Cell cycle dynamics: clustering is universal in negative feedback systems.

We study a model of cell cycle ensemble dynamics with cell-cell feedback in which cells in one fixed phase of the cycle S (Signaling) produce chemical agents that affect the growth and development rate of cells that are in another phase R (Responsive). For this type of system there are special periodic solutions that we call k-cyclic or clustered. Biologically, a k-cyclic solution represents k cohorts of synchronized cells spaced nearly evenly around the cell cycle. We show, under very general nonlinear feedback, that for a fixed k the stability of the k-cyclic solutions can be characterized completely in parameter space, a 2 dimensional triangle T. We show that T is naturally partitioned into k(2) sub-triangles on each of which the k-cyclic solutions all have the same stability type. For negative feedback we observe that while the synchronous solution (k = 1) is unstable, regions of stability of k ≥ 2 clustered solutions seem to occupy all of T. We also observe bi-stability or multi-stability for many parameter values in negative feedback systems. Thus in systems with negative feedback we should expect to observe cyclic solutions for some k. This is in contrast to the case of positive feedback, where we observe that the only asymptotically stable periodic orbit is the synchronous solution.

Simultaneous on-chip DC dielectrophoretic cell separation and quantitative separation performance characterization.

Through integration of a MOSFET-based microfluidic Coulter counter with a dc-dielectrophoretic cell sorter, we demonstrate simultaneous on-chip cell separation and sizing with three different samples including 1) binary mixtures of polystyrene beads, 2) yeast cells of continuous size distribution, and 3) mixtures of 4T1 tumor cells and murine bone marrow cells. For cells with continuous size distribution, it is found that the receiver operator characteristic analysis is an ideal method to characterize the separation performance. The characterization results indicate that dc-DEP separation performance degrades as the sorting throughput (cell sorting rate) increases, which provides insights into the design and operation of size-based microfluidic cell separation.

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.

Real-time luminescence monitoring of cell-cycle and respiratory oscillations in yeast.

The use of luciferase reporters has become a precise, noninvasive, high-throughput method for real-time monitoring of promoter activity in living cells, especially for rhythmic biological processes such as circadian rhythms. We developed a destabilized firefly luciferase as a reporter for rhythmic promoter activity in both the cell division and respiratory cycles of the budding yeast Saccharomyces cerevisiae in which real-time luminescence reporters have not been previously applied. The continuous output of light from luciferase reporters allowed us to explore the relationship between the cell division cycle and the yeast respiratory oscillation, including the observation of responses to chemicals that cause phase shifting of the respiratory oscillations. Destabilized firefly luciferase is a good reporter of cell cycle position in synchronized or partially synchronized yeast cultures, in both batch and continuous cultures. In addition, the oxygen dependence of luciferase can be used under certain conditions as a genetically encodable oxygen monitor. Finally, we use this reporter to show that there is a direct correlation between premature induction of cell division and phase resetting of the respiratory oscillation under the continuous culture conditions tested.

Measurement of the volume growth rate of single budding yeast with the MOSFET-based microfluidic Coulter counter.

We report on measurements of the volume growth rate of ten individual budding yeast cells using a recently developed MOSFET-based microfluidic Coulter counter. The MOSFET-based microfluidic Coulter counter is very sensitive, provides signals that are immune from the baseline drift, and can work with cell culture media of complex composition. These desirable features allow us to directly measure the volume growth rate of single cells of Saccharomyces cerevisiae LYH3865 strain budding yeast in YNB culture media over a whole cell cycle. Results indicate that all budding yeast follow a sigmoid volume growth profile with reduced growth rates at the initial stage before the bud emerges and the final stage after the daughter gets mature. Analysis of the data indicates that even though all piecewise linear, Gomperitz, and Hill's function models can fit the global growth profile equally well, the data strongly support local exponential growth phenomenon. Accurate volume growth measurements are important for applications in systems biology where quantitative parameters are required for modeling and simulation.

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 pilot study of the noninvasive assessment of the lung microbiota as a potential tool for the early diagnosis of ventilator-associated pneumonia.

