Eric Cytrynbaum

Associate Professor

Relevant Degree Programs


Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - May 2019)
A computationally efficient method for solving min system reaction-diffusion equations within growing and dividing domains that approximate a rod-shaped bacterium (2019)

No abstract available.

Mathematics and biophysics of cortical microtubules in plants (2011)

Microtubules confined to the two-dimensional cortex of elongating plant cells must form a parallel yet dispersed array transverse to the elongation axis for proper cell wall expansion. Collisions between microtubules, which migrate via hybrid treadmilling, can result in plus-end entrainment (zippering) or catastrophe. Here, I present (1) a cell-scale computational model of cortical microtubule organization and (2) a molecular-scale model for microtubule-cortex anchoring and collision-based interactions between microtubules. The first model treats interactions phenomenologically while the second addresses interactions by considering energetic competition between crosslinker binding, microtubule bending and microtubule polymerization. From the cell-scale model, we find that plus-end entrainment leads to self-organization of microtubules into parallel arrays, while collision-induced catastrophe does not. Catastrophe-inducing boundaries can tune the dominant orientation. Changes in dynamic-instability parameters, such as in mor1-1 mutants in Arabidopsis thaliana, can impede self-organization, in agreement with experiment. Increased entrainment, as seen in clasp-1 mutants, conserves self-organization, but delays its onset. Modulating the ability of cell edges to induce catastrophe, as the CLASP protein may do, can tune the dominant direction and regulate organization. The molecular-scale model predicts a higher probability of entrainment at lower collision angles and at longer unanchored lengths of plus-ends. The models lead to several testable predictions, including the effects of reduced microtubule severing in katanin mutants and variable membrane-anchor densities in different plants, including Arabidopsis cells and Tobacco cells.

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Master's Student Supervision (2010 - 2018)
Mathematical modeling of maltose uptake system in E. coli using nanodisc fluorescence quenching data (2013)

Recent data measured in nanodiscs conflicts with the standard theory of maltose transport in the MalE-MalFGK₂ uptake system found in E. coli. Nanodisc fluorescence quenching data suggest an alternate pathway in which unliganded MalE binds the P-open transporter, facilitating maltose acquisition. Nanodisc data also indicate that MalE regulates maltose uptake at high concentrations. We analyzed four mathematical models of the maltose uptake system: the distinct standard and alternate models, and two integrated models. Nanodisc fluorescence quenching data and nonlinear regression analysis were used to fit equilibrium constants and kinetic rates. The flux through each pathway in an integrated model was calculated using asymptotic analysis and fit parameter values. We conclude that it is likely that transport occurs when liganded MalE associates to a P-open conformation of MalFGK₂, rather than binding to the P-closed transporter as suggested by the standard model. The standard pathway was calculated to be negative, i.e. to occur in reverse as a means of regulating maltose uptake at high concentration. This analysis conflicts with the standard model in which liganded MalE binds to a closed transporter and triggers an opening of the transporter proteins which in turn open the liganded MalE. The analysis also found that a relatively small amount of maltose transport may occur through the alternate pathway involving unliganded MalE.

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A study of wave propagation in the FitzHugh Nagumo system (2012)

An excitable medium has two key properties: a sufficiently large stimulus provokes an even bigger response (excitability), and immediately following a stimulus the medium cannot be excited (refractoriness). A large class of biological systems from cardiac tissue to slime mold are examples of excitable media. FitzHugh Nagumo (FHN) is the canonical model of excitable media. Its two variables are the state of excitation and refractoriness of the one- or two-dimensional medium. Although one of the simplest models, FHN exhibits complex dynamics that have not been fully explored. For example, it supports a stable traveling pulse solution. However, this pulse can be destabilized by large perturbations. In Chapter 2 I explore a one-dimensional example where the perturbation is an increasing refractory profile. This perturbation can lead to collapse of the pulse depending on the steepness of the profile, as conjectured by Keener [Keener, J. (2004) J Theo. Bio. 230(4):459-73]. In Chapter 3 I consider a perturbation in two dimensions which can cause the stable traveling pulse to wrap around the perturbation and generate self-sustaining spiral activity.The one-dimensional example for exponential refractory profiles is explored numerically for a piecewise linear FHN system. Steep profiles lead to collapse while milder profiles allow propagation. The exponential profiles are used as bounds for more general profiles to predict where collapse and propagation will occur. I also make use of a singular FHN system in the limit ε→ 0 to provide insight into the behaviours of the full FHN system for small ε and small diffusion. I conclude this chapter by showing analytically that, in contrast to the full system, a wave in the singular system will propagate for any exponential refractory profile.The two-dimensional case is explored numerically in a FHN system. The use of a temporarily refractory region as a perturbation is a novel mechanism for generating spiral activity. Moreover, it is shown to be robust for refractory regions of a large area. This situation models the appearance of abnormal electrical activity in the heart. In particular, it models the appearance of abnormal electricity activity in undamaged cardiac tissue.

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