Noise properties of adaptation-conferring biochemical control modules.

A key goal of synthetic biology is to develop functional biochemical modules with network-independent properties. Antithetic integral feedback (AIF) is a recently developed control module in which two control species perfectly annihilate each other's biological activity. The AIF module confers robust perfect adaptation to the steady-state average level of a controlled intracellular component when subjected to sustained perturbations. Recent work has suggested that such robustness comes at the unavoidable price of increased stochastic fluctuations around average levels. We present theoretical results that support and quantify this trade-off for the commonly analyzed AIF variant in the idealized limit with perfect annihilation. However, we also show that this trade-off is a singular limit of the control module: Even minute deviations from perfect adaptation allow systems to achieve effective noise suppression as long as cells can pay the corresponding energetic cost. We further show that a variant of the AIF control module can achieve significant noise suppression even in the idealized limit with perfect adaptation. This atypical configuration may thus be preferable in synthetic biology applications.

Measuring prion propagation in single bacteria elucidates a mechanism of loss.

Prions are self-propagating protein aggregates formed by specific proteins that can adopt alternative folds. Prions were discovered as the cause of the fatal transmissible spongiform encephalopathies in mammals, but prions can also constitute nontoxic protein-based elements of inheritance in fungi and other species. Prion propagation has recently been shown to occur in bacteria for more than a hundred cell divisions, yet a fraction of cells in these lineages lost the prion through an unknown mechanism. Here, we investigate prion propagation in single bacterial cells as they divide using microfluidics and fluorescence microscopy. We show that the propagation occurs in two distinct modes. In a fraction of the population, cells had multiple small visible aggregates and lost the prion through random partitioning of aggregates to one of the two daughter cells at division. In the other subpopulation, cells had a stable large aggregate localized to the pole; upon division the mother cell retained this polar aggregate and a daughter cell was generated that contained small aggregates. Extending our findings to prion domains from two orthologous proteins, we observe similar propagation and loss properties. Our findings also provide support for the suggestion that bacterial prions can form more than one self-propagating state. We implement a stochastic version of the molecular model of prion propagation from yeast and mammals that recapitulates all the observed single-cell properties. This model highlights challenges for prion propagation that are unique to prokaryotes and illustrates the conservation of fundamental characteristics of prion propagation.

Quantifying metal-binding specificity of CcNikZ-II from Clostridium carboxidivorans in the presence of competing metal ions.

Many proteins bind transition metal ions as cofactors to carry out their biological functions. Despite binding affinities for divalent transition metal ions being predominantly dictated by the Irving-Williams series for wild-type proteins, in vivo metal ion binding specificity is ensured by intracellular mechanisms that regulate free metal ion concentrations. However, a growing area of biotechnology research considers the use of metal-binding proteins in vitro to purify specific metal ions from wastewater, where specificity is dictated by the protein's metal binding affinities. A goal of metalloprotein engineering is to modulate these affinities to improve a protein's specificity towards a particular metal; however, the quantitative relationship between the affinities and the equilibrium metal-bound protein fractions depends on the underlying binding mechanisms. Here we demonstrate a high-throughput intrinsic tryptophan fluorescence quenching method to validate binding models in multi-metal solutions for CcNikZ-II, a nickel-binding protein from Clostridium carboxidivorans. Using our validated models, we quantify the relationship between binding affinity and specificity in different classes of metal-binding models for CcNikZ-II. We further illustrate the potential relevance of data-informed models to predicting engineering targets for improved specificity.

Measuring prion propagation in single bacteria elucidates mechanism of loss.

