Stability properties of elementary dynamic models of membrane transport

Living cells are characterized by their capacity to maintain a stable steady state. For instance, cells are able to conserve their volume, internal ionic composition and electrical potential difference across the plasma membrane within values compatible with the overall cell functions. The dynamics of these cellular variables is described by complex integrated models of membrane transport. Some clues for the understanding of the processes involved in global cellular homeostasis may be obtained by the study of the local stability properties of some partial cellular processes. As an example of this approach, I perform, in this study, the neighborhood stability analysis of some elementary integrated models of membrane transport. In essence, the models describe the rate of change of the intracellular concentration of a ligand subject to active and passive transport across the plasma membrane of an ideal cell. The ligand can be ionic or nonionic, and it can affect the cell volume or the plasma membrane potential. The fundamental finding of this study is that, within the physiological range, the steady states are asymptotically stable. This basic property is a necessary consequence of the general forms of the expressions employed to describe the active and passive fluxes of the transported ligand.     

A 3-D model of tumor progression based on complex automata driven by particle dynamics

The dynamics of a growing tumor involving mechanical remodeling of healthy tissue and vasculature is neglected in most of the existing tumor models. This is due to the lack of efficient computational framework allowing for simulation of mechanical interactions. Meanwhile, just these interactions trigger critical changes in tumor growth dynamics and are responsible for its volumetric and directional progression. We describe here a novel 3-D model of tumor growth, which combines particle dynamics with cellular automata concept. The particles represent both tissue cells and fragments of the vascular network. They interact with their closest neighbors via semi-harmonic central forces simulating mechanical resistance of the cell walls. The particle dynamics is governed by both the Newtonian laws of motion and the cellular automata rules. These rules can represent cell life-cycle and other biological interactions involving smaller spatio-temporal scales. We show that our complex automata, particle based model can reproduce realistic 3-D dynamics of the entire system consisting of the tumor, normal tissue cells, blood vessels and blood flow. It can explain phenomena such as the inward cell motion in avascular tumor, stabilization of tumor growth by the external pressure, tumor vascularization due to the process of angiogenesis, trapping of healthy cells by invading tumor, and influence of external (boundary) conditions on the direction of tumor progression. We conclude that the particle model can serve as a general framework for designing advanced multiscale models of tumor dynamics and it is very competitive to the modeling approaches presented before.     

Simple Carrier Kinetics in Complex Membrane Transporters

The four-state simple carrier model (SCM) has been employed to describe facilitative transport of ligands across biological membranes. Two basic mechanisms have been invoked to account for carrier-mediated ligand translocation: (i) binding to a mobile carrier, and (ii) displacement determined by conformational changes of an integral protein. While translatory carriers may be accurately represented by a four-state diagram, it is unlikely that the transport process mediated by a complex membrane protein can be strictly described by the elementary SCM. The purpose of this article is to test whether facilitative transporters with a more complex kinetic design than the SCM can exhibit macroscopic kinetic properties indistinguishable from it. For this, I studied a ``general carrier model' (GCM), and evaluated whether the relevant kinetic parameters are subject to the same basic restrictions as in the SCM. The fundamental finding is that there is a general kinetic design embodied with SCM-like properties, that can be shared by many transporters. In particular, the classical SCM is shown here to represent a particular case of the GCM. A main conclusion of this work is therefore that the finding of a macroscopic SCM-like kinetic behavior for a particular process of facilitative transport does not represent a sufficient argument in favor of a particular type of mechanism, like the typical one involving a two-conformational single-site carrier. Received: 9 February 1998/Revised: 19 June 1998     

Relationships between the rapid axonal transport of newly synthesized proteins and membranous organelles

Rapid axonal transport is generally viewed as being exactly analogous to the secretory process in nonneuronal cells. The cell biology of rapid axonal transport is reviewed, the central concern being to explore those aspects that do not fit into the general secretory model and which may thus represent specific neuronal adaptations. Particular attention is paid to the relationship between the transport of newly synthesized proteins and of the membranous organelles that act as carriers. Sites in the transport sequence at which the behavior of axonal transport may differ from the secretory model are at the initiation of axonal transport at the trans-side of the Golgi apparatus, within the axon where molecules are deposited from the moving phase to a stationary phase, and at nerve terminals or axonal lesions where transport reversal takes place.     

