Stochastic chemical kinetics

A review of the physical principles that are the ground of the stochastic formulation of chemical kinetics is presented along with a survey of the algorithms currently used to simulate it. This review covers the main literature of the last decade and focuses on the mathematical models describing the characteristics and the behavior of systems of chemical reactions at the nano- and micro-scale. Advantages and limitations of the models are also discussed in the light of the more and more frequent use of these models and algorithms in modeling and simulating biochemical and even biological processes.     

Sequential estimation for prescribed statistical accuracy in stochastic simulation of biological systems

Stochastic simulation of biological systems proceeds by repeatedly generating sample paths or trajectories of the underlying stochastic process, from which many relevant and important system properties can be obtained. While a great deal of research is targeted towards accelerated trajectory generation, issues concerned with the variability across trajectories are often neglected. Advanced methods for properly quantifying the statistical accuracy and determining a reasonable number of trajectories are hardly addressed formally in the context of biological system simulation, though mathematical statistics provides a large body of powerful theory. We invoke this theory and show how mathematically well-founded sequential estimation approaches serve for systematically generating enough but not too many trajectories for achieving a certain prescribed accuracy. The practical applicability is demonstrated and illustrated by numerical examples through simulation studies of an immigration-death process and a gene regulatory network.     

Solving the chemical master equation for monomolecular reaction systems analytically

The stochastic dynamics of a well-stirred mixture of molecular species interacting through different biochemical reactions can be accurately modelled by the chemical master equation (CME). Research in the biology and scientific computing community has concentrated mostly on the development of numerical techniques to approximate the solution of the CME via many realizations of the associated Markov jump process. The domain of exact and/or efficient methods for directly solving the CME is still widely open, which is due to its large dimension that grows exponentially with the number of molecular species involved. In this article, we present an exact solution formula of the CME for arbitrary initial conditions in the case where the underlying system is governed by monomolecular reactions. The solution can be expressed in terms of the convolution of multinomial and product Poisson distributions with time-dependent parameters evolving according to the traditional reaction-rate equations. This very structured representation allows to deduce easily many properties of the solution. The model class includes many interesting examples. For more complex reaction systems, our results can be seen as a first step towards the construction of new numerical integrators, because solutions to the monomolecular case provide promising ansatz functions for Galerkin-type methods.     

A new reaction useful for chemical cross-linking between nucleic acids and proteins

Cytosine in nucleic acids can be converted into N4-aminocytosine by treatment with a mixture of hydrazine and bisulfite. The hydrazino group thus formed at position 4 of the pyrimidine ring can be linked to a sulhydryl group in proteins by the use of bromopyruvate as a linker. Successful use of this scheme of chemical cross-linking between nucleic acid and protein was demonstrated in the linking of poly(C) with glutathione, and of RNA with protein in the E. coli 30 S ribosomal subunit.     

Web-based applications for building, managing and analysing kinetic models of biological systems

Mathematical modelling and computational analysis play an essentialrole in improving our capability to elucidate the functionsand characteristics of complex biological systems such as metabolic,regulatory and cell signalling pathways. The modelling and concomitantsimulation render it possible to predict the cellular behaviourof systems under various genetically and/or environmentallyperturbed conditions. This motivates systems biologists/bioengineers/bioinformaticiansto develop new tools and applications, allowing non-expertsto easily conduct such modelling and analysis. However, amonga multitude of systems biology tools developed to date, onlya handful of projects have adopted a web-based approach to kineticmodelling. In this report, we evaluate the capabilities andcharacteristics of current web-based tools in systems biologyand identify desirable features, limitations and bottlenecksfor further improvements in terms of usability and functionality.A short discussion on software architecture issues involvedin web-based applications and the approaches taken by existingtools is included for those interested in developing their ownsimulation applications.      

A weighted partial likelihood approach for zero‐truncated models

Zero‐truncated data arises in various disciplines where counts are observed but the zero count category cannot be observed during sampling. Maximum likelihood estimation can be used to model these data; however, due to its nonstandard form it cannot be easily implemented using well‐known software packages, and additional programming is often required. Motivated by the Rao–Blackwell theorem, we develop a weighted partial likelihood approach to estimate model parameters for zero‐truncated binomial and Poisson data. The resulting estimating function is equivalent to a weighted score function for standard count data models, and allows for applying readily available software. We evaluate the efficiency for this new approach and show that it performs almost as well as maximum likelihood estimation. The weighted partial likelihood approach is then extended to regression modelling and variable selection. We examine the performance of the proposed methods through simulation and present two case studies using real data.     

A hybrid agent-based approach for modeling microbiological systems

Models for systems biology commonly adopt Differential Equations or Agent-Based modeling approaches for simulating the processes as a whole. Models based on differential equations presuppose phenomenological intracellular behavioral mechanisms, while models based on Multi-Agent approach often use directly translated, and quantitatively less precise if-then logical rule constructs. We propose an extendible systems model based on a hybrid agent-based approach where biological cells are modeled as individuals (agents) while molecules are represented by quantities. This hybridization in entity representation entails a combined modeling strategy with agent-based behavioral rules and differential equations, thereby balancing the requirements of extendible model granularity with computational tractability. We demonstrate the efficacy of this approach with models of chemotaxis involving an assay of 10³ cells and 1.2×10⁶ molecules. The model produces cell migration patterns that are comparable to laboratory observations.     

Threshold policies control for predator-prey systems using a control Liapunov function approach

The stability of predator-prey models, in the context of exploitation of renewable resources, subject to threshold policies (TP) is studied in this paper using the idea of backstepping and control Liapunov functions (CLF) well known in control theory, as well as the concept of virtual equilibria. TPs are defined and analysed for different types of one and two species predator-prey models. The models studied are the single species Noy-Meir herbivore-vegetation model, in a grazing management context, as well as the Rosenzweig-MacArthur two species predator-prey model, in a fishery management context. TPs are shown to be versatile and useful in managing renewable resources, being simple to design and implement, and also yielding advantages in situations of overexploitation.     

