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    Biophysical investigations of cells focusing on the utility of optical tweezers : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics at Biophysics and Soft Matter Group, School of Fundamental Science, Massey University, New Zealand
    (Massey University, 2021) Pradhan, Susav
    The aim of this thesis was to explore the utility of different biophysical techniques, particularly optical tweezers (OT), in the investigation of the mechanical properties and interactions of biological samples. Specifically, MCF7 cells and their extracted nuclei were investigated mechanically, while the adhesion property of selected bacteria to the milk fat globule was also used as an exemplar. Biological cells have the ability to actively respond to external mechanical forces exerted by the microenvironment. The cellular response can be viscous, elastic, or viscoelastic in nature depending on the nature of the applied forces and the mechanical stresses applied. Changes in the mechanical properties of cells and nuclei have emerged as a prominent hallmark of many human diseases, particularly in neurodegenerative and metastatic diseases. In this thesis, to understand the application of these techniques to biological systems better, bulk rheology and microrheology studies were first performed on a model viscoelatic fluid (PEO). Particularly, the passive and active microrheology of this model viscoelastic material was characterized using optical tweezers and video particle tracking to develop the prerequisite experimental and analytical methods. Using the experimental knowledge gained from applying optical tweezers to standard materials, a mechanistic approach was developed in order to better understand how the mechanical properties of MCF7 cells change when the amount of heterochromatin protein (HP1a) present inside the nuclei was reduced. (HP1a) is an architectural protein that establishes and maintains heterochromatin, ensuring genome fidelity and nuclear integrity. While the mechanical effects of changes in the relative amount of euchromatin and heterochromatin brought about by inhibiting chromatin modifying enzymes have been studied previously, here we measure how the material properties of the cells are modified following the knockdown HP1a. Indentation experiments using optical tweezers revealed that the knockdown cells have apparent Young’s modului significantly lower than control cells. Similarly, tether experiments performed using optical tweezers revealed that the membrane tensions of knockdown cells were lower than those of control cells. This assay led to further work on studying the mechanical properties of nuclei extracted from MCF7 cells. A combination of atomic force microscopy, optical tweezers, and techniques based on micropipette aspiration was used to characterize the mechanical properties of nuclei extracted from HP1a knockdown or matched control cells. Similar to the previous finding on cells, local indentation performed using atomic force microscopy and optical tweezers found that the knockdown nuclei have apparent Young’s modului significantly lower than control nuclei. In contrast, results from pipette-based techniques in the spirit of microaspiration, where the whole nuclei were deformed and aspirated into a conical pipette, showed considerably less variation between HP1a knockdown and control, consistent with previous studies reporting that it is predominantly the lamins in the nuclear envelope that determine the mechanical response to large whole-cell deformations. The differences in chromatin organisation observed by various microscopy techniques between the MCF7 control and HP1a knock-down nuclei correlated well with the results of our measured mechanical responses and our hypotheses regarding their origin. Finally, not just the mechanical properties of the cells but also their interactions (an interaction between the milk fat globule membrane and two bacterial strains - Lactobacillius fermentum strains - 1487 and 1485) was explored as a side project by probing with optical tweezers. The difference in bacterial cell surface properties of these two strains and its effects on intestinal epithelial barrier integrity has already been studied. This study focuses on measuring the adhesion force between membrane and bacteria using optical tweezers. The results suggested that L. fermentus AGR1487 strongly interacts with MFGM compared to AGR1485. All in all, this thesis demonstrates how biophysical techniques can provide valuable insights into understanding biological systems.