BACKGROUND: Ventilator-associated pneumonia (VAP) remains a common complication in critically ill surgical patients, and its diagnosis remains problematic. Exhaled breath contains aerosolized droplets that reflect the lung microbiota. We hypothesized that exhaled breath condensate fluid (EBCF) in hygroscopic condenser humidifier/heat and moisture exchanger (HCH/HME) filters would contain bacterial DNA that qualitatively and quantitatively correlate with pathogens isolated from quantitative BAL samples obtained for clinical suspicion of pneumonia. METHODS: Forty-eight adult patients who were mechanically ventilated and undergoing quantitative BAL (n = 51) for suspected pneumonia in the surgical ICU were enrolled. Per protocol, patients fulfilling VAP clinical criteria undergo quantitative BAL bacterial culture. Immediately prior to BAL, time-matched HCH/HME filters were collected for study of EBCF by real-time polymerase chain reaction. Additionally, convenience samples of serially collected filters in patients with BAL-diagnosed VAP were analyzed. RESULTS: Forty-nine of 51 time-matched EBCF/BAL fluid samples were fully concordant (concordance > 95% by κ statistic) relative to identified pathogens and strongly correlated with clinical cultures. Regression analysis of quantitative bacterial DNA in paired samples revealed a statistically significant positive correlation (r = 0.85). In a convenience sample, qualitative and quantitative polymerase chain reaction analysis of serial HCH/HME samples for bacterial DNA demonstrated an increase in load that preceded the suspicion of pneumonia. CONCLUSIONS: Bacterial DNA within EBCF demonstrates a high correlation with BAL fluid and clinical cultures. Bacterial DNA within EBCF increases prior to the suspicion of pneumonia. Further study of this novel approach may allow development of a noninvasive tool for the early diagnosis of VAP.

Pharmacokinetics of O6-benzylguanine (NSC637037) and its metabolite, 8-oxo-O6-benzylguanine.

O6-Benzylguanine and its metabolite, 8-oxo-O6-benzylguanine, are equally potent inhibitors of the DNA repair enzyme, O6-alkylguanine-DNA alkyltransferase. Pharmacokinetic values are derived from cancer patients participating in a phase I trial (10 or 20 mg/m2 of O6-benzylguanine in a single bolus dose or 10 to 120 mg/m2 as a 60-min constant infusion). A two-compartment model fits the plasma concentration versus time profile of O6-benzylguanine. O6-Benzylguanine is eliminated rapidly from the plasma compartment in humans (t1/2 alpha and t1/2 beta are 2 +/- 2 min and 26 +/- 15 min [mean +/- SD, n = 7], respectively), and its plasma clearance (513 +/- 148 mL/min/m2) is not dose dependent. Metabolite kinetics are evaluated using both a novel approach describing the relationship between O6-benzylguanine and 8-oxo-O6-benzylguanine and classical metabolite kinetics methods. With increasing doses of O6-benzylguanine, the plasma clearance of 8-oxo-O6-benzylguanine, decreases, prolonging elimination of the metabolite. This effect is not altered by coadministration of BCNU. The urinary excretion of drug and metabolites is minimal.

Cell cycle dynamics in a response/signalling feedback system with a gap.

We consider a dynamical model of cell cycles of n cells in a culture in which cells in one specific phase (S for signalling) of the cell cycle produce chemical agents that influence the growth/cell cycle progression of cells in another phase (R for responsive). In the case that the feedback is negative, it is known that subpopulations of cells tend to become clustered in the cell cycle; while for a positive feedback, all the cells tend to become synchronized. In this paper, we suppose that there is a gap between the two phases. The gap can be thought of as modelling the physical reality of a time delay in the production and action of the signalling agents. We completely analyse the dynamics of this system when the cells are arranged into two cell cycle clusters. We also consider the stability of certain important periodic solutions in which clusters of cells have a cyclic arrangement and there are just enough clusters to allow interactions between them. We find that the inclusion of a small gap does not greatly alter the global dynamics of the system; there are still large open sets of parameters for which clustered solutions are stable. Thus, we add to the evidence that clustering can be a robust phenomenon in biological systems. However, the gap does effect the system by enhancing the stability of the stable clustered solutions. We explain this phenomenon in terms of contraction rates (Floquet exponents) in various invariant subspaces of the system. We conclude that in systems for which these models are reasonable, a delay in signalling is advantageous to the emergence of clustering.

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.

An Analysis of Quantitative PCR Reliability Through Replicates Using the C Method.

There is considerable interest in quantitatively measuring nucleic acids from single cells to small populations. The most commonly employed laboratory method is the real-time polymerase chain reaction (PCR) analyzed with the crossing point or crossing threshold (C(t)) method. Utilizing a multiwell plate reader we have performed hundreds of replicate reactions at each of a set of initial conditions whose initial number of copies span a concentration range of ten orders of magnitude. The resultant C(t) value distributions are analyzed with standard and novel statistical techniques to assess the variability/reliability of the PCR process. Our analysis supports the following conclusions. Given sufficient replicates, the mean and/or median C(t) values are statistically distinguishable and can be rank ordered across ten orders of magnitude in initial template concentration. As expected, the variances in the C(t) distributions grow as the number of initial copies declines to 1. We demonstrate that these variances are large enough to confound quantitative classification of the initial condition at low template concentrations. The data indicate that a misclassification transition is centered around 3000 initial copies of template DNA and that the transition region correlates with independent data on the thermal wear of the TAQ polymerase enzyme. We provide data that indicate that an alternative endpoint detection strategy based on the theory of well mixing and plate filling statistics is accurate below the misclassification transition where the real time method becomes unreliable.

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.

