Michael Tabor - Academia.edu (original) (raw)

Papers by Michael Tabor

ABSTRACT The Kirchhoff equations for elastic tubes are modified to include the effect of fluid fl... more ABSTRACT The Kirchhoff equations for elastic tubes are modified to include the effect of fluid flow. Using the techniques of linear and nonlinear analysis specially developed for the Kirchhoff equations, the effect of the fluid flow on the basic twist-to-writhe instability is investigated. The results suggest an intriguing modification of the bifurcation threshold due to the flow. Beyond threshold the buckled tube acquires a slight curvature which modifies the flow rate and results in a correction to nonlinearity of the amplitude equation governing the deformation dynamics.

Applied Optimization, 2008

Growth is involved in many fundamental biological processes such as morphogenesis, physiological ... more Growth is involved in many fundamental biological processes such as morphogenesis, physiological regulation, or pathological disorders. It is, in general, a process of enormous complexity involving genetic, biochemical, and physical components at many different scales and with complex interactions. The purpose of this paper is to provide a simple introduction to the modeling of elastic growth. We first consider systems in one-dimensions (suitable to model filamentary structures)to introduce the key concepts. Second, we review the general three-dimensional theory and show how to apply it to the growth of cylindrical structures. Different possible growth mechanisms are considered.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1997

... The usual approach to modelling these structures is to assume that they can be represented as... more ... The usual approach to modelling these structures is to assume that they can be represented as an elastic filament subject to the classical laws of mechanics and elasticity theory. ... We first givea brief summary of the kinematics (ie the relationship between positions and ...

Physical Review Letters, 1998

The helix hand reversal exhibited by the tendrils of climbing plants when attached to a support i... more The helix hand reversal exhibited by the tendrils of climbing plants when attached to a support is investigated. Modeled as a thin elastic rod with intrinsic curvature, a linear and nonlinear stability analysis shows the problem to be a paradigm for curvature induced morphogenesis in which symmetry breaking is constrained by a global invariant.

Physical Review Letters, 2011

The filamentary fungus Phycomyces blakesleeanus undergoes a series of remarkable transitions duri... more The filamentary fungus Phycomyces blakesleeanus undergoes a series of remarkable transitions during aerial growth. During what is known as the stage IV growth phase, the fungus extends while rotating in a counterclockwise manner when viewed from above (stage IVa) and then, while continuing to grow, spontaneously reverses to a clockwise rotation (stage IVb). This phase lasts for 24-48 h and is sometimes followed by yet another reversal (stage IVc) before the overall growth ends. Here, we propose a continuum mechanical model of this entire process using nonlinear, anisotropic, elasticity and show how helical anisotropy associated with the cell wall structure can induce spontaneous rotation and, under appropriate circumstances, the observed reversal of rotational handedness.

Physical Review Letters, 2003

The growth of a family of filamentary microorganisms is described in terms of self-similar growth... more The growth of a family of filamentary microorganisms is described in terms of self-similar growth at the tip which is driven by pressure and sustained by a wall-building growth process. The cell wall is modeled biomechanically as a stretchable elastic membrane using large-deformation elasticity theory. Incorporation of geometry dependent elastic moduli and a self-similar ansatz shows how these equations can generate realistic tip shapes corresponding to a self-similar expansion process.

Physica D: Nonlinear Phenomena, 1995

In this study, a new perturbative scheme for nonintegrable ordinary differential equations is pro... more In this study, a new perturbative scheme for nonintegrable ordinary differential equations is proposed. These perturbative expansions are based on the singularity analysis of the unperturbed system and is performed in the neighborhood of its singularities. Under suitable conditions on the homoclinic structure of the unperturbed system, the Melnikov vector can be computed based on the knowledge of the Laurent expansions of the solutions. The existence of transverse homoclinic intersections is therefore explicitly related to the existence of critical points for the solutions in the complex plane of the independent variable.

Physica D: Nonlinear Phenomena, 1997

The Kirchhoff model provides a well-established mathematical framework to study, both computation... more The Kirchhoff model provides a well-established mathematical framework to study, both computationaly and theoretically, the dynamics of thin filaments within the approximations of linear elasticity theory. The study of static solutions to these equations has a long history and the usual approach to describing their instabilities is to study the time-dependent version of the Kirchhoff model in the Euler angle frame. Here we study the linear stability of the full, time-independent, equations by introducing a new arc length preserving perturbation scheme. As an application, we consider the instabilities of various stationary solutions, such as the planar ring and straight rod, subjected to twisting perturbations. This scheme gives a direct proof of the existence of dynamical instabilities and provides the selection mechanism for the shape of unstable filaments.