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    Functional differential equations arising in the study of a cell growth model : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand
    (Massey University, 2019) Gul, Saima
    In this thesis we study a class of functional ordinary and partial differential equations that arise in the study of a size structured cell growth model. We study first and second order pantograph equations, which arise as separable solutions, for various constant and non constant coefficients. We discuss several techniques for solving pantograph equations that use the Laplace and the Mellin transforms, including a novel technique based on Mellin convolutions. These techniques are illustrated by applying them to a simple first order equation. The use of the Mellin transform to solve pantograph equation relies upon solving the transform equation, and this can prove formidable. This motivated us to find another avenue to show the existence of a solution. We consider a simple first order pantograph equation and show the uniqueness of the solution. We extend the study to second order pantograph equations and review a few particular second order pantograph equations with constant and non constant coefficients. These equations are solved using established techniques. Among the second order equation, a cell growth model that involves the Hermite operator is a part of research problems. Two interesting features for the Hermite Problem are, the form of the Mellin transform, that is such that the inversion is formidable, and the slow decaying nature of the solution. It is shown that for a range of parameter values α, b and g, there are no pdf solutions to the Hermite Problem; however if we drop the integrability and positivity condition, then there are non trivial solutions. Although the separable solution is the prime candidate for a steady size distribution, showing analytically that it is this distribution requires more advanced techniques. We thus consider the full problem. In particular, we consider the case where cells do not divide when they are under certain size. This problem differs from earlier ones because the eigenvalue for the separable solution is not known explicitly. In order to show the uniqueness of eigenvalue and to show that there is a steady size distribution solution to this problem, we adapt the analysis of Perthame & Ryzhik, who under certain assumptions on the division rate, established the existence and the uniqueness of the solution to a first order ordinary functional differential equation for a non constant division rate. In addition, we show that we have an alternative technique to find the eigenvalue. A second order partial differential equation arises when there is stochasticity in the growth rate. The earlier studied techniques are of limited use; however, Efendiev et al. [18] developed a technique that solves these equations for constant coefficients. They proved that the solution to the problem converges to the separable solution as time goes to ∞. We adapt their analysis and solve a second order functional differential equation with linear growth rate. In addition, we show that the solution to this problem does not have an SSD solution.
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    The analysis of fragmentation type equation for special division kernels : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū, School of Fundamental Sciences, New Zealand
    (Massey University, 2020) Almalki, Adel Ahmed
    The growth fragmentation equation is a linear integro-differential equation describing the evolution of cohorts that grow, divide and die or disappear in the course of time. The general formula is of first or second order, depending whether the growth process is deterministic or stochastic, respectively. We focus on a particular choice of division kernel that models size-structured cell cohorts which divide into daughter cells of equal size. This problem reduces to an initial-boundary value type that involves a modified Fokker-Planck equation with an advanced functional term. There are no general techniques for solving these problems. The constant growth rate case has been studied by a number of researchers. In particular, it was shown that the limiting solutions converge to a special solution, the separable solution. We consider the case when the growth rate is linear and deterministic. This problem can be solved analytically for monomial splitting rates. We show that the long time dynamics for this case differ markedly from the constant growth rate case. Specifically, the solutions approach a time dependent attracting solution that is periodic in time. The qualitative features of solutions differ when the splitting rate is constant. There are two cases. The first is when the growth rate is deterministic; the second is when the growth rate is stochastic. This case involves a constant dispersion term. In both cases, the problem can be solved directly, and the classic properties of solutions can be adapted from the previous case (with non constant splitting rate). The main distinct trait is that there is no long time attracting solution in $L^1$ for probability distribution initial data. (This result in the dispersive case follows, provided the parameters $g$, $b$ and $\alpha$ satisfy a certain inequality.) The long time asymptotic behaviour of solutions proves to be formidable to evaluate analytically for these cases. We use numerical methods to elucidate possible behaviour and examine the influence of the dispersion term. We find numerical evidence that the dispersion term plays a prominent r\^{o}le as a smoothing effect on the oscillatory behaviour, spatially and time wise, encountered in the dispersion free examples with exponential growth.
