# Preamble¶

During my PhD, I decided to investigate the effect of the actin network on the mechanical properties of cells. Indeed, cell mechanics are a key parameter that has crucial impact on cellular and organisms functions. Being able to detect changes in the mechanical properties and to understand the mechanism that governs these changes is an important step in the study of cellular behavior as well as in the differentiation of healthy from cancerous cells and tissue. Understanding the mechanisms that are at the origin of cell motion and shape changes, is also a decisive step in controlling cell behavior, with the ultimate goal to prevent cancer cell invasion and division without impairing healthy cells.

During the last three years, I decided to focus on biomimetic systems and to determine the characteristics of actin networks. Actin is a highly conserved component across the living domain, and it plays a major role in cell mechanics. By interacting with a number of other components of the cell, actin is able to form various different types of networks. I decided to focus my research on such networks that were created under controlled conditions.

Along this dissertation, we will mainly focus on three systems.

First, we reconstituted an already observed actin network — the actin cortex — on a biomimetic system, then showed that a second sparse actin network, which was previously unseen, emanates from it, and finally characterized its mechanical properties. We developed the idea that the effect of this second network cannot be neglected in cells and investigated a few of the phenomena it may be involved in.

As the effect of such a sparse actin has not been demonstrated in living cells, we decided to investigate the effect of another sparse actin network found in a living cells. In collaboration with the group of Marie Hélène Verlhac at the College de France, we studied the mechanical properties in the cytosol of mouse oocytes. We saw in this system that actin-related proteins had hi-impact on the structure and the mechanics of both the cell and the actin network.

Characterizing the dynamics of a network in a living cell by controlling the conditions remains complex. In a third research stream, we characterized the dynamical changes of tension created by reconstituted actin cortices, that are linked to a lipid membrane. By studying liposome “doublets”, we could measure the variation of tension generated by the acto-myosin cortex over time. This system is composed by a liposome doublet covered with an actin network. By imaging with Spinning disk microscopy, we could reconstruct the changes in the acto-myosin network and deduce them in its properties, from the geometrical variation during the network contraction.

# Background¶

## Introduction¶

Cells are the basic components of living organisms. Understanding their individual behavior and the way they function is a key step to understand how they interact with their environment. One of the key components within most cells is the actin cytoskeleton, which is made up of actin monomers, a highly conserved protein across species, which plays an important role for cell mechanics, ranging from cell migration to cell differentiation and division. Hence the crucial role it plays for the mechanical properties of the cell and its mechanical interaction with the environment. Under the cell membrane lies a thin actin network which controls the mechanical properties of the cell: the actin cortex. The mechanical behavior of this actin cortex is itself driven by the dynamics and interactions within the actin network it is made of. So, understanding this actin network is a key to learn how the actin cortex behaves, leading to a better understanding of cells and tissue.

The properties of an actin network highly depend on its structure. The structure itself depends on many parameters that influence how the network is formed. The network structure and formation, influenced both by the physical and chemical conditions and the spatial and temporal variation of these parameters, such as mechanical stress or ion-concentration, can determine the fate of the network. It is therefore important to study these networks and their dynamic behavior in order to grasp the changing structure of the cell.

Cells are complex systems that adapt their shape, mechanical properties and biochemical conditions permanently. The spatial repartition of these properties is also variable as the cell regulates the concentration of proteins all across its cytoplasm. In order to achieve a thorough study of the effect of each component independently, it is crucial to study actin networks in a controlled environment.

Biomimetic systems allow to respond to most of these concerns. First they provide a controlled environment that mimics in-vivo phenomena. Second, biochemical conditions can be well controlled, both in space and time, hence allowing to precisely fine tune experimental conditions. Biomimetic systems are also particularly adapted to be combined with optical traps, which allow us to study local mechanical properties of actin networks with high temporal resolution. The combination of both allows us to get insight into the variation of these mechanical properties as a function of time and space, with high precision.

During my PhD, I have focused on the mechanical properties of branched actin networks polymerizing on optically trapped polystyrene beads. Such networks have already been studied before [Kawska et al. 12] but have been suspected to be highly inhomogeneous. Optical traps allow to probe the mechanics of yet inaccessible parts of the network. I further studied actin networks on other biomimetic systems constituted of liposomes, in order to better understand the effect of actin cortex polymerisation on membrane tension and to characterize network dynamics over time. Finally, I participated in a collaboration in order to understand the implication of such actin networks in living mouse oocytes.

## Living Cells¶

Cells are the basic building blocks of life, and all living beings are composed of cells, from unicellular up to multicellular organisms like us. Unicellular organisms must accomplish all their functions within a single cell. At the other end, in multicellular organisms cells differentiate in order to accomplish specialised tasks often by regrouping into organs. Despite sharing the same genetic material, for each cell to accomplish a different task often requires different mechanical properties. The variation of elasticity and other mechanical properties of cells derive from the structure they are composed of.

Cells are hence able to adapt to their environment and develop functions and behavior that may change over time. A small change of timing and/or biochemical conditions can highly injure the development of an organism: for example modification of the actin network at a given time during the cell cycle prevents symmetric division [Lenart et al. 05], [Vasilev et al. 12]. Furthermore, the mechanical properties of the substrate can govern the differentiation of cells: Soft substrate will favor brain-tissue cell, where stiff substrates increase the appearance of muscle cells [Engler et al. 06].

Nonetheless, even with all theses different behavior and phenotypes, cells have a common structure. The exterior of the cell is separated from the inside by a plasma membrane. The interior of the cell is filled with the cytoplasm which contains diverse structures such as organelles, genetic material, and a large number of proteins that the cell uses to accomplish its functions. To communicate with the outside, cells have a series of mechanisms that allow signals and cargo to pass the membrane. This communication can be chemical, but mechanics is also known to participate in the process. To sense their mechanical environment, cells often use adhesion complexes to attach to the substrate, and integrins as trans-membrane protein to transfer the force to the cell cytoskeleton situated inside the cell. Chemical signals can either cross the membrane through trans-membrane proteins, while endocytosis and exocytosis are ways for the cell to import and export proteins and chemicals through its membrane.

## Oocyte¶

A particular cell type I was interested in during my PhD are mouse oocytes. Oocytes are female germinal cells in the process of gametogenesis. Unlike somatic cells that undergoes symmetric division via mitosis which leads to two identical cells sharing the same genetic material, oocytes undergo a different process called meiosis. Meiosis in oocytes is a highly asymmetric process necessary for the specificity of being large haploid cells, containing at the end of meiosis only one chromosome of each pair that constitutes the genetic material of a mouse. The second chromosome of each pair will be provided during fertilisation of the oocyte by the male sperm.

The exact process of oocyte formation can vary among species, and in the following we will describe the main mechanisms.

The complete process of egg maturation starts with primordial germ cells that undergo mitosis to replicate until they enter the first meiosis (Meiosis I) at which state they are called primary oocytes and are still diploid, that is to say still contains two chromosomes of each pair.

The primary oocyte will start maturation and growth and then undergo a first asymmetric division just after prophase I. This first division is asymmetric both in the genetic material separation and in the unequal size of the formed daughter cell. Indeed, the primary oocyte will divide into a secondary oocyte and a polar body. Both, the secondary oocyte and the polar body are haploid and contain only half of the genetic material of the primary oocyte. The secondary oocyte can go through Meiosis II in which it undergoes a second asymmetric division and expulsion of a second polar body. These polar bodies will eventually degenerate (Fig 411).