Physica D: Nonlinear Phenomena, 1997

The linear stability analysis of the full time-dependent Kirchhoff equations for elastic filament... more The linear stability analysis of the full time-dependent Kirchhoff equations for elastic filaments gives precise information about possible dynamical instabilities. The associated dispersion relations derived in the preceding paper provides the selection mechanism for the shapes selected by highly unstable filaments. Here we perform a nonlinear analysis and derive new amplitude equations which describe the dynamics above the instability threshold. The straight filament is studied in detail and the motion is shown to be described by a pair of nonlinear Klein-Gordon equations which couple the local deformation amplitude to the twist density. Of particular interest is the effect of boundary conditions on the instability threshold. It is shown that with suitable choice of boundary conditions the threshold of instability is delayed. We also show the existence of pulse-like and front-like traveling wave solutions.

Nonlinear Dynamics, 2006

Nonlinear amplitude equations for the near-threshold behavior of twisted extensible elastic rods ... more Nonlinear amplitude equations for the near-threshold behavior of twisted extensible elastic rods under tension with inertial and dissipative dynamics are derived. In the inertial case localized solutions to the amplitude equations are derived and a linear stability criterion for the pulse solutions is obtained using the Hamiltonian formulation of the problem.

Mathematics and Mechanics of Solids, 2010

One of the possible effects of volumetric growth in elastic materials is the creation of residual... more One of the possible effects of volumetric growth in elastic materials is the creation of residual stresses. These stresses are known to change many of the classical properties of the material and have been studied extensively in the context of volumetric growth in biomechanics. Here we consider the problem of elastic cavitation in a growing compressible elastic membrane. Growth is taken to be homogeneous but anisotropic, and the membrane is assumed to remain axisymmetric during growth and deformation. We prove that neo-Hookean membranes cannot cavitate, but for Varga elastic materials we find conditions under which the material exhibits spontaneous cavitation in the absence of external loads, in marked distinction from the cavitation problem without growth.

Journal of Theoretical Biology, 2003

The tip growth of filamentary actinomycetes is investigated within the framework of large deforma... more The tip growth of filamentary actinomycetes is investigated within the framework of large deformation membrane theory in which the cell wall is represented as a growing elastic membrane with geometry-dependent elastic properties. The model exhibits realistic hyphal shapes and indicates a self-similar tip growth mechanism consistent with that observed in experiments. It also demonstrates a simple mechanism for hyphal swelling and beading that is observed in the presence of a lysing agent.

Journal of Theoretical Biology, 2006

The fungus Magnaporthe grisea, commonly referred to as the rice blast fungus, is responsible for ... more The fungus Magnaporthe grisea, commonly referred to as the rice blast fungus, is responsible for destroying from 10% to 30% of the world's rice crop each year. The fungus attaches to the rice leaf and forms a dome-shaped structure, the appressorium, in which enormous pressures are generated that are used to blast a penetration peg through the rice cell walls and infect the plant. We develop a model of the appressorial design in terms of a bioelastic shell that can explain the shape of the appressorium, and its ability to maintain that shape under the enormous increases in turgor pressure that can occur during the penetration phase.

Journal of Mathematical Biology, 2005

The growth of filamentary micro-organisms is described in terms of the geometry of evolving plana... more The growth of filamentary micro-organisms is described in terms of the geometry of evolving planar curves in which the dynamics is determined by an underlying growth process. Steadily propagating tip shapes in two and three dimensions are found that are consistent with experimentally observed growth sequences.

Mycological Research, 2006

The mechanical actions of the fungus Magnaporthe grisea raise many intriguing questions concernin... more The mechanical actions of the fungus Magnaporthe grisea raise many intriguing questions concerning the forces involved. These include: (1) the material properties of the appressorial wall; (2) the strength of the adhesive that keeps the appressorium anchored to the rice leaf surface; and (3) the forces involved in the penetration process whereby a peg is driven through the host cell wall. In this paper we give order of magnitude estimates for all three of these quantities. A simple Young-Laplace law type argument is used to show that the appressorial wall elastic modulus is of order 10-100 MPa; and an adaptation of standard adhesion theory indicates a lower bound on the strength of the appressorial adhesive to be of the order 500 J/m(2). Drawing on ideas from plasticity theory and ballistics, estimates of the penetration force raise interesting questions about experiments performed on the penetration of inert substrates by the fungus.

