Sensitivity analysis of a nonlinear lumped parameter model of HIV infection dynamics

A formal sensitivity analysis is performed on a delay differential equation model for the viral dynamics of an in vivo HIV infection during protease inhibitor therapy. We present results of both a differential analysis as well as a principle component based analysis and provide evidence that suggests the exact times at which specific parameters have the most influence over the solution. We offer insight into the pairwise mathematical relationships between the productively infected T-cell death rate δ, the viral plasma clearance rate c, and the time delay τ between infection and viral production as they relate to the viral dynamics. The results support the claim that the presence of a nonzero delay has a major impact on the model dynamics. Lastly, we comment upon the inadequacies of an alternative principle component based analysis.     

A theoretical model for water flux through the artery wall

A two layer model for water flux through the artery is studied using a mathematical model based on the theory for the consolidation of water saturated soils. The matrix is considered to be constituted by two layers with different permeabilities and different elastic constants and the two systems of equations are coupled with the condition of continuity of pressure, total stress, solid displacement and fluid seepage velocity at the interface. The luminal pressure is considered to be harmonic in time. Exact solutions are obtained for displacements and pressures in both the layers. For large consolidation times, large pressure gradients are found to exist near the boundaries and at the interface. The heterogeneous model may not only be useful to understand the mechanics of transport in the physiological system but it will also help the bioengineers to choose proper implant materials to design artificial vascular organs for the purpose of prosthesis.     

Cryopreservation procedure does not modify human carotid homografts mechanical properties: an isobaric and dynamic analysis

The viscoelastic and inertial properties of the arterial wall are responsible for the arterial functional role in the cardiovascular system. Cryopreservation is widely used to preserve blood vessels for vascular reconstruction but it is controversially suspected to affect the dynamic behaviour of these allografts. The aim of this work was to assess the cryopreservation's effects on human arteries mechanical properties. Common carotid artery (CCA) segments harvested from donors were divided into two groups: Fresh (n = 18), tested for 24–48 h after harvesting, and Cryopreserved (n = 18) for an average time of 30 days in gas-nitrogen phase, and finally defrosted. Each segment was tested in a circulation mock, and its pressure and diameter were registered at similar pump frequency, pulse and mean pressure levels, including those of normotensive and hipertensive conditions. A compliance transfer function (diameter/pressure) derived from a mathematical adaptive modelling was designed for the on line assessment of the arterial wall dynamics and its frequency response. Assessment of arterial wall dynamics was made by measuring its viscous (η), inertial (M) and elastic (E) properties, and creep and stress relaxation time constant (τ_C and τ_SR, respectively). The frequency response characterization allowed to evaluate the arterial wall filter or buffer function. Results showed that non-significant differences exist between wall dynamics and buffer function of fresh and cryopreserved segments of human CCA. In conclusion, our cryopreservation method maintains arterial wall functional properties, close to their fresh values.     

Model-free model elimination: A new step in the model-free dynamic analysis of NMR relaxation data

Model-free analysis is a technique commonly used within the field of NMR spectroscopy to extract atomic resolution, interpretable dynamic information on multiple timescales from the R ₁, R ₂, and steady state NOE. Model-free approaches employ two disparate areas of data analysis, the discipline of mathematical optimisation, specifically the minimisation of a χ² function, and the statistical field of model selection. By searching through a large number of model-free minimisations, which were setup using synthetic relaxation data whereby the true underlying dynamics is known, certain model-free models have been identified to, at times, fail. This has been characterised as either the internal correlation times, τ _e , τ _f , or τ _s , or the global correlation time parameter, local τ _m , heading towards infinity, the result being that the final parameter values are far from the true values. In a number of cases the minimised χ² value of the failed model is significantly lower than that of all other models and, hence, will be the model which is chosen by model selection techniques. If these models are not removed prior to model selection the final model-free results could be far from the truth. By implementing a series of empirical rules involving inequalities these models can be specifically isolated and removed. Model-free analysis should therefore consist of three distinct steps: model-free minimisation, model-free model elimination, and finally model-free model selection. Failure has also been identified to affect the individual Monte Carlo simulations used within error analysis. Each simulation involves an independent randomised relaxation data set and model-free minimisation, thus simulations suffer from exactly the same types of failure as model-free models. Therefore, to prevent these outliers from causing a significant overestimation of the errors the failed Monte Carlo simulations need to be culled prior to calculating the parameter standard deviations.     

