this, the cytoskeleton is an active, self-organized system [1, 2], as mentioned in the first section of this chapter.. 2.3.1. Actin filaments are highly adaptable and dynamic The actin network will be the main focus of this work, as it is a key player for cellular motility [1]. As shown in figure 2.4, the double-stranded helical filaments are built up by the spherical monomer actin, which is asymmetric due to a nucleotide binding site for ATP or guanosine triphosphate (GTP) on one side. This structural difference leads to the polymerized filament being polar, with different binding strengths depending on whether the ATP already got hydrolyzed to ADP or not [2].. Figure 2.4.: Schematic view of an actin filament (left) compared with an image from an electron microscope (right). From [1].. Opposite to the binding site is the so-called plusend, where the filament typically grows. This is caused by the binding of new monomers happening faster than the hydrolysis of ATP, which will weaken the bond between adjacent monomers and result in a faster detachment rate. On the minusend, this effect is reversed, as detachment happens faster than the binding of new monomers, resulting in shrinkage of the filament. In the steady state, the polymerization- and detachment rates on both ends are equal. As a consequence, actin monomers go through the filament from the plusto the minus-end, where they are detached and able to bind again after phosphorylation of the bound ATP. This net flux of monomers through a filament which does not vary in length is called treadmilling [1, 2, 39]. If the filament is anchored to another structure or in a viscous environment, treadmilling leads to an effective motion of the filament in the direction of its growing end, even though the sub-units themselves are not displaced [1, 2]. This characteristic makes actin important for cell motility [1], which will be discussed in more detail in chapters 2.4 and 2.6.. As the polymer is double-stranded, with a filament diameter of 5-9 nm, the monomers on one strand stabilize the ones on the other strand and vice versa [2]. Additionally, actin. 7.

(20) filaments can be bundled to fibers or branched and crosslinked into gel-like networks to increase their stability and tensile strength further [1, 2]. The persistence length, a measure of flexibility, of single actin filaments has a value of 15µm. It gives the length scale on which the movement of opposite ends of the filament becomes uncorrelated. Below this length, they can be treated as an elastic rod, whereas above it, they have an arbitrary shape due to bending already being induced by thermal fluctuations [2, 11]. In cells, the structure of the actin network is mostly dictated by cofactors of actin, a group of proteins that can branch, (de-)polymerize, stabilize, nucleate, cut or crosslink filaments. Some examples are shown in figure 2.5. Especially the branching and nucleating functions, which will be explained below, contribute to the structure, as the self-nucleation of actin is the limiting step for growing actin filaments and networks [2, 39].. Figure 2.5.: Overview of various cofactors of actin and their functions. From [40]. Out of those cofactors, we will focus on the Arp2/3- and Ena/VASP-complexes as well as formins, since these protein families are the most important ones for the actin wave dynamics discussed in chapter 2.5.. Figure 2.6.: Schematic view of the functions of Arp2/3, formins and the Ena/VASP complex. Adapted from [41]. Both formins and Arp2/3 are nucleating proteins. Figure 2.6 shows their function schematically. Arp2/3 is activated near the membrane by a protein complex called WASP [42, 41], where it binds to existing actin filaments to nucleate ones at an angle of. 8.

(21) approximately 70°[5, 41]. Likewise, formins are also activated near the membrane, acting as a nucleation site for new actin filaments while also boosting the polymerization speed by capturing and incorporating nearby actin monomers bound to profilin with its rope-like structures [2, 43, 41]. Both Arp2/3 and formins stabilize the bound end of the filament they have nucleated and protect it from capping proteins which would stop the growth [2, 5, 41]. Lastly, the Ena/VASP complex elongates actin filaments similar to formins, while having an additional bundling effect [41]. On top of these proteins which affect the actin network directly, the asymmetric or polar structure of the filaments can be utilized by the previously mentioned motor proteins. In the case of actin, the corresponding protein family is called myosins. Their structure is shown in figure 2.7. As chains of amino acids bind by connecting the carboxygroup of one with the amino-group of another amino acid, one end of the chain will end up with a free carboxy group, being called the C-terminus of the chain. Likewise, the other end with a free amino group is called the N-terminus [1]. In the case of myosin, the protein consist of two identical heavy chains which form a coiled-coil helix from their C-terminal end up to their head groups on the N-terminal side, where two light chains bind to regulate the motor activity.. Figure 2.7.: Structure of a myosin motor. Shown are the heavy chains (green and dark green) and the light chains (blue). From [1]. The C-terminal side can bind to cargo, whereas the two head groups bind to an actin filament and are able to ”walk” along it by using ATP hydrolysis to induce conformational changes, pushing the motor a bit closer to the plus-end [1]. This enables myosin to provide directional transport for the macromolecules and vesicles inside of the cytoplasm, as well as to induce tension in the actin network [2]. For the latter part, Myosin-II motors have the ability to conglomerate into so-called thick filaments, which consist of the intertwined C-terminal ends of many myosins, with the head groups sticking out and being able to bind to actin filaments [1]. As the motor head groups on opposite ends of the thick filaments point in opposite directions, making the thick filaments bipolar, this creates tension between adjacent actin (thin) filaments [2]. An example of this is seen in muscle cells, where an interplay of large thick filaments with actin bound to structures called Z-discs cause the contraction of muscle fibers. The Z-discs, as the name implies, are disc-like structures with thin filaments growing out on both sides, creating a bipolar arrangement of actin. This in unison with the bipolar thick filaments then causes contractions [2, 38, 11]. The process will be explained in more detail in chapter 2.4.. 9.

