A nonlinear structured population model of tumor growth with quiescence

A nonlinear structured cell population model of tumor growth is considered. The model distinguishes between two types of cells within the tumor: proliferating and quiescent. Within each class the behavior of individual cells depends on cell size, whereas the probabilities of becoming quiescent and returning to the proliferative cycle are in addition controlled by total tumor size. The asymptotic behavior of solutions of the full nonlinear model, as well as some linear special cases, is investigated using spectral theory of positive simigroup of operators. Supported in part by the National Science Foundation under Grant No. DMS-8722947     

A model for restriction point control of the mammalian cell cycle

Inhibition of protein synthesis by cycloheximide blocks subsequent division of a mammalian cell, but only if the cell is exposed to the drug before the "restriction point" (i.e. within the first several hours after birth). If exposed to cycloheximide after the restriction point, a cell proceeds with DNA synthesis, mitosis and cell division and halts in the next cell cycle. If cycloheximide is later removed from the culture medium, treated cells will return to the division cycle, showing a complex pattern of division times post-treatment, as first measured by Zetterberg and colleagues. We simulate these physiological responses of mammalian cells to transient inhibition of growth, using a set of nonlinear differential equations based on a realistic model of the molecular events underlying progression through the cell cycle. The model relies on our earlier work on the regulation of cyclin-dependent protein kinases during the cell division cycle of yeast. The yeast model is supplemented with equations describing the effects of retinoblastoma protein on cell growth and the synthesis of cyclins A and E, and with a primitive representation of the signaling pathway that controls synthesis of cyclin D.     

A model of tumor growth based on cell cycle kinetics

This work describes a mathematical model of growth based on the kinetics of the cell cycle. A traditional model of the cell cycle has been used, with the addition of a resting (G₀) state from which cells could reenter the reproductive cycle. The model assumes that a growth regulatory substance regulates the transition of cells to and from the resting state. Other transitions between the phases of the cycle were modeled as a first order process. Cell loss is an important feature of growth kinetics, and has been represented by a general but tractable mathematical form. The resulting model forms a system of ordinary nonlinear differential equations. Analytic methods are employed first in the study of this system. Simplifying assumptions regarding cell loss give rise to special cases for which equilibrium solutions can be found. One special case, which assumes first order loss from all cell cycle phases at equal rates, is presented here. For small time values, approximations corresponding to exponential growth were developed. The equations describing an intrinsic growth rate were derived. Simulation methods were used to further characterize the behavior of this model. Parameter values were chosen based on animal tumor cell cycle kinetic data, resulting in a set of 45 model simulations. Several tumor treatment protocols were simulated which illustrated the importance of the intrinsic growth rate and cell loss concepts. Although the qualitative behavior regarding absolute and relative growth is reasonable, this model awaits data for model fitting, parameter estimation, or revision of the equations.     

Mathematical model of proliferative activity in epidermis of the normal skin and of the skin afflicted by psoriasis

A mathematical model of the mitotic activity of epidermis in norm and psoriasis is presented, which consists of a system of two autonomous nonlinear differential equations. A qualitative analysis of the model was done, and numerical solutions at the parameter values corresponding to these state were obtained. It was shown that, in norm, the system can exist only in one stationary equilibrium state of the "focus" type, and in psoriasis, due to an increase in the growing fraction, hyperproliferation, and enhanced migration of keratinocytes, a stable limiting cycle arises from the state of an unstable focus. The existence of two stable states (focus and the limiting cycle) is regulated by a parameter that describes the inhibition of division of maturing cells of suprabasal epidermal layers by the intrinsic tissue-specific transmitters of mitosis of G1-keilon type. The model is consistent with experimental data on the kinetics of cell proliferation in the epidermis in norm and psoriasis and the clinical course of the disease.     

