Time-dependent corrections to effective rate and event statistics in Michaelis-Menten kinetics.

The authors generalise the concept of the geometric phase in stochastic kinetics to a non-cyclic evolution. Its application is demonstrated on kinetics of the Michaelis-Menten reaction. It is shown that the non-periodic geometric phase is responsible for the correction to the Michaelis-Menten law when parameters, such as a substrate concentration, are changing with time. The authors apply these ideas to a model of chemical reactions in a bacterial culture of a growing size, where the geometric correction qualitatively changes the outcome of the reaction kinetics.

Multivariate dependence and genetic networks inference.

A critical task in systems biology is the identification of genes that interact to control cellular processes by transcriptional activation of a set of target genes. Many methods have been developed that use statistical correlations in high-throughput data sets to infer such interactions. However, cellular pathways are highly cooperative, often requiring the joint effect of many molecules. Few methods have been proposed to explicitly identify such higher-order interactions, partially due to the fact that the notion of multivariate statistical dependence itself remains imprecisely defined. The authors define the concept of dependence among multiple variables using maximum entropy techniques and introduce computational tests for their identification. Synthetic network results reveal that this procedure uncovers dependencies even in undersampled regimes, when the joint probability distribution cannot be reliably estimated. Analysis of microarray data from human B cells reveals that third-order statistics, but not second-order ones, uncover relationships between genes that interact in a pathway to cooperatively regulate a common set of targets.

Quantifying evolvability in small biological networks.

The authors introduce a quantitative measure of the capacity of a small biological network to evolve. The measure is applied to a stochastic description of the experimental setup of Guet et al. (Science 2002, 296, pp. 1466), treating chemical inducers as functional inputs to biochemical networks and the expression of a reporter gene as the functional output. The authors take an information-theoretic approach, allowing the system to set parameters that optimise signal processing ability, thus enumerating each network's highest-fidelity functions. All networks studied are highly evolvable by the measure, meaning that change in function has little dependence on change in parameters. Moreover, each network's functions are connected by paths in the parameter space along which information is not significantly lowered, meaning a network may continuously change its functionality without completely losing it along the way. This property further underscores the evolvability of the networks.

Mesoscopic statistical properties of multistep enzyme-mediated reactions.

Enzyme-mediated reactions may proceed through multiple intermediate conformational states before creating a final product molecule, and one often wishes to identify such intermediate structures from observations of the product creation. In this study, the authors address this problem by solving the chemical master equations for various enzymatic reactions. A perturbation theory analogous to that used in quantum mechanics allows the determination of the first (n) and the second (σ2) cumulants of the distribution of created product molecules as a function of the substrate concentration and the kinetic rates of the intermediate processes. The mean product flux V=d(n)/dt (or 'dose-response' curve) and the Fano factor F= σ2/(n) are both realistically measurable quantities, and whereas the mean flux can often appear the same for different reaction types, the Fano factor can be quite different. This suggests both qualitative and quantitative ways to discriminate between different reaction schemes, and the authors explore this possibility in the context of four sample multistep enzymatic reactions. Measuring both the mean flux and the Fano factor can not only discriminate between reaction types, but can also provide some detailed information about the internal, unobserved kinetic rates, and this can be done without measuring single-molecule transition events.

Determinants of bistability in induction of the Escherichia coli lac operon.

The authors have developed a mathematical model of regulation of expression of the Escherichia coli lac operon, and have investigated bistability in its steady-state induction behaviour in the absence of external glucose. Numerical analysis of equations describing regulation by artificial inducers revealed two natural bistability parameters that can be used to control the range of inducer concentrations over which the model exhibits bistability. By tuning these bistability parameters, the authors found a family of biophysically reasonable systems that are consistent with an experimentally determined bistable region for induction by thio-methylgalactoside (TMG) (in Ozbudak et al. Nature, 2004, 427; p. 737). To model regulation by lactose, the authors developed similar equations in which allolactose, a metabolic intermediate in lactose metabolism and a natural inducer of lac, is the inducer. For biophysically reasonable parameter values, these equations yield no bistability in response to induction by lactose - only systems with an unphysically small permease-dependent export effect can exhibit small amounts of bistability for limited ranges of parameter values. These results cast doubt on the relevance of bistability in the lac operon within the natural context of E. coli, and help shed light on the controversy among existing theoretical studies that address this issue. The results also motivate a deeper experimental characterisation of permease-independent transport of lac inducers, and suggest an experimental approach to address the relevance of bistability in the lac operon within the natural context of E. coli. The sensitivity of lac bistability to the type of inducer emphasises the importance of metabolism in determining the functions of genetic regulatory networks.

Serially regulated biological networks fully realise a constrained set of functions.

It is shown that biological networks with serial regulation (each node regulated by at most one other node) are constrained to direct functionality, in which the sign of the effect of an environmental input on a target species depends only on the direct path from the input to the target, even when there is a feedback loop allowing for multiple interaction pathways. Using a stochastic model for a set of small transcriptional regulatory networks that have been studied experimentally, it is further found that all networks can achieve all functions permitted by this constraint under reasonable settings of biochemical parameters. This underscores the functional versatility of the networks.

Predictability, complexity, and learning.

We define predictive information I(pred)(T) as the mutual information between the past and the future of a time series. Three qualitatively different behaviors are found in the limit of large observation times T:I(pred)(T) can remain finite, grow logarithmically, or grow as a fractional power law. If the time series allows us to learn a model with a finite number of parameters, then I(pred)(T) grows logarithmically with a coefficient that counts the dimensionality of the model space. In contrast, power-law growth is associated, for example, with the learning of infinite parameter (or nonparametric) models such as continuous functions with smoothness constraints. There are connections between the predictive information and measures of complexity that have been defined both in learning theory and the analysis of physical systems through statistical mechanics and dynamical systems theory. Furthermore, in the same way that entropy provides the unique measure of available information consistent with some simple and plausible conditions, we argue that the divergent part of I(pred)(T) provides the unique measure for the complexity of dynamics underlying a time series. Finally, we discuss how these ideas may be useful in problems in physics, statistics, and biology.