Criticality and Adaptivity in Enzymatic Networks

Solving stochastic gene-expression models using queueing theory: A tutorial review

Stochastic models of gene expression are typically formulated using the chemical master equation, which can be solved exactly or approximately using a repertoire of analytical methods. Here, we provide a tutorial review of an alternative approach based on queueing theory that has rarely been used in the literature of gene expression. We discuss the interpretation of six types of infinite-server queues from the angle of stochastic single-cell biology and provide analytical expressions for the stationary and nonstationary distributions and/or moments of mRNA/protein numbers and bounds on the Fano factor. This approach may enable the solution of complex models that have hitherto evaded analytical solution.

Kullback-Leibler Divergence of an Open-Queuing Network of a Cell-Signal-Transduction Cascade

Queuing networks (QNs) are essential models in operations research, with applications in cloud computing and healthcare systems. However, few studies have analyzed the cell's biological signal transduction using QN theory. This study entailed the modeling of signal transduction as an open Jackson's QN (JQN) to theoretically determine cell signal transduction, under the assumption that the signal mediator queues in the cytoplasm, and the mediator is exchanged from one signaling molecule to another through interactions between the signaling molecules. Each signaling molecule was regarded as a network node in the JQN. The JQN Kullback-Leibler divergence (KLD) was defined using the ratio of the queuing time (λ) to the exchange time (μ), λ/μ. The mitogen-activated protein kinase (MAPK) signal-cascade model was applied, and the KLD rate per signal-transduction-period was shown to be conserved when the KLD was maximized. Our experimental study on MAPK cascade supported this conclusion. This result is similar to the entropy-rate conservation of chemical kinetics and entropy coding reported in our previous studies. Thus, JQN can be used as a novel framework to analyze signal transduction.

Switchable Biocatalytic Reactions Controlled by Interfacial pH Changes Produced by Orthogonal Biocatalytic Processes

Enzymes immobilized on a nano-structured surface were used to switch the activity of one enzyme by a local pH change produced by another enzyme. Immobilized amyloglucosidase (AMG) and trypsin were studied as examples of the pH-dependent switchable "target enzymes." The reactions catalyzed by co-immobilized urease or esterase were increasing or decreasing the local pH, respectively, thus operating as "actuator enzymes." Both kinds of the enzymes, producing local pH changes and changing biocatalytic activity with the pH variation, were orthogonal in terms of the biocatalytic reactions; however, their operation was coupled with the local pH produced near the surface with the immobilized enzymes. The "target enzymes" (AMG and trypsin) were changed reversibly between the active and inactive states by applying input signals (urea or ester, substrates for the urease or esterase operating as the "actuator enzymes") and washing them out with a new portion of the background solution. The developed approach can potentially lead to switchable operation of several enzymes, while some of them are inhibited when the others are activated upon receiving external signals processed by the "actuator enzymes." More complex systems with branched biocatalytic cascades can be controlled by orthogonal biocatalytic reactions activating selected pathways and changing the final output.

Engineering self-organized criticality in living cells

Complex dynamical fluctuations, from intracellular noise, brain dynamics or computer traffic display bursting dynamics consistent with a critical state between order and disorder. Living close to the critical point has adaptive advantages and it has been conjectured that evolution could select these critical states. Is this the case of living cells? A system can poise itself close to the critical point by means of the so-called self-organized criticality (SOC). In this paper we present an engineered gene network displaying SOC behaviour. This is achieved by exploiting the saturation of the proteolytic degradation machinery in E. coli cells by means of a negative feedback loop that reduces congestion. Our critical motif is built from a two-gene circuit, where SOC can be successfully implemented. The potential implications for both cellular dynamics and behaviour are discussed.

Magneto-Controlled Enzyme Activity with Locally Produced pH Changes

Biocatalytic activity of amyloglucosidase (AMG), immobilized on superparamagnetic nanoparticles, is dynamically and reversibly activated or inhibited by applying a magnetic field. The magnetic field triggers aggregation/deaggregation of magnetic particles that are also functionalized with urease or esterase enzymes. These enzymes produce a local pH change in the vicinity of the particles changing the AMG activity.

