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A key goal of synthetic biology is to establish 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.

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.

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 vivometal 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 proteinsin vitroto 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 kinetics. Here we demonstrate a high-throughput intrinsic tryptophan fluorescence quenching method to validate kinetic models in multi-metal solutions forCcNikZ-II, a nickel-binding protein fromClostridium carboxidivorans. Using our validated models, we quantify the relationship between binding affinity and specificity in different classes of metal-binding models forCcNikZ-II. We further demonstrate that principles for improving specificity through changes in binding affinity are qualitatively different depending on the competing metals, highlighting the power of mechanistic models to guide metalloprotein engineering targets.

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 co-variability can be exploited to rigorously infer properties of gene expression dynamics when gene expression reporters probe their upstream dynamics on separate time-scales. 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 co-regulated 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.

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.

SUMMARYWhile molecules that promote the growth of animal cells have been identified, the following question remains: How are growth promoting pathways regulated to specify a characteristic size for each of the different cell types? In 1975, Hartwell and Nurse suggested that in eukaryotes, cell size is determined by size checkpoints – mechanisms that restrict cell cycle progression from cells that aresmallerthan theirtarget size. Curiously, such checkpoint mechanisms imply a conceptual distinction between a cell’sactualsize and cell’stargetsize. In the present study, we materialize this conceptual distinction by describing experimental assays that discriminately quantify a cell’s target size value. With these assays, we show that a cell’s size and target size are distinct phenotypes that are subject to different upstream regulators. While mTORC1 promotes growth in cell size, our data suggests that a cell’s target size value is regulated by other pathways including FGFR3, ROCK2, and CDK4. For example, while rapamycin (an mTORC1 inhibitor) decreases cell size, rapamycin does not change the target size that is required for the G1/S transition. The CDK4/Rb pathway has been previously proposed as a putative regulator of target size. Yet, in lacking experimental means that discriminate perturbations of cell growth from perturbations that reprogram target size, such claims on target size were not validated. To investigate the functions of CDK4 in target size determination, we used genetic and chemical means to ‘dial’ higher and lower levels of CDK4 activity. These measurements identified functions of CDK4 on target size that are distinct from other G1 CDKs. UsingC. elegans, we further demonstrate that these influences of CDK4 on size determination functionin vivo. Finally, we propose a model whereby mTORC1, p38, and CDK4 cooperate in a manner that is analogous to the function of a thermostat. While mTORC1 promotes cellular growth as prompted by p38, CDK4 is analogous to the thermostatdialthat sets the critical target size associated with cell size homeostasis.

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 developed visual barcodes that discriminate the clonal identity of single cells by targeting different fluorescent proteins to specific subcellular locations. We demonstrate that deconvolution of these barcodes is highly accurate and robust to many cellular perturbations. We then used visual barcodes to generate ‘Signalome’ cell-lines by multiplexing live reporters to monitor the simultaneous activity in 12 branches of signaling, in live cells, at single cell resolution, over time. Using the ‘Signalome’ we identified 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 system

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.

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.

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.

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.

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.

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.

Genetically identical cells can have many variable properties. A study of correlations between cells in a lineage explains paradoxical inheritance laws, in which mother and daughter cells seem less similar than cousins. See Letter p.468

Cilia and flagella are hair-like 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., 19 pages, 5 figures

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.

Many eukaryotic cells possess cilia which are motile, whip-like appendages that can oscillate and thereby induce motion and fluid flows. These organelles contain a highly conserved structure called the axoneme, whose characteristic architecture is based on a cylindrical arrangement of nine doublets of microtubules. Complex bending waves emerge from the interplay of active internal forces generated by dynein motor proteins within the structure. These bending waves are typically chiral and often exhibit a sense of rotation. In order to study how the shape of the beat emerges from the axonemal structure, we present a three-dimensional description of ciliary dynamics based on the self-organization of dynein motors and microtubules. Taking into account both bending and twisting of the cilium, we determine self-organized beating patterns and find that modes with both a clockwise and anticlockwise sense of rotation exist. Because of the axonemal chirality, only one of these modes is selected dynamically for given parameter values and properties of dynein motors. This physical mechanism, which underlies the selection of a beating pattern with specific sense of rotation, triggers the breaking of the left–right symmetry of developing embryos which is induced by asymmetric fluid flows that are generated by rotating cilia. M This article features online multimedia enhancements

Although the superficial appearance of the vertebrate body plan is left-right symmetric, the inner organs of vertebrates exhibit a strikingly asymmetric arrangement. It has been shown that this left-right asymmetry is induced early during embryonic development and the result of a fluid flow generated by the clockwise rotation of cilia, which are are motile, hair-like cellular appendages. What determines the specific handedness of these ciliary rotations is the subject of ongoing debate. Based on a three-dimensional theoretical description of the ciliary geometry we discuss the bending modes generated by the cooperativity of force generating dynein motors working against elastic microtubules within cilia. Taking into account both bending and twisting of the ciliary structure, we find that despite the chirality of the ciliary structure, cilia can in principle generate clockwise as well as anticlockwise twirling beat patterns. However, our results show that the axoneme's chirality leads to one sense of rotation being selected dynamically for given parameter values and properties of dynein motors. This dynamic selection of asymmetric states is analogous to how the direction of motion of a motor protein moving along a filament.