During meiosis, the process of cell division also differ from mitosis. Instead of separating into two identically sized cells through the formation of a cytokinetic ring, the primary oocyte will become the secondary oocyte by expulsion a polar body. The formation expulsion of the polar body require precise positioning of the cell organelles. During prophase I the nucleus of the oocyte is carefully centered, undergoes a nuclear breakdown and spindle formation. The first meiotic spindle will migrate toward the oocyte cortex along its major axis. Once at the cortex, half of the genetic material of the spindle will be expelled through the membrane forming the first polar body of much smaller size than second oocyte.

Mouse oocyte are good model systems to study the mechanical properties inside cells. They form big spherical cells with a diameter of around 80 µm which allow to study the mechanical properties at different locations in the cytoplasm.

In the third part of my PhD I participated in a collaboration with Marie-Hélène Verlhac and Maria Almonacid at Collège de France who are interested in the effect of actin dynamics in oocyte cytoplasm during the different parts of oocyte gametogenesis.

### Cell Organelles¶

Inside the cytoplasm, cells have a number of structures with different and specialised functions which are called organelles. The position and state of organelles is of great importance for the cell to achieve its functions. Probably the most known organelle is the cell nucleus of eukariotic cells that contains the genetic material. Attached to the nucleus is the endoplasmic reticulum which is the organelle responsible for translating RNA coming from the nucleus to functional proteins that will be delivered across the cell after maturation in vesicles. Theses vesicles are transported across the cell both by dyneins and kinesins — molecular motors — that walk along microtubules originating from the centriole part of the centrosome but also by myosins walking along actin filaments. All of those processes consume energy in the form of ATP, generated within the mitocondria spread across the cytoplasm. A schematic of the cell with some organelles can be seen on figure 412

The positioning of organelles is crucial for the life of the cell. During meiotic division of cells, for example, it has been seen that the positioning of the nucleus at the center of mouse oocytes happens before its migration closer to the cortex to expel the first polar body. Failure to do so results in a incorrect amount of DNA in germinal cell that can lead to infertility.

It is already known that microtubules play a key role in organelle positioning. Microtubules emanating from centrosome position at the two ends of the cell during its division are used to fetch the correct chromosomes. Each chromosome is pulled towards the centrosome which leads to each daughter cell having the same amount of DNA.

Actin plays also an determinant role in organelle positioning process, like in drosophila oocyte maturation where it positions the nurses cell away from the dumping canal [Huelsmann et al. 13]. In a later chapter (Organelle Positioning) we will develop a few keys points where actin is indispensable in organelle positioning and how this relate to the biomimetic actin networks we reconstitute.

### The Cytoskeleton¶

The cytoskeleton, literally skeleton of the cell, is the structure which gives the shape to a cell. As for other multicellular animals that posses skeleton, its shape is often a hint on how an organism moves. As feet, fins and wings are characteristics that will tell you whether a animal prefer land, see or air, the cytoskeleton will tell you many things about a cell.

Unlike the (exo)-skeleton of animals which is rigid and static, the cytoskeleton of cell is a highly dynamic structure that keeps remodeling itself on a short time scale compared to the speed at which a cells move. Thanks to these dynamics, the cytoskeleton can achieve its functions. As vertebrates skeletons are necessary to transmit force from one part of the body to another, the cytoskeleton is responsible not only to transmit the forces the cell is exerting, but also to generate theses forces. The cytoskeleton connect a cell to its environment, both mechanically and biochemically.

We will consecrate a longer part of this work to describe the cytoskeleton.

## The Role And Composition Of The Cytoskeleton¶

We have already introduced the cell cytoskeleton in the previous part, and we will now describe its components and functionality more in detail here. The cytoskeleton has three main functions, it connects the cell both physically and biochemically to the external environment, generate and coordinate the forces that give the cell its shape and allows it to move. It is also responsible for organising spatially the cell content [Fletcher et al. 10]. The cytoskeleton is also in particular sensitive to spatial and temporal information that can affect cell fate and the assembly of the cytoskeletal structure. This can be seen for example with the bud scar of budding yeast that persists after division.

### Composition of the cell cytoskeleton¶

The cytoskeleton is mainly composed of three types of filaments. Microtubules, intermediate filament and actin filament, also known as microfilaments.

Microtubules are the widest structure with a diameter of 20nm (Fig 413) and the stiffest of the three kinds of filaments with a persistence length in the order of millimeters, much longer than the size of the usual cell. Microtubules are extensively studied [Valiron et al. 01]. Microtubules are formed by the polymerisation of a heterodimer of tubuline that leads to the formation of polar (oriented) filaments that can be walked on by molecular motors. These molecular motors can be decomposed in two families – kinesins and dyneins – depending on the end towards which the motor preferably walk. Microtubules are mostly known for their action during mitosis where they will form the majority of the mitotic spindle that drive the segregation of the chromosomes in two groups, each group ending in one of the daughter cells.

Microtubules have the characteristic of being highly dynamic by alternating between two states of rapid growth and a rapid shrinkage. The transition from microtubule growth to shrinkage is called a catastrophe, the transition from shrinkage to growth is called a rescue.

Intermediate filaments are of medium diameter in the order of around 10nm, in between actin and microtubules filaments, hence their name. Unlike microtubules and actin filaments, intermediate filaments are composed by several sub-families of proteins and are non-polar.

Intermediate filament have an important role in the mechanical properties of cells due to the fact that they are particularly resistant to stretching.

Unlike actin and microtubules, they are thought to be passive, with mechanical properties mainly deriving from how multiple filaments are linked together laterally.

Actin, is the third component of the cytoskeleton, the one on which we will focus on most of our efforts. Actin monomers, also called G-Actin for globular actin can polymerise. By polymerizing actin monomers (G-actin) into actin filaments (F-actin), the thinest of the three cytoskeletal components forms. Actin is produced in the cell as a globular protein of ~40 kDa (Fig 414) that once associated with ATP or ADP polymerises into helicoidal filament with a diameter between 7 and 9nm. The formed actin filaments are polar, where both extremities are respectively called the plus (+) or barbed end, and the minus (-) or pointed end. The polarity of the actin filament is of importance as this gives rise to a preferred direction for most processes that can happen on the filaments.

The actin protein is highly conserved across species, and is know to directly interact with hundreds of proteins [DosRemedios et al. 03].

Single undecorated filaments will behave as semi-flexible polymers at the scale of the cell with a persistence length in the order of 10 µm [Isambert et al. 95]. When they assemble into different structures and networks, or associate with other proteins and molecules the resulting mechanical and dynamic properties can be highly variable.

#### Dynamics of actin polymerisation¶

The assembly mechanisms that allow to go from single monomers of actin (also refer to as G-actin for globular actin) to actin filament (also refer as F-actin) need to be well understood to explain the different network structures created by actin filaments in the presence of other proteins.

The polymerisation of ATP/ADP actin monomers to form an actin filament need to go through the step of forming an actin proto-filament which is constituted of at least 3 actin monomers. This will most of the time be the kinetically limiting step. Once proto-filaments are present in solution, single monomers can be freely added or removed on both ends of the filament. The process of forming these proto-filaments is called nucleation and it is the rate limiting factor to form actin filaments. To circumvent this limitation experimentally one can use preformed actin filament seeds, or actin nucleators to direct the polymerisation on the cell.