A theoretical model for the control mechanisms in biochemical reactions

A mathematical representation for the analysis of control mechanisms in biochemical reactions is presented. First, the theoretical concept of concentration in biological systems is developed. Then a system consisting of two functions λ and τ is constructed as a network of single output automata. The range of λ is taken to be formed by a set of twostates qualitatively different from the “repair function” Φ _f of a mappingf: A→B in the stimulated Φ¹ and unstimulated state Φ⁰. Likewise, the range of τ is formed by the set δ={f ^o ,f ¹} wheref ¹ means the mappingf in its stimulated state andf ^o in the unstimulated one. It is demonstrated that the mathematical structure described acts as a control mechanism over thef and Φ _f , so that two biochemical components,A→B, are transformed at a controlled rate. Some of the biological applications of this model are briefly examined. The Jacob-Monod model, the enzymatic adaptation phenomenon, and the “rheon unit” hypothesis are discussed within our framework. Eventually, a concrete model for the RNA-polymerase mechanism, based on the above discussion, is presented.     

Presteady-State Currents of the Rabbit Na+/Glucose Cotransporter (SGLT1)

The rabbit Na⁺/glucose cotransporter (SGLT1) exhibits a presteady-state current after step changes in membrane voltage in the absence of sugar. These currents reflect voltage-dependent processes involved in cotransport, and provide insight on the partial reactions of the transport cycle. SGLT1 presteady-state currents were studied as a function of external Na⁺, membrane voltage V _m , phlorizin and temperature. Step changes in membrane voltage—from the holding V _h to test values, elicited transient currents that rose rapidly to a peak (at 3–4 msec), before decaying to the steady state, with time constants τ≈4–20 msec, and were blocked by phlorizin (K _i ≈30 μm). The total charge Q was equal for the application of the voltage pulse and the subsequent removal, and was a function of V _m . The Q-V curves obeyed the Boltzmann relation: the maximal charge Q _max was 4–120 nC; V _0.5, the voltage for 50% Q _max was −5 to +30 mV; and z, the apparent valence of the moveable charge, was 1. Q _max and z were independent of V _h (between 0 and −100 mV) and temperature (20–30°C), while increasing temperature shifted V _0.5 towards more negative values. Decreasing [Na⁺] _o decreased Q _max, and shifted V _0.5 to more negative voltages 9by −100 mV per 10-fold decrease in [Na⁺] _o ). The time constant τ was voltage dependent: the τ-V relations were bell-shaped, with maximal τ_max 8–20 msec. Decreasing [Na⁺] _o decreased τ_max, and shifted the τ-V curves towards more negative voltages. Increasing temperature also shifted the τ-V curves, but did not affect τ_max. The maximum temperature coefficient Q ₁₀ for τ was 3–4, and corresponds to an activation energy of 25 kcal/mole. Simulations of a 6-state ordered kinetic model for rabbit Na⁺/glucose cotransport indicate that charge-movements are due to Na⁺-binding/dissociation and a conformational change of the empty transporter. The model predicts that (i) transient currents rise to a peak before decay to steady-state; (ii) the τ-V relations are bell-shaped, and shift towards more negative voltages as [Na⁺] _o is reduced; (iii) τ_max is decreased with decreasing [Na⁺] _o ; and (iv) the Q-V relations are shifted towards negative voltages as [Na⁺] _o is reduced. In general, the kinetic properties of the presteady-state currents are qualitatively predicted by the model. Received: 12 August 1996/Revised: 30 September 1996     

Pichia pastoris fermentation optimization: energy state and testing a growth-associated model

A growth-associated model was applied to the production of recombinant ovine interferon-τ (rOvIFN-τ) with Pichia pastoris for the purpose of manufacturing preclinical and clinical active material. This model predicts that product yields will be the greatest when the specific growth of the culture is maintained at a steady and optimal rate. However, rOvIFN-τ yields did not meet the expected linear model but most closely corresponded to a polynomial relationship. After transitioning from glycerol to methanol, product accumulated for 31–45 h, and then the yield decreased. This production shift, which has been termed decoupling, was clearly related to time on methanol and not culture density. It was determined that a correlation exists between the decoupling point and a drop in energy state of the cell when expressing β-galactosidase. By assigning decoupling as a constraint that limits productivity and by reformulating the growth medium, the time prior to decoupling increased to 46.8±2.4 h, product yield improved for rOvIFN-τ from 203 to 337 mg l⁻¹, and the coefficient of variation for yield decreased from 67.9 to 23.3%. A robust and stable fermentation process was realized, resulting in a 210% improvement in total yield from 557±357 to 1,172±388 mg.     

