Computer simulation of sedimentation in the ultracentrifuge. 3. Concentration-dependent sedimentation

Advanced computational modeling for in vitro nanomaterial dosimetry

Background

Accurate and meaningful dose metrics are a basic requirement for in vitro screening to assess potential health risks of engineered nanomaterials (ENMs). Correctly and consistently quantifying what cells “see,” during an in vitro exposure requires standardized preparation of stable ENM suspensions, accurate characterizatoin of agglomerate sizes and effective densities, and predictive modeling of mass transport. Earlier transport models provided a marked improvement over administered concentration or total mass, but included assumptions that could produce sizable inaccuracies, most notably that all particles at the bottom of the well are adsorbed or taken up by cells, which would drive transport downward, resulting in overestimation of deposition.

Methods

Here we present development, validation and results of two robust computational transport models. Both three-dimensional computational fluid dynamics (CFD) and a newly-developed one-dimensional Distorted Grid (DG) model were used to estimate delivered dose metrics for industry-relevant metal oxide ENMs suspended in culture media. Both models allow simultaneous modeling of full size distributions for polydisperse ENM suspensions, and provide deposition metrics as well as concentration metrics over the extent of the well. The DG model also emulates the biokinetics at the particle-cell interface using a Langmuir isotherm, governed by a user-defined dissociation constant, K_D, and allows modeling of ENM dissolution over time.

Results

Dose metrics predicted by the two models were in remarkably close agreement. The DG model was also validated by quantitative analysis of flash-frozen, cryosectioned columns of ENM suspensions. Results of simulations based on agglomerate size distributions differed substantially from those obtained using mean sizes. The effect of cellular adsorption on delivered dose was negligible for K_D values consistent with non-specific binding (> 1 nM), whereas smaller values (≤ 1 nM) typical of specific high-affinity binding resulted in faster and eventual complete deposition of material.

Conclusions

The advanced models presented provide practical and robust tools for obtaining accurate dose metrics and concentration profiles across the well, for high-throughput screening of ENMs. The DG model allows rapid modeling that accommodates polydispersity, dissolution, and adsorption. Result of adsorption studies suggest that a reflective lower boundary condition is appropriate for modeling most in vitro ENM exposures.

Electronic supplementary material

The online version of this article (doi:10.1186/s12989-015-0109-1) contains supplementary material, which is available to authorized users.

A critical review of in vitro dosimetry for engineered nanomaterials

A major obstacle in the development of accurate cellular models for investigating nanobio interactions in vitro is determination of physiologically relevant measures of dose. Comparison of biological responses to nanoparticle exposure typically relies on administered dose metrics such as mass concentration of suspended particles, rather than the effective dose of particles that actually comes in contact with the cells over the time of exposure. Adoption of recently developed dosimetric methodologies will facilitate determination of effective dose delivered to cells in vitro, thereby improving the accuracy and reliability of in vitro screening data, validation of in vitro with in vivo data, and comparison across multiple datasets for the large variety of nanomaterials currently in the market.

Regulation of rabbit muscle phosphofructokinase by phosphorylation

Muscle phosphofructokinase is one of the glycolytic enzymes whose partitioning between the particulate and soluble fractions in skeletal muscle is linked to the biological activity of the muscle. The formation of the enzyme-actin complex is apparently regulated by phosphorylation of the enzyme. In order to understand the role of phosphorylation on the regulatory mechanism of phosphofructokinase, the self-association of the phosphorylated and dephosphorylated forms of phosphofructokinase was studied by investigating the sedimentation velocity at pH 7.0 and 23 degrees C in different solvent constituents. The results show that both the phosphorylated and dephosphorylated forms of the enzyme exhibit the same mechanism of assembly. The effects of allosteric effectors are dependent on the phosphorylation state of the enzyme. The presence of 0.2 mM fructose-6-phosphate, one of the two substrates, leads to a significant enhancement in the formation of octomers without altering the equilibrium constant for tetramerization for either phosphorylated or dephosphorylated enzyme. The presence of 10 mM citrate, an allosteric inhibitor, leads to the formation of a significant amount of dimer, an inactive form of the enzyme. Citrate decreases the propensities of the dephosphorylated and phosphorylated forms of the enzyme to tetramerize 3000 times and 100 times, respectively. Based on the mode of subunit assembly, bimodal sedimentation velocity profiles can be obtained by simulation. Furthermore, simulation showed that the seemingly very different profiles reported in the literature can be accounted for by various combinations of equilibrium constants. In summary, this study showed that the propensity of subunit assembly is affected differentially by specific metabolites and the phosphorylation state of phosphofructokinase.

