Mathematical modeling of population dynamics of unstable plasmid-containing bacteria during continuous cultivation in a chemostat

A structural approach to studying the regularities of the population dynamics of unstable recombinant bacterial strains in a chemostat was elaborated. The approach is based on the mathematical modeling of cell distribution in a population with different numbers of plasmid copies. The effect of decreased selective preference of plasmidless variants of the recombinant strain in the chemostat, which is related to a decrease in the number of plasmid copies in cells upon long-term incubation was analyzed. It is shown that the time of half-elimination of plasmids from the bacterial population in the steady state in the chemostat T1/2 does not depend on the maximum number of plasmid copies in cells N but is determined only by the mean time of generation g and the probability of the loss of one plasmid copy tau. The dependence of the preference of bacterial plasmidless variants on the efficiency of expression of genes cloned into plasmids in chemostat was analyzed using the recombinant strain E. coli Z905, whose plasmids pPHL-7 contain cloned genes for the luminescence system of marine luminescing bacteria Photobacterium leiognathi.     

Propagation of an amplifiable recombinant plasmid in Saccharomyces cerevisiae: flow cytometry studies and segregated modeling

Efficient expression of a foreign protein product by the yeastSaccharomyces cerevisiaerequires a stable recombinant vector present at a high number of copies per cell. A conditional centromere yeast plasmid was constructed which can be amplified to high copy number by a process of unequal partitioning at cell division, followed by selection for increased copy number. However, in the absence of selection pressure for plasmid amplification, copy number rapidly drops from 25 plasmids/cell to 6 plasmids/cell in less than 10 generations of growth. Copy number subsequently decreases from 6 plasmids/cell to 2 plasmids/cell over a span of 50 generations. A combination of flow cytometric measurement of copy number distributions and segregated mathematical modeling were applied to test the predictions of a conceptual model of conditional centromereplasmid propagation. Measured distributions of plasmid content displayed a significant subpopulation of cells with a copy number of 4-6, evenin a population whose mean copy number was 13.5. This type of copy number distribution was reproduced by a mathematical model which assumes that amaximum of 4-6 centromere plasmids per cell can be stably partitionedat cell division. The model also reproduces the observed biphasic kinetics of plasmid number instability. The agreement between simulation and experimental results provides support for the proposed model and demonstrates the utility of the flow cytometry/segregated modeling approach for the study of multicopy recombinant vector propagation.     

Kinetic analysis of the effects of plasmid multimerization on segregational instability of CoIE1 type plasmids in Escherichia coli B/r

The effect of plasmid multimerization on segregational instability was investigated using a structured, segregated model of genetically modified Escherichia coli cells. By including the multimerization of plasmids, the model can predict the proportion of each multimer in the total plasmid population. Simulation results suggest that the plasmid copy number is controlled by the total plasmid content (i.e., total number of plasmid origins) in the host cell and that multimerization reduces the total number of independent, monomeric segregation units. However, multimerization is found to have a minor effect on decreasing plasmid segregational stability for multicopy plasmids with average copy number per cell greater than about 25. Also model predictions were used to test whether or not a nonrandom plasmid distribution at cell fission could cause segregational instability. Even in the case of severely biased partitioning, plasmids whose copy number is above 45 per cell do not show significant segregational instability. The results suggest that when the ColE1-type plasmid does not encode and express any large or disruptive foreign proteins, the copy number of 45 per cell may be the threshold at which only growth rate-dependent instability is responsible for overall plasmid instability.     

