Adaptive coarse-grained Monte Carlo simulation of reaction and diffusion dynamics in heterogeneous plasma membranes

Background  

An adaptive coarse-grained (kinetic) Monte Carlo (ACGMC) simulation framework is applied to reaction and diffusion dynamics in inhomogeneous domains. The presented model is relevant to the diffusion and dimerization dynamics of epidermal growth factor receptor (EGFR) in the presence of plasma membrane heterogeneity and specifically receptor clustering. We perform simulations representing EGFR cluster dissipation in heterogeneous plasma membranes consisting of higher density clusters of receptors surrounded by low population areas using the ACGMC method. We further investigate the effect of key parameters on the cluster lifetime.     

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.     

Monte Carlo simulation of multiple attack mechanism of beta-amylase-catalyzed reaction

beta-Amylase (EC 3.2.1.2) produces maltose (dimer) from the nonreducing ends of alpha-1,4 glucosidic bonds of substrates like maltooligosaccharides, amylose, and amylopectin. The enzyme releases several maltose molecules from a single enzyme-substrate complex without dissociation by multiple or repetitive attack containing many branching reaction paths. The Monte Carlo method was applied to the simulation of the beta-amylase-catalyzed reaction including the multiple attack mechanism. The simulation starts from a single enzyme molecule and a finite number of substrate molecules. The selection of the substrate by the enzyme and degree of multiple attack proceeds by random numbers produced from a computer. The simulation was carried out until the whole substrate and the intermediate molecules were consumed. The simulated data were compared with experimental data of sweet potato beta-amylase using heptamer, octamer, nanomer, and 11-mer as substrates. The only adjustable parameter for odd-numbered substrates was the probability of multiple attack, while an additional adjustable parameter (a correction factor due to low reactivity of tetramer) was needed for even-numbered substrates.     

Dynamic Monte Carlo simulation of non-equilibrium Brownian diffusion of single-chain macromolecules

We performed dynamic Monte Carlo simulations of biased diffusion of 3D phantom single lattice polymer. We observed spontaneous deformation of polymer coil when the external driving forces exceed a critical strength. In addition, longer chains require lower critical strengths, at which their activated velocities deviate from Newtonian-fluid behaviours and merge into a master curve exhibiting shear-thinning followed with shear thickening. We attributed the cause of deformation to the random updating of monomers. The latter represents the dynamic heterogeneity along the real polymer chain, and raises a nonlinear asymmetric accumulation of local acceleration and then an internal tension between chain middle and chain end, as evidenced by our previous Brownian Dynamics simulations. Our results unravel a single-molecular-level source of nonlinear dynamics, which has been overlooked in current theoretical considerations on the basis of Rouse ideal-chain model.     

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     

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.     

Algorithm and data structures for efficient energy maintenance during Monte Carlo simulation of proteins.

Monte Carlo simulation (MCS) is a common methodology to compute pathways and thermodynamic properties of proteins. A simulation run is a series of random steps in conformation space, each perturbing some degrees of freedom of the molecule. A step is accepted with a probability that depends on the change in value of an energy function. Typical energy functions sum many terms. The most costly ones to compute are contributed by atom pairs closer than some cutoff distance. This paper introduces a new method that speeds up MCS by exploiting the facts that proteins are long kinematic chains and that few degrees of freedom are changed at each step. A novel data structure, called the ChainTree, captures both the kinematics and the shape of a protein at successive levels of detail. It is used to efficiently detect self-collision (steric clash between atoms) and/or find all atom pairs contributing to the energy. It also makes it possible to identify partial energy sums left unchanged by a perturbation, thus allowing the energy value to be incrementally updated. Computational tests on four proteins of sizes ranging from 68 to 755 amino acids show that MCS with the ChainTree method is significantly faster (as much as 10 times faster for the largest protein) than with the widely used grid method. They also indicate that speed-up increases with larger proteins.     

