Space as the final frontier in stochastic simulations of biological systems

Recent technological and theoretical advances are only now allowing the simulation of detailed kinetic models of biological systems that reflect the stochastic movement and reactivity of individual molecules within cellular compartments. The behavior of many systems could not be properly understood without this level of resolution, opening up new perspectives of using computer simulations to accelerate biological research. We review the modeling methodology applied to stochastic spatial models, also to the attention of non-expert potential users. Modeling choices, current limitations and perspectives of improvement of current general-purpose modeling/simulation platforms for biological systems are discussed.     

Temperature effects upon the time course of contraction and re-extension in Stentor coeruleus

A double-flash microphotographic technique has been used to follow the variation with temperature of the following kinetic parameters related to the contraction and re-extension of the ciliate Stentor coeruleus, namely the rate of contraction, the initiation time before contraction, the rate of re-extension and the initiation time before re-extension, all described by first order kinetics. Activation enthalpies, entropies and free energies related to the above mentioned parameters were calculated from the variation of the rate constants with temperature. The enthalpies and entropies appear to be of minor interest compared to the free energies. For the contraction and the initiation of contraction the delta G transition state values obtained were 14 and 15 kcal/mole, respectively, while the re-extension and the initiation of re-extension both were represented by a value of delta G transition state about 19 kcal/mole. These results are compared to activation parameters for different motile systems and for the formation and breakdown of ATP-myosin complexes. A model for the contraction and re-extension processes is proposed in accordance with the results measured.     

Fuzzy Stochastic Petri Nets for Modeling Biological Systems with Uncertain Kinetic Parameters

Stochastic Petri nets (SPNs) have been widely used to model randomness which is an inherent feature of biological systems. However, for many biological systems, some kinetic parameters may be uncertain due to incomplete, vague or missing kinetic data (often called fuzzy uncertainty), or naturally vary, e.g., between different individuals, experimental conditions, etc. (often called variability), which has prevented a wider application of SPNs that require accurate parameters. Considering the strength of fuzzy sets to deal with uncertain information, we apply a specific type of stochastic Petri nets, fuzzy stochastic Petri nets (FSPNs), to model and analyze biological systems with uncertain kinetic parameters. FSPNs combine SPNs and fuzzy sets, thereby taking into account both randomness and fuzziness of biological systems. For a biological system, SPNs model the randomness, while fuzzy sets model kinetic parameters with fuzzy uncertainty or variability by associating each parameter with a fuzzy number instead of a crisp real value. We introduce a simulation-based analysis method for FSPNs to explore the uncertainties of outputs resulting from the uncertainties associated with input parameters, which works equally well for bounded and unbounded models. We illustrate our approach using a yeast polarization model having an infinite state space, which shows the appropriateness of FSPNs in combination with simulation-based analysis for modeling and analyzing biological systems with uncertain information.     

Concerted simulations reveal how peroxidase compound III formation results in cellular oscillations

A major problem in mathematical modeling of the dynamics of complex biological systems is the frequent lack of knowledge of kinetic parameters. Here, we apply Brownian dynamics simulations, based on protein three-dimensional structures, to estimate a previously undetermined kinetic parameter, which is then used in biochemical network simulations. The peroxidase-oxidase reaction involves many elementary steps and displays oscillatory dynamics important for immune response. Brownian dynamics simulations were performed for three different peroxidases to estimate the rate constant for one of the elementary steps crucial for oscillations in the peroxidase-oxidase reaction, the association of superoxide with peroxidase. Computed second-order rate constants agree well with available experimental data and permit prediction of rate constants at physiological conditions. The simulations show that electrostatic interactions depress the rate of superoxide association with myeloperoxidase, bringing it into the range necessary for oscillatory behavior in activated neutrophils. Such negative electrostatic steering of enzyme-substrate association presents a novel control mechanism and lies in sharp contrast to the electrostatically-steered fast association of superoxide and Cu/Zn superoxide dismutase, which is also simulated here. The results demonstrate the potential of an integrated and concerted application of structure-based simulations and biochemical network simulations in cellular systems biology.     

