Estimation of diffusion constants from single molecular measurement without explicit tracking

Background

Time course measurement of single molecules on a cell surface provides detailed information about the dynamics of the molecules that would otherwise be inaccessible. To extract the quantitative information, single particle tracking (SPT) is typically performed. However, trajectories extracted by SPT inevitably have linking errors when the diffusion speed of single molecules is high compared to the scale of the particle density.

Methods

To circumvent this problem, we develop an algorithm to estimate diffusion constants without relying on SPT. The proposed algorithm is based on a probabilistic model of the distance to the nearest point in subsequent frames. This probabilistic model generalizes the model of single particle Brownian motion under an isolated environment into the one surrounded by indistinguishable multiple particles, with a mean field approximation.

Results

We demonstrate that the proposed algorithm provides reasonable estimation of diffusion constants, even when other methods suffer due to high particle density or inhomogeneous particle distribution. In addition, our algorithm can be used for visualization of time course data from single molecular measurements.

Conclusions

The proposed algorithm based on the probabilistic model of indistinguishable Brownian particles provide accurate estimation of diffusion constants even in the regime where the traditional SPT methods underestimate them due to linking errors.

     

A countercurrent mechanism with transverse diffusion

A simple treatment is given for the stationary state distribution of a substance dissolved in two liquids flowing in opposite directions in two tubes with a common wall that is permeable to the substance. The transverse diffusion is taken into account. It is shown how in the case of sufficiently fast diffusion the permeability constant of the wall can be obtained from the concentrations of the substance at the ends of the tubes, the velocities in the tubes, and the length of the tubes.     

Optical measurement of a solvent-induced isomerization in a proline-containing hexapeptide.

The hexapeptide Gly-Gly-Pro-Tyr-Gly-Gly has been synthesized and its tyrosine residue converted to nitrotyrosine by reaction with tetranitromethane. When diluted from dimethylsulfoxide into aqueous solution, the nitrated hexapeptide undergoes a slow conformational change characterized by a change in the ionization state of the nitrotyrosine group. This slow reaction is not observed with peptides containing nitrotyrosine and no proline. Also, the rate and activation enthalpy of this slow conformational change suggest that it could be due to proline cis-trans isomerization. The possibility of measuring the rate of cis-trans isomerization of proline residues in a polypeptide chain is discussed.     

Calculations suggest a pathway for the transverse diffusion of a hydrophobic peptide across a lipid bilayer

Alamethicin is a hydrophobic antibiotic peptide 20 amino acids in length. It is predominantly helical and partitions into lipid bilayers mostly in transmembrane orientations. The rate of the peptide transverse diffusion (flip-flop) in palmitoyl-oleyl-phosphatidylcholine vesicles has been measured recently and the results suggest that it involves an energy barrier, presumably due to the free energy of transfer of the peptide termini across the bilayer. We used continuum-solvent model calculations, the known x-ray crystal structure of alamethicin and a simplified representation of the lipid bilayer as a slab of low dielectric constant to calculate the flip-flop rate. We assumed that the lipids adjust rapidly to each configuration of alamethicin in the bilayer because their motions are significantly faster than the average peptide flip-flop time. Thus, we considered the process as a sequence of discrete peptide-membrane configurations, representing critical steps in the diffusion, and estimated the transmembrane flip-flop rate from the calculated free energy of the system in each configuration. Our calculations indicate that the simplest possible pathway, i.e., the rotation of the helix around the bilayer midplane, involving the simultaneous burial of the two termini in the membrane, is energetically unfavorable. The most plausible alternative is a two-step process, comprised of a rotation of alamethicin around its C-terminus residue from the initial transmembrane orientation to a surface orientation, followed by a rotation around the N-terminus residue from the surface to the final reversed transmembrane orientation. This process involves the burial of one terminus at a time and is much more likely than the rotation of the helix around the bilayer midplane. Our calculations give flip-flop rates of approximately 10(-7)/s for this pathway, in accord with the measured value of 1.7 x 10(-6)/s.     

