Pressure-sensitive nutrient consumption via dynamic normal stress in rotational bioreactors

Pressure-sensitive biological response is simulated in “rotating-cup” bioreactors with unidirectional modulations in compressive stress at the cylindrical wall that stimulate bone-tissue growth. Anchorage-dependent mammalian cells (i) adhere to a protein coating, (ii) receive nutrients and oxygen from an aqueous medium via radial diffusion toward the active surface, and (iii) respond to physiological modulations in centrifual-force-induced fluid pressure at the cell/aqueous-medium interface. This process is modeled by the classic diffusion equation (i.e., Fick's second law), with a time-dependent reaction/diffusion boundary condition at the wall. Non-reversing angular velocity modulations resemble pulsations at physiological frequencies. Computer simulations of nutrient consumption profiles suggest that rotational bioreactor designs should consider the effects of normal stress when the pressure-sensitive Damköhler number (i.e., ratio of the pressure-dependent zeroth-order rate of nutrient consumption relative to the rate of nutrient diffusion toward active cells adhered to the cylindrical wall), evaluated under steady rotation, is greater than ≈ 10–20% of the stress-free Damköhler number (i.e., β_0,1st-order = 0.025) for simple 1st-order stress-free kinetics, and ≈ 1% of the stress-free Damköhler number (i.e., β_0,2nd-order = 0.40) for complex 2nd-order stress-free nutrient consumption. When the peak-to-peak amplitude of angular velocity modulations of the cylindrical wall is the same as or larger than the angular velocity for steady rotation, the effect of non-reversing centrifugal-force-induced dynamic normal stress in rotational bioreactors, superimposed on steady rotation, can be significant when one is below the critical value of the pressure-sensitive Damköhler number that has been identified under steady rotation.     

Tubular bioreactor models that include Onsager-Curie scalar cross-phenomena to describe stress-dependent rates of cell proliferation

The theory of heterogeneous catalysis in chemical reactors is employed to simulate laminar flow through tubes at large mass transfer Peclet numbers in which anchorage-dependent cells (i) adhere to a protein coating on the inner surface at r = R_wall, (ii) receive nutrients and oxygen from an aqueous medium via transverse diffusion toward the active wall, and (iii) proliferate in the presence of viscous shear at the cell/aqueous-medium interface. This process is modeled as convective diffusion in cylindrical coordinates with chemical reaction at the boundary, where chemical reaction describes the rate of nutrient consumption. The formalism of irreversible thermodynamics is employed to describe an unusual coupling between viscous shear, or velocity gradients at the cell/aqueous-medium interface, and rates of nutrient consumption. Linear transport laws in chemically reactive systems that obey Curie's theorem predict the existence of cross-phenomena between fluxes (i.e., scalar reaction rates) and driving forces (i.e., 2nd-rank velocity gradient tensor) whose tensorial ranks differ by an even integer—in this case, two. This methodology for stress-dependent chemical reactions yields an additional zeroth-order contribution, via the magnitude of the velocity gradient tensor, to heterogeneous kinetic rate expressions because nutrient consumption and cell proliferation are stress-sensitive. Computer simulations of nutrient consumption suggest that bioreactor designs should consider stress-sensitive reactions when the shear-rate-based Damköhler number (i.e., defined for the first time in this study as the stress-dependent zeroth-order rate of nutrient consumption relative to the rate of nutrient diffusion toward active cells adhered to the tube wall) is greater than 10–20% of the stress-free Damköhler number. Models of bioreactor performance are presented for simple 1st-order, simple 2nd-order, and complex chemical kinetic rate expressions, where the latter considers adsorption/desorption equilibria via the Fowler–Guggenheim modification of the Langmuir isotherm for cell–protein docking on active sites, accompanied by cell–cell attraction. Stress sensitivity is magnified in physically realistic cell-based tubular bioreactors with complex stress-free kinetic rate expressions relative to simulations with simple 1st- and 2nd-order kinetics.     