(22) 2.3.2. Microtubules create pathways inside of the cell The second type of cytoskeletal polymers are the microtubules, which have the important roles of forming pathways for bidirectional transport as well as seperating the chromosomes during cell division. They consist of the monomers α− and β-tubulin, which alternate to form a protofilament. Between 11 and 16 of these filaments, depending on cell type and species, bind together to form a hollow cylinder. On the cylinder’s surface, the individal monomers appear in helical structures. Like actin monomers, tubulins have a pocket for nucleotide binding, commonly guanosine triphosphate (GTP), which serves the same purpose as ATP in actin, also breaking the symmetry and enabling the whole structure to be polar. The binding strength between tubulins weakens once GTP is hydrolyzed to guanosine diphosphate (GDP). If at some point, the GTP-rich growing region of the microtubule becomes entirely hydrolyzed, a so-called ”catastrophe” happens, and the tubule rapidly falls apart. This together with the strucutre of a microtubule is shown in figure 2.8. The catastrophe can be stopped (”rescued”) by phosphorylation of the leading tubulins, after which the tubule starts to regrow. This fluctuation in length coupled with wiggling of the microtubule due to thermal noise make it easier to ”find” certain targets in the cell, like the centers of the chromosomes during cell division, by letting the fluctuating end cover a larger area [2, 1]. Synthesis of new microtubules happens at microtubule organizing centers (MTOCs). An example is the centrosome in animal cells, an organelle which the entire microtubule network branches out of [1]. Due to the cylindrical shape, microtubules are significantly more rigid than actin filaments with a persistence length of roughly 6 mm [2].. Figure 2.8.: Schematic structure (left) and electron micrograph (right) of a microtubule. The two tubulins are marked in different shades of green. Taken from [1]. The polar structure of the subunits enables two families of motor proteins to walk in opposite directions on the microtubules, namely kinesins and dyneins. This facilitates directional transport and makes the microtubules the most important contributor for this task [1]. The structure of kinesins and dyneins is similar to the one of myosins.. 10.

(23) 2.3.3. Intermediary filaments are stable providers of tensile strength The last major polymer in the cytoskeleton is the family of intermediary filaments. Their diameter lies in between actin and microtubules [2], however their turnover is much slower [44]. Their main use lies in granting mechanical stability to the cell. To accomplish this, they consist of a number of very thin monomers with a length of around 48 nm. As shown in figure 2.9, two of those form a coiled-coil dimer, which in turn bind with a small offset side by side to create tetramers. Eight of these form a sheet, which is then twisted into a rope-like structure. The offset enables the subunits to slide against each other and provides flexibility, while the large number of steric interactions with neighbours in the final structure grants protection against stresses [1, 2]. Their persistence length of around 1 µm makes them the most flexible polymer in the cytoskeleton. Due to their symmetric strucuture, there are no associated motor proteins [2].. Figure 2.9.: Schematic structure (right) and electron micrograph (upper left) of intermediate filaments. Taken from [1].. 2.4. Cytoskeletal proteins generate mechanical stresses Using the cytoskeleton, cells are able to create mechanical stresses in exchange for ATPconsumption. This chapter will introduce the two main components which can then be utilized in order to create the type of motion described in chapter 2.6. 11.

(24) 2.4.1. Polymerization causes extensile stresses Both actin filaments and microtubules are able to generate mechanical stresses in the absence of motor proteins simply due to their polymerization dynamics. Two possible models for this are described in figure 2.10. This effect is sometimes called a polymerization motor [2, 45]. In order to balance the force exerted by the growing end, some part of the polymer needs to be linked to a rigid surface. If one end is fixed, the force exerted by the growing end can be measured and amounts to up to 9 pN [2, 45]. In vivo, these values can be estimated to be smaller, as filaments are not fixed perfectly rigidly. Additionally, to prevent the filaments from growing too long and buckling away from the membrane, they need to be steadily depolymerized and re-nucleated to keep pushing the membrane forward [45].. Figure 2.10.: Left: Two possible models for the pushing effect of polymerizing actin filaments. (A) Thermal fluctuations of the flexible wall make it possible for monomers to bind if the fluctuation leads away from the filament. Fluctuation towards the filament is blocked sterically. (B) Binding of a new monomer causes the filament to start buckling. As the straight configuration is energetically favorable, the filament will relax while pushing away the obstacle. Taken from [45]. Right: (a) Tread-milling dynamics of an actin filament. kon is bigger on the barbed, kof f is bigger on the pointed end. (b) Tread-milling can push obstacles around while keeping a constant filament length, as the polymerization and degradation alone cause an effective displacement. Taken from [46]. Nevertheless, polymerization of large amounts of filaments is used by cells to propel themselves using this mechanism. Two good examples of this are lamellipodia, where continuous polymerization of highly branched actin networks is used to provide persistent directional motion [6, 5], and lysteria, which are a type of bacterium that hijack a cell’s actin network to push through the plasma membrane and exit the cell by building up. 12.