ABSTRACT The Kirchhoff equations for elastic tubes are modified to include the effect of fluid fl... more ABSTRACT The Kirchhoff equations for elastic tubes are modified to include the effect of fluid flow. Using the techniques of linear and nonlinear analysis specially developed for the Kirchhoff equations, the effect of the fluid flow on the basic twist-to-writhe instability is investigated. The results suggest an intriguing modification of the bifurcation threshold due to the flow. Beyond threshold the buckled tube acquires a slight curvature which modifies the flow rate and results in a correction to nonlinearity of the amplitude equation governing the deformation dynamics.

Applied Optimization, 2008

Growth is involved in many fundamental biological processes such as morphogenesis, physiological ... more Growth is involved in many fundamental biological processes such as morphogenesis, physiological regulation, or pathological disorders. It is, in general, a process of enormous complexity involving genetic, biochemical, and physical components at many different scales and with complex interactions. The purpose of this paper is to provide a simple introduction to the modeling of elastic growth. We first consider systems in one-dimensions (suitable to model filamentary structures)to introduce the key concepts. Second, we review the general three-dimensional theory and show how to apply it to the growth of cylindrical structures. Different possible growth mechanisms are considered.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1997

... The usual approach to modelling these structures is to assume that they can be represented as... more ... The usual approach to modelling these structures is to assume that they can be represented as an elastic filament subject to the classical laws of mechanics and elasticity theory. ... We first givea brief summary of the kinematics (ie the relationship between positions and ...

Physical Review Letters, 1998

The helix hand reversal exhibited by the tendrils of climbing plants when attached to a support i... more The helix hand reversal exhibited by the tendrils of climbing plants when attached to a support is investigated. Modeled as a thin elastic rod with intrinsic curvature, a linear and nonlinear stability analysis shows the problem to be a paradigm for curvature induced morphogenesis in which symmetry breaking is constrained by a global invariant.

Physical Review Letters, 2011

The filamentary fungus Phycomyces blakesleeanus undergoes a series of remarkable transitions duri... more The filamentary fungus Phycomyces blakesleeanus undergoes a series of remarkable transitions during aerial growth. During what is known as the stage IV growth phase, the fungus extends while rotating in a counterclockwise manner when viewed from above (stage IVa) and then, while continuing to grow, spontaneously reverses to a clockwise rotation (stage IVb). This phase lasts for 24-48 h and is sometimes followed by yet another reversal (stage IVc) before the overall growth ends. Here, we propose a continuum mechanical model of this entire process using nonlinear, anisotropic, elasticity and show how helical anisotropy associated with the cell wall structure can induce spontaneous rotation and, under appropriate circumstances, the observed reversal of rotational handedness.

Physical Review Letters, 2003

The growth of a family of filamentary microorganisms is described in terms of self-similar growth... more The growth of a family of filamentary microorganisms is described in terms of self-similar growth at the tip which is driven by pressure and sustained by a wall-building growth process. The cell wall is modeled biomechanically as a stretchable elastic membrane using large-deformation elasticity theory. Incorporation of geometry dependent elastic moduli and a self-similar ansatz shows how these equations can generate realistic tip shapes corresponding to a self-similar expansion process.

Physica D: Nonlinear Phenomena, 1995

In this study, a new perturbative scheme for nonintegrable ordinary differential equations is pro... more In this study, a new perturbative scheme for nonintegrable ordinary differential equations is proposed. These perturbative expansions are based on the singularity analysis of the unperturbed system and is performed in the neighborhood of its singularities. Under suitable conditions on the homoclinic structure of the unperturbed system, the Melnikov vector can be computed based on the knowledge of the Laurent expansions of the solutions. The existence of transverse homoclinic intersections is therefore explicitly related to the existence of critical points for the solutions in the complex plane of the independent variable.

Physica D: Nonlinear Phenomena, 1997

The Kirchhoff model provides a well-established mathematical framework to study, both computation... more The Kirchhoff model provides a well-established mathematical framework to study, both computationaly and theoretically, the dynamics of thin filaments within the approximations of linear elasticity theory. The study of static solutions to these equations has a long history and the usual approach to describing their instabilities is to study the time-dependent version of the Kirchhoff model in the Euler angle frame. Here we study the linear stability of the full, time-independent, equations by introducing a new arc length preserving perturbation scheme. As an application, we consider the instabilities of various stationary solutions, such as the planar ring and straight rod, subjected to twisting perturbations. This scheme gives a direct proof of the existence of dynamical instabilities and provides the selection mechanism for the shape of unstable filaments.

