Global stability in a delayed partial differential equation describing cellular replication

Here we consider the dynamics of a population of cells that are capable of simultaneous proliferation and maturation. The equations describing the cellular population numbers are first order partial differential equations (transport equations) in which there is an explicit temporal retardation as well as a nonlocal dependence in the maturation variable due to cell replication. The behavior of this system may be considered along the characteristics, and a global stability condition is proved.     

Distribution of 1,25-dihydroxyvitamin D3 receptor immunoreactivity in the limbic system of the rat.

We used immunocytochemistry to obtain a complete cellular and subcellular mapping of the 1,25-dihydroxyvitamin D3 receptor protein (VDR) in the rat limbic system. We observed specific VDR immunostaining in the nucleus as well as in the perinuclear cytoplasm of neuronal cells. The limbic system consists of a variety of neuronal structures, and is known to have influence on memory, behavior, emotions and reproduction. In the hippocampal formation, we found strong nuclear staining as well as less distinguished cytoplasmic VDR staining in CA1, CA3 and CA4. The CA2 area showed a unique cytoplasmic predominance of VDR. The amygdala was found to exhibit specific patterns of VDR distribution in the various regions of the nucleus. We observed distinct differences of VDR localization within the limbic preoptic areas of the hypothalamus. Further parts of the brain we analyzed included the mammillary bodies, the indusium griseum and the cingulate cortex. The subcellular distribution of VDR in regions of the limbic system suggests a specific functional role of the receptor protein and indicates a role for calcitriol as a neuroactive steroid.     

Cellular automata and its advances to drug therapy for HIV infection

This paper gives an over view of the use of cellular automata (CA) model of drug therapy for HIV infection. Nonuniform CA is employed to simulate drug treatment of HIV infection, where each computational domain may contain different CA rules, in contrast to normal uniform CA models. Ordinary (or partial) differential equation models are insufficient to describe the two extreme time scales involved in HIV infection (days and decades), as well as the implicit spatial heterogeneity. Zorzenon and Coutinho [Phy Rev Lett, 16 (2001) 1] reported a cellular automata approach to simulate three-phase patterns of human immunodeficiency virus (HIV) infection consisting of primary response, clinical latency and onset of acquired immunodeficiency syndrome (AIDS). But here we present a related model, based on non-uniform CA to study the dynamics of drug therapy of HIV infection. The main aim in this model is to simulate the four phases (acute, chronic, drug treatment responds and onset of AIDS). The results shown here indicate that both simulations (with and without treatments) evolve to the relatively same steady state (characteristics of Wolfram's class II behavior). Different kinds of drug therapies can also be simulated in this model, which can be found useful for developing a proper drug therapy.     

Evaluation of dynamic behavior forecasting parameters in the process of transition rule induction of unidimensional cellular automata

The simulation of the dynamics of a cellular systems based on cellular automata (CA) can be computationally expensive. This is particularly true when such simulation is part of a procedure of rule induction to find suitable transition rules for the CA. Several efforts have been described in the literature to make this problem more treatable. This work presents a study about the efficiency of dynamic behavior forecasting parameters (DBFPs) used for the induction of transition rules of CA for a specific problem: the classification by the majority rule. A total of 8 DBFPs were analyzed for the 31 best-performing rules found in the literature. Some of these DBFPs were highly correlated each other, meaning they yield the same information. Also, most rules presented values of the DBFPs very close each other. An evolutionary algorithm, based on gene expression programming, was developed for finding transition rules according a given preestablished behavior. The simulation of the dynamic behavior of the CA is not used to evaluate candidate transition rules. Instead, the average values for the DBFPs were used as reference. Experiments were done using the DBFPs separately and together. In both cases, the best induced transition rules were not acceptable solutions for the desired behavior of the CA. We conclude that, although the DBFPs represent interesting aspects of the dynamic behavior of CAs, the transition rule induction process still requires the simulation of the dynamics and cannot rely only on the DBFPs.     

