Sheared edible oils studied using dissipative particle dynamics and ultra small angle X-ray scattering: TAGwood orientation aggregation and disaggregation.

Previous work on the computer simulation of edible fats and oils showed that triglyceride crystalline nanoplatelets (CNPs) aggregated into cylindrical structures dubbed "TAGwoods". This was experimentally verified using Ultra Small Angle X-ray Scattering experiments. In this paper, the aggregation of these TAGwoods was studied using the fluid simulation technique, Dissipative Particle Dynamics. The intent was to predict the TAGwood aggregation structures which arise via the application of a series of shear rates, [small gamma, Greek, dot above]. The effect of shear on TAGwood orientational order was also investigated. Three aggregation regimes were identified: At shear rates below a certain critical value, 0 < [small gamma, Greek, dot above] < [small gamma, Greek, dot above]t aggregation was enhanced. The value of the critical shear rate depended on the size of the CNPs. With large CNPs possessing a side length of ∼500 nm, the critical shear rate was [small gamma, Greek, dot above]t ≈ 0.6 s-1. However, if the CNPs were smaller with a side length of ∼100 nm, then [small gamma, Greek, dot above]t ≈ 75 s-1. For shear rates above the critical shear rate, [small gamma, Greek, dot above] > [small gamma, Greek, dot above]t aggregation was inhibited. The USAXS data was analyzed using the Unified Fit model and the observations were in accord with the simulation results. Three regimes were identified based on the values of the linear slope P2 of the USAXS data. P2 increased as [small gamma, Greek, dot above] increased, indicating increased aggregation of the TAGwoods as the shear rate was increased. P2 ceased increasing and began to decrease when [small gamma, Greek, dot above] ≈ [small gamma, Greek, dot above]t. With further increases in [small gamma, Greek, dot above], P2 decreased as [small gamma, Greek, dot above] increased further, which is indicative of a decrease in aggregation. The orientational quadrupole order parameter, S = 〈Q33〉 = 1/2〈cos2θ - 1〉, was computed, where θ is the angle between the axis of the TAGwood and the axis of flow, and showed that, for large [small gamma, Greek, dot above], it achieved a near-maximum value. This indicates that at high shear rates, the long axis of the cylindrical TAGwoods aligns in a direction parallel to that of the fluid flow.

Identification of quantitative trait loci for resistance to Xanthomonas campestris pv. campestris in Brassica rapa.

Resistance to six known races of black rot in crucifers caused by Xanthomonas campestris pv. campestris (Pammel) Dowson is absent or very rare in Brassica oleracea (C genome). However, race specific and broad-spectrum resistance (to type strains of all six races) does appear to occur frequently in other brassica genomes including B. rapa (A genome). Here, we report the genetics of broad spectrum resistance in the B. rapa Chinese cabbage accession B162, using QTL analysis of resistance to races 1 and 4 of the pathogen. A B. rapa linkage map comprising ten linkage groups (A01-A10) with a total map distance of 664 cM was produced, based on 223 AFLP bands and 23 microsatellites from a F(2) population of 114 plants derived from a cross between the B. rapa susceptible inbred line R-o-18 and B162. Interaction phenotypes of 125 F(2) plants were assessed using two criteria: the percentage of inoculation sites in which symptoms developed, and the severity of symptoms per plant. Resistance to both races was correlated and a cluster of highly significant QTL that explained 24-64% of the phenotypic variance was located on A06. Two additional QTLs for resistance to race 4 were found on A02 and A09. Markers closely linked to these QTL could assist in the transference of the resistance into different B. rapa cultivars or into B. oleracea.

Active microrheology of networks composed of semiflexible polymers: theory and comparison with simulations.

Based on the results of our computer simulation [Ter-Oganessian et al., Phys. Rev. E 72, 041510 (2005)], we have developed a theoretical description of the motion of a bead, embedded in a network of semiflexible polymers (filaments) and responding to an applied force. The theory reveals the existence of an osmotic restoring force, generated by the piling up of filaments in front of the moving bead and first deduced through computer simulations. The theory predicts that the bead displacement scales like x approximately t(alpha) with time, where alpha = (1/2) in an intermediate-time regime and alpha = 1 in a long-time regime. It also predicts that the compliance varies with a concentration like c(4/3) in agreement with experiment.

A model of hydrogen bond formation in phosphatidylethanolamine bilayers.