Physica D: Nonlinear Phenomena, 1997

The linear stability analysis of the full time-dependent Kirchhoff equations for elastic filament... more The linear stability analysis of the full time-dependent Kirchhoff equations for elastic filaments gives precise information about possible dynamical instabilities. The associated dispersion relations derived in the preceding paper provides the selection mechanism for the shapes selected by highly unstable filaments. Here we perform a nonlinear analysis and derive new amplitude equations which describe the dynamics above the instability threshold. The straight filament is studied in detail and the motion is shown to be described by a pair of nonlinear Klein-Gordon equations which couple the local deformation amplitude to the twist density. Of particular interest is the effect of boundary conditions on the instability threshold. It is shown that with suitable choice of boundary conditions the threshold of instability is delayed. We also show the existence of pulse-like and front-like traveling wave solutions.

Nonlinear Dynamics, 2006

Nonlinear amplitude equations for the near-threshold behavior of twisted extensible elastic rods ... more Nonlinear amplitude equations for the near-threshold behavior of twisted extensible elastic rods under tension with inertial and dissipative dynamics are derived. In the inertial case localized solutions to the amplitude equations are derived and a linear stability criterion for the pulse solutions is obtained using the Hamiltonian formulation of the problem.

Mathematics and Mechanics of Solids, 2010

One of the possible effects of volumetric growth in elastic materials is the creation of residual... more One of the possible effects of volumetric growth in elastic materials is the creation of residual stresses. These stresses are known to change many of the classical properties of the material and have been studied extensively in the context of volumetric growth in biomechanics. Here we consider the problem of elastic cavitation in a growing compressible elastic membrane. Growth is taken to be homogeneous but anisotropic, and the membrane is assumed to remain axisymmetric during growth and deformation. We prove that neo-Hookean membranes cannot cavitate, but for Varga elastic materials we find conditions under which the material exhibits spontaneous cavitation in the absence of external loads, in marked distinction from the cavitation problem without growth.

Journal of Theoretical Biology, 2003

The tip growth of filamentary actinomycetes is investigated within the framework of large deforma... more The tip growth of filamentary actinomycetes is investigated within the framework of large deformation membrane theory in which the cell wall is represented as a growing elastic membrane with geometry-dependent elastic properties. The model exhibits realistic hyphal shapes and indicates a self-similar tip growth mechanism consistent with that observed in experiments. It also demonstrates a simple mechanism for hyphal swelling and beading that is observed in the presence of a lysing agent.

Journal of Theoretical Biology, 2006

The fungus Magnaporthe grisea, commonly referred to as the rice blast fungus, is responsible for ... more The fungus Magnaporthe grisea, commonly referred to as the rice blast fungus, is responsible for destroying from 10% to 30% of the world's rice crop each year. The fungus attaches to the rice leaf and forms a dome-shaped structure, the appressorium, in which enormous pressures are generated that are used to blast a penetration peg through the rice cell walls and infect the plant. We develop a model of the appressorial design in terms of a bioelastic shell that can explain the shape of the appressorium, and its ability to maintain that shape under the enormous increases in turgor pressure that can occur during the penetration phase.

Journal of Mathematical Biology, 2005

The growth of filamentary micro-organisms is described in terms of the geometry of evolving plana... more The growth of filamentary micro-organisms is described in terms of the geometry of evolving planar curves in which the dynamics is determined by an underlying growth process. Steadily propagating tip shapes in two and three dimensions are found that are consistent with experimentally observed growth sequences.

Mycological Research, 2006

The mechanical actions of the fungus Magnaporthe grisea raise many intriguing questions concernin... more The mechanical actions of the fungus Magnaporthe grisea raise many intriguing questions concerning the forces involved. These include: (1) the material properties of the appressorial wall; (2) the strength of the adhesive that keeps the appressorium anchored to the rice leaf surface; and (3) the forces involved in the penetration process whereby a peg is driven through the host cell wall. In this paper we give order of magnitude estimates for all three of these quantities. A simple Young-Laplace law type argument is used to show that the appressorial wall elastic modulus is of order 10-100 MPa; and an adaptation of standard adhesion theory indicates a lower bound on the strength of the appressorial adhesive to be of the order 500 J/m(2). Drawing on ideas from plasticity theory and ballistics, estimates of the penetration force raise interesting questions about experiments performed on the penetration of inert substrates by the fungus.