(25) tension with continuous actin polymerization on their own surface [2, 47]. Prost et al compare this to a rubber band being stretched as the influx of actin at the cylindrical surface of the bacterium pushes the older, already crosslinked actin network outwards [48]. This tension is relieved at the tail where the actin network can relax again, pushing the bacterium forwards [48]. The mechanisms behind the keratocyte motion will be further discussed in chapter 2.6.. 2.4.2. Motor proteins cause contractile stresses Opposed to the pushing effect of the polymerization, the cytoskeleton can utilize contractile forces with an interplay of actin- and myosin-filaments. As mentioned in 2.3.1, these contractile fibers can create tension by displacing the actin filaments against each other, which is used not only in the sarcomeres of muscle cells, shown in figure 2.12, but also for example in the contractile ring that splits a cell during its division [1]. A schematic view of this for a single motor protein is presented in figure 2.11. While the details of this contraction are not of immediate concern for this work, it has to be stated that the ability of an actomyosin network to contract is not immediatly obvious. In the muscle, the symmetry is broken by the sarcomeric structure, however, that is not the case in general. By assuming that a myosin motor is point-like, undeformable and that the force it can resist before detaching is constant along the actin filament, as well as that filaments are rigid rods, all contractility is lost [49]. One of these assumptions or the isotropic structure of the actin network needs to be broken for a contraction to occur [49]. There have been numerous studies on the generation of stresses under these conditions [50, 39, 51, 52, 53]. By anchoring the actin filaments to the membrane, the contraction is transmitted to the surrounding part of the cell. In these systems, a large amount of motors amplifies a single motor’s exerted force of about 1,5 pN [2, 14].. Figure 2.11.: Two possibilities of generating contractile stresses in the actin cytoskeleton utilizing thick myosin filaments. Taken from [54].. 13.

(26) Interestingly, contractions can also be caused by depolymerization of antiparallel linked filaments. However, it is difficult to determine whether the actual motor activity or the cross-linking capabilities of myosin II are more important regarding this task [54, 55]. The depolymerization process also seems to be important for the ring closure during cell division in some cell types [55, 56]. Overall, these methods of creating stresses again make up an active system, as the continuous polymerization and degradation in the former one and the motor activity in the latter one are tied to ATP-consumption.. Figure 2.12.: Schematic view of the contraction of a sarcomere. (a) The Z-discs (called Zline here) connect actin thin filaments in a bipolar arrangement. In between them are the thick myosin filaments, aligned by the M-lines (or M-bands). Under ATP-consumption, the heads of the myosins move towards the Zdiscs, contracting the whole structure (b). Muscles use many sarcomeres in succession in order to contract. Taken from [57].. 2.5. Actin can exhibit spontaneous polymerization waves Actin polymerization waves are a feature that has first been observed in the 90s [22, 21]. Since then, they have been confirmed in numerous cell types [19, 12], such as Dic-. 14.

(27) tyostelium amoeba [58, 59, 24, 25, 60, 61, 62], human or mouse skin fibroblasts [26, 63], Xenopus frog eggs as well as embryos [27, 64] and neutrophils [15], which are a cell type in the immune system. Figure 2.13 shows an example in Dictyostelium amoeba, where waves of the actin nucleator Hem-1 are being visualized.. Figure 2.13.: Fluorescence micrograph of Hem-1 waves in leucocytes for different time points. The leading part is marked with an arrow. Taken from [15]. In order to exhibit these polymerization waves, the cortex below the membrane acts as an excitable medium [58, 23, 27, 64, 20, 65, 66, 62]. This means that it is a nonlinear dynamical system in an unstable, unexcited state, in which a perturbation above a certain threshold creates an excitation that will eventually decay back to the starting value, enabling traveling waves [67, 68]. While doing so, the system passes three characteristic stages [67, 68]. The initial response to the perturbation is a rapid auto-catalytic activation. Then, a time-delayed negative feedback sets in, leading to the decay of the wave. This decaying part is called the refractory period, during which no new excitation can happen due to the negative feedback still being active [68]. Finally, the system resumes the unexcited state again. The refractory period leads to colliding waves annihilating, as they cannot continue traveling past each other. Additionally, as the wave is amplifies itself at the front and only depends on this amplification and the baseline in the unexcited state, there is no damping observed during travel [68]. Inside of the cell, there have been numerous studies on the origin of this excitability. The characteristic self-sustaining movement of waves [27, 23, 64, 25, 24] as well as the annihilation of colliding waves [27, 64, 24] have been observed [62, 17]. In Dictyostelium amoeba, the existence of an oscillatory instability in the actin regulatory system has been proposed [69]. To explain this, the current view is that of a reaction-diffusion type system with a quickly diffusing actin-nucleating or -polymerizing cofactor [58, 23, 27, 64, 20] which can activate itself or is recruited by another protein [27, 24, 20]. The negative feedback is correlated with an increase in actin concentration [27, 24, 64, 20], possibly due to the recruitment of inhibiting cofactors at a certain threshold [27, 24]. An essential part of the excitable system is the branching cofactor Arp2/3 [24, 30, 25, 26, 27], however it also relies on other cofactors like the VASP complex or formins to. 15.