We have modelled hydrogen bond formation in phospholipid bilayers formed, in excess water, from lipids with phosphatidylethanolamine (PE) headgroups. The hydrogen bonds are formed between the NH3+ group and either of the PO2- or the (sn2 chain) C=O groups. We used a model that represented the conformational states accessible to a PE headgroup by 17 states and modelled lipid dipole-dipole interactions using a non-local electrostatics theory to include the effects of hydrogen bonding in the aqueous medium. We used Monte-Carlo simulation to calculate equilibrium thermodynamic properties of bilayers in the fluid (T = 340 K) or gel (T = 300 K) phases of the bilayer. We defined Eh to be the difference in free energy between a hydrogen bond formed between a pair of lipid groups, and the energy of hydrogen bonds formed between water and those two groups, and we required its average value, [Eh], to be approximately -0.3kcal/mol (approximately -0.2 X 10(-13) erg) as reported by T.-B. Shin, R. Leventis, J.R. Silvius, Biochemistry 30 (1991) 7491. We found: (i) Eh = -0.9 X 10(-13) erg gave [Eh] = -0.21 X 10(-13) erg (gel phase) and [Eh] = -0.19 X 10(-13) erg (fluid phase). (ii) The relative number of C=O groups on the sn2 chain calculated to take part in interlipid hydrogen bonding in the fluid phase compared to the gel is 1.06 which compares well with the experimental ratio of approximately 1.25 (R.N.A.H. Lewis, R.N. McElhaney, Biophys. J. 64 (1993) 1081). The ratio of such groups taking part in interlipid hydrogen bonding compared to water hydrogen bonding in each phase was calculated to lie between 0.16 and 0.17. (iii) We calculated the distribution of positions of the headgroup moieties, P, O, CH2(alpha), CH2(beta) and N, and found that, in both phases, the O lay furthest from the hydrocarbon chain layer (average approximately 5.3A) with the PO2 and NH3 groups lying at approximately 5A. This results in the P-N dipole lying nearly parallel to the bilayer plane in both phases. The thickness of the headgroup layer underwent essentially no change on going from the gel to the fluid phase. The 2H NMR quadrupole splittings for the alpha and beta CH2 groups were 4.9 and 5.7kHz (fluid phase) and 7.1 and 7.3 kHz (gel phase), respectively, on the assumption of sufficiently rapid rotation around the z-axis. (iv) In both phases, the location of the NH3+ group exhibited a strong peak around 5.2 A into the aqueous medium, with much smaller peaks around 2.6 and 7.8 A, the two CH2 groups exhibited narrower, double-peaked distributions and the O and the PO2 each exhibited a narrow single peak. (v) PE headgroups, in a homogeneous gel phase, exhibited dipolar orientational long-range order in the plane of the bilayer. The distribution of orientation angles exhibited a full width at half height of between approximately 40 degrees and approximately 50 degrees. In a fluid phase no such order was observed. (vi) The number of hydrogen bonds did not differ substantially between the fluid and gel phases. This model is unlikely to display any significant effect of hydrogen bonding upon the "main" hydrocarbon chain melting phase transition at Tm, except, possibly, a broadening of any hysteresis, compared to the case of PC bilayers where interlipid hydrogen bonding is absent.

Atomic force microscope measurements of long-range forces near lipid-coated surfaces in electrolytes.

The interaction of DMPC (L-alpha-dimyristoyl-1,2-diterradecanoyl-sn-glycero-3-phosphoch oli ne, C36H72NO8P) lipid-coated Si3N4 surfaces immersed in an electrolyte was investigated with an atomic force microscope. A long-range interaction was observed, even when the Si3N4 surfaces were covered with nominally neutral lipid layers. The interaction was attributed to Coulomb interactions of charges located at the lipid surface. The experimental force curves were compared with solutions for the linearized as well as with exact solutions of the Poisson-Boltzmann equation. The comparison suggested that in 0.5 mM KCl electrolyte the DMPC lipids carried about one unit of charge per 100 lipid molecules. The presence of this surface charge made it impossible to observe an effective charge density recently predicted for dipole layers near a dielectric when immersed in an electrolyte. A discrepancy between the theoretical results and the data at short separations was interpreted in terms of a decrease in the surface charge with separation distance.

Theory of electrostatic effects in soft biological interfaces using atomic force microscopy.

