Paulsson, J. ; Ehrenberg, M. Noise in a minimal regulatory network: plasmid copy number control. Quarterly Reviews of Biophysics 2001, 34, 1-59.
Berg, O. G. ; Paulsson, J. ; Ehrenberg, M. Fluctuations and quality of control in biological cells: zero-order ultrasensitivity reinvestigated. Biophysical Journal 2000, 79, 1228-36.Abstract
Living cells differ from most other chemical systems in that they involve regulation pathways that depend very nonlinearly on chemical species that are present in low copy numbers per cell. This leads to a variety of intracellular kinetic phenomena that elude macroscopic modeling, which implicitly assumes that cells are infinitely large and fluctuations negligible. It is of particular importance to assess how fluctuations affect regulation in cases where precision and reliability are required. Here, taking finite cell size and stochastic aspects into account, we reinvestigate theoretically the mechanism of zero-order ultrasensitivity for covalent modification of target enzymes ( Proc. Natl. Acad. Sci. USA. 78:6840-6844). Macroscopically, this mechanism can produce a very sharp transition in target concentrations for very small changes in the activity of the converter enzymes. This study shows that the transition is much more gradual in a finite cell or a population of finite cells. It also demonstrates that the switch is exactly analogous to a thermodynamic phase transition and that ultrasensitivity is inevitably coupled to random ultravariation. As a consequence, the average response in a large population of cells will often be much more gradual than predicted from macroscopic descriptions.
Berg, O. G. ; Paulsson, J. ; Ehrenberg, M. Fluctuations in repressor control: thermodynamic constraints on stochastic focusing. Biophysical Journal 2000, 79, 2944-53.Abstract
The influence of fluctuations in molecule numbers on genetic control circuits has received considerable attention. The consensus has been that such fluctuations will make regulation less precise. In contrast, it has more recently been shown that signal fluctuations can sharpen the response in a regulated process by the principle of stochastic focusing (SF) (, Proc. Natl. Acad. Sci. USA. 97:7148-7153). In many cases, the larger the fluctuations are, the sharper is the response. Here we investigate how fluctuations in repressor or corepressor numbers can improve the control of gene expression. Because SF is found to be constrained by detailed balance, this requires that the control loops contain driven processes out of equilibrium. Some simple and realistic out-of-equilibrium steps that will break detailed balance and make room for SF in such systems are discussed. We conclude that when the active repressors are controlled by corepressor molecules that display large ("coherent") number fluctuations or when corepressors can be irreversibly removed directly from promoter-bound repressors, the response in gene activity can become significantly sharper than without intrinsic noise. A simple experimental design to establish the possibility of SF for repressor control is suggested.
Paulsson, J. ; Berg, O. G. ; Ehrenberg, M. Stochastic focusing: fluctuation-enhanced sensitivity of intracellular regulation. Proceedings of the National Academy of Sciences of USA 2000, 97, 7148-53.Abstract
Many regulatory molecules are present in low copy numbers per cell so that significant random fluctuations emerge spontaneously. Because cell viability depends on precise regulation of key events, such signal noise has been thought to impose a threat that cells must carefully eliminate. However, the precision of control is also greatly affected by the regulatory mechanisms' capacity for sensitivity amplification. Here we show that even if signal noise reduces the capacity for sensitivity amplification of threshold mechanisms, the effect on realistic regulatory kinetics can be the opposite: stochastic focusing (SF). SF particularly exploits tails of probability distributions and can be formulated as conventional multistep sensitivity amplification where signal noise provides the degrees of freedom. When signal fluctuations are sufficiently rapid, effects of time correlations in signal-dependent rates are negligible and SF works just like conventional sensitivity amplification. This means that, quite counterintuitively, signal noise can reduce the uncertainty in regulated processes. SF is exemplified by standard hyperbolic inhibition, and all probability distributions for signal noise are first derived from underlying chemical master equations. The negative binomial is suggested as a paradigmatic distribution for intracellular kinetics, applicable to stochastic gene expression as well as simple systems with Michaelis-Menten degradation or positive feedback. SF resembles stochastic resonance in that noise facilitates signal detection in nonlinear systems, but stochastic resonance is related to how noise in threshold systems allows for detection of subthreshold signals and SF describes how fluctuations can make a gradual response mechanism work more like a threshold mechanism.