(28) elongate the branched filaments [24, 41, 30]. Recent studies in Dictyostelium amoeba suggest that an initial actin population in the cortex is made up by VASP or formins, which attract Arp2/3 to create branches off of the existing filaments, which in turn attracts more VASP/formins to elongate the newly branched actin [30]. For granulocytes, which are a type of immune cells, it has been shown that this newly generated actin in turn causes the inhibition of an activator of Arp2/3, namely Hem-1 [24, 15, 70]. This deactivation causes the refractory period [24]. In agreement to these findings on Hem-1, studies in Xenopus frog eggs show the same type of dynamics for another activator of Arp2/3, namely the Rho-GTPase [64, 27]. Inside of the wave, the branching cofactors and nucleators co-localize in overlapping clusters, a process that is likely facilitated by contractions due to the motor protein myosin-IB, which can bind to the Arp2/3 complex via another protein called CARMIL [30, 24, 71]. A study by Gerisch et al [30] comes to the conclusion that the driving part of the wave is not the polymerization of actin filaments in the direction of motion, but rather the continuous nucleation of actin at the front of the wave, making it nucleatordriven. Regarding the structure of the actin network in such a wave, it has been found that the ”mother”-filaments that Arp2/3 attaches to are mostly parallel to the membrane, whereas a significant amount of branched off filaments actually point towards it. The cortex has a typical width of around 60 nm, whereas inside of the wave it reached more than the detectable ∼175 nm[30], most likely due to the pushing force of the branched actin filaments. In the part directly behind the wave, there still was lots of leftover actin, albeit almost uniformly parallel to the membrane, with a width of around 120 nm. It’s assumed that as the nucleators are deactivated, the branches pointing towards the membrane are being degraded, while the rest is contracted by the leftover myosin-IB [30].. 2.6. Current biological models for amoeboid migration With the previous overview of stress generation methods and cortical dynamics, we can now take a look at how these processes actually enable cells to move around. As a general introduction, amoeboid migration is one of the most abundant types of crawling motion of eukaryotes. Apart from this, they can also move around utilizing cilia and flagella, much like prokaryotes [1], and do blebbing, where the membrane is partially detached from the actin cortex below, creating evaginations that can subsequently be stabilized [72]. In contrast, during the amoeboid migration, steady restructuring of the cytoskeleton is used to deform the membrane and, using friction with or adhesion to a substrate, ultimately push the cell forwards. This typically takes the shape of socalled lamellipodia, broad and flat regions of high actin polymerization due to Arp2/3 in the front of the cell, and filopodia, which are finger-like protrusions created by actin bundles nucleated by formins or similar nucleators [1, 10]. Another option are the actin. 16.

(29) polymerization waves introduced in the last chapter. Both of these possibilities will be discussed in the following sections.. 2.6.1. Persistent motion is created by steady actin polymerization in lamellipodia. Figure 2.14.: Left: Microscopic image of a moving keratocyte. The spherical bump corresponds to the nucleus; the flat, extended area in the front is the lamellipodium. Right: Fluorescent images of a moving keratocyte with labelled actin (red) and the actin degradation cofactor cofilin (green). (A) shows the entire cell, (B) gives a zoomedi-in close-up. All images taken from [1]. Figure 2.14 shows lamellipodia of moving keratocytes. They consist of flat, extended regions of highly branched actin being polymerized against the leading edge. As mentioned before, there needs to be a continuous nucleation and degradation to keep this process up, which is shown the sub-figures 2.14(A) and (B). There is a large amount of actin directly at the leading edge, which is followed by a high concentration of the actin severing cofactor cofilin in a certain distance, taking care of the degradation of older, longer filaments [6, 43]. This is also shown in more detail in figure 2.15. Apart from the electron micrograph of the highly branched network at the leading edge, the study replicated the findings for Arp2/3 and Cofilin and showed that coupled to the degradation by cofilin, there is also an inhibition of the polymerization of actin filaments by a capping protein [6]. Overall, the leading edge is able to exert forces in the range of nano-Newtons [14]. Apart from this type of motion being purely driven by the polymerization of branched actin, there is a more general description also utilizing the formation of so-called filopodia, finger-like protrusions that depend on the directed elongation by formins [73]. A schematic of a cell utilizing both of these processes is given in figure 2.16. Either mechanism needs the anchoring of the cell to the substrate via adhesion proteins to create sufficient friction with the substrate in order to move forward. As shown in subfigure 2.16 a, the lamellipodium is not fully attached, but rather depends on some deformation being caused either by the branched actin or the filopodia in order to form new adhesion. 17.