Single-channel properties of a stretch-sensitive chloride channel in the human mast cell line HMC-1

A stretch-activated (SA) Cl⁻ channel in the plasma membrane of the human mast cell line HMC-1 was identified in outside-out patch-clamp experiments. SA currents, induced by pressure applied to the pipette, exhibited voltage dependence with strong outward rectification (55.1 pS at +100 mV and an about tenfold lower conductance at −100 mV). The probability of the SA channel being open (P _o) also showed steep outward rectification and pressure dependence. The open-time distribution was fitted with three components with time constants of τ_1o = 755.1 ms, τ_2o = 166.4 ms, and τ_3o = 16.5 ms at +60 mV. The closed-time distribution also required three components with time constants of τ_1c = 661.6 ms, τ_2c = 253.2 ms, and τ_3c = 5.6 ms at +60 mV. Lowering extracellular Cl⁻ concentration reduced the conductance, shifted the reversal potential toward chloride reversal potential, and decreased the P _o at positive potentials. The SA Cl⁻ currents were reversibly blocked by the chloride channel blocker 4,4′-diisothiocyanatostilbene-2,2′-disulfonic acid (DIDS) but not by (Z)-1-(p-dimethylaminoethoxyphenyl)-1,2-diphenyl-1-butene (tamoxifen). Furthermore, in HMC-1 cells swelling due to osmotic stress, DIDS could inhibit the increase in intracellular [Ca²⁺] and degranulation. We conclude that in the HMC-1 cell line, the SA outward currents are mediated by Cl⁻ influx. The SA Cl⁻ channel might contribute to mast cell degranulation caused by mechanical stimuli or accelerate membrane fusion during the degranulation process.     

Ozone effects in a drier climate: implications for stomatal fluxes of reduced stomatal sensitivity to soil drying in a typical grassland species

The widely distributed temperate grassland species Dactylis glomerata was grown in competition with Ranunculus acris at two different watering regimes and exposed for 20 weeks to eight ozone treatments with mean concentrations ranging from 16.2 to 89.5 ppb, representing pre‐industrial to predicted post‐2100 ozone climates. Measurements of stomatal conductance were used to parameterize ozone flux models for D. glomerata. For the first time, a modification was made to the standard flux model to account for the observed decrease in sensitivity of stomatal conductance to reduced water availability with increasing ozone. Comparison of calculated cumulative ozone flux between the two versions of the model demonstrated that exclusion of the ozone effect on stomatal conductance in the standard flux model led to a large underestimation of ozone fluxes at mid‐ to high‐ozone concentrations. For example, at a mean ozone concentration of 55 ppb (as predicted for many temperate areas in the next few decades), the standard flux model underestimated ozone fluxes in D. glomerata by 30–40% under reduced water availability. Although the modified flux model does not markedly change the flux‐based critical level for D. glomerata, this study indicates that use of the standard flux model to quantify the risk of ozone damage to a widely distributed grassland species such as D. glomerata in areas where high ozone concentrations and reduced soil moisture coincide could lead to an underestimation of effects. Thus, this study has shown that under predicted future climate change and ozone scenarios, ozone effects on vegetation may be even greater than previously predicted in the drier areas of the world.     

