Recent and Ongoing Projects

Modeling plant cell walls

In early 2014, I began working with Mariya Ptashnyk on the mathematical modeling of plant cell walls. The main idea of the project is to model cell growth by explicitly taking into account the chemistry that takes place within the cell wall. An overview of the chemistry of a cell wall and how it relates to cell growth is provided by Cosgrove. A cell wall consists of a matrix of polysaccharides that is reinforced with long, inelastic cellulose microfibrils, which allows the wall to support the high turgor pressure from within the cell. The stiffness of this matrix is influenced my many factors, but Mariya and I will focus on the density of pectin-calcium cross links. When the number of cross-links decreases and the matrix softens, the turgor pressure causes the cell wall to deform, leading to growth. To prevent the wall from rupturing, new microfibrils are created and new pectin-calcium cross links are formed. The goal is to model the cell wall as a three-dimensional continuum, using homogenization techniques to account for the microstructure. Performing the homogenization in this context poses a significant challenge since the governing PDEs are coupled and nonlinear. Our work will yield a better understanding of the role of pectin in the growth of plant cells and, more generally, of plant tissue biomechanics.

Stable and unstable helices: Soap films in cylindrical tubes

Recently, Cox and Jones studied an interesting variant of the classical Plateau problem involving soap films. The geometry of their problem is as follows. (See the picture on the Home page.) Consider a cylindrical tube C of length L with circular cross-section of radius R that has line segments passing through it at each of its ends. Suppose that the line segments lie in a plane that contains the axis of C and are parallel. Cox and Jones were interested in determining the shape of the soap film whose boundary consists of the two line segments and two curves that lie on C. Since soap films want to minimize their area, determining the shape of this soap film is analogous to finding the minimal surface with the prescribed boundary conditions. It was found experimentally and numerically by Cox and Jones that when L/R is less than some critical value, the soap film takes the shape of a flat surface. However, surprisingly, when L/R is larger than the critical value, the flat soap film becomes unstable.

Together with Eliot Fried, I was able to confirm the results of Cox and Jones by analyzing an area minimization problem with the boundary conditions determined by the geometry described above [ArXiv]. Besides considering the case in which the two line segments are parallel, we investigated the case when they are not parallel. For this configuration, one can analyze the stability of certain helices. We found that (i) for sufficiently long, thin cylinders, no helicoid is stable, (ii) regardless of the dimensions of the cylinder, no helicoid that contains a little over half of a rotation is stable, and (iii) as long as the cylinder is not overly long and thin, there is a non flat stable helicoid.

Microphysical derivation of the free-energy density for a lipid bilayer

Eliot Fried and I were able to derive the Canham–Helfrich free-energy for a lipid bilayer using a microphysical method [link] [ArXiv]. We started by modeling the lipid molecules that make up the membrane as one-dimensional rigid-rods, like those Eliot and I considered when we worked on the statistical foundations of liquid-crystal theory. By assuming that (i) the lipid molecules only interact when they are within a given distance d, (ii) the lipid bilayer can be modeled as a surface, (iii) the lipid molecules remain perpendicular to the surface, and (iv) the interaction of the lipid bilayer with the ambient solution is negligible, we obtained a free-energy density for the bilayer. Following an idea of Keller and Merchant, we expanded this expression for the free-energy density in powers of a small dimensionless quantity involving d and found that the coefficient in front of the quadratic term is the Canham–Helfrich energy. Our derivation yields expressions for the bending moduli and the spontaneous curvature in terms of the interaction energy between the phospholipid molecules. It also allows us to observe that when the two leaflets that makeup the bilayer are identical, the spontaneous curvature vanishes.

Since the procedure we used gave explicit integral representations of the bending moduli in terms of the interaction potential, once a particular potential is chosen, it is possible to obtain values for the moduli [ArXiv]. Upon chosing a Gaussian potential inspired by the works of Berne and Pechuckas and Gay and Berne, Eliot and I computed the bending moduli, which were found to depend on two parameters associated with the potential. Adjusting these parameters yields values for the moduli matching most values reported in the literature. A result of note is our computations yielded a negative value for the modulus associated with the Gaussian curvature.