Nutrient diffusion and simple nth-order consumption in regenerative tissue and biocatalytic sensors

This contribution addresses intra-tissue molar density profiles for nutrients, oxygen, growth factors, and other essential ingredients that anchorage-dependent cells require for successful proliferation on biocompatible surfaces. One-dimensional transient and steady state models of the reaction-diffusion equation are solved to correct a few deficiencies in the first illustrative example of diffusion and zeroth-order rates of consumption in tissues with rectangular geometry, as discussed in Ref. [(Griffith and Swartz, 2006) 1]. The functional form of the molar density profile for each species depends on geometry and the magnitude of the species-specific intra-tissue Damk?hler number. The tissue's central core is reactant starved at high consumption rates and low rates of intra-tissue diffusion when the Damk?hler number exceeds its geometry-sensitive critical value. Ideal tissue engineering designs avoid the diffusion-limited regime such that attached cells are exposed to all of the ingredients required for proliferation everywhere within a regenerative matrix. Analytical and numerical molar density profiles that satisfy the unsteady state modified diffusion equation with pseudo-homogeneous n(th)-order rates of intra-tissue consumption (i.e., n=0,1,2) allow one to (i) predict von Kármán-Pohlhausen mass transfer boundary layer thicknesses, measured inward from the external biomaterial surface toward its central core, and, most importantly, (ii) estimate the time required to achieve steady state conditions for regenerative tissue growth and biocatalytic sensing.     

Analysis of cell growth in a polymer scaffold using a moving boundary approach

Two mathematical models of chondrocyte generation and nutrient consumption are developed to analyze the behavior of cell growth in a biodegradable polymer matrix. Substrate reaction and diffusion are analyzed in two regions: one consisting of cells and nutrients and the other consisting of only nutrients. A pseudo-steady state approximation for the transport of nutrients in these two regions is utilized. The rate of growth is determined by a moving boundary equation that equates the rate at which the interfacial region between the cells and the void space moves to a substrate dependent growth reaction. The change in the location of this interfacial region with time therefore depicts the rate at which the cells propagate. The two limiting cases discussed in this article represent extremes in how the cells will grow in the polymer matrix; one case assumes that cells grow inward from the external boundary, and the other case assumes that cells grow parallel to the external boundary. The results of both models are compared to experimental data found in the literature. It is found through these comparisons that the model parameters, including the unit cell spacing parameter L, the metabolic rate constant k, the growth rate constant k(G), and external mass transfer coefficient, K, may vary as the thickness of the polymer matrix is changed, however, unrealistic and large changes in the diffusion coefficients were required to account for the full range of experimental data. Furthermore, these results suggest modification of the functional form of the growth kinetics to include substrate or product inhibition, or death terms. Based upon diffusion/reaction concepts, these models for cell growth in a biodegradable polymer give bounds for the upper and lower limits of the cellular growth rate and nutrient consumption in a polymer matrix and will aid in the development of more extensive models. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56: 422-432, 1997.     

Pore‐network modeling of biofilm evolution in porous media

The influence of bacterial biomass on hydraulic properties of porous media (bioclogging) has been explored as a viable means for optimizing subsurface bioremediation and microbial enhanced oil recovery. In this study, we present a pore network simulator for modeling biofilm evolution in porous media including hydrodynamics and nutrient transport based on coupling of advection transport with Fickian diffusion and a reaction term to account for nutrient consumption. Biofilm has non‐zero permeability permitting liquid flow and transport through the biofilm itself. To handle simultaneous mass transfer in both liquid and biofilm in a pore element, a dual‐diffusion mass transfer model is introduced. The influence of nutrient limitation on predicted results is explored. Nutrient concentration in the network is affected by diffusion coefficient for nutrient transfer across biofilm (compared to water/water diffusion coefficient) under advection dominated transport, represented by mass transport Péclet number >1. The model correctly predicts a dependence of rate of biomass accumulation on inlet concentration. Poor network connectivity shows a significantly large reduction of permeability, for a small biomass pore volume. Biotechnol. Bioeng. 2011;108: 2413–2423. © 2011 Wiley Periodicals, Inc.     