We need to distinguish between the dynamics of polymerisation and depolymerisation on both ends of the filament. Indeed, it has been show that the association and dissociation rates are differ between the pointed (-) and barbed (+) end. The barbed end has higher dynamics than its pointed counterpart which is the reason for its (+) name. The dynamics of polymerisation is higher both in he case of ATP and ADP, though the rate constant of association and dissociation differ for both kind of filaments (Figure 415)

The equations that drive the polymerisation can be written as follow

(1)$\begin{split}\left. \frac{dn}{dt} \right|_{barbed} &= k_{+,{barbed}}.[GActin] - k_{-,{barbed}} \\ \left. \frac{dn}{dt} \right|_{pointed}&= k_{+,{pointed}}.[GActin]- k_{-,{pointed}} \\\end{split}$

Where barbed and pointed designate respectively the barbed and pointed end, $$\left.\frac{dn}{dt} \right|_{barbed|pointed}$$ represent the variation of the number of actin monomers which is due to addition or removal at the barbed (respectively the pointed) end of actin filaments. The association rate constant $$k_+$$ and dissociation rate constant $$k_-$$ are the polymerisation and de-polymerisation rate. The concentration in barbed and pointed-end denoted by $$C_{{barbed}/{pointed}}$$ , $$[GActin]$$ is the concentration of action monomers in solution. By assuming that the concentration of pointed ends is equal to the concentration of barbed ends (no capped or branch filaments), one can derive the steady state which give rise to the critical monomer concentration below which an actin filament cannot grow: $$[GActin]_c$$.

The rate constants of elongation of actin have been determined and depend on whether the monomer is bound to ADP or ATP [Pollard 86]. We should consider the fact that the ATP bound to actin will hydrolyse to ADP-Pi before releasing the inorganic phosphate. The hydrolysis and phosphate release rates also depend on whether the monomer is part of a filament or in solution. The hydrolysis of ATP-bound actin into ADP bound actin in the filament leads to an imbalance of actin (de)-polymerisation on both ends. The actin filaments preferably grow from the barbed end and shrink preferably from the pointed end.

This will lead to a phenomenon known as treadmilling where a single actin monomer bound to an ATP molecule, will be incorporated at the + end of the filament and progressively migrate toward the - end, eventually hydrolysing its ATP into ADP before detaching from the filament on the pointed end. During this process the filament will grow / shrink until it reaches the stationary state where its length would stay constant but the treadmilling continues.

Treadmilling requires an imbalance in the global rate constant on the barbed and pointed end and an energy source, in the case of actin this is provided by the hydrolysis of ATP into ADP+Pi before releasing the inorganic phosphate, without which treadmilling would not occur.

Practically, this can be approximated by having only ATP monomers at the barbed end of actin filaments while the pointed end is typically constituted only of ADP monomers, thus the critical concentration is lower at the pointed end compared to the barbed end. The growth speed of the filament on both ends depends on the monomer concentration in solution. In between the critical concentration of both ends, there exists a concentration at which the polymerisation on (+) exactly compensates the depolymerisation on (-).

#### Actin network can be controlled by a host of actin binding proteins¶

Despite the already complex process of actin polymerisation and the number of parameters that we have already introduced, the formation of an actin network is an even more complex process that involves many other components. Especially, actin monomers and filaments can interact with a high number of proteins that will affect the previously introduced dynamics. We will present some categories of such proteins in the following.

##### Formins¶

Formins are polymerase proteins that will increase the polymerisation rate of actin filaments by dimerizing and binding to the barbed end. It has the particularity of being processive, meaning that it will stay bound to the barbed end while catalysing the addition of new monomers. The processivity of formins also permits the control of the localization of actin polymerisation where formin proteins are present, like the tip of filopodia [Faix et al. 06] [Bornschlogl 13]. Formins posses domains rich in proline, capable of binding to profilin (FH1) which allows formin to elongate F-Actin using actin monomers bounds to profilin [Pruyne et al. 02] [Pring et al. 03].

##### Capping Protein¶

To regulate polymerisation, cells also have the possibility to reduce or stop the polymerisation. To achieve this, some proteins will bind to the growing end of actin filaments and prevent the addition of new monomers. Capping Protein (CP) being one particular example that will specifically bind to the barbed end of a growing filament and prevent it from growing. Capping proteins are necessary to prevent polymerisation of actin in undesired area and are essential for the structure and mechanical properties of actin gel [Kawska et al. 12]. Gelsoline is another example of Capping Protein, that unlike CP can only attached to the barbed end of an actin filament after severing it. Gelsoline is hence both a severing and a Capping Protein.

We have seen that some proteins were able to attach to actin filaments. When such a protein is able to attach to many filament at once, it can act as an attachment point between the two filament, preventing them to move with respect one to each other. Such proteins, are referred to as cross-linkers.

The amount of freedom in movement between the two filaments depends on the cross-linker used. For example , α-actinin will allow rotation of the two filament at their anchoring point whereas cross-linkers like fascine will prefer a parallel conformation of the filament and favor the formation of actin bundles.

Cross-linkers are essential for the formation of elastic network as they allow forces to be carried from one actin filament to the other. The quantity of cross link of a network will often be a key parameter for the elastic properties. The distance between the link points in the network (both cross links and entanglement points) will give the typical network mesh-size which is used to calculate the viscoelastic response of networks : [Morse 98a].

##### Stabilizing actin filaments¶

As actin networks are dynamic constructs that are changing shape and properties over time, it is convenient to be able to stabilize those networks. Tropomyosins are proteins capable to bind on the side of actin filament to stabilize them.

The use of phalloidin, a toxin extracted from fungus (Amanita phalloides), binds between F-actin subunits on the filament, and hence prevents it from de-polymerising. Though, it is known that stabilizing actin filaments with phalloidin will increase their stiffness as measure by the persistence length which can change the mechanical properties of the formed actin network.

##### Profilin¶

Profilin is a protein that will bind to the barbed end of single monomers of actin in solution. By doing so it will first prevent the association of monomers into dimers and trimmers, thus preventing the nucleation of actin filament. It thus allows a better control of localisation of actin filament both in vivo and in vitro in the presence of actin seeds of actin nucleator.

Profilin was for a long time believed to be only a sequestering protein that inhibit polymerisation [Yarmola et al. 09], though it has a more complex behavior, and if it prevent polymerisation of actin filaments by the pointed end, it can facilitate polymerisation. One of the cause of increase in polymerisation speed by profilin is the fact it binds preferably to ADP-Actin and increase the exchange rate of ADP into ATP.

##### Branching Agent¶

A type of network found of the leading edge of cells lamellipodia is dendritic network. It is characterised by tree-like structure of actin filaments in which thanks to the Arp2/3 complex branching agent a mother actin filament will form a daughter filament on its side.

We have seen previously that crosslinkers are proteins capable of linking two or more actin filaments together by binding on their side. Another mechanism involving binding on the side on actin filament is responsible for a closely related network, the branching mechanism.

The Arp2/3 complex is composed of seven subunits, two of which are highly similar to actin, forming the Arp2 and Arp3 family for Actin Related Proteins, giving the complex its name. Typically Arp2/3 will bind on the side of a pre-existing actin filament, hence initiating the growth of a daughter filament with an angle of 70° to the mother filament. The newly created daughter filament pointed end is terminated by the Arp2/3 complex that will stay attached to the mother filament, thus increasing the number of available barbed end, without changing the number of available pointed end. See Nature Review by Erin D. Goley and Matthew D. Welch [Goley et al. 06] for a longer review about the Arp2/3 complex.