Kinetic studies on the initial contraction dependent high ATPase activity of actomyosin molecules

In contracting (superprecipitating) clearing and fully contracted (previously superprecipitated) actomyosin molecules the presteady state phosphate burst was found to be 2 nanomoles inorganic phosphate (Pi) per nanomole myosin. In these muscle models a significant difference in the Mg2+ ATPase activity was found following the initial phosphate burst. Between 120 and 800 milliseconds after the commencement of the reaction the Mg2+ ATPase activity of contracting actomyosin molecules was 5-10 times greater than that of the fully contracted or clearing actomyosin molecules. In the same time interval the rate of turbidity increase of the contracting actomyosin molecules was about 10 fold greater than during the remainder of the time to reach maximal superprecipitation. This high initial ATPase activity found to be present only in the contracting actomyosin molecules and coinciding with the high rate of the velocity of contraction provides sufficient energy for contraction. We propose that this high Mg2+--ATPase activity following the initial burst and included as a part of "conventional" steady state ATPase activity is the source of energy for muscular contraction. Calculation of kinetic and thermodynamic constants indicates that the contracting actomyosin molecule is subjected to a conformational change. As a consequence of contraction the complementarity of the enzyme site to the intermediate complex decreases about 100 fold. Thus the contracted molecules temporarily become relatively refractive to provide energy for the contractile process. In our opinion these findings are important with regard to muscular contraction.     

Phenomenological and molecular-level Petri net modeling and simulation of long-term potentiation

Petri net-based modeling methods have been used in many research projects to represent biological systems. Among these, the hybrid functional Petri net (HFPN) was developed especially for biological modeling in order to provide biologists with a more intuitive Petri net-based method. In the literature, HFPNs are used to represent kinetic models at the molecular level. We present two models of long-term potentiation previously represented by differential equations which we have transformed into HFPN models: a phenomenological synapse model and a molecular-level model of the CaMKII regulation pathway. Through simulation, we obtained results similar to those of previous studies using these models. Our results open the way to a new type of modeling for systems biology where HFPNs are used to combine different levels of abstraction within one model. This approach can be useful in fully modeling a system at the molecular level when kinetic data is missing or when a full study of a system at the molecular level it is not within the scope of the research.     

Bridging the gap: linking molecular simulations and systemic descriptions of cellular compartments

Metabolic processes in biological cells are commonly either characterized at the level of individual enzymes and metabolites or at the network level. Often these two paradigms are considered as mutually exclusive because concepts from neither side are suited to describe the complete range of scales. Additionally, when modeling metabolic or regulatory cellular systems, often a large fraction of the required kinetic parameters are unknown. This even applies to such simple and extensively studied systems like the photosynthetic apparatus of purple bacteria. Using the chromatophore vesicles of Rhodobacter sphaeroides as a model system, we show that a consistent kinetic model emerges when fitting the dynamics of a molecular stochastic simulation to a set of time dependent experiments even though about two thirds of the kinetic parameters in this system are not known from experiment. Those kinetic parameters that were previously known all came out in the expected range. The simulation model was built from independent protein units composed of elementary reactions processing single metabolites. This pools-and-proteins approach naturally compiles the wealth of available molecular biological data into a systemic model and can easily be extended to describe other systems by adding new protein or nucleic acid types. The automated parameter optimization, performed with an evolutionary algorithm, reveals the sensitivity of the model to the value of each parameter and the relative importances of the experiments used. Such an analysis identifies the crucial system parameters and guides the setup of new experiments that would add most knowledge for a systemic understanding of cellular compartments. The successful combination of the molecular model and the systemic parametrization presented here on the example of the simple machinery for bacterial photosynthesis shows that it is actually possible to combine molecular and systemic modeling. This framework can now straightforwardly be applied to other currently less well characterized but biologically more relevant systems.     

Fluorescence correlation spectroscopy and quantitative cell biology

Fluorescence correlation spectroscopy (FCS) analyzes fluctuations in fluorescence within a small observation volume. Autocorrelation analysis of FCS fluctuation data can be used to measure concentrations, diffusion properties, and kinetic constants for individual fluorescent molecules. Photon count histogram analysis of fluorescence fluctuation data can be used to study oligomerization of individual fluorescent molecules. If the FCS observation volume is positioned inside a living cell, these parameters can be measured in vivo. FCS can provide the requisite quantitative data for analysis of molecular interaction networks underlying complex cell biological processes.     