Analysis of protein-protein interaction by simulation of small-zone size exclusion chromatography. Stochastic formulation of kinetic rate contributions to observed high-performance liquid chromatography elution characteristics

High-performance liquid chromatography (HPLC) procedures provide size-exclusion chromatography with sufficient speed that the elution characteristics of mixtures of interacting macromolecules are potentially determined by the kinetics of association and dissociation. However, few studies have yet addressed the consequences of interaction kinetics on HPLC analyses or evaluated the potential application of HPLC methods for the qualitative and quantitative interpretation of macromolecular interaction kinetics. An earlier simulation of small-zone chromatography of interacting molecules (Stevens, F. J. 1986. Biochemistry. 25:981-993) has been modified to incorporate the effects of association/dissociation kinetics on elution behavior. The previous assumption of instantaneous equilibration has been replaced by explicit calculation of partial relaxation of complexed and free constituent mixtures during each iteration of the simulation. In addition, a stochastically based formulation has been introduced to determine a velocity probability distribution that emulates the partial intermixing of free and complexed pools during the iteration cycle. The simulation generates bimodal elution profiles representing stable complexed and free components of mixtures for which interaction is characterized by slow kinetics relative to chromatography run times. For mixtures with rapid kinetics, a single-asymmetric peak results. When tested with a large-zone sample such that a plateau of stable concentration is generated, the simulation reproduces previous characterizations based on evaluations of solute continuity equations. Therefore, HPLC may, in many cases be an appropriate basis for techniques by which to evaluate kinetic and affinity characteristics of interacting biomolecules.

Time difference scanning gel chromatography: Computer simulations

The time difference profile method of gel scanning chromatography developed by Brumbaugh, Saffen and Chun (Biophysical Chemistry, 1979) has been examined by computer simulation. The method is found to produce values for centroid movements that mimic those of the system being examined but are not quantitatively correct. In all cases the time differential "centroid" is larger than that of the concentration derivative (true) centroid and move at a rate slightly faster than the true centroids. This faster rate slowly decreases towards the true rate but does not approach it within reasonable times. This distorted movement reflects the distorted emphasis given to the larger species in the time differential method. The time difference method has been shown to give an adequate measure of the axial dispersion coefficient, L, for single species systems.

Scanning molecular sieve chromatography of interacting protein systems. Simulation of large zone behavior for self-associating solutes undergoing rapid chemical equilibration under kinetic control

Theoretical large zone reaction boundaries for molecular sieve chromatography have been simulated by computer for a self-associating solute undergoing rapid chemical equilibration under kinetic control. These patterns show that the kinetically-controlled reaction rate between the mobile and stationary phases is the principal determinant of the elution boundary profile in molecular sieve chromatography. The overall chemical reaction rate in the mobile phase was found to have a much greater role in a rapidly equilibrating system than did the effect of axial dispersion within the gel matrix.

Isoelectric focusing of a dimerizing solute in rapid chemical equilibrium: comparison of simulation procedures

The approach to isoelectric focusing equilibrium of a rapidly dimerizing solute was simulated by two different computing procedures: a stationary-grid model developed by Cann and Stimpson and a distorted-grid technique derived from the method of Cox. The results given by the two models were virtually identical at all times during the approach to equilibrium. Of the two procedures, the distorted-grid method has an advantage in computing time, while the stationary-grid model is applicable to a broader range of transport experiments. The effect on the focusing experiment of varying the electric field was examined by distorted-grid simulations. When the field was increased, the equilibrium distribution sharpened somewhat and the peak of the concentration profile shifted toward the isoelectric position of the dimer. The rate of approach to equilibrium was approximately proportional to the field strength.

Active enzyme gel chromatography: II. Computer simulations

The behavior of an enzyme undergoing reaction while on a gel chromatography column has been studied by computer simulation using the steady state assumtion for a system with a single enzyme-substrate complex. The profiles of the enzyme-substrate complex, product, and substrate were examined varying the parameters of kcat, flow rate, partition coefficient dispersion, and time. These investigations confirm that much information about both the active enzyme and the product may be obtained by examining the product profile alone, varying the power of applying scanning gel chromatography to active enzyme systems.

Simultation of gradient and band propagation in the centrifuge

A technique is developed for simulating the behavior of both the gradient-forming solute and macromolecular bands in a centrifuge. The change with time of the density gradient due to diffusion and sedimentation of the gradient-forming solute is calculated by a finite difference method, making use of the results of earlier work on the theory of the equilibrium density gradient. Using a perturbation technique, the concentration profiles of dilute bands of macromolecules are then calculated as they sediment and diffuse through the varying supporting gradient. Results of the stimulaion techniques are compared with experiment.

Kinetic control of three-species association reactions in gel chromatography

The gel chromatographic patterns of a monomer-dimer-tetramer system under kinetic control have been studied by computer simulation. In no case do the derivative curves of the concentration profiles exhibit more than bimodality in these systems and in some cases are found to exist as a single, almost symmetric, peak. The monomer-dimer reaction affects the dimer-tetramer reaction only slightly and the same is true for the effect of the dimer-tetramer reaction on that of the monomer-dimer. All systems can be fit, to a first approximation, by an empirical formula which suggests any reaction with a first order half life within one and one half or two orders of the transport time will be under kinetic control.

Approximate solution of the differential equation for the migration of a reversibly reacting substance

A series solution is given for the differential equation describing the transport of a reversibly reacting substance in an infinite rectangular cell. The concentration dependence of the constituent transport coefficients has been approximated by polynomials. The solution converges for short time (t is less than 1 h). Several features of the leading and trailing boundary of monomer-trimer and monomer-dimer-trimer systems are discussed.