Mathematical models for exchange of substances in regional vascular beds— Measurement of the rates within vivo counters

Sangren and Sheppard developed a mathematical model for first-order processes taking place in the regional circulation, applicable—for example—to tracer studies of potassium transport. It permits calculation of specific activity at any point along a “tube of flow” or in the cuff of tissue surrounding it as a function of time following a spike injection of tracer. In efforts to relate to the exchange a rate curves obtained within vivo counters pointed at the region of interest, we developed a compartment-system model of the process. In investigating the properties of the Sangren and Sheppard model integrated over an entire circulatory bed, as thein vivo counter would see it, we found that when the distribution of transit times of the “tubes of flow” can be approximated by an exponential sum, the solution reduces to that of the compartment system model. This results in an important simplification in the calculation, and insight into the assumptions underlying the two different models. A curve-fitting computer program for the compartment model has been written and applied to double-isotope studies of potassium transport in the hind leg of the dog.     

Joint effects of mitosis and intracellular delay on viral dynamics: two-parameter bifurcation analysis

To understand joint effects of logistic growth in target cells and intracellular delay on viral dynamics in vivo, we carry out two-parameter bifurcation analysis of an in-host model that describes infections of many viruses including HIV-I, HBV and HTLV-I. The bifurcation parameters are the mitosis rate r of the target cells and an intracellular delay τ in the incidence of viral infection. We describe the stability region of the chronic-infection equilibrium E* in the two-dimensional (r, τ) parameter space, as well as the global Hopf bifurcation curves as each of τ and r varies. Our analysis shows that, while both τ and r can destabilize E* and cause Hopf bifurcations, they do behave differently. The intracellular delay τ can cause Hopf bifurcations only when r is positive and sufficiently large, while r can cause Hopf bifurcations even when τ = 0. Intracellular delay τ can cause stability switches in E* while r does not.     

Water transport in the arterial wall--a theoretical study

Water transport in the arterial wall is studied using a mathematical model based on the theory for the consolidation of water saturated soils (Biot, 1941; Kenyon, 1976a). The intimal pressure is considered to be harmonic in time. Analytical results are obtained for both large and small consolidation times since both the situations are of physiological relevance. For large consolidation times, the filtration is confined to a thin boundary layer. Large pressure gradients exist within the boundary layer while the pressure gradient is negligible in the intermediate layer. Thus, the pulsatile flow is found to be confined to the boundary layer while a smaller mean flow exists throughout the wall.     

Perturbation of endothelial junction proteins by Staphylococcus aureus α-toxin: inhibition of endothelial gap formation by adrenomedullin

Endothelial hyperpermeability is a hallmark of an inflammatory reaction and contributes to tissue damage in severe infections. Loss of endothelial cell–cell adhesion leads to intercellular gap formation allowing paracellular fluid flux. Although Staphylococcus aureus α-toxin significantly contributed to staphylococci disease, little is known about its mechanism of endothelial hyperpermeability induction. Here, we demonstrate that in a model of isolated perfused rat ileum discontinuation of capillary vascular-endothelial-cadherin (VE-cadherin) was observed after bolus application of S. aureus α-toxin being inhibited by the endogenous peptide adrenomedullin (ADM). In vitro, α-toxin exposure induced loss of immunoreactivity of VE-cadherin and occludin in human cultured umbilical vein endothelial cells. Likewise, ADM blocked α-toxin-related junctional protein disappearance from intercellular sites. Additionally, cyclic AMP elevation was shown to stabilize endothelial barrier function after α-toxin application. Although no RhoA activation was noted after endothelial α-toxin exposure, inhibition of Rho kinase and myosin light chain kinase blocked loss of immunoreactivity of VE-cadherin and occludin as well as intercellular gap formation. In summary, stabilization of endothelial junctional integrity as indicated by interendothelial immunostaining might be an interesting approach to stabilize endothelial barrier function in severe S. aureus infections.Andreas C. Hocke and Bettina Temmesfeld-Wollbrueck have contributed equally to this article.     

Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis

We present a mathematical model of the cytotoxic T lymphocyte response to the growth of an immunogenic tumor. The model exhibits a number of phenomena that are seenin vivo, including immunostimulation of tumor growth, “sneaking through” of the tumor, and formation of a tumor “dormant state”. The model is used to describe the kinetics of growth and regression of the B-lymphoma BCL₁ in the spleen of mice. By comparing the model with experimental data, numerical estimates of parameters describing processes that cannot be measuredin vivo are derived. Local and global bifurcations are calculated for realistic values of the parameters. For a large set of parameters we predict that the course of tumor growth and its clinical manifestation have a recurrent profile with a 3- to 4-month cycle, similar to patterns seen in certain leukemias.     

