Universal attenuators and their interactions with feedback loops in gene regulatory networks.

Using a combination of mathematical modelling, statistical simulation and large-scale data analysis we study the properties of linear regulatory chains (LRCs) within gene regulatory networks (GRNs). Our modelling indicates that downstream genes embedded within LRCs are highly insulated from the variation in expression of upstream genes, and thus LRCs act as attenuators. This observation implies a progressively weaker functionality of LRCs as their length increases. When analyzing the preponderance of LRCs in the GRNs of Escherichia coli K12 and several other organisms, we find that very long LRCs are essentially absent. In both E. coli and M. tuberculosis we find that four-gene LRCs are intimately linked to identical feedback loops that are involved in potentially chaotic stress response, indicating that the dynamics of these potentially destabilising motifs are strongly restrained under homeostatic conditions. The same relationship is observed in a human cancer cell line (K562), and we postulate that four-gene LRCs act as 'universal attenuators'. These findings suggest a role for long LRCs in dampening variation in gene expression, thereby protecting cell identity, and in controlling dramatic shifts in cell-wide gene expression through inhibiting chaos-generating motifs.

Inevitability and containment of replication errors for eukaryotic genome lengths spanning megabase to gigabase.

The replication of DNA is initiated at particular sites on the genome called replication origins (ROs). Understanding the constraints that regulate the distribution of ROs across different organisms is fundamental for quantifying the degree of replication errors and their downstream consequences. Using a simple probabilistic model, we generate a set of predictions on the extreme sensitivity of error rates to the distribution of ROs, and how this distribution must therefore be tuned for genomes of vastly different sizes. As genome size changes from megabases to gigabases, we predict that regularity of RO spacing is lost, that large gaps between ROs dominate error rates but are heavily constrained by the mean stalling distance of replication forks, and that, for genomes spanning ∼100 megabases to ∼10 gigabases, errors become increasingly inevitable but their number remains very small (three or less). Our theory predicts that the number of errors becomes significantly higher for genome sizes greater than ∼10 gigabases. We test these predictions against datasets in yeast, Arabidopsis, Drosophila, and human, and also through direct experimentation on two different human cell lines. Agreement of theoretical predictions with experiment and datasets is found in all cases, resulting in a picture of great simplicity, whereby the density and positioning of ROs explain the replication error rates for the entire range of eukaryotes for which data are available. The theory highlights three domains of error rates: negligible (yeast), tolerable (metazoan), and high (some plants), with the human genome at the extreme end of the middle domain.

Unreplicated DNA remaining from unperturbed S phases passes through mitosis for resolution in daughter cells.

To prevent rereplication of genomic segments, the eukaryotic cell cycle is divided into two nonoverlapping phases. During late mitosis and G1 replication origins are "licensed" by loading MCM2-7 double hexamers and during S phase licensed replication origins activate to initiate bidirectional replication forks. Replication forks can stall irreversibly, and if two converging forks stall with no intervening licensed origin-a "double fork stall" (DFS)-replication cannot be completed by conventional means. We previously showed how the distribution of replication origins in yeasts promotes complete genome replication even in the presence of irreversible fork stalling. This analysis predicts that DFSs are rare in yeasts but highly likely in large mammalian genomes. Here we show that complementary strand synthesis in early mitosis, ultrafine anaphase bridges, and G1-specific p53-binding protein 1 (53BP1) nuclear bodies provide a mechanism for resolving unreplicated DNA at DFSs in human cells. When origin number was experimentally altered, the number of these structures closely agreed with theoretical predictions of DFSs. The 53BP1 is preferentially bound to larger replicons, where the probability of DFSs is higher. Loss of 53BP1 caused hypersensitivity to licensing inhibition when replication origins were removed. These results provide a striking convergence of experimental and theoretical evidence that unreplicated DNA can pass through mitosis for resolution in the following cell cycle.

Buffered Qualitative Stability explains the robustness and evolvability of transcriptional networks.

The gene regulatory network (GRN) is the central decision-making module of the cell. We have developed a theory called Buffered Qualitative Stability (BQS) based on the hypothesis that GRNs are organised so that they remain robust in the face of unpredictable environmental and evolutionary changes. BQS makes strong and diverse predictions about the network features that allow stable responses under arbitrary perturbations, including the random addition of new connections. We show that the GRNs of E. coli, M. tuberculosis, P. aeruginosa, yeast, mouse, and human all verify the predictions of BQS. BQS explains many of the small- and large-scale properties of GRNs, provides conditions for evolvable robustness, and highlights general features of transcriptional response. BQS is severely compromised in a human cancer cell line, suggesting that loss of BQS might underlie the phenotypic plasticity of cancer cells, and highlighting a possible sequence of GRN alterations concomitant with cancer initiation.

Quantifying metastatic inefficiency: rare genotypes versus rare dynamics.