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 biospecific interactions

Numerical simulations of the stochastic time evolution of biospecific interactions are described and show that when molecular populations are large, time course predictions match those obtained using a deterministic expression. When population size is decreased the effects of stochastic noise become apparent. The significance of stochastic noise in sensitive binding-based assay systems suggests an immediate need for models of this type.     

Monte Carlo simulation of simultaneous gas flow and diffusion in an asymmetric distal pulmonary airway model

In order to evaluate the effect of anatomic asymmetries on the gas concentration distribution in the pulmonary airways, a Monte Carlo simulation of combined bulk flow and molecular diffusion was carried out in a realistic distal airway model (Parkeret al., 1971). This airway model, composed of branches distal to the 0.5-ram diameter airways, contained an upper symmetric segment consisting of four generations of conducting airways and a lower asymmetric segment of alveolar ducts and sacs arranged in five transport paths of varying lengths. In accounting for the volume increases of these ducts and sacs occurring during normal respiration, uniform alveolar filling rates and a fixed length-to-diameter ratio of all airways were assumed. For a pulse injection of inert tracer gas, the simulation was employed to determine the longitudinal concentration profiles in the conducting airways. In the alveolated airways, not only were the longitudinal profiles determined along each path, but radial transport from the core to the periphery of the airways was considered. The results of the simulations indicate that geometric asymmetries alone contribute substantially to regional concentration variations in the distal airways. For example, when a gas bolus is injected at mid*inspiration, there are concentration differences as great as 40% between two points along different transport paths located equi-distant from the proximal end of the model. As viewed from the terminal end of the model (acinus), average concentration differences as large as 6-to-1 exist between the longest and shortest transport paths respectively for gas boli introduced near the end of inspiration. The results further indicate because of large radial diffusion rates, no significant concentration differences exist between the periphery a-ld the central core of alveolated airways. Simulation of the expired concentration profiles indicate that boll injected very late during inspiration exhibit a sloping tail, unlike the earlier injected boll whose tails are virtually horizontal. Through the use of superposition teehniqnes, it was found that these sloping tails correspond to an alveolar slope of 1.5 vol% between 750 and 1250 ml expired for a continuous washing of tracer. This result is in disagreement with other transport analyses which did not directly account for the effect of geometric asymmetries.     

Monte Carlo simulation program for ecosystems

A Monte Carlo computer simulation program is designed in orderto describe the spatial and time evolution of a population ofliving individuals under preassigned environmental conditionsof energy. The simulation is inspired by previous techniquesdeveloped in physics — in particular, in molecular dynamicsand simulations of liquids — and it already provides somenew insights regarding macroscopic deterministic models in ecologyand concerning eventual control of artificial biomass productionplants. Received on July 15, 1986; accepted on October 9, 1986     

Monte Carlo simulation of LNCaP human prostate cancer cell aggregation in liquid-overlay culture

Neoplastic cells self-assemble in liquid-overlay cultures into multicellular spheroids that resemble micrometastases and avascular regions of larger tumors. A Monte Carlo simulation based on Meakin's cluster-cluster aggregation model resolved the physical mechanisms by which LNCaP human prostate cancer cells aggregate in this environment. The best-fit solution suggests that LNCaP cells aggregate with an adhesion probability of 0.5% when they migrate within a radius of influence between cell centers of 180 microm, 10 times the cell diameter. The sweeping radius of influence is indicative of cell tethering and/or chemotaxis and results in an intrinsic rate of self-aggregation that increases from k(11) = 1.5 h(-1) for single cells to k(1010) = 17.5 h(-1) for 10-mers. Similar rates are predicted by Smoluchowski's collision theory (1), suggesting that they are inherent properties of LNCaP liquid-overlay culture. Aggregates form more compact structures in culture than during simulation as measured by the fractal dimension: D(F) = 1.74 +/- 0.04 for 10-mers in culture vs D(F) = 1.25 +/- 0.10 for simulated 10-mers. Additional restructuring would further extend the radius of influence and diminish adhesion. Applications of this work include the production of highly viable spheroids for drug testing and basic oncological research.     

