Self-organisation and orderly processes by individual protein complexes in the bacterial cell

Biophysical Reviews' "Meet the Editors Series"-a profile of Germán Rivas

German Rivas is an executive editor of the IUPAB Biophysical Reviews journal based in Spain. As the head of the Department of Structural and Chemical Biology at the Center for Biological Research (CIB) Margarita Salas (one of the largest research institutes devoted to life sciences of the Spanish National Research Council (CSIC)), he leads a research program aimed at understanding the structure function relationship of large macromolecular complexes (involved in bacterial cell division) when placed in physiologically complex and "crowded" media toward their reconstitution from the bottom up in cell-like compartments. In this "Meet the Editors'" piece, he briefly describes his research interests and history.

Cell-free protein synthesis in a microchamber revealed the presence of an optimum compartment volume for high-order reactions

The application of microelectromechanical systems (MEMS) to chemistry and biochemistry allows various reactions to be performed in microscale compartments. Here, we aimed to use the glass microchamber to study the compartment size dependency of the protein synthesis, one of the most important reactions in the cell. By encapsulating the cell-free protein synthesis system with different reaction orders in femtoliter microchambers, chamber size dependency of the reaction initiated with a constant copy number of DNA was investigated. We were able to observe the properties specific to the high order reactions in microcompartments with high precision and found the presence of an optimum compartment volume for a high-order reaction using real biological molecules.

Macromolecular interactions of the bacterial division FtsZ protein: from quantitative biochemistry and crowding to reconstructing minimal divisomes in the test tube

The division of Escherichia coli is an essential process strictly regulated in time and space. It requires the association of FtsZ with other proteins to assemble a dynamic ring during septation, forming part of the functionally active division machinery, the divisome. FtsZ reversibly interacts with FtsA and ZipA at the cytoplasmic membrane to form a proto-ring, the first molecular assembly of the divisome, which is ultimately joined by the rest of the division-specific proteins. In this review we summarize the quantitative approaches used to study the activity, interactions, and assembly properties of FtsZ under well-defined solution conditions, with the aim of furthering our understanding of how the behavior of FtsZ is controlled by nucleotides and physiological ligands. The modulation of the association and assembly properties of FtsZ by excluded-volume effects, reproducing in part the natural crowded environment in which this protein has evolved to function, will be described. The subsequent studies on the reactivity of FtsZ in membrane-like systems using biochemical, biophysical, and imaging technologies are reported. Finally, we discuss the experimental challenges to be met to achieve construction of the minimum protein set needed to initiate bacterial division, without cells, in a cell-like compartment. This integrated approach, combining quantitative and synthetic strategies, will help to support (or dismiss) conclusions already derived from cellular and molecular analysis and to complete our understanding on how bacterial division works.

The effect of the signalling scheme on the robustness of pattern formation in development

Pattern formation in development is a complex process which involves spatially distributed signals called morphogens that influence gene expression and thus the phenotypic identity of cells. Usually different cell types are spatially segregated, and the boundary between them may be determined by a threshold value of some state variable. The question arises as to how sensitive the location of such a boundary is to variations in properties, such as parameter values, that characterize the system. Here, we analyse both deterministic and stochastic reaction-diffusion models of pattern formation with a view towards understanding how the signalling scheme used for patterning affects the variability of boundary determination between cell types in a developing tissue.

A new method for choosing the computational cell in stochastic reaction-diffusion systems

How to choose the computational compartment or cell size for the stochastic simulation of a reaction-diffusion system is still an open problem, and a number of criteria have been suggested. A generalized measure of the noise for finite-dimensional systems based on the largest eigenvalue of the covariance matrix of the number of molecules of all species has been suggested as a measure of the overall fluctuations in a multivariate system, and we apply it here to a discretized reaction-diffusion system. We show that for a broad class of first-order reaction networks this measure converges to the square root of the reciprocal of the smallest mean species number in a compartment at the steady state. We show that a suitably re-normalized measure stabilizes as the volume of a cell approaches zero, which leads to a criterion for the maximum volume of the compartments in a computational grid. We then derive a new criterion based on the sensitivity of the entire network, not just of the fastest step, that predicts a grid size that assures that the concentrations of all species converge to a spatially-uniform solution. This criterion applies for all orders of reactions and for reaction rate functions derived from singular perturbation or other reduction methods, and encompasses both diffusing and non-diffusing species. We show that this predicts the maximal allowable volume found in a linear problem, and we illustrate our results with an example motivated by anterior-posterior pattern formation in Drosophila, and with several other examples.

