Monte carlo simulations of enzyme reactions in two dimensions: fractal kinetics and spatial segregation

A biological interpretation of transient anomalous subdiffusion. II. Reaction kinetics

Reaction kinetics in a cell or cell membrane is modeled in terms of the first passage time for a random walker at a random initial position to reach an immobile target site in the presence of a hierarchy of nonreactive binding sites. Monte Carlo calculations are carried out for the triangular, square, and cubic lattices. The mean capture time is expressed as the product of three factors: the analytical expression of Montroll for the capture time in a system with a single target and no binding sites; an exact expression for the mean escape time from the set of lattice points; and a correction factor for the number of targets present. The correction factor, obtained from Monte Carlo calculations, is between one and two. Trapping may contribute significantly to noise in reaction rates. The statistical distribution of capture times is obtained from Monte Carlo calculations and shows a crossover from power-law to exponential behavior. The distribution is analyzed using probability generating functions; this analysis resolves the contributions of the different sources of randomness to the distribution of capture times. This analysis predicts the distribution function for a lattice with perfect mixing; deviations reflect imperfect mixing in an ordinary random walk.

Divergent regulation of protein synthesis in the cytosol and endoplasmic reticulum compartments of mammalian cells

In eukaryotic cells, mRNAs encoding signal sequence-bearing proteins undergo translation-dependent trafficking to the endoplasmic reticulum (ER), thereby restricting secretory and integral membrane protein synthesis to the ER compartment. However, recent studies demonstrating that mRNAs encoding cytosolic/nucleoplasmic proteins are represented on ER-bound polyribosomes suggest a global role for the ER in cellular protein synthesis. Here, we examined the steady-state protein synthesis rates and compartmental distribution of newly synthesized proteins in the cytosol and ER compartments. We report that ER protein synthesis rates exceed cytosolic protein synthesis rates by 2.5- to 4-fold; yet, completed proteins accumulate to similar levels in the two compartments. These data suggest that a significant fraction of cytosolic proteins undergo synthesis on ER-bound ribosomes. The compartmental differences in steady-state protein synthesis rates correlated with a divergent regulation of the tRNA aminoacylation/deacylation cycle. In the cytosol, two pathways were observed to compete for aminoacyl-tRNAs-protein synthesis and aminoacyl-tRNA hydrolysis-whereas on the ER tRNA deacylation is tightly coupled to protein synthesis. These findings identify a role for the ER in global protein synthesis, and they suggest models where compartmentalization of the tRNA acylation/deacylation cycle contributes to the regulation of global protein synthesis rates.

Mobile trap algorithm for zinc detection using protein sensors

We present a mobile trap algorithm to sense zinc ions using protein-based sensors such as carbonic anhydrase (CA). Zinc is an essential biometal required for mammalian cellular functions although its intracellular concentration is reported to be very low. Protein-based sensors like CA molecules are employed to sense rare species like zinc ions. In this study, the zinc ions are mobile targets, which are sought by the mobile traps in the form of sensors. Particle motions are modeled using random walk along with the first passage technique for efficient simulations. The association reaction between sensors and ions is incorporated using a probability (p1) upon an ion-sensor collision. The dissociation reaction of an ion-bound CA molecule is modeled using a second, independent probability (p2). The results of the algorithm are verified against the traditional simulation techniques (e.g., Gillespie's algorithm). This study demonstrates that individual sensor molecules can be characterized using the probability pair (p1,p2), which, in turn, is linked to the system level chemical kinetic constants, kon and koff. Further investigations of CA-Zn reaction using the mobile trap algorithm show that when the diffusivity of zinc ions approaches that of sensor molecules, the reaction data obtained using the static trap assumption differ from the reaction data obtained using the mobile trap formulation. This study also reveals similar behavior when the sensor molecule has higher dissociation constant. In both the cases, the reaction data obtained using the static trap formulation reach equilibrium at a higher number of complex molecules (ion-bound sensor molecules) compared to the reaction data from the mobile trap formulation. With practical limitations on the number sensors that can be inserted/expressed in a cell and stochastic nature of the intracellular ionic concentrations, fluorescence from the number of complex sensor molecules at equilibrium will be the measure of the intracellular ion concentration. For reliable detection of zinc ions, it is desirable that the sensors must not bind all the zinc ions tightly, but should rather bind and unbind. Thus for a given fluorescence and with association-dissociation reactions between ions and sensors, the static trap approach will underestimate the number of zinc ions present in the system.

