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PubMed2006

A diffusive SI model with Allee effect and application to FIV.

A minimal reaction-diffusion model for the spatiotemporal spread of an infectious disease is considered. The model is motivated by the Feline Immunodeficiency Virus (FIV) which causes AIDS in cat populations. Because the infected period is long compared with the lifespan, the model incorporates the host population growth. Two different types are considered: logistic growth and growth with a strong Allee effect. In the model with logistic growth, the introduced disease propagates in form of a travelling infection wave with a constant asymptotic rate of spread. In the model with Allee effect the spatiotemporal dynamics are more complicated and the disease has considerable impact on the host population spread. Most importantly, there are waves of extinction, which arise when the disease is introduced in the wake of the invading host population. These waves of extinction destabilize locally stable endemic coexistence states. Moreover, spatially restricted epidemics are possible as well as travelling infection pulses that correspond either to fatal epidemics with succeeding host population extinction or to epidemics with recovery of the host population. Generally, the Allee effect induces minimum viable population sizes and critical spatial lengths of the initial distribution. The local stability analysis yields bistability and the phenomenon of transient epidemics within the regime of disease-induced extinction. Sustained oscillations do not exist.

Algorithms↗

PubMed2006

Modeling relapse in infectious diseases.

An integro-differential equation is proposed to model a general relapse phenomenon in infectious diseases including herpes. The basic reproduction number R(0) for the model is identified and the threshold property of R(0) established. For the case of a constant relapse period (giving a delay differential equation), this is achieved by conducting a linear stability analysis of the model, and employing the Lyapunov-Razumikhin technique and monotone dynamical systems theory for global results. Numerical simulations, with parameters relevant for herpes, are presented to complement the theoretical results, and no evidence of sustained oscillatory solutions is found.

Computer Simulation↗

PubMed2004

Expression of HIV-1 accessory protein Vif is controlled uniquely to be low and optimal by proteasome degradation.

While the Vif protein of human immunodeficiency virus type 1 (HIV-1) is essential for viral replication in non-permissive cells, it is rapidly degraded intracellularly. We have previously suggested that the rapid turn-over of Vif is biologically meaningful to prevent detrimental effects of this protein at high expression levels. We now studied the mechanism of Vif degradation by examining the blocking effect of protease inhibitors in pulse/chase experiments and by monitoring the extent of Vif ubiquitination. The rapid turn-over of Vif could be blocked by proteasome inhibitors, and Vif was highly ubiquitinated. Cytoskeletal Vif was found to be more stable than soluble cytosolic Vif. These degradation characteristics of Vif were cell type-independent and observed in both non-permissive and permissive cells. Characterization of a series of vif deletion mutants showed that amino acids predicted to be important for formation of beta-strand structures (amino acid nos. 63-70 and 86-89) were critical for maintaining a normal expression level of Vif and for viral infectivity. Finally, we performed comparative stability analysis of the four HIV-1 accessory proteins. Vif was unique in its short half-life and in the magnitude of the degradation. Taken together, we conclude that the proteasome degradation of HIV-1 Vif is a virologically important process and crucial for the function of Vif.

Amino Acid Sequence↗

PubMed2004

Assigning vibrational polyads using relative equilibria: application to ozone.

We demonstrate how relative equilibria of a vibrating molecule, which are families of principal periodic orbits otherwise known as nonlinear normal modes, can be used to describe the global polyad structure of vibrational energy levels. The classical action integral n(E) computed along these orbits at different energies E corresponds to the polyad quantum number n so that the energy En of different relative equilibria describes the splitting of n-polyads. Further information on the internal polyad structure can be driven from the stability analysis of relative equilibria. We use the ozone molecule as a concrete example where n-polyads or "hyperpolyads" should be distinguished from the well-known polyads of the 1:1 stretching mode resonance; the stretching polyads are structural elements of hyperpolyads. We give dynamical interpretation of the relation between relative equilibria and n-polyads based on the normal form reduction in the limit of small vibrations near the equilibrium.

Models, Chemical↗

PubMed2002

Films driven by surface tension gradients.