Novel Tunable Spatio-Temporal Patterns From a Simple Genetic Oscillator Circuit

Multicellularity, the coordinated collective behavior of cell populations, gives rise to the emergence of self-organized phenomena at many different spatio-temporal scales. At the genetic scale, oscillators are ubiquitous in regulation of multicellular systems, including during their development and regeneration. Synthetic biologists have successfully created simple synthetic genetic circuits that produce oscillations in single cells. Studying and engineering synthetic oscillators in a multicellular chassis can therefore give us valuable insights into how simple genetic circuits can encode complex multicellular behaviors at different scales. Here we develop a study of the coupling between the repressilator synthetic genetic ring oscillator and constraints on cell growth in colonies. We show in silico how mechanical constraints generate characteristic patterns of growth rate inhomogeneity in growing cell colonies. Next, we develop a simple one-dimensional model which predicts that coupling the repressilator to this pattern of growth rate via protein dilution generates traveling waves of gene expression. We show that the dynamics of these spatio-temporal patterns are determined by two parameters; the protein degradation and maximum expression rates of the repressors. We derive simple relations between these parameters and the key characteristics of the traveling wave patterns: firstly, wave speed is determined by protein degradation and secondly, wavelength is determined by maximum gene expression rate. Our analytical predictions and numerical results were in close quantitative agreement with detailed individual based simulations of growing cell colonies. Confirming published experimental results we also found that static ring patterns occur when protein stability is high. Our results show that this pattern can be induced simply by growth rate dilution and does not require transition to stationary phase as previously suggested. Our method generalizes easily to other genetic circuit architectures thus providing a framework for multi-scale rational design of spatio-temporal patterns from genetic circuits. We use this method to generate testable predictions for the synthetic biology design-build-test-learn cycle.

Synchronization of stochastic expressions drives the clustering of functionally related genes

Functionally related genes tend to be chromosomally clustered in eukaryotic genomes even after the exclusion of tandem duplicates, but the biological significance of this widespread phenomenon is unclear. We propose that stochastic expression fluctuations of neighboring genes resulting from chromatin dynamics are more or less synchronized such that their expression ratio is more stable than that for unlinked genes. Consequently, chromosomal clustering could be advantageous when the expression ratio of the clustered genes needs to stay constant, for example, because of the accumulation of toxic compounds when this ratio is altered. Evidence from manipulative experiments on the yeast GAL cluster, comprising three chromosomally adjacent genes encoding enzymes catalyzing consecutive reactions in galactose catabolism, unequivocally supports this hypothesis and elucidates how disorder in one biological phenomenon-gene expression noise-could prompt the emergence of order in another-genome organization.

Signal Percolation within a Bacterial Community

Signal transmission among cells enables long-range coordination in biological systems. However, the scarcity of quantitative measurements hinders the development of theories that relate signal propagation to cellular heterogeneity and spatial organization. We address this problem in a bacterial community that employs electrochemical cell-to-cell communication. We developed a model based on percolation theory, which describes how signals propagate through a heterogeneous medium. Our model predicts that signal transmission becomes possible when the community is organized near a critical phase transition between a disconnected and a fully connected conduit of signaling cells. By measuring population-level signal transmission with single-cell resolution in wild-type and genetically modified communities, we confirm that the spatial distribution of signaling cells is organized at the predicted phase transition. Our findings suggest that at this critical point, the population-level benefit of signal transmission outweighs the single-cell level cost. The bacterial community thus appears to be organized according to a theoretically predicted spatial heterogeneity that promotes efficient signal transmission.

Dynamical predictors of an imminent phenotypic switch in bacteria

Single cells can stochastically switch across thresholds imposed by regulatory networks. Such thresholds can act as a tipping point, drastically changing global phenotypic states. In ecology and economics, imminent transitions across such tipping points can be predicted using dynamical early warning indicators. A typical example is 'flickering' of a fast variable, predicting a longer-lasting switch from a low to a high state or vice versa. Considering the different timescales between metabolite and protein fluctuations in bacteria, we hypothesized that metabolic early warning indicators predict imminent transitions across a network threshold caused by enzyme saturation. We used stochastic simulations to determine if flickering predicts phenotypic transitions, accounting for a variety of molecular physiological parameters, including enzyme affinity, burstiness of enzyme gene expression, homeostatic feedback, and rates of metabolic precursor influx. In most cases, we found that metabolic flickering rates are robustly peaked near the enzyme saturation threshold. The degree of fluctuation was amplified by product inhibition of the enzyme. We conclude that sensitivity to flickering in fast variables may be a possible natural or synthetic strategy to prepare physiological states for an imminent transition.