Modeling molecular interactions to understand spatial crowding effects on heterodimer formations

Molecular crowding occurs when the density of interacting molecules in some reaction system is sufficient to create deviations from traditional mass-action models of chemistry in diffusive systems. While there is a great deal of theory on the influence of molecular crowding on biochemistry in vivo, the effects are highly dependent on specific assumptions about the shapes, volumes, and diffusion properties of the components of an individual system and are thus difficult to predict from first principles. In this study, we use lattice Monte Carlo simulations to examine the effects on a reaction system for two limiting cases of the diffusion behavior of inert crowding agents. In cells, inert molecules might diffuse throughout a solute along with the reactant species by passive diffusion or may be anchored at fixed positions within the solute. We investigate the relative contributions of the two models to crowding effects by examining moving inert particles versus stationery inert particles on the kinetics of a heterodimer assembly system. The two models of inert crowding agents resulted in highly divergent effects on the reactant system. Stationary particles exhibited a bimodal response in the reaction rate curve that was a function of copy number and spatial arrangement and which accelerated the process at conditions not unlike those found in cellular environments. On the other hand, moving inert particles created a well mixed background that had no effect on the reaction process even under extremely compacted conditions. These results may have applications in developing more realistic simulations of reaction chemistry in crowded environments such as living cells.

Brownian motion of an asymmetrical particle in a potential field

It is well known that a free ellipsoidal Brownian particle exhibits anisotropic diffusion for short times which changes to isotropic at long times, and, that the long-time diffusion coefficient is an average of the translational diffusion coefficients along the different semiaxes of the particle. We show analytically that in the presence of external forces, the long-time diffusion coefficient is different from that of a free particle. The magnitude of the difference in the two diffusion coefficients is found to increase proportionately with the particle's asymmetry, being zero only for a perfectly spherical Brownian particle. It is also found that, for asymmetrical particles, the application of external forces can amplify the non-Gaussian character of the spatial probability distributions which consequently delays the transition to the classical behavior. We illustrate these phenomena by considering the quasi-two-dimensional Brownian motion of an ellipsoidal rigid particle in linear and harmonic potential fields. These two examples provide insight into the role played by particle asymmetry in electrophoresis and microconfinement due to a laser trap or due to intracellular macromolecular crowding.

Advanced pharmacokinetic models based on organ clearance, circulatory, and fractal concepts

Three advanced models of pharmacokinetics are described. In the first class are physiologically based pharmacokinetic models based on in vitro data on transport and metabolism. The information is translated as transporter and enzyme activities and their attendant heterogeneities into liver and intestine models. Second are circulatory models based on transit time distribution and plasma concentration time curves. The third are fractal models for nonhomogeneous systems and non-Fickian processes are presented. The usefulness of these pharmacokinetic models, with examples, is compared.

Macromolecular crowding and its influence on possible reaction mechanisms in photosynthetic electron flow

The diffusion of plastoquinol and its binding to the Qo site of the cyt bf complex in the course of photosynthetic electron transport was studied by following the sigmoidal flash-induced re-reduction kinetics of P700 after previous oxidation of the intersystem electron carriers. The data resulting from these experiments were matched with a simulation of electron transport using Monte Carlo techniques. The simulation was able to account for the experimental observations. Two different extreme cases of reaction mechanism at the Qo site were compared: a diffusion limited collisional mechanism and a non-diffusion limited tight binding mechanism. Assuming a tight binding mechanism led to best matches due to the high protein density in thylakoids. The varied parameters resulted in values well within the range of published data. The results emphasise the importance of structural characteristics of thylakoids in models of electron transport.