Paulsson, J. ; Ehrenberg, M. Molecular clocks reduce plasmid loss rates: the R1 case. Journal of Molecular Biology 2000, 297, 179-92.Abstract
Plasmids control their replication so that the replication frequency per plasmid copy responds to the number of plasmid copies per cell. High sensitivity amplification in replication response to copy number deviations generally reduces variation in copy numbers between different single cells, thereby reducing the plasmid loss rate in a cell population. However, experiments show that plasmid R1 has a gradual, insensitive replication control predicting considerable copy number variation between single cells. The critical step in R1 copy number control is regulation of synthesis of a rate-limiting cis-acting replication protein, RepA. De novo synthesis of a large number of RepA molecules is required for replication, suggesting that copy number control is exercised at multiple steps. In this theoretical kinetic study we analyse R1 multistep copy number control and show that it results in the insensitive replication response found experimentally but that it at the same time effectively prohibits the existence of only one plasmid copy in a dividing cell. In combination with the partition system of R1, this can lead to very high segregational stability. The R1 control mechanism is compared to the different multistep copy number control of plasmid ColE1 that is based on conventional sensitivity amplification. This implies that while copy number control for ColE1 efficiently corrects for fluctuations that have already occurred, R1 copy number control prevents their emergence in cells that by chance start their cycle with only one plasmid copy. We also discuss how regular, clock-like, behaviour of single plasmid copies becomes hidden in experiments probing collective properties of a population of plasmid copies because the individual copies are out of phase. The model is formulated using master equations, taking a stochastic approach to regulation, but the mathematical formalism is kept to a minimum and the model is simplified to its bare essence. This simplicity makes it possible to extend the analysis to other replicons with similar design principles.
Paulsson, J. ; Ehrenberg, M. Random signal fluctuations can reduce random fluctuations in regulated components of chemical regulatory networks. Physical Review Letters 2000, 84, 5447-50.Abstract
Many intracellular components are present in low copy numbers per cell and subject to feedback control. We use chemical master equations to analyze a negative feedback system where species X and S regulate each other's synthesis with standard intracellular kinetics. For a given number of X-molecules, S-variation can be significant. We show that this signal noise does not necessarily increase X-variation as previously thought but, surprisingly, can be necessary to reduce it below a Poissonian limit. The principle resembles Stochastic Resonance in that signal noise improves signal detection.
Paulsson, J. ; Nordström, K. ; Ehrenberg, M. Requirements for rapid plasmid ColE1 copy number adjustments: a mathematical model of inhibition modes and RNA turnover rates. Plasmid 1998, 39, 215-34.Abstract
The random distribution of ColE1 plasmids between the daughter cells at cell division introduces large copy number variations. Statistic variation associated with limited copy number in single cells also causes fluctuations to emerge spontaneously during the cell cycle. Efficient replication control out of steady state is therefore important to tame such stochastic effects of small numbers. In the present model, the dynamic features of copy number control are divided into two parts: first, how sharply the replication frequency per plasmid responds to changes in the concentration of the plasmid-coded inhibitor, RNA I, and second, how tightly RNA I and plasmid concentrations are coupled. Single (hyperbolic)- and multiple (exponential)-step inhibition mechanisms are compared out of steady state and it is shown how the response in replication frequency depends on the mode of inhibition. For both mechanisms, sensitivity of inhibition is "bought" at the expense of a rapid turnover of a replication preprimer, RNA II. Conventional, single-step, inhibition kinetics gives a sloppy replication control even at high RNA II turnover rates, whereas multiple-step inhibition has the potential of working with unlimited precision. When plasmid concentration changes rapidly, RNA I must be degraded rapidly to be "up to date" with the change. Adjustment to steady state is drastically impaired when the turnover rate constants of RNA I decrease below certain thresholds, but is basically unaffected for a corresponding increase. Several features of copy number control that are shown to be crucial for the understanding of ColE1-type plasmids still remain to be experimentally characterized. It is shown how steady-state properties reflect dynamics at the heart of regulation and therefore can be used to discriminate between fundamentally different copy number control mechanisms. The experimental tests of the predictions made require carefully planned assays, and some suggestions for suitable experiments arise naturally from the present work. It is also discussed how the presence of the Rom protein may affect dynamic qualities of copy number control.
Paulsson, J. ; Ehrenberg, M. Trade-off between segregational stability and metabolic burden: a mathematical model of plasmid ColE1 replication control. Journal of Molecular Biology 1998, 279, 73-88.Abstract
A model of ColE1 copy number control has been developed where molecular details of replication are connected both to segregational stability and metabolic burden. Efficient replication control reduces copy number variation and increases segregational stability for a given average copy number. Copy number variation is predicted to depend on the type of inhibition mechanism as well as RNA I and RNA II turnover rate constants. It is shown that when both RNA I and RNA II transcription frequencies and the rate constant for degradation of free RNA I are very large, a hyperbolic inhibition mechanism must compensate with a 1.4 times greater average copy number to obtain the same segregational stability as an exponential inhibition mechanism. How sensitively the replication frequency responds to changes in RNA I concentration depends on the type of inhibition mechanism and the number of attempts to form an RNA II replication primer per plasmid and cell cycle. If RNA I is too stable, it will not follow changes in plasmid concentration closely, and when the transcription frequency for RNA I is only slightly higher than for RNA II, RNA I concentration becomes randomized. In both these cases, the proportionality between the single cell RNA I and plasmid concentrations is lost and this impairs copy number control. Thresholds in the rate for degradation of free RNA I as well as in RNA I and RNA II transcription frequencies have been computed, where an increase in these rate constants has a negligible effect on segregational stability but a corresponding decrease leads to segregational disaster. This indicates that there exists a well defined optimal set of rate constants where the regulation system works well without excessive metabolic load. A number of new experiments are suggested to address features of particular importance for the evolution of ColE1 copy number control.