Eric Cytrynbaum

Associate Professor

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Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - May 2021)
A homotopy-minimization method for parameter estimation in differential equations and its application in unraveling the reaction mechanism of the Min system (2019)

A mathematical model of a dynamical process, often in the form of a system of differential equations, serves to elucidate underlying dynamical structure and behavior of the process that may otherwise remain opaque. However, model parameters are often unknown and may need to be estimated from data for a model to be informative. Numerical-integration-based methods, which estimate parameters in a differential equation model by fitting numerical solutions to data, can demand extensive computation, especially for large stiff systems that require implicit methods for stability. Non-numerical integration methods, which estimate parameters in a differential equation model by fitting solution approximations to data, do not provide an impartial measure of how well a model fits data, a measure required for the testability of a model. In this dissertation, I propose a new method that steps back from a numerical-integration-based method, and instead allows an optimal data-fitting numerical solution to emerge as part of an optimization process. This method bypasses the need for implicit solution methods, which can be computationally intensive, seems to be more robust than numerical-integration-based methods, and, interestingly, admits conservation principles and integral representations, which allow me to gauge the accuracy of my optimization.The Escherichia coli Min system is one of the simplest known biological systems that demonstrates diverse complex dynamic behavior or transduces local interactions into a global signal. Various mathematical models of the Min system show behaviors that are qualitatively similar to dynamic behaviors of the Min system that have been observed in experiments, but no model has been quantitatively compared to time-course data. In this dissertation, I extract time-course data for model fitting from experimental measurements of the Min system and fit established and novel biochemistry-based models to the time-course data using my parameter estimation method for differential equations. Comparing models to time-course data allows me to make precise distinctions between biochemical assumptions in the various models. My modeling and fitting supports a novel model, which suggests that a regular, ordered, stability-switching mechanism underlies the emergent, dynamic behavior of the Min system.

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Deterministic and stochastic modeling of the Min system for cell division (2019)

The Min system acts as a key regulator for cell division in E. coli, repressing cell division at either end of the cell via pole to pole oscillation. Recent in vitro experiments have demonstrated the Min system's tendency to create ``burst'' patterning under suitable concentration conditions, whereby high concentration `bursts' of Min proteins nucleate from an approximately homogeneous background, before ``freezing'' and fading away. I start this thesis by giving a quick review of some of the complexities involved in modeling chemical reactions via Partial Differential Equations - particularly in 2D surfaces such as the cell membrane. I consider a number of toy models, demonstrating discrepancies between classical Reaction-Diffusion representations of chemical systems, and the more foundational particle system. These discrepancies are in most cases minor, in some cases extreme. A simplified Min model is developed, demonstrating how particle models of Min dynamics can lead to burst formation, even in cases where differential equations predict a uniform solution.Next, I take a recently developed and parameterized ODE model of the Min system based on experimental data from Ivanov et al, and extend the model to consider finite space, both on the membrane and in the volume of the cell. This extended model allows me to map out a bifurcation diagram of the system's behavior for concentrations both higher and lower than those used in the original data fitting, and explore the conditions under which burst nucleation is predicted.Finally, I show that white noise can allow a spatially distributed reaction diffusion system to escape from a neutrally stable steady state at zero, passing to some fixed value u(0,T)>0 in finite time. The most probable path to such a state leads to a narrow sharp spike reminiscent of experimental observations. Dynamics of this kind are typical whenever a system loses stability by passing slowly through a saddle node bifurcation. Supplementary materials available at:

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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 - 2020)
Modeling milk production in the lactation period and the effect of feeding frequency on milk production (2019)

Human milk production is controlled by a variety of internal and external factors, including hormones, neurons, suckling stimulus and milk removal. One method for increasing milk production suggested to mothers who want to produce more milk is cluster feeding: splitting one feeding into a cluster of multiple feedings. Phenomenological models have been proposed to describe average weekly milk production rates in dairy cattle, but these models do not take into account the effect of the milk removal schedule used. In this thesis, two ordinary differential equation models for describing milk production and milk removal are presented: Model 1, which assumes a linear rate of milk removal during feeding, and Model 2, which uses physical models of fluid flow through a system of ducts to estimate the rate of milk removal. These models are qualitatively very similar, but Model 2 allows for examination of differences in milk dynamics between alveoli at different depths in the mammary gland. 24 hour milk production was then predicted for each model version using various cluster feeding schedules. Feeding schedules which had the most frequent and regular feedings elicited small increases in milk production during a day compared to less frequent or regular schedules. This small increase in milk production suggests that cluster feeding may not be an effective method of increasing production, as other factors that decrease production, like stress, may override this small effect. More work needs to be done to test these models in order to determine an effective method for increasing milk production.

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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 computationally efficient method for solving min system reaction-diffusion equations within growing and dividing domains that approximate a rod-shaped bacterium (2012)

Few biological systems, showing complex pattern formation that spans multiple spatial and temporal scales, have been reduced in understanding to several components. The Min system in Escherichia coli, consists of three proteins, MinD, MinE, and MinC. Through ordered, cyclic, membrane binding and unbinding, facilitated by ATP hydrolysis, the Min system regulates the site of cell division in vivo. The Min system is tightly coupled with cell growth and division. Various mathematical models have been proposed to describe specific biological phenomena, arising from the Min system, but no model has been tested in a biologically relevant space, which grows and divides. In this thesis, I develop a computationally efficient method for solving reaction-diffusion equations, in a growing and dividing geometry, which approximates growth and division in a rod-shaped bacterium.

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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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