Ultrasound propagation in anisotropic soft tissues: the application of linear elastic theory

Linear elastic theory has served well in modeling the mechanical properties of numerous materials. In modeling ultrasonic wave propagation in biological soft tissues, an isotropic model has usually been employed. Many tissues, however, possess a lower order of symmetry, and the speed of sound in muscle is known to vary with the direction of propagation. In this study, by applying linear regression to acoustic microscopic data from seven frog sartorius specimens, four observable elastic constants associated with a transversely isotropic model were obtained. The average values of these constants were c11 = 2.64, c13 = 3.39 and c33 = 4.40 Nm-2 for resting muscles and c11 = 2.65, c13 = 3.43 and c33 = 4.57 Nm-2 for muscles undergoing tetanic contraction, where '1' and '3' represent the transverse and longitudinal axes, respectively. In all cases, c44 was 0, indicating a minimal contribution from longitudinal shear. For all seven specimens, the model of transverse isotropy provided a better fit of the data than that of isotropy.     

An inexpensive microcomputer-based stopped-flow data acquisition system.

A low-cost (less than $2,500) microcomputer-controlled data acquisition system for use with a stopped-flow instrument is described. Data acquisition, reduction, signal averaging, kinetic modeling, and plotting are performed under software control. Applications to biological and inorganic systems are presented.     

计算系统生物学:理论、方法及在药物研发中的应用

计算系统生物学是一个多学科交叉的新兴领域,旨在通过整合海量数据建立其生物系统相互作用的复杂网络。数据的整合和模型的建立需要发展合适的数学方法和软件工具,这也是计算系统生物学的主要任务。生物系统模型有助于从整体上理解生物体的内在功能和特性。同时,生物网络模型在药物研发中的应用也越来越受到制药企业以及新药研发机构的重视,如用于特异性药物作用靶点的预测和药物毒性评估等。该文简要介绍计算系统生物学的常见网络和计算模型,以及建立模型所用的研究方法,并阐述其在建模和分析中的作用及面临的问题和挑战。     

Direct comparison of binding equilibrium, thermodynamic, and rate constants determined by surface- and solution-based biophysical methods

The binding interactions of small molecules with carbonic anhydrase II were used as model systems to compare the reaction constants determined from surface- and solution-based biophysical methods. Interaction data were collected for two arylsulfonamide compounds, 4-carboxybenzenesulfonamide (CBS) and 5-dimethyl-amino-1-naphthalene-sulfonamide (DNSA), binding to the enzyme using surface plasmon resonance, isothermal titration calorimetry, and stopped-flow fluorescence. We demonstrate that when the surface plasmon resonance biosensor experiments are performed with care, the equilibrium, thermodynamic, and kinetic constants determined from this surface-based technique match those acquired in solution. These results validate the use of biosensor technology to collect reliable data on small molecules binding to immobilized macromolecular targets. Binding kinetics were shown to provide more detailed information about complex formation than equilibrium constants alone. For example, although carbonic anhydrase II bound DNSA with twofold higher affinity than CBS, kinetic analysis revealed that CBS had a fourfold slower dissociation rate. Analysis of the binding and transition state thermodynamics also revealed significant differences in the enthalpy and entropy of complex formation. The lack of labeling requirements, high information content, and high throughput of surface plasmon resonance biosensors will make this technology an important tool for characterizing the interactions of small molecules with enzymes and receptors.     

The lipid/protein interface as xenobiotic target site: kinetic analysis of the nicotinic acetylcholine receptor

Membrane proteins are known to be solvated and functionally activated by a fixed number of lipid molecules whose multiple binding can be described by Adair-type binding equations. Lipophilic xenobiotics such as general anesthetics may act by competitive displacement of protein-bound lipids. A kinetic equation is now presented for various binding stoichiometries of lipid and xenobiotic, and microscopic binding constants of anesthetics and organic solvents are derived from two independent assay systems for the enhancement of agonist binding to the nicotinic acetylcholine receptor. These constants lead to the first available free energy estimate (-6.4 kcal/mol) for the binding of membrane lipid to an integral membrane protein.     

