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.
We introduce a nonintrusive method exploiting single-cell variability after cell division to validate protein localization. We found that Clp proteases, widely reported to form biologically relevant foci, were uniformly distributed in Escherichia coli cells, and that many commonly used fluorescent proteins caused severe mislocalization when fused to homo-oligomers. Retagging five other reportedly foci-forming proteins with the most monomeric fluorescent protein tested suggests that the foci were caused by the fluorescent tags.
Gene expression involves inherently probabilistic steps that create fluctuations in protein abundances. The results from many in-depth analyses and genome-scale surveys have suggested how such fluctuations arise and spread, often in ways consistent with stochastic models of transcription and translation. But fluctuations also arise during cell division when molecules are partitioned stochastically between the two daughters. Here we mathematically demonstrate how stochastic partitioning contributes to the non-genetic heterogeneity. Our results show that partitioning errors are hard to correct, and that the resulting noise profiles are remarkably difficult to separate from gene expression noise. By applying these results to common experimental strategies and distinguishing between creation versus transmission of noise, we hypothesize that much of the cell-to-cell heterogeneity that has been attributed to various aspects of gene expression instead comes from random segregation at cell division. We propose experiments to separate between these two types of fluctuations and discuss future directions.
Many RNAs, proteins, and organelles are present in such low numbers per cell that random segregation of individual copies causes large "partitioning errors" at cell division. Even symmetrically dividing cells can then by chance produce daughters with very different composition. The size of the errors depends on the segregation mechanism: Control systems can reduce low-abundance errors, but the segregation process can also be subject to upstream sources of randomness or spatial heterogeneities that create large errors despite high abundances. Here we mathematically demonstrate how partitioning errors arise for different types of segregation mechanisms and how errors can be greatly increased by upstream heterogeneity but remarkably hard to avoid through controlled partitioning. We also show that seemingly straightforward experiments cannot be straightforwardly interpreted because very different mechanisms produce identical fits and present an approach to deal with this problem by adding binomial counting noise and testing for convexity or concavity in the partitioning error as a function of the binomial thinning parameter. The results lay a conceptual groundwork for more effective studies of heterogeneity among growing and dividing cells, whether in microbes or in differentiating tissues.
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.
Negative feedback is common in biological processes and can increase a system's stability to internal and external perturbations. But at the molecular level, control loops always involve signalling steps with finite rates for random births and deaths of individual molecules. Here we show, by developing mathematical tools that merge control and information theory with physical chemistry, that seemingly mild constraints on these rates place severe limits on the ability to suppress molecular fluctuations. Specifically, the minimum standard deviation in abundances decreases with the quartic root of the number of signalling events, making it extremely expensive to increase accuracy. Our results are formulated in terms of experimental observables, and existing data show that cells use brute force when noise suppression is essential; for example, regulatory genes are transcribed tens of thousands of times per cell cycle. The theory challenges conventional beliefs about biochemical accuracy and presents an approach to the rigorous analysis of poorly characterized biological systems.
Many cellular components are present in such low numbers per cell that random births and deaths of individual molecules can cause substantial "noise" in concentrations. But biochemical events do not necessarily occur in single steps of individual molecules. Some processes are greatly randomized when synthesis or degradation occurs in large bursts of many molecules during a short time interval. Conversely, each birth or death of a macromolecule could involve several small steps, creating a memory between individual events. We present a generalized theory for stochastic gene expression, formulating the variance in protein abundance in terms of the randomness of the individual gene expression events. We show that common types of molecular mechanisms can produce gestation and senescence periods that reduce noise without requiring higher abundances, shorter lifetimes, or any concentration-dependent control loops. We also show that most single-cell experimental methods cannot distinguish between qualitatively different stochastic principles, although this in turn makes such methods better suited for identifying which components introduce fluctuations. Characterizing the random events that give rise to noise in concentrations instead requires dynamic measurements with single-molecule resolution.