Population dynamics of a recombinant culture in a chemostat under prolonged cultivation

The dynamics of a chemostat culture of Escherichia coli K12 harboring plasmid pBR322 under prolonged cultivation with a nonselective complex medium were studied. The ability of the culture to form colonies on plates supplemented with different ampicillin concentrations was monitored. It was observed that almost all cells sampled were able to grow on a high concentration of ampicillin at the beginning of the experiment. However, a subpopulation which formed colonies on intermediate-concentration (500-1000 mg/L) plates, but not on a high-concentration (2000 mg/L) plate, was detected just before the appearance of the plasmid-free cells. As time progressed, the percentage of this subpopulation increased, reached a maximum, then decreased toward the end of experiment. At this time the culture was dominated by a subpopulation which could not form colonies on the 100 mg/L ampicillin plates. These results indicate that three major processes may occur in the chemostat: a gradual shift of the higher plasmid copy number population toward a relatively lower copy number population; the complete shedding of the plasmid due to faulty segregation of plasmids during cell division; and growth competition among the subpopulations. A previously derived model is extended to account for all subpopulations. The model agrees qualitatively with the experimental results.     

A comprehensive continuous-time model for the appearance of CGH signal due to chromosomal missegregations during mitosis

Aneuploidy, the gain or loss of large regions of the genome, is a common feature in cancer cells. Irregularities in chromosomal copy number caused by missegregations of chromosomes during mitosis can be visualized by cytogenetic techniques including fluorescence in situ hybridization (FISH), spectral karyotyping (SKY) and comparative genomic hybridization (CGH). In the current work, we consider the propagation of irregular copy numbers throughout a cell population as the individual cells progress through ordinary mitotic cell cycles. We use an algebraic model to track the different copy numbers as states in a stochastic process, based on the model of chromosome instability of Gusev, Kagansky, and Dooley, and consider the average copy number of a particular chromosome within a cell population as a function of the cell division rate. We review a number of mathematical models for determining the length of the cell cycle, including the Smith-Martin transition probability model and the 'sloppy size' model of Wheals, Tyson and Diekmann. The program MITOSIM simulates the growth of a population of cells using the aforementioned models of the cell cycle. MITOSIM allows the cell population to grow, with occasional resampling, until the average copy number of a given chromosome in the population reaches a preset threshold signifying a positive copy number alteration in this region. MITOSIM calculates the relationship between the missegregation rate and the growth rate of the cell population. This allows the user to test hypotheses regarding the effect chromosomal aberrations have upon the cell cycle, cell growth rates, and time to population dominance.     

Toxic effects of excess cloned centromeres.

Plasmids carrying a Saccharomyces cerevisiae centromere have a copy number of one or two, whereas other yeast plasmids have high copy numbers. The number of CEN plasmids per yeast cell was made artificially high by transforming cells simultaneously with several different CEN plasmids carrying different, independently selectable markers. Some host cells carried five different CEN plasmids and an average total of 13 extra copies of CEN3. Several effects were noted. The copy number of each plasmid was unexpectedly high. The plasmids were mutually unstable. Cultures contained many dead cells. The viable host cells grew more slowly than control cells, even in nonselective medium. There was a pause in the cell cycle at or just before mitosis. We conclude that an excess of centromeres is toxic and that the copy number of centromere plasmids is low partly because of selection against cells carrying multiple centromere plasmids. The toxicity may be caused by competition between the centromeres for some factor present in limiting quantities, e.g., centromere-binding proteins, microtubules, or space on the spindle pole body.     

Cell cycle analysis of plasmid-containing Escherichia coli HB101 populations with flow cytometry

The relationship between cell mass and cell number dynamics for bacteria such as Escherichia coli depends on the cell cycle parameters C and D. Effects of plasmid copy number on these cell cycle parameters have been studied for Escherichia coli HB101 containing pMB1 plasmids propagated at different copy numbers ranging from 12 to 122. Determination of cell cycle and cell size parameters was accomplished using flow cytometry data on single-cell light scattering and DNA content frequency functions in conjunction with a mathematical model of cell population statistics. Two independent methods for estimating C and D intervals based on flow cytometry were developed and applied with essentially identical results. The presence of plasmids decreases the C and D periods, mean cell sizes, and initiation masses for chromosome replication by 14, 24, 38, and 18%, respectively, relative to corresponding values for plasmid-free host cells. Plasmid copy number has a negligible influence on these parameters, suggesting that host-plasmid inter actions which determine these properties are centered on the single plasmid selected for replication according to the random selection model established for ColE1-type plasmids.     