Revising Berg-Purcell for finite receptor kinetics

From nutrient uptake to chemoreception to synaptic transmission, many systems in cell biology depend on molecules diffusing and binding to membrane receptors. Mathematical analysis of such systems often neglects the fact that receptors process molecules at finite kinetic rates. A key example is the celebrated formula of Berg and Purcell for the rate that cell surface receptors capture extracellular molecules. Indeed, this influential result is only valid if receptors transport molecules through the cell wall at a rate much faster than molecules arrive at receptors. From a mathematical perspective, ignoring receptor kinetics is convenient because it makes the diffusing molecules independent. In contrast, including receptor kinetics introduces correlations between the diffusing molecules because, for example, bound receptors may be temporarily blocked from binding additional molecules. In this work, we present a modeling framework for coupling bulk diffusion to surface receptors with finite kinetic rates. The framework uses boundary homogenization to couple the diffusion equation to nonlinear ordinary differential equations on the boundary. We use this framework to derive an explicit formula for the cellular uptake rate and show that the analysis of Berg and Purcell significantly overestimates uptake in some typical biophysical scenarios. We confirm our analysis by numerical simulations of a many-particle stochastic system.     

Active Site Formation, Not Bond Kinetics, Limits Adhesion Rate between Human Neutrophils and Immobilized Vascular Cell Adhesion Molecule 1

The formation of receptor ligand bonds at the interface between different cells and between cells and substrates is a widespread phenomenon in biological systems. Physical measurements of bond formation rates between cells and substrates have been exploited to increase our understanding of the biophysical mechanisms that regulate bond formation at interfaces. Heretofore, these measurements have been interpreted in terms of simple bimolecular reaction kinetics. Discrepancies between this simple framework and the behavior of neutrophils adhering to surfaces expressing vascular cell adhesion molecule 1 (VCAM-1) motivated the development of a new kinetic framework in which the explicit formation of active bond formation sites (reaction zones) are a prerequisite for bond formation to occur. Measurements of cells interacting with surfaces having a wide range of VCAM-1 concentrations, and for different durations of contact, enabled the determination of novel kinetic rate constants for the formation of reaction zones and for the intrinsic bond kinetics. Comparison of these rates with rates determined previously for other receptor-ligand pairs points to a predominant role of extrinsic factors such as surface topography and accessibility of active molecules to regions of close contact in determining forward rates of bond formation at cell interfaces.     

An enzyme mechanism language for the mathematical modeling of metabolic pathways

MOTIVATION: As a first step toward the elucidation of the systems biology of complex biological systems, it was our goal to mathematically model common enzyme catalytic and regulatory mechanisms that repeatedly appear in biological processes such as signal transduction and metabolic pathways. RESULTS: We describe kMech, a Cellerator language extension that describes a suite of enzyme mechanisms. Each enzyme mechanism is parsed by kMech into a set of fundamental association-dissociation reactions that are translated by Cellerator into ordinary differential equations that are numerically solved by Mathematica. In addition, we present methods that use commonly available kinetic measurements to estimate rate constants required to solve these differential equations.     

Asymmetry and free energy transduction in biology

This paper is an extension of our earlier theoretical studies on the relationship between kinetic asymmetry and free-energy transductions in biological systems induced by external fluctuations. In the first part of the paper, the asymmetry conditions necessary for external-noise-induced free-energy transductions to occur are derived for a special cyclic, four-state model in which only one reaction step is perturbed by the fluctuations. The results can be used to explain the earlier findings that asymmetry in rate constants was not required in the uphill transport of ligands induced by externally fluctuating the ligand concentrations. In the second part of the paper, the coupling between two enzyme systems through direct enzyme-enzyme inter-actions is studied. The existence of kinetic asymmetry in both the driving and the driven enzyme systems is found necessary for coupling and free-energy transductions to occur.     

Distribution of the heavy isotope of carbon (13C) in biological systems

Causes conditioning fractionation of carbon isotopes in biological systems are considered. Concepts of E. M. Galimov are discussed. According to these concepts distribution of carbon isotopes between biomolecules and their fragments is quaziequilibrium, i. e. it differs from the equilibrium one by a constant multiplier, which is the same for all the biomolecules of an organism but different for various organisms. These concepts have no theoretical grounds and do not agree with the experimental evidence available. An analysis of experimental data, as well as theoretical considerations, indicate that the observed differences in isotope composition of metabolytes and their fragments in the living organisms are conditioned by the kinetic isotope effects of carbon at the stages of their enzymic transformations and by the portion of substances participating in the reaction. It means that these differences do not depend directly on the constants characterizing the equilibrium distribution of carbon isotopes between corresponding compounds and between different groups inside their molecules.