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

The circadian oscillator analysed at the single-transcript level

The circadian clock is an endogenous and self-sustained oscillator that anticipates daily environmental cycles. While rhythmic gene expression of circadian genes is well-described in populations of cells, the single-cell mRNA dynamics of multiple core clock genes remain largely unknown. Here we use single-molecule fluorescence in situ hybridisation (smFISH) at multiple time points to measure pairs of core clock transcripts, Rev-erbα (Nr1d1), Cry1 and Bmal1, in mouse fibroblasts. The mean mRNA level oscillates over 24 h for all three genes, but mRNA numbers show considerable spread between cells. We develop a probabilistic model for multivariate mRNA counts using mixtures of negative binomials, which accounts for transcriptional bursting, circadian time and cell-to-cell heterogeneity, notably in cell size. Decomposing the mRNA variability into distinct noise sources shows that clock time contributes a small fraction of the total variability in mRNA number between cells. Thus, our results highlight the intrinsic biological challenges in estimating circadian phase from single-cell mRNA counts and suggest that circadian phase in single cells is encoded post-transcriptionally.

Barriers to transmission of transcriptional noise in a c-fos c-jun pathway

We explored how transcriptional noise propagates in gene-regulatory pathways by studying the induction of two downstream genes by transcription factors c-fos and c-jun. They are produced for a brief period following serum stimulation of cells and then activate the promoters of their target genes by binding to them as heterodimers. We found that, even though they are coordinately expressed at the population level, in individual cells the expression of c-fos and c-jun is noisy and uncorrelated with each other. The expression of the downstream genes is also noisy, but there is little or no effect of the noise in the upstream genes on the expression of the downstream genes. The noise is not transmitted, because the number of heterodimers present in single cells is relatively invariant, and the induction of downstream genes is insensitive to the number of heterodimers in individual cells. Sequestration of promoters of the downstream genes within compact chromatin is a likely cause of this insensitivity. These barriers to the propagation and amplification of noise are likely to be commonplace in higher eukaryotes.

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.

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

Background

The 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.

Results

In 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 Hill's and Adair-Klotz's 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.

Conclusions

A 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.

A danger of low copy numbers for inferring incorrect cooperativity degree

BACKGROUND

A dose-response curve depicts the fraction of bound proteins as a function of unbound ligands. Dose-response curves are used to measure the cooperativity degree of a ligand binding process. Frequently, the Hill function is used to fit the experimental data. The Hill function is parameterized by the value of the dissociation constant and the Hill coefficient, which describes the cooperativity degree. The use of Hill's model and the Hill function has been heavily criticised in this context, predominantly the assumption that all ligands bind at once, which resulted in further refinements of the model. In this work, the validity of the Hill function has been studied from an entirely different point of view. In the limit of low copy numbers the dynamics of the system becomes noisy. The goal was to asses the validity of the Hill function in this limit, and to see in what ways the effects of the fluctuations change the form of the dose-response curves.

RESULTS

Dose-response curves were computed taking into account effects of fluctuations. The effects of fluctuations were described at the lowest order (the second moment of the particle number distribution) by using the previously developed Pair Approach Reaction Noise EStimator (PARNES) method. The stationary state of the system is described by nine equations with nine unknowns. To obtain fluctuation-corrected dose-response curves the equations have been investigated numerically.

CONCLUSIONS

The Hill function cannot describe dose-response curves in a low particle limit. First, dose-response curves are not solely parameterized by the dissociation constant and the Hill coefficient. In general, the shape of a dose-response curve depends on the variables that describe how an experiment (ensemble) is designed. Second, dose-response curves are multi-valued in a rather non-trivial way.