Carbonic anhydrase IX

Cell migration can be principally viewed as a chain of well-orchestrated morphological events that lead to dynamic reshaping of the cell body. However, behind the scene of such a “morphological theater” there are very complex, interrelated molecular and physiological processes that drive the cell movement. Among them, ion transport and pH regulation play a key role, with carbonic anhydrase IX (CA IX) emerging as one of the important “molecular actors.” CA IX is a highly active cell surface enzyme expressed in a broad range of solid tumors in response to hypoxia and explored as a clinically useful biomarker of hypoxia and as a therapeutic target. Its biological role is to protect tumor cells from hypoxia and acidosis in the tumor microenvironment. The study published recently by our group showed that CA IX actively contributes to cell migration and invasion. For the first time, we demonstrated CA IX accumulation in lamellipodia of migrating cells and its direct in situ interaction with bicarbonate transporters. Our findings indicate that tumor cells need CA IX not only as a pro-survival factor in hypoxia and acidosis, but also as a pro-migratory component of the cellular apparatus driving epithelial-mesenchymal transition.     

Hyperhomocysteinemia Induces Rat Memory Impairment Via Injuring Hippocampal CA3 Neurons and Downregulating cAMP Response Element-Binding Protein (CREB) Phosphorylation

Accumulated evidence demonstrated that an elevated plasma homocysteine level, hyperhomocysteinemia, induced cognitive impairment in animals, elderly and the patients with neurodegenerative diseases. To date, the underlying cellular and molecular mechanisms by which hyperhomocysteinemia induces cognitive impairment has not been clearly defined. The purpose of this study was to investigate the possible cellular and molecular mechanisms behind hyperhomocysteinemia signaling in rat memory impairment. The results from this study demonstrated that hyperhomocysteinemia induced neuronal damage and loss in hippocampal CA3 region and downregulated the cAMP response element-binding protein (CREB) phosphorylation. The findings of this study provide evidence that hyperhomocysteinemia induces rat memory impairment via injuring hippocampal CA3 neurons and downregulating CREB phosphorylation.

     

Mutation of albedo and growth response produces oscillations in a spatial Daisyworld

We extended a two-dimensional cellular automaton (CA) Daisyworld to include mutation of optimal growth temperature as well as mutation of albedo. Thus, the organisms (daisies) can adapt to prevailing environmental conditions or evolve to alter their environment. We find the resulting system oscillates with a period of hundreds of daisy generations. Weaker and less regular oscillations exist in previous daisyworld models, but they become much stronger and more regular here with mutation in the growth response. Despite the existence of a particular combination of mean albedo and optimum individual growth temperature which maximises growth, we find that this global state is unstable with respect to mutations which lower absolute growth rate, but increase marginal growth rate. The resulting system oscillates with a period that is found to decrease with increasing death rate, and to increase with increasing heat diffusion and heat capacity. We speculate that the origin of this oscillation is a Hopf bifurcation, previously predicted in a zero-dimensional system.     

Global Self-Regulation of the Cellular Metabolic Structure

Background

Different studies have shown that cellular enzymatic activities are able to self-organize spontaneously, forming a metabolic core of reactive processes that remain active under different growth conditions while the rest of the molecular catalytic reactions exhibit structural plasticity. This global cellular metabolic structure appears to be an intrinsic characteristic common to all cellular organisms. Recent work performed with dissipative metabolic networks has shown that the fundamental element for the spontaneous emergence of this global self-organized enzymatic structure could be the number of catalytic elements in the metabolic networks.

Methodology/Principal Findings

In order to investigate the factors that may affect the catalytic dynamics under a global metabolic structure characterized by the presence of metabolic cores we have studied different transitions in catalytic patterns belonging to a dissipative metabolic network. The data were analyzed using non-linear dynamics tools: power spectra, reconstructed attractors, long-term correlations, maximum Lyapunov exponent and Approximate Entropy; and we have found the emergence of self-regulation phenomena during the transitions in the metabolic activities.

Conclusions/Significance

The analysis has also shown that the chaotic numerical series analyzed correspond to the fractional Brownian motion and they exhibit long-term correlations and low Approximate Entropy indicating a high level of predictability and information during the self-regulation of the metabolic transitions. The results illustrate some aspects of the mechanisms behind the emergence of the metabolic self-regulation processes, which may constitute an important property of the global structure of the cellular metabolism.     

Variables aggregation in a time discrete linear model

In this work we extend approximate aggregation methods to deal with a very general linear time discrete model. Approximate aggregation consists in describing some features of the dynamics of a general system in terms of the dynamics of a reduced system governed by a few global variables. We present a time discrete model for a structured population (i.e., the population is subdivided in subpopulations) in which we can distinguish two processes of a general nature and whose corresponding time scales are very different from each other. We transform the general system to make the global variables appear and obtain the reduced system. These global variables are, for each subpopulation, a certain linear combination of the corresponding state variables. We show that, under quite general conditions, the asymptotic behavior of the reduced system can be known in terms of the corresponding behavior for the reduced system. The general method is applied to aggregate a multiregional Leslie model in which the demographic process is supposed to be fast with respect to migration.     

