Fitting a function to time-dependent ensemble averaged data.

Time-dependent ensemble averages, i.e., trajectory-based averages of some observable, are of importance in many fields of science. A crucial objective when interpreting such data is to fit these averages (for instance, squared displacements) with a function and extract parameters (such as diffusion constants). A commonly overlooked challenge in such function fitting procedures is that fluctuations around mean values, by construction, exhibit temporal correlations. We show that the only available general purpose function fitting methods, correlated chi-square method and the weighted least squares method (which neglects correlation), fail at either robust parameter estimation or accurate error estimation. We remedy this by deriving a new closed-form error estimation formula for weighted least square fitting. The new formula uses the full covariance matrix, i.e., rigorously includes temporal correlations, but is free of the robustness issues, inherent to the correlated chi-square method. We demonstrate its accuracy in four examples of importance in many fields: Brownian motion, damped harmonic oscillation, fractional Brownian motion and continuous time random walks. We also successfully apply our method, weighted least squares including correlation in error estimation (WLS-ICE), to particle tracking data. The WLS-ICE method is applicable to arbitrary fit functions, and we provide a publically available WLS-ICE software.

Selection Shapes Transcriptional Logic and Regulatory Specialization in Genetic Networks.

BACKGROUND: Living organisms need to regulate their gene expression in response to environmental signals and internal cues. This is a computational task where genes act as logic gates that connect to form transcriptional networks, which are shaped at all scales by evolution. Large-scale mutations such as gene duplications and deletions add and remove network components, whereas smaller mutations alter the connections between them. Selection determines what mutations are accepted, but its importance for shaping the resulting networks has been debated. METHODOLOGY: To investigate the effects of selection in the shaping of transcriptional networks, we derive transcriptional logic from a combinatorially powerful yet tractable model of the binding between DNA and transcription factors. By evolving the resulting networks based on their ability to function as either a simple decision system or a circadian clock, we obtain information on the regulation and logic rules encoded in functional transcriptional networks. Comparisons are made between networks evolved for different functions, as well as with structurally equivalent but non-functional (neutrally evolved) networks, and predictions are validated against the transcriptional network of E. coli. PRINCIPAL FINDINGS: We find that the logic rules governing gene expression depend on the function performed by the network. Unlike the decision systems, the circadian clocks show strong cooperative binding and negative regulation, which achieves tight temporal control of gene expression. Furthermore, we find that transcription factors act preferentially as either activators or repressors, both when binding multiple sites for a single target gene and globally in the transcriptional networks. This separation into positive and negative regulators requires gene duplications, which highlights the interplay between mutation and selection in shaping the transcriptional networks.

Rethinking transcriptional activation in the Arabidopsis circadian clock.

Circadian clocks are biological timekeepers that allow living cells to time their activity in anticipation of predictable daily changes in light and other environmental factors. The complexity of the circadian clock in higher plants makes it difficult to understand the role of individual genes or molecular interactions, and mathematical modelling has been useful in guiding clock research in model organisms such as Arabidopsis thaliana. We present a model of the circadian clock in Arabidopsis, based on a large corpus of published time course data. It appears from experimental evidence in the literature that most interactions in the clock are repressive. Hence, we remove all transcriptional activation found in previous models of this system, and instead extend the system by including two new components, the morning-expressed activator RVE8 and the nightly repressor/activator NOX. Our modelling results demonstrate that the clock does not need a large number of activators in order to reproduce the observed gene expression patterns. For example, the sequential expression of the PRR genes does not require the genes to be connected as a series of activators. In the presented model, transcriptional activation is exclusively the task of RVE8. Predictions of how strongly RVE8 affects its targets are found to agree with earlier interpretations of the experimental data, but generally we find that the many negative feedbacks in the system should discourage intuitive interpretations of mutant phenotypes. The dynamics of the clock are difficult to predict without mathematical modelling, and the clock is better viewed as a tangled web than as a series of loops.