Some comments on the conductance of slow inward current in cardiac muscle

Voltage clamp experiments, which determine the kinetic parameters of calcium conductance of cardiac muscle, (d _∞,f _∞, τ _d and τ _f ) are analyzed with a generally accepted expression for slow inward currentI _s=g _sdf (E-E _R). Activation (d) and inactivation (f) reach the final valuesd _∞ andf _∞ with time constants τ _d and τ _f respectively. The analysis indicates that the measuredf _∞ agrees with the theoreticalf _∞, but the measuredd _∞ differs from the theoreticald _∞ by a factor which depends on τ _d . The peak tension can be made to correlate closely with the theoreticald _∞ after a correction factor is applied to the raw measurements of activation. It can be shown that experiments designed to measure τ _f can also be used to determine τ _d with greater accuracy.     

Changes in the mechanical properties of fibroblasts during spreading: a micromanipulation study

Cell morphology is controlled in part by physical forces. If the main mechanical properties of cells have been identified and quantitated, the question remains of how the cell structure specifically contributes to these properties. In this context, we addressed the issue of whether cell rheology was altered during cell spreading, taken as a fundamental morphological change. On the experimental side, we used a novel dual micromanipulation system. Individual chick fibroblasts were allowed to spread for varying amounts of time on glass microplates, then their free extremity was aspirated into a micropipet at given pressure levels. Control experiments were also done on suspended cells. On the theoretical side, the cell was modeled as a fluid drop of viscosity μ, bounded by a contractile cortex whose tension above a resting value was taken to be linearly dependent on surface area expansion. The pipet negative pressure was first adjusted to an equilibrium value, corresponding to formation of a static hemispherical cap into the pipet. This allowed computation, through Laplace's law, of the resting tension (τ ₀), on the order of 3×10^–4 N/m. No difference in τ ₀ was found between the different groups of cells studied (suspended, adherent for 5 min, spread for 0.5 h, and spread for 3 h). However, τ ₀ was significantly decreased upon treatment of fibroblasts with inhibitors of actin polymerization or myosin function. Then, the pressure was set at 30 mmH₂O above the equilibrium pressure. All cells showed a biphasic behavior: (1) a rapid initial entrance corresponding to an increase in surface area, which was used to extract an area expansion elastic modulus (K), in the range of 10^–2 N/m; this coefficient was found to increase up to 40% with cell spreading; (2) a more progressive penetration into the pipet, linear with time; this phase, attributed to viscous behavior of the cytoplasm, was used to compute the apparent viscosity (μ, in the range of 2–5×10⁴ Pa s) which was found to increase by as much as twofold with cell spreading. In some experiments the basal force at the cell-microplate interface was quantitated with flexible microplates and found to be around 1 nN, in agreement with values calculated from the model. Taken together, our results indicate a stiffening of fibroblasts upon spreading, possibly correlated with structural organization of the cytoskeleton during this process. This study may help understand better the morphology of fibroblasts and their mechanical role in connective tissue integrity. Received: 22 June 1998 / Revised version: 14 October 1998 / Accepted: 15 October 1998     

A Generalised Age- and Phase-Structured Model of Human Tumour Cell Populations Both Unperturbed and Exposed to a Range of Cancer Therapies

We develop a general mathematical model for a population of cells differentiated by their position within the cell division cycle. A system of partial differential equations governs the kinetics of cell densities in certain phases of the cell division cycle dependent on time t (hours) and an age-like variable τ (hours) describing the time since arrival in a particular phase of the cell division cycle. Transition rate functions control the transfer of cells between phases. We first obtain a theoretical solution on the infinite domain −∞ < t < ∞. We then assume that age distributions at time t=0 are known and write our solution in terms of these age distributions on t=0. In practice, of course, these age distributions are unknown. All is not lost, however, because a cell line before treatment usually lies in a state of asynchronous balanced growth where the proportion of cells in each phase of the cell cycle remain constant. We assume that an unperturbed cell line has four distinct phases and that the rate of transition between phases is constant within a short period of observation (‘short’ relative to the whole history of the tumour growth) and we show that under certain conditions, this is equivalent to exponential growth or decline. We can then gain expressions for the age distributions. So, in short, our approach is to assume that we have an unperturbed cell line on t ≤ 0, and then, at t=0 the cell line is exposed to cancer therapy. This corresponds to a change in the transition rate functions and perhaps incorporation of additional phases of the cell cycle. We discuss a number of these cancer therapies and applications of the model.     

