Lateral diffusion and aggregation. A Monte Carlo study.

Aggregation in a lipid bilayer is modeled as cluster-cluster aggregation on a square lattice. In the model, clusters carry out a random walk on the lattice, with a diffusion coefficient inversely proportional to mass. On contact, they adhere with a prescribed probability, rigidly and irreversibly. Monte Carlo calculations show that, as expected, rotational diffusion of the aggregating species is highly sensitive to the initial stages of aggregation. Lateral diffusion of an inert tracer obstructed by the aggregate is a sensitive probe of the later stages of aggregation. Cluster-cluster aggregates are much more effective barriers to lateral diffusion of an inert tracer than the same area fraction of random point obstacles is, but random point obstacles are more effective barriers than the same area fraction of compact obstacles. The effectiveness of aggregates as obstacles is discussed in terms of particle-particle correlation functions and fractal dimensions. Results are applicable to aggregation of membrane proteins, and at least qualitatively to aggregation of gel-phase lipid during lateral phase separation.     

Anomalous diffusion due to binding: a Monte Carlo study.

In classical diffusion, the mean-square displacement increases linearly with time. But in the presence of obstacles or binding sites, anomalous diffusion may occur, in which the mean-square displacement is proportional to a nonintegral power of time for some or all times. Anomalous diffusion is discussed for various models of binding, including an obstruction/binding model in which immobile membrane proteins are represented by obstacles that bind diffusing particles in nearest-neighbor sites. The classification of binding models is considered, including the distinction between valley and mountain models and the distinction between singular and nonsingular distributions of binding energies. Anomalous diffusion is sensitive to the initial conditions of the measurement. In valley models, diffusion is anomalous if the diffusing particles start at random positions but normal if the particles start at thermal equilibrium positions. Thermal equilibration leads to normal diffusion, or to diffusion as normal as the obstacles allow.     

Sources of anomalous diffusion on cell membranes: a Monte Carlo study

A stochastic random walk model of protein molecule diffusion on a cell membrane was used to investigate the fundamental causes of anomalous diffusion in two-dimensional biological media. Three different interactions were considered: collisions with fixed obstacles, picket fence posts, and capture by, or exclusion from, lipid rafts. If motion is impeded by randomly placed, fixed obstacles, we find that diffusion can be highly anomalous, in agreement with previous studies. In contrast, collision with picket fence posts has a negligible effect on the anomalous exponent at realistic picket fence parameters. The effects of lipid rafts are more complex. If proteins partition into lipid rafts there is a small to moderate effect on the anomalous exponent, whereas if proteins are excluded from rafts there is a large effect on the anomalous exponent. In combination, these mechanisms can explain the level of anomaly in experimentally observed membrane diffusion, suggesting that anomalous diffusion is caused by multiple mechanisms whose effects are approximately additive. Finally, we show that the long-range diffusion rate, D(macro), estimated from fluorescence recovery after photobleaching studies, can be much smaller than D(micro), the small-scale diffusion rate, and is highly sensitive to obstacle densities and other impeding structures.     

Monte Carlo simulation of diffusion and reaction in two-dimensional cell structures.

Diffusion and reaction processes control the dynamics of many different biological systems. For example, tissue respiration can be limited by the delivery of oxygen to the cells and to the mitochondria. In this case, oxygen is small and travels quickly compared with the mitochondria, which can be considered as immobile reactive traps in the cell cytoplasm. A Monte Carlo theoretical investigation quantifying the interplay of diffusion, reaction, and structure on the reaction rate constant is reported here for diffusible particles in two-dimensional, reactive traps. The placement of traps in overlapping, nonoverlapping, and clustered spatial arrangements can have a large effect on the rate constant when the process is diffusion limited. However, under reaction-limited conditions the structure has little effect on the rate constant. For the same trap fractions and reactivities, nonoverlapping traps have the highest rate constants, overlapping traps yield intermediate rate constants, and clustered traps have the lowest rate constants. An increase in the particle diffusivity in the traps can increase the rate constant by reducing the time required by the particles to reach reactive sites. Various diffusive, reactive, and structural conditions are evaluated here, exemplifying the versatility of the Monte Carlo technique.     

Lateral diffusion in an archipelago. The effect of mobile obstacles.