Effects of stiffness and volume on the transit time of an erythrocyte through a slit

By using a fully coupled fluid–cell interaction model, we numerically simulate the dynamic process of a red blood cell passing through a slit driven by an incoming flow. The model is achieved by combining a multiscale model of the composite cell membrane with a boundary element fluid dynamics model based on the Stokes flow assumption. Our concentration is on the correlation between the transit time (the time it takes to finish the whole translocation process) and different conditions (flow speed, cell orientation, cell stiffness, cell volume, etc.) that are involved. According to the numerical prediction (with some exceptions), the transit time rises as the cell is stiffened. It is also highly sensitive to volume increase inside the cell. In general, even slightly swollen cells (i.e., the internal volume is increased while the surface area of the cell kept unchanged) travel dramatically slower through the slit. For these cells, there is also an increased chance of blockage.     

A data-assimilation approach to predict population dynamics during epithelial-mesenchymal transition

Epithelial-mesenchymal transition (EMT) is a biological process that plays a central role in embryonic development, tissue regeneration, and cancer metastasis. Transforming growth factor-β (TGFβ) is a potent inducer of this cellular transition, comprising transitions from an epithelial state to partial or hybrid EMT state(s), to a mesenchymal state. Recent experimental studies have shown that, within a population of epithelial cells, heterogeneous phenotypical profiles arise in response to different time- and TGFβ dose-dependent stimuli. This offers a challenge for computational models, as most model parameters are generally obtained to represent typical cell responses, not necessarily specific responses nor to capture population variability. In this study, we applied a data-assimilation approach that combines limited noisy observations with predictions from a computational model, paired with parameter estimation. Synthetic experiments mimic the biological heterogeneity in cell states that is observed in epithelial cell populations by generating a large population of model parameter sets. Analysis of the parameters for virtual epithelial cells with biologically significant characteristics (e.g., EMT prone or resistant) illustrates that these sub-populations have identifiable critical model parameters. We perform a series of in silico experiments in which a forecasting system reconstructs the EMT dynamics of each virtual cell within a heterogeneous population exposed to time-dependent exogenous TGFβ dose and either an EMT-suppressing or EMT-promoting perturbation. We find that estimating population-specific critical parameters significantly improved the prediction accuracy of cell responses. Thus, with appropriate protocol design, we demonstrate that a data-assimilation approach successfully reconstructs and predicts the dynamics of a heterogeneous virtual epithelial cell population in the presence of physiological model error and parameter uncertainty.     

Feed-Forward versus Feedback Inhibition in a Basic Olfactory Circuit

Inhibitory interneurons play critical roles in shaping the firing patterns of principal neurons in many brain systems. Despite difference in the anatomy or functions of neuronal circuits containing inhibition, two basic motifs repeatedly emerge: feed-forward and feedback. In the locust, it was proposed that a subset of lateral horn interneurons (LHNs), provide feed-forward inhibition onto Kenyon cells (KCs) to maintain their sparse firing—a property critical for olfactory learning and memory. But recently it was established that a single inhibitory cell, the giant GABAergic neuron (GGN), is the main and perhaps sole source of inhibition in the mushroom body, and that inhibition from this cell is mediated by a feedback (FB) loop including KCs and the GGN. To clarify basic differences in the effects of feedback vs. feed-forward inhibition in circuit dynamics we here use a model of the locust olfactory system. We found both inhibitory motifs were able to maintain sparse KCs responses and provide optimal odor discrimination. However, we further found that only FB inhibition could create a phase response consistent with data recorded in vivo. These findings describe general rules for feed-forward versus feedback inhibition and suggest GGN is potentially capable of providing the primary source of inhibition to the KCs. A better understanding of how inhibitory motifs impact post-synaptic neuronal activity could be used to reveal unknown inhibitory structures within biological networks.     