We calculated the electrostatic force between a planar interface, such as a planar-supported lipid bilayer membrane, and the tip of a stylus on which another lipid bilayer or some other biomacromolecular system might be deposited. We considered styli with rounded tips as well as conical tips. To take into account the effect of dynamical hydrogen-bonded structures in the aqueous phase, we used a theory of nonlocal electrostatics. We used the Derjaguin approximation and identified the systems for which its use is valid. We pointed out where our approach differs from previous calculations and to what extent the latter are inadequate. We found that 1) the nonlocal interactions have significant effects over distances of 10-15 A from the polar zone and that, at the surface of this zone, the effect on the calculated force can be some orders of magnitude; 2) the lipid dipoles and charges are located a distance L from the hydrophobic layer in the aqueous medium and this can have consequences that may not be appreciated if it is ignored; 3) dipoles, located in the aqueous region, can give rise to forces even though the polar layer is unchanged, and if this is ignored the interpretation of force data can be erroneous if an attempt is made to rationalize an observed force with a knowledge of an uncharged surface; 4) the shape of the stylus tip can be very important, and a failure to take this into account can result in incorrect conclusions, a point made by other workers; and 5) when L is nonzero, the presence of charges and dipoles can yield a force that can be nonmonotonic as a function of ionic concentration.

Intersecting polymers in lipid bilayers: cliques, static order parameters and lateral diffusion.

We have modelled a macrolipid polymer composed of lipid molecules (monomers) embedded in a lipid bilayer or monolayer and polymerized via their polar groups. Because of fluctuations perpendicular to the plane of the bilayer, the polar region occupied by the polymer chain possesses sufficient space so that the polymer might exhibit 'self-intersection' if its conformational state is projected onto the plane of the bilayer/monolayer. We represent the plane of the bilayer/monolayer by a triangular lattice. Each site can be occupied by a monomer or be empty (and thus occupied by one of the unpolymerizable lipids which make up the bilayer/monolayer). A macrolipid is represented by a sequence of N monomers connected by N-1 bonds. Bonds may be either short (connecting nearest neighbour monomers) or long (between second neighbour monomers), in accord with the average properties of the spacers between the polymerized lipids. We have carried out computer simulation of this system using the Carmesin-Kremer bond stretching algorithm. Although no two monomers can occupy the same site, bonds may cross each other. We analyzed the dependence of and approximately N2vc and + approximately N2 sigma c, where Nsc and Nmc are the number of bond-crossings in the same macrolipid ('self-crossing') or in two different macrolipids ('mutual-crossing'). For single macrolipids, we confirmed that vc = 3/4 and have found that sigma c approximately 0.52, which we consider supports that sigma c = 1/2. For the dense case with monomer concentration, c = 0.72, we found that vc = 1/2 and that sigma c approximately 0.52 supports that sigma c = 1/2. In the semi-dilute regime (c = 0.2) we found crossover behaviour, although sigma c = 1/2. The total number of bond crossings thus scale like N, independent of concentration. We studied the connectivity of the system by calculating the weight averaged cluster, or 'clique', size. Cliques are defined as being composed of all macrolipids which exhibit at least one crossing bond with one other member of the clique. We found that while the average clique contains about two macrolipids at low concentrations, the clique size approaches the maximum possible value at high concentrations if the macrolipids are sufficiently long. In the latter case a transition appears to occur as the macrolipid length increases. This transition occurs at length = 40 when c = 0.72. These observations should have experimental consequences for the viscoelastic properties of the system.(ABSTRACT TRUNCATED AT 400 WORDS)

Computer simulation of lipid diffusion in a two-component bilayer. The effect of adsorbing macromolecules.