(30) Figure 2.15.: Light- and elecron micrographs of moving keratocytes. The leading edge shows short, highly branched actin, with the length increasing and the number decreasing with the distance. Areas of cofactor activity are highlighted. Taken from [6]. sites at the leading edge (subfigure b), which are then linked to the existing network via the creation of new actin bundles, the stress fibers (c). After these connections have been stabilized, the adhesion sites in the back of the cell are detached, which leads to the stress fibers there becoming contractile and pulling in the back of the cell body (d). This leads to an effective displacement of the cell and the process starts anew at the leading edge [47, 74, 8].. 2.6.2. The balance of Arp2/3 and formins determines the shape of the leading edge and cell persistence As mentioned before, the lamellipodium is mostly formed by branched actin nucleated by Arp2/3, whereas filopodia depend mostly on formins. Studies show that disturbing the balance between those nucleators leads to drastic deformations of the leading edge and even the cells. In particular, overexpression of formins in mammalian cells leads to the spontaneous formation of filopodia [43]. However, these are not necessarily tied to movement, as having the lamellipodium alone is already sufficient for a number of cell types [6, 43]. A study in 2014 checked the effect of branched vs elongated actin networks on cellular motility and came to the conclusion that increased Arp2/3 activation by overexpression of WASP lead to a slower, but more homogeneous leading edge creating. 18.

(31) Figure 2.16.: Schematic view on the basic dynamics for motion with lamellipodia. Taken from [74]. a very persistent type of motion. On the other hand, overexpression of elongation factors lead to more frequent inhomogeneous protrusions forming at the leading edge, coupled to a faster but less persistent type of motion [40]. These results are summarized in figure 2.17. A higher filament count generally translated to an increase in the force that can be generated, but more branching leads to a decrease in the amount of available actin monomers, which slows down the dynamics.. 2.6.3. Polymerization waves can spontaneously polarize the cell So far, the motion of cells has been attributed to continuous actin polymerization in lamellipodia. On top of this mechanism, due to actin wave dynamics as described in 2.5, the cytoskeleton can cause a spontaneous polarization and change thereof in the absence of external cues [75, 76, 60, 25, 26]. This polarization process can even help to react to external chemotactic signals [76]. These signals only need to tune the excitability of the cortex when detected [17, 20]. They might even directly influence motility, as even cells which typically rely on lamellipodia, but have been modified to not form them anymore, are still motile, hinting at other possible mechanisms [78].. 19.

(32) Figure 2.17.: Comparison and schematic view of the effects of branched vs elongated actin networks on cellular motility. Taken from [40].. Figure 2.18.: Fluorescence micrograph of a component of the WAVE-complex, RAC (A,B), Arp2/3 (C) and actin (D) in leucocytes during polymerization waves. Leading edge is marked with an arrow. Taken from [15]. Furthermore, it has been shown that actin waves can deform membranes on their own [15, 24, 79, 60, 25, 26] thus leading to an effective displacement of the cells, which is shown in figure 2.18 for multiple components of the system. Here, the WAVE complex is localized at the propagating front of the wave, whereas Arp2/3 and actin show a more homogeneous distribution with maxima still being present around the wave front. The wave then causes a leading edge to appear and moves the cell towards that direction. Similar results have been found for self-sustaining wave-like protrusions of dictyostelium amoeba [25]. Besides that, actin polymerization waves in filopodia, thin protrusions of the cell, can act as seeds for adhesion and lamellipodia formation, thereby also changing. 20.

(33) the cell polarization [26]. On top of these lateral effects, the waves also push into the substrate [30, 79]. Related to the waves’ refractory period, studies by Weiner et al in neutrophils indicate that actin polymerization waves annihilating at obstacles lead to repulsive interactions with them due to polymerization only continuing in the part that did not make contact, as is shown in figure 2.19 [15]. For fibroblasts, the waves seem to facilitate movement along fibrous structures [63].. Figure 2.19.: Upper part: Collision of a cell with an obstacle (red). Fluorescence micrograph of labeled Hem-1. Lower part: Schematic view of the cell-obstacle interaction due to the polymerization wave. Taken from [15].. 2.6.4. Myosin motors enhance motility, but are not essential In both the polymerization waves and the lamellipodium-driven migration, myosin has the important role of creating contractility, to retract the back of the cell during migration and compactify the actin network when a polymerization wave has passed [47, 30, 6]. During migration, the motor proteins can generate forces comparable to those in the leading edge, in the order of nano-Newtons, in the actin-rich back of the cell [14], with a single motor being able to exert a force of around 1 pN [2]. However, studies indicate that cells can still move in the abscence of myosin, albeit much slower than normally [80, 77]. In actin waves, myosin does not influence the wave dynamics themselves, however the loss of compactification seems to reduce the force that can be exerted by the waves and thus again the migration speed [24].. 2.6.5. The role of microtubules and intermediate filaments After all of the explanations regarding the importance of actin for cell motility, the other parts of the cytoskeleton should not be neglected. Regading microtubules, which are important for growth and movement of certain cells in the brain, experiments have shown that they do not influence the movement in keratocytes at all [13]. In epithelial cells and fibroblasts, they have proven to be necessary for the polarization of the cell and. 21.