Probing the nanoscale viscoelasticity of intracellular fluids in living cells

We have used fluorescence correlation spectroscopy to determine the anomalous diffusion properties of fluorescently tagged gold beads in the cytoplasm and the nucleus of living cells. From the extracted mean-square displacement v(tau) approximately tau(alpha), we have determined the complex shear modulus G(omega) approximately omega(alpha) for both compartments. Without treatment, all tested cell lines showed a strong viscoelastic behavior of the cytoplasm and the nucleoplasm, highlighting the crowdedness of these intracellular fluids. We also found a similar viscoelastic response in frog egg extract, which tended toward a solely viscous behavior upon dilution. When cells were osmotically stressed, the diffusion became less anomalous and the viscoelastic response changed. In particular, the anomality changed from alpha approximately 0.55 to alpha approximately 0.66, which indicates that the Zimm model for polymer solutions under varying solvent conditions is a good empirical description of the material properties of the cytoplasm and the nucleoplasm. Since osmotic stress may eventually trigger cell death, we propose, on the basis of our observations, that intracellular fluids are maintained in a state similar to crowded polymer solutions under good solvent conditions to keep the cell viable.

Modeling fractal-like drug elimination kinetics using an interacting random-walk model

We introduce an interacting random-walk model to describe the residence time of drug molecules undergoing a series of sojourn times in the body before being permanently eliminated under either homogeneous or heterogeneous conditions. We show that short-term correlations between drug molecules lead to Michaelis-Menten kinetics while long-term correlations lead to transient fractal-like kinetics. By combining both types of correlation, fractal-like Michaelis-Menten kinetics are achieved, and the simulations confirm previous analytical results.

Finite element analysis of the time-dependent Smoluchowski equation for acetylcholinesterase reaction rate calculations

This article describes the numerical solution of the time-dependent Smoluchowski equation to study diffusion in biomolecular systems. Specifically, finite element methods have been developed to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to the mouse acetylcholinesterase (mAChE) monomer and several tetramers. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different time steps. Calculated rates show very good agreement with experimental and theoretical steady-state studies. Furthermore, these finite element methods require significantly fewer computational resources than existing particle-based Brownian dynamics methods and are robust for complicated geometries. The key finding of biological importance is that the rate accelerations of the monomeric and tetrameric mAChE that result from electrostatic steering are preserved under the non-steady-state conditions that are expected to occur in physiological circumstances.

Extracting biochemical parameters for cellular modeling: A mean-field approach

Recent developments in molecular biology have made it feasible to carry out experimental verification of mathematical models for biochemical processes, offering the eventual prospect of creating a detailed, validated picture of gene expression. A persistent difficulty with this long-term goal is the incompleteness of the kinetic information available in the literature: Many rate constants cannot or have not yet been measured. Here, we present a method of filling in missing parameters using an approach conceptually analogous to mean-field approaches in statistical mechanics: When studying a particular gene, we extract key parameters by considering the averaged effect of all other genes in the system, analogously to considering the averaged magnetic field in a physical spin model. This methodology has been applied to account for the effect of the presence of the Escherichia coli genome on the availability of key enzymes involved in gene expression (RNA polymerases and ribosomes), yielding the number of free enzymes as a function of cellular growth rate. These conclusions have been obtained by deriving genome-wide averages and matching them to bulk literature values of E. coli K-12 and B/r. Average rate constants have been found for RNA polymerases and ribosomes binding to promoter and ribosome-binding sites, respectively; these results suggest that cells vary not only their production rates of RNA polymerase and ribosomes under different growth-rate conditions but also change their global level of transcriptional/translational activation and repression, thus altering the average binding rate constants for these enzymes. To test the mean-field method, the results from the genome-wide averages have been applied to the induced lac operon, where our derived on-rate for binding of RNA polymerase to the promoter is in good agreement with previous experimental results.

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.

Effect of crowding by dextrans and Ficolls on the rate of alkaline phosphatase-catalyzed hydrolysis: A size-dependent investigation

The cell cytosol is crowded with macromolecules such as proteins, nucleic acids, and membranes. The consequences of such crowding remain unclear. How is the rate of a typical enzymatic reaction, involving a freely diffusing enzyme and substrate, affected by the presence of macromolecules of different sizes, shapes, and concentrations? Here, we mimic the cytosolic crowding in vitro, using dextrans and Ficolls, for the first time in a variety of sizes ranging from 15 to 500 kDa, in a concentration range 0-30% w/w. Alkaline phosphatase-catalyzed hydrolysis of p-nitrophenyl phosphate (PNPP) was chosen as the model reaction. A pronounced decrease in the rate with increase in fractional volume occupancy of dextran is observed for larger dextrans (200 and 500 kDa) in contrast to smaller dextrans (15-70 kDa). Our results indicate that, at 20% w/w, smaller dextrans (15-70 kDa) reduce the initial rate moderately (1.4- to 2.4-fold slowing), while larger dextrans (>200 kDa) slow the reaction considerably (>5-fold). Ficolls (70 and 400 kDa) slow the reaction moderately (1.3- to 2.3-fold). The influence of smaller dextrans was accounted by a combination of increase in viscosity as sensed by PNPP and a minor offsetting increase in enzyme activity due to crowding. Larger dextrans apparently reduce the frequency of enzyme substrate encounter. The reduced influence of Ficolls is attributed to their compact and quasispherical shape, much unlike the dextrans.