Fingering instabilities are often observed at the contact line of drops of surfactant solutions spreading spontaneously on solid surfaces. It has been recognised recently that a usual linear stability analysis predicts stable behaviour in contrast to the observed instability. It now seems the instability arises from short-time transients, where the thickness of the film ahead of the main drop is a crucial parameter for amplification. We reconsidered previous experiments and performed new ones along these lines. The strengths and weaknesses of the available models were analysed.

Journal Article↗

PubMed1969

Oscillations in controlled biochemical systems.

Stability analysis of equations describing certain biochemical control mechanisms involving negative feedback suggests that limit cycle behavior might be possible if the control system involves a sufficient number of intermediate chemical steps. For the example considered in this paper, digital simulation of the non-linear control system illustrates that limit cycle behavior actually arises for a sixth-order system. On the other hand, the corresponding fourth- and fifth-order systems are asymptotically stable.

Computers↗

PubMed1980

Subunit interactions and the allosteric response in phosphorylase.

The contribution of intersubunit interactions to allosterically induced conformational changes in phosphorylase are considered. Phosphorylase a, Pa (phosphorylated at Ser-14), is significantly in the active (R) conformation, while phosphorylase b, Pb (nonphosphorylated), is predominantly in the inactive (T) conformation. The structure of glucose-inhibited (T) Pa has been determined at 2.5-A resolution and atomic coordinates have been measured. These data have been used to calculate the solvent accessible surface area at the subunit interface and map noncovalent interactions between protomers. The subunit contact involves only 6% of the Pa monomer surface, but withdraws an area of 4,600 A2 from solvent. The contact region is confined to the N-terminal (regulatory) domain of the subunit. Half of the residues involved are among the 70 N-terminal peptides. A total of approximately 100 atoms take part in polar or nonpolar contacts of less than 4.0 A with atoms of the symmetry-related monomer. The contact surface surrounds a central cavity at the core of the interface of sufficient volume to accommodate 150-180 solvent molecules. There are four intersubunit salt bridges. Two of these (Arg 10/Asp 32, Ser-14-P/Arg 43) are interactions between the N-terminus of one protomer with an alpha-helix loop segment near the N-terminus of the symmetry-related molecule. These two are relatively solvent accessible. The remainder (Arg 49/Glu 195, Arg 184/Asp 251) are nearer the interface core and are less accessible. The salt bridges at the N-terminus are surrounded by the polar and nonpolar contacts which may contribute to their stability. Analysis of the difference electron density between the isomorphous Pa and Pb crystal structures reveals that the N-terminal 17 residues of Pb are disordered. Pb thus lacks two intermolecular and one intersubunit (Ser-14-P/Arg 69) salt linkage present in Pa. The absence of these interactions in Pb is manifested in the difference in the free energy of T leads to R activation, which is 4 kcal more than that for Pa. Difference Fourier analysis of the T leads to R transition in substrate-activated crystals of Pa suggests that the 70 N-terminal residues undergo a concerted shift towards the molecular core; salt bridges are probably conserved in the transition. It is proposed that the N-terminus, when "activated" by phosphorylation (via a specific kinase) behaves as an intramolecular "effector" of the R state in phosphorylase and serves as the vehicle of homotropic cooperativity between subunits of the dimer.

Allosteric Regulation↗

PubMed1999

Vesicle deformation by an axial load: from elongated shapes to tethered vesicles.

A sufficiently large force acting on a single point of the fluid membrane of a flaccid phospholipid vesicle is known to cause the formation of a narrow bilayer tube (tether). We analyze this phenomenon by means of general mathematical methods allowing us to determine the shapes of strongly deformed vesicles including their stability. Starting from a free vesicle with an axisymmetric, prolate equilibrium shape, we consider an axial load that pulls (or pushes) the poles of the vesicle apart. Arranging the resulting shapes of strained vesicles in dependence of the axial deformation and of the area difference of monolayers, phase diagrams of stable shapes are presented comprising prolate shapes with or without equatorial mirror symmetry. For realistic values of membrane parameters, we study the force-extension relation of strained vesicles, and we demonstrate in detail how the initially elongated shape of an axially stretched vesicle transforms into a shape involving a membrane tether. This tethering transition may be continuous or discontinuous. If the free vesicle is mirror symmetric, the mirror symmetry is broken as the tether forms. The stability analysis of tethered shapes reveals that, for the considered vesicles, the stable shape is always asymmetric (polar), i.e., it involves only a single tether on one side of the main vesicle body. Although a bilayer tube formed from a closed vesicle is not an ideal cylinder, we show that, for most practical purposes, it is safe to assume a cylindrical geometry of tethers. This analysis is supplemented by the documentation of a prototype experiment supporting our theoretical predictions. It shows that the currently accepted model for the description of lipid-bilayer elasticity (generalized bilayer couple model) properly accounts for the tethering phenomenon.