Lateral diffusion of mobile proteins and lipids (tracers) in a membrane is hindered by the presence of proteins (obstacles) in the membrane. If the obstacles are immobile, their effect may be described by percolation theory, which states that the long-range diffusion constant of the tracers goes to zero when the area fraction of obstacles is greater than the percolation threshold. If the obstacles are themselves mobile, the diffusion constant of the tracers depends on the area fraction of obstacles and the relative jump rate of tracers and obstacles. This paper presents Monte Carlo calculations of diffusion constants on square and triangular lattices as a function of the concentration of obstacles and the relative jump rate. The diffusion constant for particles of various sizes is also obtained. Calculated values of the concentration-dependent diffusion constant are compared with observed values for gramicidin and bacteriorhodopsin. The effect of the proteins as inert obstacles is significant, but other factors, such as protein-protein interactions and perturbation of lipid viscosity by proteins, are of comparable importance. Potential applications include the diffusion of proteins at high concentrations (such as rhodopsin in rod outer segments), the modulation of diffusion by release of membrane proteins from cytoskeletal attachment, and the diffusion of mobile redox carriers in mitochondria, chloroplasts, and endoplasmic reticulum.     

Diffusion influenced binary reactive processes in membranes involving identical particles: a Monte Carlo study

Simple random walk simulations on triangular lattices were performed in order to obtain a basic quantitative understanding of the kinetics of diffusion influenced binary reactive processes of membrane associated peptides or proteins within the two dimensionality of lipid bilayers. The results of the Monte Carlo simulations are compared with various formal approximate steady-state approaches, such as presented by Keizer [Acc. Chem. Res., 18 (1985) 235-241] in the context of statistical nonequilibrium thermodynamics or by Hardt [Biophys. Chem., 10 (1979) 239-243], based on the well known work of Delbrück and Adam. For diffusion controlled binary reactions of identical particles, nice agreement with the numerically simulated values is found in the low concentration limit for both Hardt's and Keizer's approach. For the latter a fluctuating steady-state particle source has to be considered. The dependence of the steady-state rate coefficient on system size is investigated, and the results are compared to the work of Swartz and Peacock-López [J. Chem. Phys., 95 (4) (1991) 2727-2731]. In order to elucidate the results, a practical application is considered. An application to a dimerization reaction on vesicles of typical experimental dimensions is given.     

Monte Carlo simulation of diffusion of adsorbed proteins

We present the results of three-dimensional lattice Monte Carlo simulations of protein diffusion on the liquid-solid interface in a wide temperature range including the most interesting temperatures (from slightly below T(f) and up to T(c), where T(f) and T(c) are the folding and collapse temperatures). For the model under consideration (27 monomers of two types), the temperature dependence of the diffusion coefficient is found to obey the Arrhenius law with the normal value (approximately 10(-2)-10(-3) cm(2)/s) of the preexponential factor. Proteins 2000;39:76-81.     

The structure of polymer chains in confinement. A Monte Carlo study

A coarse-grained model of polymer star chains confined in two parallel impenetrable surfaces, which were attractive for polymer beads was studied. The flexible homopolymer chains were built of united atoms whose positions in space were restricted to vertices of a simple cubic lattice. The chains were modeled in good solvent conditions and, thus, there were no long-range specific interactions between polymer beads—only the excluded volume was present. The influence of the polymer density and the distances between the confining surfaces on the properties of star-branched polymers was studied. It is shown that the chains adsorbed on one surface could change their position so that they swap between both surfaces with frequency depending on the size of the slit and on the density of the system only. The increase of the polymer density diminished the frequency of jumps and caused that chains became only partially adsorbed. The analysis of structural elements of chains showed that the increase of the density of the system leads to increase of the number of bridges connecting the two adsorbing surfaces, thus, the frequency of jumps between them decreases.     

A Monte Carlo study of the dynamics of G-protein activation.

To link quantitatively the cell surface binding of ligand to receptor with the production of cellular responses, it may be necessary to explore early events in signal transduction such as G-protein activation. Two different model frameworks relating receptor/ligand binding to G-protein activation are examined. In the first framework, a simple ordinary differential equation model is used to describe receptor/ligand binding and G-protein activation. In the second framework, the events leading to G-protein activation are simulated using a dynamic Monte Carlo model. In both models, reactions between ligand-bound receptors and G-proteins are assumed to be diffusion-limited. The Monte Carlo model predicts two regimes of G-protein activation, depending upon whether the lifetime of a receptor/ligand complex is long or short compared with the time needed for diffusional encounters of complexes and G-proteins. When the lifetime of a complex is relatively short compared with the diffusion time, the movement of ligand among free receptors by binding and unbinding ("switching") significantly enhances G-protein activation. Receptor antagonists dramatically reduce G-protein activation and, thus, signal transduction in this case, and significant clustering of active G-proteins near receptor/ligand complexes results. The simple ordinary differential equation model poorly predicts G-protein activation for this situation. In the alternative case, when diffusion is relatively fast, ligand movement among receptors is less important and the simple ordinary differential equation model and Monte Carlo model results are similar. In this case, there is little clustering of active G-proteins near receptor/ligand complexes. Results also indicate that as the GTPase activity of the alpha-subunit decreases, the steady-state level of alpha-GTP increases, although temporal sensitivity is compromised.     