Fractal michaelis-menten kinetics under steady state conditions: Application to mibefradil

PURPOSE

To provide the first application of fractal kinetics under steady state conditions to pharmacokinetics as a model for the enzymatic elimination of a drug from the body.

MATERIALS AND METHODS

A one-compartment model following fractal Michaelis-Menten kinetics under a steady state is developed and applied to concentration-time data for the cardiac drug mibefradil in dogs. The model predicts a fractal reaction order and a power law asymptotic time-dependence of the drug concentration, therefore a mathematical relationship between the fractal reaction order and the power law exponent is derived. The goodness-of-fit of the model is assessed and compared to that of four other models suggested in the literature.

RESULTS

The proposed model provided the best fit to the data. In addition, it correctly predicted the power law shape of the tail of the concentration-time curve.

CONCLUSION

A simple one-compartment model with steady state fractal Michaelis-Menten kinetics describing drug elimination from the body most accurately describes the pharmacokinetics of mibefradil in dogs. The new fractal reaction order can be explained in terms of the complex geometry of the liver, the organ responsible for eliminating the drug.

An integrated agent-mathematical model of the effect of intercellular signalling via the epidermal growth factor receptor on cell proliferation

We have previously developed Epitheliome, a software agent representation of the growth and repair characteristics of epithelial cell populations, where cell behaviour is governed by a number of simple rules. In this paper, we describe how this model has been extended to incorporate an example of a molecular 'mechanism' behind a rule-in this case, how signalling by both endogenous and exogenous ligands of the epidermal growth factor receptor (EGFR) can impact on the proliferation of cell agents. We have developed a mathematical model representing release of endogenous ligand by cells, three-dimensional diffusion of the secreted molecules through a volume of cell culture medium, ligand-receptor binding, and bound receptor internalization and trafficking. Information relating to quantities of molecular species associated with each cell agent is frequently exchanged between the agent and signalling models, and the ratio of bound to free receptors determines cell cycle progression and hence the proliferative behaviour of the cell agents. We have applied this integrated model to examine the effect of plating density on tissue growth via autocrine/paracrine signalling. This predicts that cell growth is dependent on the concentration of exogenous ligand, but where this is limited, then growth becomes dependent on cell density and the availability of endogenous ligand. We have further modified the calcium concentration of the medium to modulate the formation of intercellular bonds between cells and shown that the increased propensity for cells to form colonies in physiological calcium does not result in significantly different patterns of receptor occupancy. In conclusion, our approach demonstrates that by combining agent-based and mathematical modelling paradigms, it is possible to probe the complex feedback relationship between the behaviour of individual cells and their interaction with one another and their environment.

A systematic investigation of the rate laws valid in intracellular environments

Recently there has been significant interest in deducing the form of the rate laws for chemical reactions occurring in the intracellular environment. This environment is typically characterized by low-dimensionality and a high macromolecular content; this leads to a spatial heterogeneity not typical of the well stirred in vitro environments. For this reason, the classical law of mass action has been presumed to be invalid for modeling intracellular reactions. Using lattice-gas automata models, it has recently been postulated [H. Berry, Monte Carlo simulations of enzyme reactions in two dimensions: Fractal kinetics and spatial segregation, Biophys. J. 83 (2002) 1891-1901; S. Schnell, T.E. Turner, Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws, Prog. Biophys. Mol. Biol. 85 (2004) 235-260] that the reaction kinetics is fractal-like. In this article we systematically investigate for the first time how the rate laws describing intracellular reactions vary as a function of: the geometry and size of the intracellular surface on which the reactions occur, the mobility of the macromolecules responsible for the crowding effects, the initial reactant concentrations and the probability of reaction between two reactant molecules. We also compare the rate laws valid in heterogeneous environments in which there is an underlying spatial lattice, for example crystalline alloys, with the rate laws valid in heterogeneous environments where there is no such natural lattice, for example in intracellular environments. Our simulations indicate that: (i) in intracellular environments both fractal kinetics and mass action can be valid, the major determinant being the probability of reaction, (ii) the geometry and size of the intracellular surface on which reactions are occurring does not significantly affect the rate law, (iii) there are considerable differences between the rate laws valid in heterogeneous non-living structures such as crystals and those valid in intracellular environments. Deviations from mass action are less pronounced in intracellular environments than in a crystalline material of similar heterogeneity.

