Final Report Summary - FLAGELLA (Fluid mechanics of flagellar propulsion)
Objectives
The initial objective of this project was to study the flagellar locomotion of microorganisms in fluids, a theme of major importance in biology. Indeed, this swimming mode affects many processes such as mammalian reproduction, the marine life ecosystem, and the dynamics of bacterial infection. For these microorganisms, locomotion is typically achieved by the periodic deformation of flagella (short and flexible organelles) that drive the fluid motion around the microorganisms, and generate propulsive forces. The shape of the flagella is, in turn, affected by the fluid dynamic forces generated by the organisms. The understanding of this complex fluid-structure interaction calls for a multidisciplinary approach, at the intersection of physics, mechanics, biology and applied mathematics.
During the course of this research and training project the initial objectives have been widened to two other examples of fluid-structure interactions in biological systems: the fluid mechanics of undulatory swimming as performed by most of the known aquatic animals and the constraints on tree architectures imposed by wind-induced loads.
Main results
Flagellar propulsion
A theoretical model has been developed to take into account the internal mechanics into the energy budget of flagellar and ciliary kinematics. This model assumes that eukaryotic flagella and cilia cannot harvest energy from the surrounding fluid, similarly to muscles. This novel assumption allows us to calculate optimal shapes and kinematics without resorting to artificial constraints.
Tree architecture
In his notebooks, Leonardo da Vinci observed that 'all the branches of a tree at every stage of its height when put together are equal in thickness to the trunk', which means that the cross-sectional area of branches is conserved across branching nodes. The usual explanation for this rule involves vascular transport of sap, but this argument is questionable because the portion of wood devoted to transport varies across species and can be as low as 5 %. It has been proposed in this project that Leonardo's rule is a consequence of the tree skeleton having a self-similar structure and the branch diameters being adjusted to resist wind-induced loads. To address this problem, a continuous model has been first considered by neglecting the geometrical details of branching and wind incident angles. The robustness of this analytical model has then been assessed with numerical simulations on tree skeletons generated with a simple branching rule producing self-similar structures.
Undulatory swimming
Most aquatic vertebrates swim by passing a bending wave down their bodies, a swimming mode known as undulatory propulsion. Except for very elongated swimmers like eels and lampreys, these animals have evolved to a similar shape. However, the link between this particular shape and the hydrodynamical constraints remains to be explored. In this project, this question has been addressed by seeking the optimal design for undulatory swimmers with an evolutionary algorithm. Animals of varying elliptic cross-section have been considered whose motions are prescribed by arbitrary periodic curvature laws. A bi-objective optimisation then consists in finding body shapes and corresponding motions associated to the highest swimming velocities, the lowest energetic costs, or any trade-offs between the two. For biologically relevant parameters, this optimisation calculation yields two distinct 'species': one specialised in economical swimming and the other in fast swimming. By comparing, the attributes and performances of these numerically-obtained swimmers with data on undulatory-swimming animals, it is argued that evolution likely favoured low energetic costs.
The initial objective of this project was to study the flagellar locomotion of microorganisms in fluids, a theme of major importance in biology. Indeed, this swimming mode affects many processes such as mammalian reproduction, the marine life ecosystem, and the dynamics of bacterial infection. For these microorganisms, locomotion is typically achieved by the periodic deformation of flagella (short and flexible organelles) that drive the fluid motion around the microorganisms, and generate propulsive forces. The shape of the flagella is, in turn, affected by the fluid dynamic forces generated by the organisms. The understanding of this complex fluid-structure interaction calls for a multidisciplinary approach, at the intersection of physics, mechanics, biology and applied mathematics.
During the course of this research and training project the initial objectives have been widened to two other examples of fluid-structure interactions in biological systems: the fluid mechanics of undulatory swimming as performed by most of the known aquatic animals and the constraints on tree architectures imposed by wind-induced loads.
Main results
Flagellar propulsion
A theoretical model has been developed to take into account the internal mechanics into the energy budget of flagellar and ciliary kinematics. This model assumes that eukaryotic flagella and cilia cannot harvest energy from the surrounding fluid, similarly to muscles. This novel assumption allows us to calculate optimal shapes and kinematics without resorting to artificial constraints.
Tree architecture
In his notebooks, Leonardo da Vinci observed that 'all the branches of a tree at every stage of its height when put together are equal in thickness to the trunk', which means that the cross-sectional area of branches is conserved across branching nodes. The usual explanation for this rule involves vascular transport of sap, but this argument is questionable because the portion of wood devoted to transport varies across species and can be as low as 5 %. It has been proposed in this project that Leonardo's rule is a consequence of the tree skeleton having a self-similar structure and the branch diameters being adjusted to resist wind-induced loads. To address this problem, a continuous model has been first considered by neglecting the geometrical details of branching and wind incident angles. The robustness of this analytical model has then been assessed with numerical simulations on tree skeletons generated with a simple branching rule producing self-similar structures.
Undulatory swimming
Most aquatic vertebrates swim by passing a bending wave down their bodies, a swimming mode known as undulatory propulsion. Except for very elongated swimmers like eels and lampreys, these animals have evolved to a similar shape. However, the link between this particular shape and the hydrodynamical constraints remains to be explored. In this project, this question has been addressed by seeking the optimal design for undulatory swimmers with an evolutionary algorithm. Animals of varying elliptic cross-section have been considered whose motions are prescribed by arbitrary periodic curvature laws. A bi-objective optimisation then consists in finding body shapes and corresponding motions associated to the highest swimming velocities, the lowest energetic costs, or any trade-offs between the two. For biologically relevant parameters, this optimisation calculation yields two distinct 'species': one specialised in economical swimming and the other in fast swimming. By comparing, the attributes and performances of these numerically-obtained swimmers with data on undulatory-swimming animals, it is argued that evolution likely favoured low energetic costs.