Interaction of interleukin 2 with cells: quantitative analysis of effects

The response of T lymphocytes to interleukin 2 (IL 2) is accurately described by a four-parameter logistic function. Both data generated by a theoretical model of IL 2-driven proliferation and experimental data conformed to this function for all doses of IL 2. Assays measuring either the rate of DNA synthesis or cellular metabolism were well described. The variance of response was not constant but increased in a predictable way. Weighting was therefore included in deriving a nonlinear curve-fitting program. The effects on response of cell density, time, and the T lymphocyte line used were examined. Assays gave reproducible estimates of potency when test preparations were compared with a standard preparation, but not otherwise. A model for IL 2 proliferation was derived on the basis of the two-state model of the cell cycle, with cells leaving a quiescent state randomly and then traversing the other stages of the cell cycle in a determinate way.     

From START to FINISH: The Influence of Osmotic Stress on the Cell Cycle

The cell cycle is a sequence of biochemical events that are controlled by complex but robust molecular machinery. This enables cells to achieve accurate self-reproduction under a broad range of different conditions. Environmental changes are transmitted by molecular signalling networks, which coordinate their action with the cell cycle. The cell cycle process and its responses to environmental stresses arise from intertwined nonlinear interactions among large numbers of simpler components. Yet, understanding of how these pieces fit together into a coherent whole requires a systems biology approach. Here, we present a novel mathematical model that describes the influence of osmotic stress on the entire cell cycle of S. cerevisiae for the first time. Our model incorporates all recently known and several proposed interactions between the osmotic stress response pathway and the cell cycle. This model unveils the mechanisms that emerge as a consequence of the interaction between the cell cycle and stress response networks. Furthermore, it characterises the role of individual components. Moreover, it predicts different phenotypical responses for cells depending on the phase of cells at the onset of the stress. The key predictions of the model are: (i) exposure of cells to osmotic stress during the late S and the early G2/M phase can induce DNA re-replication before cell division occurs, (ii) cells stressed at the late G2/M phase display accelerated exit from mitosis and arrest in the next cell cycle, (iii) osmotic stress delays the G1-to-S and G2-to-M transitions in a dose dependent manner, whereas it accelerates the M-to-G1 transition independently of the stress dose and (iv) the Hog MAPK network compensates the role of the MEN network during cell division of MEN mutant cells. These model predictions are supported by independent experiments in S. cerevisiae and, moreover, have recently been observed in other eukaryotes.     

A stochastic limit cycle oscillator model of the EEG

We present an empirical model of the electroencephalogram (EEG) signal based on the construction of a stochastic limit cycle oscillator using Itô calculus. This formulation, where the noise influences actually interact with the dynamics, is substantially different from the usual definition of measurement noise. Analysis of model data is compared with actual EEG data using both traditional methods and modern techniques from nonlinear time series analysis. The model demonstrates visually displayed patterns and statistics that are similar to actual EEG data. In addition, the nonlinear mechanisms underlying the dynamics of the model do not manifest themselves in nonlinear time series analysis, paralleling the situation with real, non-pathological EEG data. This modeling exercise suggests that the EEG is optimally described by stochastic limit cycle behavior.     

Mathematical description and analysis of cell cycle kinetics and the application to Ehrlich ascites tumor

The linear and nonlinear aspects of the dynamics of the cell cycle kinetics of cell populations are studied. The dynamics are represented by difference equations. The characteristics of cell population systems are analyzed by applying the model to Ehrlich ascites tumor. The model applied for the simulations of the growth of Ehrlich ascites tumor cells incorporates processes of cell division, cell death, transition of cells to resting states and clearance of dead cells. Comparison of the results obtained with the model and the experimental data suggests that the duration of the mean generation time of the proliferating EAT cells increases with aging of the tumor. An attempt is made to relate the prolongation of cell mean generation time with processes of cell death and dead cell clearance. Studying the transition of cells to the resting states, it becomes apparent that in fact transition of proliferating cells to the resting states occurs somewhere close to the end of the cell cycle and with a rate that varies with the age of the tumor. Time course behavior of the cell age, cell size, and cell DNA distribution with aging of the tumor are obtained. Variations in average size and average DNA contents are determined.     