Artificial chaperone-assisted refolding in a microchannel

Protein refolding using a simple dilution method in a microchannel often led to the formation of protein aggregates, which bound to the microchannel wall, resulting in low refolding yields. To inhibit aggregation and improve refolding yields, an artificial chaperone-assisted (ACA) refolding, which employed detergents and β-cyclodextrin was used. Model proteins, hen egg white lysozyme and yeast α-glucosidase, were successfully refolded in a microchannel. The microscopic observation showed that the ACA method suppressed protein aggregation and facilitated the refolding of lysozyme, whereas significant aggregation was observed when a simple dilution method was employed. The ACA method increased the lysozyme refolding yield by 40% over the simple dilution approach. Similarly, for α-glucosidase, the refolding yield using the ACA method (ca. 50%) was approximately three times compared with the simple dilution method. The ACA refolding method is a suitable approach to use in the refolding of proteins using a microfluidic system.     

Direct measurement of diffusion in olfactory cilia using a modified FRAP approach

The diffusion coefficient of fluorescein in detached cilia of Xenopus laevis olfactory receptor neurons was measured using spatially-resolved FRAP, where the dye along half of the ciliary length was photobleached and its spatiotemporal fluorescence redistribution recorded. Fitting a one-dimensional numerical simulation of diffusion and photobleaching for 35 cilia resulted in a mean value of the diffusion coefficient (1.20 ± 0.23) · 10(-10)m(2)/s and thus a reduction by a factor of 3.4 compared to free diffusion in aqueous solution.     

Characterizing molecular diffusion in the lens capsule

The lens capsule compartmentalizes the cells of the avascular lens from other ocular tissues. Small molecules required for lens cell metabolism, such as glucose, salts, and waste products, freely pass through the capsule. However, the lens capsule is selectively permeable to proteins such as growth hormones and substrate carriers which are required for proper lens growth and development. We used fluorescence recovery after photobleaching (FRAP) to characterize the diffusional behavior of various sized dextrans (3, 10, 40, 150, and 250 kDa) and proteins endogenous to the lens environment (EGF, γD-crystallin, BSA, transferrin, ceruloplasmin, and IgG) within the capsules of whole living lenses. We found that proteins had dramatically different diffusion and partition coefficients as well as capsule matrix binding affinities than similar sized dextrans, but they had comparable permeabilities. We also found ionic interactions between proteins and the capsule matrix significantly influence permeability and binding affinity, while hydrophobic interactions had less of an effect. The removal of a single anionic residue from the surface of a protein, γD-crystallin [E107A], significantly altered its permeability and matrix binding affinity in the capsule. Our data indicated that permeabilities and binding affinities in the lens capsule varied between individual proteins and cannot be predicted by isoelectric points or molecular size alone.     

Optical measurement of conduction in single demyelinated axons

Demyelination was initiated in Xenopus sciatic nerves by an intraneural injection of lysolecithin over a 2-3-mm region. During the next week macrophages and Schwann cells removed all remaining damaged myelin by phagocytosis. Proliferating Schwann cells then began to remyelinate the axons, with the first few lamellae appearing 13 d after surgery. Action potentials were recorded optically through the use of a potential-sensitive dye. Signals could be detected both at normal nodes of Ranvier and within demyelinated segments. Before remyelination, conduction through the lesion occurred in only a small fraction of the fibers. However, in these particular cases we could demonstrate continuous (nonsaltatory) conduction at very low velocities over long (greater than one internode) lengths of demyelinated axons. We have previously found through loose patch clamp experiments that the internodal axolemma contains voltage-dependent Na+ channels at a density approximately 4% of that at the nodes. These channels alone, however, are insufficient for successful conduction past the transition point between myelinated and demyelinated regions. Small improvements in the passive cable properties of the axon, adequate for propagation at this site, can be realized through the close apposition of macrophages and Schwann cells. As the initial lamellae of myelin appear, the probability of success at the transition zone increases rapidly, though the conduction velocity through the demyelinated segment is not appreciably changed. A detailed computational model is used to test the relative roles of the internodal Na+ channels and the new extracellular layer. The results suggest a possible mechanism that may contribute to the spontaneous recovery of function often seen in demyelinating disease.     