Effects of quenched and annealed macromolecular crowding elements on a simple model for signaling in T lymphocytes

Biochemical reactions in cells occur in an environment that is crowded in the sense that various macromolecular species and organelles occupy much of the space. The effects of molecular crowding on biochemical reactions have usually been studied in the past in a spatially homogeneous environment. However, signal transduction in cells is often initiated by the binding of receptors and ligands in two apposed cell membranes, and the pertinent biochemical reactions occur in a spatially inhomogeneous environment. We have studied the effects of crowding on biochemical reactions that involve both membrane proteins and cytosolic molecules by investigating a simplified version of signaling in T lymphocytes using a Monte Carlo algorithm. We find that, if signal transduction occurs on time scales that are slow compared to the motility of the molecules and organelles that constitute the crowding elements, the effects of crowding are qualitatively the same as in a homogeneous three-dimensional (3D) medium. In contrast, if signal transduction occurs on a time scale that is much faster than the time over which the crowding elements move, then the effects of varying the extent of crowding are very different when reactions occur in both 2- and 3D space. We discuss these differences and their origin. Since many signaling reactions are fast, our results may be useful for diverse situations in cell biology.

Multifractality in intracellular enzymatic reactions

Enzymatic kinetics adjust well to the Michaelis-Menten paradigm in homogeneous media with dilute, perfectly mixed reactants. These conditions are quite different from the highly structured cell plasm, so applications of the classic kinetics theory to this environment are rather limited. Cytoplasmic structure produces molecular crowding and anomalous diffusion of substances, modifying the mass action kinetic laws. The reaction coefficients are no longer constant but time-variant, as stated in the fractal kinetics theory. Fractal kinetics assumes that enzymatic reactions on such heterogeneous media occur within a non-Euclidian space characterized by a certain fractal dimension, this fractal dimension gives the dependence on time of the kinetic coefficients. In this work, stochastic simulations of enzymatic reactions under molecular crowding have been completed, and kinetic coefficients for the reactions, including the Michaelis-Menten parameter KM, were calculated. The simulations results led us to confirm the time dependence of michaelian kinetic parameter for the enzymatic catalysis. Besides, other chaos related phenomena were pointed out from the obtained KM time series, such as the emergence of strange attractors and multifractality.

Identifying optimal lipid raft characteristics required to promote nanoscale protein-protein interactions on the plasma membrane

The dynamic lateral segregation of signaling proteins into microdomains is proposed to facilitate signal transduction, but the constraints on microdomain size, mobility, and diffusion that might realize this function are undefined. Here we interrogate a stochastic spatial model of the plasma membrane to determine how microdomains affect protein dynamics. Taking lipid rafts as representative microdomains, we show that reduced protein mobility in rafts segregates dynamically partitioning proteins, but the equilibrium concentration is largely independent of raft size and mobility. Rafts weakly impede small-scale protein diffusion but more strongly impede long-range protein mobility. The long-range mobility of raft-partitioning and raft-excluded proteins, however, is reduced to a similar extent. Dynamic partitioning into rafts increases specific interprotein collision rates, but to maximize this critical, biologically relevant function, rafts must be small (diameter, 6 to 14 nm) and mobile. Intermolecular collisions can also be favored by the selective capture and exclusion of proteins by rafts, although this mechanism is generally less efficient than simple dynamic partitioning. Generalizing these results, we conclude that microdomains can readily operate as protein concentrators or isolators but there appear to be significant constraints on size and mobility if microdomains are also required to function as reaction chambers that facilitate nanoscale protein-protein interactions. These results may have significant implications for the many signaling cascades that are scaffolded or assembled in plasma membrane microdomains.

