Zoran Konkoli

Controlling Chemistry by Geometry in Nanoscale Systems

Scientific literature dealing with the rates, mechanisms, and thermodynamic properties of chemical reactions in condensed media almost exclusively assumes that reactions take place in volumes that do not change over time. The reaction volumes are compact (such as a sphere, a cube, or a cylinder) and do not vary in shape. In this review article, we discuss two important systems at small length scales (approximately 10 nm to 5 microm), in which these basic assumptions are violated. The first system exists in cell biology and is represented by the tiniest functional components (i.e., single cells, organelles, and other physically delineated cellular microenvironments). The second system comprises nanofluidic devices, in particular devices made from soft-matter materials such as lipid nanotube-vesicle networks. In these two systems, transport, mixing, and shape changes can be achieved at or very close to thermal energy levels. In further contrast to macroscopic systems, mixing by diffusion is extremely efficient, and kinetics can be controlled by shape and volume changes.

Diffusive transport in networks built of containers and tubes.

We have developed analytical and numerical methods to study the transport of noninteracting particles in large networks consisting of M d -dimensional containers C1,...,C(M) with radii R(i) linked together by tubes of length l(ij) and radii a(ij) where i,j = 1,2,...,M. Tubes may join directly with each other, forming junctions. It is possible that some links are absent. Instead of solving the diffusion equation for the full problem we formulated an approach that is computationally more efficient. We derived a set of rate equations that govern the time dependence of the number of particles in each container, N1(t), N2(t),...,N(M)(t). In such a way the complicated transport problem is reduced to a set of M first-order integro-differential equations in time, which can be solved efficiently by the algorithm presented here. The workings of the method have been demonstrated on a couple of examples: networks involving three, four, and seven containers and one network with a three-point junction. Already simple networks with relatively few containers exhibit interesting transport behavior. For example, we showed that it is possible to adjust the geometry of the networks so that the particle concentration varies in time in a wave-like manner. Such behavior deviates from simple exponential growth and decay occurring in the two-container system.

Interplay between chemical reactions and transport in structured spaces.

The main motivation behind this study is to understand the interplay between the reactions and transport in a geometries that are not compact. Typical examples of compact geometries are a box or a sphere. A network made of containers C(1) , C(2),..., C(N) and tubes is an example of the space that is structured and noncompact. In containers, particles react with the rate lambda. Tubes connecting containers allow for the exchange of chemicals with the transport rate D. A situation is considered where a number of reactants is small and kinetics is noise dominated. A method is presented that can be used to calculate the average and higher moments of the reaction time. A number of different chemical reactions are studied and their performance compared in various ways. Two schemes are discussed in general, the reaction on a fixed geometry ensemble (ROGE) and the geometry on a fixed reaction ensemble, examples are given in the ROGE case. The most important findings are as follows. (i) There is a large number of reactions that run faster in a networklike geometry. Such reactions contain antagonistic catalytic influences in the intermediate stages of a reaction scheme that are best dealt with in a networklike structure. (ii) Antagonistic catalytic influences are hard to identify since they are strongly connected to the pattern of injected molecules (inject pattern) and depend on the choice of molecules that have to be synthesized at the end (task pattern). (iii) The reaction time depends strongly on the details of the inject and task patterns.

Modeling reaction noise with a desired accuracy by using the X level approach reaction noise estimator (XARNES) method

A novel computational method for modeling reaction noise characteristics has been suggested. The method can be classified as a moment closure method. The approach is based on the concept of correlation forms which are used for describing spatially extended many body problems where particle numbers change in space and time. In here, it was shown how the formalism of spatially extended correlation forms can be adapted to study well mixed reaction systems. Stochastic fluctuations in particle numbers are described by selectively capturing correlation effects up to the desired order, ξ. The method is referred to as the ξ-level Approximation Reaction Noise Estimator method (XARNES). For example, the ξ=1 description is equivalent to the mean field theory (first-order effects), the ξ=2 case corresponds to the previously developed PARNES method (pair effects), etc. The main idea is that inclusion of higher order correlation effects should lead to better (more accurate) results. Several models were used to test the method, two versions of a simple complex formation model, the Michaelis-Menten model of enzymatic kinetics, the smallest bistable reaction network, a gene expression network with negative feedback, and a random large network. It was explicitly demonstrated that increase in ξ indeed improves accuracy in all cases investigated. The approach has been implemented as automatic software using the Mathematica programming language. The user only needs to input reaction rates, stoichiometry coefficients, and the desired level of computation ξ.

Diffusion-controlled reactions in small and structured spaces as a tool for describing living cell biochemistry

A simplified model of the living cell is studied. The reaction space is divided into compartments and the structured (non-compact) geometry is described in terms of a network consisting of containers connected by tubes. By assumption, reactions in the containers ( tubes) are allowed ( forbidden). It is assumed that the number of reactants is low, leading to stochastic (noisy) dynamics. By varying the transport rate among various containers D relative to the reaction rate within each container lambda, using either D >> lambda or D sigma(0)). Opposite cases are possible where reactions become slower (tau(n) > tau(0)) but more accurate (sigma(n) < sigma(0)).