Enhanced dibenzothiophene biodesulfurization in a microchannel reactor

A microchannel reactor system was used in a biodesulfurization process in which the rate of biodesulfurization in the oil/water phase of the microchannel reaction was more than nine-fold that in a batch (control) reaction. In addition, the microchannel reaction system using a bacterial cell suspension degraded alkylated dibenzothiophene that was not degraded by the batch reaction system. This work provides a foundation for the application of a microchannel reactor system consisting of biological catalysts using an oil/water phase reaction.     

Optical characteristics of the plasma of a transverse volume discharge in a Ne/Ar/SiH4 mixture

Results are presented from studies of the characteristics of a transverse volume discharge in a Ne/Ar/SiH₄ mixture at pressures of 5–35 kPa. It is shown that SiI 288.2-nm, H_β 486.1-nm, and NeI 585.3-nm lines and H₂. Lyman bands can be used to monitor the process of destruction of silane molecules. The obtained porous film, consisting of the products of SiH₄ destruction, is of interest for yielding siliceous fullerenes and for application in optoelectronics.     

Label-free determination of the number of biomolecules attached to cells by measurement of the cell's electrophoretic mobility in a microchannel

We developed a label-free method for a determination of the number of biomolecules attached to individual cells by measuring the electrophoretic mobility of the cells in a microchannel. The surface of a biological cell, which is dispersed in aqueous solution, is normally electrically charged and the charge quantity at the cell's surface is slightly changed once antibody molecules are attached to the cell, based on which we detect the attachment of antibody molecules to the surface of individual red blood cells by electrophoretic mobility measurement. We also analyzed the number of antibody molecules attached to the cell's surface using a flow cytometer. We found that there is a clear correlation between the number of antibody molecules attached to the individual cells and the electrophoretic mobility of the cells. The present technique may well be utilized not only in the field of cell biology but also in the medical and pharmaceutical industries.     

Optical measurement of aqueous potassium concentration by a hydrophobic indicator in lipid vesicles

An assay was developed for K+ in aqueous solution at neutral pH. The method was based on the change in optical absorbance of the hydrophobic indicator 7-(n-decyl)-2-methyl-4-(3',5'-dichlorophen-4'-one)indonaphthl++ +-1-ol (MEDPIN) in phospholipid vesicles. Formation of a ternary complex between a valinomycin-K+ pair and the anionic form of MEDPIN in the bilayer resulted in an absorption band at 584 nm. K+ concentration was determined by monitoring the MEDPIN absorbance at 584 nm and MEDPIN quenching of lissamine rhodamine B sulfonylphosphatidylethanolamine (L-RhB-PE) fluorescence by an energy-transfer mechanism. Both the fluorescence intensity and lifetime of L-RhB-PE decreased by more than 25% upon addition of 50 mM K+. Kinetic studies using stopped-flow photometry showed a single-exponential reaction of MEDPIN and valinomycin in vesicles with aqueous K+ (maximum rate 1.7 s-1) that was dependent upon [valinomycin] and [K+]. The lipid surface charge was shown to influence the ratio of anionic to neutral MEDPIN at constant pH, and to alter the sensitivity of MEDPIN absorbance to aqueous [K+]. A 1:20 neutral/negative lipid mole ratio was optimal for K+ detection at pH 7.4. Spectroscopic and kinetic data suggest that the optical response of MEDPIN to K+ involves the formation of a ternary complex between K+, valinomycin and MEDPIN.     

Coarse-grained molecular simulation of diffusion and reaction kinetics in a crowded virtual cytoplasm

We present a general-purpose model for biomolecular simulations at the molecular level that incorporates stochasticity, spatial dependence, and volume exclusion, using diffusing and reacting particles with physical dimensions. To validate the model, we first established the formal relationship between the microscopic model parameters (timestep, move length, and reaction probabilities) and the macroscopic coefficients for diffusion and reaction rate. We then compared simulation results with Smoluchowski theory for diffusion-limited irreversible reactions and the best available approximation for diffusion-influenced reversible reactions. To simulate the volumetric effects of a crowded intracellular environment, we created a virtual cytoplasm composed of a heterogeneous population of particles diffusing at rates appropriate to their size. The particle-size distribution was estimated from the relative abundance, mass, and stoichiometries of protein complexes using an experimentally derived proteome catalog from Escherichia coli K12. Simulated diffusion constants exhibited anomalous behavior as a function of time and crowding. Although significant, the volumetric impact of crowding on diffusion cannot fully account for retarded protein mobility in vivo, suggesting that other biophysical factors are at play. The simulated effect of crowding on barnase-barstar dimerization, an experimentally characterized example of a bimolecular association reaction, reveals a biphasic time course, indicating that crowding exerts different effects over different timescales. These observations illustrate that quantitative realism in biosimulation will depend to some extent on mesoscale phenomena that are not currently well understood.     

