The effects of cell density and metabolite flux on cellular dynamics

Summary Density-dependent regulation of cell growth in tissue culture is a well-known phenomenon but the mechanism of regulation remains obscure. Here we explore the effects of cell density and metabolite flux on the collective dynamics of a cell population. The intracellular dynamics are modelled by positive feedback kinetic mechanisms of the kind known to apply to yeast cells. Several experimental observations related to glycolytic oscillations are predicted and it is suggested that the general conclusions may be applicable in a broader context.     

Instabilities,bifurcations and fluctuations in chemical systems : L. E. Reichl and W. C. Schieve University of Texas Press, 1982, xi+427 pp, $35.00

The occupancy principle and the mean-transit-time theorem are derived for the passage of a tracer through a system that can be described by a general pool model. It is proved, using matrix theory, that if (and only if) tracer entering the system labels equally all tracee fluxes into the system, then the integral of the tracer concentration is the same in all the pools. It is also proved that if, in addition, all flow out of the system is through the observation point, the first moment of the tracer concentration at the observation point can be used to calculate the total amount of trace in the system. The necessity of this condition is analyzed. Examples are given of models in which the occupancy principle and the mean-transit-time theorem hold or do not hold.     

A Hybrid Model of Tumor–Stromal Interactions in Breast Cancer

Ductal carcinoma in situ (DCIS) is an early stage noninvasive breast cancer that originates in the epithelial lining of the milk ducts, but it can evolve into comedo DCIS and ultimately, into the most common type of breast cancer, invasive ductal carcinoma. Understanding the progression and how to effectively intervene in it presents a major scientific challenge. The extracellular matrix (ECM) surrounding a duct contains several types of cells and several types of growth factors that are known to individually affect tumor growth, but at present the complex biochemical and mechanical interactions of these stromal cells and growth factors with tumor cells is poorly understood. Here we develop a mathematical model that incorporates the cross-talk between stromal and tumor cells, which can predict how perturbations of the local biochemical and mechanical state influence tumor evolution. We focus on the EGF and TGF-β signaling pathways and show how up- or down-regulation of components in these pathways affects cell growth and proliferation. We then study a hybrid model for the interaction of cells with the tumor microenvironment (TME), in which epithelial cells (ECs) are modeled individually while the ECM is treated as a continuum, and show how these interactions affect the early development of tumors. Finally, we incorporate breakdown of the epithelium into the model and predict the early stages of tumor invasion into the stroma. Our results shed light on the interactions between growth factors, mechanical properties of the ECM, and feedback signaling loops between stromal and tumor cells, and suggest how epigenetic changes in transformed cells affect tumor progression.     

A stochastic analysis of actin polymerization in the presence of twinfilin and gelsolin

We develop an efficient stochastic simulation algorithm for analyzing actin filament growth and decay in the presence of various actin-binding proteins. The evolution of nucleotide profiles of filaments can be tracked and the resulting feedback to actin-binding proteins is incorporated. The computational efficiency of the new method enables us to focus on experimentally realistic problems, and as one example we use it to analyze the experimental data of Helfer et al. [(2006). Mammalian twinfilin sequesters ADP-G-actin and caps filament barbed ends: implications in motility. EMBO J. 25, 1184-1195] on the capping and G-actin sequestering activity of twinfilin. We show that the binding specificity of twinfilin for ADP-G-actin is crucial for the observed biphasic evolution of the filament length distribution in the presence of twinfilin, and we demonstrate that twinfilin can be an essential part of the molecular machinery for regulating filament lengths after a short burst of polymerization. Significantly, our simulations indicate that the pyrenyl-actin fluorescence experiments would fail to report the emergence of large filaments under certain experimental conditions.     