We introduce and solve a 'null model' of stochastic metastatic colonization. The model is described by a single parameter θ: the ratio of the rate of cell division to the rate of cell death for a disseminated tumour cell in a given secondary tissue environment. We are primarily interested in the case in which colonizing cells are poorly adapted for proliferation in the local tissue environment, so that cell death is more likely than cell division, i.e. θ < 1. We quantify the rare event statistics for the successful establishment of a metastatic colony of size N. For N >> 1, we find that the probability of establishment is exponentially rare, as expected, and yet the mean time for such rare events is of the form ~log (N)/(1 - θ) while the standard deviation of colonization times is ~1/(1 - θ). Thus, counter to naive expectation, for θ < 1, the average time for establishment of successful metastatic colonies decreases with decreasing cell fitness, and colonies seeded from lower fitness cells show less stochastic variation in their growth. These results indicate that metastatic growth from poorly adapted cells is rare, exponentially explosive and essentially deterministic. These statements are brought into sharper focus by the finding that the temporal statistics of the early stages of metastatic colonization from low-fitness cells (θ < 1) are statistically indistinguishable from those initiated from high-fitness cells (θ > 1), i.e. the statistics show a duality mapping (1 - θ) --> (θ - 1). We conclude our analysis with a study of heterogeneity in the fitness of colonising cells, and describe a phase diagram delineating parameter regions in which metastatic colonization is dominated either by low or high fitness cells, showing that both are plausible given our current knowledge of physiological conditions in human cancer.

Replisome stall events have shaped the distribution of replication origins in the genomes of yeasts.

During S phase, the entire genome must be precisely duplicated, with no sections of DNA left unreplicated. Here, we develop a simple mathematical model to describe the probability of replication failing due to the irreversible stalling of replication forks. We show that the probability of complete genome replication is maximized if replication origins are evenly spaced, the largest inter-origin distances are minimized, and the end-most origins are positioned close to chromosome ends. We show that origin positions in the yeast Saccharomyces cerevisiae genome conform to all three predictions thereby maximizing the probability of complete replication if replication forks stall. Origin positions in four other yeasts-Kluyveromyces lactis, Lachancea kluyveri, Lachancea waltii and Schizosaccharomyces pombe-also conform to these predictions. Equating failure rates at chromosome ends with those in chromosome interiors gives a mean per nucleotide fork stall rate of ∼5 × 10(-8), which is consistent with experimental estimates. Using this value in our theoretical predictions gives replication failure rates that are consistent with data from replication origin knockout experiments. Our theory also predicts that significantly larger genomes, such as those of mammals, will experience a much greater probability of replication failure genome-wide, and therefore will likely require additional compensatory mechanisms.

Correlating cell behavior with tissue topology in embryonic epithelia.

Measurements on embryonic epithelial tissues in a diverse range of organisms have shown that the statistics of cell neighbor numbers are universal in tissues where cell proliferation is the primary cell activity. Highly simplified non-spatial models of proliferation are claimed to accurately reproduce these statistics. Using a systematic critical analysis, we show that non-spatial models are not capable of robustly describing the universal statistics observed in proliferating epithelia, indicating strong spatial correlations between cells. Furthermore we show that spatial simulations using the Subcellular Element Model are able to robustly reproduce the universal histogram. In addition these simulations are able to unify ostensibly divergent experimental data in the literature. We also analyze cell neighbor statistics in early stages of chick embryo development in which cell behaviors other than proliferation are important. We find from experimental observation that cell neighbor statistics in the primitive streak region, where cell motility and ingression are also important, show a much broader distribution. A non-spatial Markov process model provides excellent agreement with this broader histogram indicating that cells in the primitive streak may have significantly weaker spatial correlations. These findings show that cell neighbor statistics provide a potentially useful signature of collective cell behavior.

Modeling cell rheology with the Subcellular Element Model.

Recently, the Subcellular Element Model (SEM) has been introduced, primarily to compute the dynamics of large numbers of three-dimensional deformable cells in multicellular systems. Within this model framework, each cell is represented by a collection of elastically coupled elements, interacting with one another via short-range potentials, and dynamically updated using over-damped Langevin dynamics. The SEM can also be used to represent a single cell in more detail, by using a larger number of subcellular elements exclusively identified with that cell. We have tested whether, in this context, the SEM yields viscoelastic properties consistent with those measured on single living cells. Employing virtual methods of bulk rheology and microrheology we find that the SEM successfully captures many cellular rheological properties at intermediate time scales and moderate strains, including weak power law rheology. In its simplest guise, the SEM cannot describe long-time/large-strain cell responses. Capturing these cellular properties requires extensions of the SEM which incorporate active cytoskeletal rearrangement. Such extensions will be the subject of a future publication.

Grid-free models of multicellular systems, with an application to large-scale vortices accompanying primitive streak formation.

This paper is comprised of two parts. In the first we provide a brief overview of grid-free methods for modeling multicellular systems. We focus on an approach based on Langevin equations, in which inertia is ignored, and stochastic effects on cell motion are included. The discussion starts with simpler models, in which cells are modeled as adhesive spheres. We then turn to more sophisticated approaches in which nontrivial cell shape is accommodated, including the recently introduced Subcellular Element Model, in which each cell is described as a cluster of adhesively coupled over-damped subcellular elements, representing patches of cytoskeleton. In the second part of the paper we illustrate the use of a standard grid-free cell-based model to computationally probe interesting new features associated with primitive streak formation in the chick embryo. Streak formation is a key developmental step in amniotes (i.e., birds, reptiles, and mammals), and can be observed in detail in the chick embryo, where the streak extends across a tightly-packed two-dimensional sheet (the epiblast) comprised of about 50,000 cells. The Weijer group [Cui, Yang, Chuai, Glazier, and Weijer, Dev. Biol. 284 (2005) 37-47] recently observed that streak formation is accompanied by coordinated cell movement lateral to the streak, resulting in two large counter-rotating vortices. We study a mechanism based on cell polarity (in the plane of the epiblast) that provides an explanation for these vortices, and test it successfully using computer simulations. This mechanism is robust, since the emergent vortex formation depends only on the gross features of the initial spatial distribution of planar polarity in the epiblast.