Prions are self-propagating protein aggregates formed by specific proteins that can adopt alternative folds. Prions were discovered as the cause of the fatal transmissible spongiform encephalopathies in mammals, but prions can also constitute non-toxic protein-based elements of inheritance in fungi and other species. Prion propagation has recently been shown to occur in bacteria for more than a hundred cell divisions, yet a fraction of cells in these lineages lost the prion through an unknown mechanism. Here, we investigate prion propagation in single bacterial cells as they divide using microfluidics and fluorescence microscopy. We show that the propagation occurs in two distinct modes with distinct stability and inheritance characteristics. We find that the prion is lost through random partitioning of aggregates to one of the two daughter cells at division. Extending our findings to prion domains from two orthologous proteins, we observe similar propagation and loss properties. Our findings also provide support for the suggestion that bacterial prions can form more than one self-propagating state. We implement a stochastic version of the molecular model of prion propagation from yeast and mammals that recapitulates all the observed single-cell properties. This model highlights challenges for prion propagation that are unique to prokaryotes and illustrates the conservation of fundamental characteristics of prion propagation across domains of life.

The effect of microRNA on protein variability and gene expression fidelity.

Small regulatory RNA molecules such as microRNA modulate gene expression through inhibiting the translation of messenger RNA (mRNA). Such posttranscriptional regulation has been recently hypothesized to reduce the stochastic variability of gene expression around average levels. Here, we quantify noise in stochastic gene expression models with and without such regulation. Our results suggest that silencing mRNA posttranscriptionally will always increase, rather than decrease, gene expression noise when the silencing of mRNA also increases its degradation, as is expected for microRNA interactions with mRNA. In that regime, we also find that silencing mRNA generally reduces the fidelity of signal transmission from deterministically varying upstream factors to protein levels. These findings suggest that microRNA binding to mRNA does not generically confer precision to protein expression.

Quantifying biochemical reaction rates from static population variability within incompletely observed complex networks.

Quantifying biochemical reaction rates within complex cellular processes remains a key challenge of systems biology even as high-throughput single-cell data have become available to characterize snapshots of population variability. That is because complex systems with stochastic and non-linear interactions are difficult to analyze when not all components can be observed simultaneously and systems cannot be followed over time. Instead of using descriptive statistical models, we show that incompletely specified mechanistic models can be used to translate qualitative knowledge of interactions into reaction rate functions from covariability data between pairs of components. This promises to turn a globally intractable problem into a sequence of solvable inference problems to quantify complex interaction networks from incomplete snapshots of their stochastic fluctuations.

Visual barcodes for clonal-multiplexing of live microscopy-based assays.

While multiplexing samples using DNA barcoding revolutionized the pace of biomedical discovery, multiplexing of live imaging-based applications has been limited by the number of fluorescent proteins that can be deconvoluted using common microscopy equipment. To address this limitation, we develop visual barcodes that discriminate the clonal identity of single cells by different fluorescent proteins that are targeted to specific subcellular locations. We demonstrate that deconvolution of these barcodes is highly accurate and robust to many cellular perturbations. We then use visual barcodes to generate 'Signalome' cell-lines by mixing 12 clones of different live reporters into a single population, allowing simultaneous monitoring of the activity in 12 branches of signaling, at clonal resolution, over time. Using the 'Signalome' we identify two distinct clusters of signaling pathways that balance growth and proliferation, emphasizing the importance of growth homeostasis as a central organizing principle in cancer signaling. The ability to multiplex samples in live imaging applications, both in vitro and in vivo may allow better high-content characterization of complex biological systems.

Inferring gene regulation dynamics from static snapshots of gene expression variability.

Inferring functional relationships within complex networks from static snapshots of a subset of variables is a ubiquitous problem in science. For example, a key challenge of systems biology is to translate cellular heterogeneity data obtained from single-cell sequencing or flow-cytometry experiments into regulatory dynamics. We show how static population snapshots of covariability can be exploited to rigorously infer properties of gene expression dynamics when gene expression reporters probe their upstream dynamics on separate timescales. This can be experimentally exploited in dual-reporter experiments with fluorescent proteins of unequal maturation times, thus turning an experimental bug into an analysis feature. We derive correlation conditions that detect the presence of closed-loop feedback regulation in gene regulatory networks. Furthermore, we show how genes with cell-cycle-dependent transcription rates can be identified from the variability of coregulated fluorescent proteins. Similar correlation constraints might prove useful in other areas of science in which static correlation snapshots are used to infer causal connections between dynamically interacting components.