Spontaneous noise reduction in a strongly cooperative reaction model

Noise characteristics of a simple multiparticle reaction model were investigated. The model can describe strongly cooperative reactions and is defined as follows. Particles A react in clusters of size k and each reaction forms a product molecule P. The back reaction is also allowed, and each cluster can dissociate into k reactants A. To describe a situation where the reaction is part of a pathway, the system is made open by assuming that particles A are injected and that particles P decay. This is a continuation study. The model is studied by using the same technique as employed previously, the pair approach reaction noise estimator (PARNES) method. Several new issues have been addressed. (i) In the previous work closeness to the Poisson distribution was used as a noise measure. In this work a more traditional noise measure, the ratio of the variance and the mean, was used to analyze stochastic features of the problem. (ii) The dependence of the new noise measure on k has been analyzed in detail, with an emphasis of investigating reactions with large k values. (iii) The previous study focused on understanding time-dependent issues, while this study focuses on describing the equilibrium state of the system. (iv) An exact solution (published elsewhere), available for closed system only, was used to re-investigate the validity of the PARNES method for describing reactions with large k. It was found that the PARNES method cannot describe fine details of the noise characteristics of such reactions.

Kinetic analysis of β-galactosidase and β-glucuronidase tetramerization coupled with protein translation

Both β-galactosidase (GAL) and β-glucuronidase (GUS) are tetrameric enzymes used widely as reporter proteins. However, little is known about the folding and assembly of these enzymes. Although the refolding kinetics of GAL from a denatured enzyme have been reported, it is not known how the kinetics differ when coupled with a protein translation reaction. Elucidating the assembly kinetics of GAL and GUS when coupled with protein translation will illustrate the differences between these two reporter proteins and also the assembly process under conditions more relevant to those in vivo. In this study, we used an in vitro translation/transcription system to synthesize GAL and GUS, measured the time development of the activity and oligomerization state of these enzymes, and determined the rate constants of the monomer to tetramer assembly process. We found that at similar concentrations, GAL assembles into tetramers faster than GUS. The rate constant of monomer to dimer assembly of GAL was 50-fold faster when coupled with protein translation than that of refolding from the denatured state. Furthermore, GAL synthesis was found to lack the rate-limiting step in the assembly process, whereas GUS has two rate-limiting steps: monomer to dimer assembly and dimer to tetramer assembly. The consequence of these differences when used as reporter proteins is discussed.

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. Kirkwood's 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.

Exact equilibrium-state solution of an intracellular complex formation model: kA↔P reaction in a small volume

A generic model of complex formation in small volumes was studied under the assumption of perfect mixing. Particles A react in clusters, and each reaction converts k A particles into a P particle. The back reaction is also allowed. The equilibrium state of the model is solved exactly. Fluctuations in product particle number are reduced by increasing the degree of cooperativity k. Three qualitatively distinct reactant fluctuation characteristics emerge.

Biochemical simulations: stochastic, approximate stochastic and hybrid approaches

Computer simulations have become an invaluable tool to study the sometimes counterintuitive temporal dynamics of (bio-)chemical systems. In particular, stochastic simulation methods have attracted increasing interest recently. In contrast to the well-known deterministic approach based on ordinary differential equations, they can capture effects that occur due to the underlying discreteness of the systems and random fluctuations in molecular numbers. Numerous stochastic, approximate stochastic and hybrid simulation methods have been proposed in the literature. In this article, they are systematically reviewed in order to guide the researcher and help her find the appropriate method for a specific problem.

A variational approach to the stochastic aspects of cellular signal transduction

Cellular signaling networks have evolved to cope with intrinsic fluctuations, coming from the small numbers of constituents, and the environmental noise. Stochastic chemical kinetics equations govern the way biochemical networks process noisy signals. The essential difficulty associated with the master equation approach to solving the stochastic chemical kinetics problem is the enormous number of ordinary differential equations involved. In this work, we show how to achieve tremendous reduction in the dimensionality of specific reaction cascade dynamics by solving variationally an equivalent quantum field theoretic formulation of stochastic chemical kinetics. The present formulation avoids cumbersome commutator computations in the derivation of evolution equations, making the physical significance of the variational method more transparent. We propose novel time-dependent basis functions which work well over a wide range of rate parameters. We apply the new basis functions to describe stochastic signaling in several enzymatic cascades and compare the results so obtained with those from alternative solution techniques. The variational Ansatz gives probability distributions that agree well with the exact ones, even when fluctuations are large and discreteness and nonlinearity are important. A numerical implementation of our technique is many orders of magnitude more efficient computationally compared with the traditional Monte Carlo simulation algorithms or the Langevin simulations.