In cells, the Arp2/3 complex needs to be activated by a Nucleation Promoting Factor (NFP). Among them is the WASp protein (Wiskott-Aldrich Syndrome protein) and its neural homologue N-WASP which are from the same family as SCAR/WAVE [Machesky et al. 99]. All these activators of Arp2/3 have in common a WCA motif. The wild type protein need to be activated in order to activate Arp2/3. The activation is done by a change in conformation that exposes the active region and provides the first actin monomer necessary for nucleation of the daughter filament (Figure 416). To circumvent the activation process of these proteins, we use a reconstructed version of the protein that cut all region before the poly-proline. This confer to pVCA the ability to be permanently active. This region can also be replaced by streptavidin in order to selectively bind pVCA to selected regions. Characterisation and more detailed description of pVCA can be found in [Noguera 12].

Unlike Cells that are able to control the localisation of actin nucleation processes thanks to activation of WASp and its homologue, the ‘in vitro’ control of localisation of actin polymerisation is directly done by the localisation of pVCA.

The network formed by Arp2/3 is called a dendritic network, and is in particular found at the leading edge of the cell in the lamellipodia. It is such a network that is present in the bead system we will study hereafter.

As for crosslinkers, dendritic networks are able to carry forces across single actin filaments by the intermediary of Arp2/3. Two dendritic network of Arp2/3 can also entangle and allow forces to be carried across them [Kawska et al. 12].

A schematic that recapitulate the interaction of actin with other protein and the formation of a dendritic network at the leading edge of the cell is presented on figure 417.

##### Molecular Motor¶

A particular kind of protein that can bind to the cytoskeletal filaments are molecular motors. Molecular motors are proteins that will consume energy in the form of ATP, hydrolyse it to change conformation and produce forces.

The motors that move along actin filaments are part of the myosin superfamily, they are both responsible for the transport of cargo along filaments, cell motility, division, and muscle contraction. They acquire their name from their discovery in 1864 by Willy Kühne who extracted the first myosin II extract from muscle cell [Hartman et al. 12].

The myosin super family is divided into subfamilies numbered with roman literals. As of today we count more than 30 families of myosin [Berridge 12]. Muscle myosin is part of the myosin II family and is often referred to as conventional myosin for historical reason as being the first discovered. Non-muscle myosin are also referred to as unconventional myosin.

Myosin motors seem to be shared among the living domain, hinting for an early emerging of myosin in the evolution. All the myosin motors move on actin filaments toward the barbed end, with the exception of myosin VI which moves towards the pointed end [Buss et al. 08].

Different subfamily of myosin are used for different function in cells. Even in subfamilies each type of myosin can have specific functions. For example, conventional myosin found in muscle cells are use for large scale cell contraction. In contrast, myosin V is known to transport cargo and is found to be responsible for actin network dynamics and vesicle positioning [Holubcova et al. 13].

###### Myosin II¶

As stated before, the myosin II family both encompass conventional myosin as well as Non-muscle myosin II (NMII). Both have a similar structure (Fig 418).

All myosin IIs are dimers constituted of two heavy chains and light chains. The heavy chains are held together by a coil-coiled alpha helix referred to as the tail. On the other side of the protein sequence is a globular head, which is responsible for ATP hydrolysis and is able to convert the energy from the hydrolysis into mechanical force. It is also the part that will bind to the actin filaments. In between the tail and head is the neck domain that acts as a lever to transmit the force generated by the head to the tail. The length of the neck influences the length of the movement done by the cargo at each step of the myosin as well as the size of the step the myosin can effect. The two light chains are situated in the neck region and are responsible for the myosin activity regulation.

Myosin II dimers can align and assemble by the tail region, forming myosin minifilaments. These minifilaments are bipolar, having numbers of myosin head with the same orientation at each extremity.

In the myosin II family, conventional myosin and NMII differentiate by the size of the minifilaments they form. Muscle myosin will form minifilaments aggregating around 200 dimers, where NMII minifilaments will be composed only of 10 to 20 minifilaments. The other characteristic of unconventional myosin with muscle myosin is the mode of activation. Conventional myosin activity is regulated by the amount of $$Ca^{2+}$$ available, which frees the actin filaments to let the myosin motor bind. However, its counterparts are typically activated by the phosphorilation of the Myosin Light Chain (MLC).

Another parameter that discriminates muscle from non-cell myosin is their duty ratio. The duty ratio is define as the ratio of the time the myosin stays attached to an actin filament over the typical time of a contraction cycle. By noting $$\tau_{on}$$ and $$\tau_{off}$$ the time the myosin head spent attached/detached from the filament, the duty-ratio or duty-cycle can be noted :

(2)$r = \frac{\tau_{on}}{\tau_{on}+\tau_{off}}$

We will see in the following that the duty-ratio might have an important effect on the processivity of the myosin.

It should be noted that as minifilaments can attach to actin filaments on both ends, they can also act as a bridge that holds two points close to each other, though having the properties of crosslinkers.

###### Myosin V¶

Myosin V is an unconventional myosin. Unlike myosin II it does not aggregate into minifilaments. Though, myosin V has a similar structure to myosin II but with a longer neck, this confers to myosin V the ability to realize longer steps on actin filaments. Indeed, the myosin V step size is of 36nm, which is close to the twisting length of actin filaments. This allows myosin V motors to walk along actin filament without having to rotate around it with the helix they form. At the end the tail domain myosin V posses another globular domain capable of binding to its cargo, and the variability of this region is what mostly define the difference between the different type of myosin V.

Myosin V also has a high duty-ratio, this leads to dimers having almost always one of the two head of the myosin to be bound to actin. It grants to the myosin V the ability to walk in a processive manner toward the barbed end of the actin filaments, both head successively binding 36 nm in front of the other head.

###### Myosin cycle¶

We saw earlier that the duty ratio of myosin was the ratio of time the head of the myosin spent attached to the actin filament. Indeed, myosin can generate displacement through a cycle of ATP hydrolysis and attachment/detachment described below for a Myosin II motor:

The cycle can be decomposed in 5 steps, last of which will be responsible for the forced exerted on the myosin cargo.

• The myosin start in the ‘rigor’ conformation where it is lightly bound to the actin filament.
• An ATP molecule binds to the myosin head inducing the detachment of the myosin from the actin filament.
• ATP molecule is hydrolysed into ADP+Pi, providing energy which is stored into a conformational change of the myosin which effects a recovery stroke.
• Inorganic phosphate is released as the myosin head attaches to the actin filament.
• The actin-bound myosin change conformation, applying forces on it’s cargo. This step is known as the power-stroke and is responsible for most of the applied forces or displacements of the myosin. During the power-stroke the ADP bound to the myosin head is released, leading back to first step of the cycle.

This principle is the same for all kinds of myosins. In the case of Myosin II the duty-ratio is only of about 5%, which leave Myosin II detached from the actin filament most of the time. A single dimer cannot achieve processivity. The tail of myosin II can bundle itself with the tail of other myosin II motors. They from large bipolar thick filaments of hundreds of dimers. As each myosin dimer attaches and detaches independently from the actin network the effective attachment of of the filament increases with the number of motors in the minifilaments. Indeed the probability of having at least one motor attached increases with the number of motors. The constant attachment of at least one myosin II head in minifilaments insure that the filament does not displace with respect to the actin network when others myosin heads recover from their power stroke and reattach, thus conferring processivity to myosin II minifilaments.