Cell size homeostasis is maintained by CDK4-dependent activation of p38 MAPK.

While molecules that promote the growth of animal cells have been identified, it remains unclear how such signals are orchestrated to determine a characteristic target size for different cell types. It is increasingly clear that cell size is determined by size checkpoints-mechanisms that restrict the cell cycle progression of cells that are smaller than their target size. Previously, we described a p38 MAPK-dependent cell size checkpoint mechanism whereby p38 is selectively activated and prevents cell cycle progression in cells that are smaller than a given target size. In this study, we show that the specific target size required for inactivation of p38 and transition through the cell cycle is determined by CDK4 activity. Our data suggest a model whereby p38 and CDK4 cooperate analogously to the function of a thermostat: while p38 senses irregularities in size, CDK4 corresponds to the thermostat dial that sets the target size.

Kinetic Uncertainty Relations for the Control of Stochastic Reaction Networks.

Nonequilibrium stochastic reaction networks are commonly found in both biological and nonbiological systems, but have remained hard to analyze because small differences in rate functions or topology can change the dynamics drastically. Here, we conjecture exact quantitative inequalities that relate the extent of fluctuations in connected components, for various network topologies. Specifically, we find that regardless of how two components affect each other's production rates, it is impossible to suppress fluctuations below the uncontrolled equivalents for both components: one must increase its fluctuations for the other to be suppressed. For systems in which components control each other in ringlike structures, it appears that fluctuations can only be suppressed in one component if all other components instead increase fluctuations, compared to the case without control. Even the general N-component system-with arbitrary connections and parameters-must have at least one component with increased fluctuations to reduce fluctuations in others. In connected reaction networks it thus appears impossible to reduce the statistical uncertainty in all components, regardless of the control mechanisms or energy dissipation.

Exploiting Natural Fluctuations to Identify Kinetic Mechanisms in Sparsely Characterized Systems.

From biochemistry to ecology, many biological systems are stochastic, complex, and sparsely characterized. In such systems, each component may respond to changes in any directly or indirectly connected components, thus requiring knowledge of the whole to predict the dynamics of the parts. Here, we address this challenge by deriving relations between properties of fluctuations that only reflect local interactions between a subset of components but are invariant to all indirectly connected dynamics. This greatly reduces the number of assumptions when evaluating dynamic models experimentally. We illustrate the approach by revisiting systematic single-cell gene expression data, and we show that the observed fluctuations contradict the assumptions made in most published models of stochastic gene expression, even when accounting for the possibility of systematic experimental artifacts.

Constraints on Fluctuations in Sparsely Characterized Biological Systems.

Biochemical processes are inherently stochastic, creating molecular fluctuations in otherwise identical cells. Such "noise" is widespread but has proven difficult to analyze because most systems are sparsely characterized at the single cell level and because nonlinear stochastic models are analytically intractable. Here, we exactly relate average abundances, lifetimes, step sizes, and covariances for any pair of components in complex stochastic reaction systems even when the dynamics of other components are left unspecified. Using basic mathematical inequalities, we then establish bounds for whole classes of systems. These bounds highlight fundamental trade-offs that show how efficient assembly processes must invariably exhibit large fluctuations in subunit levels and how eliminating fluctuations in one cellular component requires creating heterogeneity in another.

Using temporal correlations and full distributions to separate intrinsic and extrinsic fluctuations in biological systems.