Noise in gene expression is generated at multiple levels, such as transcription and translation, chromatin remodeling and pathway-specific regulation. Studies of individual promoters have suggested different dominating noise sources, raising the question of whether a general trend exists across a large number of genes and conditions. We examined the variation in the expression levels of 43 Saccharomyces cerevisiae proteins, in cells grown under 11 experimental conditions. For all classes of genes and under all conditions, the expression variance was approximately proportional to the mean; the same scaling was observed at steady state and during the transient responses to the perturbations. Theoretical analysis suggests that this scaling behavior reflects variability in mRNA copy number, resulting from random 'birth and death' of mRNA molecules or from promoter fluctuations. Deviation of coexpressed genes from this general trend, including high noise in stress-related genes and low noise in proteasomal genes, may indicate fluctuations in pathway-specific regulators or a differential activation pattern of the underlying gene promoters.
Chromatin-based repression is a major mechanism for epigenetically heritable variation. Work in the July 21 issue of Molecular Cell quantitatively examines transcriptional silencing in individual yeast cells, demonstrating locus-specific effects and finding that different silencing mutants exhibit qualitatively distinct single-cell defects.
Initiation of DNA replication is a highly regulated process in all organisms. Proteins that are required to recruit DNA polymerase - initiator proteins - are often used to regulate the timing or frequency of initiation in the cell cycle by limiting either their own synthesis or availability. Studies of the Escherichia coli chromosome and of bacterial plasmids with iterated initiator binding sites (iterons) have revealed that, in addition to initiator limitation, replication origin inactivation is used to prevent replication that is untimely or excessive. Our recent studies of plasmid P1 revealed that this additional mode of control becomes a requirement when initiator availability is limited only by autoregulation. Thus, although initiator limitation appears to be a well-conserved and central mode of replication control, optimal replication might require additional control mechanisms. This review gives examples of how the multiple mechanisms can act synergistically, antagonistically or be partially redundant to guarantee low frequency events. The lessons learned are likely to help understand many other regulatory systems in the bacterial cell.
Many organisms control initiation of DNA replication by limiting supply or activity of initiator proteins. In plasmids, such as P1, initiators are limited primarily by transcription and dimerization. However, the relevance of initiator limitation to plasmid copy number control has appeared doubtful, because initiator oversupply increases the copy number only marginally. Copy number control instead has been attributed to initiator-mediated plasmid pairing ("handcuffing"), because initiator mutations to handcuffing deficiency elevates the copy number significantly. Here, we present genetic evidence of a role for initiator limitation in plasmid copy number control by showing that autorepression-defective initiator mutants also can elevate the plasmid copy number. We further show, by quantitative modeling, that initiator dimerization is a homeostatic mechanism that dampens active monomer increase when the protein is oversupplied. This finding implies that oversupplied initiator proteins are largely dimeric, partly accounting for their limited ability to increase copy number. A combination of autorepression, dimerization, and handcuffing appears to account fully for control of P1 plasmid copy number.
Protein levels have been shown to vary substantially between individual cells in clonal populations. In prokaryotes, the contribution to such fluctuations from the inherent randomness of gene expression has largely been attributed to having just a few transcripts of the corresponding mRNAs. By contrast, eukaryotic studies tend to emphasize chromatin remodeling and burst-like transcription. Here, we study single-cell transcription in Escherichia coli by measuring mRNA levels in individual living cells. The results directly demonstrate transcriptional bursting, similar to that indirectly inferred for eukaryotes. We also measure mRNA partitioning at cell division and correlate mRNA and protein levels in single cells. Partitioning is approximately binomial, and mRNA-protein correlations are weaker earlier in the cell cycle, where cell division has recently randomized the relative concentrations. Our methods further extend protein-based approaches by counting the integer-valued number of transcript with single-molecule resolution. This greatly facilitates kinetic interpretations in terms of the integer-valued random processes that produce the fluctuations.