Anomalous diffusion of proteins due to molecular crowding

We have studied the diffusion of tracer proteins in highly concentrated random-coil polymer and globular protein solutions imitating the crowded conditions encountered in cellular environments. Using fluorescence correlation spectroscopy, we measured the anomalous diffusion exponent alpha characterizing the dependence of the mean-square displacement of the tracer proteins on time, r(2)(t) approximately t(alpha). We observed that the diffusion of proteins in dextran solutions with concentrations up to 400 g/l is subdiffusive (alpha < 1) even at low obstacle concentration. The anomalous diffusion exponent alpha decreases continuously with increasing obstacle concentration and molecular weight, but does not depend on buffer ionic strength, and neither does it depend strongly on solution temperature. At very high random-coil polymer concentrations, alpha reaches a limit value of alpha(l) approximately 3/4, which we take to be the signature of a coupling between the motions of the tracer proteins and the segments of the dextran chains. A similar, although less pronounced, subdiffusive behavior is observed for the diffusion of streptavidin in concentrated globular protein solutions. These observations indicate that protein diffusion in the cell cytoplasm and nucleus should be anomalous as well, with consequences for measurements of solute diffusion coefficients in cells and for the modeling of cellular processes relying on diffusion.

Reactions on cell membranes: comparison of continuum theory and Brownian dynamics simulations

Biochemical transduction of signals received by living cells typically involves molecular interactions and enzyme-mediated reactions at the cell membrane, a problem that is analogous to reacting species on a catalyst surface or interface. We have developed an efficient Brownian dynamics algorithm that is especially suited for such systems and have compared the simulation results with various continuum theories through prediction of effective enzymatic rate constant values. We specifically consider reaction versus diffusion limitation, the effect of increasing enzyme density, and the spontaneous membrane association/dissociation of enzyme molecules. In all cases, we find the theory and simulations to be in quantitative agreement. This algorithm may be readily adapted for the stochastic simulation of more complex cell signaling systems.

Space in systems biology of signaling pathways--towards intracellular molecular crowding in silico

How cells utilize intracellular spatial features to optimize their signaling characteristics is still not clearly understood. The physical distance between the cell-surface receptor and the gene expression machinery, fast reactions, and slow protein diffusion coefficients are some of the properties that contribute to their intricacy. This article reviews computational frameworks that can help biologists to elucidate the implications of space in signaling pathways. We argue that intracellular macromolecular crowding is an important modeling issue, and describe how recent simulation methods can reproduce this phenomenon in either implicit, semi-explicit or fully explicit representation.

Multi-scale modelling and the IUPS physiome project

We review the development of models of cellular and tissue function and in particular address issues of multi-scale modelling, including the transition from stochastic models to continuum models and the incorporation of cell and tissue structure. The heart is used as an example of linking models at the molecular level to cell, tissue and organ level function.

Michaelis-Menten kinetics under spatially constrained conditions: application to mibefradil pharmacokinetics

Two different approaches were used to study the kinetics of the enzymatic reaction under heterogeneous conditions to interpret the unusual nonlinear pharmacokinetics of mibefradil. Firstly, a detailed model based on the kinetic differential equations is proposed to study the enzymatic reaction under spatial constraints and in vivo conditions. Secondly, Monte Carlo simulations of the enzyme reaction in a two-dimensional square lattice, placing special emphasis on the input and output of the substrate were applied to mimic in vivo conditions. Both the mathematical model and the Monte Carlo simulations for the enzymatic reaction reproduced the classical Michaelis-Menten (MM) kinetics in homogeneous media and unusual kinetics in fractal media. Based on these findings, a time-dependent version of the classic MM equation was developed for the rate of change of the substrate concentration in disordered media and was successfully used to describe the experimental plasma concentration-time data of mibefradil and derive estimates for the model parameters. The unusual nonlinear pharmacokinetics of mibefradil originates from the heterogeneous conditions in the reaction space of the enzymatic reaction. The modified MM equation can describe the pharmacokinetics of mibefradil as it is able to capture the heterogeneity of the enzymatic reaction in disordered media.