Recombinant protein synthesis and plasmid instability in continuous cultures of Escherichia coli JM103 harboring a high copy number plasmid

Escherichia coli JM103[pUC8] was employed as a model to investigate the behavior of a recombinant microbial system harboring a plasmid at high copy numbers. Experiments with batch and continuous cultures of recombinant and plasmid-free cells were conducted in a well-controlled bio-reactor. In batch experiments, plasmid copy number varied typically from an average of 500 during the exponential growth phase to as high as 1250 during the stationary phase. While the segregational plasmid instability was negligible in batch experiments, severe segregational instability occurred in continuous experiments conducted over a range of dilution rates, resulting in complete loss of plasmid-bearing cells from the continuous cultures within few residence times after transition to continuous operation. The profound differences in the specific growth rates and mass yields of the plasmid-free and plasmid-bearing cells resulting from the extra metabolic burden on the plasmid-bearing cells mainly due to excessive plasmid DNA content was the major cause for the plasmid instability. Plasmid multirnerization was detected in batch and continuous cultures and was found to have significant influence on the effective copy number and was partially responsible for the severe segregational instability in continuous cultures. A quasi-steady state representative of plasmid-bearing cells was established in the initial portion of each continuous culture experiment. Due to the profound growth rate differential between the two types of cells, transients of considerable duration were observed in each continuous culture experiment (initiated with a pure culture of plasmid bearing cells) following the slow accumulation of plasmid-free cells near the end of the quasi-steady state. Significant variations in various culture parameters (including a rapid decline in the plasmid-bearing fraction of the total cell population) occurred during this period, leading ultimately to a steady state for a culture dominated entirely by plasmid-free cells. In continuous cultures, plasmid copy number during the quasi-steady states increased with decreasing dilution rate from 50 (at 0.409 h(-1)) to 941 (at 0.233 h(-1)). Production of the plasmid-encoded protein (beta-lactamase) in these experiments was maximized at an intermediate dilution rate, corresponding to an optimum copy number of about 450. A similar optimum copy number was observed in batch cultures. Significant excretion of beta-lactamase was observed at both low and high dilution rates.     

Clustering versus random segregation of plasmids lacking a partitioning function: a plasmid paradox?

Plasmids lacking a functional partition system are randomly distributed to the daughter cells; plasmid-free daughter cells are formed with a frequency of (1/2)2n per cell and cell generation where 2n is the (average) copy number at cell division. Hence, the unit of segregation is one plasmid copy. However, plasmids form clusters in the cells. A putative solution to this potential paradox is presented: one plasmid copy at a time is recruited from the plasmid clusters to the replication factories that are located in the cell centres. Hence, replication offers the means of declustering that is necessary in a growing host population. The daughter copies diffuse freely and each copy may with equal probability end up in either of the two cell halves. In this way, the random segregation of the plasmids is coupled to replication and occurs continuously during the cell cycle, and is not linked to cell division. The unit of segregation is the plasmid copy and not the plasmid clusters. In contrast, the two daughters of a Par+ plasmid are directed in opposite directions by the plasmid-encoded partition system, thereby assuring that each daughter cell receives the plasmid.     

Random diffusion can account for topA-dependent suppression of partition defects in low-copy-number plasmids.

