free-7.2. OUTLOOK 113
via FEM it might by more convenient to use a discrete network of nonlin-ear springs. This was done for instance to describe polymer gels [Yashin &
Balazs, 2006].
Moreover it should be clarified if the two-dimensional results can really be mapped to a three-dimensional system. Using Dirichlet-boundary condition for the displacement is not exact. The boundary actually moves in real ex-periments. Only the bottom part of the protoplasmic droplet is fixed to the substrate, the remaining parts of the membrane or extracellular matrix are free to move.
Another important issue mentioned in Chapter 6 is the type of viscoelastic constitutive law. In many of the recent models the cytoskeleton is assumed to behave according to the Maxwell model, i.e. it behaves like a fluid on long time scales (see e.g. [Joanny et al., 2007]). The consequences to use this instead of the Kelvin-Voigt model are discussed in Chapter 6 with re-spect to the dispersion relation but numerical simulations have not been performed in this work. Another model that accounts for stress relaxation but describes a solid on long time scales is the standard linear solid model and was used in [Romanovsky & Teplov, 1995].
Using the Maxwell model, i.e. treating the cytoskeleton as a fluid, it is possible to consider amoeboid movement in the framework of the theory de-veloped in this thesis. In such a model the sol fraction is not considered as a constant but as a field that obeys a local kinetics ˙ρgel =g(ρgel, nc, ...). It should be noted, that ρsol is the density in the reference coordinate system (ρsol = ˜ρsoldet(F), ˜ρsol denotes the sol density in lab frame). The func-tiong describes sol-gel transformations and depends on the sol fraction, the Ca2+concentration but can also depend on the flow velocity [Guy et al., 2011] or an oriental order parameter. For the description of the filamentous as a viscoelastic material it may be important to account for the polar order.
A theory of active polar gels was developed and it was found that the polar orientation field leads to topological defects like asters and vortices [Kruse et al., 2005]. It is not clear if these effects are relevant for Physarum. The cytoskeleton is assumed to be isotropic in this work but further experiments, e.g using birefringence to visualize the filament orientation, may show if this simplification is realistic.
The above discussed extension of the model leads to a free boundary prob-lem. These are inconvenient numerical problems, which one can overcome by using a phase field model as an approximation. Such a model was re-cently applied to successfully model the migration of keratocyte fragments [Ziebertet al., 2012]. This also involves treadmilling of filaments that leads to the additional generation of tension at the moving front.
Some of the mechanisms discussed in this work remain qualitative, since experimental data is missing to provide feedback to the model. It would be of great benefit if the flow field and the tension in protoplasmic droplet would be recorded together. These measurement were performed
indepen-7.2. OUTLOOK 115 dently for different cellular systems (see e.g. [Matsumoto et al., 2008; Betz et al., 2011]). To acquire information about the spatiotemporal distribution of the free intracellular Ca2+, e.g. via fluorescence imaging is another pos-sibility to test if the height field and theCa2+concentration obey a relation as predicted by the presented model.
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125
List of Publications
• M. Radszuweit, H. Engel and M. B¨ar, A model for oscillations and pattern formation in protoplasmic droplets of Physarum polycephalum, Eur. Phys. J. - Special Topics, Springer Berlin/Heidelberg, 2010,191, 159-172
• M. Radszuweit, S. Alonso, H. Engel and M. B¨ar,Intracellular Mechano-chemical Waves in an Active Poroelastic Model, Phys. Rev. Lett., 2013, 110, 138102
Acknowledgements
I acknowledge the financial support from the German Science Foundation DFG within the GRK1558 “Nonequilibrium Collective Dynamics in Con-densed Matter and Biological Systems”. I would like to thank my super-visors Markus B¨ar and Harald Engel for their mentoring and for giving me the opportunity to freely develop my ideas. Furthermore, I would like to thank my colleagues at the TU-Berlin for fruitful discussions in numer-ous coffee breaks, especially David Strehober and Christian Otto for giving me some insight in bifurcation theory, Sergio Alonso from the PTB for his critical and enlightening remarks, as well as Stefan Fruhner and Arash Az-hand for helping me with Linux and scripting issues. Ulrike Strachauer and Marcus Hauser from the University of Magdeburg was so kind to send me unpublished data of experiments with protoplasmic droplets. Finally, I ac-knowledge J. R. Shewchuk for providing the scientific community with the nice and free mesh generator Triangle.
