Single-cell analysis of the physiology of mechanosensation in bacteria

Escherichia coli is one of the best studied living organisms and a model system for many biophysical investigations. Despite countless discoveries of the details of its physiology, we still lack a holistic understanding of how these bacteria react to changes in their environment. One of the most important examples is their response to osmotic shock. One of the mechanistic elements protecting cell integrity upon exposure to sudden changes of osmolarity is the presence of mechanosensitive channels in the cell membrane. These channels are believed to act as tension release valves protecting the inner membrane from rupturing. This thesis presents an experimental study of various aspects of mechanosensation in bacteria. We examine cell survival after osmotic shock and how the number of MscL (Mechanosensitive channel of Large conductance) channels expressed in a cell influences its physiology. We developed an assay that allows real-time monitoring of the rate of the osmotic challenge and direct observation of cell morphology during and after the exposure to osmolarity change. The work described in this thesis introduces tools that can be used to quantitatively determine at the single-cell level the number of expressed proteins (in this case MscL channels) as a function of, e.g., growth conditions. The improvement in our quantitative description of mechanosensation in bacteria allows us to address many, so far unsolved, problems, like the minimal number of channels needed for survival, and can begin to paint a clearer picture of why there are so many distinct types of mechanosensitive channels.

On the Cesari fixed point method in a Banach space

In a paper published in 1961, L. Cesari [1] introduces a method which extends certain earlier existence theorems of Cesari and Hale ([2] to [6]) for perturbation problems to strictly nonlinear problems. Various authors ([1], [7] to [15]) have now applied this method to nonlinear ordinary and partial differential equations. The basic idea of the method is to use the contraction principle to reduce an infinite-dimensional fixed point problem to a finite-dimensional problem which may be attacked using the methods of fixed point indexes.

The following is my formulation of the Cesari fixed point method:

Let B be a Banach space and let S be a finite-dimensional linear subspace of B. Let P be a projection of B onto S and suppose Г≤B such that pГ is compact and such that for every x in PГ, P^-1x∩Г is closed. Let W be a continuous mapping from Г into B. The Cesari method gives sufficient conditions for the existence of a fixed point of W in Г.

Let I denote the identity mapping in B. Clearly y = Wy for some y in Г if and only if both of the following conditions hold:

(i) Py = PWy.

(ii) y = (P + (I - P)W)y.

Definition. The Cesari fixed paint method applies to (Г, W, P) if and only if the following three conditions are satisfied:

(1) For each x in PГ, P + (I - P)W is a contraction from P^-1x∩Г into itself. Let y(x) be that element (uniqueness follows from the contraction principle) of P^-1x∩Г which satisfies the equation y(x) = Py(x) + (I-P)Wy(x).

(2) The function y just defined is continuous from PГ into B.

(3) There are no fixed points of PWy on the boundary of PГ, so that the (finite- dimensional) fixed point index i(PWy, int PГ) is defined.

Definition. If the Cesari fixed point method applies to (Г, W, P) then define i(Г, W, P) to be the index i(PWy, int PГ).

The three theorems of this thesis can now be easily stated.

Theorem 1 (Cesari). If i(Г, W, P) is defined and i(Г, W, P) ≠0, then there is a fixed point of W in Г.

Theorem 2. Let the Cesari fixed point method apply to both (Г, W, P₁) and (Г, W, P₂). Assume that P₂P₁=P₁P₂=P₁ and assume that either of the following two conditions holds:

(1) For every b in B and every z in the range of P₂, we have that ‖b=P₂b‖ ≤ ‖b-z‖

(2)P₂Г is convex.

Then i(Г, W, P₁) = i(Г, W, P₂).

Theorem 3. If Ω is a bounded open set and W is a compact operator defined on Ω so that the (infinite-dimensional) Leray-Schauder index i_LS(W, Ω) is defined, and if the Cesari fixed point method applies to (Ω, W, P), then i(Ω, W, P) = i_LS(W, Ω).

Theorems 2 and 3 are proved using mainly a homotopy theorem and a reduction theorem for the finite-dimensional and the Leray-Schauder indexes. These and other properties of indexes will be listed before the theorem in which they are used.

Endoscopic optical coherence tomography : design and application

This thesis presents an investigation on endoscopic optical coherence tomography (OCT). As a noninvasive imaging modality, OCT emerges as an increasingly important diagnostic tool for many clinical applications. Despite of many of its merits, such as high resolution and depth resolvability, a major limitation is the relatively shallow penetration depth in tissue (about 2∼3 mm). This is mainly due to tissue scattering and absorption. To overcome this limitation, people have been developing many different endoscopic OCT systems. By utilizing a minimally invasive endoscope, the OCT probing beam can be brought to the close vicinity of the tissue of interest and bypass the scattering of intervening tissues so that it can collect the reflected light signal from desired depth and provide a clear image representing the physiological structure of the region, which can not be disclosed by traditional OCT. In this thesis, three endoscope designs have been studied. While they rely on vastly different principles, they all converge to solve this long-standing problem.

A hand-held endoscope with manual scanning is first explored. When a user is holding a hand- held endoscope to examine samples, the movement of the device provides a natural scanning. We proposed and implemented an optical tracking system to estimate and record the trajectory of the device. By registering the OCT axial scan with the spatial information obtained from the tracking system, one can use this system to simply ‘paint’ a desired volume and get any arbitrary scanning pattern by manually waving the endoscope over the region of interest. The accuracy of the tracking system was measured to be about 10 microns, which is comparable to the lateral resolution of most OCT system. Targeted phantom sample and biological samples were manually scanned and the reconstructed images verified the method.

Next, we investigated a mechanical way to steer the beam in an OCT endoscope, which is termed as Paired-angle-rotation scanning (PARS). This concept was proposed by my colleague and we further developed this technology by enhancing the longevity of the device, reducing the diameter of the probe, and shrinking down the form factor of the hand-piece. Several families of probes have been designed and fabricated with various optical performances. They have been applied to different applications, including the collector channel examination for glaucoma stent implantation, and vitreous remnant detection during live animal vitrectomy.

Lastly a novel non-moving scanning method has been devised. This approach is based on the EO effect of a KTN crystal. With Ohmic contact of the electrodes, the KTN crystal can exhibit a special mode of EO effect, termed as space-charge-controlled electro-optic effect, where the carrier electron will be injected into the material via the Ohmic contact. By applying a high voltage across the material, a linear phase profile can be built under this mode, which in turn deflects the light beam passing through. We constructed a relay telescope to adapt the KTN deflector into a bench top OCT scanning system. One of major technical challenges for this system is the strong chromatic dispersion of KTN crystal within the wavelength band of OCT system. We investigated its impact on the acquired OCT images and proposed a new approach to estimate and compensate the actual dispersion. Comparing with traditional methods, the new method is more computational efficient and accurate. Some biological samples were scanned by this KTN based system. The acquired images justified the feasibility of the usage of this system into a endoscopy setting. My research above all aims to provide solutions to implement an OCT endoscope. As technology evolves from manual, to mechanical, and to electrical approaches, different solutions are presented. Since all have their own advantages and disadvantages, one has to determine the actual requirements and select the best fit for a specific application.