Periodic fermentor yield and enhanced product enrichment from autonomous oscillations.

Four decades of work have clearly established the existence of autonomous oscillations in budding yeast culture across a range of operational parameters and in a few strains. Autonomous oscillations impact substrate conversion to biomass and products. Relatively little work has been done to quantify yield in this case. We have analyzed the yield of autonomously oscillating systems, grown under different conditions, and demonstrate that it too oscillates. Using experimental data and mathematical models of yeast growth and division, we demonstrate strategies to increase the efficient recovery of products. The analysis makes advantage of the population structure and synchrony of the system and our ability to target production within the cell cycle. While oscillatory phenomena in culture have generally been regarded with trepidation in the engineering art of bioprocess control, our results provide further evidence that autonomously oscillating systems can be a powerful tool, rather than an obstruction.

Molecular seismology: an inverse problem in nanobiology.

The density profile of an elastic fiber like DNA will change in space and time as ligands associate with it. This observation affords a new direction in single molecule studies provided that density profiles can be measured in space and time. In fact, this is precisely the objective of seismology, where the mathematics of inverse problems have been employed with success. We argue that inverse problems in elastic media can be directly applied to biophysical problems of fiber-ligand association, and demonstrate that robust algorithms exist to perform density reconstruction in the condensed phase.

Reliable cell disruption in yeast.

We report the results of an optical assay to determine the degree of cell wall disruption in yeast. The results indicate that cell wall disruption with glass beads yields reproducible results that can be modelled with an integral measure of time to failure that implies a decreasing failure rate. It is shown that a standard protocol results in only 60% disruption, with a relatively large coefficient of variation. The data show that the yield of total RNA harvested is proportional to the degree of cellular disruption, and that there is no loss of RNA quality with > 90% disruption. The data also show that cell disruption of a synchronous culture varies with the cell cycle. We speculate that the decreasing failure rate is related to the cell cycle phase-dependent disruptability.

Dynamics of a simple regulatory switch.

We consider the dynamics of a model toggle switch abstracted from the genetic interactions operative in a fungal stress response circuit. The switch transduces an external signal and propagates it forward by mediating the transport between compartments of two interacting gene products. The transport between compartments is assumed to be related to the degree of association between the interacting proteins, a fact for which there exists a wealth of biological evidence. The ubiquity and modularity of this cellular control mechanism warrants a detailed study of the dynamics entailed by various modelling assumptions. Specifically, we consider a general gate model in which both of the associating proteins are freely transportable between compartments. A more restrictive, but biologically supported model, is considered in which only one of the two proteins undergoes transport. Under the strong assumption that the disassociation of the interacting proteins is unidirectional we show that the qualitative dynamics of the two models are similar; that is they both converge to unique periodic orbits. From a biophysical perspective the assumption of unidirectional dissociation is unrealistic. We show that the same result holds for the more restrictive model when one weakens the assumption of unidirectional binding or disassociation. We speculate that this is not true for the more general model. This difference in dynamics may have important biological implications and certainly points to promising avenues of research.

Connecting the dots between genes, biochemistry, and disease susceptibility: systems biology modeling in human genetics.

Understanding how DNA sequence variations impact human health through a hierarchy of biochemical and physiological systems is expected to improve the diagnosis, prevention, and treatment of common, complex human diseases. We have previously developed a hierarchical dynamic systems approach based on Petri nets for generating biochemical network models that are consistent with genetic models of disease susceptibility. This modeling approach uses an evolutionary computation approach called grammatical evolution as a search strategy for optimal Petri net models. We have previously demonstrated that this approach routinely identifies biochemical network models that are consistent with a variety of genetic models in which disease susceptibility is determined by nonlinear interactions between two or more DNA sequence variations. We review here this approach and then discuss how it can be used to model biochemical and metabolic data in the context of genetic studies of human disease susceptibility.

Structure theorems and the dynamics of nitrogen catabolite repression in yeast.

By using current biological understanding, a conceptually simple, but mathematically complex, model is proposed for the dynamics of the gene circuit responsible for regulating nitrogen catabolite repression (NCR) in yeast. A variety of mathematical "structure" theorems are described that allow one to determine the asymptotic dynamics of complicated systems under very weak hypotheses. It is shown that these theorems apply to several subcircuits of the full NCR circuit, most importantly to the URE2-GLN3 subcircuit that is independent of the other constituents but governs the switching behavior of the full NCR circuit under changes in nitrogen source. Under hypotheses that are fully consistent with biological data, it is proven that the dynamics of this subcircuit is simple periodic behavior in synchrony with the cell cycle. Although the current mathematical structure theorems do not apply to the full NCR circuit, extensive simulations suggest that the dynamics is constrained in much the same way as that of the URE2-GLN3 subcircuit. This finding leads to the proposal that mathematicians study genetic circuits to find new geometries for which structure theorems may exist.