Steady accretion of an elastic body on a hard spherical surface and the notion of a four-dimensional reference space

Abstract

Taking the cue from experiments on actin growth on spherical beads, we formulate and solve a model problem describing the accretion of an incompressible elastic solid on a rigid sphere due to attachment of diffusing free particles. One of the peculiar characteristics of this problem is that accretion takes place on the interior surface that separates the body from its support rather than on its exterior surface, and hence is responsible for stress accumulation. Simultaneously, ablation takes place at the outer surface where material is removed from the body. As the body grows, mechanical effects associated with the build-up of stress and strain energy slow down accretion and promote ablation. Eventually, the system reaches a point where internal accretion is balanced by external ablation. The present study is concerned with this stationary regime called "treadmilling".The principal ingredients of our model are: a nonstandard choice of the reference configuration, which allows us to cope with the continually evolving material structure; and a driving force and a kinetic law for accretion/ablation that involves the difference in chemical potential, strain energy and the radial stress. By combining these ingredients we arrive at an algebraic system which governs the stationary treadmilling state. We establish the conditions under which this system has a solution and we show that this solution is unique. Moreover, by an asymptotic analysis we show that for small beads the thickness of the solid is proportional to the radius of the support and is strongly affected by the stiffness of the solid, whereas for large beads the stiffness of the solid is essentially irrelevant, the thickness being proportional to a characteristic length that depends on the parameters that govern diffusion and accretion kinetics.

Nonlinear elasticity of incompatible surface growth

“…Consider now accretion on a rigid spherical surface, mimicking the process of inwards actin polymerization stimulated biochemically on a specially treated spherical bead [18,48,74], see Fig.7b. A similar process with cylindrical symmetry would be the growth of a tree where new mass is deposited between the existing trunk and the bark [1,41].…”

“…In deviation from the previous example, we assume that the pressure control is not direct but is rather an outcome of the control of the Eulerian velocity of the arriving materialχ(ψ t ) =ṁ (¯ α). Essentially this means the control of a volumetric inflow rate which in the context actin polymerization appear to be more realistic than the full control of the attachment stress [18,48]. On the exterior surface of the growing body we assume the no-tractions condition, s r (ψ 0 , t) = 0.…”

Kinetics of surface growth with coupled diffusion and the emergence of a universal growth path

“…The absence of a global natural reference configuration limits the application of usual continuum representations of the solid body, and probably explains the focus on kinematics in available studies of surface growth (Skalak et al, 1982(Skalak et al, , 1997Menzel and Kuhl, 2012;Moulton et al, 2012). Recent studies (Tomassetti et al, 2016;Sozio and Yavari, 2017;Ganghoffer and Goda, 2018) have explored new ways to examine surface growth. Specifically, Tomassetti et al (2016) identified that for the particular surface growth scenario considered therein, the grown body possesses a natural global reference configuration if relaxed into a fourdimensional space.…”

Treadmilling stability of a one-dimensional actin growth model

“…Further refinements of the model would be relevant for applications, especially experimental ones: accounting for the density and stiffness increase of actin under higher external load and developing analytical relationships between growth velocity and applied force are two of them. Finally, extension of the present work to the stability of two dimensional treadmilling states previously studied in Tomassetti et al (2016) A second example of a suitable strain energy density is…”

“…Conversely, growth speed could be obtained as a result of mechanical and biochemical local conditions. These in turn may be expressed by a suitable kinetic law once the thermodynamical force driving growth has been consistently defined (Abeyaratne and Knowles, 1990;Tomassetti et al, 2016).…”