The maintenance of partition-defective (Par-) mini-P1 and mini-F plasmids was studied in topA strains of Escherichia coli, which are defective in topoisomerase I activity. The partition defects were substantially but not completely suppressed in broth-grown cultures. This suppression was not due to a large increase in copy number. However, the absolute number of copies of Par- mini-P1 plasmids per average dividing cell is sufficiently high to account for the modest stability observed if a random distribution of the copies to daughter cells is assumed. The similar number of Par- plasmid copies in wild-type cells are distributed in a considerably worse-than-random fashion. Thus, it is unnecessary to propose, as was suggested previously, that an active, par-independent pathway operates in topA strains to ensure proper segregation of the plasmids to daughter cells. Rather, it seems likely that the lack of topoisomerase I activity aids the random distribution of the partition-defective plasmids, perhaps by facilitating their separation after replication. The results of studies carried out at reduced growth rates were consistent with this view; when topA cells containing Par- mini-P1 plasmids were cultured in minimal medium, in which the copy number of the plasmids per average cell is sharply reduced, very little suppression of the partition defect was observed.     

Growth kinetics of the granulosa cell population in ovarian follicles: an approach by mathematical modelling

This paper describes, from a mathematical viewpoint, the cellular changes in the granulosa of ovarian follicles during their terminal development. A dynamic model takes into account the processes of (1) cell division, (2) exit from the cell cycle towards differentiation, and (3) apoptotic cell death. Proliferative cells leave the cycle in an irreversible way. The risk of entering apoptosis applies to non-cycling cells. Changes in the cell numbers and in the growth fraction are derived from differential equations. The transitions between the different cell states are ruled by time-dependent rates. Numerical applications of the model concern ovulating and degenerating ovarian follicles in the ewe. The main feature of the ovulating case is the progressive exhaustion of the proliferating compartment for the benefit of the non-cycling cells. From an initial mainly proliferative state the granulosa progressively switches to a highly differentiated state, so that the growth fraction continuously decreases. In the atretic cases, the pattern of changes in the total viable cell number is influenced by the follicular age at the onset of the apoptotic process and by the intensity of the cell death rate. As apoptosis affects the non-cycling cells, the growth fraction is no longer strictly decreasing. The sensitivity of the model to the parameters is studied in a more general framework than the granulosa cell population.     

Pedigree analysis of plasmid segregation in yeast

We have used pedigree analysis to investigate the mitotic segregation of circular and linear DNA plasmids in Saccharomyces cerevisae. Circular ARS plasmids, which bear putative chromosomal replication origins, have a high segregation frequency and a strong bias to segregate to the mother cell at mitosis. The segregation bias explains how the fraction of plasmid-bearing cells can be small despite the high average copy number of circular ARS plasmids. Linear ARS plasmids do not show strong segregation bias, nor does the 2 mu ori-containing plasmid YEp 13, when it is present in strains containing intact 2 mu circles. In the absence of endogenous 2 mu circles, YEp 13 behaves like an ARS plasmid, showing a strong maternal segregation bias. The presence of a centromere on circular ARS plasmids eliminates segregation bias. We discuss a model for plasmid segregation, which explains these findings and the possible biological significance of mother-daughter segregation bias.     

Role of the rom protein in copy number control of plasmid pBR322 at different growth rates in Escherichia coli K-12

The copy number per cell mass of plasmid pBR322 and a rom- derivative was measured as a function of generation time. In fast growing cells the copy number per cell mass was virtually identical for rom+ and rom- derivatives. However, the copy number of pBR322 only increased 3- to 4-fold from a 20- to 80-min generation time, whereas the copy number of the rom- derivative increased 7- to 10-fold. The copy number stayed constant for the rom+ and rom- plasmids at generation times longer than 80-100 min. Thus, the presence of the rom gene decreased the copy number of plasmid pBR322 in slowly growing cells at least 2-fold when compared with the rom- plasmid. To study the effect of the rom gene in trans we cloned the gene into the compatible P15A-derived rom- plasmid pACYC184. In cells carrying both pACYC184 rom+ and pBR322 rom- the presence of the rom gene in trans had little effect on the copy number of pBR322 rom- at fast growth, but it decreased its copy number at slow growth to the same level as found for pBR322, i.e., complemented the pBR322 rom- plasmid. The pACYC184 plasmid and its rom+ derivatives showed copy numbers similar to those of pBR322 rom- and pBR322 itself, respectively, at fast and slow growth. We conclude that the rom gene product-the Rom protein-is an important element in copy number control of ColE1-type plasmids especially in slowly growing cells.     

