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.
AbstractThe 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.
AbstractA 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.