Stomatal control of transpiration in the canopy of a clonal Eucalyptus grandis plantation

 Predawn leaf water potential, stomatal conductance and microclimatic variables were measured on 13 sampling days from November 1995 through August 1996 to determine how environmental and physiological factors affect water use at the canopy scale in a plantation of mature clonal Eucalyptus grandis Hill ex-Maiden hybrids in the State of Espirito Santo, Brazil. The simple ”big leaf” Penman-Monteith model was used to estimate canopy transpiration. During the study period the predawn leaf water potential varied from –0.4 to –1.3 MPa, with the minimum values observed in the winter months (June and August 1996), while the average estimated values for canopy conductance and canopy transpiration fell from 17.3 to 5.8 mm s^–1 and from 0.54 to 0.18 mm h^–1, respectively. On the basis of all measurements, the average value of the decoupling coefficient was 0.25. During continuous soil water shortage a proportional reduction was observed in predawn leaf water potential and in daily maximum values of stomatal conductance, canopy transpiration and decoupling coefficient. The results showed that water vapour exchange in this canopy is strongly dominated by the regional vapour pressure deficit and that canopy transpiration is controlled mainly by stomatal conductance. On a seasonal basis, stomatal conductance and canopy transpiration were mainly related to predawn leaf water potential and, thus, to soil moisture and rainfall. Good results were obtained with a multiplicative empirical model that uses values of photosynthetically active radiation, vapour pressure deficit and predawn leaf water potential to estimate stomatal conductance. Received: 10 June 1998 / Accepted: 20 July 1998     

Determination of time-constants in cables of finite length

Current flow in cylindrical nerve and muscle fibre has been analysed in terms of a mathematical model leading to a linear partial differential equation for the voltage as a function of both position and time. In the case of a one-dimensional cable subject to a step input of current, the solution will consist of a steady-state behaviour preceded by an initial transient. The electrical properties of the fibre or cable itself determine a length-constant, λ, which can be determined experimentally from the steady-state response, and a time-constant, τ, which must be found from the initial transient. When the cable is infinite and when there is a single input electrode, an exact solution can be produced which enables ready determination of the time-constant τ. Two complications arise in experimental practice, however. In the first place, the fibre has finite length, and in the second, two spatially separated stimulation electrodes are often required. We thus analyse a more complicated and more general situation. The linearity of the membrane properties, however, allows the solution to the more general case to be built up by superposition of solutions from the simpler case (equivalent to the classical method of images). We also approximate the Hodgkin and Rushton solution by asymptotic formulae in order to allow more tractable expressions for the exact solution. We are thus able to give a method for the ready evaluation of the time constant τ under more general conditions.     

A stochastic model of the effects of ionizing radiation on mammalian cellsin vitro

A new, more realistic model of the action of ionizing radiation on mammalian cells growingin vitro is presented. Although this model requires a large number of parameters, these are linked to biologically observable quantities rather than being abstract sensitivities, as had previously been the case. Three different stochastic processes are required: {X(t);t ∈ [0, τ]}, representing damage alterations during irradiation; {(X(t), S(t));t ∈ [τ, τ+T _D]}, representing changes in both damageX(t) and cell cycle positionS(t) during the post-irradiation cell cycle; and {N _x(t);t ∈ [0,T _G]}, representing the subsequent colony growth process conditioned on the value ofX(τ+T _D). The assumptions used to define these processes extend a previous model of short term DNA damage formation and repair (Nelson S. J. 1982,Radiat. Res. 92, 120–145) to include the influence of cell cycle progression on damage in the irradiated cell and the effect of permanent inherited damage on the daughter cells' colony growth pattern. Expressions corresponding to commonly measured radiation effects are derived from the model and compared with predictions from previous models. It is found that these previous models oversimplified the mechanism of radiation action because they did not adequately represent repair during irradiation, the influence of radiation-induced cycle delays and damage inheritance by any daughter cells. Suggestions are then made for ways in which the new model can be used to test the importance of these effects.