Chemotaxis in densely populated tissue determines germinal center anatomy and cell motility: a new paradigm for the development of complex tissues

Germinal centers (GCs) are complex dynamic structures that form within lymph nodes as an essential process in the humoral immune response. They represent a paradigm for studying the regulation of cell movement in the development of complex anatomical structures. We have developed a simulation of a modified cyclic re-entry model of GC dynamics which successfully employs chemotaxis to recapitulate the anatomy of the primary follicle and the development of a mature GC, including correctly structured mantle, dark and light zones. We then show that correct single cell movement dynamics (including persistent random walk and inter-zonal crossing) arise from this simulation as purely emergent properties. The major insight of our study is that chemotaxis can only achieve this when constrained by the known biological properties that cells are incompressible, exist in a densely packed environment, and must therefore compete for space. It is this interplay of chemotaxis and competition for limited space that generates all the complex and biologically accurate behaviors described here. Thus, from a single simple mechanism that is well documented in the biological literature, we can explain both higher level structure and single cell movement behaviors. To our knowledge this is the first GC model that is able to recapitulate both correctly detailed anatomy and single cell movement. This mechanism may have wide application for modeling other biological systems where cells undergo complex patterns of movement to produce defined anatomical structures with sharp tissue boundaries.     

Cinnamoyl nitrogen mustard derivatives of pyrazole analogues of tallimustine modified at the amidino moiety: design, synthesis, molecular modeling and antitumor activity studies

The design, synthesis and in vitro activities of a series of cinnamoyl nitrogen mustard pyrazole analogues of tallimustine 8-13, in which the amidino moiety has been replaced by moieties of different physico-chemical features are described, and the structure-activity relationships are discussed. In spite of the relevance of these modifications on the amidino moiety, these derivatives showed significant growth inhibitory activity against mouse leukemia L1210 cells. A selected series of compounds have been evaluated for their sequence selective alkylating properties and cytotoxicity against human K562 leukemia cells. Therefore, the presence of the amidino moiety, and in general of a basic moiety, is not an absolute requirement for biological activity. Our preliminary results indicated that the compounds of this series have a pattern of alkylation similar to that of tallimustine, but they seem to be less reactive overall in alkylating naked DNA.     

An abstract cell model that describes the self-organization of cell function in living systems

The principal aim of systems biology is to search for general principles that govern living systems. We develop an abstract dynamic model of a cell, rooted in Mesarovi? and Takahara's general systems theory. In this conceptual framework the function of the cell is delineated by the dynamic processes it can realize. We abstract basic cellular processes, i.e., metabolism, signalling, gene expression, into a mapping and consider cell functions, i.e., cell differentiation, proliferation, etc. as processes that determine the basic cellular processes that realize a particular cell function. We then postulate the existence of a 'coordination principle' that determines cell function. These ideas are condensed into a theorem: If basic cellular processes for the control and regulation of cell functions are present, then the coordination of cell functions is realized autonomously from within the system. Inspired by Robert Rosen's notion of closure to efficient causation, introduced as a necessary condition for a natural system to be an organism, we show that for a mathematical model of a self-organizing cell the associated category must be cartesian closed. Although the semantics of our cell model differ from Rosen's (M,R)-systems, the proof of our theorem supports (in parts) Rosen's argument that living cells have non-simulable properties. Whereas models that form cartesian closed categories can capture self-organization (which is a, if not the, fundamental property of living systems), conventional computer simulations of these models (such as virtual cells) cannot. Simulations can mimic living systems, but they are not like living systems.     

New physical concepts for cell amoeboid motion.

