Steven Heilman

A hydrogel-based microfluidic device for the studies of directed cell migration

We have developed a hydrogel-based microfluidic device that is capable of generating a steady and long term linear chemical concentration gradient with no through flow in a microfluidic channel. Using this device, we successfully monitored the chemotactic responses of wildtype Escherichia coli (suspension cells) to alpha-methyl-DL-aspartate (attractant) and differentiated HL-60 cells (a human neutrophil-like cell line that is adherent) to formyl-Met-Leu-Phe (f-MLP, attractant). This device advances the current state of the art in microchemotaxis devices in that (1) it demonstrates the validity of using hydrogels as the building material for a microchemotaxis device; (2) it demonstrates the potential of the hydrogel based microfluidic device in biological experiments since most of the proteins and nutrients essential for cell survival are readily diffusible in hydrogel; (3) it is capable of applying chemical stimuli independently of mechanical stimuli; (4) it is straightforward to make, and requires very basic tools that are commonly available in biological labs. This device will also be useful in controlling the chemical and mechanical environment during the formation of tissue engineered constructs.

A three-channel microfluidic device for generating static linear gradients and its application to the quantitative analysis of bacterial chemotaxis

We have developed a prototype three-channel microfluidic chip that is capable of generating a linear concentration gradient within a microfluidic channel and is useful in the study of bacterial chemotaxis. The linear chemical gradient is established by diffusing a chemical through a porous membrane located in the side wall of the channel and can be established without through-flow in the channel where cells reside. As a result, movement of the cells in the center channel is caused solely by the cells chemotactic response and not by variations in fluid flow. The advantages of this microfluidic chemical linear gradient generator are (i) its ability to produce a static chemical gradient, (ii) its rapid implementation, and (iii) its potential for highly parallel sample processing. Using this device, wildtype Escherichia coli strain RP437 was observed to move towards an attractant (e.g., l-asparate) and away from a repellent (e.g., glycerol) while derivatives of RP437 that were incapable of motility or chemotaxis showed no bias of the bacterias distribution. Additionally, the degree of chemotaxis could be easily quantified using this assay in conjunction with fluorescence imaging techniques, allowing for estimation of the chemotactic partition coefficient (CPC) and the chemotactic migration coefficient (CMC). Finally, using this approach we demonstrate that E. coli deficient in autoinducer-2-mediated quorum sensing respond to the chemoattractant l-aspartate in a manner that is indistinguishable from wildtype cells suggesting that chemotaxis is insulated from this mode of cell-cell communication.

Orthogonal Polynomials with Respect to Self-Similar Measures

We study experimentally systems of orthogonal polynomials with respect to self-similar measures. When the support of the measure is a Cantor set, we observe some interesting properties of the polynomials, both on the Cantor set and in the gaps of the Cantor set. We introduce an effective method to visualize the graph of a function on a Cantor set. We suggest a new perspective, based on the theory of dynamical systems, for studying families Pn (x) of orthogonal functions as functions of n for fixed values of x.

Standard Simplices and Pluralities are Not the Most Noise Stable

The Standard Simplex Conjecture and the Plurality is Stablest Conjecture are two conjectures stating that certain partitions are optimal with respect to Gaussian and discretenoise stability respectively. These two conjectures are natural generalizations of the Gaussian noise stability result by Borell (1985) and the Majority is Stablest Theorem (2004). Here we show that the standard simplex is not the most stable partition in Gaussian space and that Plurality is not the most stable low inuence partition in discrete space for every number of parts k > 3, for every value ρ ≠ of the noise and for every prescribed measures for the different parts as long as they are not all equal to 1/k. Our results do not contradict the original statements of the Plurality is Stablest and Standard Simplex Conjectures concerning partitions into sets of equal measure. However, they indicate that if these conjectures are true, their veracity and their proofs will crucially rely on assuming that the sets are of equal measures, in stark contrast to Borells result, the Majority is Stablest Theorem and many other results in isoperimetric theory. In other words, the optimal partitions for noise stability are of a different nature than the ones considered for partitions into three parts in isoperimetric theory. In the latter case, the standard simplex is the partition of the plane into three sets of smallest Gaussian perimeter, where the sets are restricted to have Gaussian measures a1, a2, a3 > 0 respectively, with a1 + a2 + a3 = 1 and |ai --1/3| < .04 for all i ∈ {1, 2, 3}. Thus, we now know that the extension of noise stability theory from two to three or more parts is very much different than the extension of isoperimetric theory from two to three or more parts. Moreover, all existing proofs which optimize noise stability of two sets must fail for more than three sets, since these proofs rely on the fact that a half-space optimizes noise stability with respect to any measure restriction. Given our results it is natural to ask for (conjectured) partitions achieving the optimum noise stability. The main new ingredient in our work shows that the Ornstein-Uhlenbeck operator applied to the indicator function of a simplicial cone becomes holomorphic when restricted to certain lines. This holomorphicity condition, when combined with a first variation argument (i.e. an innite dimensional perturbative argument of the rst order), then shows that any simplicial cone can be perturbed in a volume-preserving manner to improve its noise stability. Such a holomorphicity argument seems unavailable for the is operimetric problem, since this argument uses the inherent non-locality of the Ornstein-Uhlenbeck semigroup. A full version of the paper is available at arXiv:1403.0885

Solution of the propeller conjecture in R 3

It is shown that every measurable partition {A₁,..., A_k} of R³ satisfies: ∑_i=1^k|int_{A_i} xe^-1/2|x|₂²dx|₂²≤ 9π². Let P₁,P₂,P₃ be the partition of R² into 120^o sectors centered at the origin. The bound (1) is sharp, with equality holding if A_i=P_i x R for i∈ {1,2,3} and A_i=∅ for i∈ {4,...,k}. This settles positively the 3-dimensional Propeller Conjecture of Khot and Naor (FOCS 2008). The proof of (1) reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of (1) is complexity-theoretic: the Unique Games hardness threshold of the Kernel Clustering problem with 4 x 4 centered and spherical hypothesis matrix equals 2π/3.

HOMOTOPIES OF EIGENFUNCTIONS AND THE SPECTRUM OF THE LAPLACIAN ON THE SIERPINSKI CARPET

Consider a family of bounded domains Ωt in the plane (or more generally any Euclidean space) that depend analytically on the parameter t, and consider the ordinary Neumann Laplacian Δt on each of them. Then we can organize all the eigenfunctions into continuous families

Localized Eigenfunctions: Here You See Them, There You Don't

u_{t}^{(j)}

Outer Approximation of the Spectrum of a Fractal Laplacian

with eigenvalues

Euclidean partitions optimizing noise stability

\lambda_{t}^{(j)}

Standard simplices and pluralities are not the most noise stable

also varying continuously with t, although the relative sizes of the eigenvalues will change with t at crossings where