Random response of nonlinear continuous systems

This thesis presents a technique for obtaining the stochastic response of a nonlinear continuous system. First, the general method of nonstationary continuous equivalent linearization is developed. This technique allows replacement of the original nonlinear system with a time-varying linear continuous system. Next, a numerical implementation is described which allows solution of complex problems on a digital computer. In this procedure, the linear replacement system is discretized by the finite element method. Application of this method to systems satisfying the one-dimensional wave equation with two different types of constitutive nonlinearities is described. Results are discussed for nonlinear stress-strain laws of both hardening and softening types.

Nonlinear oscillations in discrete and continuous systems

The first thesis topic is a perturbation method for resonantly coupled nonlinear oscillators. By successive near-identity transformations of the original equations, one obtains new equations with simple structure that describe the long time evolution of the motion. This technique is related to two-timing in that secular terms are suppressed in the transformation equations. The method has some important advantages. Appropriate time scalings are generated naturally by the method, and don't need to be guessed as in two-timing. Furthermore, by continuing the procedure to higher order, one extends (formally) the time scale of valid approximation. Examples illustrate these claims. Using this method, we investigate resonance in conservative, non-conservative and time dependent problems. Each example is chosen to highlight a certain aspect of the method.

The second thesis topic concerns the coupling of nonlinear chemical oscillators. The first problem is the propagation of chemical waves of an oscillating reaction in a diffusive medium. Using two-timing, we derive a nonlinear equation that determines how spatial variations in the phase of the oscillations evolves in time. This result is the key to understanding the propagation of chemical waves. In particular, we use it to account for certain experimental observations on the Belusov-Zhabotinskii reaction.

Next, we analyse the interaction between a pair of coupled chemical oscillators. This time, we derive an equation for the phase shift, which measures how much the oscillators are out of phase. This result is the key to understanding M. Marek's and I. Stuchl's results on coupled reactor systems. In particular, our model accounts for synchronization and its bifurcation into rhythm splitting.

Finally, we analyse large systems of coupled chemical oscillators. Using a continuum approximation, we demonstrate mechanisms that cause auto-synchronization in such systems.

Response of multiple simultaneously recorded macaque area LIP neurons in a memory saccade task

Cells in the lateral intraparietal cortex (LIP) of rhesus macaques respond vigorously and in spatially-tuned fashion to briefly memorized visual stimuli. Responses to stimulus presentation, memory maintenance, and task completion are seen, in varying combination from neuron to neuron. To help elucidate this functional segmentation a new system for simultaneous recording from multiple neighboring neurons was developed. The two parts of this dissertation discuss the technical achievements and scientific discoveries, respectively.

Technology. Simultanous recordings from multiple neighboring neurons were made with four-wire bundle electrodes, or tetrodes, which were adapted to the awake behaving primate preparation. Signals from these electrodes were partitionable into a background process with a 1/f-like spectrum and foreground spiking activity spanning 300-6000 Hz. Continuous voltage recordings were sorted into spike trains using a state-of-the-art clustering algorithm, producing a mean of 3 cells per site. The algorithm classified 96% of spikes correctly when tetrode recordings were confirmed with simultaneous intracellular signals. Recording locations were verified with a new technique that creates electrolytic lesions visible in magnetic resonance imaging, eliminating the need for histological processing. In anticipation of future multi-tetrode work, the chronic chamber microdrive, a device for long-term tetrode delivery, was developed.

Science. Simultaneously recorded neighboring LIP neurons were found to have similar preferred targets in the memory saccade paradigm, but dissimilar peristimulus time histograms, PSTH). A majority of neighboring cell pairs had a difference in preferred directions of under 45° while the trial time of maximal response showed a broader distribution, suggesting homogeneity of tuning with het erogeneity of function. A continuum of response characteristics was present, rather than a set of specific response types; however, a mapping experiment suggests this may be because a given cell's PSTH changes shape as well as amplitude through the response field. Spike train autocovariance was tuned over target and changed through trial epoch, suggesting different mechanisms during memory versus background periods. Mean frequency-domain spike-to-spike coherence was concentrated below 50 Hz with a significant maximum of 0.08; mean time-domain coherence had a narrow peak in the range ±10 ms with a significant maximum of 0.03. Time-domain coherence was found to be untuned for short lags (10 ms), but significantly tuned at larger lags (50 ms).