Cellular automata for contact ecoepidemic processes in predator–prey systems

Spatial ecoepidemic models, in which diseases affect interacting populations, are often explored through reaction-diffusion equations. However, cellular automata (CA) are a widely recognized tool for modelling spatial pattern formation that are broadly analagous to reaction diffusion equations, but provide greater flexibility in defining population dynamics. In this work we present a CA defined to mimic the prey–predators interactions while a pathogen is affecting, in turn, one population. We explore system equilibria, given different initial conditions and local interaction neighborhoods. Furthermore, in the various ecoepidemic systems considered we report the formation of waves and spirals: a key summary of how diseases may spread among individuals. Some inferences on the predators and infection eradication strategies are presented and supported by simulations results.     

Detection of cellular rhythms and global stability within interlocked feedback systems

In this paper, we show how to detect cellular rhythm and its global stability by extending the techniques from the recently developed theory of monotone systems. We establish theoretical results for globally asymptotic stability with consideration of delay by a discrete map. The relationship between positive, negative elements and delay in a general class of interlocked feedback networks can be understood in a system level. Moreover, the correspondence of attractors between a network and its reduced map is obtained and can be used to detect cellular rhythm, and further control the dynamics of the network. We show that global cellular rhythms can always be obtained, thereby enhancing robustness against perturbations of initial conditions and avoiding chaotic oscillations or complete abolishment of oscillations. In this paper, we focus on analyzing the circadian oscillator in Drosophila as an example to detect the occurrence of cellular rhythm and its global stability.     

The malignant social network: Cell-cell adhesion and communication in cancer stem cells

Tumors contain a vastly complicated cellular network that relies on local communication to execute malignant programs. The molecular cues that are involved in cell-cell adhesion orchestrate large-scale tumor behaviors such as proliferation and invasion. We have recently begun to appreciate that many tumors contain a high degree of cellular heterogeneity and are organized in a cellular hierarchy, with a cancer stem cell (CSC) population identified at the apex in multiple cancer types. CSCs reside in unique microenvironments or niches that are responsible for directing their behavior through cellular interactions between CSCs and stromal cells, generating a malignant social network. Identifying cell-cell adhesion mechanisms in this network has implications for the basic understanding of tumorigenesis and the development of more effective therapies. In this review, we will discuss our current understanding of cell-cell adhesion mechanisms used by CSCs and how these local interactions have global consequences for tumor biology.     

The malignant social network

Tumors contain a vastly complicated cellular network that relies on local communication to execute malignant programs. The molecular cues that are involved in cell-cell adhesion orchestrate large-scale tumor behaviors such as proliferation and invasion. We have recently begun to appreciate that many tumors contain a high degree of cellular heterogeneity and are organized in a cellular hierarchy, with a cancer stem cell (CSC) population identified at the apex in multiple cancer types. CSCs reside in unique microenvironments or niches that are responsible for directing their behavior through cellular interactions between CSCs and stromal cells, generating a malignant social network. Identifying cell-cell adhesion mechanisms in this network has implications for the basic understanding of tumorigenesis and the development of more effective therapies. In this review, we will discuss our current understanding of cell-cell adhesion mechanisms used by CSCs and how these local interactions have global consequences for tumor biology.     

Global analysis of dynamical decision-making models through local computation around the hidden saddle

Bistable dynamical switches are frequently encountered in mathematical modeling of biological systems because binary decisions are at the core of many cellular processes. Bistable switches present two stable steady-states, each of them corresponding to a distinct decision. In response to a transient signal, the system can flip back and forth between these two stable steady-states, switching between both decisions. Understanding which parameters and states affect this switch between stable states may shed light on the mechanisms underlying the decision-making process. Yet, answering such a question involves analyzing the global dynamical (i.e., transient) behavior of a nonlinear, possibly high dimensional model. In this paper, we show how a local analysis at a particular equilibrium point of bistable systems is highly relevant to understand the global properties of the switching system. The local analysis is performed at the saddle point, an often disregarded equilibrium point of bistable models but which is shown to be a key ruler of the decision-making process. Results are illustrated on three previously published models of biological switches: two models of apoptosis, the programmed cell death and one model of long-term potentiation, a phenomenon underlying synaptic plasticity.     