We have modelled the effects of macromolecular adsorption upon lipid lateral diffusion in a two-component lipid bilayer or monolayer, which is at a temperature above both of the main transition temperatures. One set of lipids (binders, b) can bind to the macromolecules with a free energy of binding, FB, while the other set does not bind (non-binders, nb). We assumed that no phase separation of the lipids occurs in the absence of adsorbed macromolecules. We represented the lipid bilayer/monolayer by a triangular lattice, each site of which is occupied by a lipid molecule. Adsorbed macromolecules were represented by hexagons covering nH sites, and we defined a probability per unit of time, p, that a hexagon attempts to adsorb onto the lattice. We considered two sizes of hexagons, nH = 7 (Size-1) and nH = 19 (Size-2) and disallowed or permitted adsorbed hexagons to move laterally on the lattice. We calculate the lipid relative diffusion coefficients, Dnb, and Db, for three characteristic time-regimes, (i) tau c << tau a, (ii) tau c approximately tau a and (iii) tau c >> tau a, where tau c and tau a are the times for proteins to adsorb/desorb or for lipids to move from site to site, respectively. We obtain analytical expressions for Dnb and Db in the first case and calculate them using computer simulation in the other two cases. We found that (i) D alpha (iii) < or = D alpha (ii) < or = D alpha (i) (alpha = nb, b); (ii) D alpha could display a shoulder as a function of FB for low values of p; (iii) compared to cases in which lateral diffusion was disallowed, the lateral diffusion of absorbed hexagons appeared to have little effect on Dnb, but could cause Db to increase by 50%. (iv) Scatter in the calculated values of D via simulation appeared to be largest for Size-1 hexagons, and could be understood as a consequence of the large interfacial region between areas free of hexagons and areas 'covered' by hexagons. Our results suggest that it is advisable to measure Db, since Dnb might show little change from 1.0 for the values of F and p appropriate to the system being studied.

Cell surfaces and fractal dimensions.

The perimeters of the surface membranes of some different cell types have been digitized from electron micrographs and the data analysed in order to discover whether the perimeter can be described by a fractal dimension, df. Micrographs obtained at various magnifications and subsequently enlarged by different amounts have been used. Values of df ranging from 1.02 to 1.34 were manifested over a scale length of about one order of magnitude. Values of df were independent of the magnification, and were the same for cells of the same type. Possible implications of these results are discussed.

Line defects in gel phase lipid monolayers.

We investigate various models of the hydrolysis of gel-phase phosphatidylcholine monolayers by phospholipase A2 (Grainger et al. (1989) FEBS Lett. 252, 73-82). We assume that the probability of hydrolysis of a given lipid depends only upon how many of its nearest neighbour lipids have already been hydrolysed. We find that the experimental data are consistent with a model in which line defects exist in the gel phase and that lipids on such defects are more easily hydrolysed than the other gel-phase lipids. Based on this model, we calculate the course of hydrolysis of a gel-phase region possessing line defects, and we suggest how such a structure might be made and the model tested. An experiment, similar to that proposed by us, has been carried out by Grainger et al. (1990) Biochim. Biophys. Acta 1023, 365-379). We also calculate the fractal dimension, df, of the interface created by the hydrolytic process and show that a measurement of df might identify how this process proceeds.

Peak locater for multicomponent spectra.

A sliding pivot technique capable of locating component peak maxima of multicomponent spectra is presented. The locations of peak maxima obtained in this way are shown to be the same as those of the minima in the second derivative. A major advantage over the second-derivative test is simply that derivative spectra are not needed. The sliding pivot technique requires only the original spectrum to locate the component peak maxima and consequently reduces the noise enhancement factor. The deconvoluted Fourier transform infrared spectrum of purple membrane is analyzed and compared to a Gaussian analysis and a second-derivative analysis. The sliding pivot technique identifies a band missed by second-derivative analysis.

Models of protein lateral arrangements in lipid bilayer membranes. Application to electron spin resonance studies of cytochrome c oxidase.

We consider the situation of integral membrane proteins in a lipid bilayer matrix where the size of the polar group of the protein is important in determining the lateral packing of the proteins. We represent the cross-section of the protein hydrophobic core as a hexagon moving on a lattice, and represent the projection of the polar group onto the plane of the bilayer as a shape, parts of which overlap the hexagon. Lattice sites represent lipid molecules. We calculate the fraction of lipid molecules which are adjacent to the hydrophobic core of at least one protein. We use this data to consider the "motion restricted" spectrum observed in electron spin resonance (ESR) probe studies, and compute the dependence of the "motion restricted" fraction upon protein concentration. The resulting curves can be used to analyse ESR data in order to deduce the size and shape of the proteins' polar segment. We have used the range of models examined to study the dependence upon protein concentration of the particular case of the "motion restricted" spectrum of a spin-labelled lipid freely diffusing or, alternatively, covalently bound to cytochrome c oxidase. We find that our calculations are in accord with a model where approximately 60 lipid molecules can fit around an isolated such protein in both halves of the bilayer, and where the polar segment is substantially anisotropic and extends laterally beyond the limits of the hydrophobic core. The latter is in accord with what is known about the structure of cytochrome c oxidase. We indicate further measurements that should be performed in order to establish more definitively the dependence of the "motion restricted" component upon protein concentration, giving the lipid protein ratios at which they should be performed, and we make predictions concerning the results. Finally we argue for a particular unified way of plotting experimental data.