Exploration of new drug delivery pathways: I. Mechanism of folate uptake in cultured human cells. II. Role of nuclear localization signal peptides in the delivery of oligonucleotide into reconstituted nuclei

The roles of the folate receptor and an anion carrier in the uptake of 5- methyltetrahydrofolate (5-MeH_4folate) were studied in cultured human (KB) cells using radioactive 5-MeH_4folate. Binding of the 5-MeH_4folate was inhibited by folic acid, but not by probenecid, an anion carrier inhibitor. The internalization of 5-MeH_4folate was inhibited by low temperature, folic acid, probenecid and methotrexate. Prolonged incubation of cells in the presence of high concentrations of probenecid appeared to inhibit endocytosis of folatereceptors as well as the anion carrier. The V_(max) and K_M values for the carrier were 8.65 ± 0.55 pmol/min/mg cell protein and 3.74 ± 0.54µM, respectively. The transport of 5-MeH4folate was competitively inhibited by folic acid, probenecid and methotrexate. The carrier dissociation constants for folic acid, probenecid and methotreate were 641 µM, 2.23 mM and 13.8 µM, respectively. Kinetic analysis suggests that 5-MeH_4folate at physiological concentration is transported through an anion carrier with the characteristics of the reduced-folate carrier after 5-MeH_4folate is endocytosed by folate receptors in KB cells. Our data with KB cells suggest that folate receptors and probenecid-sensitive carriers work in tandem to transport 5-MeH_4folate to the cytoplasm of cells, based upon the assumption that 1 mM probenecid does not interfere with the acidification of the vesicle where the folate receptors are endocytosed.

Oligodeoxynucleotides designed to hybridize to specific mRNA sequences (antisense oligonucleotides) or double stranded DNA sequences have been used to inhibit the synthesis of a number of cellular and viral proteins (Crooke, S. T. (1993) FASEB J. 7, 533-539; Carter, G. and Lemoine, N. R. (1993) Br. J. Cacer 67, 869-876; Stein, C. A. and cohen, J. S. (1988) Cancer Res. 48, 2659-2668). However, the distribution of the delivered oligonucleotides in the cell, i.e., in the cytoplasm or in the nucleus has not been clearly defined. We studied the kinetics of oligonucleotide transport into the cell nucleus using reconstituted cell nuclei as a model system. We present evidences here that oligonucleotides can freely diffuse into reconstituted nuclei. Our results are consistent with the reports by Leonetti et al. (Proc. Natl. Acad. Sci. USA, Vol. 88, pp. 2702-2706, April 1991), which were published while we were carrying this research independently. We also investigated whether a synthetic nuclear localization signal (NLS) peptide of SV40 T antigen could be used for the nuclear targeting of oligonucleotides. We synthesized a nuclear localization signal peptide-conjugated oligonucleotide to see if a nuclear localization signal peptide can enhance the uptake of oligonucleotides into reconstituted nuclei of Xenopus. Uptake of the NLS peptide-conjugated oligonucleotide was comparable to the control oligonucleotide at similar concentrations, suggesting that the NLS signal peptide does not significantly enhance the nuclear accumulation of oligonucleotides. This result is probably due to the small size of the oligonucleotide.

Continuous stochastic processes

A general review of stochastic processes is given in the introduction; definitions, properties and a rough classification are presented together with the position and scope of the author's work as it fits into the general scheme.

The first section presents a brief summary of the pertinent analytical properties of continuous stochastic processes and their probability-theoretic foundations which are used in the sequel.

The remaining two sections (II and III), comprising the body of the work, are the author's contribution to the theory. It turns out that a very inclusive class of continuous stochastic processes are characterized by a fundamental partial differential equation and its adjoint (the Fokker-Planck equations). The coefficients appearing in those equations assimilate, in a most concise way, all the salient properties of the process, freed from boundary value considerations. The writer’s work consists in characterizing the processes through these coefficients without recourse to solving the partial differential equations.