Transport theorem for irregular domains

During my time as a postdoc working with Eliot Fried, we proved a transport theorem for evolving irregular domains. A transport theorem refers to a formula for the rate of change of an integral in which both the domain of integration and the integrand may depend upon time. If the time variable represents a variational parameter, than a transport theorem is equivalent to calculating a first variation. Of course, results of this type are nothing new. The result I obtained with Eliot differs from all previous transport theorems in that ours holds for evolving domains that can, among other things, develop holes, split into pieces, or whose fractal dimension may evolve with time. This result was proven using Harrison's theory of differential chains. A differential chain generalizes the concept of a domain of integration and can represent classical domains—that is, manifolds—as well as fractal domains. Thus, a time-dependent differential chain generalizes the concept of an evolving domain. By considering a certain set of time-dependent differential chains, Eliot and I were able to prove a generalize transport theorem that holds for evolving domains with the kind of irregularities mentioned above and, importantly, contains a term associated with the evolution of the boundary of the domain. Having such a boundary term is important when the transport relation is used in either the calculus of variations or continuum physics. Domains that evolve with the kind of irregularities mentioned above occur in the study of phase transitions, fracture mechanics, diffusion, and heat conduction. Our result could be useful in dealing with problems in the calculus of variations in which the domain of integration is to be extremized.

Together with Denis Hinz, Eliot and I wrote a paper that presented the ideas and results related to our work on the generalized transport theorem in a way that makes them accessible to researchers working in the field of continuum physics. This was achieved in part by considering three concrete, illustrative examples in which the terms appearing in the generalized transport theorem were calculated numerically. Eliot and I's desire to establish the transport theorem was motivated by the existence of evolving irregular domains that occur in nature, but the proof is a work in pure mathematics. Thus, many researchers who could potentially benefit from our work might not become exposed to it and, so, presenting our results specifically to this audience could prove useful.

Publications



Ptashnyk, M., Seguin, B.: Multiscale analysis of a system of elastic-viscolelastic and reaction-diffusion equations modelling plant cell wall biomechanics. arXiv:1410.6911 [math.AP]

Seguin, B., Fried, E.: Stable and unstable helices: Soap films in cylindrical tubes. accepted at Calculus of Variations and Partial Differential Equations [ArXiv]

Seguin, B., Fried, E.: Calculating the bending moduli of the Canham–Helfrich free-energy density from a particular potential. ICMS Workshop: Differential Geometry and Continuum Mechanics. Proceedings in Mathematics and Statistics. Springer (forthcoming) [ArXiv]

Seguin, B., Hinz, D. F., Fried, E.: Extending the Transport Theorem to Rough Domains of Integration. Applied Mechanics Reviews 66, 050802 (2014) [link]

Seguin, B., Fried, E.: Roughening it — Evolving irregular domains and transport theorems. Mathematical Models and Methods in Applied Sciences 24, 1729 (2014) [link]

Seguin, B., Fried, E.: Microphysical derivation of the Canham–Helfrich free-energy density. Journal of Mathematical Biology 68, 647–665 (2014) [link] [ArXiv]

Maleki, M., Seguin, B., Fried, E.: Kinematics, material symmetry, and energy densities for lipid bilayers with spontaneous curvature. Biomechanics and Modeling in Mechanobiology 12, 997–1017 (2013) [link]

Seguin, B., Fried, E.: Statistical foundations of liquid-crystal Theory II. Continuum-level balances. Archive for Rational Mechanics and Analysis 207, 1–37 (2013) [link]

Seguin, B., Fried, E.: Statistical foundations of liquid-crystal Theory I. Discrete systems of rod-like molecules. Archive for Rational Mechanics and Analysis 206, 1039–1072 (2012) [link]

Capriz, G., Fried, E., Seguin, B.: Constrained ephemeral continua. Rendiconti Lincei–Matematica e Applicazioni 23, 157–195 (2012) [link]

Seguin, B.: Simple thermomechanical materials with memory. Journal of Elasticity 105, 207–252 (2011) [link]

Seguin, B.: Thermoelasto-viscous materials. Journal of Elasticity 101, 153–177 (2010) [link]

Noll, W., Seguin, B.: Basic concepts in thermomechanics. Journal of Elasticity 101, 121–151 (2010) [link]

Noll, W., Seguin, B.: Plugs in viscometric flows of simple semi-liquids. Journal of the Society of Rheology Japan 37, 1–10 (2009) [link]

Noll, W., Seguin, B.: Monoids, boolean algebras, and materially ordered sets. International Journal of Pure and Applied Mathematics 37, 187–202 (2007) [link]