Plasmid R1 is a low-copy-number plasmid that is present at a level of about four or five copies per average cell. The copy number is controlled posttranscriptionally at the level of synthesis of the rate-limiting initiator protein RepA. In addition to this, R1 has an auxiliary system that derepresses a second promoter at low copy numbers, leading to increased repA mRNA synthesis. This promoter is normally switched off by a constitutively synthesized plasmid-encoded repressor protein, CopB; in cells with low copy numbers, the concentration of CopB is low and the promoter is derepressed. Here we show that the rate of loss of a Par(+) derivative of the basic replicon of R1 increased about sevenfold when the cells contained a high concentration of the CopB protein formed from a compatible plasmid.
Random fluctuations in genetic networks are inevitable as chemical reactions are probabilistic and many genes, RNAs and proteins are present in low numbers per cell. Such 'noise' affects all life processes and has recently been measured using green fluorescent protein (GFP). Two studies show that negative feedback suppresses noise, and three others identify the sources of noise in gene expression. Here I critically analyse these studies and present a simple equation that unifies and extends both the mathematical and biological perspectives.
The supply and consumption of metabolites in living cells are catalyzed by enzymes. Here we consider two of the simplest schemes where one substrate is eliminated through Michaelis-Menten kinetics, and where two types of substrates are joined together by an enzyme. It is demonstrated how steady-state substrate concentrations can change ultrasensitively in response to changes in their supply rates and how this is coupled to slow relaxation back to steady state after a perturbation. In the one-substrate system, such near-critical behavior occurs when the supply rate approaches the maximal elimination rate, and in the two-substrate system it occurs when the rates of substrate supply are almost balanced. As systems that operate near criticality tend to display large random fluctuations, we also carried out a stochastic analysis using analytical approximations of master equations and compared the results with molecular-level Monte Carlo simulations. It was found that the significance of random fluctuations was directly coupled to the steady-state sensitivity and that the two substrates can fluctuate greatly because they are anticorrelated in such a way that the product formation rate displays only small variation. Basic relations are highlighted and biological implications are discussed.
The replication control genes of bacterial plasmids face selection at two conflicting levels. Plasmid copies that systematically overreplicate relative to their cell mates have a higher chance of fixing in descendant cells, but these cells typically have a lower chance of fixing in the population. Apart from identifying the conflict, this mathematical discussion characterizes the efficiency of the selection levels and suggests how they drive the evolution of kinetic mechanisms. In particular it is hypothesized that: (1) tighter replication control is more vulnerable to selfishness; (2) cis-acting replication activators are relics of a conflict where a plasmid outreplicated its intracellular competitors by monopolizing activators; (3) high-copy plasmids with sloppy replication control arise because intracellular selection favors overreplication, thereby relieving intercellular selection for lower loss rates; (4) the excessive synthesis of cis-acting replication activators and trans-acting inhibitors is the result of an arms race between cis selfishness and trans retaliations; (5) site-specific recombination of plasmid dimers is equivalent to self-policing; and (6) plasmids modify their horizontal transfer to spread without promoting selfishness. It is also discussed how replication control may be subject to a third level of selection acting on the entire population of plasmid-containing cells.
In one family of bacterial plasmids, multiple initiator binding sites, called iterons, are used for initiation of plasmid replication as well as for the control of plasmid copy number. Iterons can also pair in vitro via the bound initiators. This pairing, called handcuffing, has been suggested to cause steric hindrance to initiation and thereby control the copy number. To test this hypothesis, we have compared copy numbers of isogenic miniP1 plasmid monomer and dimer. The dimer copy number was only one-quarter that of the monomer, suggesting that the higher local concentration of origins in the dimer facilitated their pairing. Physical evidence consistent with iteron-mediated pairing of origins preferentially in the dimer was obtained in vivo. Thus, origin handcuffing can be a mechanism to control P1 plasmid replication.