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    Theoretical investigation into the origins of multicellularity : a thesis presented in partial fulfilment of the requirements for the degree of PhD in Theoretical Biology at Massey University, Albany, New Zealand
    (Massey University, 2015) Pichugin, Yuriy
    Evolution of multicellularity is a major event in the history of life. The first step is the emergence of collectives of cooperating cells. Cooperation is generally costly to cooperators, thus, non-cooperators have a selective advantage. I investigated the evolution of cooperation in a population in which cells may migrate between collectives. Four different modes of migration were considered and for each mode I identified the set of multiplayer games in which cooperation has a higher fixation probability than defection. I showed that weak altruism may evolve without coordination among cells. However, the evolution of strong altruism requires the coordination of actions among cells. The second step in the emergence of multicellularity is the transition in Darwinian individuality. A likely hallmark of the transition is fitness decoupling. In the second part of my thesis, I present a method for characterizing fitness (de-)coupling which involves an analysis of the correlation between cell and collective fitnesses. In a population with coupled fitnesses, this correlation is close to one. As a population evolves towards multicellularity, collective fitness starts to rely more on the interactions between cells rather than the individual performance of cells, so the correlation between particle and collective fitnesses decreases. This metric makes it possible to detect fitness decoupling. I used the suggested metric to investigate under which conditions fitness decoupling occurs. I constructed a model of a population defined by a linear traits-to-fitness function and used this to identify those functions that promote fitness decoupling. In this model, the fitness correlation is equal to the cosine of the angle between the gradients of fitnesses. Therefore, my results allow an estimation of the fitness (de-)coupling state before selection takes place. In the third section of my thesis, the accuracy of this estimation was tested on available experimental data and using a model simulating an experimental selection regime, which featured non-linear traits-to-fitness functions. The results obtained from the estimation of fitness correlations showed a close approximation to the fitness correlation calculated from experimental data and from simulations in a range of selection regimes.
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    The evolution of multicellularity : a thesis presented in partial fulfillment of the requirements for the degree of PhD in Evolutionary Biology at Massey University, Albany, New Zealand
    (Massey University, 2014) Rose, Caroline
    Major evolutionary transitions in Darwinian individuality are central to the emergence of biological complexity. The key to understanding the evolutionary transition to multicellularity is to explain how a collective becomes a single entity capable of self-reproduction – a Darwinian individual. During the transition from single cells to multicellular life, populations of cells acquire the capacity for collective reproduction; however, the selective causes and underlying mechanisms are unclear. This thesis presents long-term evolution experiments using a single-celled model system to address fundamental questions arising during the evolution of multicellularity. Populations of the cooperating bacterium Pseudomonas fluorescens were subjected to experimental regimes that directly selected on the capacity for collectives to differentially reproduce – an essential requirement for the evolution of collectives by natural selection. A crucial stage during an evolutionary transition to multicellularity occurs when the fitness of the multicellular collective becomes ‘decoupled’ from the fitness of its constituent cells. Before this stage, any differences in collective fitness are due to selection at the cellular level. In the present study, collectives that competed to reproduce via a cooperative propagule cell attained high levels of cooperation and also reached high levels of collective fitness. However, these improvements were shown to be a consequence of selection acting at the cell-level. In contrast, Darwinian individuality emerged in collectives that reproduced via a primitive life cycle that was fueled by conflict between cooperating cells and cheating cells that did not bear the cost of cooperation. Cheats were analogous to a germ line, acting as propagules to seed new collectives. Enhanced fitness of evolved collectives was attributable to a property selected at the collective-level, namely, the capacity to transition through phases of the life cycle, and was not explained by improvement in individual cell fitness. Indeed, the fitness of individual cells declined. In addition to providing the first experimental evidence of a major evolutionary transition in individuality, the work presented in this thesis highlights the possibility that the prevalence of complex life cycles among extant multicellular organisms reflects the fact that such cycles, on first emergence, had the greatest propensity to participate in Darwinian evolution.
