From graph topology to ODE models for gene regulatory networks.

A gene regulatory network can be described at a high level by a directed graph with signed edges, and at a more detailed level by a system of ordinary differential equations (ODEs). The former qualitatively models the causal regulatory interactions between ordered pairs of genes, while the latter quantitatively models the time-varying concentrations of mRNA and proteins. This paper clarifies the connection between the two types of models. We propose a property, called the constant sign property, for a general class of ODE models. The constant sign property characterizes the set of conditions (system parameters, external signals, or internal states) under which an ODE model is consistent with a signed, directed graph. If the constant sign property for an ODE model holds globally for all conditions, then the ODE model has a single signed, directed graph. If the constant sign property for an ODE model only holds locally, which may be more typical, then the ODE model corresponds to different graphs under different sets of conditions. In addition, two versions of constant sign property are given and a relationship between them is proved. As an example, the ODE models that capture the effect of cis-regulatory elements involving protein complex binding, based on the model in the GeneNetWeaver source code, are described in detail and shown to satisfy the global constant sign property with a unique consistent gene regulatory graph. Even a single gene regulatory graph is shown to have many ODE models of GeneNetWeaver type consistent with it due to combinatorial complexity and continuous parameters. Finally the question of how closely data generated by one ODE model can be fit by another ODE model is explored. It is observed that the fit is better if the two models come from the same graph.

Transcriptome-Enabled Network Inference Revealed the GmCOL1 Feed-Forward Loop and Its Roles in Photoperiodic Flowering of Soybean.

Photoperiodic flowering, a plant response to seasonal photoperiod changes in the control of reproductive transition, is an important agronomic trait that has been a central target of crop domestication and modern breeding programs. However, our understanding about the molecular mechanisms of photoperiodic flowering regulation in crop species is lagging behind. To better understand the regulatory gene networks controlling photoperiodic flowering of soybeans, we elucidated global gene expression patterns under different photoperiod regimes using the near isogenic lines (NILs) of maturity loci (E loci). Transcriptome signatures identified the unique roles of the E loci in photoperiodic flowering and a set of genes controlled by these loci. To elucidate the regulatory gene networks underlying photoperiodic flowering regulation, we developed the network inference algorithmic package CausNet that integrates sparse linear regression and Granger causality heuristics, with Gaussian approximation of bootstrapping to provide reliability scores for predicted regulatory interactions. Using the transcriptome data, CausNet inferred regulatory interactions among soybean flowering genes. Published reports in the literature provided empirical verification for several of CausNet's inferred regulatory interactions. We further confirmed the inferred regulatory roles of the flowering suppressors GmCOL1a and GmCOL1b using GmCOL1 RNAi transgenic soybean plants. Combinations of the alleles of GmCOL1 and the major maturity locus E1 demonstrated positive interaction between these genes, leading to enhanced suppression of flowering transition. Our work provides novel insights and testable hypotheses in the complex molecular mechanisms of photoperiodic flowering control in soybean and lays a framework for de novo prediction of biological networks controlling important agronomic traits in crops.

Time series experimental design under one-shot sampling: The importance of condition diversity.

Many biological data sets are prepared using one-shot sampling, in which each individual organism is sampled at most once. Time series therefore do not follow trajectories of individuals over time. However, samples collected at different times from individuals grown under the same conditions share the same perturbations of the biological processes, and hence behave as surrogates for multiple samples from a single individual at different times. This implies the importance of growing individuals under multiple conditions if one-shot sampling is used. This paper models the condition effect explicitly by using condition-dependent nominal mRNA production amounts for each gene, it quantifies the performance of network structure estimators both analytically and numerically, and it illustrates the difficulty in network reconstruction under one-shot sampling when the condition effect is absent. A case study of an Arabidopsis circadian clock network model is also included.

Kernel representations for flux and concentration in ion channel models with time-varying concentrations.

This paper explores stochastic models for the study of ion transport in biological cells. It considers one-dimensional models with time-varying concentrations at the boundaries. The average concentration and flux in the channel are obtained as kernel representations, where the kernel functions have a probabilistic interpretation which contributes to a better understanding of the models. In particular, the kernel representation is given for the flux at a boundary point, providing a correct version of a representation found in the literature. This requires special attention because one of the kernel functions exhibits a singularity. This kernel representation is feasible due to the linearity of the system that arises from the assumed independence between ions.

Equivalence of trans paths in ion channels.

We explore stochastic models for the study of ion transport in biological cells. Analysis of these models explains and explores an interesting feature of ion transport observed by biophysicists. Namely, the average time it takes ions to cross certain ion channels is the same in either direction, even if there is an electric potential difference across the channels. It is shown for simple single ion models that the distribution of a path (i.e., the history of location versus time) of an ion crossing the channel in one direction has the same distribution as the time-reversed path of an ion crossing the channel in the reverse direction. Therefore, not only is the mean duration of these paths equal, but other measures, such as the variance of passage time or the mean time a path spends within a specified section of the channel, are also the same for both directions of traversal. The feature is also explored for channels with interacting ions. If a system of interacting ions is in reversible equilibrium (net flux is zero), then the equivalence of the left-to-right trans paths with the time-reversed right-to-left trans paths still holds. However, if the system is in equilibrium, but not reversible equilibrium, then such equivalence need not hold.