Lateral diffusion of gramicidin S, M-13 coat protein and glycophorin in bilayers of saturated phospholipids. Mean field and Monte Carlo studies

We have developed a general model that relates the lateral diffusion coefficient of one isolated large intrinsic molecule (mol. wt. greater than or approximately 1000) in a phosphatidylcholine bilayer to the static lipid hydrocarbon chain order. We have studied how protein lateral diffusion can depend upon protein-lipid interactions but have not investigated possible non-specific contributions from gel-state lattice defects. The model has been used in Monte Carlo simulations or in mean-field approximations to study the lateral diffusion coefficients of Gramicidin S, the M-13 coat protein and glycophorin in dimyristoyl- and dipalmitoylphosphatidylcholine (DMPC and DPPC) bilayers as functions of temperature. Our calculated lateral diffusion coefficients for Gramicidin S and the M-13 coat protein are in good agreement with what has been observed and suggest that Gramicidin S is in a dimeric form in DMPC bilayers. In the case of glycophorin we find that the 'ice breaker' effect can be understood as a consequence of perturbation of the lipid polar region around the protein. In order to understand this effect is necessary that the protein hydrophilic section perturb the polar regions of at least approx. 24 lipid molecules, in good agreement with the numbers of 29-30 measured using 31P-NMR. Because of lipid-lipid interactions this effect extends itself out to four or five lipid layers away from the protein so that the hydrocarbon chains of between approx. 74 and approx. 108 lipid molecules are more disordered in the gel phase, so contributing less to the transition enthalpy, in agreement with the numbers of 80-100 deduced from differential scanning calorimetry (DSC). An understanding of the abrupt change in the diffusion coefficient at a temperature below the main bilayer transition temperature requires an additional mechanism. We propose that this change may be a consequence of a 'coupling-uncoupling' transition involving the protein hydrophilic section and the lipid polar regions, which may be triggered by the lipid bilayer pretransition. Our calculation of the average number of gauche bonds per lipid chain as a function of temperature and distance away from an isolated polypeptide or integral protein shows the extent of statically disordered lipid around such molecules. The range of this disorder depends upon temperature, particularly near the main transition.     

Monte Carlo analysis of obstructed diffusion in three dimensions: application to molecular diffusion in organelles.

Molecular transport in the aqueous lumen of organelles involves diffusion in a confined compartment with complex geometry. Monte Carlo simulations of particle diffusion in three dimensions were carried out to evaluate the influence of organelle structure on diffusive transport and to relate experimental photobleaching data to intrinsic diffusion coefficients. Two organelle structures were modeled: a mitochondria-like long closed cylinder containing fixed luminal obstructions of variable number and size, and an endoplasmic reticulum-like network of interconnected cylinders of variable diameter and density. Trajectories were computed in each simulation for >10(5) particles, generally for >10(5) time steps. Computed time-dependent concentration profiles agreed quantitatively with analytical solutions of the diffusion equation for simple geometries. For mitochondria-like cylinders, significant slowing of diffusion required large or wide single obstacles, or multiple obstacles. In simulated spot photobleaching experiments, a approximately 25% decrease in apparent diffusive transport rate (defined by the time to 75% fluorescence recovery) was found for a single thin transverse obstacle occluding 93% of lumen area, a single 53%-occluding obstacle of width 16 lattice points (8% of cylinder length), 10 equally spaced 53% obstacles alternately occluding opposite halves of the cylinder lumen, or particle binding to walls (with mean residence time = 10 time steps). Recovery curve shape with obstacles showed long tails indicating anomalous diffusion. Simulations also demonstrated the utility of measurement of fluorescence depletion at a spot distant from the bleach zone. For a reticulum-like network, particle diffusive transport was mildly reduced from that in unobstructed three-dimensional space. In simulated photobleaching experiments, apparent diffusive transport was decreased by 39-60% in reticular structures in which 90-97% of space was occluded. These computations provide an approach to analyzing photobleaching data in terms of microscopic diffusive properties and support the paradigm that organellar barriers must be quite severe to seriously impede solute diffusion.     