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    Mathematics of cell growth : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand
    (Massey University, 2014) Zaidi, Ali Ashher
    We present a model that describes growth, division and death of cells structured by size. Here, size can be interpreted as DNA content or physical size. The model is an extension of that studied by Hall and Wake [24] and incorporates the symmetric as well as the asymmetric division of cells. We first consider the case of symmetric cell division. This leads to an initial boundary value problem that involves a first-order linear PDE with a functional term. We study the separable solution to this problem which plays an important role in the long term behaviour of solutions. We also derive a solution to the problem for arbitrary initial cell distributions. The method employed exploits the hyperbolic character of the underlying differential operator, and the advanced nature of the functional argument to reduce the problem to a sequence of simple Cauchy problems. The existence of solutions for arbitrary initial distributions is established along with uniqueness. The asymptotic relationship with the separable solution is established, and because the solution is known explicitly, higher order terms in the asymptotics can be obtained. Adding variability to the growth rate of cells leads to a modified Fokker-Planck equation with a functional term. We find the steady size distribution solution to this equation. We also obtain a constructive existence and uniqueness theorem for this equation with an arbitrary initial size-distribution and with a no-flux condition. We then proceed to study the binary asymmetric division of cells. This leads to an initial boundary value problem that involves a first-order linear PDE with two functional terms. We find and prove the unimodality of the steady size distribution solution to this equation. The existence of higher eigenfunctions is also discussed. Adding stochasticity to the growth rate of cells yields a second-order functional differential equation with two non-local terms. These problems, being a particular kind of functional differential equations exhibit unusual characteristics. Although the associated boundary value problems are well-posed, the spectral problems that arise by separating the variables, cannot be easily shown to have a complete set of eigenfunctions or the usual orthogonality properties.
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    Dynamical modelling of the effect of insulin-like growth factor 1 on human cell growth : a thesis presented in fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University, Albany, New Zealand
    (Massey University, 2013) Phillips, Gemma
    Insulin-like Growth Factor-1 (IGF-1) plays a vital role in human growth and development. Interactions with IGF-1 receptors and IGF-1 binding proteins (IGFBPs) regulate IGF-1 function. Boroujerdi et al. (1997) published a mathematical model describing dynamic regulation of IGF-1. We extended the Boroujerdi et al. (1997) model to evaluate the role of cyclic Gly-Pro (CGP) in dynamic regulation of IGF-1 function. Recent research from the Liggins Institute suggests that a metabolite of IGF-1, CGP, may have a role in regulating IGF-1 homeostasis, possibly through competitive binding to IGFBPs. The goal of the research was to understand the kinetics of IGF-1, IGFBPs and CGP, along with their interactions with IGF-1 receptors. This goal and an understanding of how the kinetics mediate IGF-1 function was achieved through consideration of the nonlinear dynamics of the physiology using a modelling approach. The resulting models were directly focused on three central theories. The first is that CGP can either inhibit, stimulate or maintain IGF-1 function based on the extent of receptor binding. The other theories are that CGP regulates IGF-1 through competitive binding to IGFBPs and that CGP does not directly interact with the IGF-1 receptors. Four in vitro models were developed and fitted to experimental data. These included two implicit models which relied on two feedback terms in the equations. The second model was an alteration of the first to produce a reduction in cell number levels for high doses of CGP added to the system. The other two models were explicit models, the first of which could not express the IGF-1 dynamics well (it showed no CGP response). Although the models incorporated these theories, there are other mechanisms influencing the system which will have an effect on the data. Therefore the fourth model was introduced as a simplified version of the third. This was aimed at resembling cell culture situations more closely and was designed to have the receptor bound IGF-1 dependent on IGF-1 and CGP production rates. The models can be used to predict cellular response in an in vitro situation, or as a basis for further research in this field.