The topology of phase response curves induced by single and paired stimuli in spontaneously oscillating chick heart cell aggregates.

The topological properties of the phase resetting of biological oscillators by an isolated stimulus delivered at various phases of the cycle depend on whether the stimulus is "weak" or "strong." When multiple stimuli are delivered to the oscillator, the response to stimulation also depends on the time between the stimuli, and the rate at which the oscillator returns to an underlying limit cycle attractor. If the time between two consecutive "weak" stimuli is sufficiently short, the effects produced by the pair of stimuli may be characteristic of a single "strong" stimulus. These results are demonstrated in a model experimental system, spontaneously beating aggregates of cells derived from embryonic chick heart, and are illustrated by consideration of a simple theoretical model of nonlinear oscillators, the Poincaré oscillator.     

Model of DNA Dynamics and Replication

Before DNA replication can be initiated a definite number of adenosine triphosphate (ATP) containing pre-replication protein complexes (pre-RCs) must be assembled and bound to DNA like in a super-critical mass. A chemically driven dynamics of the Ginzburg-Landau (GL) type is derived, using the non-equilibrium equation for binding of pre-RCs to DNA and a probabilistic conformational distribution of these protein complexes. This dynamics, in which the DNA-protein system behaves like a nonlinear elastically braced string (NEBS), can control the cell cycle via conformational transitions such that G₂ cells contain exactly twice as much DNA as G₁ cells. After adjustment of previously-made derivations, the model is compared with cell growth data from the T lymphocyte MLA-144.     

A bioeconomic model of a fishery with saturated catch and variable price: Stabilizing effect of marine reserves on fishery dynamics

In this paper, we study a mathematical bio-economic model of a fishery with varying price. The three dimensional model describes the time evolution of the resource, the fishing effort and the price. The model is original because it considers a nonlinear harvesting function assumed to depend upon stock size and fishing effort with a saturation effect with respect to the resource as well as a price equation depending on demand and supply which is in addition proportional to price. Assuming that the price varies at a fast time scale, we are able to use ”aggregation of variables methods” in order to reduce the model in a two dimensional model at a slow time scale. This aggregated (reduced) model is analyzed. Several numerical simulations of the model are performed to substantiate our analytical findings. The existence of nonlinear harvesting makes the dynamics of the model more complicated, including multiple equilibria, bi-stability and limit cycle. Such large amplitude cycle variations are not desirable because they generate periods of overfishing at periods of very low activity. We then study the effects of marine reserves on the dynamics of the fishery, showing that for an adequate number of small reserves, limit cycle oscillations are switched off.     

A constitutive model for the time-dependent,nonlinear stress response of fibrin networks

Blood clot formation is important to prevent blood loss in case of a vascular injury but disastrous when it occludes the vessel. As the mechanical properties of the clot are reported to be related to many diseases, it is important to have a good understanding of their characteristics. In this study, a constitutive model is presented that describes the nonlinear viscoelastic properties of the fibrin network, the main structural component of blood clots. The model is developed using results of experiments in which the fibrin network is subjected to a large amplitude oscillatory shear (LAOS) deformation. The results show three dominating nonlinear features: softening over multiple deformation cycles, strain stiffening and increasing viscous dissipation during a deformation cycle. These features are incorporated in a constitutive model based on the Kelvin–Voigt model. A network state parameter is introduced that takes into account the influence of the deformation history of the network. Furthermore, in the period following the LAOS deformation, the stiffness of the networks increases which is also incorporated in the model. The influence of cross-links created by factor XIII is investigated by comparing fibrin networks that have polymerized for 1 and 2 h. A sensitivity analysis provides insights into the influence of the eight fit parameters. The model developed is able to describe the rich, time-dependent, nonlinear behavior of the fibrin network. The model is relatively simple which makes it suitable for computational simulations of blood clot formation and is general enough to be used for other materials showing similar behavior.     