Optical density measurement of ultramicrovolumes of liquid

Measurement of the absorbancy of ultramicrovolumes of liquids in necessary to perform in the ultramicro-scale important analytical procedures like determination of the concentrations of substances, of enzymatic activities, etc. The proposed procedure of serial analysis is based on the application of capillary flow cells and of sensitive double-wavelength microphotometer designed in this Institute. Droplets of the sample solution, of standard solution and of the solvent are sucked by means of precision microsyringe into an Uviol glass capillary tube 50–100 μm i.d. with extended tips of smaller diameter which serve the flow cells. the droplets are separated from each other with air or pentadecane layers. During the extrusion of the droplets through the capillary across light probe of the instrument, the absorbancy is recorded. Minimum length of the droplets is 150 μm, and the minimum amount of nucleotide material necessary for the determination is 1·10^?10–5·10^?11 g.     

Optical measurement of presynaptic calcium currents.

Measurements of presynaptic calcium currents are vital to understanding the control of transmitter release. However, most presynaptic boutons in the vertebrate central nervous system are too small to allow electrical recordings of presynaptic calcium currents (I(Ca)pre). We therefore tested the possibility of measuring I(Ca)pre optically in boutons loaded with calcium-sensitive fluorophores. From a theoretical treatment of a system containing an endogenous buffer and an indicator, we determined the conditions necessary for the derivative of the stimulus-evoked change in indicator fluorescence to report I(Ca)pre accurately. Matching the calcium dissociation rates of the endogenous buffer and indicator allows the most precise optical measurements of I(Ca)pre. We tested our ability to measure I(Ca)pre in granule cells in rat cerebellar slices. The derivatives of stimulus-evoked fluorescence transients from slices loaded with the low-affinity calcium indicators magnesium green and mag-fura-5 had the same time courses and were unaffected by changes in calcium influx or indicator concentration. Thus both of these indicators were well suited to measuring I(Ca)pre. In contrast, the high-affinity indicator fura-2 distorted I(Ca)pre. The optically determined I(Ca)pre was well approximated by a Gaussian with a half-width of 650 micros at 24 degrees C and 340 micros at 34 degrees C.     

Optical measurement of age-related calcification in human blood vessels

Vascular calcification is commonly associated with aging. Quantification of calcium accumulation in vessel walls is important in understanding the mechanisms of vascular calcification. To elucidate age-related change of calcification, site dependence of calcification, and the effect of hemodynamic stress on calcification, we measured calcium contents in various blood vessels with atomic emission spectrometry and simulated blood flow in the vessels by computational fluid dynamics. The content of calcium in the arteries increased progressively with aging while there is no change in the veins. The higher accumulation of calcium occurred in the arteries of the lower limb in comparison to the arteries of the upper limb. In the arterial bifurcation, there was the correlation at hemodynamic stress distribution and calcium content. The results of this study quantitatively support clinical findings of nonuniform calcification, and suggest that hemodynamic stress affects vascular calcification.     

Optical verification of a technique for in situ ultrasonic measurement of articular cartilage thickness

Several investigators have used pulse-echo ultrasonics to measure the thickness of articular cartilage in situ. The underlying assumption in all measurements was that the second reflection used in thickness calculations was from the calcified-cartilage/cartilage boundary (tidemark). To investigate this assumption, the thickness of 24 cartilage plugs excised from a human femoral head was measured both ultrasonically and optically. Measurements established that the second reflection was from the tidemark and validated the ultrasonic technique as a method of mapping the thickness distribution of articular cartilage in synovial joints in situ.