The bipolar nature of myosin II minifilaments also allow them to act as force dipoles, each of the extremity pulling the surrounding actin network or filament towards the center of the minifilaments. This is the mechanism at the origin of muscle contraction and can allow to build-up tension in actin network.

### The actin cortex¶

The actin cortex is a thin layer of between 200 to 500 nm that can be found just underneath the plasma membrane of a cell (Fig 419) . The properties of the actin cortex makes it a key component to diverse processes. Its capacity to resit to, and transmit forces is indispensable for locomotion of many cells by allowing the retraction of the rear of the migrating cell and will be describe in more detail in the next section. Its structure is also essential for the cellular division as contractility is necessary to generate cortical tension and achieve the separation of the two daughter cells.

The actin cortex is constituted of actin filaments that can be parallel or orthogonal to the membrane as one can see using electron microscopy on cells [Morone et al. 06].

We saw through the bud scar of budding yeast that the full cytoskeleton could retain memory of past events. It is also the case for simple actin networks as show in [Parekh et al. 05] who describe how actin-network growth can be determined by network history, showing actin cortex could also act as a memory for cell.

### Cell Motility¶

The way cells move highly depends on their environment and the cell type. We can distinguish several strategies of movement, mainly categorised into amoeboid and mesenchymal movement. The type of motility for certain cells can be characteristic for malignant tissue, and plays a significant role in the ability of the cells to invade nearby tissues.

#### Lamellipodium based Motility¶

We can ave a first look into the mesenchymal mode of locomotion of cells, which is also often referred to as crawling. To understand how a cell is able to crawl, to move itself, we will in particular take the example of the lamellipodium. The lamellipodium is a characteristic structure found in cells moving on a 2D substrate. By its nature, motion using lamellipodia is one of the easiest to study using microscopy which might explain why it is one of the best know process of cell displacement. None the less, it does not diminish its importance in tissues behavior as all epithelial cell can be considered as moving on a 2D substrate. Beyond lamellipodia, further structures that are responsible for cell motion are filopodia and pseudopodia. They mainly differ from lamellipodia by their shape and the organisation of the actin structure inside (Fig 420). Lamellipodia-based motion can move a cell up to a few micrometers per minute.

The action necessary to move in an mesenchymal way can be decomposed into three steps. First the cell needs to grow a protrusion. Growing this protrusion is typically governed by actin polymerisation just underneath the plasma membrane. The lamellipodium is such a protrusion which is constituted by a 2D dendritic actin network that polymerize at the leading edge. Second the cell’s protrusion need to attach to the surface. This is done through trans membrane proteins that are bound to the actin cortex on the inside of the cell. The actin cortex will act as a scaffold to transmit the force across the cellular to these anchor points. The last part is the generation of traction in which the rest of the cell is pulled toward the attached protrusion. The traction force is mediated through the cytoskeleton and actin cortex while the contraction force themselves can origin from actin network contraction and reorganisation due to myosin motors (Fig 421).

#### Blebbing based Motility¶

The second mode of motility which is known as amoeboid is more characteristic of 3D displacement of cells. In this mode, the cell will also form protrusions but will not rely on adhesion to move its body. This motility rely on blebs, that are blister-like protrusion that appear on the cell surface. A bleb forms on the surface of cell when the membrane detach from the actin cytoskeleton underneath it, or when the cortex ruptures (Fig 422). The small protrusions are formed, quickly grow as they lack the force supporting layer that the actin cortex provides. While growing, the bleb fills with cytosol. The actin cortex can rapidly reform on the bleb slowing down its growth. In some cases, the reformation of the actin cortex in the bleb and the rebuilding of the tension inside the bleb by myosins mediated contraction is enough to reverse the bleb. Though, the content of the cell can also drain itself into the bleb as it grows and while the main body of the cell contract and empties, thus moving the cell from its old position to a new one in the direction of the initial growth of the bleb.

At their initial state, blebs are simple membrane protrusions filled with cytosol and empty of organelles. The stop of their growth is due to the spontaneous formation of an actin cortex on the inner side of the bare membrane.

By their relative simplicity to the rest of the cells, blebs are the perfect system to be reconstituted in vitro in liposomes.

### Organelle Positioning¶

We have seen previously that organelle positioning plays an important role in cell function. Several mechanisms involving actin are at the origin of structure positioning in cells. The positioning of organelles by actin can have a wide impact from being necessary for the correct cell division, to allowing locust eyes to adapt in the dark by repositioning mitocondrion [Sturmer et al. 95].

We already know that the actin cortex is a necessary element in cell motility. It also plays a determinant role in organelle positioning. It has been shown [Chaigne et al. 13] that the correct range of elasticity of the actin cortex during oocyte division is needed for proper spindle positioning. The correct spatial position of this spindle is necessary to perform a viable division of the cell.

The actin cortex is not the only actin structure in the cell, beyond the thin and dense layer just below the membrane lies a softer and sparser actin structure that has a crucial role in organelle positioning.

During cell division, there are several stages that require actin structures. As shown previously [Azoury et al. 11] the expulsion of polar body during oocyte asymmetric division is strongly dependent on the time evolution of a sparse actin network that can be found in the cell. Actin structures are also required at a later stage to permit the correct capture of chromosomes by microtubules and achieve correct haploid division. [Schuh et al. 08] also shows that a similar sparse actin network contracted by myosins is necessary for spindle migration.

Especially in oocyte that are typically large, the effect of gravity is not negligible. The presence of a sparse “actin scaffold” is discussed in [Feric et al. 13], where it is found that an actin network is present to balance the gravitational force.

In drosophila, nurses cell need to expel their content into oocytes. It has been observed [Huelsmann et al. 13] that during this phase, the nurse cells’ nucleus is pushed away from the dumping canal by single actin filaments polymerising from the membrane and forming a soft and sparse actin network.

## In vitro reconstituted actin networks¶

Living cells are complex organisms, for which each function requires a number of interacting proteins and components. To understand the action of each individual component, it is necessary to isolate or modify their actions independently.

In order to achieve the precise tuning of each component independently, two approaches are possible. First, an approach referred to as “Top-Down”, where starting from the full system — in our case the cell — we will modify or remove single or multiple components and study the global change of behavior. This is a complex process that might be difficult to interpret, as biological systems often have multiple pathways and feedback loops to regulate each of their processes. Taking into account the large number of components that constitute a living cell, it is also difficult to come up with the minimal system required to replicate certain behaviors.

The other approach, also referred to as the bottom-up approach, requires the reconstitution of the system part by part until it replicates the expected behavior. This is also a complex process, as there is a large number of potential components likely to be added to the reconstituted system. This vast complexity often leads to a wide range of testable parameters. These controlled systems allow in principle for a deeper understanding of the governing working mechanisms, and often permit access to a wider range of accessible conditions and individual tweaking of components.

In our lab, we are mainly interested in the bottom-up approach and the use of biomimetic systems. We try to reconstitute biologically relevant behavior within minimal systems, constituted from pure protein components.

I n this manuscript in particular, we focus on mimicking the motility process by which the listeria pathogen is able to hijack cellular mechanisms, by recruiting proteins responsible for actin polymerisation at the leading edge of the cell, and use them to polymerize actin on the pathogen surface. This is what allows ‘listeria’ to propel itself fast enough (1.5 to 2 µm /min) [Dabiri et al. 90] to be able to penetrate the cell membrane to move from one cell to the other.

