GROWINGRODS

Mathematics and Mechanics of growth and remodelling of bio-filaments

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 Obiettivo del progetto (Objective)

'This project aims at developing a theory of growth, dynamics and mechanics of bio-filaments including branching processes. It will then be applied to a variety of physically and biologically interesting systems.

At the mathematical level, the growth of biological filaments involves interesting aspects of both kinematics and dynamics of curves. A natural starting point for the modeling of biofilaments is to consider them as thin elastic rods subjected to external constraints. The basic idea is to cross-sectionally average all stresses along the space curve representing the centreline of the rod. This leads to a set of equations, the Kirchhoff equations, relating averaged forces and moments to the curve’s strains (characterized by the curvatures, shears, and extension). These equations provide the starting point for much theoretical analysis and numerical modeling. They are valid for a large range of scales and have been used to model many different structures including DNA, vines and plants.

Early work will be the basis for establishing a general and rigorous theory of growing elastic rods that we referred to as morphoelastic rods. Various microscopic mechanisms for the generation of intrinsic macroscopic curvature, torsion and twist will be studied. Furthermore, the initiation of branching points will be described. Simultaneously, we will consider various applications in life sciences to motivate these general theories. In particular, in collaboration with various biologists, we will model the growth of neurons and plants.'

Introduzione (Teaser)

Many systems from DNA to telephone cables to the braided magnetic flux tubes of solar flares have a filamentary structure. New mathematical descriptions of mechanics and growth enhance understanding and potential applications.

Descrizione progetto (Article)

Despite the apparent differences in filamentary systems, their growth, movement and plasticity appear to follow universal physical laws. However, important gaps still exist in our understanding.

With EU support of the GROWINGRODS (Mathematics and mechanics of growth and remodelling of bio-filaments) project, scientists have developed unifying mathematical frameworks correlating mechanics with growth.

When a growing body attempts to expand against geometrical constraints of its environment, non-linear growth regimes can cause emergent mechanical properties of the whole to be much more than the sum of its parts. GROWINGRODS' general theory for filament growth of arbitrary material properties that explains this phenomenon was used to explain the diversity and evolution of sea shells.

Structures made of bundles of filaments are another type of system for which no general theory was previously able to predict behaviours. Taking a simplified system composed of two filaments interacting elastically, scientists developed a theory of mechanics and growth allowing elucidation of the effective properties of the structure from that of the subfilaments. Significantly, the framework demonstrates that the conventional laws relating averaged forces and moments to strains for single rods (Kirchoff's laws) are not appropriate for bundles. Instead, scientists developed a definition of generalised stresses and solutions of the corresponding balance equations.

Most importantly, scientists overcame a confounding barrier in descriptions of the stability of mechanical equilibria of 1D systems. Many equilibrium states can exist but, if they are not stable, they are rarely observed. Until now, theoretical descriptions could only identify a few cases of equilibrium. The team developed a formulation that now enables finding most if not all equilibrium states with ease. This opens the door to applications in engineering such that one can modify a system to stabilise particular equilibria.

Researchers successfully applied the new mathematical frameworks to a natural system, a hybrid and an engineered device. GROWINGRODS has contributed important and widely useful new mathematical formulations describing the mechanics and growth of ubiquitous filamentary structures. The resulting three publications with advanced descriptions will have far-reaching implications on understanding and engineering innovative devices in many fields.