Confinement Regulates Complex Biochemical Networks: Initiation of Blood Clotting by “Diffusion Acting”

This study shows that environmental confinement strongly affects the activation of nonlinear reaction networks, such as blood coagulation (clotting), by small quantities of activators. Blood coagulation is sensitive to the local concentration of soluble activators, initiating only when the activators surpass a threshold concentration, and therefore is regulated by mass transport phenomena such as flow and diffusion. Here, diffusion was limited by decreasing the size of microfluidic chambers, and it was found that microparticles carrying either the classical stimulus, tissue factor, or a bacterial stimulus, Bacillus cereus, initiated coagulation of human platelet-poor plasma only when confined. A simple analytical argument and numerical model were used to describe the mechanism for this phenomenon: confinement causes diffusible activators to accumulate locally and surpass the threshold concentration. To interpret the results, a dimensionless confinement number, Cn, was used to describe whether a stimulus was confined, and a Damköhler number, Da₂, was used to describe whether a subthreshold stimulus could initiate coagulation. In the context of initiation of coagulation by bacteria, this mechanism can be thought of as “diffusion acting”, which is distinct from “diffusion sensing”. The ability of confinement and diffusion acting to change the outcome of coagulation suggests that confinement should also regulate other biological “on” and “off” processes that are controlled by thresholds.     

Improvement of the yield of physiologically active oligosaccharides in continuous hydrolysis of chitosan using immobilized chitosanases

The continuous production of chitosan oligosaccharides using a packed-bed enzyme reactor was investigated as to the effects of the operation conditions on the yield of pentamers and hexamers of chitosan oligosaccharides. A column reactor packed with immobilized chitosanases prepared by the multipoint attachment method was used for continuous hydrolysis of chitosan. In this reactor, the decrease of the yield of the target intermediate oligosaccharides due to axial mixing was negligible. The surface enzyme density of the support and flow rate of the substrate solution significantly affected the maximum yield of pentamers and hexamers. These effects were summarized as a correlation with the Damk?hler number (Da), defined as the ratio of the maximum reaction rate to the maximum mass transfer rate. The optimum condition was determined based on Da. Under the optimized condition (Da = 0.12), pentamers and hexamers could be produced continuously for a month with a yield of over 35% (7 kg/m(3) in concentration).     

Aboveground biomass and soil nutrient pools of a Scalesia pedunculata montane forest on Santa Cruz, Galápagos

We estimated the live aboveground biomass (AGB) and soil nutrient pools of the Scalesia pedunculata monodominant tropical montane forest at 600 m above sea level on Santa Cruz, Galápagos, an isolated oceanic island. The estimated AGB was 60.4 Mg ha^–1, which was considerably lower than that of other montane forests of similar climates elsewhere. Nutrient pools were ample for inorganic N, soluble P, and exchangeable cations. We suggest that the low AGB, in spite of the ample nutrients, is related to the absence of tall-statured climax species, that have high demands for nutrients (particularly N) to fix C, due to the isolation.     

Transport effects on surface reaction arrays: biosensor applications

The ubiquity of surface-volume reactions in biological and industrial processes makes knowledge of their kinetics critical. This has spurred technological advances in several biosensors designed to measure rate constants, such as the Flexchip and the dotLab. These biosensors have multiple reacting zones in a single flow channel, and hence they also serve as good model systems for biochemical systems with multiple reacting zones, such as cell membranes. A correct mathematical model for such systems must incorporate the effects of transport and zone position. A basic unidirectional flow model is developed in general and solved for typical experimental parameters using perturbation methods. The effect of zone placement along the channel can be quantified in terms of an effective Damköhler number based upon position. Moreover, it is established that zone placement across the channel does not affect the measurements.     

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.     