A heteronuclear correlation experiment for simultaneous determination of 15N longitudinal decay and chemical exchange rates of systems in slow equilibrium

Summary A heteronuclear correlation experiment is described which permits simultaneous characterization of both ¹⁵N longitudinal decay rates and slow conformational exchange rates. Data pertaining to the exchange between folded and unfolded forms of an SH3 domain is used to illustrate the technique. Because the unfolded form of the molecule, on average, shows significantly higher NH exchange rates than the folded form, and approach which minimizes the degree of water saturation is employed, enabling the extraction of accurate rate constants.     

A new process for the esterification of wood by reaction with vinyl esters

A novel route to wood modification by transesterification of vinyl esters is developed in the current study. The reaction between varied saturated and unsaturated vinyl esters and the hydroxyl groups of maritime pine sapwood (Pinus pinaster Soland) was examined using potassium carbonate as a catalyst. The esterification of wood was investigated by weight percent gain calculations (WPG), Fourier-transform infrared spectroscopy (FTIR) and ¹³C cross-polarization with magic-angle spinning nuclear magnetic resonance spectroscopy (¹³C CP–MAS NMR). Differences in the rates of modification were noted, depending on the vinyl ester used, but relatively high yields were obtained in all cases. The infrared and NMR spectra of the different esterified samples were analysed in detail and the assignment of the signals corresponding to the grafted acyl groups confirmed that esterification occurred.     

A simple and highly accurate numerical differentiation method for sensitivity analysis of large-scale metabolic reaction systems

Numerical differentiation is known to be one of the most difficult numerical calculation methods to obtain reliable calculated values at all times. A simple numerical differentiation method using a combination of finite-difference formulas, derived by approximation of Taylor-series equations, is investigated in order to efficiently perform the sensitivity analysis of large-scale metabolic reaction systems. A result of the application to four basic mathematical functions reveals that the use of the eight-point differentiation formula with a non-dimensionalized stepsize close to 0.01 mostly provides more than 14 digits of accuracy in double precision for the numerical derivatives. Moreover, a result of the application to the modified TCA cycle model indicates that the numerical differentiation method gives the calculated values of steady-state metabolite concentrations within a range of round-off error and also makes it possible to transform the Michaelis-Menten equations into the S-system equations having the kinetic orders whose accuracies are mostly more than 14 significant digits. Because of the simple structure of the numerical differentiation formula and its promising high accuracy, it is evident that the present numerical differentiation method is useful for the analysis of large-scale metabolic reaction systems according to the systematic procedure of BST.     

A systems biology approach for elucidating the interaction of curcumin with Fanconi anemia FANC G protein and the key disease targets of leukemia

Fanconi anemia (FA) is an autosomal recessive disorder with a high risk of malignancies including acute myeloid leukemia and squamous cell carcinoma. There is a constant search out of new potential therapeutic molecule to combat this disorder. In most cases, patients with FA develop haematological malignancies with acute myeloid leukemia and acute lymphoblastic leukemia. Identifying drugs which can efficiently block the pathways of both these disorders can be an ideal and novel strategy to treat FA. The curcumin, a natural compound obtained from turmeric is an interesting therapeutic molecule as it has been reported in the literature to combat both FA as well as leukemia. However, its complete mechanism is not elucidated. Herein, a systems biology approach for elucidating the therapeutic potential of curcumin against FA and leukemia is investigated by analyzing the computational molecular interactions of curcumin ligand with FANC G of FA and seven other key disease targets of leukemia. The proteins namely DOT1L, farnesyl transferase (FDPS), histone decetylase (EP3000), Polo-like kinase (PLK-2), aurora-like kinase (AUKRB), tyrosine kinase (ABL1), and retinoic acid receptor alpha (RARA) were chosen as disease targets for leukemia and modeled structure of FANC G protein as the disease target for FA. The docking investigations showed that curcumin had a very high binding affinity of ?8.1?kcal/mol with FANC G protein. The key disease targets of leukemia namely tyrosine kinase (ABL1), aurora-like kinase (AUKRB), and polo-like kinase (PLK-2) showed that they had the comparable binding affinities of ?9.7 k cal/mol, ?8.7 k cal/mol, and ?8.6 k cal/mol, respectively with curcumin. Further, the percentage similarity scores obtained from PAM50 using EMBOSS MATCHER was shown to provide a clue to understand the structural relationships to an extent and to predict the binding affinity. This investigation shows that curcumin effectively interacts with the disease targets of both FA and leukemia.     

A density-based approach for the modelling of root architecture: application to Maritime pine (Pinus pinaster Ait.) root systems

Root morphology influences strongly plant/soil interactions. However, the complexity of root architecture is a major barrier when analysing many phenomena, e.g. anchorage, water or nutrient uptake. Therefore, we have developed a new approach for the representation and modelling of root architecture based on branching density. A general root branching density in a space of finite dimension was used and enabled us to consider various morphological properties. A root system model was then constructed which minimizes the difference between measured and simulated root systems, expressed with functions which map root density in the soil. The model was tested in 2D using data from Maritime pine Pinus pinaster Ait. structural roots as input. We showed that simulated and real root systems had similar root distributions in terms of radial distance, depth, branching angle and branching order. These results indicate that general density functions are not only a powerful basis for constructing models of architecture, but can also be used to represent such structures when considering root/soil interaction. These models are particularly useful in that they provide a local morphological characterization which is aggregated in a given unit of soil volume.