Clustering in cell cycle dynamics with general response/signaling feedback.

Motivated by experimental and theoretical work on autonomous oscillations in yeast, we analyze ordinary differential equations models of large populations of cells with cell-cycle dependent feedback. We assume a particular type of feedback that we call responsive/signaling (RS), but do not specify a functional form of the feedback. We study the dynamics and emergent behavior of solutions, particularly temporal clustering and stability of clustered solutions. We establish the existence of certain periodic clustered solutions as well as "uniform" solutions and add to the evidence that cell-cycle dependent feedback robustly leads to cell-cycle clustering. We highlight the fundamental differences in dynamics between systems with negative and positive feedback. For positive feedback systems the most important mechanism seems to be the stability of individual isolated clusters. On the other hand we find that in negative feedback systems, clusters must interact with each other to reinforce coherence. We conclude from various details of the mathematical analysis that negative feedback is most consistent with observations in yeast experiments.

Finite amplitude, horizontal motion of a load symmetrically supported between isotropic hyperelastic springs.

The undamped, finite amplitude horizontal motion of a load supported symmetrically between identical incompressible, isotropic hyperelastic springs, each subjected to an initial finite uniaxial static stretch, is formulated in general terms. The small amplitude motion of the load about the deformed static state is discussed; and the periodicity of the arbitrary finite amplitude motion is established for all such elastic materials for which certain conditions on the engineering stress and the strain energy function hold. The exact solution for the finite vibration of the load is then derived for the classical neo-Hookean model. The vibrational period is obtained in terms of the complete Heuman lambda-function whose properties are well-known. Dependence of the period and hence the frequency on the physical parameters of the system is investigated and the results are displayed graphically.

THE HOPF BIFURCATION WITH BOUNDED NOISE.

We study Hopf-Andronov bifurcations in a class of random differential equations (RDEs) with bounded noise. We observe that when an ordinary differential equation that undergoes a Hopf bifurcation is subjected to bounded noise then the bifurcation that occurs involves a discontinuous change in the Minimal Forward Invariant set.

Universality of stable multi-cluster periodic solutions in a population model of the cell cycle with negative feedback.

We study a population model where cells in one part of the cell cycle may affect the progress of cells in another part. If the influence, or feedback, from one part to another is negative, simulations of the model almost always result in multiple temporal clusters formed by groups of cells. We study regions in parameter space where periodic 'k-cyclic' solutions are stable. The regions of stability coincide with sub-triangles on which certain events occur in a fixed order. For boundary sub-triangles with order 'rs1', we prove that the k-cyclic periodic solution is asymptotically stable if the index of the sub-triangle is relatively prime with respect to the number of clusters k and neutrally stable otherwise. For negative linear feedback, we prove that the interior of the parameter set is covered by stable sub-triangles, i.e. a stable k-cyclic solution always exists for some k. We observe numerically that the result also holds for many forms of nonlinear feedback, but may break down in extreme cases.

Early treatment gains for antibiotic administration and within human host time series data.

As technological improvements continue to infiltrate and impact medical practice, it has become possible to non-invasively collect dense physiological time series data from individual patients in real time. These advances continue to improve physicians' ability to detect and to treat infections early. One important benefit of early detection and treatment of nascent infections is that it leads to earlier resolution. In response to current and anticipated advances in data capture, we introduce the Early Treatment Gain (ETG) as a measure to quantify this benefit. Roughly, we define the gain to be the limiting ratio: ETG=differential change in time of resolutiondifferential change in treatment time.We study the gain using standard dynamical models and demonstrate its use with time series data from Surgical Intensive Care Unit (SICU) patients facing ventilator associated pneumonia. The main conclusion from the mathematical modelling is that the ETG is always greater than one unless there is an effective immune response, in which case the ETG can be less than one. Using real patient time series data, we observe that the formula derived for a linear model can be applied and that this produces a ETG greater than one.

A low dimensional dynamical model of the initial pulmonary innate response to infection.

In order to gain a deeper understanding of the onset and progression of pulmonary infections we present and analyze a low dimensional, phenomenological model of infection and the innate immune response in the lungs. Because pulmonary innate immunity has features unique to itself, general mathematical models of the immune system may not be appropriate. The differential equations model that we propose is based on current knowledge of the biology of pulmonary innate immunity and accurately reproduces known features of the initial phase of the dynamics of pulmonary innate system as exhibited in recent experiments. Further, we propose to use the model as a starting point for interrogation with clinical data from a new noninvasive technique for sampling alveolar lining fluid.

Non-invasive detection of pulmonary pathogens in ventilator-circuit filters by PCR.

Ventilator associated pneumonia is a common and costly complication in critically ill and injured surgical patients. The diagnosis of pneumonia remains problematic and non-specific. Using clinical criteria, a diagnosis of pneumonia is typically not made until an infection is well established. Semi-quantitative cultures of endotracheal aspirate and broncho-alveolar lavage are employed to improve the accuracy of diagnosis but are invasive and require time for culture results to become available. We report data that show that an inexpensive, rapid and non-invasive alternative may exist. In particular we show that: 1). Bio-aerosols evolved in the breath of ventilated patients and captured in the hygroscopic condenser humidifier filter of the ventilator circuit contain pathogenic micro-organisms. 2). The number (CFU/ml) and identity (Genus, species) of the pathogens in the aerosol samples can rapidly and inexpensively be determined by PCR. 3). Data from a convenience sample of filters correlate with clinical findings from standard microbiological methods such as broncho-alveolar lavage. The evaluation of the bacterial load evolved in exhaled breath by PCR is amenable to repeated sampling. Since increasing bacterial burden is believed to correlate with the establishment of infection, the use of quantitative PCR may provide a method to rapidly, inexpensively, and effectively detect and diagnose the early onset of pneumonia and identify pathogens involved.

BIFURCATIONS OF RANDOM DIFFERENTIAL EQUATIONS WITH BOUNDED NOISE ON SURFACES.

In random differential equations with bounded noise minimal forward invariant (MFI) sets play a central role since they support stationary measures. We study the stability and possible bifurcations of MFI sets. In dimensions 1 and 2 we classify all minimal forward invariant sets and their codimension one bifurcations in bounded noise random differential equations.

ODE, RDE and SDE models of cell cycle dynamics and clustering in yeast.

Biologists have long observed periodic-like oxygen consumption oscillations in yeast populations under certain conditions, and several unsatisfactory explanations for this phenomenon have been proposed. These 'autonomous oscillations' have often appeared with periods that are nearly integer divisors of the calculated doubling time of the culture. We hypothesize that these oscillations could be caused by a form of cell cycle synchronization that we call clustering. We develop some novel ordinary differential equation models of the cell cycle. For these models, and for random and stochastic perturbations, we give both rigorous proofs and simulations showing that both positive and negative growth rate feedback within the cell cycle are possible agents that can cause clustering of populations within the cell cycle. It occurs for a variety of models and for a broad selection of parameter values. These results suggest that the clustering phenomenon is robust and is likely to be observed in nature. Since there are necessarily an integer number of clusters, clustering would lead to periodic-like behaviour with periods that are nearly integer divisors of the period of the cell cycle. Related experiments have shown conclusively that cell cycle clustering occurs in some oscillating yeast cultures.