The interplay between discrete noise and nonlinear chemical kinetics in a signal amplification cascade

We used various analytical and numerical techniques to elucidate signal propagation in a small enzymatic cascade which is subjected to external and internal noises. The nonlinear character of catalytic reactions, which underlie protein signal transduction cascades, renders stochastic signaling dynamics in cytosol biochemical networks distinct from the usual description of stochastic dynamics in gene regulatory networks. For a simple two-step enzymatic cascade which underlies many important protein signaling pathways, we demonstrated that the commonly used techniques such as the linear noise approximation and the Langevin equation become inadequate when the number of proteins becomes too low. Consequently, we developed a new analytical approximation, based on mixing the generating function and distribution function approaches, to the solution of the master equation that describes nonlinear chemical signaling kinetics for this important class of biochemical reactions. Our techniques work in a much wider range of protein number fluctuations than the methods used previously. We found that under certain conditions the burst phase noise may be injected into the downstream signaling network dynamics, resulting possibly in unusually large macroscopic fluctuations. In addition to computing first and second moments, which is the goal of commonly used analytical techniques, our new approach provides the full time-dependent probability distributions of the colored non-Gaussian processes in a nonlinear signal transduction cascade.

The hard cell: From proteomics to a whole cell model

Proteomics has provided a wealth of data related to the nature of the proteome, including subcellular location, copy number, interaction partners and protein complexes. This raises the question of whether it is feasible to combine these data, together with other data related to overall cellular structure, to construct a static picture of the cell. In this minireview, we discuss these data, and the issues of turning them into whole cell models.

A stochastic analysis of first-order reaction networks

A stochastic model for a general system of first-order reactions in which each reaction may be either a conversion reaction or a catalytic reaction is derived. The governing master equation is formulated in a manner that explicitly separates the effects of network topology from other aspects, and the evolution equations for the first two moments are derived. We find the surprising, and apparently unknown, result that the time evolution of the second moments can be represented explicitly in terms of the eigenvalues and projections of the matrix that governs the evolution of the means. The model is used to analyze the effects of network topology and the reaction type on the moments of the probability distribution. In particular, it is shown that for an open system of first-order conversion reactions, the distribution of all the system components is a Poisson distribution at steady state. Two different measures of the noise have been used previously, and it is shown that different qualitative and quantitative conclusions can result, depending on which measure is used. The effect of catalytic reactions on the variance of the system components is also analyzed, and the master equation for a coupled system of first-order reactions and diffusion is derived.

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.

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.

The Engineering of Gene Regulatory Networks

The rapid accumulation of genetic information and advancement of experimental techniques have opened a new frontier in biomedical engineering. With the availability of well-characterized components from natural gene networks, the stage has been set for the engineering of artificial gene regulatory networks with sophisticated computational and functional capabilities. In these efforts, the ability to construct, analyze, and interpret qualitative and quantitative models is becoming increasingly important. In this review, we consider the current state of gene network engineering from a combined experimental and modeling perspective. We discuss how networks with increased complexity are being constructed from simple modular components and how quantitative deterministic and stochastic modeling of these modules may provide the foundation for accurate in silico representations of gene regulatory network function in vivo.

Modeling network dynamics: the lac operon, a case study

We use the lac operon in Escherichia coli as a prototype system to illustrate the current state, applicability, and limitations of modeling the dynamics of cellular networks. We integrate three different levels of description (molecular, cellular, and that of cell population) into a single model, which seems to capture many experimental aspects of the system.

A mathematical model of single target site location by Brownian movement in subcellular compartments

The location of distinct sites is mandatory for many cellular processes. In the subcompartments of the cell nucleus, only very small numbers of diffusing macromolecules and specific target sites of some types may be present. In this case, we are faced with the Brownian movement of individual macromolecules and their "random search" for single/few specific target sites, rather than bulk-averaged diffusion and multiple sites. In this article, I consider the location of a distant central target site, e.g. a globular protein, by individual macromolecules executing unbiased (i.e. drift-free) random walks in a spherical compartment. For this walk-and-capture model, the closed-form analytic solution of the first passage time probability density function (p.d.f.) has been obtained as well as the first and second moment. In the limit of a large ratio of the radii of the spherical diffusion space and central target, well-known relations for the variance and the first two moments for the exponential p.d.f. were found to hold with high accuracy. These calculations reinforce earlier numerical results and Monte Carlo simulations. A major implication derivable from the model is that non-directed random movement is an effective means for locating single sites in submicron-sized compartments, even when the diffusion coefficients are comparatively small and the diffusing species are present in one copy only. These theoretical conclusions are underscored numerically for effective diffusion constants ranging from 0.5 to 10.0 microm(2) s(-1), which have been reported for a couple of nuclear proteins in their physiological environment. Spherical compartments of submicron size are, for example, the Cajal bodies (size: 0.1-1.0 microm), which are present in 1-5 copies in the cell nucleus. Within a small Cajal body of radius 0.1 microm a single diffusing protein molecule (with D=0.5 microm(2) s(-1)) would encounter a medium-sized protein of radius 2.5 nm within 1 s with a probability near certainty (p=0.98).