Spatiotemporal Patterning Enabled by Gene Regulatory Networks.

Spatiotemporal pattern formation plays a key role in various biological phenomena including embryogenesis and neural network formation. Though the reaction-diffusion systems enabling pattern formation have been studied phenomenologically, the biomolecular mechanisms behind these processes have not been modeled in detail. Here, we study the emergence of spatiotemporal patterns due to simple, synthetic and commonly observed two- and three-node gene regulatory network motifs coupled with their molecular diffusion in one- and two-dimensional space. We investigate the patterns formed due to the coupling of inherent multistable and oscillatory behavior of the toggle switch, toggle switch with double self-activation, toggle triad, and repressilator with the effect of spatial diffusion of these molecules. We probe multiple parameter regimes corresponding to different regions of stability (monostable, multistable, oscillatory) and assess the impact of varying diffusion coefficients. This analysis offers valuable insights into the design principles of pattern formation facilitated by these network motifs, and it suggests the mechanistic underpinnings of biological pattern formation.

Precision of Protein Thermometry.

Temperature sensing is a ubiquitous cell behavior, but the fundamental limits to the precision of temperature sensing are poorly understood. Unlike in chemical concentration sensing, the precision of temperature sensing is not limited by extrinsic fluctuations in the temperature field itself. Instead, we find that precision is limited by the intrinsic copy number, turnover, and binding kinetics of temperature-sensitive proteins. Developing a model based on the canonical TlpA protein, we find that a cell can estimate temperature to within 2%. We compare this prediction with in vivo data on temperature sensing in bacteria.

Intermediate adhesion maximizes migration velocity of multicellular clusters.

Collections of cells exhibit coherent migration during morphogenesis, cancer metastasis, and wound healing. In many cases, bigger clusters split, smaller subclusters collide and reassemble, and gaps continually emerge. The connections between cell-level adhesion and cluster-level dynamics, as well as the resulting consequences for cluster properties such as migration velocity, remain poorly understood. Here we investigate collective migration of one- and two-dimensional cell clusters that collectively track chemical gradients using a mechanism based on contact inhibition of locomotion. We develop both a minimal description based on the lattice gas model of statistical physics and a more realistic framework based on the cellular Potts model which captures cell shape changes and cluster rearrangement. In both cases, we find that cells have an optimal adhesion strength that maximizes cluster migration speed. The optimum negotiates a tradeoff between maintaining cell-cell contact and maintaining configurational freedom, and we identify maximal variability in the cluster aspect ratio as a revealing signature. Our results suggest a collective benefit for intermediate cell-cell adhesion.

Statistics of correlated percolation in a bacterial community.

Signal propagation over long distances is a ubiquitous feature of multicellular communities, but cell-to-cell variability can cause propagation to be highly heterogeneous. Simple models of signal propagation in heterogenous media, such as percolation theory, can potentially provide a quantitative understanding of these processes, but it is unclear whether these simple models properly capture the complexities of multicellular systems. We recently discovered that in biofilms of the bacterium Bacillus subtilis, the propagation of an electrical signal is statistically consistent with percolation theory, and yet it is reasonable to suspect that key features of this system go beyond the simple assumptions of basic percolation theory. Indeed, we find here that the probability for a cell to signal is not independent from other cells as assumed in percolation theory, but instead is correlated with its nearby neighbors. We develop a mechanistic model, in which correlated signaling emerges from cell division, phenotypic inheritance, and cell displacement, that reproduces the experimentally observed correlations. We find that the correlations do not significantly affect the spatial statistics, which we rationalize using a renormalization argument. Moreover, the fraction of signaling cells is not constant in space, as assumed in percolation theory, but instead varies within and across biofilms. We find that this feature lowers the fraction of signaling cells at which one observes the characteristic power-law statistics of cluster sizes, consistent with our experimental results. We validate the model using a mutant biofilm whose signaling probability decays along the propagation direction. Our results reveal key statistical features of a correlated signaling process in a multicellular community. More broadly, our results identify extensions to percolation theory that do or do not alter its predictions and may be more appropriate for biological systems.

Ultrasensitivity and fluctuations in the Barkai-Leibler model of chemotaxis receptors in Escherichia coli.

A stochastic version of the Barkai-Leibler model of chemotaxis receptors in Escherichia coli is studied here with the goal of elucidating the effects of intrinsic network noise in their conformational dynamics. The model was originally proposed to explain the robust and near-perfect adaptation of E. coli observed across a wide range of spatially uniform attractant/repellent (ligand) concentrations. In the model, a receptor is either active or inactive and can stochastically switch between the two states. The enzyme CheR methylates inactive receptors while CheB demethylates active receptors and the probability for a receptor to be active depends on its level of methylation and ligand occupation. In a simple version of the model with two methylation sites per receptor (M = 2), we show rigorously, under a quasi-steady state approximation, that the mean active fraction of receptors is an ultrasensitive function of [CheR]/[CheB] in the limit of saturating receptor concentration. Hence the model shows zero-order ultrasensitivity (ZOU), similar to the classical two-state model of covalent modification studied by Goldbeter and Koshland (GK). We also find that in the limits of extremely small and extremely large ligand concentrations, the system reduces to two different two-state GK modules. A quantitative measure of the spontaneous fluctuations in activity is provided by the variance [Formula: see text] in the active fraction, which is estimated mathematically under linear noise approximation (LNA). It is found that [Formula: see text] peaks near the ZOU transition. The variance is a non-monotonic, but weak function of ligand concentration and a decreasing function of receptor concentration. Gillespie simulations are also performed in models with M = 2, 3 and 4. For M = 2, simulations show excellent agreement with analytical results obtained under LNA. Numerical results for M = 3 and M = 4 are qualitatively similar to our mathematical results in M = 2; while all the models show ZOU in mean activity, the variance is found to be smaller for larger M. The magnitude of receptor noise deduced from available experimental data is consistent with our predictions. A simple analysis of the downstream signaling pathway shows that this noise is large enough to affect the motility of the organism, and may have a beneficial effect on it. The response of mean receptor activity to small time-dependent changes in the external ligand concentration is computed within linear response theory, and found to have a bilobe form, in agreement with earlier experimental observations.

Zero-order ultrasensitivity: a study of criticality and fluctuations under the total quasi-steady state approximation in the linear noise regime.

Zero-order ultrasensitivity (ZOU) is a long known and interesting phenomenon in enzyme networks. Here, a substrate is reversibly modified by two antagonistic enzymes (a 'push-pull' system) and the fraction in modified state undergoes a sharp switching from near-zero to near-unity at a critical value of the ratio of the enzyme concentrations, under saturation conditions. ZOU and its extensions have been studied for several decades now, ever since the seminal paper of Goldbeter and Koshland (1981); however, a complete probabilistic treatment, important for the study of fluctuations in finite populations, is still lacking. In this paper, we study ZOU using a modular approach, akin to the total quasi-steady state approximation (tQSSA). This approach leads to a set of Fokker-Planck (drift-diffusion) equations for the probability distributions of the intermediate enzyme-bound complexes, as well as the modified/unmodified fractions of substrate molecules. We obtain explicit expressions for various average fractions and their fluctuations in the linear noise approximation (LNA). The emergence of a 'critical point' for the switching transition is rigorously established. New analytical results are derived for the average and variance of the fractional substrate concentration in various chemical states in the near-critical regime. For the total fraction in the modified state, the variance is shown to be a maximum near the critical point and decays algebraically away from it, similar to a second-order phase transition. The new analytical results are compared with existing ones as well as detailed numerical simulations using a Gillespie algorithm.