A mathematical model for plasmid replication and distribution in microbial populations

A mathematical model and a computer program for its implementation have been developed to predict the distribution of plasmid copy numbers in the individual cells of a microbial population. The kinetics of accumulation of plasmid-free cells. the copy number distribution within the population and the mean copy number can all be calculated using the computer program. The model has been shown to accurately predict these parameters for recombinant plasmids in yeast populations.     

Analysis of unstable recombinant Saccharomyces cerevisiae population growth in selective medium

Widely applied selection strategies for plasmid-containing cells in unstable recombinant populations are based upon synthesis in those cells of an essential, selection gene product. Regular partitioning of this gene product combined with asymmetric plasmid segregation produces plasmid-free cells which retain for some time the ability to grow in selective medium. This theory is elaborated here in terms of a segregated model for an unstable recombinant population which predicts population growth characteristics and composition based upon experimental data for stable strain growth kinetics, plasmid content, and selection gene product stability. Analytical solutions from this model are compared with an unsegregated phenomenological model to evaluate the effective specific growth rate of plasmid-free cells in selective medium. Model predictions have been validated using experimental growth kinetics and flow cytometry data for Saccharomyces cerevisiae D603 populations containing one of the plasmids YCpG1ARS1, YCpG1DeltaR8, YCpG1DeltaR88, YCpG1DeltaH103, YCpG1DeltaH200, pLGARS1, and pLGSD5. The recombinant strains investigated encompass a broad range of plasmid content (from one to 18 plasmids per cell) and probability alpha of plasmid loss at division (0.05 相似文献   

Optimizing initial plasmid copy number distribution for improved protein activity in a recombinant fermentation

Recombinant bacterial cells in a fermentation broth rarely contain the same number of plasmids, even though this simplification is often used. Recent work has however indicated limitations of the simplified approach. Based on these studies, the distribution of plasmid copy numbers per cell has been represented macroscopically here in a Gaussian form for the fraction of biomass as a function of the copy number. Applying this distribution and an experimentally validated kinetic model to glyceraldehyde-3-phosphate dehydrogenase (GAPDH) synthesis by Escherichia coli containing the plasmid pBR Eco gap, it is seen that GAPDH production in a batch fermentation is maximized by a particular initial (non-zero) copy number variance and an optimal duration. To implement this distribution in a bioreactor, it is suggested that the profile may be discretized, inocula corresponding to the mean copy number of each fraction prepared, and then combined to obtain the seed culture.     

Transformation of Kluyveromyces lactis by killer plasmid DNA

A single plasmid of 55 kilobases was found in crude cell lysates of each of nine strains of Rhodospirillum rubrum. Restriction endonuclease analysis showed identical fragment patterns with a given nuclease for all plasmids except one, for which an additional EcoRI site was observed. Elimination of the plasmid required that the cells be passaged several times in 25 mM calcium-containing medium, followed by at least two passages under photosynthetic growth conditions in low-calcium medium before treatment with ethyl methanesulfonate. The resulting plasmidless mutants only grew aerobically and were all incapable of pigment formation and photosynthetic growth, suggesting that plasmid DNA is required for photosynthetic competence in R. rubrum.     

A kinetic model for plasmid curing

A simple mathematical model of drug-induced plasmid elimination (curing) considering density-dependent growth rates and plasmid transfers is presented. It describes nonlinear population dynamics of conjugative plasmids during in vitro curing experiments in batch culture. The model was tested on kinetics of acridine orange curing of F'lac plasmid. Effects of density dependence, plasmid elimination, selection for plasmidless segregants, conjugation, initial and maximal population density, and postsegregational killing on curing kinetics are simulated and discussed.