The lateral distribution of cholesterol in the plane of lipid multibilayers.

We consider three models of cholesterol distribution in the plane of a bilayer of DMPC. We analyse recent 2H-NMR data obtained from deuterated fluorescent probes and show that, on the characteristic time-scale of 2H-NMR, it is in accord with a random distribution of cholesterol in a fluid-like DMPC bilayer in a single phase at least for T greater than or approximately equal to 35 degrees C and for 0 less than or equal to c less than or equal to 0.42.

Thermodynamics of glycophorin in phospholipid bilayer membranes.

We have developed a model of glycophorin in a phospholipid bilayer membrane in order to study the thermodynamics of this system and to understand the detailed behavior of recent calorimetric data. We assume that the larger glycophorin polar group can be considered as either adopting a pancakelike conformation at the bilayer interface (D state) or be directed generally away from the interface (U state) [Ruppel, D., Kapitza, H.G., Galla, H.J., Sixl, F., & Sackmann, E. (1982) Biochim. Biophys. Acta 692, 1-17]. Lipid hydrocarbon chains are described either as excited (e state) with high energy and relatively many gauche conformers or as generally extended (g state) with low energy. We performed a Monte-Carlo simulation using the Glauber and Kawasaki procedures on a triangular lattice which represents the plane of half of the bilayer. Lattice sites can be occupied either by lipid hydrocarbon chains or by model glycophorin alpha-helical hydrophobic cores. The states D and U are represented by hexagons of different sizes in the plane of the lattice, and the hard core repulsion between two such polar groups is accounted for by forbidding hexagon-hexagon overlap. We have studied the effect of having the glycophorin polar group interact in various ways with the lipid bilayer. We find that the protein polar group in its D state interacts, either directly or indirectly, with the lipid bilayer so as to reduce the effective lateral pressure acting on the lipid hydrocarbon chains by about 3 dyn/cm. Polar groups in their U states do not reduce this lateral pressure.(ABSTRACT TRUNCATED AT 250 WORDS)

Protein lateral distribution in lipid bilayer membranes. Applications to ESR studies.

We analyse recent ESR measurements on Ca2+ ATPase and Myelin proteolipid apoprotein reconstituted in phosphatidylcholine bilayer membranes. Our intention is to discover whether the measurements indicate significant protein-protein repulsive or attractive interactions. In order to do so we have studied a model of a lipid bilayer membrane containing transbilayer proteins. It represents the proteins by hexagons moving on a triangular lattice interacting via an energy E0 which can be attractive, repulsive or zero. The last-named represents that all of the Ca2+ ATPase data is best described either by the "random" model or, possibly, by one in which there is a small repulsive interaction, but not by the "annulus" model or one in which there is always at least one layer of lipid chains between every pair of proteins. We find that all of the Myelin PLA data is best described by a "random" distribution of hexamers and not by an "annulus" model of hexamers. We suggest measurements that can be done in order to unambiguously settle the question of whether these systems are best described by a "random"-type model or an "annulus"-type model.

Electron spin resonance and steady-state fluorescence polarization studies of lipid bilayers containing integral proteins.

We derive equations that describe changes in the steady-state fluorescence polarization of the probe 1,6-diphenyl-1,3,5-hexatriene (DPH) or in the spectrum of electron spin resonance (ESR) nitroxide spin-labeled lipid probes as a function of the intrinsic molecule concentration in lipid bilayer membranes. We make use of an assumption used by us in an earlier paper. The equations are independent of any membrane model. They are valid when a DPH probe or a spin-labeled chain is equivalent to an unlabeled lipid hydrocarbon chain only as far as their general space-filling properties are concerned. We consider cases where the bilayer is either in a single homogeneous phase or in a two-phase region. We apply our equations to analyze ESR data from delipidated sarcoplasmic reticulum membranes and from egg yolk phosphatidylcholine bilayers containing Ca2+-ATPase, and DPH data from dipalmitoylphosphatidylcholine (DPPC) bilayers containing Ca2+-ATPase, both for T greater than Tc. The following conclusions were derived: (i) Ca2+-ATPase oligomers are "randomly" distributed, for the concentrations studied, in the fluid phase. (ii) There is no fixed stoichiometric ratio of "boundary" lipids and oligomers. (iii) Between 24k and 28k lipid molecules are able to surround each isolated oligomer composed of k Ca2+-ATPase monomers. Finally, we apply our equations to analyze DPH studies on DPPC bilayers containing Ca2+-ATPase for T less than Tc. We find that the results reported are in accord with the predictions of the model. In the Appendix, we show that an analytical expression for probabilities used by us is in very good agreement with the results of computer simulation.