The bead motility system is a minimal in-vitro system capable of replicating the listeria motility.

The Listeria pathogen is a 1.5 to 5 micrometer cylindrical bacteria that enters cells, hijacks its actin polymerisation machinery to propel itself and infect neighbour cells. It does so by the recruitment of a single protein on its surface : ActA, that activates the Arp2/3 complex. By the recruitment of Arp2/3, a dense branched and entangled actin network grows that will eventually form a comet behind the bacteria propelling it at the speed of actin comet polymerisation. Listeria comets are composed of a wide range of proteins. It has been shown [Loisel et al. 99] that the number of required components can be highly reduced, still maintaining the motility features.

A simpler system replicating the listeria motility is the bead motility assay. Consisting of a micrometer-sized bead covered with a nucleation promoting factor (NPF), it will activate the Arp2/3, present in the solution.

In the case of listeria, this NPF can be ActA, but such other NPF such as N-WASp or pVCA can also be used. We chose pVCA in the experiments presented in this work. The NPF covered bead is mixed with a G-Actin solution. Capping Protein is added to prevent polymerisation from happening away from the bead surface as well as the components required for actin polymerisation (ATP, Salt..., see Material and methods)

Due to the presence of Capping Protein in the solution and NPF on the surface of the bead, the polymerisation of actin will only happen on the bead surface, forming a thin and dense actin gel capable of sustaining stress, depending on the different protein concentrations. Unlike listeria, which seems to control on which of its sides the nucleation process happens, this is not controlled in bead motility assays. In the right condition, though, [Kawska et al. 12] the dense actin gel formed on the bead surface can accumulate the stress induced by the inner layer polymerisation, until symmetry breaking occurs. The gel ruptures on one side of the bead, leading to the formation of a comet on the opposite side (Figure 423).

Due to the further polymerisation of the actin network on the surface of the bead, the comet will grow, propelling the bead forward. This is the reason why the bead system is a biomimetic system replicating the listeria motion.

It should be noted that during the movement of this system, two phases can be distinguished. In the first phase, the system presents a spherical symmetry with an homogeneous actin network around the bead. The gel is growing from the surface and is accumulating stress, due to the polymerisation of inner layers.

If the gel has accumulated sufficient stress by polymerisation, the symmetry breaking event happens, and the system enters into a second phase with the formation of a comet.

The conditions that lead to symmetry breaking have been investigated in detail [Kawska et al. 12]. In the absence of Capping Protein, the actin polymerisation does not seem to be restricted enough near the surface of the bead, and thus the formed network is not able to generate or sustain enough stress to achieve symmetry breaking. At high Capping Protein concentration, the growth of the gel is heavily impaired, thus preventing symmetry breaking. The concentration of Arp2/3 is also critical, as Arp2/3 forms branched networks, and these branched networks are primordial essential for the ability to sustain stress.

In the rest of this chapter, we use the bead motility system, but only consider it during the first phase, where the symmetry breaking has not yet occurred, or in condition where it should not occur. In particular, we will investigate a condition at 25 nM Arp2/3 with a concentration of Capping Protein varying from 0 to 50 nM. As shown in fig 424 this range corresponds to conditions where no symmetry breaking occurs, but also to conditions in which symmetry breaking is expected. It should be noted that unlike other studies that also characterize actin network growing on beads [Pujol et al. 12], our system is still dynamically polymerising and thus changing with time.

### Liposomes¶

Beads are used as a model biomimetic system that replicates the polymerisation mechanism happening on the leading edge of cells. Because of their composition and rigidity, the phenomenon observed on beads cannot necessarily reproduce all the interactions and processes that take place on the cell membrane. Cells are finite compartments with a limited amount of actin that acts on the dynamics of polymerisation. The fact that cell size is in the order of the persistence length of actin filaments also plays a role on the structure of actin networks. Indeed, at these scales, a single filament can never reach the length at which it can be considered fully flexible.

Liposomes are one of the biomimetic systems able to capture some interactions between cell membranes. Liposomes are lipid bilayers that imprison an aqueous compartment and exhibit many characteristics similar to cells. The inside of liposomes can act as a biochemical reactor of limited size, with the lipid bilayer acting as a separation from the outside, like the cell membrane. The composition of the lipid layer can be varied in order to reflect the composition of a cell membrane. In particular, it is possible to attach proteins to the liposome membrane. Finally, the size of the liposomes can be varied, leading to actin networks, the size and shape of which are similar to those found in cells.

It is possible to mimic the cellular actin cortex using liposomes, and especially its contractility. A crosslinked actin network can be formed and attached to the outer leaflet of liposomes, and contractility can be triggered by injecting molecular motors. The behavior of the system will depend on the attachment between the reconstituted actin cortex and the liposome membrane. A weak attachment leads to a favorable rupture of the actin cortex during the increase of tension, implying a symmetry breaking, as in the bead motility system. In the case of strong attachment, the liposome actin-cortex will accumulate tension until it has enough force to crush the supporting lipid layer, thus collapsing the liposome [Carvalho et al. 13b],(Figure 425). This system also allows the observation over time, giving extra insight into the dynamics of the actin network (Figure 426).

## Membrane Physics¶

The cell’s plasma membrane is a biological membrane that separates the cell from its outside environment. It consists of a lipid bilayer containing a high number of proteins. A lipid bilayer is formed by two layers of lipids and has a thickness of a few nm. The classical theoretical description of these bilayers has been done by W. Helfrich [Helfrich 73] in 1973 in a model based on the elasticity and fluidity of lipid bilayers as well as the self assembly properties of lipids.

In the case of a close lipid bilayer, the potential energy stored by the deformation of a lipid bilayer by unit area can be written as

(3)$H = H_{ext} + H_{curv}$

In which $$H_{ext}$$ is due to the extension/compression of the membrane, and $$H_{curv}$$ is due to the local curvature of the membrane.

The density of energy cost to extend the membrane $$H_{ext}$$ can be written as a function of the elastic area compressibility modulus $$K_a$$ and the relative variation surface of the membrane $$A$$ :

(4)$H_{ext} = \frac 1 2 K_a \left(\frac{\Delta A}{A}\right)^2$

$$K_a$$ expresses how much energy is required to expand the surface of the lipid bilayer and is due to the exposition of more hydrophobic surface to water when expanding it. $$K_a$$ is expressed in $$J.m^{-2}$$, or $$N/m$$ and is close to twice the surface tension between the lipids and water.

For closed lipid bilayers, the total curvature energy can be expressed as the sum of the curvature energy $$H_{curv}$$ :

(5)$H_{cur} = \frac 1 2 \kappa (c_1 + c_2 -c_0)^2$

In which $$kappa$$ is the bending modulus of the membrane and $$c_1,c_2$$ are the principal curvatures of the membrane. $$c_0$$ is the spontaneous curvature of the membrane, which is defined as the curvature the membrane would adopt, when free of external constraints.

An important parameter introduced in membrane mechanics is the membrane tension, $$\sigma$$ which is the stress associated with an increase in the membrane surface. The tension $$\sigma$$ is linked to the energy required to expand the membrane $$H_{ext}$$ by :

(6)$\begin{split}\sigma &= \frac {\partial H} {\partial \left(\frac{\Delta A}{A}\right)} \\\end{split}$

i.e.

