measured along the central line and the width of the gels is averaged from a set of equidistant lines. Droplet radii are determined from the projected area, whereby samples with deformed non-spherical shapes are discarded. Asym-metric, partly attached gel volumes are estimations. Merged data points in the phase diagram represent one to five samples.
2.7 Simulation
Numerical simulations and analytical computation is performed by Mathematica (Wolfram), using the routines Solve and NDSolve. In brief, the network com-position and its temporal development is derived from the kinetic equations of protein interactions by implementation of the corresponding system of partial dif-ferential equations. The main assumption of the model is, that all activity takes place in an active region at the membrane, whereby the back part of the network remains passive. If not stated otherwise, the standard parameter set given by Table 2.2 is used.
Parameter Definition Value Source
k1 Actin-VCA binding 43/µM/s (i)
k2 Actin-VCA unbinding 30 s−1 (i)
k3 Arp2/3-(Actin-VCA) binding 0.8/µM/s (i) k4 Arp2/3-(Actin-VCA) unbinding 0.6 s−1 (i) kon Polymerisation rate (plus end) 11.6/µM/s (ii) koff Depolymerisation rate (plus end) 1.4 s−1 (ii)
kCP Capping rate 12/µM/s (iii)
kArp Actin-Arp2/3 interaction rate 0.03µm2/s (iv)
G Actin concentration 3µM (v)
C Capping Protein concentration 0.04µM (v)
A Arp2/3 concentration 0.3µM (v)
V Activator (VCA) density 1200/µm2 (v)
Table 2.2: Binding constants and initial values. Parameter values are either taken from (i) Marchand et al. (2001), (ii) Pollard et al. (2000), (iii) Shekhar et al. (2015), or (iv) set to allow steady network growth in a reasonable concentration regime. Initial values (v) are describing typical conditions from currentin vitro work.
Chapter 3
Extensile microtubule-kinesin gels
A remarkable, yet often neglected constraint on spatial organisation is set by topology. It is easy to see, that lines cannot be drawn in parallel on a sphere without creating points where the resulting pattern is ill defined. A common ex-ample are the lines of longitude and latitude on a globe, which form aster and ring patterns that diverge at the poles. These necessary singular points are known as topological defects (Mermin, 1979). Naturally all assemblies of matter have to obey this mathematical law. A closer look reveals this concept behind various phenomena like the ridge patterns of fingerprints (Penrose, 1965), the organisation of elongated cells (Elsdale & Wasoff, 1976; Gruler et al., 1999) or the alignment of liquid crystals on microscopic length scales (Chandrasekhar, 1992). They all can be described as an ensemble of rod-like particles that spon-taneously align along a preferred orientation that is locally defined by the dir-ector line field. By this, a so called nematic phase is formed, where there is order in the directionality of the particles, however their positions are unordered (Fig. 3.1a)(Gramsbergen et al., 1986). The physics underlying the formation of the nematic phase are of entropic nature and originate from excluded volume effects, e.g. described by Onsager theory (Onsager, 1949; Frenkel, 2015).
Mathematically, the nematic phase is quantified by an orientational order para-meterSwhich is the average of the second Legendre polynomial:
S =hP2(cos(θ))i=
3 cos2(θ)−1 2
(3.1) where θ is the angle between the particle axis and the director. Therewith a perfect alignment is characterised byS=1 and a total disorder is found at S=0.
Furthermore, a topological charge is assigned to the nematic defects, which denotes the rotation of the director field when following a closed path encircling the defect (Vitelli & Nelson, 2006; Lopez-Leon et al., 2011). Thus a charge s rotates the director field by2πs. The basic nematic defects have charges of +½ or –½, corresponding to aπ rotation of the director field (Fig. 3.1b). According to the Poincaré-Hopf theorem, the charges on a spherical surface add up to +2 (Fig. 3.1c).
In material science, the nematic ordering has become a promising building principle to drive self-organisation. The topological constraint forces particles to assemble into highly complex and tunable spatial arrangements, which
en-b
θ d isotropic
nematic
a c
+1 +1
+½ -½
charge sum = +2
Figure 3.1: Schematics of nematic order and defects. (a) Isotropic and nematic phase have in common that the positions of rod-like particles are not ordered. However in the nematic phase the axis are aligned on average parallel to a directord.(b) Discon-tinuities in the director field are called defects and described by a topological charge, where +1 means a360° rotation of the director field. Only +1 and +½ charges are ob-served in the active nematic vesicles. (c)Schematic of a nematic phase confined in a spherical topology. A continuous non-vanishing director field is not possible, expressed by the Poincaré-Hopf-theorem: the sum of the charges has to be +2.
able intriguing higher-order hierarchical materials (Poulin, 1997; Musevic et al., 2006). Previous work in this field has focused on equilibrium materials confined on rigid surfaces of varying topology (Bausch et al., 2003; Moreno-Razo et al., 2012). As their assembly process relies on equilibration, these systems only show dynamics during their formation while the final state is quasi static (Irvine et al., 2010; Lipowsky et al., 2005). Subsequent recent studies indeed have created non-equilibrium active nematic liquid crystals, that are propelled by the continuous conversion of chemical to mechanical energy by the rod-like building blocks (Narayan et al., 2007). Therefore they make use of biological matter, util-ising rod-like swimming bacteria (Mushenheim et al., 2013; Zhou et al., 2014), elongated crawling cells (Duclos et al., 2014; Kemkemer et al., 2000), or driving motion by cytoskeletal motors (Sanchez et al., 2012). The resulting dynamic systems expose out-of-equilibrium phenomena, such as chaotic flows with con-tinuous defect pair generation and annihilation, that are also subject to current theoretical research (Thampi et al., 2013; Gao et al., 2015).
In this chapter, a novel system that merges active nematics with topological constraints is investigated. An extensile microtubule-kinesin gel is confined onto the spherical surface of a lipid vesicle. Consequently the nematic microtubule layer has to expose defects with a total charge of +2. The steady energy input by the kinesin motors drives defect motion and thereby prohibits relaxation of the nematic to an equilibrium configuration. In contrast an oscillatory steady state is created where the defects move on fixed trajectories passing extremal configurations. Furthermore the flexible membrane allows the nematic to ex-pand, causing dynamic shape changes of the vesicles as well as the formation of protrusions. When an additional geometrical confinement is set by the ves-icle diameter, various morphologies appear, as different defect configurations become energetically favourable. Finally a spatial constraint by a surrounding network is shown to be able to stall defect motion.