(34) especially the actin network, however they did not influence movement as such [11, 81]. Thus, assuming normal functionality, these polymers do not have to be incorporated into our system explicitly. Next, the intermediary filaments are vital for the mechanical stability of cells, on top of that they serve important regulatory purposes. They also actively influence cell adhesion [82, 83]. Nevertheless, there is no direct correlation with the actin network, so again, assuming an undisturbed intermediary filament network, they do not need to be described explicitly.. 2.7. Summary and migration of immature dendritic cells In the previous sections, we have seen the current concepts of cell motility. The interior of the cell is filled with proteins and other molecules, which makes it highly viscous and overdamped. The main contributor to motility for many cell types is the actin cytoskeleton, a very dynamic, active, self-organized system of monomers, filaments and many different cofactors that lead to systematic nucleation, polymerization, cross-linking, contraction and degradation. In the cortex of some cell types, this regulatory system of actin is excitable, leading to the emergence of traveling excitation waves. These are able to deform the cell and possibly help to polarize the actin network, leading to directed migration by continuous polymerization at the leading edge. It is now time to look at the benefits of different types of motility and their regulation in living tissues.. Figure 2.20.: Schematic overview of a dendritic cell’s function. The immature state has up-regulated protein expression for uptake of antigens through endocytosis and reduced expression for antigen-presenting sites for interaction with T-cells. The mature one becomes more motile, has reduced endocytosis and increased T-cell-interaction sites, which it needs to get to the nearest lymph node efficiently and initiate the adaptive immune response. Taken from [84]. While this is a vast subject with lots of possible candidates, for the context of this work we will focus on cells of the immune system. Dendritic cells (DCs) are one of the most important cell types for triggering the adaptive immune response to infections with pathogens. They densely populate all outer and inner surfaces of the body as well as. 22.

(35) most organs and act as a bridging factor, being very efficient at taking up antigens and presenting those to T helper cells to initiate further countermeasures. In order to do this, they have two distinct phenotypes, one being an ”immature”, scanning one in the tissue in search of antigens, and the other one being a ”mature”, persistently moving one to get back to the nearest lymph node as quickly as possible. There, they initiate the differentiation of lymphocytes into different effectors, depending on the type and amount of antigen that has been detected previously [85, 86]. A scheme of this is shown in figure 2.20. In more detail, the migration behaviour of DCs drastically changes once they are activated. Upon maturation, both speed and persistence of dendritic cells increase significantly in one- and two-dimensional in-vitro confinements [3, 4]. An overview of this for a 2D confinement is given in figure 2.21. The data suggests that immature dendritic cells turn around more often, likely to scan their surroundings more efficiently, while mature ones engage in more persistent motion with a longer ballistic phase and faster diffusion thereafter, which makes them reach lymphatic capillaries and vessels more quickly [4].. Figure 2.21.: Comparison of the movement of immature DCs and DCs activated with LPS. Shown are example trajectories (a), speed (b) and persistence (c) distributions as well as a plot for the mean square displacement (d). Adapted from [4]. As mentioned in the introduction, dendritic cells do not need adhesive bonds with the surrounding tissue in order to move and only use those to scan cells for pathogens [4], making them easy to study ex vivo. The erratic migration they exhibit during their search state when being immature makes them a promising specimen to study. Especially the possible coupling of actin wave dynamics to their motility is an intriguing concept.. 23.

(36) 3. Theoretical approaches to studying the cytoskeleton Before defining our system, this chapter will give a general overview of the possible approaches one can use to do so. In principle there are three different length- and time scales that can be chosen when describing the cytoskeleton. Each of the approaches has a different focus, as will be discussed in the following.. 3.1. Stochastic reaction kinetics and laws of motion To arrive at a theoretical description of the systems introduced in the previous chapter, one approach is to derive one based on the microscopic constituents of said systems. In the context of the cytoskeleton, this means that all of the important parts of the system, such as monomers, filaments and cofactors/motor proteins, are described explicitly and individually. The advantage is that all of the interaction rates of individual components as well as possible structural changes directly translate to processes happening in an experimental system without making simplifications and thus can possibly be measured directly. One basic numerical strategy is to use a Monte-Carlo method, a class of algorithms which utilize random sampling to arrive at a result [87]. An example that helps to understand this concept is a Gillespie-algorithm [88]. First, all of the molecules that need to be described as well as all possible reaction rates are initialized. Then, the rates are summed up to give a total reaction rate rT for the system. To advance, a time step is randomly chosen from an exponential distribution with an expected value of τ = 1/rT . The event that happens is chosen at random as well, with all events having a probability scaling with their corresponding reaction rate. All dynamic processes are updated, then the algorithm goes back to the step of summing up all of the current reaction rates and repeats until a threshold time is reached. This has been used to study the interplay of different types of molecules in the cytoskeleton [53, 89, 90]. Another way would be to describe a many-body system following the Langevinequation [91, 87] ẍ(t). = µ F (x, t) + f (t). (3.1). where x is a vector of dimension nd for n particles in d spatial dimensions, µ is a mobility coefficient matrix, the force tensor F (x, t) contains all particle interactions and possible. 24.