The variety of cytosolic calcium responses and possible roles of PLC and PKC

A model of ligand-induced intracellular calcium (Ca2+) responses incorporating phospholipase C (PLC) and protein kinase C (PKC) is developed for the purpose of understanding the mechanisms underlying the observed temporal patterns of intracellular calcium (Ca(i)2+) under sustained agonist stimulation. Some studies have suggested that inhibition of ligand receptors and PLC by PKC could generate sinusoidal Ca2+ oscillations, while PKC-independent Ca2+-induced Ca2+ release (CICR) via IP(3)-gated Ca2+ channels on the endoplasmic reticulum (ER) is believed to be responsible for baseline spiking. However, some evidence also indicates that baseline spiking can be observed under high-PKC activity, or under low-PKC activity with low agonist stimulus, as well. Insight into the basis of these observations regarding the role of PKC in Ca(i)2+ response patterns can be gained by developing and analyzing a mathematical model of Ca(i)2+ responses. We do this herein and find that (1) interaction of CICR and the sarcoplasmic/endoplasmic reticulum calcium ATPase (SERCA) pump is enough to generate both types of Ca(i)2+ oscillations, (2) there exist four possible Ca(i)2+ response patterns under sustained agonist stimulus: a sub-threshold response (SR), baseline spiking, sinusoidal oscillations (SO) and transient with plateau, and (3) the IP(3) concentration, which is controlled by the strength of the interaction between PKC and PLC, can be used to predict the Ca(i)2+ response patterns. From this analysis we conclude that the different patterns of Ca(i)2+ oscillations can be understood as a generic consequence of the interactions between CICR via the IP(3)-gated Ca(2+) channels in response to changes in the level of IP(3), and re-uptake into the ER/SR via the SERCA pump. PKC, in conjunction with PLC, can act as a switch between different Ca(i)2+ response patterns by modulating the cytosolic IP(3) level, which determines the Ca(i)2+ patterns.     

The role of the microenvironment in tumor growth and invasion

Mathematical modeling and computational analysis are essential for understanding the dynamics of the complex gene networks that control normal development and homeostasis, and can help to understand how circumvention of that control leads to abnormal outcomes such as cancer. Our objectives here are to discuss the different mechanisms by which the local biochemical and mechanical microenvironment, which is comprised of various signaling molecules, cell types and the extracellular matrix (ECM), affects the progression of potentially-cancerous cells, and to present new results on two aspects of these effects. We first deal with the major processes involved in the progression from a normal cell to a cancerous cell at a level accessible to a general scientific readership, and we then outline a number of mathematical and computational issues that arise in cancer modeling. In Section 2 we present results from a model that deals with the effects of the mechanical properties of the environment on tumor growth, and in Section 3 we report results from a model of the signaling pathways and the tumor microenvironment (TME), and how their interactions affect the development of breast cancer. The results emphasize anew the complexities of the interactions within the TME and their effect on tumor growth, and show that tumor progression is not solely determined by the presence of a clone of mutated immortal cells, but rather that it can be ‘community-controlled’.     

The BMP-binding protein Crossveinless 2 is a short-range, concentration-dependent, biphasic modulator of BMP signaling in Drosophila

In Drosophila, the secreted BMP-binding protein Short gastrulation (Sog) inhibits signaling by sequestering BMPs from receptors, but enhances signaling by transporting BMPs through tissues. We show that Crossveinless 2 (Cv-2) is also a secreted BMP-binding protein that enhances or inhibits BMP signaling. Unlike Sog, however, Cv-2 does not promote signaling by transporting BMPs. Rather, Cv-2 binds cell surfaces and heparan sulfate proteoglygans and acts over a short range. Cv-2 binds the type I BMP receptor Thickveins (Tkv), and we demonstrate how the exchange of BMPs between Cv-2 and receptor can produce the observed biphasic response to Cv-2 concentration, where low levels promote and high levels inhibit signaling. Importantly, we show also how the concentration or type of BMP present can determine whether Cv-2 promotes or inhibits signaling. We also find that Cv-2 expression is controlled by BMP signaling, and these combined properties enable Cv-2 to exquisitely tune BMP signaling.     

Simplification and analysis of models of calcium dynamics based on IP3-sensitive calcium channel kinetics.

We study the models for calcium (Ca) dynamics developed in earlier studies, in each of which the key component is the kinetics of intracellular inositol-1,4,5-trisphosphate-sensitive Ca channels. After rapidly equilibrating steps are eliminated, the channel kinetics in these models are represented by a single differential equation that is linear in the state of the channel. In the reduced kinetic model, the graph of the steady-state fraction of conducting channels as a function of log10(Ca) is a bell-shaped curve. Dynamically, a step increase in inositol-1,4,5-trisphosphate induces an incremental increase in the fraction of conducting channels, whereas a step increase in Ca can either potentiate or inhibit channel activation, depending on the Ca level before and after the increase. The relationships among these models are discussed, and experimental tests to distinguish between them are given. Under certain conditions the models for intracellular calcium dynamics are reduced to the singular perturbed form epsilon dx/d tau = f(x, y, p), dy/d tau = g(x, y, p). Phase-plane analysis is applied to a generic form of these simplified models to show how different types of Ca response, such as excitability, oscillations, and a sustained elevation of Ca, can arise. The generic model can also be used to study frequency encoding of hormonal stimuli, to determine the conditions for stable traveling Ca waves, and to understand the effect of channel properties on the wave speed.