Pore water N and P concentration in a floodplain marsh of the Lower Paraná River*

Inorganic nitrogen and soluble reactive phosphate (o-P) concentrations were measured in the water of a marsh and in its interstitial water at two sites, and in the river water of a floodplain marsh of the Lower Paraná River. These values were compared with the N and P concentration in sediments and macrophyte biomass in order to assess nutrient availability, fate and storage capacity. High variability was found in the interstitital water using a 1 cm resolution device. Nitrate was never detected in the pore water. Depth averaged NH₄ ⁺ concentrations in the upper 30 cm layer often ranged from N = 1.5 to 1.8 mg l^-1, but showed a pronounced minimum (0.5–0.7 mg l^-1), close to (March 95), or relatively soon after (May 94) the end of the macrophyte growing season. Soluble phosphate showed a large variation between P = 0.1–1.1 mg l^-1 without any discernible seasonal pattern. NH₄ ⁺ depletion in the pore water concentration and low N/P ratios (3.7 by weight) within the macrophyte biomass at the end of the growing period suggest that available N limits plant growth. NH₄ ⁺ and o-P concentrations were 35 and 7 times higher, respectively, in the pore water than in the overlying marsh, suggesting a permanent flux of nutrients from the sediments. o-P accumulate in the marsh leading to higher concentrations than in the incoming river. NH₄ ⁺ did not accumulate in the marsh, and no significant differences were observed between the river and the marsh water, while the NO₃ ^- contributed by the river water was depleted within the marsh, caused probably by coupled nitrification-denitrification at the sediment–water interface. Although an order of magnitude smaller, the pore water pool can supply enough nutrients to build up the macrophyte biomass pool, but only if a fast turnover is attained. The Paraná floodplain marsh retains a large amount of nutrients being stored mainly in the sediment compartment.     

A 3D Hybrid Model for Tissue Growth: The Interplay between Cell Population and Mass Transport Dynamics

To provide theoretical guidance for the design and in vitro cultivation of bioartificial tissues, we have developed a multiscale computational model that can describe the complex interplay between cell population and mass transport dynamics that governs the growth of tissues in three-dimensional scaffolds. The model has three components: a transient partial differential equation for the simultaneous diffusion and consumption of a limiting nutrient; a cellular automaton describing cell migration, proliferation, and collision; and equations that quantify how the varying nutrient concentration modulates cell division and migration. The hybrid discrete-continuous model was parallelized and solved on a distributed-memory multicomputer to study how transport limitations affect tissue regeneration rates under conditions encountered in typical bioreactors. Simulation results show that the severity of transport limitations can be estimated by the magnitude of two dimensionless groups: the Thiele modulus and the Biot number. Key parameters including the initial seeding mode, cell migration speed, and the hydrodynamic conditions in the bioreactor are shown to affect not only the overall rate, but also the pattern of tissue growth. This study lays the groundwork for more comprehensive models that can handle mixed cell cultures, multiple nutrients and growth factors, and other cellular processes, such as cell death.     

Spatial modeling of dimerization reaction dynamics in the plasma membrane: Monte Carlo vs. continuum differential equations