(7)$\begin{split}H_{ext} &= \sigma\left( \frac {\Delta A} A \right)\end{split}$

In which

(8)$\sigma = K_a \left( \frac {\Delta A} A \right)$

Membrane tension is a key parameter as it can be measured in cells. It is one of the parameters responsible for cell sorting [Maitre et al. 12]. In particular between cells, the tension of the couple (membrane+actin cortex) can be determined by using the contact angle between cells which is the angle between interfaces, as defined in figure 427.

In a later part, we use a reconstituted biomimetic system made of liposomes. The injection of myosin motors changes the tension of the acto-myosin cortex attached to a membrane. By determining the geometrical parameters of this system, and in particular the evolution of the contact angle with time, we are able to measure the variation of tension of the acto-myosin cortex due to the contraction by molecular motors.

## Actin networks as viscoelastic material¶

We have previously seen that while polymerising, G-actin assembles into F-actin filaments. The stiffness of filaments can be measured by a characteristic number called the persistence length ($$l_p$$). More precisely, the persistence length characterizes the average loss of correlation between the tangents along the considered polymer. With $$s$$ the curvilinear abscissae along the polymer, and $$\Theta_{(x,y)}$$ the angle between the two tangents at two different abscissae (Figure 428):

(9)$\begin{split}\left<\Theta_{(s,s+l)}\right> = exp\left(\frac{-l}{l_p}\right)\end{split}$

For actin filaments, the persistence length is in the order of 10 µm [Isambert et al. 95]. This means that for much smaller scales, the actin filament can be considered as rigid. This is the case in the cell cortex where the meshwork has a typical size, smaller than 250 nm. In the other extreme, at length scale much bigger than $$l_p$$, filaments can be considered as flexible. While in typical cells, the filament length is rarely much bigger than the persistence length of actin, Xenopus eggs can be reach 1 mm, so hundreds fold the actin persistence length. Still, for the majority of cells, the typical size we are interested in is about the persistence length of an actin filament, thus at this scale, the filament can neither be considered purely rigid nor completely flexible.

For the above reasons, actin solutions are often compared to semi-flexible polymers, and models that predict the behavior of actin networks often take foundation on polymers physics [Morse 98b] [Morse 98a]. Still, if these models rely on local microscopic parameters, experimental methods only have access to bulk properties of the studied material, and it is from these properties, and through the models, that we can deduce possible values for the microscopic models [MacKintosh et al. 95].

### Elastic Modulus¶

The elastic moduli are probably the easiest to understand. They are characteristic of how a material will deform non-permanently under an applied force. The stiffer something is, the higher its elastic moduli will be. There are two specific elastic moduli of interest in this manuscript, Young’s Modulus and shear modulus. The first one describes how a material will react to compression or extension, while the second describes how a material resists shearing. For isotropic and homogeneous materials, the Young’s modulus (E) and the shear models (G) are related by the Poisson’s ratio ($$\nu$$):

(10)$G = \frac{E}{2(1+\nu)}$

Both G and E units are homogeneous to $$N/m^2$$ or $$Pa$$. It is instructive to have an idea of the order of magnitude of a few usual materials. Aluminium will have an elastic modulus $$G_{Al}\simeq 70~GPa$$ while rubber will be more in the order of $$G_{rubber}\simeq 0.1~GPa$$. The elastic modulus of muscle cell is in the order of $$G_{muscle} \sim 10~kPa$$ and brain tissues around $$G_{brain} \sim 0.1~\text{to}~1~kPa$$ [Engler et al. 06].

A more formal definition of the Young’s modulus is the ratio between the stress $$\sigma$$ along the direction of the deformation and the relative deformation $$\epsilon$$.

(11)$\begin{split}E &= \frac{\sigma}{\epsilon} \\ & = \frac{ F/S }{ \Delta L / L_0 }\end{split}$

In which $$F$$ is the applied force, $$S$$ is the cross section of the material, $$\Delta L$$ is the elongation and $$L_0$$ is the initial length of the considered material. (Figure 429 A):

The shear modulus is defined for a deformation, parallel to the surface on which it is applied :

(12)$\begin{split}G &= \frac{\tau_{xy}}{\gamma_{xy}} \\ & = \frac{ F/S }{ \Delta x / l }\end{split}$

In which $$\tau_{xy}$$ is the shear stress, $$\gamma_{xy}$$ is the shear strain, $$F$$ is the applied force on the cross section of the material $$S$$. $$l$$ is the thickness of the material and $$\Delta x$$ is the transverse displacement (Figure 429 B).

Other characteristic numbers can also be defined, such as the bulk modulus. In the case of isotropic elastic materials, only two of those parameters are required to completely define the properties of the material.

### Poisson’s Ratio¶

We have seen that the shear modulus is linked to the Young modulus using the Poisson’s ratio. It is another characteristic of a material that defines how much a material will compress/expand in the orthogonal directions to its elongation. The Poisson’s ratio is the negative ratio of transverse to axial strain :

(13)$\nu = - \frac{ d \epsilon_{trans} }{ d \epsilon_{axial} }$

In which $$\epsilon_{axial}$$ is the relative deformation along one of the axis of compression/elongation and $$\epsilon_{trans}$$ corresponds to the relative deformation along an axis, orthogonal to the axis of deformation.

Volume conservation during compression or elongation requires a Poisson’s ratio of 0.5. Such values have been found in bulk measurements of actin networks at actin concentrations of 21.5 µM in G-actin [Gardel et al. 03]. Materials with a Poisson’s ratio of 0.5 are said to be incompressible. A Poisson’s ratio lower than 0.5 corresponds to materials expanding less than incompressible materials, and some cells and tissues are known to have a Poisson’s ratio lower than 0.5 [Mahaffy et al. 04]. Another critical value is 0, where the materials only expand or contract in the direction of the main stress.

Materials with a Poisson’s ratio superior to 0.5 would show a bigger deformation in the orthogonal direction than incompressible materials, leading to a global volume increase, if compressed.

### Viscosity¶

Like elasticity, viscosity is something tangible we are used to work with in everyday life. The more viscous a material is, the more difficult it is to move something in it, at high speed. And indeed, viscosity is the pendant of the elastic modulus, but considering forces induced by the deformation rate instead of displacement

(14)$\begin{split}\frac{F}{S} &= \tau_{xy} \\ &= \eta \frac{\partial v}{\partial z}\end{split}$

In which $$\tau_{xy}$$ is the shear stress, $$F$$ is the force exerted on the surface $$S$$. $$\eta$$ is the viscosity, and is expressed in $$Pa.s$$, $$v$$ is the deformation rate along the direction $$z$$ .

At room temperature, water has a viscosity of around 1 mPa.s, and honey of 10 Pa.s. The consideration of viscosity in problems will often depend on the timescale and deformation rate. At a short timescale, tissue often behaves elastic, whereas at a long timescale, the effect of viscosity will be seen [Thoumine et al. 97]. In actin networks, the effect of viscosity at short time scale can be similar to elasticity [Gardel et al. 03].

### Viscoelasticity¶

Typically, no material is purely elastic or purely viscous. While glaciers seem purely solid at the time scale of a few days, observation on a longer time scale, ranging from months to years, show that ice is not only a solid, but can also flow. Of course, ice in its solid form is not the only material which is both solid and viscous. In order to describe such behaviour, one can use the theory of viscoelastic materials. A number of models have been and are still developed to describe viscoelastic behavior. The Kelvin-Voigt and Maxwell models are two of the simpler ones (Figure 430). A thought experiment, conducted to understand each of these models, consists of putting a spring and a dash pot in parallel or series. Such model systems exhibit viscoelastic behavior.