The nonlinear pathway of Y ganglion cells in the cat retina

Retinal ganglion cells of the Y type in the cat retina produce two different types of response: linear and nonlinear. The nonlinear responses are generated by a separate and independent nonlinear pathway. The functional connectivity in this pathway is analyzed here by comparing the observed second-order frequency responses of Y cells with predictions of a "sandwich model" in which a static nonlinear stage is sandwiched between two linear filters. The model agrees well with the qualitative and quantitative features of the second-order responses. The prefilter in the model may well be the bipolar cells and the nonlinearity and postfilter in the model are probably associated with amacrine cells.     

Metabolic engineering in silico

This review briefs on the main directions in the field of mathematical modeling of metabolic processes aimed at a rational design of genetically modified organisms. The class of generalized Hill functions is described, and their application to modeling of nonlinear processes in Escherichia coli metabolic systems is illustrated by several examples. A model for the pyrimidine biosynthesis in E. coli, taking into account the nonlinear effects of a negative allosteric regulation of enzyme activities involved in the control of the subsequent stages by the end products of synthesis, is considered. It has been shown that the model displays its own continuous oscillation mode of functioning with a period of approximately 50 min, which is close to the duration of E. coli cell cycle. The need in considering the nonlinear effects in the models as essential elements in the function of metabolic systems far from equilibrium is discussed.     

Mathematical modeling of hyphal tip growth

The mathematical modelling of growing filamentous cells has been approached in a variety of ways ranging from simple geometric to biomechanically based models using exact, nonlinear, elasticity theory for shells and membranes in which a growth mechanism is included, and alternative approaches using visco-plasticity theory. We describe how the nonlinear elastic model is able to capture essential biomechanical mechanical features of the growth of a broad array of filamentous cells including fungi, actinomycetes, pollen tubes, and root hairs. A comparison between this approach and visco-plasticity based models is made.     

Dynamical properties of the two-process model for sleep-wake cycles in infantile autism

The two-process model is a scheme for the timing of sleep that consists of homeostatic (Process S) and circadian (Process C) variables. The two-process model exhibits abnormal sleep patterns such as internal desynchronization or sleep fragmentation. Early infants with autism often experience sleep difficulties. Large day-by-day changes are found in the sleep onset and waking times in autistic children. Frequent night waking is a prominent property of their sleep. Further, the sleep duration of autistic children is often fragmented. These sleep patterns in infants with autism are not fully understood yet. In the present study, the sleep patterns in autistic children were reproduced by a modified two-process model using nonlinear analysis. A nap term was introduced into the original two-process model to reproduce the sleep patterns in early infants. The nap term and the time course of Process S are mentioned in the present study. Those parameters led to bifurcation of the sleep-wake cycle in the modified two-process model. In a certain range of these parameter sets, a small external noise was amplified, and an irregular sleep-wake cycle appeared. The short duration of sleep led to another irregular sleep onset or waking. Consequently, an irregular sleep-wake cycle appeared in early infantile autism.     

Theory of distributed quiescent state in the cell cycle

A generalization of the familiar two-compartment or G₀ model of the cell cycle is described. Instead of reserving the quiescent state strictly to newly born cells, it is distributed throughout the cell cycle. A cell may cease its proliferative activities anywhere in the cycle with a probability depending on its maturity. The probability of returning to cycle is also a function of maturity. Analytical expressions for cycle time distributions, growth rates, wave frequency and relative damping rates are derived for certain cases. A stable, diffusion-free numerical algorithm is used to work out some examples.     

A stochastic model of cell cycle desynchronization

A general branching process model is suggested to describe cell cycle desynchronization. Cell cycle phase times are modeled as random variables and a formula for the expected fraction of cells in S phase as a function of time is established. The model is compared to data from the literature and is also compared to previously suggested deterministic and stochastic models.