A generic simulator for large networks of memristive elements.

This work develops a generic software tool for simulation of the dynamics of sparse multi-terminal memristive networks. The simulator is coded on one platform and relatively compact for easy work-flow and future extensions. Due to its relatively small size it should be easy to transfer (re-program) it using other computer languages. Future applications of the software include studies of various information processing scenarios.

Safe uses of Hill's model: an exact comparison with the Adair-Klotz model

BackgroundThe Hill function and the related Hill model are used frequently to study processes in the living cell. There are very few studies investigating the situations in which the model can be safely used. For example, it has been shown, at the mean field level, that the dose response curve obtained from a Hill model agrees well with the dose response curves obtained from a more complicated Adair-Klotz model, provided that the parameters of the Adair-Klotz model describe strongly cooperative binding. However, it has not been established whether such findings can be extended to other properties and non-mean field (stochastic) versions of the same, or other, models.ResultsIn this work a rather generic quantitative framework for approaching such a problem is suggested. The main idea is to focus on comparing the particle number distribution functions for Hills and Adair-Klotzs models instead of investigating a particular property (e.g. the dose response curve). The approach is valid for any model that can be mathematically related to the Hill model. The Adair-Klotz model is used to illustrate the technique. One main and two auxiliary similarity measures were introduced to compare the distributions in a quantitative way. Both time dependent and the equilibrium properties of the similarity measures were studied.ConclusionsA strongly cooperative Adair-Klotz model can be replaced by a suitable Hill model in such a way that any property computed from the two models, even the one describing stochastic features, is approximately the same. The quantitative analysis showed that boundaries of the regions in the parameter space where the models behave in the same way exhibit a rather rich structure.

Application of Bogolyubov's theory of weakly nonideal Bose gases to the A+A, A+B, B+B reaction-diffusion system

Theoretical methods for dealing with diffusion-controlled reactions inevitably rely on some kind of approximation, and to find the one that works on a particular problem is not always easy. Here the approximation used by Bogolyubov to study a weakly nonideal Bose gas, referred to as the weakly nonideal Bose gas approximation (WBGA), is applied in the analysis of three reaction-diffusion models: (i) A+A-->Ø, (ii) A+B-->Ø, and (iii) A+A,B+B,A+B-->Ø (the ABBA model). Two types of WBGA are considered, the simpler WBGA-I and the more complicated WBGA-II. All models are defined on the lattice to facilitate comparison with computer experiment (simulation). It is found that the WBGA describes the A+B reaction well, it reproduces the correct d/4 density decay exponent. However, it fails in the case of the A+A reaction and the ABBA model. (To cure the deficiency of WBGA in dealing with the A+A model, a hybrid of the WBGA and Kirkwood superposition approximations is suggested.) It is shown that the WBGA-I is identical to the dressed-tree calculation suggested by Lee [J. Phys. A 27, 2633 (1994)], and that the dressed-tree calculation does not lead to the d/2 density decay exponent when applied to the A+A reaction, as normally believed, but it predicts the d/4 decay exponent. Last, the usage of the small n(0) approximation suggested by Mattis and Glasser [Rev. Mod. Phys. 70, 979 (1998)] is questioned if used beyond the A+B reaction-diffusion model.

Diffusion controlled reactions, fluctuation dominated kinetics, and living cell biochemistry

In recent years considerable portion of the computer science community has focused its attention on understanding living cell biochemistry and efforts to understand such complication reaction environment have spread over wide front, ranging from systems biology approaches, through network analysis (motif identification) towards developing language and simulators for low level biochemical processes. Apart from simulation work, much of the efforts are directed to using mean field equations (equivalent to the equations of classical chemical kinetics) to address various problems (stability, robustness, sensitivity analysis, etc.). Rarely is the use of mean field equations questioned. This review will provide a brief overview of the situations when mean field equations fail and should not be used. These equations can be derived from the theory of diffusion controlled reactions, and emerge when assumption of perfect mixing is used.

Multiparticle reaction noise characteristics

A simple multiparticle reaction model was studied where reactants A react with (a possibly large) stoichiometric coefficient k. Each reaction forms a product molecule P, and every product molecule can be split into k A particles through the back reaction. To study the fluctuations in particle numbers a novel approach has been developed; to be referred to as the Pair approach based Reaction Noise EStimator (PARNES) method. The PARNES method is based on the full Kirkwood superposition approximation implemented at the pair level. Kirkwoods method has been adapted to study stochastic properties of an arbitrary reaction network in a perfectly mixed reaction volume. PARNES works well for large particle numbers. It provides qualitative description when particle numbers are low. The PARNES method can easily augment mean field calculations. Extension of the method beyond the pair approach level is straightforward. Both stationary and non-stationary properties of the model were investigated, and the findings of this work point to two possible scenarios of intracellular noise control. When k is increased, the fluctuations in the number of product molecules become smaller (sub-Poissonian) in a stationary state, and relaxation to a stationary state becomes faster.