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    DNA synthesis in mammary epithelial cells of Swiss mice during lacation : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Animal Science at Massey University, Palmerston North, New Zealand
    (Massey University, 2003) Croke-Auldist, Danielle E
    Proliferation of extra secretory epithelial cells in mammary glands during lactation could potentially increase milk production, with flow-on benefits such as improved weaning weights of young or increased exports from the dairy cow industry. The primary objective of the research reported in this thesis was to increase proliferation of mammary epithelial cells during lactation so that the mechanisms associated with this phenomenon could be studied. Induction of increased proliferation in mammary glands was attempted by applying challenges to mice which were used as a laboratory model for agriculturally important species such as cows and pigs. The experiments reported in this thesis also included refinement of methodologies developed to study proliferation of secretory cells in mammary glands during lactation. The first was to improve techniques for describing the chemical composition of mammary glands collected during lactation. This was achieved by collecting and analysing the composition of mouse milk at 3 stages of lactation (Chapter 3). While milk protein and milk fat remained constant throughout, the concentration of lactose increased with time. These data were critically important for correcting the weights of mammary glands for milk content. A second investigation was carried out to compare different methods of calculating the milk production of mice (Chapter 3). Three methods were evaluated with the best based on calculating the maintenance energy requirements of the metabolic weight of the litter which was added to the energy required for measured litter growth. The total energy required was then converted to a quantity of milk. The third methodology developed during the course of the work in this thesis was sample preparation for analysis of lactating mammary cells using flow cytometry. One approach to increasing proliferation of mammary epithelial cells during lactation was to increase the suckling intensity of the mice. This challenge was accomplished by either increasing litter size (Chapter 6) or by increasing the ratio of pups per gland by taping over 5 of the 10 glands (Chapter 5), Suckling intensity was increased to 2 pups per gland but the effect was to accelerate mammary gland development in terms of cell number and milk synthesis status. Once a suckling intensity of >1 pup per gland was reached, there was no additive effect on the size of mammary glands or milk production at mid lactation. Mammary glands appeared to have a limit on their size and output which is reached at a suckling intensity of 1 pup per gland. Manipulation of suckling intensity did not produce a suitable model of elevated proliferation of mammary epithelial cells during lactation. Another approach tested was to use exogenous steroids as these had previously caused increased proliferation in mammary glands (Nagasawa and Yanai, 1978; Knight and Peaker, 1982d). The work reported herein showed that the response of mammary glands of mice to administration of steroids was dependent on stage of lactation and the dose (Chapter 4). In mid lactation, mammary glands were unresponsive for the parameters measured but in late lactation, incorporation of [3H] thymidine into DNA increased and milk production decreased in response to higher doses of estrogen. The high estrogen dose did not however yield a suitable model for the study because the elevated incorporation of [3H] thymidine was associated with early involution of mammary glands rather than proliferation leading to a net increase of epithelial cells. The most promising method of analysis came from histological studies of lactating glands of mice labelled for DNA synthesis. Labelling indices of epithelial cells were >1.5 times greater on the edges of glands on D1 of lactation compared to the inner zones of glands. This within mouse variation was much greater than any between mouse variation arising from the suckling intensity and steroid experiments. An attractive feature is that tissues are derived from the same gland and have therefore been exposed to the same factors such as systemic mitogens and nutrition. In addition, the differences in labelling indices were measured in glands of mice suckling litters of 10 pups which is an easily repeatable treatment compared to some of the more complicated treatments tested during the course of this thesis. Dissection of mammary glands into outer and inner zones could provide useful tissue for the study of local factors involved with increased DNA synthesis of epithelial cells during lactation. Histological studies also revealed that following labelling of mammary epithelial cells for DNA synthesis on the day after parturition, the proportion of cells labelled decrease at a constant rate over the next 23 days (Chapter 8). This project has increased the knowledge of manipulations of mouse mammary glands during lactation. It was found that growth of mouse mammary glands during lactation is difficult to increase experimentally and may have limited application as a model system to study regulation of growth of mammary glands during lactation. However, the work completed in this thesis will allow similar work to continue, with a high chance of success of investigating factors involved in mitosis of epithelial cells in lactating mammary glands.