The idea for more complex models is similar: any material can be considered as an (infinite) combination of springs (for elasticity), and dash-pots, (for viscosity).

The theory of viscoelastic materials explains the mechanical properties of a system by using a single parameter: the viscoelasticity of a material. This can be done by describing $$E$$ as a relaxation modulus depending on time. In the case of a linear system, we can express the strain on the material at a given time, as a function of its history :

(15)$\sigma (t) = \int_{-\infty}^t E(t-\tau) \frac{du}{d\tau} d\tau$

In which $$\sigma(t)$$ is the time dependent stress, and $$u(t)$$ is the known strain.

Through the use of rheology, it is common to measure the properties of a material using a sinusoidal strain of known amplitude $$u_0$$ and frequency $$f = \omega/ 2.\pi$$ : $$u(t) = u_0.cos(\omega t)$$, which also implies a sinusoidal strain rate. Using the complex notation $$\dot u = u_0 i\omega e^{i\omega t}$$ in equation (15), and operating the change of variable $$t-\tau \to t'$$ leads to :

(16)$\sigma(t) = u_0\int_0^\infty E(t') i\omega e^{i\omega(t-t')}dt'$

By factoring out the time dependent part, the rest can be rewritten as two integrals with respectively a real and an imaginary prefactor:

(17)$\sigma(t) = u_0e^{i\omega t}\times\left( \omega \int_0^\infty E(t') sin(\omega t) dt' + i \omega \int_0^\infty E(t') cos(\omega t) dt' \right)$

The two integrals in brackets only depend on the pulsation $$\omega$$ and the properties of the considered material. They are both in factors of the complex strain $$u(t) = u_0 e^{i\omega t}$$. We thus define the storage modulus of the material as the real part of ((17) in bracket) $$E'$$ :

(18)$E'(\omega) = \omega \int_0^\infty E(t') sin(\omega t) dt'$

And the loss modulus as the imaginary part of ((17) in bracket)

(19)$\begin{split}E"(\omega) = \omega \int_0^\infty E(t') cos(\omega t) dt'\end{split}$

And define the complex frequency dependent Young’s modulus as :

(20)$\begin{split}E^*(\omega) = E'(\omega) + i.E"(\omega)\end{split}$

Thus we can write (17) as :

(21)$\sigma(\omega) = E^*(\omega).u(\omega)$

In this representation of $$E^*(\omega)$$, the real part will correspond to the elastic response of the material (in-phase response under oscillatory strain) and the imaginary part corresponds to the viscous response of the system (out of phase under sinusoidal strain). The complete knowledge of $$E^*(\omega)$$ at all frequencies completely characterizes the material.

.

Models for actin networks have been extensively studied as viscoelastic material both theoretically [Morse 98a], [Kruse et al. 05] , and experimentally [Mizuno et al. 07]. Actin networks have also been shown to exhibit linear characteristic behavior, but at certain concentration ranges, a non-linear behavior has also been observed [Yao et al. 11], [Gardel et al. 03].

The actin networks we will study hereafter, being in the condition where a linear behavior is expected, we will thus use the viscoelastic theory to interpret the observed relation stress/strain, in order to determine the mechanical properties of the formed actin gels.

## Optical tweezer¶

Optical tweezers, or optical traps, are a technique that allows to trap objects near the focal plane of a microscope, at the focal point of a high-power laser. It is a versatile technique that allows to trap both fabricated objects and parts of living cells. Optical traps typically allow to apply forces up to a few tenth of pico Newton.

In order to understand that light can trap an object, it is instructive to keep in mind that, despite having no mass, photons carry momentum and that, as for any massive object, changing the trajectory requires a force. According to Newton’s third law, when applying a force via a photon on an object, the object will in turn exert the opposite force on the photon, thus changing its trajectory. If a photon changes its trajectory in a material, the material has to apply a force on it(Figure 431), meaning that the photon also applies a force on the material. In particular, the higher the refractive index of a material, the more light beams are deviated, and hence the more photons apply forces on material.

More specifically, it can be shown that objects with a higher refractive index than the surrounding medium, are attracted towards higher light intensities (Figure 431). In particular, laser beams with a Gaussian intensity profile, will lead to the object being attracted towards their center.

In addition to the lateral trapping, the laser focus leads to another intensity gradient along the direction of beam propagation, the intensity being at its maximum at the laser waist.

So, a laser coupled into a microscope objective acts as a three dimensional potential that traps particles, similar to a tweezer. Usually the trapping in a parallel to the laser direction, is only about half as strong if compared to the trapping in the lateral direction.

Among the qualities of optical traps, one is that in principle, multiple traps can be obtained. A simple method to generate two traps is to split the incoming light into two orthogonally polarized independent beams. Instead of sharing the laser power between the different traps by using polarisation, one can use what is known as multiplexing by time sharing. This is achieved by rapidly switching the laser between different positions at a much faster speed than the diffusion of the particles. By using this method, it is possible to virtually achieve multiple traps on the same sample.

In this work, we use a multiplexed system, where the rapid switching is achieved, by means of Accousto Optic Deflectors (aka AODs). An AOD consists of a crystal in which a high frequency sound-wave propagates perpendicular to the incoming laser beam. This sound-wave generates local changes in the refractive index of the material, which acts as a diffraction grating. In the right conditions, a laser passing through the crystal will be deflected by this grating under the Bragg angle.

In practice, rapidly controlling the frequency and amplitude of the sound-wave in the crystal, allow direct adjustment of laser deflection and hence the trap position. Not only does the use of AODs offer the advantage of controlling the number and position of multiple traps, but also the individual power allocated to each trap, hence their stiffness. .. _ots:

A schematic of the optical setup used to trap beads in the focal plane on the microscope, can be found in figure figure #ots. The scheme also contains the detection part of the setup used to measure the force exerted on each bead, technique which is explained in the following part.

### Determination of trapping forces and bead displacement¶

In addition to allowing the objects to be held in place, the use of a QDP (Quadrant Photo Diode, a precise position detector) with optical traps, has the advantage to acquire the high frequency quantitative measurements of the displacement and force exerted on an object. Indeed, when the trapped particle is not in the trap center, the laser applies a force on the object. Reciprocally, the object applies the opposite force on the light beam, thus deflecting it. With a proper use of optics and lenses correctly placed on the Fourier plane of the sample, it is hence possible to translate this orientation change of the light beam into a displacement of a light spot, onto a photo detector with high sensitivity to applied forces.

Through a careful calibration of the trap, which gives the force/displacement relationship, [Jahnel et al. 11], [Vermeulen et al. 06], one can then also recover the sample displacement inside the optical trap.

Using an optical tweezer, not only aiming at holding a particle in position, but also at getting a quantitative measurement of its displacement and the exerted force, requires to calibrated each probe particle. Polystyrene beads are common artificial probes, used to achieve such a goal.

The use of polystyrene beads has multiple advantages. First, one can obtain mono-dispersed beads, leading to reproducible and predictable trap stiffness. Secondly, theory can predict the shape of the potential felt by such a bead in a Gaussian beam [Nieminen et al. 07].

The third advantage is that beads can be fictionalized, allowing specific interaction to be controlled, both in vitro and in vivo. Of course, the calibration is essential for the correct measurement of the different systems mechanical property , and the choice of the bead diameter has an impact both on the biological side and in the measurement physics.