Studies of stochastic biological dynamics typically compare observed fluctuations to theoretically predicted variances, sometimes after separating the intrinsic randomness of the system from the enslaving influence of changing environments. But variances have been shown to discriminate surprisingly poorly between alternative mechanisms, while for other system properties no approaches exist that rigorously disentangle environmental influences from intrinsic effects. Here, we apply the theory of generalized random walks in random environments to derive exact rules for decomposing time series and higher statistics, rather than just variances. We show for which properties and for which classes of systems intrinsic fluctuations can be analyzed without accounting for extrinsic stochasticity and vice versa. We derive two independent experimental methods to measure the separate noise contributions and show how to use the additional information in temporal correlations to detect multiplicative effects in dynamical systems.

Separating intrinsic from extrinsic fluctuations in dynamic biological systems.

From molecules in cells to organisms in ecosystems, biological populations fluctuate due to the intrinsic randomness of individual events and the extrinsic influence of changing environments. The combined effect is often too complex for effective analysis, and many studies therefore make simplifying assumptions, for example ignoring either intrinsic or extrinsic effects to reduce the number of model assumptions. Here we mathematically demonstrate how two identical and independent reporters embedded in a shared fluctuating environment can be used to identify intrinsic and extrinsic noise terms, but also how these contributions are qualitatively and quantitatively different from what has been previously reported. Furthermore, we show for which classes of biological systems the noise contributions identified by dual-reporter methods correspond to the noise contributions predicted by correct stochastic models of either intrinsic or extrinsic mechanisms. We find that for broad classes of systems, the extrinsic noise from the dual-reporter method can be rigorously analyzed using models that ignore intrinsic stochasticity. In contrast, the intrinsic noise can be rigorously analyzed using models that ignore extrinsic stochasticity only under very special conditions that rarely hold in biology. Testing whether the conditions are met is rarely possible and the dual-reporter method may thus produce flawed conclusions about the properties of the system, particularly about the intrinsic noise. Our results contribute toward establishing a rigorous framework to analyze dynamically fluctuating biological systems.

Nonlinear dynamics of cilia and flagella.

Cilia and flagella are hairlike extensions of eukaryotic cells which generate oscillatory beat patterns that can propel micro-organisms and create fluid flows near cellular surfaces. The evolutionary highly conserved core of cilia and flagella consists of a cylindrical arrangement of nine microtubule doublets, called the axoneme. The axoneme is an actively bending structure whose motility results from the action of dynein motor proteins cross-linking microtubule doublets and generating stresses that induce bending deformations. The periodic beat patterns are the result of a mechanical feedback that leads to self-organized bending waves along the axoneme. Using a theoretical framework to describe planar beating motion, we derive a nonlinear wave equation that describes the fundamental Fourier mode of the axonemal beat. We study the role of nonlinearities and investigate how the amplitude of oscillations increases in the vicinity of an oscillatory instability. We furthermore present numerical solutions of the nonlinear wave equation for different boundary conditions. We find that the nonlinear waves are well approximated by the linearly unstable modes for amplitudes of beat patterns similar to those observed experimentally.

How molecular motors shape the flagellar beat.

Cilia and eukaryotic flagella are slender cellular appendages whose regular beating propels cells and microorganisms through aqueous media. The beat is an oscillating pattern of propagating bends generated by dynein motor proteins. A key open question is how the activity of the motors is coordinated in space and time. To elucidate the nature of this coordination we inferred the mechanical properties of the motors by analyzing the shape of beating sperm: Steadily beating bull sperm were imaged and their shapes were measured with high precision using a Fourier averaging technique. Comparing our experimental data with wave forms calculated for different scenarios of motor coordination we found that only the scenario of interdoublet sliding regulating motor activity gives rise to satisfactory fits. We propose that the microscopic origin of such "sliding control" is the load dependent detachment rate of motors. Agreement between observed and calculated wave forms was obtained only if significant sliding between microtubules occurred at the base. This suggests a novel mechanism by which changes in basal compliance could reverse the direction of beat propagation. We conclude that the flagellar beat patterns are determined by an interplay of the basal properties of the axoneme and the mechanical feedback of dynein motors.