Bimolecular reactions in the plasma membrane, such as receptor dimerization, are a key signaling step for many signaling systems. For receptors to dimerize, they must first diffuse until a collision happens, upon which a dimerization reaction may occur. Therefore, study of the dynamics of cell signaling on the membrane may require the use of a spatial modeling framework. Despite the availability of spatial simulation methods, e.g., stochastic spatial Monte Carlo (MC) simulation and partial differential equation (PDE) based approaches, many biological models invoke well-mixed assumptions without completely evaluating the importance of spatial organization. Whether one is to utilize a spatial or non-spatial simulation framework is therefore an important decision. In order to evaluate the importance of spatial effects a priori, i.e., without performing simulations, we have assessed the applicability of a dimensionless number, known as second Damköhler number (Da), defined here as the ratio of time scales of collision and reaction, for 2-dimensional bimolecular reactions. Our study shows that dimerization reactions in the plasma membrane with Da ∼> 0.1 (tested in the receptor density range of 10²–10⁵/μm²) require spatial modeling. We also evaluated the effective reaction rate constants of MC and simple deterministic PDEs. Our simulations show that the effective reaction rate constant decreases with time due to time dependent changes in the spatial distribution of receptors. As a result, the effective reaction rate constant of simple PDEs can differ from that of MC by up to two orders of magnitude. Furthermore, we show that the fluctuations in the number of copies of signaling proteins (noise) may also depend on the diffusion properties of the system. Finally, we used the spatial MC model to explore the effect of plasma membrane heterogeneities, such as receptor localization and reduced diffusivity, on the dimerization rate. Interestingly, our simulations show that localization of epidermal growth factor receptor (EGFR) can cause the diffusion limited dimerization rate to be up to two orders of magnitude higher at higher average receptor densities reported for cancer cells, as compared to a normal cell.     

A method to evaluate the isothermal effectiveness factor for dynamic oxygen into mycelial pellets in submerged cultures

Several models have been developed simulating O2 transfer in bioreactors, but three limitations are often found: (i) an inadequate kinetic representation of O2 consumption or wrong boundary conditions, (ii) unrealistic parameter values, and (iii) inadequate experimental systems. In our study we minimized those possible sources of error. Oxygen uptake rate, void fraction of the pellet, and external O2 mass transfer coefficient were experimentally obtained from bioreactor studies in which pellets of Gibberella fujikuroi were naturally formed. Michaelis-Menten kinetics and diffusion equations were used to describe the O2 consumption rate and to evaluate the effectiveness factor in dynamic mode. The nonlinear mathematical model proposed was solved by the orthogonal collocation technique. The O2 consumption rate in pellets of G. fujikuroi of 1.7-2.0 mm is only marginally inhibited by diffusion constraints under conditions tested. Simulation analysis showed that the effectiveness factor decreased as the Thiele modulus and pellet diameter increased. The proposed model was applied to experimental data reported for other fungal pellets and allowed to predict optimal conditions for O2 transfer into mycelial pellets.     

Effects of plant nutrient availability and host plant species on the performance of two Pieris butterflies (Lepidoptera: Pieridae)

We assayed the interaction on the availability of plant nutrient and species of host plant on the performance of two species of Pieris butterfly. The results indicated that constant application of different levels of fertilizers to the four different host plants resulted to an increase in their content of plant nutrients. The chemical analysis showed that the added nutrients increased foliar nitrogen and water contents, but there was no effect on the level of glucosinolates. Larvae that fed on highly-nutritious foliage increased their growth rates and showed a shorter development period. The results of feeding trials revealed that the 4th-instar larvae, which had fed on host plants with higher levels of fertilization had a shorter duration of development, less consumption rate, higher growth rate and food processing efficiency. To summarize, this research revealed that both the availability of plant nutrient and species of host plant can strongly influence the physiology and foliar chemistry of host plants. Moreover, the changes of phytochemical in the host plants may play an important role in affecting the performance (growth and food utilization efficiency) of both species of Pieris butterflies.     

Mechanistic simulation models of nutrient uptake: A review

Mechanistic models of nutrient uptake consider diffusion and mass flow acting simultaneously to supply nutrients to the sorbing root surface. Plant parameters that determine nutrient uptake include those describing changes in root geometry and size due to root growth and others describing kinetics of the nutrient uptake process. Mechanistic models generally assume that nutrient uptake occurs evenly along the roots that are uniformly distributed in homogeneous and isotropic soil having no temporal and spatial gradients in volumetric moisture content. Uptake of immobile nutrients (like P and K) is mainly determined by the soil-supply parameters and is well predicted by the simulation models. In contrast, uptake of mobile nutrients (e.g. Ca and Mg) that usually accumulate at the root surface is determined mainly by the plant-uptake parameters; prediction of uptake of those nutrients is subject to a much wider error due to uncertainties of applying kinetic parameters measured on hydroponically-grown plants to soil-grown plants. Comparison of model-predicted and experimentally-observed uptake values should be done by calculating the mean squares of deviates instead of performing regression analysis, especially if data that encompass a relatively wide range in root length are considered. Complementary-ion effects occurring at the soil-root interface raise the need for developing a multi-nutrient uptake model that will simultaneously calculate uptake of several essential nutrients taking into account interactions among them.     