Amoeboid motion of cells is an essential mechanism in the function of many biological organisms (e.g., the regiment of scavenger cells in the immune defense system of animals). This process involves rapid chemical polymerization (with numerous protein constituents) to create a musclelike contractile network that advances the cell over the surface. Significant progress has been made in the biology and biochemistry of motile cells, but the physical dynamics of cell spreading and contraction are not well understood. The reason is that general approaches are formulated from complex mass, momentum, and chemical reaction equations for multiphase-multicomponent flow with the nontrivial difficulty of moving boundaries. However, there are strong clues to the dynamics that allow bold steps to be taken in simplifying the physics of motion. First, amoeboid cells often exhibit exceptional kinematics, i.e., steady advance and retraction of local fixed-shape patterns. Second, recent evidence has shown that cell projections "grow" by polymerization along the advancing boundary of the cell. Together, these characteristics represent a local growth process pinned to the interfacial contour of a contractile network. As such, the moving boundary becomes tractable, but subtle features of the motion lead to specific requirements for the chemical nature of the boundary polymerization process. To demonstrate these features, simple examples for limiting conditions of substrate interaction (i.e., "strong" and "weak" adhesion) are compared with data from experimental studies of yeast particle engulfment by blood granulocytes and actin network dynamics in fishscale keratocytes.     

Modelling biological processes using workflow and Petri Net models

MOTIVATION: Biological processes can be considered at many levels of detail, ranging from atomic mechanism to general processes such as cell division, cell adhesion or cell invasion. The experimental study of protein function and gene regulation typically provides information at many levels. The representation of hierarchical process knowledge in biology is therefore a major challenge for bioinformatics. To represent high-level processes in the context of their component functions, we have developed a graphical knowledge model for biological processes that supports methods for qualitative reasoning. RESULTS: We assessed eleven diverse models that were developed in the fields of software engineering, business, and biology, to evaluate their suitability for representing and simulating biological processes. Based on this assessment, we combined the best aspects of two models: Workflow/Petri Net and a biological concept model. The Workflow model can represent nesting and ordering of processes, the structural components that participate in the processes, and the roles that they play. It also maps to Petri Nets, which allow verification of formal properties and qualitative simulation. The biological concept model, TAMBIS, provides a framework for describing biological entities that can be mapped to the workflow model. We tested our model by representing malaria parasites invading host erythrocytes, and composed queries, in five general classes, to discover relationships among processes and structural components. We used reachability analysis to answer queries about the dynamic aspects of the model. AVAILABILITY: The model is available at http://smi.stanford.edu/projects/helix/pubs/process-model/.     

The art of cellular communication: tunneling nanotubes bridge the divide

The ability of cells to receive, process, and respond to information is essential for a variety of biological processes. This is true for the simplest single cell entity as it is for the highly specialized cells of multicellular organisms. In the latter, most cells do not exist as independent units, but are organized into specialized tissues. Within these functional assemblies, cells communicate with each other in different ways to coordinate physiological processes. Recently, a new type of cell-to-cell communication was discovered, based on de novo formation of membranous nanotubes between cells. These F-actin-rich structures, referred to as tunneling nanotubes (TNT), were shown to mediate membrane continuity between connected cells and facilitate the intercellular transport of various cellular components. The subsequent identification of TNT-like structures in numerous cell types revealed some structural diversity. At the same time it emerged that the direct transfer of cargo between cells is a common functional property, suggesting a general role of TNT-like structures in selective, long-range cell-to-cell communication. Due to the growing number of documented thin and long cell protrusions in tissue implicated in cell-to-cell signaling, it is intriguing to speculate that TNT-like structures also exist in vivo and participate in important physiological processes.     