Biophysical Phenomena↗

PubMed2002

From linear to nonlinear control means: a practical progression.

With the rapid advance of digital control hardware, it is time to take the simple but effective proportional-integral-derivative (PID) control technology to the next level of performance and robustness. For this purpose, a nonlinear PID and active disturbance rejection framework are introduced in this paper. It complements the existing theory in that (1) it actively and systematically explores the use of nonlinear control mechanisms for better performance, even for linear plants; (2) it represents a control strategy that is rather independent of mathematical models of the plants, thus achieving inherent robustness and reducing design complexity. Stability analysis, as well as software/hardware test results, are presented. It is evident that the proposed framework lends itself well in seeking innovative solutions to practical problems while maintaining the simplicity and the intuitiveness of the existing technology.

Algorithms↗

PubMed2002

An effective frequency domain approach to tuning non-PID controllers for high performance.

In this paper, a new tuning method is proposed for the design of non-PID controllers for complex processes to achieve high performance. Compared with the existing PID tuning methods, the proposed non-PID controller design method can yield better performance for a wide range of complex processes. A suitable objective transfer function for the closed-loop system is chosen according to process characteristics. The corresponding ideal controller is derived. Model reduction is applied to fit the ideal controller into a much simpler and realizable form. Stability analysis is given and simulation examples are provided to demonstrate the effectiveness of the proposed method.

Computer Simulation↗

PubMed2006

Nominal and robust stability regions of optimization-based PID controllers.

In recent decades, several optimization-based methods have been developed for the proportional-integral-derivative (PID) controller design, and the common feature of these methods is that the controller has only one adjustable parameter. To keep the closed-loop systems stable is an essential requirement for the optimization-based PID controllers. In almost all these methods, however, no exact stability region for the single adjustable parameter was sketched. In this paper, using the proposed analytical procedure based on the dual-locus diagram technique, explicit stability regions of the optimization-based PID controllers are derived for stable, integrating, and unstable processes with time delay in the nominal and perturbed cases, respectively. It is revealed that the proposed analytical procedure is effective for the determination of the nominal and robust stability regions and it offers simplicity and ease of mathematical calculations over other available stability analysis methods. The results in this paper provide some insight into the tuning of the optimization-based PID controllers.

Algorithms↗

PubMed2003

Stable drop shapes under disjoining pressure. III. Sharp contact lines.

The shape and stability of a drop with a sharp contact line are analyzed. The disjoining pressure is included. The choice made for the latter is special in that it admits a sharp contact angle as appropriate in this case. The stability analysis is considered for two scenarios, one where the contact line cannot be moved which mode turns out to be stable to small perturbations, and the other where it can be moved but the contact angle is fixed at the equilibrium value. This second case turns out to be unstable to small perturbations due to the curvature effect of the drop.

Journal Article↗

PubMed2003

Surface patterning via evaporation of ultrathin films containing nanoparticles.

The dewetting dynamics of ultrathin films containing potentially surface-active nanoparticles is considered in the presence of evaporation. Evolution equations for the film height and particle surface and bulk concentration are derived using a lubrication model coupled by a constitutive relation for the dependence of the viscosity on local particle concentration. A linear stability analysis and numerical simulations are used to determine how particle mass distribution depends on the various physical parameters such as equilibrium film separation distance, initial packing concentration, rate of evaporation, and particle surface activity. Our results show that when starting from an initially uniform distribution the particles become aligned into distinct "bands" in rectilinear geometry, or "rings" in cylindrical geometry. The functional dependence of the pattern spacing on relevant system parameters is studied and detailed herein.