Theoretical studies of phospholipid bilayers and monolayers. Perturbing probes, monolayer phase transitions, and computer simulations of lipid-protein bilayers.

This paper describes some mathematical models for studying properties of lipid bilayer membranes. It is shown that there is evidence from fluorescent probe and electron spin resonance studies that some integral proteins are "randomly" distributed in the plane of a lipid bilayer for T greater than Tc, so that protein-protein "contacts" can occur. This implies that there is no permanent annulus of lipids around these proteins, so that there is no unique stoichiometric ratio of "boundary lipids" to protein. Computer simulation techniques are reviewed and comments are made about some recently introduced methods. Dynamical models of lipid--integral protein bilayers are outlined and it is shown that much differential scanning calorimetry, freeze-fracture, X-ray, and 2H nuclear magnetic resonance data can be understood. It is predicted that specific heat curves should show a rise at a temperature, TK less than Tc, at which a protein-rich phase starts to "melt"; that dipalmitoyl phosphatidylcholine (DPPC) hydrocarbon chains in such a protein-rich phase should exhibit some static disorder down to approximately -10 degrees C, around which they should freeze noncooperatively; and that the crossing of phase boundaries, as protein concentration expressed as mole fraction X changes just below Tc, should be reflected in a complicated dependence of protein lateral motion upon X. Models to study the liquid condensed-liquid expanded (LC-LE) transition of phosphatidylcholine (PC) monolayers at the air-water interface are described. The rounding in the LC phase of pressure-area isotherms are understood as the melting of microscopic gel-phase "domains" into macroscopic regions of fluid-phase lipids. There is some experimental support for this. Finally, to model the difference between a monolayer and the corresponding bilayer, an interaction is introduced between the two halves of a bilayer. It is shown to be very weak in the case of PC bilayers and it is outlined in the paper how monolayer and bilayer thermodynamics can be related. It was found that the internal lateral pressure of a DPPC bilayer is approximately 30 dyn/cm (1 dyn = 10 microN).

The thermodynamic properties of mixed phospholipid bilayers: a theoretical analysis.

A theoretical model is presented with the intention of describing lateral phase separations in binary lipid mixtures in which the acyl chains of the components differ in their length. The model includes explicitly interactions between the acyl chains and between polar heads of the lipid molecules. Phase diagrams and thermodynamic properties of binary lipid mixtures were calculated using a wide range of interaction parameters. It is shown that the occurrence of immiscibility in the gel phase is related to the interactions between the polar heads of the lipid molecules. The calculated results for binary lipid mixtures are compared with the available experimental data. In particular, the calculated specific heat for dilauroyl phosphatidylcholine-distearoyl phosphatidylcholine is in reasonable agreement with experimental results obtained from differential scanning calorimetry measurements.

On computer simulation methods used to study models of two-component lipid bilayers.

We show that the use of a computer simulation method introduced to calculate the equilibrium thermodynamic properties of a model of a two-component lipid bilayer membrane [Freire, E., & Snyder, B. (1980) Biochemistry 19, 88-94] is incorrect. This is done by comparing the method to that of Metropolis, which has been proven to generate equilibrium distribution of that model, and by showing that back-processes have been omitted in the implicit master equation of Freire and Snyder. We have illustrated this explicitly by first generating distributions according to the method of Freire and Snyder and then allowing the system to relax via the Kawasaki method, which uses the technique of Metropolis. We show that relaxation to a different distribution occurs. We also remark that the cluster distributions generated by the Freire-Snyder method are substantially different from those occurring in equilibrium distributions. Thus, conclusions about equilibrium thermodynamic properties such as specific heats and transition enthalpies or about transport properties or cluster properties at equilibrium cannot be drawn from the results obtained by using this method. Finally, we point out that the method of Freire and Snyder is appropriate to so-called aggregation models, which have been used to study irreversible growth; and we suggest biological systems that might be simulated by their method.