A Monte Carlo study of the self-assembly of bacteriorhodopsin

Bacteriorhodopsin (BR) and specific lipid molecules self-assemble into a quasi two-dimensional lattice structure known as the purple membrane (PM). In the PM, BR molecules exist in a trimeric form with lipid molecules present in the space enclosed by each trimeric unit and in the inter-trimer space. These trimeric units, which have a roughly circular cross-section, are arranged in hexagonal patterns with long-ranged crystalline order. In this work, we investigate the self-assembly of BR in the PM via Monte Carlo simulations of a two-dimensional model of the membrane and proteins. The protein molecules are modeled as 120 degrees sectors of a circle and the lipid molecules enter into the model through effective protein-protein interactions. The sectors cannot overlap with each other, and in addition to this excluded volume interaction there are site-site attractive interactions between specific points of the proteins to mimic interactions between helices on the proteins and lipid-induced interactions. At low values of the attractive well depth, the proteins are found in the monomeric form at all concentrations. At moderate and high values of the attractive well depth, trimers are formed as the concentration increases, and with a further increase in concentration the trimers organize into a hexagonal lattice. The interactions between the proteins and those induced by the intra-trimer lipids play an equally important role in the formation of trimers and the lattice. The lipids in the inter-trimer space cause the trimers to orient in a specific direction in the hexagonal crystal lattice.     

A multiple-start Monte Carlo docking method.

We present a method to search for possible binding modes of molecular fragments at a specific site of a potential drug target of known structure. Our method is based on a Monte Carlo (MC) algorithm applied to the translational and rotational degrees of freedom of the probe fragment. Starting from a randomly generated initial configuration, favorable binding modes are generated using a two-step process. An MC run is first performed in which the energy in the Metropolis algorithm is substituted by a score function that measures the average distance of the probe to the target surface. This has the effect of making buried probes move toward the target surface and also allows enhanced sampling of deep pockets. In a second MC run, a pairwise atom potential function is used, and the temperature parameter is slowly lowered during the run (Simulated Annealing). We repeat this procedure starting from a large number of different randomly generated initial configurations in order to find all energetically favorable docking modes in a specified region around the target. We test this method using two inhibitor-receptor systems: Streptomyces griseus proteinase B in complex with the third domain of the ovomucoid inhibitor from turkey, and dihydrofolate reductase from E. coli in complex with methotrexate. The method could consistently reproduce the complex found in the crystal structure searching from random initial positions in cubes ranging from 25 A to 50 A about the binding site. In the case of SGPB, we were also successful in docking to the native structure. In addition, we were successful in docking small probes in a search that included the entire protein surface.     

Monte Carlo simulation of diffusion and reaction in water radiolysis – a study of reactant `jump through' and jump distances

In Monte Carlo simulations of water radiolysis, the diffusion of reactants can be approximated by “jumping” all species randomly, to represent the passage of a short period of time, and then checking their separations. If, at the end of a jump, two reactant species are within a distance equal to the reaction radius for the pair, they are allowed to react in the model. In principle, the possibility exists that two reactants could “jump through” one another and end up with a separation larger than the reaction radius with no reaction being scored. Ignoring this possibility would thus reduce the rate of reaction below that intended by such a model. By making the jump times and jump distances shorter, any error introduced by `jump through' is made smaller. This paper reports numerical results of a systematic study of `jump through' in Monte Carlo simulations of water radiolysis. With a nominal jump time of 3 ps, it is found that more than 40% of the reactions of the hydrated electron with itself and of the H atom with itself occur when reactions during `jump through' are allowed. For all other reactions, for which the effect is smaller, the contributions of `jump through' lie in the range l%–16% of the total. Corrections to computed rate constants for two reactions are evaluated for jump times between 0.1 and 30 ps. It is concluded that jump-through corrections are desirable in such models for jump times that exceed about 1 ps or even less. In a separate study, we find that giving all species of a given type the same size jump in a random direction yields results that are indistinguishable from those when the jump sizes are selected from a Gaussian distribution. In this comparison, the constant jump size is taken to be the root-mean-square jump size from the Gaussian distribution. Received: 8 September 1997 / Accepted in revised form: 27 October 1997     

Insertion and hairpin formation of membrane proteins: a Monte Carlo study.

Some particular effects of a lipid membrane on the partitioning and the concomitant folding processes of model proteins have been investigated using Monte Carlo methods. It is observed that orientational order and lateral density fluctuations of the lipid matrix stabilize the orientation of helical proteins and induce a tendency of spontaneous formation of helical hairpins for helices longer than the width of the membrane. The lateral compression of the lipids on a hairpin leads to the extrusion of a loop at the trans side of the membrane. The stability of the hairpin can be increased by the design of appropriate groups of hydrophilic and hydrophobic residues at the extruded loop. It is shown that in the absence of lipids the orientation of proteins is not stable and the formation of hairpins is absent. Some analogies between the formation of helical hairpins in membranes and the formation of hairpins in polymer liquid crystals are discussed. The simulations indicate that the insertion process follows a well-defined pattern of kinetic steps.     