Journal Article↗

PubMed1984

Cell membranes after malignant transformation. Part I: Dynamic stability at low surface tension.

A hydrodynamic cell model is introduced to analyze the dynamic stability of the cell membrane after malignant transformation. The cell membrane is considered as a two-dimensional charged interface between intra- and extra-cellular fluids. Employing a first order stability analysis, conditions are established under which growth of surface fluctuations can occur (leading to microvilli formation or cell division). The system is unstable if the total surface tension, i.e. the pure surface tension plus the free energy of formation of the double layers, is negative. Following that criterion, cell division is promoted in cancer cells; moreover, as cancer cells are more fluid than normal cells, they will divide more rapidly. The model also predicts that microvilli (protrusions of the cell membrane) will have a diameter of the order of the dominant wavelengths of perturbation (0.1 - 1 mu) which supports the view that such protrusions are consequences of amplified cell surface fluctuations.

Cell Line↗

PubMed1985

Stability of multienzyme systems with feedback regulation: a graph theoretical approach.

Stability of multienzyme systems with feedback regulation has been analyzed on the basis of the Lienard-Chipart criteria. The rules governing the topological graph construction for multienzyme systems have been developed. A theorem about correspondence of the graph constructed and coefficients of the characteristic polynomial of linearized kinetic equations is proved. The graph-theoretical stability analysis proposed is illustrated by a number of examples of multienzyme systems with feedback regulation.

Feedback↗

PubMed1988

Mathematical modelling of dynamics and control in metabolic networks: VI. Dynamic bifurcations in single biochemical control loops.

A dynamic stability analysis of an extended form of the Goodwin equations is presented. The Goodwin equations are extended to include Michaelis-Menten kinetics for the removal of the end-product. Inclusion of saturation kinetic behavior substantially increases the likelihood of dynamic instability in this model control loop. Oscillations are found for reaction chains of low order, as low as second order, and low degrees of co-operativity, as low as v = 2, simultaneously, thus indicating that dynamic instability in this system exists for physiologically realistic parameter values. The branches of bifurcated solutions are computed numerically and unstable Hopf bifurcations are found. Further, solution branches from stable Hopf bifurcation points are found to "fold back", i.e. have periodic limit points, producing situations where multiple stable limit cycles exist.

Kinetics↗

PubMed1989

Diffusion effects in calcium-regulated strain fields.

The Goodwin-Trainor equations for cellular morphogenesis, based on calcium ion regulation of the visco-elastic properties of the cellular cortex, are generalized to the situation where the concentration of calcium ions (free plus bound) is allowed to change locally. A stability analysis is presented which shows that the generalized equations are also stable against perturbations at low and high wave lengths and may, for certain parameter values, develop instabilities at intermediate wave lengths.

Animals↗

PubMed2000

Subcritical endemic steady states in mathematical models for animal infections with incomplete immunity.

Many classical mathematical models for animal infections assume that all infected animals transmit the infection at the same rate, all are equally susceptible, and the course of the infection is the same in all animals. However for some infections there is evidence that seropositives may still transmit the infection, albeit at a lower rate. Animals can also experience more than one episode of the infection although those who have already experienced it have a partial immune resistance. Animals who experience a second or subsequent period of infection may not necessarily exhibit clinical symptoms. The main example discussed is bovine respiratory syncytial virus (BRSV) amongst cattle. We consider simple models with vaccination and homogeneous and proportional mixing between seropositives and seronegatives. We derive an expression for the basic reproduction number, R(o), and perform an equilibrium and stability analysis. We find that it may be possible for there to be two endemic equilibria (one stable and one unstable) for R(o)<1 and in this case at R(o)=1 there is a backwards bifurcation of an unstable endemic equilibrium from the infection-free equilibrium. Then the implications for control strategies are considered. Finally applications to Aujesky's disease (pseudorabies virus) in pigs are discussed.

Animals↗