In a mirror dimly: tracing the movements of molecules in living cells

The random movement of molecules (diffusion) is fundamental to most cellular processes, including enzymatic reactions, signalling, protein-protein interaction, as well as domain and pattern formation. Despite playing a central role, diffusion is, to a large extent, under-appreciated in the cell biology community. One reason for this is that diffusion is rather challenging to study in living cells. This article is intended to explain, at least in part, how we can go about studying diffusion of molecules in living cells, why it is important and how it provides us with important clues about biological systems. As the title 'In a mirror dimly' suggests, we do this by monitoring faint light emitted by fluorescent probes or proteins using advanced optics (e.g. mirrors) and electronics. The data are then fitted and interpreted with mathematical and physical models, providing a glimpse into the world of molecules.

Finite element solution of the steady-state Smoluchowski equation for rate constant calculations

This article describes the development and implementation of algorithms to study diffusion in biomolecular systems using continuum mechanics equations. Specifically, finite element methods have been developed to solve the steady-state Smoluchowski equation to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to mouse acetylcholinesterase. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different reaction criteria. The calculated rates were compared with experimental data and show very good agreement when the correct reaction criterion is used. Additionally, these finite element methods require significantly less computational resources than existing particle-based Brownian dynamics methods.

Mathematical and computational techniques to deduce complex biochemical reaction mechanisms

Time series data can now be routinely collected for biochemical reaction pathways, and recently, several methods have been proposed to infer reaction mechanisms for metabolic pathways and networks. In this paper we provide a survey of mathematical techniques for determining reaction mechanisms for time series data on the concentration or abundance of different reacting components, with little prior information about the pathways involved.

Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells

Macromolecular crowding dramatically affects cellular processes such as protein folding and assembly, regulation of metabolic pathways, and condensation of DNA. Despite increased attention, we still lack a definition for how crowded a heterogeneous environment is at the molecular scale and how this manifests in basic physical phenomena like diffusion. Here, we show by means of fluorescence correlation spectroscopy and computer simulations that crowding manifests itself through the emergence of anomalous subdiffusion of cytoplasmic macromolecules. In other words, the mean square displacement of a protein will grow less than linear in time and the degree of this anomality depends on the size and conformation of the traced particle and on the total protein concentration of the solution. We therefore propose that the anomality of the diffusion can be used as a quantifiable measure for the crowdedness of the cytoplasm at the molecular scale.

Stochastic approaches for modelling in vivo reactions

In recent years, stochastic modelling has emerged as a physically more realistic alternative for modelling in vivo reactions. There are numerous stochastic approaches available in the literature; most of these assume that observed random fluctuations are a consequence of the small number of reacting molecules. We review some important developments of the stochastic approach and consider its suitability for modelling intracellular reactions. We then describe recent efforts to include the fluctuation effects caused by the structural organisation of the cytoplasm and the limited diffusion of molecules due to macromolecular crowding.

Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws

We review recent evidence illustrating the fundamental difference between cytoplasmic and test tube biochemical kinetics and thermodynamics, and showing the breakdown of the law of mass action and power-law approximation in in vivo conditions. Simulations of biochemical reactions in non-homogeneous media show that as a result of anomalous diffusion and mixing of the biochemical species, reactions follow a fractal-like kinetics. Consequently, the conventional equations for biochemical pathways fail to describe the reactions in in vivo conditions. We present a modification to fractal-like kinetics following the Zipf-Mandelbrot distribution which will enable the modelling and analysis of biochemical reactions occurring in crowded intracellular environments.

Synergistic cellulose hydrolysis can be described in terms of fractal-like kinetics

A fractal-like kinetics model was used to describe the synergistic hydrolysis of bacterial cellulose by Trichoderma reesei cellulases. The synergistic action of intact cellobiohydrolase Cel7A and endoglucanase Cel5A at low enzyme-to-substrate ratios showed an apparent substrate inhibition consistent with a case where two-dimensional (2-D) surface diffusion of the cellobiohydrolase is rate-limiting. The action of Cel7A core and Cel5A was instead consistent with a three-dimensional (3-D) diffusion-based mode of action. The synergistic action of intact Cel7A was far superior to that of the core at a high enzyme-to-substrate ratio, but this effect was gradually reduced at lower enzyme-to-substrate ratios. The apparent fractal kinetics exponent h obtained by nonlinear fit of hydrolysis data to the fractal-like kinetics analogue of a first-order reaction was a useful empirical parameter for assessing the rate retardation and its dependence on the reaction conditions.