Leader cell positioning drives wound-directed collective migration in TGFβ-stimulated epithelial sheets

During wound healing and cancer metastasis, cells are frequently observed to migrate in collective groups. This mode of migration relies on the cooperative guidance of leader and follower cells throughout the collective group. The upstream determinants and molecular mechanisms behind such cellular guidance remain poorly understood. We use live-cell imaging to track the behavior of epithelial sheets of keratinocytes in response to transforming growth factor β (TGFβ), which stimulates collective migration primarily through extracellular regulated kinase 1/2 (Erk1/2) activation. TGFβ-treated sheets display a spatial pattern of Erk1/2 activation in which the highest levels of Erk1/2 activity are concentrated toward the leading edge of a sheet. We show that Erk1/2 activity is modulated by cellular density and that this functional relationship drives the formation of patterns of Erk1/2 activity throughout sheets. In addition, we determine that a spatially constrained pattern of Erk1/2 activity results in collective migration that is primarily wound directed. Conversely, global elevation of Erk1/2 throughout sheets leads to stochastically directed collective migration throughout sheets. Our study highlights how the spatial patterning of leader cells (cells with elevated Erk1/2 activity) can influence the guidance of a collective group of cells during wound healing.     

Dickkopf-1 (DKK-1) interrupts FAK/PI3K/mTOR pathway by interaction of carbonic anhydrase IX (CA9) in tumorigenesis

Recently, we found that carbonic anhydrase IX (CA9) modulates tumor-associated cell migration and invasion, and then identified dickkopf-1 (DKK-1) as a novel CA9-interacting protein. In this study, we have determined the binding regions that are required for interaction between CA9 and DKK-1 through in vitro and in vivo. The N-terminal domain of CA9 is participated to interact with the Val(60)-Tyr(168) site of DKK-1. We also observed that DKK-1 inhibits endothelial cell angiogenesis of CA9 in tumorigenesis. Furthermore, induction of CA9-mediated mTOR phosphorylation and angiogenesis was significantly inhibited by over-expression of DKK-1. Taken together, these findings identify DKK-1 as a potential factor in the regulation of CA9 cellular homeostasis and also suggest a new possible role for DKK1-1 in tumorigenesis.     

Cellular Automata Modeling of Stem‐Cell‐Driven Development of Tissue in the Nervous System

Mathematical and computational modeling enables biologists to integrate data from observations and experiments into a theoretical framework. In this review, we describe how developmental processes associated with stem‐cell‐driven growth of tissue in both the embryonic and adult nervous system can be modeled using cellular automata (CA). A cellular automaton is defined by its discrete nature in time, space, and state. The discrete space is represented by a uniform grid or lattice containing agents that interact with other agents within their local neighborhood. This possibility of local interactions of agents makes the cellular automata approach particularly well suited for studying through modeling how complex patterns at the tissue level emerge from fundamental developmental processes (such as proliferation, migration, differentiation, and death) at the single‐cell level. As part of this review, we provide a primer for how to define biologically inspired rules governing these processes so that they can be implemented into a CA model. We then demonstrate the power of the CA approach by presenting simulations (in the form of figures and movies) based on building models of three developmental systems: the formation of the enteric nervous system through invasion by neural crest cells; the growth of normal and tumorous neurospheres induced by proliferation of adult neural stem/progenitor cells; and the neural fate specification through lateral inhibition of embryonic stem cells in the neurogenic region of Drosophila.     

Switching control of an essential gene for reprogramming of cellular phenotypes in Escherichia coli

Biological systems designs require various dynamic controllers capable of modulating cellular phenotypes to adapt to changing environments. Cellular phenotypes are simultaneously affected by combinations of multiple genes that are controlled by global regulators. However, it is difficult to intentionally control the expression of these global regulators dynamically because they are essential for cell survival and are involved in regulatory networks clustered in operons. Here, we designed a platform that allows dynamic modulation of the expression of an essential gene. Using this system, comprising of on/off switches that respond to an extracellular stimulus, we successfully demonstrated the switching control of the expression of fusA encoding elongation factor G (EF-G). An additional control module in this system that responds to changed external signals was shown to provide the capacity to "switch gears" and reprogram cellular phenotypes with desired timing.