Monte Carlo simulation of intercalated carbon nanotubes

Monte Carlo simulations of the single- and double-walled carbon nanotubes (CNT) intercalated with different metals have been carried out. The interrelation between the length of a CNT, the number and type of metal atoms has also been established. This research is aimed at studying intercalated systems based on CNTs and d-metals such as Fe and Co. Factors influencing the stability of these composites have been determined theoretically by the Monte Carlo method with the Tersoff potential. The modeling of CNTs intercalated with metals by the Monte Carlo method has proved that there is a correlation between the length of a CNT and the number of endo-atoms of specific type. Thus, in the case of a metallic CNT (9,0) with length 17 bands (3.60 nm), in contrast to Co atoms, Fe atoms are extruded out of the CNT if the number of atoms in the CNT is not less than eight. Thus, this paper shows that a CNT of a certain size can be intercalated with no more than eight Fe atoms. The systems investigated are stabilized by coordination of 3d-atoms close to the CNT wall with a radius-vector of (0.18–0.20) nm. Another characteristic feature is that, within the temperature range of (400–700) K, small systems exhibit ground-state stabilization which is not characteristic of the higher ones. The behavior of Fe and Co endo-atoms between the walls of a double-walled carbon nanotube (DW CNT) is explained by a dominating van der Waals interaction between the Co atoms themselves, which is not true for the Fe atoms.     

A Monte Carlo simulation of chemical reactions

A computer-based algorithm to solve complex chemical rate equations is introduced. A simple Monte Carlo sampling method is used to generate chemical reactions in numbers proportional to reaction probabilities, and a second-order Runge-Kutta method is used to calculate time. The method is compared with a closed form mathematical solution for a simple chemical system, and it is compared with a numerical integration of the rate equations for a more complicated system.     

A Monte Carlo simulation of Auger cascades

The energy imparted to biological tissue after the decay of incorporated Auger emitters stems from two sources: (a) energy deposition by the Auger and Coster-Kronig electrons and (b) the charge potential which remains on the multiple ionized atom after the end of the cascade. For the numerical assessment of both the kinetic energy of the released electrons and the charge potential, a new and--for purposes of microdosimetry--precise method is presented. Based on relativistic Dirac-Fock calculations and a rigorous bookkeeping, this method provides a perfect energy balance of the considered atomic system when applied to Monte Carlo simulations of Auger cascades. By comparing the results for charge distribution for krypton and iodine with experimental data and the electron spectrum of 125I with theoretical data, it can be shown that the approach followed in this work is reasonable and appropriate for the determination of the energy deposited by incorporated Auger emitters in small volumes of condensed matter. The total energy deposited by 125I in a volume of 20-nm diameter is 2.03 keV which is made up by multiple ionization (1.07 keV) and energy deposition by the emitted Auger electrons (0.96 keV).     

Track structures, DNA targets and radiation effects in the biophysical Monte Carlo simulation code PARTRAC

This review describes the PARTRAC suite of comprehensive Monte Carlo simulation tools for calculations of track structures of a variety of ionizing radiation qualities and their biological effects. A multi-scale target model characterizes essential structures of the whole genomic DNA within human fibroblasts and lymphocytes in atomic resolution. Calculation methods and essential results are recapitulated regarding the physical, physico-chemical and chemical stage of track structure development of radiation damage induction. Recent model extension towards DNA repair processes extends the time dimension by about 12 orders of magnitude and paves the way for superior predictions of radiation risks.     

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.     

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.     

Monte Carlo simulation of the water in a channel with charges.

A Monte Carlo simulation of water in a channel with charges suggests the existence of water in immobile, high density, essentially glasslike form near the charges. The channel model has a conical section with an opening through which water molecules can pass, at the narrow end of the cone, and a cylindrical section at the other end. When the charges are placed near the narrow section of the model, the "glass" effectively blocks the channel; with the charges removed, the channel opens. The effect can be determined from the rate of passage of the water molecules through the pore, from the average orientation of the water molecule, and from distortion of the distribution of molecules. In the simulations carried out to date, no external ions have been considered. In addition to the energy, the Helmholtz free energy has been calculated.