A Monte Carlo study of the amylosic chain conformation

Monte Carlo studies of the unperturbed amylosic chain conformation have been carried out in the approximation of separable chain configuration energies. Sample chains of arbitrary chain length have been generated so as to be distributed consistent with refined estimates of the configuration energy and thus suitable for evaluation of averages of the desired configuration-dependent properties. Perspective drawings of representative chains from the Monte Carlo sample have been made for comparison with standard idealizations of amylosic chain conformation. He molecular model employed generates a randomly coiling chain possessing perceptible regions of left-handed pseudohelical backbone trajectory. Distribution functions for the end-to-end distance of short amylosic chains disclose some propensity for the chain to suffer self-intersections at sort range in the chain sequence, which may vitiate the usual amylosic chain models based on the assumed independence of sets of glycosidic linkage torsion angles. The amylosic persistence vector and persistence length have been calculated as a function of chain length for the chain model employed.     

Lateral diffusion in biological membranes. A normal-mode analysis of diffusion on a spherical surface.

A new approach is described for the analysis of lateral diffusion in biological membranes. It is shown that a suitably defined first moment of the concentration distribution on a spherical surface decays as a single exponential with a relaxation rate proportional to the diffusion coefficient and inversely proportional to the square of the radius of the sphere. The approach is illustrated with an example of fluorescence redistribution after photobleaching of membrane proteins in a spectrin-deficient spherocytic mouse erythrocyte membrane.     

Monte Carlo study of cycloamylose: chain conformation, radius of gyration, and diffusion coefficient

Cyclic (1 --> 4)-alpha-D-glucan chains with or without excluded volume have been collected from a huge number (about 10(7)) of linear amylosic chains generated by the Monte Carlo method with a conformational energy map for maltose, and their mean-square radii of gyration and translational diffusion coefficients D (based on the Kirkwood formula) have been computed as functions of x (the number of glucose residues in a range from 7 to 300) and the excluded-volume strength represented by the effective hard-core radius. Both /x and D in the unperturbed state weakly oscillate for x < 30 and the helical nature of amylose appears more pronouncedly in cyclic chains than in linear chains. As x increases, these properties approach the values expected for Gaussian rings. Though excluded-volume effects on them are always larger in cycloamylose than in the corresponding linear amylose, the ratios of and the hydrodynamic radius of the former to the respective properties of the latter in good solvents can be slightly lower than or comparable to the (asymptotic) Gaussian-chain values when x is not sufficiently large. An interpolation expression is constructed for the relation between the gyration-radius expansion factors for linear and cyclic chains from the present Monte Carlo data and the early proposed asymptotic relation with the aid of the first-order perturbation theories.     

Monte Carlo studies of a model for lipid-gramicidin A bilayers.

This paper presents results of Monte Carlo simulations of a full bilayer of 200 lipid chains and one gramicidin A dimer. Simulations are described for systems with lipid chains of 14, 16, and 18 carbons, respectively. Using accepted potential functions to calculate interactions between all non-hydrogen atoms a Monte Carlo configuration sampling is generated from which order parameter profiles are calculated and specific configurations are displayed. Results are compared with experimental data for lipid-gramicidin bilayers.     

Vacancy clustering and diffusion in germanium using kinetic lattice Monte Carlo simulations

We investigated vacancy-assisted self-diffusion in germanium by means of kinetic lattice Monte Carlo (KLMC) simulations below the melting temperature, for a vacancy concentration of 1 × 10¹⁸/cm³. At higher temperatures, fewer clusters formed, but there was less variation in the number of clusters than at lower temperatures as the time increased. Equilibrium diffusivities in the clustering region were 10² lower than those of free vacancies in the initial stage of KLMC simulations. They were expressed according to three temperature regimes: 6.5 × 10^? 4 exp(–0.35/k _B T) cm²/s at temperatures above 1100 K, 5.2 × 10⁵ exp(–2.32/k _B T) cm²/s at temperatures of 900–1100 K and 6.0 × 0^–7 exp(–0.19/k _B T) cm²/s at temperatures below 900 K. The effective mean migration energy, 1.1 eV, closely coincided with that of the 1.0–1.2 eV in experiments and was very different from the migration energy of the free vacancy.