First, a class of coefficients leading to a unique, continuous process is presented, and several facts are proven to show why this class is restricted. Then, in terms of the coefficients, the unconditional statistics are deduced, these being the mean, variance and covariance. The most general class of coefficients leading to the Gaussian distribution is deduced, and a complete characterization of these processes is presented. By specializing the coefficients, all the known stochastic processes may be readily studied, and some examples of these are presented; viz. the Einstein process, Bachelier process, Ornstein-Uhlenbeck process, etc. The calculations are effectively reduced down to ordinary first order differential equations, and in addition to giving a comprehensive characterization, the derivations are materially simplified over the solution to the original partial differential equations.

In the last section the properties of the integral process are presented. After an expository section on the definition, meaning, and importance of the integral process, a particular example is carried through starting from basic definition. This illustrates the fundamental properties, and an inherent paradox. Next the basic coefficients of the integral process are studied in terms of the original coefficients, and the integral process is uniquely characterized. It is shown that the integral process, with a slight modification, is a continuous Markoff process.

The elementary statistics of the integral process are deduced: means, variances, and covariances, in terms of the original coefficients. It is shown that an integral process is never temporally homogeneous in a non-degenerate process.

Finally, in terms of the original class of admissible coefficients, the statistics of the integral process are explicitly presented, and the integral process of all known continuous processes are specified.

Transmission matrices and lumped parameter models for continuous systems

The use of transmission matrices and lumped parameter models for describing continuous systems is the subject of this study. Non-uniform continuous systems which play important roles in practical vibration problems, e.g., torsional oscillations in bars, transverse bending vibrations of beams, etc., are of primary importance.

A new approach for deriving closed form transmission matrices is applied to several classes of non-uniform continuous segments of one dimensional and beam systems. A power series expansion method is presented for determining approximate transmission matrices of any order for segments of non-uniform systems whose solutions cannot be found in closed form. This direct series method is shown to give results comparable to those of the improved lumped parameter models for one dimensional systems.

Four types of lumped parameter models are evaluated on the basis of the uniform continuous one dimensional system by comparing the behavior of the frequency root errors. The lumped parameter models which are based upon a close fit to the low frequency approximation of the exact transmission matrix, at the segment level, are shown to be superior. On this basis an improved lumped parameter model is recommended for approximating non-uniform segments. This new model is compared to a uniform segment approximation and error curves are presented for systems whose areas very quadratically and linearly. The effect of varying segment lengths is investigated for one dimensional systems and results indicate very little improvement in comparison to the use of equal length segments. For purposes of completeness, a brief summary of various lumped parameter models and other techniques which have previously been used to approximate the uniform Bernoulli-Euler beam is a given.

Delivery of Targeted Nanoparticles Across the Blood-Brain Barrier Using a Detachable Targeting Ligand

Chronic diseases of the central nervous system are poorly treated due to the inability of most therapeutics to cross the blood-brain barrier. The blood-brain barrier is an anatomical and physiological barrier that severely restricts solute influx, including most drugs, from the blood to the brain. One promising method to overcome this obstacle is to use endogenous solute influx systems at the blood-brain barrier to transport drugs. Therapeutics designed to enter the brain through transcytosis by binding the transferrin receptor, however, are restricted within endothelial cells. The focus of this work was to develop a method to increase uptake of transferrin-containing nanoparticles into the brain by overcoming these restrictive processes.

To accomplish this goal, nanoparticles were prepared with surface transferrin molecules bound through various liable chemical bonds. These nanoparticles were designed to shed the targeting molecule during transcytosis to allow increased accumulation of nanoparticles within the brain.

Transferrin was added to the surface of nanoparticles through either redox or pH sensitive chemistry. First, nanoparticles with transferrin bound through disulfide bonds were prepared. These nanoparticles showed decreased avidity for the transferrin receptor after exposure to reducing agents and increased ability to enter the brain in vivo compared to those lacking the disulfide link.

Next, transferrin was attached through a chemical bond that cleaves at mildly acidic pH. Nanoparticles containing a cleavable link between transferrin and gold nanoparticle cores were found to both cross an in vitro model of the blood-brain barrier and accumulate within the brain in significantly higher numbers than similar nanoparticles lacking the cleavable bond. Also, this increased accumulation was not seen when using this same strategy with an antibody to transferrin receptor, indicating that behavior of nanoparticles at the blood-brain barrier varies depending on what type of targeting ligand is used.