Effects of simultaneous changes in light, nutrients, and herbivory levels, on the structure and function of a subtropical turtlegrass meadow

The interactive effects of light, nutrients, and simulated herbivory on the structure and functioning of a subtropical turtlegrass bed were analyzed monthly from May to October 2001 in Perdido Bay, FL. For each of the three factors, two levels were evaluated in a factorial design with four replicates per treatment. The variables included: light, at ambient and 40% reduction; nutrients, at ambient and 2× ambient concentrations; and herbivory, with no herbivory and simulated effects of a density of 15 sea urchins/m². In practice, light levels turned out to be 40% of surface PAR for ambient conditions, and 16% for shaded plots. Biomass removed as herbivory represented, on average, slightly less than 20% of the above-ground biomass. Separate three-way ANOVAs found no significant three-way interactions for any of the response variables, and few two-way interactions. There were no significant nutrient effects on turtlegrass above-ground biomass, although nutrient additions produced significant decreases in epibiont biomass, and net above-ground primary production (NAPP); significant increases in below-ground biomass during the peak of the growing season. Shoot density and average number of leaves per shoot increased significantly, while the C/N ratio of the oldest leaf in the enriched plots decreased significantly. Light reduction significantly negatively affected all response variables, except below-ground biomass, shoot density and leaf length. Herbivory had isolated and inconsistent significant effects on below-ground biomass, shoot density, average number of leaves per shoot, and leaf length and width. Overall, our results indicate that nutrients are not limiting in Perdido Bay, and that nutrient additions had mostly detrimental effects. Light appeared to be the most important variable limiting seagrasses growth and abundance, and as with terrestrial plants, seagrasses seemed to respond more to light and nutrients than to herbivory. However, it is essential that additional tests of the single and interactive effects of the three key factors of light, nutrients and herbivory be done to evaluate the generality of our work, since our study is the first of its kind in seagrass meadows.     

Buffalograss decreases ramet propagation in infertile patches to enhance interconnected ramet proliferation in fertile patches

Buffalograss (Buchloë dactyloides) is known for its low-nutrient tolerance. However, in natural habitats, nutrients are usually patchily distributed. For clonal plants like buffalograss, physiological integration is an important strategy to cope with adverse environmental conditions. In order to examine how integration helps buffalograss to survive in patchy conditions, a greenhouse experiment was conducted for 91 days. Interconnected ramet pairs of stoloniferous buffalograss were planted in two partitioned same-sized containers, and subjected to identical or contrasting nutrient supply. In contrast to normally perceived resource-sharing concepts, results showed that buffalograss genets reduced production of new ramets in nutrient-poor patches promoting at the same time propagation of interconnected ramets in nutrient-rich patches. Ramets in nutrient-rich patches gained significant benefit from heterogeneous treatments, whereas nutrient-poor ramets performed even worse than in uniform low-nutrient treatment. Younger ramets developed more biomass than elder ramets with the same amounts of nutrient supply under homogeneous treatment, while elder ramets were more tolerant when nutrients were scarce. Heterogeneity had a particular strong effect on stolons and new ramet production in nutrient-rich patches. Rooted ramets in nutrient-poor patches suffered from a by-pass of nutrients to interconnected ramets on nutrient-rich substrate that probably resulted from different transpiration rates. We conclude that this resource-sharing strategy is advantageous for buffalograss to concentrate more ramets in fertile patches, and facilitate the survivorship of more buffalograss ramets in adverse environments with uneven nutrient supply.