(37) external forces, and f (t) introduces white noise as a fluctuating force with hf (t)i = 0 and hf (t1 )f (t2 )i = 2Dδ(t1 − t2 ) where D is a diffusive constant. This system can then be integrated [94]. Either way, due to microscopic systems being noisy, a stochastic description is needed. Overall, these approaches led to new findings in the areas of molecular dynamics, polymer properties and basic network features. To name a few examples, studies by Nédélec et al showed how microtubules interacting with an array of motors could form asters, replicating in vitro results [92]. Placing two of these asters next to each other, which is a description of microtubule organisation during cell division [1], led to antiparallel overlaps in between them. These regions induced stable interaction forces in the presence of hetero-complexes of motors, motor dimers whose ends move towards opposite ends of the microtubule [93]. For actin, there have been simulations of the interaction dynamics of actin and a number of its various cofactors to study important feedback loops [90]. As the actin filaments are modeled by individual monomers, the monomer fluxes through the treadmilling filament as well as the percentage of hydrolized actin monomers in the filament could be determined in dependence of the effects of the cofactors. Through replication of experimental data, the authors could provide additional insight into the underlying molecular dynamics happening on the actin filament [90]. Additionally, there are studies of contractile fibers and the contractile ring during cell division, where imposing an anti-symmetric order or the existence of bipolar actin filaments - two filaments growing out of opposite ends of a single nucleator or nucleator complex - causes contractions [53]. However, because the complexity of these simulations increases exponentially with the number of parts simulated, the study of big structures like lamellipodia and other systems with lots of individual components is numerically unfeasible. For these cases, it is typically beneficial to switch to a bigger length scale.. 3.2. Physical principles, symmetries and conservation laws When taking into account structures on length - and time scales which are large compared to the molecular ones, a hydrodynamic approach similar to the derivation of the Navier-Stokes-equations can be used. With this, the underlying molecular properties of the constituents of the system can be largely ignored, while only focusing on the conserved macroscopic quantities and broken symmetries [95]. In the case of actin, the network on the scale of a cell consists of polar, active filaments which are cross-linked by cofactors, creating a polar, active gel. As the active part means that the system is continuously spending energy, here for (de-)polymerizing or contracting the network, a non-equilibrium description is needed. One of the possible ways to derive it is perturbing the system near an equilibrium state [95, 96]. By doing so,. 25.

(38) a reference frame is given and non-equilibrium thermodynamics can be used. However, this approach is only valid for small perturbations, which is not always justifiable [95, 96]. As the total mass of actin is assumed to be constant, the mass density ρ satisfies the continuity equation ∂t ρ. = −∂i ji. (3.2). where j is the flow of the gel. When treadmilling and active stresses are considered to induce flows with a local velocity v, j assumes the form j = vρ [95, 96]. The second conservation law is given by momentum conservation. Due to the overdamped interior of cells, inertial terms can be neglected and momentum conservation becomes a force balance equation, which reads tot ∂i σij + fj. =. 0. (3.3). tot and external force density f [95, 96]. In general, this with the total stress tensor σij j system of equations is determined if the flux j is known. For more complicated systems with additional constituents, the fluxes of each corresponding continuity equation need to be known as well [95, 96].. This is where next-to-equilibrium thermodynamics can be utilized. The general strategy here is to use the rate of change in the free energy F to read off conjugate fluxes and forces. Subsequently, linear response theory is used to derive the relations between them and Onsager reciprocity relations reduce the amount of parameters by taking into account the symmetries of the system [95, 96]. Before writing down the rate of change for an actin gel, it is important to remember the polar aspect of the description. Since this means that aligning actin filaments create a local order, a director field is needed to describe this. Due to this, we introduce the actin polarity field p. The resulting rate of change in the free energy is [96] Z D ∂t F = − dV σij uij + hi pi + ∆µr (3.4) Dt Here, the generalized flux D/Dt pi = ∂t pi + vj ∂j pi has the conjugated generalized force hi = −δF p /δpi . h corresponds to the minimization of the free energy of the gel, F p , at thermal equilibrium [96]. Apart from that, we have the generalized force that drives the system out of equilibrium, which is the difference in the chemical potentials of ATP and its hydrolysis products ADP and phosphate, ∆µ = µAT P − µADP − µP . If ∆µ > 0, an equal amount of free energy is consumed with each ATP [96]. Its conjugated flux is the ATP consumption rate r. Lastly, the symmetric part of the stress tensor, σij , is conjugate to the velocity gradient uij = 1/2 (∂i vj − ∂j vi ) [96].. 26.