A population balance model describing the cell cycle dynamics of myeloma cell cultivation

A multi-staged population balance model is proposed to describe the cell cycle dynamics of myeloma cell cultivation. In this model, the cell cycle is divided into three stages, i.e., G1, S, and G2M phases. Both DNA content and cell volume are used to differentiate each cell from other cells of the population. The probabilities of transition from G1 to S and division of G2M are assumed to be dependent on cell volume, and transition probability from S to G2M is determined by DNA content. The model can be used to simulate the dynamics of DNA content and cell volume distributions, phase fractions, and substrate and byproduct concentrations, as well as cell densities. Measurements from myeloma cell cultivations, especially the FACS data with respect to DNA distribution and cell fractions in different stages, are employed for model validation.     

Analysis of cell growth kinetics and substrate diffusion in a polymer scaffold.

The cultivation of cartilage cells (chondrocytes) in polymer scaffolds leads to implants that may potentially be used to repair damaged joint cartilage or for reconstructive surgery. For this technique to be medically applicable, the physical parameters that govern cell growth in a polymer scaffold must be understood. This understanding of cell behavior under in vitro conditions, where diffusion is the primary mode of transport of nutrients, may aid in the scale-up of the cartilage generation process. A mathematical model of chondrocyte generation and nutrient consumption is developed here to analyze the behavior of cell growth in a biodegradable polymer matrix for a series of different thickness polymers. Recent literature has implied that the diffusion of nutrients is a major factor that limits cell growth (Freed et al., 1994). In the present paper, a mathematical model is developed to directly relate the effects of increasing cell mass in the polymer matrix on the transport of nutrients. Reaction and diffusion of nutrients in the cell-polymer system are described using the fundamental species continuity equations and the volume averaging method. The volume averaging method is utilized to derive a single averaged nutrient continuity equation that includes the effective transport properties. This approach allows for the derivation of effective diffusion and rate coefficients as functions of the cell volume fraction. The cell volume fraction as a function of time is determined by solution of a material balance on cell mass. Growth functions including the Moser, a modified Contois, and an nth-order heterogeneous growth kinetic model are evaluated through a parameter analysis, and the results are compared to experimental data found in the literature. The results indicate that cellular functions in conjunction with mass transfer processes can account partially for the general trends in the cell growth behavior for various thickness polymers. The Contois growth function appeared to describe the data more accurately in terms of the lag period at early times and the long time limits. However, all kinetic growth functions required variations in the kinetic parameters to fully describe the effects of polymer thickness. This result implies that restricted diffusion of nutrients is not the sole factor limiting cell growth when the thickness of the polymer is changed. Therefore, further experimental data and model improvements are needed to accurately describe the cell growth process.     

Headgroup behaviour of an uncharged complex glycolipid

A globoside spin labelled on the terminal sugar residue has been synthesized, and employed in model membranes to study headgroup behaviour of complex uncharged glycolipids. The labelled headgroup demonstrated a high degree of motional freedom limited to the aqueous region of the interface between lipid bilayer and surrounding medium. This observation was unaltered by the presence of a dense, tightly-bound surface layer of peripheral proteins or polysaccharide—which might be expected to reproduce conditions present at a cell surface. Headgroup dynamics were only very modestly correlated with the physical state (i.e., fluidity) of the membrane itself. In spite of the absence of charged sugar residues in globoside, the aspects of its headgroup behaviour monitored here we found to be similar to those of oligosaccharide chains on gangliosides and several sialic acid-rich glycoproteins.     

A case of apparent active transport. I

When an experimenter determines the “internal concentration” of a substance in a cell (or cell suspension) it is in general the average concentration (quantity of substance divided by cell volume or volume of cell water) which is measured. When this concentration is less than that in the ambient medium but there is either no flow into the cell or flow from the cell into the medium, then (under the usually tacit assumption of spatial uniformity in the cell) the possibility of active transport is considered. The possibility that lack of spatial uniformity could lead to apparent active transport was early proposed by A. Bierman and later examined quantitatively by N. Rashevsky for a special case. In this paper spherical cells are treated but under quite general conditions regarding the metabolic aspects of the problem. It is shown that apparent active transport can result for a metabolite which is a reactant in one set of reactions and a product in another provided the sites of these sets of reactions are spatially separated in the cell.