Finally, polymeric nanoparticles loaded with dopamine and utilizing a superior acid-cleavable targeting chemistry were investigated as a potential treatment for Parkinson’s disease. These nanoparticles were capable of increasing dopamine quantities in the brains of healthy mice, highlighting the therapeutic potential of this design. Overall, this work describes a novel method to increase targeted nanoparticle accumulation in the brain.

Development of a Cationic Mucic Acid Polymer-Based Nanoparticle siRNA Delivery System

Cancer chemotherapy has advanced from highly toxic drugs to more targeted treatments in the last 70 years. Chapter 1 opens with an introduction to targeted therapy for cancer. The benefits of using a nanoparticle to deliver therapeutics are discussed. We move on to siRNA in particular, and why it would be advantageous as a therapy. Specific to siRNA delivery are some challenges, such as nuclease degradation, quick clearance from circulation, needing to enter cells, and getting to the cytosol. We propose the development of a nanoparticle delivery system to tackle these challenges so that siRNA can be effective.

Chapter 2 of this thesis discusses the synthesis and analysis of a cationic mucic acid polymer (cMAP) which condenses siRNA to form a nanoparticle. Various methods to add polyethylene glycol (PEG) for stabilizing the nanoparticle in physiologic solutions, including using a boronic acid binding to diols on mucic acid, forming a copolymer of cMAP with PEG, and creating a triblock with mPEG on both ends of cMAP. The goal of these various pegylation strategies was to increase the circulation time of the siRNA nanoparticle in the bloodstream to allow more of the nanoparticle to reach tumor tissue by the enhanced permeation and retention effect. We found that the triblock mPEG-cMAP-PEGm polymer condensed siRNA to form very stable 30-40 nm particles that circulated for the longest time – almost 10% of the formulation remained in the bloodstream of mice 1 h after intravenous injection.

Chapter 3 explores the use of an antibody as a targeting agent for nanoparticles. Some antibodies of the IgG1 subtype are able to recruit natural killer cells that effect antibody dependent cellular cytotoxicity (ADCC) to kill the targeted cell to which the antibody is bound. There is evidence that the ADCC effect remains in antibody-drug conjugates, so we wanted to know whether the ADCC effect is preserved when the antibody is bound to a nanoparticle, which is a much larger and complex entity. We utilized antibodies against epidermal growth factor receptor with similar binding and pharmacokinetics, cetuximab and panitumumab, which differ in that cetuximab is an IgG1 and panitumumab is an IgG2 (which does not cause ADCC). Although a natural killer cell culture model showed that gold nanoparticles with a full antibody targeting agent can elicit target cell lysis, we found that this effect was not preserved in vivo. Whether this is due to the antibody not being accessible to immune cells or whether the natural killer cells are inactivated in a tumor xenograft remains unknown. It is possible that using a full antibody still has value if there are immune functions which are altered in a complex in vivo environment that are intact in an in vitro system, so the value of using a full antibody as a targeting agent versus using an antibody fragment or a protein such as transferrin is still open to further exploration.

In chapter 4, nanoparticle targeting and endosomal escape are further discussed with respect to the cMAP nanoparticle system. A diboronic acid entity, which gives an order of magnitude greater binding (than boronic acid) to cMAP due to the vicinal diols in mucic acid, was synthesized, attached to 5kD or 10kD PEG, and conjugated to either transferrin or cetuximab. A histidine was incorporated into the triblock polymer between cMAP and the PEG blocks to allow for siRNA endosomal escape. Nanoparticle size remained 30-40 nm with a slightly negative ca. -3 mV zeta potential with the triblock polymer containing histidine and when targeting agents were added. Greater mRNA knockdown was seen with the endosomal escape mechanism than without. The nanoparticle formulations were able to knock down the targeted mRNA in vitro. Mixed effects suggesting function were seen in vivo.

Chapter 5 summarizes the project and provides an outlook on siRNA delivery as well as targeted combination therapies for the future of personalized medicine in cancer treatment.