Appendix A
Finite deformation equations and the linear limit
A.0.1 Covariant formulation
In the finite theory of elasticity one has to distinguish carefully between the different version of the stress tensor living on different tangent spaces.
Let TX(B) the tangent space at point X in the lab frame and Tx( ˜B) the corresponding tangent space in the body frame. In the formulation of the force balance using the body reference coordinate system the first Piola-Kirchhoff tensor P :Tx( ˜B)× TX(B)→Ris used:
∇·P +f = 0,f ∈ TX(B) (A.1) Since the constitutive relations should be either given in the physical or body reference frame (P maps from mixed spaces and is not necessary symmetric) one introduces the second Piola-Kirchhoff tensor
S :Tx( ˜B)× Tx( ˜B)→R,S =F−1P. Then:
∇·(F S) +f = 0. (A.2)
Actually, in this continuum formulation some quantities like e.g. the stresses are tensor densities. And thus, to get the stress tensorT :TX(B)×TX(B)→ R in the physical lab frame we have T =F SFT/detF.
The sol and gel fractions are tensor densities of rank zero:
ρgel =detF̺gel
ρsol =detF̺sol, (A.3)
where ̺sol,gel are the sol/gel fractions in the physical lab frame. They are related by
̺sol+̺gel = 1
ρsol+ρgel = detF. (A.4)
127
It is not taken into account that there are sol-gel transformations, hence it is assumed that ρgel =: ρ0gel = const. Furthermore we define the constant ρ0sol := 1 −ρ0gel. that corresponds to the sol fractions of the undeformed configuration.
We are looking for a covariant formulation of the force balance equations.
LetSvegel the viscoelastic gel stress andSvissol the viscous sol stress. Then the force balance equations read
∇·
ρ0gelF(Svegel+ (Ta−p)C−1)
+ρ0sol̺gelβ(v−u) = 0˙
∇· ρ0solF(Svissol −pC−1)
−̺solρ0gelβ(v−u)˙ = 0 (A.5) The form of the total stresses is justified by the fact that it should trans-form like a tensor density of rank two: Ttot =F StotFT/detF. (Here, the subscript indicates the total stress including active and passive part.) The special form of the drag-force density ensures that it is a tensor density of rank one: f =Ff˜/detF.
In addition to the force balance equation we write the incompressibility con-dition valid for finite deformations. It is derived by the following arguments:
The outflux of fluid has to be replaced by the same volume of solid material:
(Bis the volume element in the lab frame and ˜Bt=χ−1t (B)) I
∂B
̺solvdA= Z
B
˙
̺geldX (A.6)
Transformation to body reference frame:
I
∂B˜t
̺solvdetF F−Tda= Z
B˜t
˙
̺geldetFdv
ρgel =const.⇒0 = dtd(̺geldetF) = ˙̺geldetF +̺geldtddetF
⇒̺˙geldetF =−tr(F−1∇u)detF˙ ̺gel ⇒
and Gauss’s theorem then finally gives the incompressibility condition
∇· ρsolvF−T
+tr(F−1∇u)ρ˙ gel = 0. (A.7) Now, it is shown that to first order the sol fraction obeys a continuity equa-tion similar to the free Ca2+concentration when diffusion and reaction are omitted. Usingρsol=detF −ρ0gel and computing the time derivative gives
∂tρsol =∂t(detF) =tr(F−1∇u)detF˙
= (̺sol+̺gel)detFtr(F−1∇u)˙
= (ρsol+ρgel)tr(F−1∇u)˙ With the incompressibility constraint (A.7) this yields
∂tρsol =−∇· ρsolvF−T
+ρsoltr(∇uF˙ −T)
=−∇· ρsol(v−u)F˙ −T
+tr ∇(ρsolF−1) ˙u
129 The second term is of higher order and gives an equation analogue to (3.28) for the free Ca2+:
∂tρsol+∇· ρsol(v−u)F˙ −T
+O(∇uu) = 0.˙ (A.8)