(39) Finally, writing down the fluxes as linear functions of the forces leads to ν1 2ηuij = σij − (pi hj + pj hi ) − ν̄1 pk hk δij 2 ¯ ij + ξpi pj + ξ 0 pk pk δij )∆µξ +(ξδ Dpi 1 = hi + λ1 pi ∆µ − ν1 pj uij − ν̄1 ujj pi Dt γ ¯ i pj uij + ξ 0 pi pi ujj r = Λ∆µ + λ1 pi hi + ξp. (3.5). (3.6) (3.7). ¯ ξ 0 , γ and Λ. With these equations, the with phenomenological coefficients η, ν1 , ν̄1 , ξ, ξ, system is fully determined. Using this type of approach, it is possible to describe the retrograde flow of actin in lamellipodia [97] as well as the formation of asters and spirals in pure actin networks [98]. Simplifying the system by assuming an asymmetric structure like in muscle sarcomeres could be used to study their contraction behavior [99]. The authors replicate oscillations of the sarcomeres by assuming force-dependent binding kinetics for the myosins. Apart from these processes involving polar order, growing filaments align and aquire orientational ordering if the density is high enough. In those geometries, the polarity of the filaments does not matter, only their elongated rod-like shape. They can then be described like a liquid crystal using the nematic tensor Qi,j = S(ni nj − 0.5δij ) [100], where S is the nematic order parameter, describing how well filaments are aligned locally, and n is the director field pointing in the direction of filament elongation. Q is traceless and symmetric, so the sign of n does not matter, unlike before for the polarity vector p. This description can be used to study experiments with motor assays, where filaments are displaced by motors bound to a substrate. Studies show that active stresses in nematic substances are already sufficient to create excitable states and oscillations without the need for polar order if S is allowed to fluctuate [100]. Additionally, if one assumes that the nematic material phase-seperates from the medium to form an active nematic droplet, these droplets have been shown to exhibit behavior like swimming motility and division by generating flows in the surrounding fluid [101]. Concluding these results, the provided approach can be used to identify the most important macroscopic physical processes needed to create the phenomena observed in experiments [102]. The trade-off comes with the fact that there is no immediate connection between the macro- and the micro-scale, so knowledge of the essential physical processes does not necessarily lead to a better understanding of the microscopic components. Especially compared to experiments, the phenomenological coefficients like in equations (3.5-3.7) are often abstract and not directly relatable to biological quantities. Lastly, since the description of flows and fluxes is linear, possible nonlinear effects need to be captured separately through the introduction of additional terms. In order to avoid these disadvantages while keeping the benefits of larger scales, it is feasible to use a mesoscopic description instead.. 27.

(40) 3.3. Mean-field descriptions of microscopic processes The mesoscopic- or mean-field-approach makes use of the fact that when looking at a large number of individual particles, the effects of noise will average out in a number of systems. It presents a middle ground between the microscopic and the macroscopic descriptions, as the terms in the equation are motivated by individual particle interactions. The simplification lies in assuming that these interactions, when observed in large numbers over long time scales, result in an average particle-particle interaction. This effectively reduces a many-body problem to a one-body problem and decreases the degrees of freedom in the system by doing so. One example of this is given in the previously presented description of molecular dynamics by the Langevin equation, which can be transformed into an equivalent, mesoscopic Fokker-Planck equation. As an example, equation 3.1 for one particle in one spatial dimension becomes [103] ∂P (x, t) ∂t. = −µ. ∂ ∂2 (F (x, t)P (x, t)) + D 2 P (x, t) ∂x ∂x. (3.8). with the probability P (x, t) to find the particle at position x at time t and the other quantities like before. The noise is now contained in the diffusion term. Generally, each microscopic description has an equivalent mesoscopic one and vice versa. While the Langevin equation results in random particle fluctuations and interactions over time, the Fokker-Planck equation gives the average probability of finding a particle in a certain state at a certain position. Another way of describing such a system is not with a probability -, but with a mass density, typically resulting in convection-diffusion equations. The simplifications are the same. For a sufficiently high particle count, the interactions can be averaged and only result in mean changes of the mass density. Start of the derivation in this case is typically a force balance equation for the described constituent. As an example, consider a bundle of action filaments of mixed polarity together with myosins. The structure of the bundle is effectively one-dimensional. By assuming a constant total density for the filaments of both orientations, the number of filament centers at a given position x at time t, c± (x, t) with their plus ends pointing in positive or negative x-direction, satisfy the continuity equation [39, 104] ∂c± (x, t) ∂t. = −. ∂ v± c± ∂x. (3.9). The speed can then be determined from a local force balance of the actin filaments. In the absence of an external force, the friction due to the displacement with speed v± and stochastic fluctuations due to diffusion, D∂x c± , needs to be balanced by the integral. 28.